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0704.0536

Binaries, microquasars and GLAST

Binaries and GLAST Guillaume Dubus Laboratoire d’Astrophysique de Grenoble, Université J. Fourier, CNRS, BP 53, 38041 Grenoble, France Abstract. Radio and X-ray observations of the relativistic jets of microquasars show evidence for the acceleration of particles to very high energies. Signatures of non-thermal processes occurring closer in to the compact object can also be found. In addition, three binaries are now established emitters of high (>100 MeV) and/or very high (>100 GeV) energy gamma-rays. High-energy emission can originate from a microquasar jet (accretion-powered) or from a shocked pulsar wind (rotation- powered). I discuss the impact GLAST will have in the very near future on studies of such binaries. GLAST is expected to shed new light on the link between accretion and ejection in microquasars and to enable to probe pulsar winds on small scales in rotation-powered binaries. Keywords: Gamma-ray; Pulsars; Black holes; X-ray binaries; Infall and accretion; Jets, outflows and bipolar flows PACS: 95.85.Pw, 97.60.Gb, 97.60.Lf, 97.80.Jp, 98.35.Mp, 98.58.Fd INTRODUCTION Binaries composed of a black hole or neutron star in orbit with a stellar companion are prominent sources of the X-ray sky. X-ray binaries are usually powered by accretion of matter from the companion. The gravitational energy released by accretion heats the plasma to temperatures in the range 1 keV < kT < 50 MeV for a stellar-mass black hole, which is someway below the GLAST energy range. However, part of the power is also emitted non-thermally by particles of much greater energies. The radio emission from X-ray binaries, due to synchrotron radiation from electrons located in relativistic outflows, provides clear evidence for this. These relativistic jets are the most striking demonstration of the analogy between accretion onto the stellar-mass compact objects in X-ray binaries and onto the supermassive black holes in Active Galactic Nuclei (AGN). X-ray binaries with relativistic jets have thus been denominated ‘microquasars’. Similarities also exist in timing and spectral characteristics. The conjecture, at least for black holes, is that the underlying scaling factor of the physical processes is the gravitational radius Rg=GM/c 2 (and its associated timescale Rg/c). The similarities have prompted speculation that some X-ray binaries may be analogs of blazars, AGNs dominated by non-thermal output because their relativistic jet is fortuitously aligned with the line-of-sight. Blazars emit a large fraction of their power in high energy (HE) gamma-rays (>100 MeV). Notwithstanding the possible issue that the microquasar population may be too small for a chance alignment to occur, finding genuine ‘microblazars’ could also prove exceedingly difficult if the scaling strictly holds, since the observed sub-hour TeV flaring in blazars translates to milliseconds for a microblazar. But even if the jet is misaligned, the high particle energies implied by the radio to X-ray observations make it likely, if not unavoidable, that HE gamma-ray emission should be present at some level in generic microquasars. Cataclysmic variables (AE Aqr) and colliding winds in Wolf-Rayet binaries have also been proposed to emit HE gamma-rays - but these systems will not be addressed here. GLAST observations will soon open a new window into non-thermal processes in accreting binaries. Three compact binaries are presently known sources of HE gamma-rays. These three gamma-ray binaries (in that most of their radiative output is in gamma-rays, regardless of the underlying physics) are probably all rotation-powered by the spin- down of a young pulsar rather than accretion-powered. The current observational status and GLAST prospects for both types of sources are reviewed. GLAST STUDIES OF ACCRETING BINARIES X-ray jets The best evidence for particle acceleration to high energies in accreting binaries comes from the observation of X-ray emission from localized regions in the relativistic jet of two microquasars, XTE J1550-563 and H1743-322 http://arxiv.org/abs/0704.0536v1 [1]. The radio spectrum from these regions connects remarkably well with the X-ray spectrum across nine decades in energy. The overall spectral slope α ≈−0.6 points to synchrotron emission from a canonical non-thermal distribution of electrons dN ∝ N−2.2dE over nine orders-of-magnitude. The emission zone is resolved (≈1”). The equipartition magnetic field is ≈ 500 µG and the X-ray emitting electrons have energies ≈ 10 TeV [1]. Electrons of such energies necessarily emit in the GLAST energy range. However, the prospects for detecting this emission are not favorable. Synchrotron losses limit the maximum possible electron energy to a few PeV, in which case the spectrum could extend to a 100 MeV at a level ≈ 10−12 ergs s−1 cm−2, too faint for GLAST observations in the Galactic Plane. Moreover, the short radiative timescale (≈ 5 days) of PeV electrons would lead to a break or cutoff in the spectrum below 100 MeV. (The timescale also shows the particle acceleration has to occur in situ, although why and how it occurs is not understood but could involve internal shocks, magnetic energy dissipation or a shock with the ISM.) Self-Compton or inverse Compton on CMB photons are also unlikely to lead to detectable levels of emission for GLAST. In both cases an inverse Compton luminosity greater or equal to the synchrotron luminosity requires a magnetic field < 3 µG, a hundred times below equipartition. Therefore, although these observations undoubtedly show the presence of particles of high-energies in microquasar jets, the gamma-rays emitted in those conditions are not likely to be detected by GLAST. Large scale jet-ISM interaction The dissipation of the jet power in the ISM is a possible source of gamma-ray emission for GLAST. The energy involved can be substantial. For example, radio observations show the a.u.-scale compact jet of Cyg X-1 is prolonged into a parsec-scale structure associated with the termination shock. The inferred power inconspicuously transported to the large scales is comparable to the bolometric luminosity of the binary [2]. Jets can therefore quietly inject large amounts of high energy particles into the ISM, which might be traced by their gamma-ray emission. Heinz & Sunyaev [3] estimate their contribution could reach 10% of the Galactic cosmic ray luminosity. If there is indeed a significant component of high energy nucleons, a jet interacting with a nearby molecular cloud (effectively modeled as an accelerator + beam dump in [4]) would create pions. The subsequent decay, bremsstrahlung and inverse Compton emission can be detected by GLAST, depending upon the jet power and composition, but also upon the duty cycle of the ejection process, the distance of the cloud to the source etc. As an example, the emission predicted in Fig. 12 of [4] is detectable within a year for a microquasar at a distance of 1 kpc (e.g. A0620-00, XTE J1118+480, the latter having the advantage of a large Galactic latitude b=+62.3o). Such observations would give new clues as to the content and power of relativistic jets in binaries. Gamma-ray spectral states and major ejections Gamma-ray emission originating closer in to the compact object can also be expected. There is reasonable evidence from CGRO observations for soft power-law tails (spectral slope α ≈-1.5) extending beyond 100 keV, up to several MeV in some X-ray binaries [5]. These soft tails are a defining property of the very high state (or steep power law state) together with significant thermal emission (around a keV) and fast variability (QPOs) [6]. This is most clearly seen in the high state of Cyg X-1 where the power-law extends to 10 MeV. Extrapolating shows the 100 MeV emission was beyond the reach of EGRET but should be detectable by GLAST within days. Cyg X-1 spends 90% of its time in the hard state where there is a hint for a similar, but fainter, power-law component that could be detected in a year by GLAST. These power-laws can be produced in plasmas where a fraction of the accretion energy goes into non-thermal channels [7]. Models predict a cutoff in the GLAST energy range either because there is a maximum electron energy or because pair production sets in when the plasma compacity is high. Changes in X-ray spectral states can therefore be surmised to be associated with changes in HE gamma-ray luminosity, the very high X-ray state (resp. hard X-ray state) involving high (resp. low) levels of gamma-ray emission. Interestingly, spectral state changes from hard to very high X-ray states have been conjectured to be associated with major relativistic ejections [8] and one should note, perhaps naively, that if jets are composed of e+e− pairs then there has to be some gamma-ray emission linked to the pair production, if only at a few MeV. Radio/IR observations of discrete ejections in GRS 1915+105 show non-thermal emission cooling with expansion. Extrapolating back to early times, the fluence expected in a day by Atoyan & Aharonian [9] in HE gamma-rays is detectable by GLAST. Gamma- ray monitoring of outbursting binaries or sources such as GRS 1915+105, a task that is well-suited to GLAST, can therefore shed light on how spectral state changes relate to ejection events. THE OBSERVATIONAL STATUS: GAMMA-RAY BINARIES The view from space Although the above (should) demonstrate that there are reasonable grounds to expect HE gamma-rays from compact binaries, observational confirmation has proved elusive and when found, arguably disconcerting. Several tentative associations of binaries with EGRET sources were made based on positional coincidence and/or the detection of variability, with the source 2CG 135+01 figuring prominently as the first and most secure: follow-up observations carried out after the initial COS B discovery had revealed a high-mass X-ray binary, LS I+61 303, in a 26 day elliptical orbit showing periodic radio outbursts [10]. The latter feature being rare, this highlighted the system as a plausible counterpart. Yet, although variability was reported in the EGRET data, neither this nor the position were enough to formally identify the two (the HE variation not being tied to variability at other wavelengths and the stellar counterpart being localized only in between the 95% and 99% confidence contours of the HE source). The limited angular resolution combined with the strong underlying Galactic diffuse HE emission resulted in error boxes of a several tens of arcmins, too large to pick up the needle in the haystack of possible Galactic Plane counterparts. The view from the ground Breakthrough observations were obtained by the ground-based Cherenkov telescopes operated by the HESS and MAGIC collaborations. These observe at a higher threshold (≥ 100 GeV) but benefit from a larger collecting area and a better angular resolution. Three binaries were detected: PSR B1259-63, LS 5039 and LS I+61 303 [11, 12, 13]. The latter two had tentative EGRET associations but not PSR B1259-63, possibly because the HE gamma-ray emission is highly variable along the 3.5 year orbit and confined to a short period around periastron passage. The next passage occurs this year, too early for GLAST so searches may have to wait 2010 (unfortunately, periastron passage cannot be observed by HESS before 2014). The localizations are much more precise: for instance, LS 5039 is coincident with HESS J1826-148 within the positional uncertainty of 30”, excluding a nearby SNR and a pulsar that were within the error box of the EGRET source 3EG 1824-1514. More importantly, all three binaries display variability. In LS 5039, it was demonstrated that the >100 GeV flux is strongly modulated on the orbital period [14] and there is little doubt that the fluxes measured in PSR B1259-63 and LS I+61 303 also depend on orbital phase. Gamma-ray binaries All three binaries have high mass O or Be type companions with compact objects in eccentric orbits. The X-ray output from these binaries is about 1034 erg/s, rather low in itself compared to typical X-ray binaries, and smaller or comparable to the emission above 100 MeV. These systems are therefore gamma-ray loud, a first surprise. All of them display radio emission, resolved in LS 5039 and LS I+61 303 as collimated outflows on milliarcsecond scales, immediately suggesting a microquasar nature [15]. X-ray and radio variability, when (and if) present is of limited amplitude (odd in accreting binaries) and occurs on the orbital timescale. The overall spectral energy distributions are similar for all three systems, showing (in νFν ) a rising spectrum from radio to X-rays flattening around an MeV and reaching energies of several TeV. However, PSR B1259-63 is not a microquasar but a young 48 ms pulsar with a spin-down power ≈ 1036 erg s−1. The relativistic pulsar wind is sufficient to quench any wind accretion and the emission is thought to arise from particles accelerated where the pulsar and stellar wind interact [16]. This paints a rather different picture than accretion-powered scenarios. Gamma-ray binaries as compact pulsar wind nebulae Why the observational properties of the three gamma-ray binaries should bear any resemblance is disconcerting unless all are actually rotation-powered by a young pulsar [17]. This can explain the low, steady level of emission, conceivably modulated as the pulsar moves around its orbit. Pulsed emission would be absorbed in the stellar wind because of the smaller orbital separations in LS 5039 and LS I+61 303 (as observed near periastron in PSR B1259-63). The pulsar wind is confined by the stellar wind to a cometary nebula pointing away from the massive companion, producing the collimated radio outflow. Radio VLBI observations of LS I+61 303 recently reported by [18] are consistent with this picture, showing a periodic sweep of the radio tail with orbital phase which appears irreconcilable with an accretion-powered jet: LS I+61 303 is almost certainly powered by a pulsar. The small-scale radio morphology of LS 5039 has been successfully modeled as a pulsar wind nebula [17] but this does not provide the same level of certainty in the absence of observations at other orbital phases. Other models have proposed the emission arises in a relativistic jet powered by accretion [19, 20], but that hypothesis seems rather uneconomical to this author. At this stage, it seems more probable that HE gamma-ray emission from accretion-powered binaries has yet to be detected and that, when this will be achieved, their observational properties will be clearly different from those of the rotation- powered binaries. GLAST STUDIES OF ROTATION-POWERED BINARIES Gamma-ray orbital modulation That all three gamma-ray binaries have high-mass stellar companions may be instrumental to generate the HE emission, as these will provide copious amounts of seed photons for inverse Compton scattering. On the other hand, the large photon densities at UV energies also imply pair production with TeV photons can be important. The starlight both provides a source and a sink for gamma-rays. Gamma-ray absorption has a strong orbital dependence as the cross- section for pair production depends on the angle between the two photons. The effect can be dramatic on gamma-rays emitted towards the observer and crossing head-on the path of stellar photons. In LS 5039, the tight 4-day orbit brings the compact object to within a stellar radius from the O6V star. Gamma-rays of >30 GeV emitted close to the compact object are modulated with peak attenuation at superior conjunction (when the compact object is behind the star as seen by the observer) [21]. Such an orbital modulation has been observed in LS 5039 by HESS with a peak and trough at the predicted orbital phases [14]. However, the flux at superior conjunction is not completely absorbed. Furthermore, the >100 GeV spectrum varies from a soft power law at superior conjunction (low flux) to a hard power law at inferior conjunction (high flux), whereas pure absorption predicts a dip in the spectrum around a TeV. Other effects must play a role. One is that a pair cascade is initiated when the newly created e+e− up-scatter star photons back into the absorption range. The magnetic field has to be low enough (≤10 G) to prevent synchrotron losses from dominating. The radiated energy is redistributed below the pair production threshold (<30 GeV) i.e. in the GLAST range. Calculations show an anti-correlation of the HESS and GLAST light-curves [22], detectable within a year [23], that would prove the existence of a cascade. A second effect is that inverse Compton scattering is also anisotropic. For example, this will decrease the gamma- ray flux around inferior conjunction, when the (incoming) star photons and (outgoing) gamma-ray photons both go towards the observer: in this configuration the energy of the outgoing photon and the cross-section for IC are small. There actually is a hint of a dip in the HESS light-curve at this phase. Probing pulsar winds Besides these geometrical effects, the efficiency with which gamma-rays are emitted may also change along the orbit. The variations in the >100 GeV flux observed by HESS and MAGIC in LS I+61 303 and PSR B1259-63 do not match the expected light-curve for pure absorption (whose effect is marginal due to the wider orbits) and so must be intrinsic to the emission process [24]. LS I+61 303 may prove a Rosetta stone for this problem as its orbit is both wide enough (0.2-0.7 a.u.) to avoid most cascading and absorption, but short enough (26 days) to allow for detailed studies over many orbits. MAGIC reports a minimum in flux close to periastron (which is close to inferior conjunction, where absorption is minimal) and a maximum towards apastron. The explanation is straightforward with a compact pulsar wind nebula. The pulsar wind is contained by the stellar wind of the Be companion. At periastron, the ram pressure of the dense equatorial wind crushes the PWN nebula to a small distance from the pulsar. The magnetic field at the shock location is strong and particles lose energy rapidly to synchrotron emission without radiating much inverse Compton above >10 GeV. At apastron, the stellar wind is polar and diffuse, implying a large shock distance and weaker magnetic field, allowing for higher particle energies emitting more HE inverse Compton gamma-rays. Hence, phase-resolved spectral energy distributions from X-rays to VHE gamma-rays can yield information on the magnetic field at different locations, forming a probe of the relativistic wind as a function of distance to the pulsar. Population studies of gamma-ray binaries The pulsar spin-down timescale is short (∼ 105 years for PSR B1259-63) and can only power the binary emission for a brief period of time. Accretion from the stellar wind is then no longer quenched by a pulsar wind and an X-ray pulsar turns on. Hence, gamma-ray binaries are the progenitors of the longer-lived, accretion-powered high-mass X- ray binaries (HMXBs). Because most of their output is in gamma-rays, GLAST all-sky observations provide a unique way of identifying these progenitors and studying them as a population, constraining the birth rate and evolution of HMXBs. Population synthesis calculations find there should be around 30 active gamma-ray binaries in order to match the present-day population of HMXBs, assuming the rotation-powered phase lasts only 104 years [25]. They should be visible throughout the Galaxy if their luminosity in the GLAST band is comparable to that of the three known systems (∼ 1035 erg s−1). This estimate does not take into account the reduction in sensitivity to be expected in the Galactic Plane due to the diffuse emission. In this respect, the Magellanic Clouds, which harbor a comparatively very large population of HMXBs, might also make for good dedicated studies despite being more distant. ACKNOWLEDGMENTS I thank the organizers of the 1st GLAST symposium for their invitation and acknowledge support from the Agence Nationale de la Recherche. REFERENCES 1. S. Corbel et al., Science 298, 196–199 (2002). 2. E. Gallo, R. Fender, C. Kaiser, D. Russell, R. Morganti, T. Oosterloo, and S. Heinz, Nature 436, 819–821 (2005). 3. S. Heinz, and R. Sunyaev, A&A 390, 751–766 (2002). 4. V. Bosch-Ramon, F. A. Aharonian, and J. M. Paredes, A&A 432, 609–618 (2005). 5. J. E. Grove et al., ApJ 500, 899 (1998). 6. R. A. Remillard, and J. E. McClintock, ARA&A 44, 49–92 (2006). 7. A. A. Zdziarski, and M. Gierliński, Progress of Theoretical Physics Supplement 155, 99–119 (2004). 8. R. P. Fender, T. M. Belloni, and E. Gallo, MNRAS 355, 1105–1118 (2004). 9. A. M. Atoyan, and F. A. Aharonian, MNRAS 302, 253–276 (1999). 10. P. C. Gregory, and A. R. Taylor, Nature 272, 704–706 (1978). 11. F. Aharonian et al. (HESS collaboration), A&A 442, 1–10 (2005). 12. F. Aharonian et al. (HESS collaboration), Science 309, 746–749 (2005). 13. J. Albert et al. (MAGIC collaboration), Science 312, 1771–1773 (2006). 14. F. Aharonian et al. (HESS collaboration), A&A 460, 743–749 (2006). 15. J. M. Paredes, J. Martí, M. Ribó, and M. Massi, Science 288, 2340–2342 (2000). 16. M. Tavani, and J. Arons, ApJ 477, 439 (1997). 17. G. Dubus, A&A 456, 801–817 (2006). 18. V. Dhawan et al., in Proc. of the VI Microquasar Workshop, Sep. 18-22, 2006, Como, Italy, PoS (MQW6) 52 (2006). 19. C. D. Dermer, and M. Böttcher, ApJ 643, 1081–1097 (2006). 20. J. M. Paredes, V. Bosch-Ramon, and G. E. Romero, A&A 451, 259–266 (2006). 21. G. Dubus, A&A 451, 9–18 (2006). 22. W. Bednarek, A&A 464, 259–262 (2007). 23. R. Dubois, in Proc. of the VI Microquasar Workshop, Sep. 18-22, 2006, Como, Italy, PoS (MQW6) 68 (2006). 24. G. Dubus, in Proc. Vulcano workshop Frontier Objects in Astrophysics and Particle Physics, May 22-27, 2006, F. Giovannelli & G. Mannocchi (eds.), Italian Physical Society, Editrice Compositori, Bologna, Italy, (astro-ph/0608262) (2006). 25. E. J. A. Meurs, and E. P. J. van den Heuvel, A&A 226, 88–107 (1989). Introduction GLAST studies of accreting binaries X-ray jets Large scale jet-ISM interaction Gamma-ray spectral states and major ejections The observational status: gamma-ray binaries The view from space The view from the ground Gamma-ray binaries Gamma-ray binaries as compact pulsar wind nebulae GLAST studies of rotation-powered binaries Gamma-ray orbital modulation Probing pulsar winds Population studies of gamma-ray binaries

0704.0537

Linearisation of finite abelian subgroups of the Cremona group of the plane

Linearisation of finite Abelian subgroups of the Cremona group of the plane Jérémy Blanc November 4, 2018 Abstract Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transforma- tions of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to Z/2Z×Z/4Z, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles. 1 Introduction 1.1 The main questions and results In this paper, every surface will be complex, rational, algebraic and smooth, and except for C2, will also be projective. By an automorphism of a surface we mean a biregular algebraic morphism from the surface to itself. The group of automor- phisms (respectively of birational transformations) of a surface S will be denoted by Aut(S) (respectively by Bir(S)). The group Bir(P2) is classically called the Cremona group. Taking some sur- face S, any birational map S 99K P2 conjugates Bir(S) to Bir(P2); any subgroup of Bir(S) may therefore be viewed as a subgroup of the Cremona group, up to conjugacy. The minimal surfaces are P2, P1×P1 and the Hirzebruch surfaces Fn for n ≥ 2; their groups of automorphisms are a classical object of study, and their structures are well known (see for example [Bea1]). These groups are in fact the maximal connected algebraic subgroups of the Cremona group (see [Mu-Um], [Um]). Given some group acting birationally on a surface, we would like to determine some geometric properties that allow us to decide whether the group is conjugate to a group of automorphisms of a minimal surface, or equivalently to decide whether http://arxiv.org/abs/0704.0537v2 it belongs to a maximal connected algebraic subgroup of the Cremona group. This conjugation looks like a linearisation, as we will see below, and explains our title. We observe that the set of points of a minimal surface which are fixed by a non-trivial automorphism is the union of a finite number of points and rational curves. Given a group G of birational transformations of a surface, the following properties are thus related (note that for us the genus is the geometric genus, so that a curve has positive genus if and only if it is not rational); property (F ) is our candidate for the geometric property for which we require: (F ) No non-trivial element of G fixes (pointwise) a curve of positive genus. (M) The group G is birationally conjugate to a group of automorphisms of a minimal surface. The fact that a curve of positive genus is not collapsed by a birational trans- formation of surfaces implies that property (F ) is a conjugacy invariant; it is clear that the same is true of property (M). The above discussion implies that (M) ⇒ (F ); we would like to prove the converse. The implication (F ) ⇒ (M) is true for finite cyclic groups of prime order (see [Be-Bl]). The present article describes precisely the case of finite Abelian groups. We prove that (F ) ⇒ (M) is true for finite cyclic groups of any order, and that we may restrict the minimal surfaces to P2 or P1 × P1. In the case of finite Abelian groups, there exists, up to conjugation, only one counterexample to the implication, which is represented by a group isomorphic to Z/2Z × Z/4Z acting biregularly on a special conic bundle. Precisely, we will prove the following results, announced without proof as Theorems 4.4 and 4.5 in [Bla3]: Theorem 1. Let G be a finite cyclic subgroup of order n of the Cremona group. The following conditions are equivalent: • If g ∈ G, g 6= 1, then g does not fix a curve of positive genus. • G is birationally conjugate to a subgroup of Aut(P2). • G is birationally conjugate to a subgroup of Aut(P1 × P1). • G is birationally conjugate to the group of automorphisms of P2 generated by (x : y : z) 7→ (x : y : e2iπ/nz). Theorem 2. Let G be a finite Abelian subgroup of the Cremona group. The following conditions are equivalent: • If g ∈ G, g 6= 1, then g does not fix a curve of positive genus. • G is birationally conjugate to a subgroup of Aut(P2), or to a subgroup of Aut(P1 × P1) or to the group Cs24 isomorphic to Z/2Z × Z/4Z, generated by the two elements (x : y : z) 99K (yz : xy : −xz), (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)). Moreover, this last group is conjugate neither to a subgroup of Aut(P2), nor to a subgroup of Aut(P1 × P1). Then, we discuss the case in which the group is infinite, respectively non- Abelian (Section 11) and provide many examples of groups satisfying (F ) but not Note that many finite groups which contain elements that fix a non-rational curve are known, see for example [Wim] or more recently [Bla2] and [Do-Iz]. This can also occur if the group is infinite, see [BPV] and [Bla5]. In fact, the set of non-rational curves fixed by the elements of a group is a conjugacy invariant very useful in describing conjugacy classes (see [Ba-Be], [dFe], [Bla4]). 1.2 How to decide Given a finite Abelian group of birational transformations of a (rational) surface, we thus have a good way to determine whether the group is birationally conjugate to a group of automorphisms of a minimal surface (in fact to P2 or P1 × P1). If some non-trivial element fixes a curve of positive genus (i.e. if condition (F ) is not satisfied), this is false. Otherwise, if the group is not isomorphic to Z/2Z×Z/4Z, it is birationally conjugate to a subgroup of Aut(P2) or of Aut(P1 × P1). There are exactly four conjugacy classes of groups isomorphic to Z/2Z×Z/4Z satisfying condition (F ) (see Theorem 5); three are conjugate to a subgroup of Aut(P2) or Aut(P1 × P1), and the fourth (the group Cs24 of Theorem 2, described in detail in Section 7) is not. 1.3 Linearisation of birational actions Our question is related to that of linearisation of birational actions on C2. This latter question has been studied intensively for holomorphic or polynomial actions, see for example [De-Ku], [Kra] and [vdE]. Taking some group acting birationally on C2, we would like to know if we may birationally conjugate this action to have a linear action. Note that working on P2 or C2 is the same for this question. Theorem 1 implies that for finite cyclic groups, being linearisable is equivalent to fulfilling condition (F ). This is not true for finite Abelian groups in general, since some groups acting biregularly on P1 × P1 are not birationally conjugate to groups of automorphisms of P2. Note that Theorem 1 implies the following result on linearisation, also announced in [Bla3] (as Theorem 4.2): Theorem 3. Any birational map which is a root of a non-trivial linear automor- phism of finite order of the plane is conjugate to a linear automorphism of the plane. 1.4 The approach and other results Our approach – followed in all the modern articles on the subject – is to view the finite subgroups of the Cremona group as groups of (biregular) automorphisms of smooth projective rational surfaces and then to assume that the action is minimal (i.e. that it is not possible to blow-down some curves and obtain once again a biregular action on a smooth surface). Manin and Iskovskikh ([Man] and [Isk2]) proved that the only possible cases are action on del Pezzo surfaces or conic bun- dles. We will clarify this classification, for finite Abelian groups fillfulling (F), by proving the following result: Theorem 4. Let S be some smooth projective rational surface and let G ⊂ Aut(S) be a finite Abelian group of automorphisms of S such that • the pair (G,S) is minimal; • if g ∈ G, g 6= 1, then g does not fix a curve of positive genus. Then, one of the following occurs: 1. The surface S is minimal, i.e. S ∼= P2, or S ∼= Fn for some integer n 6= 1. 2. The surface S is a del Pezzo surface of degree 5 and G ∼= Z/5Z. 3. The surface S is a del Pezzo surface of degree 6 and G ∼= Z/6Z. 4. The pair (G,S) is isomorphic to the pair (Cs24, Ŝ4) defined in Section 7. We will then prove that all the pairs in cases 1, 2 and 3 are birationally equiv- alent to a group of automorphisms of P1 × P1 or P2, and that this is not true for case 4. In fact, we are able to provide the precise description of all conjugacy classes of finite Abelian subgroups of Bir(P2) satisfying (F ): Theorem 5. Let G be a finite Abelian subgroup of the Cremona group such that no non-trivial element of G fixes a curve of positive genus. Then, G is birationally conjugate to one and only one of the following: [1] G ∼= Z/nZ× Z/mZ g.b. (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy) [2] G ∼= Z/2Z× Z/2nZ g.b. (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζ2ny) [3] G ∼= (Z/2Z)2 × Z/2nZ g.b. (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζ2ny) [4] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1) [5] G ∼= (Z/2Z)4 g.b. (x, y) 7→ (±x±1,±y±1) [6] G ∼= Z/2Z× Z/4Z g.b. (x, y) 7→ (x−1, y−1) and (x, y) 7→ (−y, x) [7] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (−x,−y), (x, y) 7→ (x−1, y−1), and (x, y) 7→ (y, x) [8] G ∼= (Z/2Z)× (Z/4Z) g.b. (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)) and (x : y : z) 99K (yz : xy : −xz) [9] G ∼= (Z/3Z)2 g.b. (x : y : z) 7→ (x : ζ3y : (ζ3) and (x : y : z) 7→ (y : z : x) (where n,m are positive integers, n divides m and ζn = e 2iπ/n). Furthermore, the groups in cases [1] through [7] are birationally conjugate to sub- groups of Aut(P1×P1), but the others are not. The groups in cases [1] and [9] are birationally conjugate to subgroups of Aut(P2), but the others are not. To prove these results, we will need a number of geometric results on automor- phisms of rational surfaces, and in particular on automorphisms of conic bundles and del Pezzo surfaces (Sections 3 to 9). We give for example the classification of all the twisting elements (that exchange the two components of a singular fibre) acting on conic bundles in Proposition 6.5 (for the elements of finite order) and Proposition 6.8 (for those of infinite order); these are the most important elements in this context (see Lemma 3.8). We also prove that actions of (possibly infinite) Abelian groups on del Pezzo surfaces satifying (F ) are minimal only if the degree is at least 5 (Section 9) and describe these cases precisely (Sections 4, 5 and 9). We also show that a finite Abelian group acting on a projective smooth surface S such that (KS) 2 ≥ 5 is birationally conjugate to a group of automorphisms of P1 × P1 or P2 (Corollary 9.10) and in particular satisfies (F ). 1.5 Comparison with other work Many authors have considered the finite subgroups of Bir(P2). Among them, S. Kantor [Kan] gave a classification of the finite subgroups, which was incomplete and included some mistakes; A. Wiman [Wim] and then I.V. Dolgachev and V.A. Iskovskikh [Do-Iz] successively improved Kantor’s results. The long paper [Do-Iz] expounds the general theory of finite subgroups of Bir(P2) according to the modern techniques of algebraic geometry, and will be for years to come the reference on the subject. Our viewpoint and aim differ from those of [Do-Iz]: we are only interested in Abelian groups in relation with the above conditions (F) and (M); this gives a restricted setting in which the theoretical approach is simplified and the results obtained are more accurate. In the study of del Pezzo surfaces, using the classification [Do-Iz] of subgroups of automorphisms would require the examination of many cases; for the sake of readibility we prefered a direct proof. The two main theorems of [Do-Iz] on automorphism of conic bundles (Proposition 5.3 and Theorem 5.7(2)) do not exclude groups satisfying property (F ) and do not give explicit forms for the generators of the groups or the surfaces. 1.6 Aknowledgements This article is part of my PhD thesis [Bla2]; I am grateful to my advisor T. Vust for his invaluable help during these years, to I. Dolgachev for helpful discussions, and thank J.-P. Serre and the referees for their useful remarks on this paper. 2 Automorphisms of P2 or P1 × P1 Note that a linear automorphism of C2 may be extended to an automorphism of either P2 or P1 × P1. Moreover, the automorphisms of finite order of these three surfaces are birationally conjugate. For finite Abelian groups, the situation is quite different. We give here the birational equivalence of these groups. Notation 2.1. The element [a : b : c] denotes the diagonal automorphism (x : y : z) 7→ (ax : by : cz) of P2, and ζm = e 2iπ/m. Proposition 2.2 (Finite Abelian subgroups of Aut(P2)). Every finite Abelian subgroup of Aut(P2) = PGL(3,C) is conjugate, in the Cremona group Bir(P2), to one and only one of the following: 1. A diagonal group, isomorphic to Z/nZ×Z/mZ, where n divides m, generated by [1 : ζn : 1] and [ζm : 1 : 1]. (The case n = 1 gives the cyclic groups). 2. The special group V9, isomorphic to Z/3Z×Z/3Z, generated by [1 : ζ3 : (ζ3) and (x : y : z) 7→ (y : z : x). Thus, except for the group V9, two isomorphic finite Abelian subgroups of PGL(3,C) are conjugate in Bir(P2). Proof. First of all, a simple calculation shows that every finite Abelian subgroup of PGL(3,C) is either diagonalisable or conjugate to the group V9. Furthermore, since this last group does not fix any point, it is not diagonalisable, even in Bir(P2) [Ko-Sz, Proposition A.2]. Let T denote the torus of PGL(3,C) constituted by diagonal automorphisms of P2. Let G be a finite subgroup of T ; as an abstract group it is isomorphic to Z/nZ× Z/mZ, where n divides m. Now we can conjugate G by a birational map of the form h : (x, y) 99K (xayb, xcyd) so that it contains [ζm : 1 : 1] (see [Be-Bl] and [Bla1]). Since h normalizes the torus T , the group G remains diagonal and contains the n-torsion of T , hence it contains [1 : ζn : 1]. Corollary 2.3. Every finite Abelian group of linear automorphisms of C2 is bi- rationally conjugate to a diagonal group, isomorphic to Z/nZ × Z/mZ, where n divides m, generated by (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy). Proof. This follows from the fact that the group GL(2,C) of linear automorphisms of C2 extends to a group of automorphisms of P2 that leaves the line at infinity invariant and fixes one point. Example 2.4. Note that Aut(P1 × P1) contains the group (C∗)2 ⋊ Z/2Z, where (C∗)2 is the group of automorphisms of the form (x, y) 7→ (αx, βy), α, β ∈ C∗, and Z/2Z is generated by the automorphism (x, y) 7→ (y, x). The birational map (x, y) 99K (x : y : 1) from P1×P1 to P2 conjugates (C∗)2⋊ Z/2Z to the group of automorphisms of P2 generated by (x : y : z) 7→ (αx : βy : z), α, β ∈ C∗ and (x : y : z) 7→ (y : x : z). Proposition 2.5 (Finite Abelian subgroups of Aut(P1 × P1)). Up to birational conjugation, every finite Abelian subgroup of Aut(P1×P1) is conjugate to one and only one of the following: [1] G ∼= Z/nZ× Z/mZ g.b. (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy) [2] G ∼= Z/2Z× Z/2nZ g.b. (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζ2ny) [3] G ∼= (Z/2Z)2 × Z/2nZ g.b. (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζ2ny) [4] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1) [5] G ∼= (Z/2Z)4 g.b. (x, y) 7→ (±x±1,±y±1) [6] G ∼= Z/2Z× Z/4Z g.b. (x, y) 7→ (x−1, y−1) and (x, y) 7→ (−y, x) [7] G ∼= (Z/2Z)3 g.b. (x, y) 7→ (−x,−y), (x, y) 7→ (x−1, y−1), and (x, y) 7→ (y, x) (where n,m are positive integers, n divides m and ζn = e 2iπ/n). Furthermore, the groups in [1] are conjugate to subgroups of Aut(P2), but the others are not. Proof. Recall that Aut(P1 × P1) = (PGL(2,C) × PGL(2,C)) ⋊ Z/2Z. Let G be some finite Abelian subgroup of Aut(P1 × P1); we now prove that G is conjugate to one of the groups in cases [1] through [7]. First of all, if G is a subgroup of the group (C∗)2⋊Z/2Z given in Example 2.4, then it is conjugate to a subgroup of Aut(P2) and hence to a group in case [1]. Assume that G ⊂ PGL(2,C)×PGL(2,C) and denote by π1 and π2 the projec- tions πi : PGL(2,C)×PGL(2,C) → PGL(2,C) on the i-th factor. Since π1(G) and π2(G) are finite Abelian subgroups of PGL(2,C) each is conjugate to a diagonal cyclic group or to the group x 99K ±x±1, isomorphic to (Z/2Z)2. We enumerate the possible cases. If both groups π1(G) and π2(G) are cyclic, the group G is conjugate to a subgroup of the diagonal torus (C∗)2 of automorphisms of the form (x, y) 7→ (αx, βy), α, β ∈ C∗. If exactly one of the two groups π1(G) and π2(G) is cyclic we may assume, up to conjugation in Aut(P1 × P1), that π2(G) is cyclic, generated by y 7→ ζmy, for some integer m ≥ 1, and that π1(G) is the group x 99K ±x ±1. We use the exact sequence 1 → G ∩ kerπ2 → G → π2(G) → 1 and find, up to conjugation, two possibilities for G: (a) G is generated by (x, y) 7→ (x−1, y) and (x, y) 7→ (−x, ζmy). (b) G is generated by (x, y) 7→ (±x±1, y) and (x, y) 7→ (x, ζmy). If m is even, we obtain respectively [2] and [3] for n = m/2. If m is odd, the two groups are equal; conjugating by ϕ : (x, y) 99K (x, y(x + x−1)) (which conjugates (x, y) 7→ (−x, y) to (x, y) 7→ (−x,−y)) we obtain the group [2] for n = m. If both groups π1(G) and π2(G) are isomorphic to (Z/2Z) 2, then up to conju- gation, we obtain three groups, namely (a) G is generated by (x, y) 7→ (−x,−y) and (x, y) 7→ (x−1, y−1). (b) G is generated by (x, y) 7→ (±x,±y) and (x, y) 7→ (x−1, y−1). (c) G is given by (x, y) 7→ (±x±1,±y±1). The group [2] with n = 1 is conjugate to (a) by (x, y) 99K (x, x y+x y+x−1 ). The groups (b) and (c) are respectively equal to [4] and [5]. We now suppose that the group G is not contained in PGL(2,C)×PGL(2,C). Any element ϕ ∈ Aut(P1×P1) not contained in PGL(2,C)×PGL(2,C) is conjugate to ϕ : (x, y) 7→ (α(y), x), where α ∈ Aut(P1), and if ϕ is of finite order, α may be chosen to be y 7→ λy with λ ∈ C∗ a root of unity. Thus, up to conjugation, G is generated by the group H = G ∩ (PGL(2,C)× PGL(2,C)) and one element (x, y) 7→ (λy, x), for some λ ∈ C∗ of finite order. Since the group G is Abelian, every element of H is of the form (x, y) 7→ (β(x), β(y)), for some β ∈ PGL(2,C) satisfying β(λx) = λβ(x). Three possibilities occur, depending on the value of λ which may be 1, −1 or something else. If λ = 1, we conjugate the group by some element (x, y) 7→ (γ(x), γ(y)) so that H is either diagonal or equal to the group generated by (x, y) 7→ (−x,−y) and (x, y) 7→ (x−1, y−1). In the first situation, the group is contained in (C∗)2 ⋊Z/2Z (which gives [1]); the second situation gives [7]. If λ = −1, the group H contains the square of (x, y) 7→ (−y, x), which is (x, y) 7→ (−x,−y) and is either cyclic or generated by (x, y) 7→ (−x,−y) and (x, y) 7→ (x−1, y−1). If H is cyclic, it is diagonal, since it contains (x, y) 7→ (−x,−y), so G is contained in (C∗)2 ⋊ Z/2Z. The second possibility gives [6]. If λ 6= ±1, the group H is diagonal and then G is contained in (C∗)2 ⋊ Z/2Z. We now prove that distinct groups of the list are not birationally conjugate. First of all, each group of case [1] fixes at least one point of P1 × P1. Since the other groups of the list don’t fix any point, they are not conjugate to [1] [Ko-Sz, Proposition A.2]. Consider the other groups. The set of isomorphic groups are those of cases [3] (with n = 1), [4] and [7] (isomorphic to (Z/2Z)3), and of cases [2] (with n = 2) and [6] (isomorphic to Z/2Z× Z/4Z). The groups of cases [2] to [5] leave two pencils of rational curves invariant (the fibres of the two projections P1 × P1 → P1) which intersect freely in exactly one point. We prove that this is not the case for [6] and [7]; this shows that these two groups are not birationally conjugate to any of the previous groups. Take G ⊂ Aut(P1×P1) to be either [6] or [7]. We have then Pic(P1×P1)G = Zd, where d = − 1 KP1×P1 is the diagonal of P 1×P1. Suppose that there exist two G-invariant pencils Λ1 = n1d and Λ2 = n2d of rational curves, for some positive integers n1, n2 (we identify here a pencil with the class of its elements in Pic(P1 × P1)G). The intersection Λ1 · Λ2 = 2n1n2 is an even integer. Note that the fixed part of the intersection is also even, since G is of order 8 and acts without fixed points on P1 × P1. The free part of the intersection is then also an even integer and hence is not 1. Let us now prove that [4] is not birationally conjugate to [3] (with n = 1). This follows from the fact that [4] contains three subgroups that are fixed-point free (the groups generated by (x, y) 7→ (x−1, y−1) and one of the three involutions of the group (x, y) 7→ (±x,±y)), whereas [3] (with n = 1) contains only one such subgroup, which is (x, y) 7→ (±x±1, y). We now prove the last assertion. The finite Abelian groups of automorphisms of P2 are conjugate either to [1] or to the group V9, isomorphic to (Z/3Z) 2 (see Proposition 2.2). As no group of the list [2] through [7] is isomorphic to (Z/3Z)2, we are done. Summary of this section. We have found that the groups common to the three surfaces C2,P2 and P1 × P1 are the ”diagonal” ones (generated by (x, y) 7→ (ζnx, y) and (x, y) 7→ (x, ζmy)). On P 2 there is only one more group, which is the special group V9, and on P 1 × P1 there are 2 families ([2] and [3]) and 4 special groups ([4], [5], [6] and [7]). 3 Some facts about automorphisms of conic bun- We first consider conic bundles without mentioning any group action on them. We recall some classical definitions: Definition 3.1. Let S be a rational surface and π : S → P1 be a morphism. We say that the pair (S, π) is a conic bundle if a general fibre of π is isomorphic to P1, with a finite number of exceptions: these singular fibres are the union of smooth rational curves F1 and F2 such that (F1) 2 = (F2) 2 = −1 and F1 · F2 = 1. Let (S, π) and (S̃, π̃) be two conic bundles. We say that ϕ : S 99K S̃ is a birational map of conic bundles if ϕ is a birational map which sends a general fibre of π on a general fibre of π̃. We say that a conic bundle (S, π) is minimal if any birational morphism of conic bundles (S, π) → (S̃, π̃) is an isomorphism. We remind the reader of the following well-known result: Lemma 3.2. Let (S, π) be a conic bundle. The following conditions are equivalent: • (S, π) is minimal. • The fibration π is smooth, i.e. no fibre of π is singular. • S is a Hirzebruch surface Fm, for some integer m ≥ 0. � Blowing-down one irreducible component in any singular fibre of a conic bundle (S, π), we obtain a birational morphism of conic bundles S → Fm for some integer m ≥ 0. Note that m depends on the choice of the blown-down components. The following lemma gives some information on the possibilities. Note first that since the sections of Fm have self-intersection≥ −m, the self-intersections of the sections of π are also bounded from below. Lemma 3.3. Let (S, π) be a conic bundle on a surface S 6∼= P1 × P1. Let −n be the minimal self-intersection of sections of π and let r be the number of singular fibres of π. Then n ≥ 1 and: 1. There exists a birational morphism of conic bundles p− : S → Fn such that: (a) p− is the blow-up of r points of Fn, none of which lies on the exceptional section En. (b) The strict pull-back Ẽn of En by p− is a section of π with self-intersection 2. If there exist two different sections of π with self-intersection −n, then r ≥ 2n. In this case, there exist birational morphisms of conic bundles p0 : S → F0 = P 1 × P1 and p1 : S → F1. Proof. We denote by s a section of π of minimal self-intersection −n, for some integer n (this integer is in fact positive, as will appear in the proof). Note that this curve intersects exactly one irreducible component of each singular fibre. If r = 0, the lemma is trivially true: take p− to be the identity map. We now suppose that r ≥ 1, and denote by F1, ..., Fr the irreducible components of the singular fibres which do not intersect s. Blowing these down, we get a birational morphism of conic bundles p− : S → Fm, for some integerm ≥ 0. The image of the section s by p− is a section of the conic bundle of Fm of minimal self-intersection, so we get m = n, and n ≥ 0. If we had n = 0, then taking some section s̃ of P1×P1 of self-intersection 0 passing through at least one blown-up point, its strict pull-back by p− would be a section of negative self-intersection, which contradicts the minimality of s2 = −n = 0. We find finally that m = n > 0, and that p−(s) is the unique section Fn of self-intersection −n. This proves the first assertion. We now prove the second assertion. Suppose that some section t 6= s has self- intersection −n. The Picard group of S is generated by s = p∗−(En), the divisor f of a fibre of π and F1, ..., Fr. Write t as t = s+ bf − i=1 aiFi, for some integers b, a1, ..., ar, with a1, ..., ar ≥ 0. We have t 2 = −n and t · (t+KS) = −2 (adjunction formula), where KS = p −(KFn) + i=1 Fi = −(n + 2)f − 2s + i=1 Fi. These relations give: s2 = t2 = s2 − i=1 a i + 2b, n− 2 = t ·KS = −(n+ 2) + 2n− 2b+ i=1 ai, whence i=1 ai = i=1 a i = 2b, so each ai is equal to 0 or 1 and consequently 2b ≤ r. Since s · t = b− n ≥ 0, we find that r ≥ 2n, as announced. Finally, by contracting f − F1, f − F2, ..., f − Fn, Fn+1, Fn+2, ..., Fr, we obtain a birational morphism p0 of conic bundles which sends s on a section of self- intersection 0 and whose image is thus F0. Similarly, the morphism p1 : S → F1 is given by the contraction of f − F1, f − F2, ..., f − Fn−1, Fn, Fn+1, ..., Fr. We now add some group actions on the conic bundles, and give natural defi- nitions (note that we will restrict ourselves to finite or Abelian groups only when this is needed and will then say so): Definition 3.4. Let (S, π) be some conic bundle. • We denote by Aut(S, π) ⊂ Aut(S) the group of automorphisms of the conic bundle, i.e. automorphisms of S that send a general fibre of π on another general fibre. Let G ⊂ Aut(S, π) be some group of automorphisms of the conic bundle (S, π). • We say that a birational map of conic bundles ϕ : S 99K S̃ is G-equivariant if the G-action on S̃ induced by ϕ is biregular (it is clear that it preserves the conic bundle structure). • We say that the triple (G,S, π) is minimal if any G-equivariant birational morphism of conic bundles ϕ : S → S̃ is an isomorphism. Remark 3.5. We insist on the fact that since a conic bundle is for us a pair (S, π), an automorphism of S is not necessarily an automorphism of the conic bundle (i.e. Aut(S) 6= Aut(S, π) in general). One should be aware that in the literature, conic bundle sometimes means ”a variety admitting a conic bundle structure”. Remark 3.6. If G ⊂ Aut(S, π) is such that the pair (G,S) is minimal, so is the triple (G,S, π). The converse is not true in general (see Remark 4.7). Note that any automorphism of the conic bundle acts on the set of singular fibres and on its irreducible components. The permutation of the two components of a singular fibre is very important (Lemma 3.8). For this reason, we introduce some terminology: Definition 3.7. Let g ∈ Aut(S, π) be an automorphism of the conic bundle (S, π). Let F = {F1, F2} be a singular fibre. We say that g twists the singular fibre F if g(F1) = F2 (and consequently g(F2) = F1). If g twists at least one singular fibre of π, we will say that g twists the conic bundle (S, π), or simply (if the conic bundle is implicit) that g is a twisting element. Here is a simple but very important observation: Lemma 3.8. Let G ⊂ Aut(S, π) be a group of automorphisms of a conic bundle. The following conditions are equivalent: 1. The triple (G,S, π) is minimal. 2. Any singular fibre of π is twisted by some element of G. � Remark 3.9. An automorphism of a conic bundle with a non-trivial action on the basis of the fibration may twist at most two singular fibres. However, an automorphism with a trivial action on the basis of the fibration may twist a large number of fibres. We will give in Propositions 6.5 and 6.8 a precise description of all twisting elements. The following lemma is a direct consequence of Lemma 3.3; it provides infor- mation on the structure of the underlying variety of a conic bundle admitting a twisting automorphism. Lemma 3.10. Suppose that some automorphism of the conic bundle (S, π) twists at least one singular fibre. Then, the following occur. 1. There exist two birational morphisms of conic bundles p0 : S → F0 and p1 : S → F1 (which are not g-equivariant). 2. Let −n be the minimal self-intersection of sections of π and let r be the number of singular fibres of π. Then, r ≥ 2n ≥ 2. Proof. Note that any section of π touches exactly one component of each singular fibre. Since g twists some singular fibre, its action on the set of sections of S is fixed-point-free. The number of sections of minimal self-intersection is then greater than 1 and we apply Lemma 3.3 to get the result. Remark 3.11. A result of the same kind can be found in [Isk1], Theorem 1.1. Lemma 3.12. Let G ⊂ Aut(S, π) be a group of automorphisms of the conic bundle (S, π), such that: • π has at most 3 singular fibres (or equivalently (KS) 2 ≥ 5); • the triple (G,S, π) is minimal. Then, S is either a Hirzeburch surface or a del Pezzo surface of degree 5 or 6, depending on whether the number of singular fibres is 0, 3 or 2 respectively. Proof. Let −n be the minimal self-intersection of sections of π and let r ≤ 3 be the number of singular fibres of π. If r = 0, we are done, so we may suppose that r > 0. Since (G,S, π) is minimal, every singular fibre is twisted by some element of G (Lemma 3.8). From Lemma 3.10, we get r ≥ 2n ≥ 2, whence r = 2 or 3 and n = 1, and we obtain the existence of some birational morphism of conic bundles (not G-equivariant) p1 : S → F1. So the surface S is obtained by the blow-up of 2 or 3 points of F1, not on the exceptional section (Lemma 3.3), and thus by blowing-up 3 or 4 points of P2, no 3 of which are collinear (otherwise we would have a section of self-intersection ≤ −2). The surface is then a del Pezzo surface of degree 6 or 5. Remark 3.13. We conclude this section by mentioning an important exact se- quence. Let G ⊂ Aut(S, π) be some group of automorphisms of a conic bundle (S, π). We have a natural homomorphism π : G → Aut(P1) = PGL(2,C) that satisfies π(g)π = πg, for every g ∈ G. We observe that the group G′ = kerπ of automorphisms that leave every fibre invariant embeds in the group PGL(2,C(x)) of automorphisms of the generic fibre P1(C(x)). Then we get the exact sequence 1 → G′ → G → π(G) → 1. (1) This restricts the structure of G; for example if G is Abelian and finite, so are G′ and π(G), and we know that the finite Abelian subgroups of PGL(2,C) and PGL(2,C(x)) are either cyclic or isomorphic to (Z/2Z)2. We also see that the group G is birationally conjugate to a subgroup of the group of birational transformations of P1 × P1 of the form (written in affine coor- dinates): (x, y) 99K ax+ b cx+ d α(x)y + β(x) γ(x)y + δ(x) where a, b, c, d ∈ C, α, β, γ, δ ∈ C(x), and (ad− bc)(αδ − βγ) 6= 0. This group, called the de Jonquières group, is the group of birational transfor- mations of P1 ×P1 that preserve the fibration induced by the first projection, and is isomorphic to PGL(2,C(x))⋊ PGL(2,C). The subgroups of this group can be studied algebraically (as in [Bea2] and [Bla4]) but we will not adopt this point of view here. 4 The del Pezzo surface of degree 6 There is a single isomorphism class of del Pezzo surfaces of degree 6, since all sets of three non-collinear points of P2 are equivalent under the action of linear automorphisms. Consider the surface S6 of degree 6 defined by the blow-up of the points A1 = (1 : 0 : 0), A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1). We may view it in P2 × P2, defined as { (x : y : z), (u : v : w) | ux = vy = wz}, where the blow-down p : S6 → P 2 is the restriction of the projection on one copy of P2, explicitly p : (x : y : z), (u : v : w) 7→ (x : y : z). There are exactly 6 exceptional divisors, which are the pull-backs of the Ai’s by the two projection morphisms. We write Ei = p −1(Ai) and denote by Dij the strict pull-back by p of the line of P2 passing through Ai and Aj . The group of automorphisms of S6 is well known (see for example [Wim], [Do-Iz]). It is isomorphic to (C∗)2 ⋊ (Sym3 × Z/2Z), where (C ∗)2 ⋊ Sym3 is the lift on S6 of the group of automorphisms of P 2 that leave the set {A1, A2, A3} invariant, and Z/2Z is generated by the permutation of the two factors (it is the lift of the standard quadratic transformation (x : y : z) 99K (yz : xz : xy) of P2); the action of Z/2Z on (C∗)2 sends an element on its inverse. There are three conic bundle structures on the surface S6. Let π1 : S6 → P be the morphism defined by (x : y : z), (u : v : w) (y : z) if (x : y : z) 6= (1 : 0 : 0), (w : v) if (u : v : w) 6= (1 : 0 : 0). Note that p sends the fibres of π1 on lines of P 2 passing through A1. There are exactly two singular fibres of this fibration, namely π−11 (1 : 0) = {E2, D12} and π 1 (0 : 1) = {E3, D13}; and E1, D23 are sections of π1. Lemma 4.1. The group Aut(S6, π1) of automorphisms of the conic bundle (S6, π1) acts on the hexagon {E1, E2, E3, D12, D13, D23} and leaves the set {E1, D23} in- variant. 1. The action on the hexagon gives rise to the exact sequence 1 → (C∗)2 → Aut(S6, π1) → (Z/2Z) 2 → 1. 2. This exact sequence is split and Aut(S6, π1) = (C ∗)2 ⋊ (Z/2Z)2, where (a) (C∗)2 is the group of automorphisms of the form( (x : y : z), (u : v : w) (x : αy : βz), (αβu : βv : αw) , α, β ∈ C∗. (b) The group (Z/2Z)2 is generated by the automorphisms (x : y : z), (u : v : w) (x : z : y), (u : w : v) whose action on the set of exceptional divisors is (E2 E3)(D12 D13); (x : y : z), (u : v : w) (u : v : w), (x : y : z) whose action is (E1 D23)(E2 D13)(E3 D12). (c) The action of (Z/2Z)2 on (C∗)2 is generated by permutation of the coordinates and inversion. Proof. Since Aut(S6) acts on the hexagon, so does Aut(S6, π1) ⊂ Aut(S6). Since the group Aut(S6, π1) sends a section on a section, the set {E1, D23} is invariant. The group (C∗)2 leaves the conic bundle invariant, and is the kernel of the action of Aut(S6, π1) on the hexagon. As the set {E1, D23} is invariant, the image is contained in the group (Z/2Z)2 generated by (E2 E3)(D12 D13) and (E1 D23)(E2 D13)(E3 D12). The rest of the lemma follows directly. By permuting coordinates, we have two other conic bundle structures on the surface S6, given by the following morphisms π2, π3 : S6 → P (x : y : z), (u : v : w) (x : z) if (x : y : z) 6= (0 : 1 : 0), (w : u) if (u : v : w) 6= (0 : 1 : 0). (x : y : z), (u : v : w) (x : y) if (x : y : z) 6= (0 : 0 : 1), (v : u) if (u : v : w) 6= (0 : 0 : 1). The description of the exceptional divisors on S6 shows that π1, π2 and π3 are the only conic bundle structures on S6. Lemma 4.2. For i = 1, 2, 3, the pair (Aut(S6, πi), S6) is not minimal. More precisely the morphism πj×πk : S6 → P 1×P1 conjugates Aut(S6, πi) to a subgroup of Aut(P1 × P1), where {i, j, k} = {1, 2, 3}. Proof. The union of the sections E1 and D23 is invariant by the action of the whole group Aut(S6, π1). Since these two exceptional divisors don’t intersect, we can contract both and get a birational Aut(S6, π1)-equivariant morphism from S6 to P1×P1: the pair (Aut(S6, π1), S6) is thus not minimal; explicitly, the birational morphism is given by q 7→ (π2(q), π3(q)), as stated in the lemma. We obtain the other cases by permuting coordinates. Remark 4.3. The subgroup of Aut(P1×P1) obtained in this manner doesn’t leave any of the two fibrations of P1 × P1 invariant. Corollary 4.4. If (G,S6) is a minimal pair (where G ⊂ Aut(S6)), then G does not preserve any conic bundle structure. � We conclude this section with a fundamental example; we will use several times the following automorphism κα,β of (S6, π1): Example 4.5. For any α, β ∈ C∗, we define κα,β to be the following automorphism of (S6, π1): κα,β : (x : y : z), (u : v : w) (u : αw : βv), (x : α−1z : β−1y) Note that κα,β twists the two singular fibres of π1 (see Lemma 4.6 below); its action on the basis of the fibration is (x1 : x2) 7→ (αx1 : βx2) and κ2α,β( (x : y : z), (u : v : w) (x : αβ−1y : α−1βz), (u : α−1βv : αβ−1w) So κα,β is an involution if and only if its action on the basis of the fibration is trivial. Lemma 4.6. Let g ∈ Aut(S6, π1) be an automorphism of the conic bundle (S6, π1). The following conditions are equivalent: • the triple (< g >, S6, π1) is minimal; • g twists the two singular fibres of π1; • the action of g on the exceptional divisors of S6 is (E1 D23)(E2 D12)(E3 D13); • g = κα,β for some α, β ∈ C Proof. According to Lemma 4.1 the action of Aut(S6, π1) on the exceptional curves is isomorphic to (Z/2Z)2 and hence the possible actions of g 6= 1 are these: 1. id, 2. (E2 E3)(D12 D13), 3. (E1 D23)(E2 D13)(E3 D12), 4. (E1 D23)(E2 D12)(E3 D13). In the first three cases, the triple (< g >, S6, π1) is not minimal. Indeed, the blow-down of {E2, E3} or {E2, D13} gives a g-equivariant birational morphism of conic bundles. Hence, if (< g >, S6, π1) is minimal, its action on the exceptional curves is the fourth one above, as stated in the lemma, and it then twists the two singular fibres of π1. Conversely if g twists the two singular fibres of π1, the triple (< g >, S6, π1) is minimal (by Lemma 3.8). It remains to see that the last assertion is equivalent to the others. This follows from Lemma 4.1; indeed this lemma implies that (C∗)2κ1,1 is the set of elements of Aut(S6, π1) inducing the permutation (E1 D23)(E2 D12)(E3 D13). Remark 4.7. The pair (Aut(S6, π1), S6) is not minimal (Lemma 4.2). Consequently < κα,β > is an example of a group whose action on the surface is not minimal, but whose action on a conic bundle is minimal. 5 The del Pezzo surface of degree 5 As for the del Pezzo surface of degree 6, there is a single isomorphism class of del Pezzo surfaces of degree 5. Consider the del Pezzo surface S5 of degree 5 defined by the blow-up p : S5 → P 2 of the points A1 = (1 : 0 : 0), A2 = (0 : 1 : 0), A3 = (0 : 0 : 1) and A4 = (1 : 1 : 1). There are 10 exceptional divisors on S5, namely the divisor Ei = p −1(Ai), for i = 1, ..., 4, and the strict pull-back Dij of the line of P2 passing through Ai and Aj , for 1 ≤ i < j ≤ 4. There are 5 sets of 4 skew exceptional divisors on S5, namely F1 = {E1, D23, D24, D34}, F2 = {E2, D13, D14, D34}, F3 = {E3, D12, D14, D24}, F4 = {E4, D12, D13, D23}, F5 = {E1, E2, E3, E4}. Proposition 5.1. The action of Aut(S5) on the five sets F1, ..., F5 of four skew exceptional divisors of S5 gives rise to an isomomorphism ρ : Aut(S5) → Sym5. Furthermore, the actions of Symn, Altm ⊂ Aut(S5) on S5 given by the canonical embedding of these groups into Sym5 are fixed-point free if and only if n = 3, 4, 5, respectively m = 4, 5. Proof. Since any automorphism in the kernel of ρ leaves E1, E2, E3 and E4 invari- ant and hence is the lift of an automorphism of P2 that fixes the 4 points, the homomorphism ρ is injective. We now prove that ρ is also surjective. Firstly, the lift of the group of au- tomorphisms of P2 that leave the set {A1, A2, A3, A4} invariant is sent by ρ on Sym4 = Sym{F1,F2,F3,F4}. Secondly, the lift of the standard quadratic transforma- tion (x : y : z) 99K (yz : xz : xy) is an automorphism of S5, as its lift on S6 is an automorphism, and as it fixes the point A4; its image by ρ is (F4 F5). It remains to prove the last assertion. First of all, it is clear that the actions of the cyclic groups Alt3 and Sym2 fix some points. The group Sym3 ⊂ Aut(P of permutations of A1, A2 and A3 fixes exactly one point, namely (1 : 1 : 1). The blow-up of this point gives a fixed-point free action on F1, and thus its lift on S5 is also fixed-point free. The group Alt4 ⊂ Aut(P 2) contains the element (x : y : z) 7→ (z : x : y) (which corresponds to (1 2 3)) that fixes exactly three points, i.e. (1 : a : a2) for a3 = 1. It also contains the element (x : y : z) 7→ (z − y : z − x : z) (which corresponds to (1 2)(3 4)) that does not fix (1 : a : a2) for a3 = 1. Thus, the action of Alt4 on P 2 is fixed-point free and the same is true on S5. Remark 5.2. The structure of Aut(S5) is classical and can be found for example in [Wim] and [Do-Iz]. Lemma 5.3. Let π : S5 → P 1 be some morphism inducing a conic bundle (S5, π). There are exactly four exceptional curves of S5 which are sections of π; the blow- down of these curves gives rise to a birational morphism p : S5 → P 2 which conjugates the group Aut(S5, π) ∼= Sym4 to the subgroup of Aut(P 2) that leaves invariant the four points blown-up by p. In particular, the pair (Aut(S5, π), S5) is not minimal. Proof. Blowing-down one component in any singular fibre, we obtain a birational morphism of conic bundles (not Aut(S5, π)-equivariant) from S5 to some Hirze- bruch surface Fn. Since S5 does not contain any curves of self-intersection ≤ −2, n is equal to 0 or 1. Changing the component blown-down in a singular fibre performs an elementary link Fn 99K Fn±1; we may then assume that n = 1, and that F1 is the blow-up of A1 ∈ P 2. Consequently, the fibres of the conic bundles correspond to the lines passing through A1. Denoting by A2, A3, A4 the other points blown-up by the constructed birational morphism S5 → P 2 and using the same notation as before, the three singular fibres are {Ei, D1i} for i = 2, ..., 4, and the other excep- tional curves are four skew sections of the conic bundle, namely the elements of F1 = {E1, D23, D24, D34}. The blow-down of F1 gives an Aut(S5, π)-equivariant birational morphism (that is not a morphism of conic bundles) p : S5 → P 2 and conjugates Aut(S5, π) to a subgroup of the group Sym4 ⊂ Aut(P 2) of automor- phisms that leaves the four points blown-up by p invariant. The fibres of π are sent on the conics passing through the four points, so the lift of the whole group Sym4 belongs to Aut(S5, π). Corollary 5.4. Let G be some group of automorphisms of a conic bundle (S, π) such that the pair (G,S) is minimal and (KS) 2 ≥ 5 (or equivalently such that the number of singular fibres of π is at most 3). Then, the fibration is smooth, i.e. S is a Hirzebruch surface. Proof. Since (G,S) is minimal, so is the triple (G,S, π). By Lemma 3.12, the surface S is either a Hirzebruch surface, or a del Pezzo surface of degree 5 or 6. Corollary 4.4 shows that the del Pezzo surface of degree 6 is not possible and Lemma 5.3 eliminates the possibility of the del Pezzo surface of degree 5. 6 Description of twisting elements In this section, we describe the twisting automorphisms of conic bundles, which are the most important automorphisms (see Lemma 3.8). Lemma 6.1 (Involutions twisting a conic bundle). Let g ∈ Aut(S, π) be a twist- ing automorphism of the conic bundle (S, π). Then, the following properties are equivalent: 1. g is an involution; 2. π(g) = 1, i.e. g has a trivial action on the basis of the fibration; 3. the set of points of S fixed by g is an irreducible hyperelliptic curve of genus (k − 1) – a double covering of P1 by means of π, ramified over 2k points – plus perhaps a finite number of isolated points, which are the singular points of the singular fibres not twisted by g. Furthermore, if the three conditions above are satisfied, the number of singular fibres of π twisted by g is 2k ≥ 2. Proof. 1 ⇒ 2: By contracting some exceptional curves, we may assume that the triple (< g >, S, π) is minimal. Suppose that g is an involution and π(g) 6= 1. Then g may twist only two singular fibres, which are the fibres of the two points of P1 fixed by π(g). Hence, the number of singular fibres is ≤ 2. Lemma 3.12 tells us that S is a del Pezzo surface of degree 6 and then Lemma 4.6 shows that g = κα,β (Example 4.5) for some α, β ∈ C ∗. But such an element is an involution if and only if it acts trivially on the basis of the fibration. (1 and 2) ⇒ 3: Suppose first that (< g >, S, π) is minimal. This implies that g twists every singular fibre of π. Therefore, since π(g) = 1 and g2 = 1, on a singular fibre there is one point fixed by g (the singular point of the fibre) and on a general fibre there are two fixed points. The set of points of S fixed by g is thus a smooth irreducible curve. The projection π gives it as a double covering of P1 ramified over the points whose fibres are singular and twisted by g. By the Riemann-Hurwitz formula, this number is even, equal to 2k and the genus of the curve is k − 1. The situation when (< g >, S, π) is not minimal is obtained from this one, by blowing-up some fixed points. This adds in each new singular fibre (not twisted by the involution) an isolated point, which is the singular point of the singular fibre. We then get the third assertion and the final remark. 3 ⇒ 2: This implication is clear. 2 ⇒ 1: If π(g) = 1, then, g2 leaves every component of every singular fibre of π invariant. Let p1 : S → F1 be the birational morphism of conic bundles given by Lemma 3.10; it is a g2-equivariant birational morphism which conjugates g2 to an automorphism of F1 that necessarily fixes the exceptional section. The pull-back by p1 of this section is a section C of π, fixed by g 2. Since C touches exactly one component of each singular fibre (in particular those that are twisted by g), g sends C on another section D also fixed by g2. The union of the sections D and C intersects a general fibre in two points, which are exchanged by the action of g. This implies that g has order 2. We now give some further simple results on twisting involutions. Corollary 6.2. Let (S, π) be some conic bundle. No involution twisting (S, π) has a root in Aut(S, π) which acts trivially on the basis of the fibration. Proof. Such a root must twist a singular fibre and so (Lemma 6.1) is an involution. Remark 6.3. There may exist some roots in Aut(S, π) of twisting involutions which act non trivially on the basis of the fibration. Take for example four general points A1, ..., A4 of the plane and denote by g ∈ Aut(P2) the element of order 4 that permutes these points cyclically. The blow-up of these points conjugates g to an automorphism of the del Pezzo surface S5 of degree 5 (see Section 5). The pencil of conics of P2 passing through the four points induces a conic bundle structure on S5, with three singular fibres which are the lift of the pairs of two lines passing through the points. The lift on S5 of g is an automorphism of the conic bundle whose square is a twisting involution. Corollary 6.4. Let (S, π) be some conic bundle and let g ∈ Aut(S, π). The following conditions are equivalent. 1. g twists more than 2 singular fibres of π. 2. g fixes a curve of positive genus. And these conditions imply that g is an involution which acts trivially on the basis of the fibration and twists at least 4 singular fibres. Proof. The first condition implies that g acts trivially on the basis of the fibration, and thus (by Lemma 6.1) that g is an involution which fixes a curve of positive genus. Suppose that g fixes a curve of positive genus. Then, g acts trivially on the basis of the fibration, and fixes 2 points on a general fibre. Consequently, the curve fixed by g is a smooth hyperelliptic curve; we get the remaining assertions from Lemma 6.1. As we mentioned above, the automorphisms that twist some singular fibre are fundamental (Lemma 3.8). We now describe these elements and prove that the only possibilities are twisting involutions, roots of twisting involutions (of even or odd order) and elements of the form κα,β (see Example 4.5): Proposition 6.5 (Classification of twisting elements of finite order). Let g ∈ Aut(S, π) be a twisting automorphism of finite order of a conic bundle (S, π). Let n be the order of its action on the basis. Then gn is an involution that acts trivially on the basis of the fibration and twists an even number 2k of singular fibres; furthermore, exactly one of the fol- lowing situations occurs: 1. n = 1. 2. n > 1 and k = 0; in this case n is even and there exists a g-equivariant bi- rational morphism of conic bundles η : S → S6 (where S6 is the del Pezzo surface of degree 6) such that ηgη−1 = κα,β for some α, β ∈ C ∗ (see Exam- ple 4.5). 3. n > 1 is odd and k > 0; here g twists 1 or 2 fibres, which are the fibres twisted by gn that are invariant by g. 4. n is even and k > 0; here g twists r = 1 or 2 singular fibres; none of them are twisted by gn; moreover the action of g on the set of 2k fibres twisted by gn is fixed-point free; furthermore, n divides 2k, and 2k/n ≡ r (mod 2). Proof. Lemma 6.1 describes the situation when n = 1. We now assume that n > 1; by blowing-down some components of singular fibres we may also suppose that the triple (G,S, π) is minimal. Denote by a1, a2 ∈ P 1 the two points fixed by π(g) ∈ Aut(P1). For i 6≡ 0 (mod n) the element π(gi) fixes only two points of P1, namely a1 and a2 (since π(g) has order n); the only possible fibres twisted by gi are thus π−1(a1), π −1(a2). Suppose that gn does not twist any singular fibre. By minimality there are at most 2 singular fibres (π−1(a1) and/or π −1(a2)) of π and g twists each one. Lemma 3.12 tells us that S is a del Pezzo surface of degree 6 and Lemma 4.6 shows that g = κα,β : (x : y : z), (u : v : w) (u : αw : βv) , (x : α−1z : β−1y) for some α, β ∈ C∗. We compute the square of g and find (x : y : z), (u : v : w) (x : αβ−1y : α−1βz) , (u : α−1βv : αβ−1w) Consequently, the order of g is 2n. The fact that gi twists π−1(a1) and π −1(a2) when i is odd implies that n is even. Case 2 is complete. If gn twists at least one singular fibre, it twists an even number of singular fibres (Lemma 6.1) which we denote by 2k, and gn is an involution. If n is odd, each fibre twisted by gn is twisted by g, and conversely; this yields case 3. It remains to consider the more difficult case when n is even. Firstly we observe that there are r + 2k singular fibres with r ∈ {1, 2}, cor- responding to the points a1 and/or a2, c1, ..., c2k of P 1, the first r of them be- ing twisted by g and the 2k others by gn. Under the permutation π(g), the set {c1, ..., c2k} decomposes into disjoint cycles of length n (this action is fixed- point-free); this shows that n divides 2k. We write t = 2k/n ∈ N and set {c1, ..., c2k} = ∪ i=1Ci, where each Ci ⊂ P 1 is an orbit of π(g) of size n. To deduce the congruence r ≡ t (mod 2), we study the action of g on Pic(S). For i ∈ {1, ..., t}, choose Fi to be a component in the fibre of the singular fibre of some point of Ci, and for i ∈ {1, r} choose Li to be a component in the fibre of ai. Let us write i=1(Fi + g(Fi) + ...+ g n−1(Fi)) + i=1 Li ∈ Pic(S). Denoting by f ⊂ S a general fibre of π, we find the equalities g(Li) = f − Li and gn(Fi) = f − Fi in Pic(S), which yield (once again in Pic(S)): g(R) = R+ (r + t)f − 2( i=1 Li + i=1 Fi). The contraction of the divisor R gives rise to a birational morphism of conic bundles (not g-equivariant) ν : S → Fm for some integer m ≥ 0. Denote by s ⊂ S the pull-back by ν of a general section of Fm of self-intersection m (which does not pass through any of the base-points of ν−1). The canonical divisor KS of S is then equal in Pic(S) to the divisor −2s+ (m − 2)f + R. We compute g(2s) and 2(g(s)− s) = g(2s)− 2s in Pic(S): g(2s) = g(−KS + (m− 2)f +R) = −KS + (m− 2)f + g(R); g(2s)− 2s = g(R)−R = (r + t)f − 2( i=1 Li + i=1 Fi). This shows that (r + t)f ∈ 2Pic(S), which implies that r ≡ t (mod 2). Case 4 is complete. Corollary 6.6. If g ∈ Aut(S, π) is a root of a twisting involution h that fixes a rational curve (i.e. that twists 2 singular fibres) and if g twists at least one fibre not twisted by h, then g2 = h, g twists exactly one singular fibre, and it exchanges the two fibres twisted by h. Proof. We apply Proposition 6.5 and obtain case 4 with k = 1. Corollary 6.6 and the following result will be useful in the sequel. Lemma 6.7. Let g ∈ Aut(S, π) be a non-trivial automorphism of finite order that leaves every component of every singular fibre of π invariant (i.e. that acts trivially on Pic(S)) and let h ∈ Aut(S, π) be an element that commutes with g. Then, either no singular fibre of π is twisted by h or each singular fibre of π which is invariant by h is twisted by h. Proof. If no twisting element belongs to Aut(S, π), we are done. Otherwise, the birational morphism of conic bundles p0 : S → P 1 × P1 given by Lemma 3.10 conjugates g to an element of finite order of Aut(P1 × P1, π1) whose set of fixed points is the union of two rational curves. The set of points of S fixed by g is thus the union of two sections and a finite number of points (which are the singular points of the singular fibres of π). Any element h ∈ Aut(S, π) that commutes with g leaves the set of these two sections invariant. More precisely, the action on one invariant singular fibre F implies the action on the two sections: h exchanges the two sections if and only if it twists F . Since the situation is the same at any other singular fibre, we obtain the result. We conclude this section with some results on automorphisms of infinite order of conic bundles, which will not help us directly here but seem interesting to observe. Proposition 6.8 (Classification of twisting elements of infinite order). Let (S, π) be a conic bundle and g ∈ Aut(S, π) be a twisting automorphism of infinite order. Then g twists exactly two fibres of π and there exists some g-equivariant bira- tional morphism of conic bundles η : S → S6, where S6 is the del Pezzo surface of degree 6 and ηgη−1 = κα,β for some α, β ∈ C Proof. Assume that the triple (< g >, S, π) is minimal. Lemma 6.1 shows that no twisting element of infinite order acts trivially on the basis of the fibration. Consequently, gk acts trivially on the basis if and only if k = 0, whence gk twists a fibre F if and only if k is odd and g twists F . There thus exist at most 2 singular fibres of π, and Lemma 3.12 tells us that S is a del Pezzo surface of degree 6. Lemma 4.6 shows that g = κα,β for some α, β ∈ C Corollary 6.9. Let g ∈ Aut(S, π) be an element of infinite order; then a birational morphism conjugates g to an automorphism of a Hirzebruch surface. Proof. Assume that the triple (< g >, S, π) is minimal. If the fibration is smooth, we are done. Otherwise, a birational morphism conjugates g to an automorphism κα,β ∈ Aut(S6) of a conic bundle on the del Pezzo surface of degree 6 (Lemma 6.8). We conclude by using Lemma 4.2. 7 The example Cs24 We now give the most important example of this paper. This is the only finite Abelian subgroup of the Cremona group which is not conjugate to a group of automorphisms of P2 or P1 × P1 but whose non-trivial elements do not fix any curve of positive genus (Theorem 2). Let S6 ⊂ P 2 × P2 be the del Pezzo surface of degree 6 (see Section 4) defined S6 = { (x : y : z), (u : v : w) | ux = yv = zw}; we keep the notation of Section 4. We denote by η : Ŝ4 → S6 the blow-up of A4, A5 ∈ S6 defined by (0 : 1 : 1) , (1 : 0 : 0) ∈ D23, (1 : 0 : 0) , (0 : 1 : −1) ∈ E1. We again denote by E1, E2, E3, D12, D13, D23 the total pull-backs by η of these divisors of S6. We denote by Ẽ1 and D̃23 the strict pull-backs of E1 and D23 by η. (Note that for the other exceptional divisors, the strict and total pull-backs are the same.) Let us illustrate the situations on the surfaces S6 and Ŝ4 respectively: E2 D15 E4 E3 D12 E5 D14 D13 Let π1 denote the morphism S6 → P 1 defined in Section 4. The morphism π = π1 ◦ η gives the surface Ŝ4 a conic bundle structure (Ŝ4, π). It has 4 singular fibres, which are the fibres of (−1 : 1), (0 : 1), (1 : 1) and (1 : 0). We denote by f the divisor of Ŝ4 corresponding to a fibre of π and set E4 = η −1(A4), E5 = η −1(A5). Note that E4 is one of the components of the singular fibre of (1 : 1); we denote by D14 = f−E4 the other component, which is the strict pull-back by η of π 1 (1 : 1). Similarly, we denote by D15 the divisor f−E5, so that the singular fibre of (−1 : 1) is {E5, D15}. Lemma 7.1. On the surface Ŝ4 there are exactly 10 irreducible rational smooth curves of negative self-intersection. Explicitly, the 8 curves E2, E3, E4, E5, D12, D13, D14, D15 have self-intersection −1 and the two curves Ẽ1 = E1 − E5 and D̃23 = D23 − E4 have self-intersection −2. Proof. The difficult part is to show that every rational irreducible smooth curve of negative self-intersection is one of the ten given above. Let C be such a curve. Denote by L the pull-back of a general line of P2 by the blow-up pr1 ◦ η : Ŝ4 → P2 of the five points. If C is collapsed by pr1 ◦ η, then C is one of the curves Ẽ1, E2, E3, E4, E5. Otherwise, C = mL − i=1 aiEi, where m, a1, ..., a5 are non- negative integers, and m > 0. Since C is rational we have C · (C + K ) = −2, and by hypothesis C2 = −r for some positive integer r. The relations C2 = −r and C ·K = r − 2 imply (since K = −3L+ i=1 Ei) the equations i=1 a i = m 2 + r, i=1 ai = 3m+ r − 2. If m = r = 1, we find that C is the pull-back of a line passing through two of the points, so C = D1i for some i ∈ {2, ..., 5}. If m = 2 and r = 1, C is the pull-back of a conic passing through each blown-up point. The configuration of the points eliminates this possibility. If m = 1 and r = 2, we obtain a line passing through three blown-up points, so C = D̃23. We now prove that if there is no integral solution to (2) for m, r ≥ 2. Since i=1 ai) 2 ≤ 5( i=1 a i ) (by the Cauchy-Schwarz inequality with the vectors (1, ..., 1) and (a1, ..., a5)), we obtain (3m+ (r − 2)) 2 ≤ 5(m2 + r), and this gives 4m2 − 10 + (r − 2) · (6m+ r − 7) ≤ 0. But this is not possible if m, r ≥ 2, since in this case 4m2 > 10 and 6m+r > 7. Note that (K )2 = 4, which is why we denote this surface by Ŝ4; the hat is here because the surface is not a del Pezzo surface, since it contains irreducible divisors of self-intersection −2. Corollary 7.2. There is only one conic bundle structure on Ŝ4, which is the one induced by π = π1 ◦ η. Proof. Since (K )2 = 4, the number of singular fibres of any conic bundle is 4, and thus it consists of eight (−1)-curves C1, ..., C8. The divisor of a fibre of the conic bundle is thus equal to 1 i=1 Ci. Since there are exactly eight (−1)-curves on Ŝ4, there is no choice. The group of automorphisms of Ŝ4 that leave every curve of negative self- intersection invariant is isomorphic to C∗ and corresponds to automorphisms of P2 of the form (x : y : z) 7→ (αx : y : z), for α ∈ C∗. Indeed, such automorphisms are the lifts of automorphisms of S6 leaving invariant every exceptional curve (which are of the form (x : y : z), (u : v : w) (x : αy : βz), (u : α−1v : β−1w) , for α, β ∈ C∗) and which fix both points A4 and A5. Definition 7.3. Let h1 and h2 be the following birational transformations of P h1 : (x : y : z) 99K (yz : xy : −xz) h2 : (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)) and denote respectively by g1, g2 the lift of these elements on Ŝ4 and by Cs24 the group generated by g1 and g2. The following lemma shows that Cs24 ⊂ Aut(Ŝ4, π) and describes some of the properties of this group. Lemma 7.4. Let h1, h2, g1, g2,Cs24 be as in Definition 7.3. Then: 1. The group Cs24 is a group of automorphisms of Ŝ4 that preserve the conic bundle (Ŝ4, π), i.e. Cs24 ⊂ Aut(Ŝ4, π). 2. The action of g1 and g2 on the set of irreducible rational curves of negative self-intersection is respectively: (Ẽ1 D̃23)(E2 D12)(E3 D13)(E4 E5)(D14 D15), (Ẽ1 D̃23)(E2 D13)(E3 D12)(E4 D14)(E5 D15). In particular, both g1 and g2 twist the conic bundle (Ŝ4, π). 3. Both g1 and g2 are elements of order 4 and 2 = (h2) 2 = (x : y : z) 7→ (−x : y : z). Thus (g1) 2 = (g2) 2 ∈ kerπ is an automorphism of Ŝ4 which leaves every divisor of negative self-intersection invariant. 4. The group Cs24 is isomorphic to Z/2Z × Z/4Z and the action on the basis of the fibration π yields the exact sequence 1 →< (h1) 2 >∼= Z/2Z → Cs24 →< π(h1), π(h2) >∼= (Z/2Z) 2 → 1. 5. The group Cs24 contains no involution that twists the conic bundle (Ŝ4, π). In particular, no element of Cs24 fixes a curve of positive genus. 6. The pair (Cs24, Ŝ4) and the triple (Cs24, Ŝ4, π) are both minimal. Proof. Observe first that h1 and h2 preserve the pencil of lines of P 2 passing through the point A1 = (1 : 0 : 0), so g1, g2 are birational transformations of Ŝ4 that send a general fibre of π on another fibre. Then, we compute (h1) 2 = (h2) (x : y : z) 7→ (−x : y : z). This implies that both h1 and h2 are birational maps of order 4. Note that the lift of h1 on the surface S6 is the automorphism κ1,−1 : (x : y : z), (u : v : w) (u : w : −v), (x : z : −y) (see Example 4.5). Since this automorphism permutes A4 and A5, its lift on Ŝ4 is biregular. The action on the divisors with negative self-intersection is deduced from that of κ1,−1 (see Lemma 4.6). Compute the involution h3 = h1h2 = (x : y : z) 99K (x(y + z) : z(y − z) : −y(y − z)). Its linear system is {ax(y + z) + (by + cz)(y − z) = 0 | (a : b : c) ∈ P2}, which is the linear sytem of conics passing through (0 : 1 : 1) and A1 = (1 : 0 : 0), with tangent y + z = 0 at this point (i.e. passing through A5). Blowing-up these three points (two on P2 and one in the blow-up of A1), we get an automorphism g3 of some rational surface. As the points A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1) are permuted by h3, we can also blow them up and again get an automorphism. The isomorphism class of the surface obtained is independent of the order of the blown-up points. We may first blow-up A1, A2, A3 and get S6. Then, we blow-up the two other base-points of h3, which are in fact A4 (the point (0 : 1 : −1)) and A5 (the point infinitely near to A1 corresponding to the tangent y + z = 0). This shows that g3, and therefore g2, belong to Aut(Ŝ4, π). Since h3 permutes the points A2 and A3, g3 = g1g2 permutes the divisors E2 and E3. It also permutes D12 and D13, since h3 leaves the pencil of lines passing through A1 invariant. It therefore leaves Ẽ1 and D̃23 invariant, since E2 and E3 touch D̃23 but not E1. The remaining exceptional divisors are E4, E5, D14, D15. Either g1g2 leaves all four invariant, or it acts as (E4 D15)(E5 D14) (using the intersection with Ẽ1 and D̃23). Since A4 and A5 are base-points of h1h2, E4 and E5 are not invariant. Thus, g1g2 acts on the irreducible rational curves of negative self-intersection as (E2 E3)(D12 D13)(E4 D15)(E5 D14). We obtain the action of g2 by composing that of g1g2 with that of g1 and thus have proved assertions 1 through 3. Assertion 4 follows from assertion 3 and the fact that g1 and g2 commute. Let us prove that Cs24 contains no involution that twists the conic bundle (Ŝ4, π). Recall that such elements are involutions acting trivially on the basis of the fibration (see Lemma 6.1). Note that the 2-torsion of Cs24 is equal to {1, g21, g1g2, g1g 2 }. The elements g1g2 and g1g 2 do not act trivially on the basis of the fibration, and the element (g1) 2 does not twist any singular fibre since it leaves every curve of negative self-intersection invariant. This proves assertion 5. It remains to prove the last assertion. Observe that the orbits of the action of Cs24 on the exceptional divisors of Ŝ4 are {E2, E3, D12, D13} and {E4, E5, D14, D15}. Since these orbits cannot be contracted, the pair (Cs24, Ŝ4) is minimal, and so is the triple (Cs24, Ŝ4, π). Remark 7.5. The pair (Cs24, Ŝ4) was introduced in [Bla2] and was called Cs.24 because it is a group acting on a conic bundle, which is special, and isomorphic to Z/2Z× Z/4Z. 8 Finite Abelian groups of automorphisms of conic bundles - birational representative elements In this section we use the tools prepared in the previous sections to describe the finite Abelian groups of automorphisms of conic bundles such that no non-trivial element fixes a curve of positive genus. We first treat the case in which no involution twisting the conic bundle belongs to the group: Proposition 8.1. Let G ⊂ Aut(S, π) be a finite Abelian group of automorphisms of the conic bundle (S, π) such that: • no involution that twists the conic bundle (S, π) belongs to G; • the triple (G,S, π) is minimal. Then, one of the following occurs: • The fibration is smooth, i.e. S is a Hirzebruch surface. • S is the del Pezzo surface of degree 6. • The triple (G,S, π) is isomorphic to the triple (Cs24, Ŝ4, π) of Section 7. Proof. We assume that the fibration is not smooth. Recall that since the triple (G,S, π) is minimal, any singular fibre of π is twisted by an element of G (by Lemma 3.8). Since no twisting involution belongs to G, any element g ∈ G that twists a fibre corresponds to case 2 of Proposition 6.5. In particular, g is the lift on S of an automorphism of the form κα,β of the del Pezzo surface of degree 6 and it twists 2 singular fibres, which correspond to the fibres of the two fixed points of π(g) ∈ PGL(2,C). Furthermore, g is the root of an involution that leaves every component of every singular fibre of π invariant. If the number of singular fibres is exactly two, then S is the del Pezzo surface of degree 6, and we are done. Now suppose that the number of singular fibres is larger than two. This implies that π(G) is not a cyclic group (otherwise the non-trivial elements of π(G) would have the same two fixed points: there would then be at most two singular fibres); therefore, π(G) is isomorphic to (Z/2Z)2. By a judicious choice of coordinates we may suppose that π(G) = Since a singular fibre corresponds to a fixed point of one of the three elements of order 2 of π(G), only the fibres of (0 : 1), (1 : 0), (1 : 1), (−1 : 1), (i : 1), (−i : 1) can be singular. Since the group π(G) acts transitively on the sets {(1 : 0), (0 : 1)}, {(1 : ±1)} and {(1 : ±i)}, there are 4 or 6 singular fibres. We denote by g1 an element of G which twists the two singular fibres of (1 : 0) and (0 : 1). Let η : S → S6 denote the birational g1-equivariant morphism given by Proposition 6.5, which conjugates g1 to the automorphism −1 = κα,β : (x : y : z), (u : v : w) (u : αw : βv), (x : α−1z : β−1y) of the del Pezzo surface S6 of degree 6, for some α, β ∈ C ∗. In fact, since π(g1) has order 2, we have β = −α, so ηg1η −1 = κα,−α. The points blown-up by η are fixed by η(g1) 2η−1 = (κα,−α) (x : y : z), (u : v : w) (x : −y : −z), (u : −v : −w) and therefore belong to the curves E1 = { (1 : 0 : 0), (0 : a : b) | (a : b) ∈ P1} and D23 = { (0 : a : b), (1 : 0 : 0) | (a : b) ∈ P1}. Since these points consist of orbits of ηg1η −1, half of them lie in E1 and the other half in D23. In fact, up to a change of coordinates, (x, y, z), (u, v, w) (u, v, w), (x, y, z) , the points that may be blown-up by η are (0 : 1 : 1) , (1 : 0 : 0) ∈ D23, κα,−α(A4) = A5 = (1 : 0 : 0) , (0 : 1 : −1) ∈ E1, (0 : 1 : i) , (1 : 0 : 0) ∈ D23, κα,−α(A6) = A7 = (1 : 0 : 0) , (0 : 1 : i) ∈ E1. The strict pull-backs Ẽ1 and D̃23 by η of E1 and D23 respectively thus have self- intersection −2 or −3 in S, depending on the number of points blown-up. By convention we again denote by E1, E2, E3, D12, D13, D23 the total pull-backs by η of these divisors. (Note that for E2, E3, D12, D13, the strict and the total pull- backs are the same.) We set E4 = η −1(A4),..., E7 = η −1(A7) and denote by f the divisor class of the fibre of the conic bundle. (a) Suppose that η is the blow-up of A4 and A5, which implies that S is the surface Ŝ4 of Section 7. The Picard group of S is then generated by E1, E2, ..., E5 and f . Since we assumed that (G,S, π) is minimal, the singular fibres of (1 : 1) and (−1 : 1) must be twisted. One element g2 twists these two singular fibres and acts with order 2 on the basis of the fibration, with action (x1 : x2) 7→ (x2 : x1). Since g1 and g2 twist some singular fibre, both must invert the two curves of self- intersection −2, namely Ẽ1 and D̃23. The action of g1 and g2 on the irreducible rational curves of negative self-intersection is then respectively (Ẽ1 D̃23)(E2 D12)(E3 D13)(E4 E5)(D14 D15), (Ẽ1 D̃23)(E2 D13)(E3 D12)(E4 D14)(E5 D15). The elements g1 and g2 thus have the same action on Pic(S) = Pic(Ŝ4) as the two automorphisms with the same name in Definition 7.3 and Lemma 7.4, which generate Cs24. Note that the group H of automorphisms of S that leave every curve of negative self-intersection invariant is isomorphic to C∗ and corresponds to automorphisms of P2 of the form (x : y : z) 7→ (αx : y : z), for any α ∈ C∗. Then, g1 and g2 are equal to the lift of the the following birational maps of P h1 : (x : y : z) 99K (µyz : xy : −xz), h2 : (x : y : z) 99K (νyz(y − z) : xz(y + z) : xy(y + z)), for some µ, ν ∈ C∗. As h1h2(x : y : z) = (µx(y + z) : νz(y − z) : −νy(y − z)) and h2h1(x : y : z) = (νx(y + z) : µz(y − z) : −µy(y − z)) must be the same by hypothesis, we get µ2 = ν2. We observe that π(g1) and π(g2) generate π(G) ∼= (Z/2Z) 2; on the other hand, by hypothesis an element of G′ does not twist a singular fibre and hence belongs to H . As the only elements of H which commute with g1 are id and (g1) 2 (which is the lift of (h1) 2 : (x : y : z) 7→ (−x : y : z)), we see that g1 and g2 generate the whole group G. Conjugating h1 and h2 by (x : y : z) 7→ (αx : y : z), where α ∈ C ∗, α2 = µ, we may suppose that µ = 1. So ν = ±1 and we get in both cases the same group, because (h1) 2(x : y : z) = (−x : y : z). The triple (G,S, π) is hence isomorphic to the triple (Cs24, Ŝ4, π) of Section 7. (b) Suppose that η is the blow-up of A6 and A7. We get a case isomorphic to the previous one, using the automorphism (x : y : z), (u : v : w) (x : y : iz), (u : v : −iw) of S6. (c) Suppose that η is the blow-up of A4, A5, A6 and A7. The Picard group of S is then generated by E1, E2, ..., E6, E7 and f . Since (G,S, π) is minimal, there must be two elements g2, g3 ∈ G that twist respectively the fibres of (±1 : 1) and those of (±i : 1). As in the previous example, the three actions of these elements on the basis are of order 2, and the three elements transpose Ẽ1 and D̃23. The actions of g1, g2 and g3 on the set of irreducible components of the singular fibres of π are then respectively (E2 D12)(E3 D13)(E4 E5)(D14 D15)(E6 E7)(D16 D17), (E2 D13)(E3 D12)(E4 D14)(E5 D15)(E6 E7)(D16 D17), (E2 D13)(E3 D12)(E4 E5)(D14 D15)(E6 D16)(E7 D17). This implies that the action of the element g1g2g3 is (E2 D12)(E3 D13)(E4 D14)(E5 D15)(E6 D17)(E7 D16), and thus it twists six singular fibres of the conic bundle and fixes a curve of genus 2 (Lemma 6.1), which contradicts the hypothesis. (In fact, one can also show that the group generated by g1, g2 and g3 is not Abelian, see [Bla2], page 66.) After studying the groups that do not contain a twisting involution, we now study those which contain such elements. Since these twisting involutions cannot fix a curve of positive genus, they twist exactly two fibres (Lemma 6.1). Proposition 8.2. Let G ⊂ Aut(S, π) be a finite Abelian group of automorphisms of a conic bundle (S, π) such that: 1. If g ∈ G, g 6= 1, then g does not fix a curve of positive genus. 2. The group G contains at least one involution that twists the conic bundle (S, π). 3. The triple (G,S, π) is minimal. Then, S is a del Pezzo surface of degree 5 or 6. Proof. If the number of singular fibres is at most 3, then the surface is a del Pezzo surface of degree 5 or 6 (Lemma 3.12). We now assume that the number of singular fibres is at least 4 and show that this situation is not compatible with the hypotheses. We recall once again the exact sequence of Remark 3.13 1 → G′ → G → π(G) → 1, (1) and prove the following important assertions: (a) No element of G twists more than two singular fibres. (b) Any twisting involution that belongs to G belongs to G′ and twists exactly two singular fibres. (c) Any singular fibre is twisted by an element of G. (d) No non-trivial element preserves every component of every singular fibre. (e) Any twisting element of G is a root of (or equal to) a twisting involution that belongs to G′. Corollary 6.4 shows that an element that twists more than two fibres fixes a curve of positive genus; since this possibility is excluded by hypothesis, we obtain assertion (a). Lemma 6.1 shows that any twisting involution contained in G be- longs to G′ and twists an even number of fibres; using assertion (a), we thus obtain assertion (b). Assertion (c) follows from the minimality of the triple (G,S, π) (see Lemma 3.8). Let us prove assertion (d). Suppose that there exists a non-trivial element g ∈ G that leaves every component of every singular fibre invariant, and denote by h ∈ G′ a twisting involution (which exists by hypothesis). Since g and h commute, Lemma 6.7 shows that each singular fibre invariant by h – there are at least 4 – is twisted by h, which contradicts assertion (a). Therefore, such an element g doesn’t exist and assertion (d) is proved. Finally, Proposition 6.5 shows that any twisting element that does not act trivially on the basis of the fibration is a root of an involution that belongs to G′, and assertion (d) shows that this involution is twisting, and we obtain assertion (e). Now that assertions (a) through (e) are proved, we deduce the proposition from them. Let us denote by σ ∈ G′ a twisting involution, which twists two singular fibres that we denote by F1 and F2. There are at least two other singular fibres F3 and F4 that are twisted by other elements of G. If G′ =< σ >, the fibres F3 and F4 are twisted by roots of σ belonging to G (assertions (c) and (e)). The description of these elements (Proposition 6.5, and in particular Corollary 6.6) shows that the roots must be square roots that twist exactly one singular fibre and permute the two fibres F1 and F2 twisted by σ. There thus exist two elements h3, h4 ∈ G that twist respectively the fibres F3 and F4. Since h3 commutes with h4, it must leave invariant the unique fibre twisted by h4, i.e. F4. Similarly, h4 must leave F3 invariant. Therefore, h3h4 leaves the four fibres F1,...,F4 invariant and twists the two fibres F3 and F4; it is thus an involution that belongs to G′, which contradicts the fact that G′ =< σ >. If G′ 6=< σ >, since σ has no root in G′ (Corollary 6.2), the Abelian groupG′ ⊂ PGL(2,C(x)) is isomorphic to (Z/2Z)2 and contains (using (d)) three twisting involutions σ, ρ and σρ. Note that two of these three involutions do not twist singular fibres which are all distinct, otherwise the product of the two involutions would give an involution that twists 4 singular fibres, contradicting (a). We may thus suppose that ρ twists F1 and F3, which implies that σρ twists F2 and F3. The fibre F4 is then twisted by an element which is a square root of one of the three twisting involutions (assertion (e) and Corollary 6.6). Denote this square root by h and suppose that h2 6= σ. Note that h exchanges the two singular fibres twisted by h2. One of these is twisted by σ and the other is not, so h and σ do not commute. The only remaining possible finite Abelian groups of automorphisms of conic bundles satisfying property (F ) are thus del Pezzo surfaces of degree 6 or 5 (studied in Sections 4 and 5), the triple (Cs24, Ŝ4, π) studied in Section 7, and Hirzebruch surfaces. We now describe this last case and prove that it is birationally reduced to the case of P1 × P1. Proposition 8.3. Let G ⊂ Aut(Fn) be a finite Abelian subgroup of automorphisms of Fn, for some integer n ≥ 1. Then, a birational map of conic bundles conjugates G to a finite group of automorphisms of F0 = P 1 × P1 that leaves one ruling invariant. Proof. Let G ⊂ Aut(Fn) be a finite Abelian group, with n ≥ 1. Note that G preserves the unique ruling of Fn. We denote by E ⊂ Fn the unique section of self-intersection −n, which is necessarily invariant by G. We have the exact sequence (see Remark 3.13) 1 → G′ → G → π(G) → 1. (1) Since the group π(G) ⊂ PGL(2,C) is Abelian, it is isomorphic to a cyclic group or to (Z/2Z)2. If π(G) is a cyclic group, at least two fibres are invariant by G. The group G fixes two points in one such fibre. We can blow-up the point that does not lie on E and blow-down the corresponding fibre to get a group of automorphisms of Fn−1. We do this n times and finally obtain a birational map of conic bundles that conjugates G to a group of automorphisms of F0 = P 1 × P1. If π(G) is isomorphic to (Z/2Z)2, there exist two fibres F, F ′ of π whose union is invariant by G. Let GF ⊂ G be the subgroup of G of elements that leave F invariant. This group is of index 2 in G and hence is normal. Since GF fixes the point F ∩ E in F , it acts cyclically on F . There exists another point P ∈ F , P /∈ E, which is fixed by GF . The orbit of P by G consists of two points, P and P ′, such that P ′ ∈ F ′, P ′ /∈ E. We blow-up these two points and blow-down the strict transforms of F and F ′ to get a group of automorphisms of Fn−2. We do this ⌊n/2⌋ times to obtain G as a group of automorphisms of F0 or F1. If n is even, we get in this manner a group of automorphisms of F0 = P 1 × P1. Note that n cannot be odd, if the group π(G) is not cyclic. Otherwise, we could conjugate G to a group of automorphisms of F1 and then to a group of automorphisms of P2 by blowing-down the exceptional section on a point Q ∈ P2. We would get an Abelian subgroup of PGL(3,C) that fixes Q, and thus a group with at least three fixed points. In this case, the action on the set of lines passing through Q would be cyclic (see Proposition 2.2), which contradicts our hypothesis. We can now prove the main result of this section: Proposition 8.4. Let G ⊂ Aut(S, π) be some finite Abelian group of automor- phisms of the conic bundle (S, π) such that the triple (G,S, π) is minimal and no non-trivial element of G fixes a curve of positive genus. Then, one of the following situations occurs: 1. S is a Hirzebruch surface Fn; 2. S is a del Pezzo surface of degree 5 or 6; 3. The triple (G,S, π) is isomorphic to the triple (Cs24, Ŝ4, π) of Section 7. If we suppose that the pair (G,S) is minimal, then we are in case 1 with n 6= 1 or in case 3. Moreover, cases 1 and 2 are birationally conjugate to automorphisms of P1 × P1 whereas the third is not. Proof. The fact that one of the three cases occurs follows directly from Proposi- tions 8.1 and 8.2. Case 1 is clearly minimal if and only if n 6= 1 and Proposition 8.3 shows that it is conjugate to automorphisms of P1 × P1. In the case of del Pezzo surfaces of degree 5 and 6, the pair (G,S) is not minimal and the group is respectively birationally conjugate to a subgroup of Sym4 ⊂ Aut(P 2) (Lemma 5.3) or Aut(P1× P1) (Lemma 4.2). If the first situation occurs, since the group is Abelian and not isomorphic to (Z/3Z)2 it is diagonalisable and conjugate to a subgroup of Aut(P1 × P1) (Proposition 2.2). Thus, we are done with case 2. It remains to show that the pair (Cs24, Ŝ4) is not birationally conjugate to a group of automorphisms of P1 × P1. Let us suppose the contrary, i.e. that there exists some Cs24-equivariant birational map ϕ : Ŝ4 99K P 1 × P1 (that conjugates Cs24 to a group of automorphisms). Then, ϕ is the composition of Cs24-equivariant elementary links (see for example [Isk3, Theorem 2.5], or [Do-Iz, Theorem 7.7]). Since our group preserves the conic bundle, the first link is of type II, III or IV (in the classical notation of Mori theory). We now study these possibilities and show that it is not possible to go to P1 × P1. Link of type II - In our case, this link is a birational map of conic bundles, which is the composition of the blow-up of an orbit of Cs24, no two points on the same fibre, with the blow-down of the strict transforms of the fibres of the points blown-up. The points must be fixed by the elements of Cs24 that act trivially on the basis of the fibration, and thus an orbit has 4 points, two on Ẽ1 and two on D̃23. This link conjugates the triple (Cs24, Ŝ4, π) to a triple isomorphic to it, by Proposition 8.1. Link of type III - It is the contraction of some set of skew exceptional curves, in- variant by Cs24. This is impossible since the pair (Cs24, Ŝ4) is minimal (Lemma 7.4). Link of type IV - It is a change of the fibration. This is not possible since the surface Ŝ4 admits only one conic bundle fibration (Corollary 7.2). 9 Actions on del Pezzo surfaces with fixed part of the Picard group of rank one In this section we prove the following result (note that finiteness is not required and that minimality of the action is implied by the condition on Pic(S)G). Proposition 9.1. Let S be a del Pezzo surface, and let G ⊂ Aut(S) be an Abelian group such that rk Pic(S)G = 1 and no non-trivial element of G fixes a curve of positive genus. Then, one of the following occurs: 1. S ∼= P2 or S ∼= P1 × P1; 2. S is a del Pezzo surface of degree 5 and G ∼= Z/5Z; 3. S is a del Pezzo surface of degree 6 and G ∼= Z/6Z. Furthermore, in cases 2 and 3, the group G is birationally conjugate to a diagonal cyclic subgroup of Aut(P2). This will be proved separately for each degree, in Lemmas 9.7, 9.8, 9.13, 9.15, 9.16 and 9.17. Remark 9.2. A del Pezzo surface S is either P1 × P1 or the blow-up of 0 ≤ r ≤ 8 points in general position on P2 (i.e. such that no irreducible curve of self- intersection ≤ −2 appears on S). The group Pic(S) has dimension r + 1, and its intersection form gives a decomposition Pic(S) ⊗ Q = QKS ⊕K S ; the signature is (1,−1, ...,−1). The group Aut(S) of automorphisms of a del Pezzo surface S acts on Pic(S) and preserves the intersection form. This gives an homomorphism of Aut(S) → Aut(Pic(S)) which is injective if and only if r > 3, since the kernel is the lift of automorphisms of P2 that fix the r blown-up points. Furthermore, the image is contained in the Weyl group and is finite (see [Dol]). In particular, the group Aut(S) is finite if and only if r > 3. When we have some group action on a del Pezzo surface, we would like to determine the rank of the fixed part of the Picard group. Here are some tools to this end. Lemma 9.3 (Size of the orbits). Let S be a del Pezzo surface, which is the blow-up of 1 ≤ r ≤ 8 points of P2 in general position, and let G ⊂ Aut(S) be a subgroup of automorphisms with rk Pic(S)G = 1. Then: • G 6= {1}; • the size of any orbit of the action of G on the set of exceptional divisors is divisible by the degree of S, which is 9− r; • in particular, if the order of G is finite, it is divisible by the degree of S. Proof. It is clear that G 6= {1}, since rk Pic(S) > 1. Let D1, D2, ..., Dk be k exceptional divisors of S, forming an orbit of G (the orbit is finite, see Remark 9.2). The divisor i=1 Di is fixed by G and thus is a multiple of KS . We can write∑k i=1 Di = aKS, for some a ∈ Q. In fact, since aKS is effective, we have a < 0 and a ∈ Z. Since the Di’s are irreducible and rational, we deduce from the adjunction formula Di(KS +Di) = −2 that Di ·KS = −1. Hence i=1 Di = i=1 KS ·Di = −k = KS · aKS = a(9− r). Consequently, the degree 9− r divides the size k of the orbit. Remark 9.4. This lemma shows in particular that rk Pic(S)G > 1 if S is the blow- up of r = 1, 2 points of P2, a result which is obvious when r = 1, and is clear when r = 2, since the line joining the two blown-up points is invariant by any automorphism. Lemma 9.5. Let S be some (smooth projective rational) surface, and let g ∈ Aut(S) be some automorphism of finite order. Then, the trace of g acting on Pic(S) is equal to χ(Fix(g)) − 2, where Fix(g) ⊂ S is the set of fixed points of g and χ is the Euler characteristic. Proof. This follows from the topological Lefschetz fixed-point formula, which as- serts that the trace of g acting on H∗(S,Z) is equal to χ(Fix(g)) (this uses the fact that g is an homeomorphism of finite order). Since S is a complex rational surface, H0(S,Z) and H4(S,Z) have dimension 1, H2(S,Z) ∼= Pic(S), and Hi(S,Z) = 0 for i 6= 0, 2, 4. Since the trace on H2 and H4 is 1, we obtain the result. Remark 9.6. This lemma is false if the order of g is infinite. Take for example the automorphism (x : y : z) 7→ (λx : y : z + y) of P2, for any λ ∈ C∗, λ 6= 1. It fixes exactly two points, namely (1 : 0 : 0) and (0 : 0 : 1), but its trace on Pic(P2) = Z is 1. We now start the proof of Proposition 9.1 by studying the cases of del Pezzo surfaces of degree 6 or 5. Lemma 9.7 (Actions on the del Pezzo surface of degree 6). Let S6 = { (x : y : z), (u : v : w) | ux = vy = wz} ⊂ P2 × P2 be the del Pezzo surface of degree 6 and let G ⊂ Aut(S6) be an Abelian group such that rk Pic(S6) G = 1. Then, G is conjugate in Aut(S6) to the cyclic group of order 6 generated by (x : y : z), (u : v : w) (v : w : u), (y : z : x) . Furthermore, G is birationally conjugate to a diagonal subgroup of Aut(P2). Proof. Lemma 9.3 implies that the sizes of the orbits of the action of G on the ex- ceptional divisors are divisible by 6. The action of G on the hexagon of exceptional divisors is thus transitive, so G contains an element of the form (x : y : z), (u : v : w) (αv : βw : u), (βy : αz : αβx) where α, β ∈ C∗. As the only element of (C∗)2 that commutes with g is the identity (see the description of Aut(S6) = (C ∗)2 ⋊ (Sym3 × Z/2Z) in Section 4), G must be cyclic, generated by g. Conjugating it by (x : y : z), (u : v : w) (βx : y : αz), (αu : αβv : βw) we may assume that α = β = 1, as stated in the lemma (this shows in particular thatG is of finite order). It remains to prove that this automorphism is birationally conjugate to a linear automorphism of the plane. Denote by p : S → P2 the restriction of the projection on the first factor. This is a birational morphism which is the blow-up of the three diagonal points A1, A2, A3 of P 2. Consider the birational map ĝ = pgp−1 of P2, which is explicitly ĝ : (x : y : z) 99K (xz : xy : yz). Since g is an automorphism of the surface, it fixes the canonical divisor KS , so the birational map ĝ leaves the linear system of cubics of P2 passing through A1, A2 and A3 invariant (this can also be verified directly). Note that ĝ fixes exactly one point of P2, namely P = (1 : 1 : 1), and that its action on the projective tangent space P(TP (P 2)) of P2 at P is of order 3, with two fixed points, corresponding to the lines (x− y) + ωk(z − y) = 0, where ω = e2iπ/3, k = 1, 2. Hence, the birational map ĝ preserves the linear system of cubics of P2 passing through A1, A2 and A3, which have a double point at P and are tangent to the line (x− y) + ω(z − y) = 0 at this point. This linear system thus induces a birational transformation of P2 that conjugates ĝ to a linear automorphism. Lemma 9.8 (Actions on the del Pezzo surface of degree 5). Let S5 be the del Pezzo surface of degree 5 and let G ∈ Aut(S5) = Sym5 be an Abelian group such that rk Pic(S5) G = 1. Then, G is cyclic of order 5. Furthermore, G is birationally conjugate to a diagonal subgroup of Aut(P2). Proof. We use the description of the surface S5 and its automorphisms group Aut(S5) = Sym5 given in Section 5. Lemma 9.3 implies that the order of G is divisible by 5, and thus that G is a cyclic subgroup of Sym5 of order 5. Since all such subgroups are conjugate in Aut(S5) = Sym5, we may suppose that G is generated by the lift of the birational transformation h : (x : y : z) 99K (xy : y(x− z) : x(y− z)) of P2, that fixes two points of P2, namely (ζ +1 : ζ : 1), where ζ2 − ζ − 1 = 0. Denoting one of them by P , the linear system of cubics passing through the four blown-up points and having a double point at P is invariant by h. The birational transformation associated to this system thus conjugates h to a linear automorphism of P2. Remark 9.9. The fact that (x : y : z) 99K (xy : y(x− z) : x(y − z)) is linearisable was proved in [Be-Bl], using the same argument as above. Corollary 9.10. Let S be a rational surface with (KS) 2 ≥ 5 and let G ⊂ Aut(S) be a finite Abelian group. Then G is birationally conjugate to a subgroup of Aut(P2) or Aut(P1 × P1). Proof. We may assume that the pair (G,S) is minimal; consequently there are two possibilities (see [Man], [Isk2] or [Do-Iz]): 1. S is a del Pezzo surface and rk Pic(S)G = 1. Then S is either P2, P1×P1 or a del Pezzo surface of degree 6 or 5 (Remark 9.4); we apply Lemmas 9.7 and 9.8 to conclude. 2. G preserves a conic bundle structure on S. Here the number of fibres is at most 3, hence no element of G fixes a curve of positive genus (Corollary 6.4); we apply Proposition 8.4 to conclude. To study del Pezzo surfaces of degree 4, let us describe their group of auto- morphisms (note that we do not use the notation Sd for the del Pezzo surfaces of degree d ≤ 4, because there are many different surfaces of the same degree): Lemma 9.11 (Automorphism group of del Pezzo surfaces of degree 4). Let S be a del Pezzo surface of degree 4 given by the blow-up η : S → P2 of five points A1, ..., A5 ∈ P 2 such that no three are collinear. Setting Ei = η −1(Ai) and denoting by L the pull-back by η of a general line of P2, we have: 1. There are exactly 10 conic bundle structures on S, whose fibres are respec- tively L− Ei, −KS − (L− Ei), for i = 1, ..., 5. 2. The action of Aut(S) on the five pairs of divisors {L−Ei,−KS− (L−Ei)}, i = 1, ..., 5 gives rise to a split exact sequence 0 → F → Aut(S) → Sym5, where F = {(a1, ..., a5) ∈ (F2) ai = 0} ∼= (F2) 4, and the automorphism (a1, ..., a5) permutes the pair {L−Ei,−KS − (L−Ei)} if and only if ai = 1. 3. We have Aut(S) = F⋊Aut(S, η), where Aut(S, η) is the lift of the group of automorphisms of P2 that leave the set {A1, ..., A5} invariant, and Aut(S, η) acts on F = {(a1, ..., a5) ∈ ai = 0} by permutation of the ai’s, as it acts on {A1, ..., A5}, and as ρ(Aut(S)) = ρ(Aut(S, η)) ⊂ Sym5 acts on the exceptional pairs. 4. The elements of F with two ”ones” correspond to quadratic involutions of P2 and fix exactly 4 points of S. 5. The elements of F with four ”ones” correspond to cubic involutions of P2 and the points of S fixed by these elements form a smooth elliptic curve. Remark 9.12. The group F ⊂ Aut(S) has been studied intensively since 1895 (see [Kan], Theorem XXXIII). A modern description of the group as the 2-torsion of PGL(5,C) can be found in [Bea2, (4.1)], together with a study of the conjugacy classes of such groups in the Cremona group. For further descriptions of the auto- morphism groups of these surfaces, see [Do-Iz, section 6.4] and [Bla2, section 8.1]. Proof. Let A = mL − i=1 aiEi be the divisor of the fibre of some conic bundle structure on S, for some m, a1, ..., a5 ∈ Z. From the relations A 2 = 0 (the fibres are disjoint) and AKS = −2 (adjunction formula) we get: i=1 ai 2 = m2, i=1 ai = 3m− 2. As in Lemma 7.1, we have ( i=1 ai) 2 ≤ 5 i=1 ai 2, which implies here that (3m− 2)2 ≤ 5m2, that is 4(m2 − 3m+ 1) ≤ 0. As m is an integer, we must have 1 ≤ m ≤ 2. If m = 1, we replace it in (3) and see that there exists i ∈ {1, ..., 5} such that A = L − Ei. Otherwise, taking m = 2 and replacing it in (3), we see that four of the aj ’s are equal to 1, and one is equal to 0. This gives the ten conic bundles of assertion 1, which are the lift on S of the lines of P2 passing through one of the Ai’s or of the conic passing through four of the Ai’s. The group Aut(S) acts on the set ∪5i=1{L−Ei,−KS − (L−Ei)}; since KS is fixed, this induces an action on the set of five pairs {L−Ei,−KS − (L−Ei)}. We denote by ρ : Aut(S) → Sym5 the corresponding homomorphism. The action of the kernel of ρ on the pairs of conic bundles gives a natural embedding of Ker(ρ) into (F2) We now prove that Ker(ρ) = {(a1, ..., a5) | ai = 0} = F. Acting by a linear automorphism of P2, we may assume that the points blown-up by η are A1 = (1 : 0 : 0), A2 = (0 : 1 : 0), A3 = (0 : 0 : 1), A4 = (1 : 1 : 1), A5 = (a : b : c), for some a, b, c ∈ C∗. Then, the birational involution τ : (x0 : x1 : x2) 99K (ax1x2 : bx0x2 : cx0x1) of P 2 lifts as an automorphism η−1τη ∈ Aut(S) that acts on Pic(S)  0 −1 −1 0 0 −1 −1 0 −1 0 0 −1 −1 −1 0 0 0 −1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 2  with respect to the basis (E1, E2, E3, E4, E5, L). It follows from this observation that η−1τη belongs to the kernel of ρ, and acts on the pairs of conic bundles as (0, 0, 0, 1, 1) ∈ (F2) 5. Permuting the roles of the points A1, ..., A5, we get 10 involutions whose representations in (F2) 5 have two ”ones” and three ”zeros”. These involutions generate the group {(a1, ..., a5) | ai = 0} = F. To prove that this group is equal to Ker(ρ), it suffices to show that (1, 1, 1, 1, 1) does not belong to Ker(ρ). This follows from the fact that (1, 1, 1, 1, 1) would send L = 1 (KS +∑5 i=1(L − Ei)) on the divisor (KS + i=1(−KS − L + Ei)) = (−2L − 3KS), which doesn’t belong to Pic(S). This concludes the proof of assertion 2 (except the fact that the exact sequence is split, which will be proved by assertion 3). We now prove assertion 3. Let σ ∈ Sym5 be a permutation of the set {1, ..., 5} in the image of ρ and g be an automorphism of S such that ρ(g) = σ. Let α be the element of Aut(Pic(S)) that sends Ei on Eσ(i) and fixes L. Viewing Aut(S) as a subgroup of Aut(Pic(S)), the element gα−1 ∈ Aut(Pic(S)) fixes the five pairs of conic bundles. There exists some element h ∈ F ⊂ Aut(S) such that hgα−1 either fixes the divisor of every conic bundle or permutes the divisors of conic bundles in each pair. The same argument as in the above paragraph shows that this latter possibility cannot occur. Hence hgα−1 fixes L−E1, ..., L−E5 and KS. It follows that hgα−1 acts trivially on Pic(S), so α = hg ∈ Aut(S), and α is by construction the lift of an automorphism of P2 that acts on the set {A1, ..., A5} as σ does on {1, ..., 5}. Conversely, it is clear that every automorphism r of P2 which leaves the set {A1, A2, A3, A4, A5} invariant lifts to the automorphism η −1rη of S whose action on the pairs of conic bundles is the same as that of r on the set {A1, A2, A3, A4, A5}. This gives assertion 3. Assertion 4 follows from the above description of some element of F ⊂ Aut(S) with two ”ones” as the lift of a birational map of the form τ : (x0 : x1 : x2) 99K (ax1x2 : bx0x2 : cx0x1). As the automorphism η −1τη ∈ Aut(S) does not leave any exceptional divisor invariant, its fixed points are the same as those of τ , which are the four points (α : β : γ), where α2 = a, β2 = b, γ2 = c. It remains to prove the last assertion. Note that the element h = (0, 1, 1, 1, 1) ∈ Aut(S) fixes the divisor L−E1, hence acts on the associated conic bundle structure. Furthermore, the four singular fibres of this conic bundle, {L − E1 − Ei, Ei}, for i = 2, ..., 5, are invariant by h and this element switches the two components of each fibre. This shows that the action of h on the basis of the fibration is trivial, so the restriction of h on each fibre is an involution of P1 which fixes two points. On each singular fibre, exactly one point is fixed, which is the singular point of the fibre. The situation is similar for the other elements with four ”ones” (in fact, the involutions described here are twisting involutions, see Lemma 6.1). Lemma 9.13 (Actions on the del Pezzo surfaces of degree 4). Let S be a del Pezzo surface of degree 4, and let G ∈ Aut(S) be an Abelian group such that rk Pic(S)G = 1. Then, G contains an involution that fixes an elliptic curve. Proof. We keep the notation of Lemma 9.11 for η : S → P2,Aut(S, η), ρ,F, ... and denote by H the group G ∩ F = G ∩ Kerρ. We will prove that H contains an element of F with four ”ones”, which is an involution that fixes an elliptic curve (Lemma 9.11). The group ρ(G) ⊂ ρ(Aut(S)) ∼= Aut(S, η) is isomorphic to a subgroup of Aut(S, η). The group Aut(S, η) is the lift of the group of automorphisms of P2 that leave the set {A1, ..., A5} invariant (Lemma 9.11). The restriction of this group to the conic of P2 passing through the five points is a subgroup of PGL(2,C) that leaves five points invariant. Since ρ(G) is finite and Abelian, it is cyclic, of order at most 5. We consider the different possibilities. The order of ρ(G) is 1. This implies that G ⊂ F. If G contains an element with four ”ones”, we are done. Otherwise, up to conjugation G is a subset of the group generated by (1, 1, 0, 0, 0) and (1, 0, 1, 0, 0), and fixes L − E4 and L − E5 (thus rk Pic(S)G > 1). The order of ρ(G) is 2. Up to a change of numbering, ρ(G) is generated by (1 2)(3 4); since G is Abelian, we find that H ⊂ V = {(a, a, b, b, 0) | a, b ∈ F2}. Let g = ((a, b, c, d, e), (1 2)(3 4)) ∈ G be such that ρ(g) = (1 2)(3 4). We may suppose that e = 1 (otherwise, the group G would fix L − E5 and we would have rk Pic(S)G ≥ 2.) Conjugating by ((0, b, 0, d, b + c), id) we may assume that g = ((a+ b, 0, c+ d, 0, 1), (1 2)(3 4)). In fact, since a+ b+ c+ d+ e = 0, we have g = ((α, 0, 1+α, 0, 1), (1 2)(3 4)), where α = a+b = c+d+1 ∈ F2. If α = 1, then g has order 4 and fixes the divisor 2L− E3 − E4, thus G cannot be equal to < g > and it follows that V ⊂ G; in particular the element (1, 1, 1, 1, 0) is contained in G. If α = 0, then < g > fixes 2L− E1 − E2, so once again G contains V . The order of ρ(G) is 3. In this case, ρ(G) is generated by a 3-cycle, namely (1 2 3); then H must be a subgroup of V = {(a, a, a, b, a + b) | a, b ∈ F2}. The order of G must be a multiple of 4, by Lemma 9.3, hence H = V , and thus G contains the element (1, 1, 1, 1, 0). The order of ρ(G) is 4. Then ρ(G) is generated by (1 2 3 4), so H must be a subgroup of V =< (1, 1, 1, 1, 0) >. Let g = ((a, b, c, d, e), (1 2 3 4)) ∈ G be such that ρ(g) = (1 3 2 4). Conjugating the group by ((a, a+b, a+b+c, 0, a+c), id), we may suppose that g = ((0, 0, 0, e, e), (1 3 2 4)). If e = 1, then g4 = (1, 1, 1, 1, 0) ∈ G. If e = 0, the element g belongs to HS , so it fixes the divisors L and E5. As the group V fixes L− E5, the rank of Pic(S) G cannot be 1. The order of ρ(G) is 5. Then, ρ(G) is generated by a 5-cycle and H = {1}. The rank of Pic(S)H cannot be 1, by Lemma 9.3. Before studying the case of del Pezzo surfaces of degree ≤ 3, we remind the reader of some classical embeddings of these surfaces. Remark 9.14. Recall ([Kol], Theorem III.3.5) that a del Pezzo surface of degree 3 (respectively 2, 1) is isomorphic to a smooth hypersurface of degree 3 (respectively 4, 6) in the projective space P3 (respectively in P(1, 1, 1, 2), P(1, 1, 2, 3)). Further- more, in each of the 3 cases, any automorphism of the surface is the restriction of an automorphism of the ambient space. We will use these classical embeddings, take w, x, y, z as the variables on the projective spaces, and denote by [α : β : γ : δ] the automorphism (w : x : y : z) 7→ (αw : βx : γy : δz). Note that a del Pezzo surface of degree 4 is isomorphic to the intersection of two quadrics in P4, but we will not use this here. Lemma 9.15 (Actions on the del Pezzo surfaces of degree 3). Let S be a del Pezzo surface of degree 3, and let G ∈ Aut(S) be an Abelian group such that rk Pic(S)G = 1. Then, G contains an element of order 2 or 3 that fixes an elliptic curve of S. Proof. Lemma 9.3 implies that the order of G is divisible by 3, so G contains an element of order 3. We view S as a cubic surface in P3, and Aut(S) as a subgroup of PGL(4,C) (see Remark 9.14). There are three kinds of elements of order 3 in PGL(4,C), depending on the nature of their eigenvalues. Setting ω = e2iπ/3, there are elements with one eigenvalue of multiplicity 3 (conjugate to [1 : 1 : 1 : ω], or its inverse), elements with two eigenvalues of multiplicity 2 (conjugate to [1 : 1 : ω : ω]) and elements with three distinct eigenvalues (conjugate to [1 : 1 : ω : ω2]). We consider the three possibilities. Case a: G contains an element of order 3 with one eigenvalue of multiplicity 3. The element [1 : 1 : 1 : ω] fixes the hyperplane z = 0, whose intersection with the surface S is an elliptic curve (because Fix(g) ⊂ S is smooth). Thus, we are done. Case b: G contains an element of order 3 with two eigenvalues of multiplicity 2. With a suitable choice of coordinates, we may assume that this element is g = [1 : 1 : ω : ω]. Since S is smooth, its equation F is of degree at least 2 in each variable, which implies that F (w, x, ωy, ωz) = F (w, x, y, z) (the eigenvalue is 1); up to a change of coordinates F = w3 + x3 + y3 + z3, which means that S is the Fermat cubic surface. The group of automorphisms of S is (Z/3Z)3 ⋊ Sym4 and the centraliser of g in it is (Z/3Z)3 ⋊ V , where V ∼= (Z/2Z)2 is the subgroup of Sym4 generated by the two transpositions (w, x) and (y, z). The structure of the centraliser gives rise to an exact sequence 1 → (Z/3Z)3 → (Z/3Z)3 ⋊ V → V → 1 ∪ ∪ ∪ 1 → G ∩ (Z/3Z)3 → G → γ(G) → 1. We may suppose that G contains no element of order 3 with an eigenvalue of multiplicity 3, since this case has been studied above (case a). There are then three possibilities for G ∩ (Z/3Z)3, namely < g >, < g, [1 : ω : 1 : ω] > and < g, [1 : ω : ω : 1] >. The last is conjugate to the second by the automorphism (y, z). Note that g preserves exactly 9 of the 27 lines on the surface; these are {w + ωix = y + ωjz = 0}, for 0 ≤ i, j ≤ 2. If G ∩ (Z/3Z)3 is equal to < g >, then G/ < g >∼= γ(G) has order 1, 2 or 4 and thus G leaves at least one of the 9 lines invariant, whence rk Pic(S)G > 1. If G ∩ (Z/3Z)3 is the group H =< g, [1 : ω : 1 : ω] > we have G = H , since the centraliser of H in (Z/3Z)3⋊V is the group (Z/3Z)3. As the set of three skew lines {w + ωix = y + ωiz = 0} for 0 ≤ i ≤ 2 is an orbit of H , the rank of Pic(S)G is strictly larger than 1. Case c: G contains an element g of order 3 with three distinct eigenvalues. We may suppose that g = [1 : 1 : ω : ω2]. Note that the action of g on P3 fixes the line Lyz of equation y = z = 0 and thus the whole group G leaves this line invariant. If Lyz ⊂ S, the rank of rk Pic(S) G is at least 2. Otherwise, the equation of S is of the form L3(w, x)+L1(w, x)yz+ y 3+ z3 = 0, where L3 and L1 are homogeneous forms of degree respectively 3 and 1, and L3 has three distinct roots, so Fix(g) = S ∩ Lyz. Since g fixes exactly three points, the trace of its action on Pic(S) ∼= Z7 is 1 (Lemma 9.5) and thus rk Pic(S)g > 1, which implies that G 6=< g >. Note that every subgroup of PGL(4,C) isomorphic to (Z/3Z)2 contains an element with only two distinct eigenvalues, so we may assume that G contains only two elements of order 3, which are g and g2. This implies that the action of G on the three points of Lyz ∩ S gives an exact sequence 1 →< g >→ G → Sym3, where the image on the right is a transposition. The group G thus contains an element of order 2, that we may assume to be diagonal of the form (w : x : y : z) 7→ (−w : x : y : z) and that fixes the elliptic curve which is the trace on S of the plane w = 0. Lemma 9.16 (Actions on the del Pezzo surfaces of degree 2). Let S be a del Pezzo surface of degree 2, and let G ∈ Aut(S) be an Abelian group such that rk Pic(S)G = 1. Then, G contains either the Geiser involution (that fixes a curve isomorphic to a smooth quartic curve) or an element of order 2 or 3 that fixes an elliptic curve. Proof. We view S as a surface of degree four in the weighted projective space P(2, 1, 1, 1) (see Remark 9.14). Note that the projection on the last three coordi- nates gives S as a double covering of P2 ramified over a smooth quartic curve Q. Lemma 9.3 implies that the order of G is divisible by 2, so G contains an element g of order 2. If the element g is the involution induced by the double covering (classically called the Geiser involution), we are done; otherwise we may assume that g acts on P(2, 1, 1, 1) as g : (w : x : y : z) 7→ (ǫw : x : y : −z), where ǫ = ±1, and the equation of S is w2 = z4 + L2(x, y)z 2 + L4(x, y), where Li is a form of degree i, and L4 has four distinct roots. The trace on S of the equation z = 0 defines an elliptic curve Lz ⊂ S. If ǫ = 1, then g fixes the curve Lz and we are done; we therefore assume that ǫ = −1. If G contains another involution, we diagonalise the group generated by these two involutions and see that one element of the group fixes either an elliptic curve or the smooth quartic curve, so we may assume that g is the only involution of G. Note that g fixes exactly four points of S, which are the points of intersection of Lz with the quartic Q (of equation w = 0). The trace of g on Pic(S) ∼= Z thus equal to 2 (Lemma 9.5), whence rk Pic(S)g = 5 and G 6=< g >. The group G acts on the line z = 0 of P2 and on the four points of Lz ∩ Q. Since g is the only element of order 2 of G, the action of G on these four aligned points has order 3 and thus, we may assume that L4(x, y) = x(x 3 +λy3) and that there exists an element h of G that acts as (w : x : y : z) 7→ (αw : x : e2iπ/3y : βz), with α2 = β4 = 1. We find that h4 is an element of order 3 that fixes the elliptic curve which is the trace on S of the equation y = 0. Lemma 9.17 (Actions on the del Pezzo surfaces of degree 1). Let S be a del Pezzo surface of degree 1, and let G ∈ Aut(S) be an Abelian group such that rk Pic(S)G = 1. Then, some non-trivial element of G fixes a curve of S of positive genus. Proof. We view S as a surface of degree six in the weighted projective space P(3, 1, 1, 2) (see Remark 9.14). Up to a change of coordinates, we may assume that the equation is w2 = z3 + zL4(x, y) + L6(x, y), where L4 and L6 are homogeneous forms of degree 4 and 6 respectively. The embedding of S into P(3, 1, 1, 2) is given by | − 3KS| × | −KS | × | − 2KS|, which implies that G is a subgroup of P (GL(1,C)×GL(2,C)×GL(1,C)). The projection (w : x : y : z) 99K (x : y) is an elliptic fibration generated by | − KS |, and has one base-point, namely (1 : 0 : 0 : 1), which is fixed by Aut(S). This projection induces an homomorphism ρ : Aut(S) → Aut(P1) = PGL(2,C). Note that the kernel of ρ is generated by the Bertini involution w 7→ −w (and the element z 7→ ωz (ω = e2iπ/3) if L4 = 0) and is hence cyclic of order 2 (or 6). Furthermore, any element of this kernel fixes a curve of positive genus. We assume that no non-trivial element of G fixes a curve of positive genus. This implies that G is isomorphic to ρ(G) ⊂ Aut(P1), and thus is either cyclic or isomorphic to (Z/2Z)2. Since the lift of this latter group in Aut(S) is not Abelian, G is cyclic. We use the Lefschetz fixed-point formula (Lemma 9.5) to deduce the eigenvalues of the action of elements of G on Pic(S) ∼= Z9. For any element g ∈ G, g 6= 1, Fix(g) contains the point (1 : 0 : 0 : 1) and is the disjoint union of points and lines. Thus χ(Fix(g)) ≥ 1 and so the trace of g on Pic(S) is at least −1 (Lemma 9.5). Elements of order 2: The eigenvalues are < 1a, (−1)b > with a ≥ 4, b ≤ 5. Elements of order 3: The eigenvalues are < 1a, (ω)b, (ω2)b > with a ≥ 3, b ≤ 3. Elements of order 4: The eigenvalues are < 1a, (−1)b, (i)c, (−i)c > with a ≥ b−1. Furthermore, the information on the square induces that a+b ≥ 4, so a ≥ 2. Elements of order 5: The eigenvalues are < 15, l1, l2, l3, l4 >, where l1, ..., l4 are the four primitive 5-th roots of unity. Elements of order 6: The eigenvalues are< 1a, (−1)b, (ω)c, (ω2)c, (−ω)d, (−ω2)d >, where a − b − c + d ≥ −1. Computing the square and the third power, we find respectively a + b ≥ 3, c + d ≤ 3 and a + 2c ≥ 4, b + 2d ≤ 5. This implies that a ≥ 2. Indeed, if a = 1, we get b, c ≥ 2 and thus d ≤ 1, which contradicts the fact that the trace a− b− c+ d is at least −1. Since rk Pic(S)G = 1, the order of the cyclic group G is at least 7. As the action of G leaves L4 and L6 invariant, both L6 and L4 are monomials. If some double root of L6 is a root of L4, the surface is singular, so up to an exchange of coordinates we may suppose that L4 = x 4 and either L6 = xy 5 or L6 = y In the first case, the equation of the surface is w2 = z3+x4z+xy5 whose group of automorphisms Aut(S) is isomorphic to Z/20Z, generated by [i : 1 : ζ10 : −1], and contains the Bertini involution. No subgroup of Aut(S) fullfills our hypotheses. In the second case, the equation of the surface is w2 = z3 + x4z + y6, whose group of automorphisms is isomorphic to Z/2Z×Z/12Z, generated by the Bertini involution and g = [i : 1 : ζ12 : −1]. The only possibility for G is to be equal to < g >. Since g4 = [1 : 1 : ω : 1] fixes an elliptic curve, we are done. Proposition 9.1 now follows, using all the lemmas proved above. 10 The results We now prove the five theorems stated in the introduction. Proof of Theorem 4. Since the pair (G,S) is minimal, either rk Pic(S)G = 1 and S is a del Pezzo surface, or G preserves a conic bundle structure (see [Man], [Isk2] or [Do-Iz]). In the first case, either S ∼= P2, or S ∼= P1 × P1 or S is a del Pezzo surface of degree d = 5 or 6 and G ∼= Z/dZ (Proposition 9.1). In the second case, either S is a Hirzebruch surface or the pair (G,S) is the pair (Cs24, Ŝ4) of Section 7 (Proposition 8.4). Proof of Theorem 2. No non-trivial element of Aut(P2),Aut(P1 × P1) or Cs24 fixes a non-rational curve (the first two cases are clear, the last one follows from Lemma 7.4). Conversely, suppose that G is a finite Abelian subgroup of the Cremona group such that no non-trivial element fixes a curve of positive genus. Since G is finite, it is birationally conjugate to a group of automorphisms of a rational surface S (see for example [dF-Ei, Theorem 1.4] or [Do-Iz]). Then, we assume that the pair (G,S) is minimal and use the classification of Theorem 4. If S is an Hirzebruch surface, the group is birationally conjugate to a subgroup of Aut(P1 ×P1) (Proposition 8.3). If S is a del Pezzo surface, the group G is bira- tionally conjugate to a subgroup of Aut(P1 × P1) or Aut(P2), by Proposition 9.1. Otherwise, the pair (G,S) is isomorphic to the pair (Cs24, Ŝ4). It remains to show that the group Cs24 is not birationally conjugate to a subgroup of Aut(P1 × P1) or Aut(P2). Since the group is isomorphic to Z/2Z × Z/4Z, only the case of Aut(P1 × P1) need be considered (see Section 2). This was proved in Proposition 8.4. Proof of Theorem 5. By Theorem 2, G is birationally conjugate either to a sub- group of Aut(P2), or of Aut(P1 × P1), or to the group Cs24. The group Cs24 is case [8]. The finite Abelian subgroups of Aut(P 2) are conju- gate to the groups of case [1] or [9] (Proposition 2.2). The finite Abelian subgroups of Aut(P1 × P1) are conjugate to the groups of cases [1] through [7] (Proposi- tion 2.5). It was proved in Proposition 2.5 that cases [1] through [7] are distinct. In Proposition 8.4 we showed that [8] (Cs24) is not birationally conjugate to any groups of cases [1] through [7]. Finally, the group [9] is isomorphic only to [1], but is not birationally conjugate to it (Proposition 2.2). This completes the proof that the distincts cases given above are not birationally conjugate. The proof of Theorem 1 follows directly from Theorem 5, and Theorem 3 is a corollary of Theorem 1. 11 Other kinds of groups Our main interest up to now was in finite Abelian subgroups of the Cremona group. In this section, we give some examples in the other cases, in order to show why the hypothesis ”finite”, respectively ”Abelian”, is necessary to ensure that condition (F ) (no curve of positive genus is fixed by a non-trivial element) implies condition (M) (the group is birationally conjugate to a group of automorphisms of a minimal surface). We refer to the introduction for more details. Finiteness is important since it imposes that the group is conjugate to a group of automorphisms of a projective rational surface. This is not the case if the group is not finite (see for example [Bla2], Proposition 2.2.4). Lemma 11.1. Let ϕ : P2 99K P2 be a quadratic birational transformation with three proper base-points, and such that deg(ϕn) = 2n for each integer n ≥ 1. Then, the following occur: 1. no pencil of curves is invariant by ϕ; 2. ϕ is not birationally conjugate to an automorphism of P2 or of P1 × P1. Proof. Denote by A1, A2, A3 the three base-points of ϕ and by B1, B2, B3 those of ϕ−1. Up to a change of coordinates, we may suppose that A1 = (1 : 0 : 0), A2 = (0 : 1 : 0) and A3 = (0 : 0 : 1). The birational transformation ϕ is thus the composition of the standard quadratic transformation σ : (x : y : z) 99K (yz : xz : xy) with a linear automorphism τ ∈ Aut(P2) that sends Ai on Bi for i = 1, 2, 3. Let Λ be some pencil of curves, and assume that ϕ(Λ) = Λ. We will prove that some base-point of Λ is sent by ϕ on an orbit of infinite order. The con- dition deg(ϕn) = 2n is equivalent to saying that for i = 1, 2, 3, the sequence Bi, ϕ(Bi), ..., ϕ n(Bi), ... is well-defined, i.e. that ϕ m(Bi) is not equal to Aj for any i, j ∈ {1, 2, 3},m ∈ N. Denote by α1, α2, α3, β1, β2, β3 the multiplicity of Λ at respectively A1, A2, A3, B1, B2, B3 and by n the degree of the curves of Λ. The curves of the pencil ϕ(Λ) thus have degree 2n−α1−α2−α3. Since Λ is invariant, n = α1 + α2 + α3, so at least one of the αi’s is not equal to zero. The equality n = α1+α2+α3 implies that the curves of σ(Λ) have multiplicity αi at Ai, so the curves of ϕ(Λ) have multiplicity αi at Bi, whence αi = βi for i = 1, 2, 3. Since Λ passes through Bi with multiplicity αi, the pencil ϕ(Λ) = Λ passes through ϕ(Bi) with multiplicity αi for i = 1, 2, 3. Continuing in this way, we see that Λ passes through ϕn(Bi) with multiplicity αi for each n ∈ N. Consequently, Λ has infinitely many base-points, which is not possible. This establishes the first assertion. The second assertion follows directly, as each automorphism of P2 or P1 × P1 leaves a pencil of rational curves invariant. Corollary 11.2. The group generated by a very general quadratic transformation is a infinite cyclic group satisfying (F ) but not (M). Proof. The condition deg(ϕn) = 2n, n ∈ N is satisfied for all quadratic transfor- mations, except for a countable set of proper subvarieties. Consequently condition (F ) is not satisfied (Lemma 11.1) for a very general quadratic transformation. Let n be some positive integer and write ϕn : (x : y : z) 99K (f1(x, y, z) : f2(x, y, z) : f3(x, y, z)), for some homogeneous polynomials fi of degree 2 n. The set of points fixed by ϕn belongs to the intersection of the curves with equations xf2 − yf1, xf3 − zf1 and yf3 − zf2. In general, there is only a finite number of points; this yields condition (F ). In fact, the argument of Lemma 11.1 works for any very general birational transformation of P2, since this is a composition of quadratic transformations. We thus find infinitely many cyclic subgroups of the Cremona group that are not birationally conjugate to a group of automorphisms of a minimal surface although none of their non-trivial elements fixes a non-rational curve. The implication (F ) ⇒ (M) is therefore false for general cyclic groups. We now study the finite non-Abelian subgroups and provide, in this case, many examples satisfying (F ) but not (M): Lemma 11.3. Let S6 = { (x : y : z), (u : v : w) | ux = vy = wz} ⊂ P2 × P2 be the del Pezzo surface of degree 6. Let G ∼= Sym3 × Z/2Z be the subgroup of automorphisms of S6 generated by (x : y : z), (u : v : w) (u : v : w), (x : y : z) (x : y : z), (u : v : w) (y : x : z), (v : u : w) (x : y : z), (u : v : w) (z : y : x), (w : v : u) Then no non-trivial element of G fixes a curve of positive genus, and G is not birationally conjugate to a group of automorphisms of a minimal surface. Proof. Since every non-trivial element of finite order of Aut(S6) is birationally conjugate to a linear automorphism of P2 (Corollary 9.10), no such element fixes a curve of positive genus. The description of every G-equivariant elementary link starting from S6 was given by Iskovskikh in [Isk4]. This shows that this group is not birationally conjugate to a group of automorphisms of a minimal surface. Lemma 11.4. Let S5 be the del Pezzo surface of degree 5. Let G ∼= Sym5 be the whole group Aut(S5). Then no non-trivial element of G fixes a curve of posi- tive genus, and G is not birationally conjugate to a group of automorphisms of a minimal surface. Proof. Since every non-trivial element of Aut(S5) is birationally conjugate to a linear automorphism of P2 (Corollary 9.10), such an element does not fix a curve of positive genus. Suppose that there exists some G-equivariant birational trans- formation ϕ : S5 99K S̃ where S̃ is equal to P 2 or P1 × P1. We decompose ϕ into G-equivariant elementary links (see for example [Isk3], Theorem 2.5). The classi- fication of elementary links ([Isk3], Theorem 2.6) shows that a link S5 99K S either a Bertini or a Geiser involution (and in this case S′ = S5, and thus this link conjugates G to itself), or the composition of the blow-up of one or two points, and the contraction of 5 curves to respectively P1 × P1 or P2. It remains to show that no orbit of G has size 2 or 1, to conclude that these links are not possible. This follows from the fact that the actions of Sym5,Alt5 ⊂ G on S5 are fixed-point free (Proposition 5.1). Finally, the way to find more counterexamples is to look at groups acting on conic bundles. The generalisation of the example Cs24 gives many examples of non-Abelian finite groups. Here is the simplest family: Lemma 11.5. Let n be some positive integer, and let G be the group of birational transformations of P2 generated by g1 : (x : y : z) 99K (yz : xy : −xz), g2 : (x : y : z) 99K (yz(y − z) : xz(y + z) : xy(y + z)), h : (x : y : z) 99K (e2iπ/2nx : y : z). Then, G preserves the pencil Λ of lines passing through (1 : 0 : 0) and the corre- sponding action gives rise to a non-split exact sequence 1 →< h >∼= Z/2nZ → G → (Z/2Z)2 → 1. In particular, the group G has order 8n. Furthermore, no non-trivial element of G fixes a curve of positive genus, and G is not birationally conjugate to a group of automorphisms of a minimal surface. Proof. Firstly, since g1 and g2 generate the group Cs24, which is not birationally conjugate to a group of automorphisms of a minimal surface, this is also the case for G. Secondly, we compute that (g1) 2 = (g2) 2 = (h)n is the birational transforma- tion (x : y : z) 7→ (−x : y : z). The maps g1 and g2 thus have order 4 and h has order 2n. Thirdly, every generator of G preserves the pencil Λ of lines passing through (1 : 0 : 0). The action of g1, g2 and h on this pencil is respectively (y : z) 7→ (−y : z), (y : z) 7→ (z : y) and (y : z) 7→ (y : z). The action of G on the pencil thus gives an exact sequence 1 → G′ → G → (Z/2Z)2 → 0, where G′ is the subgroup of elements of G that act trivially on the pencil Λ. It is clear that < h >∼= Z/2nZ is a subgroup of G′. Since g1h(g1) −1 = g2h(g2) −1 = h−1 and g1 and g2 commute, the group < h > is equal to G Finally, any element of G that fixes a curve of positive genus must act trivially on the pencil Λ and thus belongs to < h >. Hence, only the identity is possible. References [Ba-Be] L. Bayle, A. Beauville, Birational involutions of P2. Asian J. Math. 4 (2000), no. 1, 11–17. [Be-Bl] A. Beauville, J. Blanc, On Cremona transformations of prime order. C.R. Acad. Sci. Paris, Sér. I 339 (2004), 257-259. [Bea1] A. Beauville, Complex Algebraic Surfaces. LondonMathematical Society Student Texts, 34, 1996. [Bea2] A. Beauville, p-elementary subgroups of the Cremona group. J. Algebra 314 (2007), no. 2, 553-564 [Bla1] J. Blanc, Conjugacy classes of affine automorphisms of Kn and linear automorphisms of Pn in the Cremona groups. Manuscripta Math., 119 (2006), no.2 , 225-241. [Bla2] J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, PhD Thesis, University of Geneva, 2006. Available online at http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.html [Bla3] J. Blanc, Finite Abelian subgroups of the Cremona group of the plane. C.R. Acad. Sci. Paris, Sér. I 344 (2007), 21-26. [Bla4] J. Blanc, The number of conjugacy classes of elements of the Cremona group of some given finite order. Bull. Soc. Math. France 135 (2007), no. 3, 419-434. [Bla5] J. Blanc, On the inertia group of elliptic curves in the Cremona group of the plane. Michigan Math. J. (to appear) math.AG/0703804 http://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.html http://arxiv.org/abs/math/0703804 [BPV] J. Blanc, I. Pan, T. Vust, Sur un théorème de Castelnuovo. Bull. Braz. Math. Soc. 39 (2008), no. 1, 61-80. [De-Ku] H. Derksen, F. Kutzschebauch, Nonlinearizable holomorphic group ac- tions. Math. Ann. 311 (1998), no. 1, 41–53. [dFe] T. de Fernex, On planar Cremona maps of prime order. Nagoya Math. J. 174 (2004), 1–28. [dF-Ei] T. de Fernex, L. Ein, Resolution of indeterminacy of pairs. Algebraic geometry, 165-177, de Gruyter, Berlin (2002). [Dol] I.V. Dolgachev, Weyl groups and Cremona transformations. Singulari- ties I, 283–294, Proc. Sympos. Pure Math. 40, AMS, Providence (1983). [Do-Iz] I.V. 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Ann., vol. 48, (1896), 497-498, 195- Introduction The main questions and results How to decide Linearisation of birational actions The approach and other results Comparison with other work Aknowledgements Automorphisms of P2 or P1P1 Some facts about automorphisms of conic bundles The del Pezzo surface of degree 6 The del Pezzo surface of degree 5 Description of twisting elements The example Cs24 Finite Abelian groups of automorphisms of conic bundles - birational representative elements Actions on del Pezzo surfaces with fixed part of the Picard group of rank one The results Other kinds of groups

0704.0538

Oriented growth of pentacene films on vacuum-deposited polytetrafluoroethylene layers aligned by rubbing technique

Microsoft Word - Oriented_growth___for_publish.doc Oriented growth of pentacene films on vacuum-deposited polytetrafluoroethylene layers aligned by rubbing technique Marius Prelipceanu∗,1, Otilia – Sanda Prelipceanu1, Ovidiu-Gelu Tudose1, Konstantin Grytsenko1,2 , Sigurd Schrader 1 1University of Applied Sciences Wildau, Faculty of Engineering, Department of Engineering Physics, Friedrich-Engels-Strasse 63, D-15475 Wildau, Germany. 2Institute of Semiconductor Physics, 45 Nauki pr., Kyiv, 03028, Ukraine. Abstract A new method for preparation of high quality dielectric thin films made of polytetrafluoroethylene (PTFE) is described. This method includes film formation by means of a special kind of vacuum deposition polymerization (VDP) of PTFE, assisted by electron cloud activation. Rubbing of those layers makes them orienting substrate materials which induce spontaneous ordering of deposited chromophore layers. We investigated structure and morphology of PTFE layers deposited by vacuum process in dependence on deposition parameters: deposition rate, deposition temperature, electron activation energy and activation current. Pentacene (PnC) layers deposited on top of those PTFE films are used as a tool to demonstrate the orienting ability of the PTFE layers. The molecular structure of the PTFE films was investigated by use of infrared spectroscopy. By means of ellipsometry, values of refractive index between 1.33 and 1.36 have been obtained for PTFE films in dependence on deposition conditions. Using the cold friction technique orienting PTFE layers with unidirectional grooves are obtained. On top of these PTFE films oriented PnC layers were grown. The obtained order depends both on the PTFE layer thickness and on PnC growth temperature. Keywords: polytetrafluoroethylene, vacuum deposition polymerisation, orientation, organic insulator, semiconductor, pentacene. ∗ Marius Prelipceanu, Tel: +49 3375 508 524, Fax: +49 3375 508 503, E-Mail: [email protected] 1. Introduction Organic electronics, in particular, organic field effect transistors (OFET) is a fast developing field of research and technological development [1-3]. Pentacene (PnC) is one of the most extensively studied organic semiconductors for OFETs due to its relatively high carrier mobility [2]. Ordered molecular materials are used in electronic and photonic organic devices for obtaining anisotropic properties. Therefore, techniques for formation of high-quality films play an important role in the development of organic thin film devices. For such applications, uniform films with the thickness range from nanometers to submicrons are required. For electronic applications, film purity and interface characteristics influence the charge transport and energy transfer processes. For optical applications, controlling of dipole orientation is required as well as uniform thickness and low scattering loss. It is not easy to fulfill all these requirements by the wet processing. On the other hand, stable polymers like polytetrafluoroethylene (PTFE) do not dissolve in any solvent. Therefore, vacuum-based dry processing is the only possible method for deposition of such polymers. Some polymers can be evaporated by heating in vacuum, but for complex polymers low temperature plasma polymerisation should be used. Primary polymer degradation products are generated by the scission of the molecular chain at various sites and/or the cleavage of side groups or atoms. Depending on the nature of the polymer structure, the scission of polymer chains can occur either randomly or in an ordered depolymerisation mechanism. PTFE films were deposited in vacuum, but with a modified technique, which includes electron cloud activation of the decomposition products [4]. Since the discovery of the friction transfer method of PTFE hot friction transfer has been used extensively to prepare substrates materials on top of which deposited chromophores form oriented layers by self- organization [5-7]. Recently it was found that vacuum deposited and rubbed PTFE films also support growth of oriented dye layers [8-10]. Using a series of measuring techniques (e.g. ellipsometry, optical and infrared spectroscopy and atomic force microscopy) we investigated physical and optical properties of vacuum deposited PTFE and PnC thin films formed on top of these PTFE layers in order to find optimal conditions for deposition of highly oriented PnC films. 2. Experimental 2.1. Description of PTFE and PnC PTFE is a linear polymer having the chemical structure shown in figure 1a. (a) polytetrafluoroethylene (PTFE) (b) pentacene Fig.1. Chemical structure of: (a) polytetrafluoroethylene (PTFE), (b) pentacene (PnC). PTFE can be considered to be a suitable organic material serving as gate dielectric in organic field-effect transistor (OFET) devices because of its physical and chemical properties: very good chemical, photochemical and thermal stability, low dielectric constant, very low conductivity and high breakdown filed strength. PTFE is one of the most thermally stable plastic materials manifesting no appreciable decompositions below 260°C. The chemical structure of pentacene which consists of five annulated benzene rings is shown in fig. 1b. Due to its flat conformation it can easily form crystals, which show highly anisotropic transport properties. Pentacene has a molar mass of 278.35 grams. The melting point is at about 300°C and the heat of vaporization is 74.4 kJ/mol. 2.2. Deposition technique The preparation of PTFE films was carried out by use of a special vacuum deposition technique. The films were obtained by evaporation of bulk PTFE pellets in the temperature range between 300° and 450°C with electron cloud-assisted activation with typical process pressure of 10-2 Pa, an accelerating voltage of 1-3 kV and an electron activation current of 0 – 5 mA as proposed before [4,11]. The electron cloud was produced by an electron gun with a ring cathode. A computer equipped with a quartz oscillator card Sigma SQM-242 was monitoring the film thickness and deposition rate. The temperature of the crucible was monitored by a chromel-alumel thermocouple. The deposition rate depends both on the electron current used for activation and on the temperature of the crucible. At the start of a deposition run, the increase of PTFE temperature in the evaporator causes an increase of both pressure and deposition rate. Fragments are colliding with each other before reaching the substrate, losing their chemical reactivity by forming stable gaseous species, which will not be incorporated into the deposited layer on the substrate. The evaporation rate is limited by the fact that the pressure can rise only to a certain value at which a breakdown of the electrical gun occurs. Hence, there exists an operation heating temperature [4, 11], which strongly depends on the pumping speed of the vacuum system and should be determined for each installation. This method gives the possibility to have a fast control of the evaporation rate, that in general is limited by the thermal inertia of the crucible, but in this method it is controlled instantly by the electrical power, that produces the activating cloud of electrons and, therefore, changing the quantity of active species. All PTFE films were deposited at a substrate temperature of 200C. Fig. 2 shows schematically the deposition installation used for the PTFE film deposition. Fig.2. Deposition set-up used for PTFE and pentacene deposition. PnC for fluorescence was purchased from Sigma-Aldrich and used as received. PnC films were deposited onto rubbed PTFE layers using conventional a tantalum boat heated by electric current. Important parameter which governs film formation is the temperature of the substrate. Related to this temperature, kinetic limitations such as molecular mobility, crystallization speed, and other thermodynamic factors are controlling the structure and morphology of the film. The substrate temperature was kept constant at room or elevated temperature and monitored by a chromel-alumel thermocouple. The deposition rate was chosen in the range between 0.05 and 0.2 nm/s. The distance between evaporators and substrate was 0.15 m. 2.3. Mechanical rubbing method The PTFE layers deposited onto different substrates were rubbed in an unidirectional mode on a cotton surface used to clean optical systems. The cotton for friction was placed in a fixed position on an optical table. The samples were rubbed 3- 6 times on a cotton surface with a constant force and speed. The scheme for cold friction is shown in Fig. 3. Fig.3. Schematic representation of the cold friction technique applied to a PTFE layer. 2.4. Studies of the deposited films The surface morphology of the films was obtained using an Atomic Force Microscope (Autoprobe VP 2 Park Scientific Instruments), operating in non-contact mode in air at room temperature. The mean thickness and index of refraction (n) of the PTFE films were determined by means of ellipsometry using a Plasmos SD2000 Automatic Ellipsometer operating at a wavelength of 632,8 nm. The thickness of the investigated thin films was measured using a Dektak Profilometer (DEKTAK 3 from Veeco Instruments) device, which has the capability of measuring the step height down to a few nm. Polarized absorption spectra of pentacene films were obtained with a UV/VIS Spectrometer (Lambda 16 Perkin Elmer). Measurements of infrared spectra of PTFE films have been carried out by use of a Perkin-Elmer Spectrum 2000 fourier transform infrared (FT-IR) spectrometer. 3. Results and discussions The analysis of the results, obtained using electron cloud assisted activation evaporation revealed that only a few important processes determine the film properties. Fig. 4 shows the influence of the evaporator temperature on the layer thickness at constant deposition time of 10 minutes. The presented curve stops well below the limiting pressure above which a decrease of deposition rate occurs due to the reason described above [4]. With increase of electron activation current the deposition rate and resulting film thickness are increasing. The maximum deposition rate obtained at a limiting pressure of 5-6 x10–2 Pa was 0.18 nm/s at an activation current of 10 mA and a voltage of 3 kV. The surface relief of PTFE films deposited at different conditions onto silicon substrate is shown in Fig. 5. Fig.4. Dependence of layer thickness on crucible temperature after 10 minutes of deposition, I=2mA, V=1.5 kV. Fig.5. Surface morphology and profile of PFTE films: (a) 2 mA and (b) 1 mA electron current activation, respectively. The surface of all PTFE films is smooth. For smaller electron activation current, a larger granular structure on the surface is detected. Root mean square (RMS) roughness is 1 nm and 3 nm, respectively. The obtained RMS values indicate a smoother surface occurs at higher electron activation energy. Ellipsometry results confirm the AFM investigations: thicknesses determined by both methods are comparable. In addition, a change of refractive index in dependence on electron activation energy and on current density was found, as determined by means of ellipsometry. Thus, electron activation parameters affect the surface morphology and refractive index of the PTFE films. Table 1. Refractive index of PTFE films versus activation conditions. The IR spectra of deposited films under different activation are depicted in Fig. 6. The bands at 1161 and 1258 cm-1 was assigned to the -CF2- groups, the band at 1350 cm- 1 to groups with a double bond. The intensity of the bands at 524 and 556 cm-1 is lower, than the intensity of the band at 736 cm-1, thus indicating, that the material of the films is almost amorphous [11-13]. Normally at low electron activation PTFE layers are crystalline [4, 13]. An increase of electron activation current makes the films amorphous and increases the content of double bonds and side branches. Here at low activation power almost amorphous films with some double bonds but almost without branches were deposited. Fig.6. IR-spectra of a 500 nm-thick films, deposited by PTFE evaporation under following conditions: 1 with activation current 1,5 mA; 2 with activation current 2 mA. Inset: the magnification of IR spectra in the range from 500 to 1000 cm-1 is shown. After rubbing with a cotton cloth, the film surfaces were investigated by AFM and profilometry techniques. The film surface acquired ordered relief oriented in the direction of friction. Fig. 7 shows the relief and the profile of the PTFE layer after friction. The PTFE grooves have 10 -100 µm length, and about 300 nm height. Also, we can see that the spectral lines show exactly the linear structure of PTFE. Fig.7. The 1 µm × 1 µm AFM scan of rubbed PTFE films: (a) 3D image of the film relief; (b) profile of a series of grooves. The groove length is about 100 nm, and the height is 300 nm. Fig.8. Polarized absorption spectra of pentacene films for the parallel (dotted curves) and perpendicular (solid curves) orientation of the electric vector of light in respect to the PTFE layer alignment. Films were deposited at the following substrate conditions: a – onto 36 nm PTFE at 200C, b – onto50 nm PTFE at 200C, c – onto 90 nm PTFE at 750C, d – onto50 nm PTFE at 750C. Film thicknesses by quartz monitor: a) and b) – 75 nm, c) and d) – 80 nm. Band splitting at the main absorption is 30, 34, 40 and 40 nm for a), b), c) and d) respectively. Fig.9. Surface relief of PnC film onto rubbed PTFE sublayer. Measurements of electronic absorption spectra of the pentacene films deposited onto rubbed PTFE layers of different thickness have shown that orientation of the PnC films depend on both the PTFE film thickness and on substrate temperature. Optical spectra of some PnC films are presented in Fig.8. They are in a good agreement with spectra of α- and β- phase of PnC films, deposited onto both inorganic and polymer substrates, including PTFE [7, 14]. Rubbed PTFE films of about 50 nm thickness lead to the best oriented PnC films. Absorption measurements with polarized light have shown that the deposited PnC films show a pronounced dichroism. A dichroic ratio of about 2 was measured even at deposition temperature of 20°C. This dichroic ratio is larger than obtained for PnC films deposited onto friction transferred PTFE layers, and for deposition at 20°C there no dichroism was observed at all. The temperature elevation from 20°C to 75°C slightly enhances the PnC film orientation and changes the spectral shape. The latter two effects can be explained by the PnC molecular mobility enhancement. The former one is subject for further detailed studies. A little difference in the spectral shape indicates different molecular interactions inside of the PnC crystals dependent on deposition conditions. The crystal size and structure is also sensitive to the deposition conditions and results in modification of the absorption spectra. The optical spectra of the PnC films deposited at 75°C show a small shift of all bands towards to red region and an increase of band splitting in comparison with bands of the films, deposited at 20°C, thus evidencing better intermolecular interactions in films, deposited at elevated temperature. Both PTFE film thickness and substrate temperature allow controlling this parameter in order to deposit PnC films with predetermined properties. The absorption of films deposited at 75°C is smaller than the absorption of films deposited at 20°C, although the quartz monitor thickness was the same for both samples. Obviously, a re- evaporation took place already at 75°C as mentioned before. By AFM no preferred crystal orientation was found in all PnC films. The typical relief of a PnC film on a PTFE aligned layer is shown in Fig.9a. The crystal size is in the range of 80 to 200 nm, depending on deposition conditions. Sometimes freely distributed needle-like PnC crystal with long axis up to 500 nm appeared (Fig.9b). Such films have low optical anisotropy. Therefore, the optical anisotropy of PnC films is due to the unidirectional arrangement of PnC molecules inside all crystals. Comparison of the obtained PnC crystals with those grown on friction-transferred PTFE layers shows that the crystals grown on the vacuum deposited, rubbed PTFE layers have smaller size and more round shape. The like effect was found for the growth of squarylium dyes on such vacuum deposited PTFE films [10]. This effect is caused by a smaller relief of the surface of the vacuum-deposited and rubbed PTFE film in comparison to the friction transferred films. In addition, some differences in the structure of friction-transferred and vacuum- deposited PTFE also plays a role. The PnC nucleation directed by PTFE edges are the main mechanism of growth of oriented PnC film as it was proposed by Brinkmann et. al.[7]. They observed that the top material domains have been enforced to grow parallel to the ledge direction due to the confinement by the PTFE nanofibrils. Only when the height of these domains exceeds that of the ledge the lateral growth of the domains is possible. The opinion about the prevailing influence of PTFE aligned on the molecular level onto dye oriented growth was expressed previously by Tanaka et. al. [8] and Wittmann et al. [5, 6, 7]. Our results seem to support the latter opinion, but the amorphous structure of our PTFE films should be taken into account. Perhaps, both mechanisms are taking place with different contributions in dependence on both the sublayer properties and deposition conditions. But even this suggestion does not explain all peculiarities, so further research should be carried out. 4. Conclusions Amorphous PTFE films with RMS roughness of 1-3 nm were deposited by electron cloud-assisted deposition in vacuum. Aligned grooves and ridges on the PTFE film surface were obtained by rubbing with a cotton cloth. PTFE film thickness and growth temperature elevation influence anisotropy of pentacene film. A dichroic ratio about of 2 was obtained even when the substrate was held at room temperature. The pentacene film is oriented on the molecular level. The strength of this technique is that the vacuum deposited, and rubbed PTFE layers have a higher orienting power than friction transferred PTFE layers so that they may favorably be used in OFETs as bottom gate dielectric which induces enhanced order in the channel material deposited on top of them. In addition, the vacuum deposited PTFE layers can also be used in top gate geometry, i.e. by deposition on top of OFET channel materials on plastic substrates. 5. Acknowledgements The authors would like to thanks to Dagmar Stabenow (University of Potsdam), Ramakrishna Velagapudi (University of Applied Sciences Wildau) for the AFM and optical measurements and Dr. Oleg Dimitiriev (Institute of Semiconductor Physics, Kyiv) for the fruitful discussions. Financial support of the European Commission under contract number: HPRN-CT-2002-00327-RTN EUROFET and of Federal Ministry of Education and Research (BMBF) Project under no. Ukr 04/004 is gratefully acknowledged. 6. References 1. Daraktchlev M, von Muchlenen A, Nuesch F. New J. of Physics 2005; 7:113. 2. Mattis BA, Pei Y, Subramanian V. Appl.Phys. Lett. 2005; 86: 033113. 3. Misaki M, Ueda Y. Appl. Phys. Lett. 2005; 87:243503. 4. Gritsenko KP, Krasovsky AM. Chem. Rev. 2004; 103(9):3607. 5. Wittmann JC, Smith P. Nature 1991; 352:414. 6. Moulin JF, Brinkmann M, Thierry A, Wittmann JC. Adv. Mater.2002; 14(6):436. 7. Brinkmann M, Graff S, Straupe C, Wittmann JC. J.Phys.Chem. 2003; B107:10531. 8. Tanaka T, Honda Y, Ishitobi M. Langmuir 2002; 17:2192. 9. Gritsenko KP, Slominski Yu L, Tolmachev AI, Tanaka T, Schrader S. Proc.SPIE 2002; 4833: 482. 10. Gritsenko KP, Grinko DO, Dimitrev OP, Schrader S, Thierry A, Wittmann JC. Optical Memory and Neutral Networks 2004; N3:135. 11. Roeges NP. G. A Guide to the Complete Interpretation of Infrared Spectra of Organic Structures, Wiley: New York (1994). 12. Liang CY, Krimm S J. J. Chem. Phys. 1956; 25:563. 13. Gritsenko KP, Lantoukh GV. J. Applied Spectroscopy. 1990; 52:677. 14. Brinkmann M, Videva VS, Bieber A. J. Phys. Chem. 2004; A108:8170. 15. Ruiz R, Chouldhary D, Nickel B. Chem. Mater. 2004; 16:4497. 16. Pratontep S, Nüesch F, Zuppiroli L, Brinkmann M. Phys. Rev.2005; B 72:085211.

0704.0539

Integral representations for convolutions of non-central multivariate gamma distributions

INTEGRAL REPRESENTATIONS FOR CONVOLUTIONS OF NON–CENTRAL MULTIVARIATE GAMMA DISTRIBUTIONS T. ROYEN Fachhochschule Bingen, University of applied sciences, Berlinstrasse 109, D–55411 Bingen, Germany E–mail: [email protected] Abstract. Three types of integral representations for the cumulative distribution functions of convolutions of Γp(αk,Σk,∆k)–distributions with non–centrality matri- ces ∆k are given by integration of products of simple complex functions over the p–cube (−π, π]p. In particular, the joint distribution of the diagonal elements of a generalized quadratic form XAX ′ with n independentNp(µk,Σ)–distributed columns in Xp×n and a fixed A ≥ 0 is obtained. For a single Γp(α,Σ,∆)–cdf (p− 1)–variate integrals over (−π, π]p−1 are derived. The integrals are numerically more favourable than integrals obtained from the Fourier– or Laplace inversion formula. Key words and phrases: Convolutions of multivariate distributions, generalized quadratic forms of normal random vectors, multivariate chi–square distribution, Mul- tivariate gamma distribution AMS 2000 subject classifications: 62H10, 62E15 1. Introduction The following notations are used: (n) stands for n1+...+np=n n1, . . . , np ∈ N0 and without any indices means (n). The notation D ≥ 0 is also used for non–symmetrical matrices Dp×p with only non–negative eigenvalues. The spectral norm of a p× p–matrix B is denoted by ‖B‖, I or Ip is always an identity matrix and Cp is the p–cube (−π, π]p. The Laplace transform (L.t.) of a p–variate non–central Γp(α,Σ,∆)–density with α > 0, Σ > 0 and a non–centrality matrix ∆ ≥ 0 was originally obtained from the L.t. of a non-central Wp(2α,Σ,∆)–Wishart distribution (with an additional scale factor 2) and is given by f̂(t1, . . . , tp;α,Σ,∆) = |Ip +ΣT |−αetr(−ΣT (I +ΣT )−1∆), (1) T = diag(t1, . . . , tp), t1, . . . , tp ≥ 0. This function f̂ is generally the L.t. of the density of a real measure on (0,∞)p which is not always a probability measure. The term ”Γp(α,Σ,∆)–distribution” is used here in this general sense. The exact set of values α, leading to a probability density (pdf) f(x1, . . . , xp;α,Σ,∆), depends on Σ and presumedly on ∆. To obtain a pdf, all positive integers 2α (degrees of freedom) are admissible and all 2α > p − 1. Moreover, in the central case all non–integer values 2α > p− 2 ≥ 0 are allowed. For p− 2 < 2α < p− 1 http://arxiv.org/abs/0704.0539v1 see Royen (1997). Furthermore all α > 0 are admissible if |I + ΣT |−1 is infinitely divisible. Two characterizations of infinite divisibility of a Γp(α,Σ)–distribution are found in Griffiths (1984) and Bapat (1989). Further conditions for admissible non– integer 2α < p− 2 are given in Royen (1997), (2006). Three integral representations by integration over Cp are provided by theorem 2 in section 4 for the functions F (x1, . . . , xp;α1, . . . , αn,Σ1, . . . ,Σn,∆1, . . . ,∆n) (2) . . . f(ξ1, . . . , ξp;α1, . . . , αn,Σ1, . . . ,Σn,∆1, . . . ,∆n)dξ1 . . . dξp, where f has the L.t. |Ip +ΣkT |−αketr(−ΣkT (I +ΣkT )−1∆k), (3) α1, . . . , αn > 0,Σ1, . . . ,Σn > 0,∆1, . . . ,∆n ≥ 0. Thus, F is not always the cumulative distribution function (cdf) of a probability measure. In particular letXp×n be aNp×n(Mp×n,Σp×p⊗In)–random matrix and An×n ≥ 0 of rank q with T ′AT = Λ = diag(λ1, . . . , λn), λ1 ≥ . . . ≥ λn. Then the joint distribution of the diagonal elements of the generalized quadratic form 1 XAX ′ equals the distribution of the diagonal of 1 Y ΛY ′ with a Np×n(MT,Σp×p⊗In)–distributed Y = XT . This is the distribution of a sum of q independent Γp( , λkΣ,∆k = −1)–random vectors, where µ∗k is the k–th column of M ∗ = MT . This joint distribution of p quadratic forms of normal random vectors is comprised within theorem 2 as a special case with , Σk = λkΣ, k = 1, . . . , q. For methods under more general assumptions see also Blacher (2003). For a survey of univariate quadratic forms of normal random variables see chapter 4 in Mathai and Provost (1992). For several quadratic forms of skew elliptical distributions see B.Q. Fang (2005). In Royen (1991), (1992) three different types of series expansions for the χ2p(2α,Σ)– cdf were derived from three different representations of the χ2p(2α,Σ)–L.t. which are extended to the general Γp(α,Σ,∆)–L.t. in section 3 in a similar way as in Royen (1995). Some series expansions, closely related to the first two types, are already found in Khatri, Krishnaiah and Sen (1977). The third type was introduced because of its superior convergence properties. The simple method to transform many series expansions into integrals over Cp is explained in more detail in section 2 and summarized in theorem 1. The idea is as follows: If A(z1, . . . , zp) and B(z1, . . . , zp) are analytical functions whose power series have the coefficients a(m1, . . . ,mp) and b(n1, . . . , np) and which are absolutely convergent for max |zj | < rA and max |zj | < rB respectively, where r−1B < rA, then (2π)−p A(y1, . . . , yp)B(y 1 , . . . , y p )dϕ1 . . . dϕp (4) a(n1, . . . , np)b(n1, . . . , np) holds with yj = re iϕj , −π < ϕj ≤ π, j = 1, . . . , p and r−1B < r < rA. The integrals in (4) might be more economical than the series if the generating functions A and B are simple available functions and if the series are slowly convergent with very intricate coefficients. For non–central multivariate gamma distributions series expansions are practically not feasible. The integral representations in theorem 2 of section 4 are of the type in (4). As long as no elementary density formulas are availale it should be a reasonable way to obtain the joint cdf by integration of elementary terms only over Cp and not over Rp as by the Fourier or Laplace inversion formula. A single Γp(α,Σ,∆)–cdf is represented by a (p− 1)–variate integral over Cp−1 in section 5. A totally different –variate integral representation of the Γp(α,Σ)–cdf has been given recently by Royen (2006), which is based on m–factorial decompositions∑ p×p = D −BB′, where D is a real or complex diagonal matrix minimizing the rank m of Σ−1 −D. Approximations to a Γp(α,Σ)–cdf are obtained by m–factorial approx- imations to Σ with a low value of m. These approximations are improved further by successive correction terms. 2. The method Theorem 1 in this section can be generalized in many ways, e.g. for Fourier trans- forms, but the version below is sufficient for the purpose of the underlying paper. Let f̂(t1, . . . , tp), t1, . . . , tp ≥ 0, be a given L.t. of an unknown function f(x1, . . . , xp) with f = 0 for minxj < 0. It is assumed that there are univariate L.t. ĝj0(t) of some probability densities gj0(x) on (0,∞) and further functions hj(t) with |hj(t)| ≤ 1, uniformly for t ≥ 0, which enable a representation f̂(t1, . . . , tp) = ĝj0(tj) B (h1(t1), . . . , hp(tp)) (5) with an analytical function B(z1, . . . , zp) whose power series expansion b(n1, . . . , np) j (6) is absolutely convergent for |z1|, . . . , |zp| < rB with a certain value rB > 1. Furthermore, the products ĝj0(t)(hj(t)) n are supposed to be the L.t. of continuous functions gjn(x), x > 0, which satisfy the conditions |gjn(x)| ≤ nck(x) with a constant c and (7)∫ ∞ k(x)e−txdx < ∞ for all t > 0 . Hence, the generating functions (generators) gj(x, y) = gjn(x)y n, j = 1, . . . , p, (8) are defined for all x > 0 and |y| < 1, and they have the L.t. ĝj(t, y) = ĝj0(t) 1− yhj(t) , t ≥ 0. (9) Theorem 1. Under the assumptions from (6) and (7) f̂ in (5) is the L.t. of f(x1, . . . , xp) = (2π) B(y−11 , . . . , y gj(xj , yj)dϕj (10) with yj = re iϕj , −π < ϕ ≤ π, r−1B < r < 1, gj from (8). Proof. The integral in (10) is evaluated by (2π)−p b(m1, . . . ,mp) gjnj (xj)y  dϕ1 . . . dϕp b(n1, . . . , np) j=1 gjnj (xj) and this series has the L.t. from (5). Some further remarks: With Gj(xj , yj) = gj(ξ, yj)dξ (11) instead of the gj in (10), the corresponding representation arises for F (x1, . . . , xp) = . . . f(ξ1, . . . , ξp)dξ1 . . . dξp. (12) If the series in (8) are absolutely convergent for all y ∈ C then additionally hj(t) = 0 (13) is supposed to hold. Then the rhs of (9) is the L.t. of gj(x, y) for any fixed y and all sufficiently large t. In some cases the functions gj0 and their L.t. ĝj0 are known from univariate marginal distributions apart from some scale factors. If the functions uj = hj(t) are explicitly invertible then B(u1, . . . , up) = h−11 (u1), . . . , h p (up) j=1 ĝj0 h−1j (uj) ) (14) can sometimes be found easily from the given f̂ . 3. Three representations for the Γp(α,Σ,∆)–Laplace transform and the related generators With any v > 0 we define zj = (1 + v −1tj) −1, tj ≥ 0, uj = 1− zj = v−1tjzj , ωj = zj − uj , Z = diag(z1, . . . , zp), U = diag(u1, . . . , up), Ω = diag(ω1, . . . , ωp). The scale factor v is introduced to obtain ‖B‖ < 1 for the matrices B defined in (20) below and to effect the convergence of some series expansions. For a more general scaling see remarks following theorem 2 in section 4. From the relations v−1T = UZ−1, Ip = Z + U, Ω = Z − U, (16) it follows for the matrices I +ΣT in the L.t. (1): I +ΣT = I + vΣUZ−1 = (Z + vΣU)Z−1 (17) Z + vΣU = I + (vΣ− I)U, (18a) vΣ(I + (v−1Σ−1 − I)Z), (18b) (I + vΣ)(I + (2(I + vΣ)−1 − I)Ω), (18c) and therefore |I +ΣT |−α = cα|Z|α|I +BY |−α (19) Y = U, B = vΣ− I, c = 1, (20a) Y = Z, B = (vΣ)−1 − I, c = |I +B|, (20b) Y = Ω, B = 2(I + vΣ)−1 − I, c = |I +B|. (20c) It should be noticed that ‖B‖ < 1 in (20c) for every v > 0 and Σ > 0. Now, using (16), by a straightforward calculation the L.t. in (1) can be represented f̂(t1, . . . , tp;α,Σ,∆) = |Z|α|I +BU |−αetr(−(I +B)U(I +BU)−1∆), (21a) |I +B|αetr(−∆)|Z|α|I +BZ|−αetr(Z(I +BZ)−1(I +B)∆), (21b) |I +B|αetr(− 1 ∆(I −B)) (21c) ·|Z|α|I +BΩ|−αetr(1 Ω(I +BΩ)−1(I +B)∆(I −B)), with the corresponding matrices B from (20) and Z,U,Ω from (15). For the former series expansions the following relations were used: Laplace transform f̂(t): f(x): F (x) = f(ξ)dξ: zαun vg α+n(vx) G α+n(vx) (22a) zα+n vgα+n(vx) Gα+n(vx) (22b) zαωn vhα,n(vx) Hα,n(vx) (22c) where z = (1 + v−1t)−1, gα+n(x) = e −xxα−1+n/Γ(α+ n), α+n(x) = gα+n(x) = α− 1 + n L(α−1)n (x)gα(x) with the generalized Laguerre polynomials L (α−1) n and hα,n(x) = (−1)n α− 1 + n L(α−1)n (2x)gα(x). The last identity is verified by L.t. The following bounds are derived from (22.14.13) in Abramowitz and Stegun (1965): ∣∣∣g(n)α+n(x) ∣∣∣ ≤ ex/2gα(x), α ≥ 1 2nα−1ex/2gα(x), 0 < α < 1 |hα,n(x)| ≤ xα−1/Γ(α), α ≥ 1 2nxα−1/Γ(α+ 1), 0 < α < 1 , (24) matching with the conditions in (7). The following generators (generating functions) with the Γ(α+n)–cdf Gα+n(x) are required for the formulas in theorem 2: Fα(x, y) = n=0 G α+n(x)y n = 1 , |y| < 1, (25a) n=0 Gα+n(x)y n = Gα(x, y), y ∈ C, (25b) n=0 Hα,n(x)y n = 1 x, 2y , |y| < 1 (25c) The identities (a) and (c) are verified by the L.t. of fα(x, y) = Fα(x, y). A short calculation shows Gα(x, y) = Gα(x) − y1−αe(y−1)x Gα(xy) , y 6= 1, α > 0 Gα−1(x) − y1−αe(y−1)x Gα−1(xy) , α ≥ 1, G0 := 1 xgα(x) + (1 + x− α) Gα(x), y = 1 gα(x, y) = Gα(x, y) = gα(x) + y 1−αe(y−1)x Gα(xy), α > 0 y1−αe(y−1)x Gα−1(xy), α ≥ 1 The functions Fα(x, y) are especially simple for α ∈ N since Gα(z) = 1−e−z j=0 z j/j!, α ∈ N. Besides, Gk+1/2(z) = erf(z 1/2)− e−z zj−1/2 Γ(j + 1/2) , k ∈ N0. The following simple lemma is used for the proof of theorem 2. Lemma 1. If B is a symmetrical p× p–matrix with ‖B‖ < 1 and Y = diag(y1, . . . , yp) then the power series expansion |I +BY |−α = b(n1, . . . , np) is absolutely convergent for max |yj | < rB = ‖B‖−1. This follows from (n) |b(n1, . . . , np)| = O(ϑn) with any ϑ > ‖B‖, which has been already shown in (2.1.16) . . . (2.1.18) in Royen (1991) (with the notation −C instead of 4. The integral representations In theorem 2 below the functions F (x1, . . . , xp;α1, . . . , αn,Σ1, . . . ,Σn,∆1, . . . ,∆n) from (2) are represented by three different integrals over Cp = (−π, π]p. Together with the generators Fα from (25), α = k=1 αk, the following matrices are used with a scale factor v to enforce ‖Bk‖ < 1: Bk = vΣk − I, Dk = ∆k(I +Bk), Fα from (25a), (27a) Bk = (vΣk) −1 − I, Dk = (I +Bk)∆k, Fα from (25b), (27b) Bk = 2(I + vΣk) −1 − I, Dk = 12 (I +Bk)∆k(I −Bk), Fα from (25c). (27c) Furthermore, we define λmax = max ‖Σk‖ , λ−1min = max ‖Σ k ‖, yj = reiϕj , −π < ϕj ≤ π, Y = diag(y1, . . . , yp), K = K(y1, . . . , yp) = etr(±(Y +Bk)−1Dk)|I +BkY −1|−αk , where the negative sign occurs only with Bk, Dk from (27a), and Fαdϕ = j=1 Fα(vxj , yj)dϕj . Theorem 2. With the above notations the functions F from (2) are respresentable by each of the following three integrals: (2π)−p KFαdϕ, (28) Fα from (25a), Bk, Dk from (27a), ‖Bk‖ < 1 if v < 2λ−1max, max ‖Bk‖ < r < 1, etr(−∆k)|I +Bk|αk (2π)−p KFαdϕ, (29) Fα from (25b), Bk, Dk from (27b), ‖Bk‖ < 1 if v > 12λ min, max ‖Bk‖ < r, ∆k(I −Bk) |I +Bk|αk (2π)−p KFαdϕ, (30) Fα from (25c), Bk, Dk from (27c), v > 0, max ‖Bk‖ < r < 1. Proof. Because of lemma 1 the assumptions of theorem 1 are satisfied with ĝj0(t) = z j = (1 + v −1tj) −α and hj(tj) corresponding to zj or uj = v −1tjzj = 1 − zj or ωj = zj − uj respectively. The functions ĝj0(t)(hj(t))n are the L.t. of the functions in the second column of (22) from which type (a) and (c) have the bounds in (23), (24), satisfying the condition (7) for theorem 1. The series n=0 Gα+n(x)y n = Gα(x, y) in (25b) is absolutely convergent for every y ∈ C. Thus, all r > max ‖Bk‖ are admissible in (29). In (30) we have max ‖Bk‖ < 1 for every v > 0. Hence, theorem 1 together with the respresentations of the L.t. in (21) implies (28), (29) and (30). The univariate case of (29) provides F (x;α1, . . . , αn, σ 1 , . . . , σ 1 , . . . , δ n) = (31)( σ−2αkk e Gα(vx, e δ2k/(1 + vσ iϕ − 1)) 1 + (v−1σ−2k − 1)e−iϕ with 2v > maxσ−2k , r = 1, Gα from (26). With p = 1 similar formulas arise from (28) or (30). The cdf of a quadratic form 1 x′Ax with T ′AT = diag(λ1, . . . , λn) ≥ 0 of rank q and a N (µ, σ2In)–random vector x is a special case of (31) with αk = 12 , σ k = λkσ 2 and non–centrality parameters δ2k = µ∗2k /σ 2, k = 1, . . . , q, µ∗ = T ′µ. Some further remarks: In (29) also ‖Bk‖ > 1 is allowed since every r = ‖Y ‖ > max ‖Bk‖ is admissible, which entails max ‖BkY −1‖ < 1. With ϑ = λmax/λmin it follows with special values of v: max ‖Bk‖ ≤ in (28) with v = 2(λmin + λmax) max ‖Bk‖ ≤ in (29) with v = (λ−1min + λ max), max ‖Bk‖ ≤ ϑ− 1√ in (30) with v = (λminλmax) −1/2. More generally, the scale factor v = w2 can be replaced by a scale matrix W 2 = diag(w21 , . . . , w p) > 0. Then with Tw = W −1TW−1, Σw = WΣW , ∆w = W∆W the L.t. (1) equals |I +ΣwTw|−α etr(−ΣwTw(I +ΣwTw)−1∆w). (32) Consequently, besides the substitutions vΣk → WΣkW , ∆k → W∆kW−1, the matrices I + Bk in theorem 2 must be replaced by WΣkW , (WΣkW ) −1 and 2(I + WΣkW ) respectively, and the generators Fα(vxj , yj) by Fα(w jxj , yj). In particular for a single Γp(α,Σ,∆)–distribution this more general scaling can be used to minimize ‖B‖ or for a ”natural scaling” i.e. to standardize I+B to a correlation matrix. However, ‖B‖ < 1 must be taken into account in (28), whereas this condition is satisfied in (30) for every scaling. It was shown in Royen (1991) that natural scaling can always be accomplished also in I +B = 2(I +WΣW )−1 by a unique W 2. 5. Representations of the Γp(α,Σ,∆) distribution function by (p− 1)–variate integrals For a single Γp(α,Σ,∆)–cdf it is always possible to perform the integration over a single variable ϕj within the integrals from theorem 2. We use the following functions Gα(x, y) = e−y Gα+n(x) α+n(x) (−y)n x, y ∈ C, Gα+n, G(n)α+n from (22), and G∗α(x, y) = eyGα(x, y). For positive half integers α = 1/2 + k these functions can also be computed by the erf–function and a sum of k terms which are essentially given by the modified Bessel functions Ij−1/2(2(xy) 1/2), j = 1, . . . , k, (see e.g. Royen (1995) or (2006)). Now let be W 2 = diag(w21 , . . . , w p) a general scale matrix, Y = diag(y1, . . . , yp), yj = re iϕj , −π < ϕj ≤ π, Bpp bp b′p bpp WΣW − I, (34a) (WΣW )−1 − I, (34b) 2(I +WΣW )−1 − I, (34c) Dpp dp dp dpp W∆ΣW, (35a) W−1Σ−1∆W−1, (35b) 2(I +WΣW )−1W∆W−1(I − (I +WΣW )−1), (35c) y0 = y0(y1, . . . , yp−1) = b p(Ypp +Bpp) −1bp − bpp (36) q = q(y1, . . . , yp−1) = (b p(Ypp +Bpp) −1,−1)D (Ypp +Bpp) Kα = Kα(y1, . . . , yp−1) = etr(±(Ypp +Bpp)−1Dpp)|I +BppY −1pp |−α, where the negative sign is only taken for Bpp from (34a). Theorem 3. With the above notations the Γp(α,Σ,∆)–cdf F (x1, . . . , xp;α,Σ,∆) is given by each of the following three integrals: (2π)p−1 w2pxp 1− y0 1− y0 1− yj w2jxj , yj − 1 dϕj , (38) B from (34a), D from (35a), ‖B‖ < r < 1, etr(−W∆W−1) |WΣW |α · (2π)p−1 (1− y0)−α G∗α (1− y0)w2pxp, 1− y0 jxj , yj)dϕj , B from (34b), D from (35b), ‖B‖ < r, 2αpetr(− 1 W∆W−1(I −B)) |I +WΣW |α · (2π)p−1 1− y0 (1− y0)−αGα 1− y0 1 + y0 w2pxp, 1− y20 1 + yj w2jxj , yj + 1 dϕj , B from (34c), D from (35c), ‖B‖ < r < 1. For the proof of theorem 3 the following two lemmas are required. Lemma 2. With Y = diag(y1, . . . , yp), yj = re iϕj , ‖B‖ < r, B,D, y0, q from (34), (35), (36), (37) the following decomposition is obtained etr((Y +B)−1D)|Y +B|−α = etr (Ypp +Bpp) −1Dpp |Ypp +Bpp|−α exp yp − y0 (yp − y0)−α. Proof. From frequently used formulas for p× p–matrices, (see e.g. complements and problems 2.4, 2.7 in chapter 1b of Rao (1973)) it follows for A = Y +B = App bp b′p yp + bpp |A| = |App|(yp + bpp − b′pA−1pp bp) = |Ypp +Bpp|(yp − y0), A−1 = A−1pp + yp−y0 A−1pp bpb pp − 1yp−y0A pp bp yp−y0 yp−y0 trace(A−1D) = trace A−1pp Dpp + yp − y0 A−1pp bpb pp Dpp −A−1pp bpdp yp − y0 (dpp − b′pA−1pp dp) = trace(A−1pp Dpp) + yp − y0 , which implies (41). Lemma 3. Let be q any number, Sr = {y ∈ C ∣∣|y| = r}, y0 any number with |y0| < r, then with Fα from (25), Gα,G∗α from (33), and the negative sign in ±q only for (42a) y − y0 Fα(x, y)(y − y0)−αyα−1dy Fα from (25a), r < 1, (42a) (1− y0)−α G∗α (1 − y0)x, q1−y0 , Fα from (25b), (42b) (1− y0)−α Gα x, 2q , Fα from (25c), r < 1, (42c) Proof. It is sufficient to verify (42) for the corresponding derivatives fα = At first, (42a) is shown: With Fα from (25a) and the binomial series for (1− y0/y)−(α+n) we obtain fα(x, y)(y − y0)−(α+n)yα−1dy α+m(x)y α+ n+ k − 1 y−n−1dy. With z = (1 + t)−1, u = tz, the last integral has the L.t. (uy)m α+ n+ k − 1 y−n−1dy m=n+k α+ n+ k − 1 yk0 = z αun(1− uy0)−(α+n). Multiplication by (−q)n/n! and summation over n leads to the L.t. (1− uy0)α 1− uy0 (1 + (1− y0)t)α (1− y0)t 1 + (1− y0)t and this is the L.t. of ∂ To verify (42b) we obtain with Fα from (25b): fα(x, y)(y − y0)−(α+n)yα−1dy gα+m(x)y Γ(α+ n+ k) Γ(α+ n)k! y−n−1dy m=n+k gα+m(x) Γ(α + n+ k) Γ(α+ n)k! yk0 = gα+n(x)e = (1− y0)−(α+n)(1− y0)gα+n((1 − y0)x). Multiplication by qn/n! and summation provides (1− y0)−α ∂∂x G (1− y0)x, q1−y0 (42c) can be shown by L.t. in a similar way as (42a). Proof of theorem 3. Without loss of generality yp is selected from the variables yj = re iϕj in Y = diag(y1, . . . , yp) with any fixed r > ‖B‖. If yp is replaced by a variable y with any |y| then the equation |Y +B| = |Ypp +Bpp|(yp + bpp − b′p(Ypp +Bpp)−1bp) = 0 has always a unique solution y = y0 = b p(Ypp +Bpp) −1bp − bpp with |y0| < r since ‖Bpp‖ ≤ ‖B‖. Hence, with lemma 2 and lemma 3, theorem 3 is obtained by integration over ϕp in the integrals of theorem 2 with n = 1. References Abramowitz, M. and Stegun, I.A. (1968). Handbook of Mathematical Functions, Dover, New York. Bapat, R.B. (1989). Infinite divisibility of multivariate gamma distributions and M–matrices, Sankhyā, Series A 51, 73–78. Blacher, R. (2003). Multivariate quadratic forms of random vectors, Journal of Multivariate Analysis 87, 2–23. Fang, B.Q. (2005). Noncentral quadratic forms of the skew elliptical variables, Journal of Multivariate Analysis 95, 410–430. Griffiths, R.C. (1984). Characterization of infinitely divisible multivariate gamma distributions, Journal of Multivariate Analysis 15, 13–20. Khatri, C.G., Krishnaiah, P.R. and Sen, P.K. (1977). A note on the joint distribution of correlated quadratic forms, Journal of Statistical Planning and Inference 1, 299–307. Krishnamoorthy, A.S. and Parthasarathy, M. (1951). A multivariate gamma type distribution, Annals of Mathematical Statistics 22, 549–557 (correction: ibid. (1960), 31, p. 229). Mathai, A.M. and Provost, S.B. (1992). Quadratic forms in random variables: Theory and applications, Marcel Dekker, New York. Rao, C.R. (1973). Linear Statistical Inference and its Applications, 2nd edition, Wiley, New York. Royen, T. (1991). Expansions for the multivariate chi–square distribution, Journal of Multivariate Analysis 38, 213–232. Royen, T. (1992). On representation and computation of multivariate gamma distributions, in: Data Analysis and Statistical Inference - Festschrift in Honour of Friedhelm Eicker, 201–216, Verlag Josef Eul, Bergisch Gladbach, Köln. Royen, T. (1995). On some central and non–central multivariate chi–square distributions, Statistica Sinica 5, 373–397. Royen, T. (1997). Multivariate gamma distributions (Update), Encyclopedia of Statistical Sciences, Update Volume 1, 419–425, Wiley, New York. Royen, T. (2006). Integral representations and approximations for multivariate gamma distributions, Annals of the Institute of Statistical Mathematics, DOI 10.1007/s10463-006-0057-5. Introduction The method Three representations for the p(,,)–Laplace transform and the related generators The integral representations Representations of the p(,,) distribution function by (p-1)–variate integrals

0704.0540

On the Achievable Rate Regions for Interference Channels with Degraded Message Sets

On the Achievable Rate Regions for Interference Channels with Degraded Message Sets† Jinhua Jiang and Yan Xin Department of Electrical and Computer Engineering National University of Singapore, Singapore 117576 Email: { jinhua.jiang, elexy }@nus.edu.sg Abstract The interference channel with degraded message sets (IC-DMS) refers to a communication model in which two senders attempt to communicate with their respective receivers simultaneously through a common medium, and one of the senders has complete and a priori (non-causal) knowledge about the message being transmitted by the other. A coding scheme that collectively has advantages of cooperative coding, collaborative coding, and dirty paper coding, is developed for such a channel. With resorting to this coding scheme, achievable rate regions of the IC-DMS in both discrete memoryless and Gaussian cases are derived, which, in general, include several previously known rate regions. Numerical examples for the Gaussian case demonstrate that in the high-interference-gain regime, the derived achievable rate regions offer considerable improvements over these existing results. Index Terms Cognitive radio, cooperative communication, degrade message sets, dirty paper coding, Gel’fand- Pinsker coding, interference channels, superposition coding. † The work is supported by the National University of Singapore (NUS) under start-up grants R-263-000-314-101 and R- 263-000-314-112 and by a NUS Research Scholarship. The correspondence author of the paper is Dr. Yan Xin (tel. no. +65 6516-5513 and fax no. +65 6779-1103). November 4, 2018 DRAFT http://arxiv.org/abs/0704.0540v2 IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 1 I. INTRODUCTION The interference channel with degraded message sets (IC-DMS) refers a communication model in which two senders attempt to communicate with their respective receivers simultaneously through a common medium, and one of the senders has complete and a priori (non-causal) knowledge about the message being transmitted by the other. Such a model generically characterizes some realistic communication scenarios taking place in cognitive radio channels [1], [2] or in wireless sensor networks over a correlated field [3], [4], which we illustrate in Figs. 1(a) and 1(b). From an information-theoretic perspective, the IC-DMS have been investigated in [1]–[4]. Specifically, several achievable rate results have been obtained in [1]–[4], and the capacity regions for two special cases have been characterized in [2]–[4]. The main achievable rate region in [1] was obtained by incorporating the Gel’fand-Pinsker coding [5] into the well-known coding scheme applied to the interference channel (IC) [6], [7]. In this coding scheme, each of the two senders splits its message into two sub-messages, and allows its non-pairing receiver to decode one of the sub-messages. Knowing the two sub-messages and the corresponding codewords which sender 1 wishes to transmit, sender 2 applies the Gel’fand-Pinsker coding to encode its own sub- messages by treating the codewords of sender 1 as known interferences. It has been also shown in [1, Corollary 2] that, an improved achievable rate region can be attained by time-sharing between the early derived rate region and a so called fully-cooperative rate point achieved by letting sender 2 use all its power to transmit sender 1’ messages. A different coding scheme was adopted in [2] and [3], in which neither of the senders splits its message into two sub-messages, and receiver 2 does not decode any transmitted information from sender 1. Since sender 2 knows what sender 1 wishes to transmit, sender 2 is allowed to: 1) apply the Gel’fand-Pinsker coding to encode its own message; and 2) partially cooperate with sender 1 using superposition coding. It has been proven in [2], [3] that, this is the capacity-achieving scheme for the Gaussian IC-DMS in the low-interference-gain regime, in which the normalized link gain between sender 2 and receiver 1 is less than or equal to 1. However, in practice, due to the mobility of the users or random distributions of the sensors, Genie (a) (b) Receiver 1 Receiver 2 Receiver 1Primary user Cognitive user Receiver 2 Sender 2 Sender 1 PSfrag replacements Fig. 1. (a) A genie-aided cognitive radio channel [1], in which the Genie informs the cognitive user of what the primary user will transmit; (b) A four-node wireless sensor network [3], in which sender 2 senses a larger area such that it knows what information sender 1 obtains. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 2 Sender 2 Receiver 2 Receiver 1 Sender 1 PSfrag replacements Fig. 2. An interference channel with degraded message sets in which sender 2 is close to receiver 1. sender 2 may be geographically located near to receiver 1, as illustrated in Fig. 2. It is likely, in such a situation, that the Gaussian IC-DMS is in the high-interference-gain regime, in which the normalized link gain between sender 2 and receiver 1 is greater than 1. In fact, the findings in this paper reveal that the achievable rate region, which was proven to be the capacity region in the low-interference-gain regime in [2] and [3], is strictly non-optimal for the Gaussian IC-DMS in the high-interference-gain regime. In this paper, we develop a new coding scheme for the IC-DMS to improve existing achievable rate regions. Our coding scheme differs from one proposed in [2], [3] in the way that, sender 2 splits its message into two sub-messages, and encodes both sub-messages using Gel’fand- Pinsker coding. Moreover, receiver 1 is required to jointly decode the message from sender 1 and one sub-message from sender 2. With this coding scheme, we derive our main achievable rate region for the discrete memoryless case. For comparison purpose, we compromise either the coding flexibility (fixing an auxiliary random variable as a constant), or the advantage of simultaneous decoding [7], to obtain two subregions of the main achievable rate region. The obtained subregions are shown to either include or be the same as the existing ones. We further extend the obtained regions from the discrete memoryless case to the Gaussian case, and show by numerical examples that our Gaussian achievable rate results strictly improve the existing ones in the high-interference-gain regime. The rest of the paper is organized as follows. In Section II, we introduce the channel model of the IC-DMS, and the related terminologies. In Section III, we present the main achievable result for the discrete memoryless case with a detailed proof. In Section IV, we derive two subregions of the main achievable rate region, and we show that the derived subregions include several existing results as special cases. Lastly, in Section V, we extend our rate regions from the discrete memoryless case to the Gaussian case, and compare them with the existing results. Notations: Random variables and their realizations are denoted by upper case letters and lower case letters respectively, e.g., X and x. Bold lower (upper) case letters are used to denote vectors (matrices), e.g., x and Σ. Calligraphic fonts are used to denote sets, e.g., X and R. II. THE CHANNEL MODEL Consider the IC-DMS (also termed as the genie-aided cognitive radio channel in [1]) depicted in Fig. 3, in which sender 1 wishes to transmit a message (or message index), w1 ∈ M1 := {1, ...,M1}, to receiver 1 and sender 2 wishes to transmit its message, November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 3 Sender 1 Channel Receiver 1 Receiver 2Sender 2 PSfrag replacements p(y1, y2|x1, x2) f1(·) f2(·) g1(·) g2(·) Fig. 3. An interference channel with degraded message sets. w2 ∈ M2 := {1, ...,M2}, to receiver 2. Typically, this discrete memoryless IC-DMS is described by a tuple (X1,X2,Y1,Y2, p(y1, y2|x1, x2)), where X1 and X2 are the channel input alphabets, Y1 and Y2 are the channel output alphabets, and p(y1, y2|x1, x2) denotes the conditional probability of (y1, y2) ∈ Y1×Y2 given (x1, x2) ∈ X1×X2. The channel is discrete memoryless in the sense p(y1,t, y2,t|x1,t, x2,t, x1,t−1, x2,t−1, ...) = p(y1,t, y2,t|x1,t, x2,t), (1) for every discrete time instant t in a synchronous transmission. In terms of the channel input- output relationship, the IC-DMS is the same as the IC. However, in the IC-DMS, sender 2 is able to noncausally obtain the knowledge of the message w1, which will be transmitted from sender 1. This is the key difference between the IC-DMS and IC in terms of the information flow. We next present the following standard definitions with regard to the existence of codes and achievable rates for the discrete memoryless IC-DMS channel. Definition 1: An (M1,M2, n, Pe) code exists for the discrete memoryless IC-DMS, if and only if there exist two encoding functions f1 : M1 → X n1 , f2 : M1 ×M2 → X n2 , and two decoding functions g1 : Yn1 → M1, g2 : Yn2 → M2, such that max{P (n)e,1 , P e,2 } ≤ Pe, where P e,1 and P e,2 denote the respective average probabilities of error at decoders 1 and 2, and are computed as e,1 = p(ŵ1 6= w1|(w1, w2) were sent), e,2 = p(ŵ2 6= w2|(w1, w2) were sent). November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 4 Definition 2: A non-negative rate pair (R1, R2) is achievable for the IC-DMS, if for any given 0 < Pe < 1 and any sufficiently large n, there exists a (2 nR1 , 2nR2, n, Pe) code for the channel. The capacity region of the IC-DMS is the set of all the achievable rate pairs for the channel, and an achievable rate region is a subset of the capacity region. It should be noted that from an information-theoretic standpoint, the IC can not be simply treated as a special case of the IC-DMS in the sense that the capacity region of the IC-DMS, if any, does not imply a capacity region of the IC. III. AN ACHIEVABLE RATE REGION FOR THE DISCRETE MEMORYLESS IC-DMS In this section, we present the main achievable rate region for the discrete memoryless IC- DMS, which is the primary result in this paper. Consider auxiliary random variables W , U , Ũ , V , Ṽ and a time-sharing random variable Q, defined on arbitrary finite sets W , U , Ũ , V , Ṽ and Q respectively. Let P denote the set of all joint probability distributions p(·) that factor in the form of p(q, w, x1, u, ũ, v, ṽ, x2, y1, y2) =p(q)p(w, x1|q)p(u, ũ|w, q)p(v, ṽ|w, q) · p(x2|ũ, ṽ, w, q)p(y1, y2|x1, x2), (2) where w, u, ũ, v, ṽ, and q are realizations of random variables W , U , Ũ , V , Ṽ and Q. Let R(p) denote the set of all non-negative rate pairs (R1, R2) such that the following inequalities hold simultaneously R1 ≤ I(W ; Y1U |Q), (3) R2 ≤ I(UV ; Y2|Q)− I(U ;W |Q)− I(V ;W |Q), (4) R1 + R2 ≤ I(UW ; Y1|Q) + I(V ; Y2U |Q)− I(U ;W |Q)− I(V ;W |Q); (5) 0 ≤ I(UW ; Y1|Q)− I(U ;W |Q), (6) 0 ≤ I(U ; Y2V |Q)− I(U ;W |Q), (7) 0 ≤ I(V ; Y2U |Q)− I(V ;W |Q), (8) 0 ≤ I(UV ; Y2|Q)− I(U ;W |Q)− I(V ;W |Q), (9) for a given joint distribution p(·) ∈ P . Let C denote the capacity region of the discrete memoryless IC-DMS, and let p(·)∈P R(p). Theorem 1: The region R is an achievable rate region for the discrete memoryless IC-DMS, i.e., R ⊆ C. Proof: Before presenting the proof of Theorem 1, we state the following lemma as it will be frequently used in the proof. Lemma 1 ( [8, Theorem 14.2.3]): Let A ǫ denote the typical set for the probability mass distribution p(s1, s2, s3), and let P (S 1 = s1,S 2 = s2,S 3 = s3) = p(s1i|s3i)p(s2i|s3i)p(s3i), then P{(S′1,S′2,S′3) ∈ A ǫ } .= 2−n(I(S′1;S′2|S′3)±6ǫ). To prove this theorem we apply the notion of the asymptotic equipartition property (APE) [8]. Our coding scheme is mainly based on the arguments of superposition coding [9] and November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 5 Gel’fand-Pinsker coding [5]. Specifically, sender 1 independently encodes its message w1 as a whole; while sender 2 needs split its message into two parts, i.e., w2 = (w21, w22), and encode them separately. Both w21 and w22 are encoded using the Gel’fand-Pinsker approach, but they are processed differently at the receivers. The message w22 will be decoded by receiver 2 only, while w21 will be decoded by both receivers. Moreover, knowing the message and codeword which sender 1 is going to transmit, sender 2 not only can apply Gel’fand-Pinsker coding to deal with the known interference, but also can cooperate with sender 1 to transmit w1 using superposition coding. Let R21 and R22 denote the rates of w21 and w22 respectively, i.e., w21 ∈ {1, . . . , 2nR21} and w22 ∈ {1, . . . , 2nR22}. If receiver 1 can decode w1 and receiver 2 can decode both w21 and w22 with vanishing probabilities of error, then (R1, R21 +R22) is an achievable rate pair for the IC-DMS. To prove that the entire region R is achievable for the channel, it is sufficient to prove that R(p) is achievable for a fixed joint probability distribution p(·) ∈ P . A. Random Codebook Generation Consider a fixed joint distribution p(·) ∈ P , and a random time-sharing codeword q of length n, which is given to both senders and receivers. The codeword q is assumed to be generated according to i=1 p(qi). Generate 2nR1 independent codewords w(j), j ∈ {1, . . . , 2nR1}, according to i=1 p(wi|qi); and for each w(j) generate one x1(j), according to i=1 p(x1i|wiqi). Similarly, generate 2nR̃21 independent codewords u(l1), l1 ∈ {1, . . . , 2nR̃21}, according to p(ui|qi), and generate 2nR̃22 independent codewords v(l2), l2 ∈ {1, . . . , 2nR̃22}, according to i=1 p(vi|qi). For each codeword pair (u(l1),w(j)), generate one codeword ũ(l1, j) according i=1 p(ũi|ui(l1)wi(j)qi), and similarly for each codeword pair (v(l2),w(j)), gen- erate one codeword ṽ(l2, j) according to i=1 p(ṽi|vi(l2)wi(j)qi). Lastly, for each codeword triple (u(l1),v(l2),w(j)), generate one codeword x2(l1, l2, j) according to i=1 p(x2i|ũi(l1)ṽi(l2)wi(j)qi). Now uniformly distribute 2nR̃21 codewords u(l1) into 2 nR21 bins indexed by k1 ∈ {1, . . . , 2nR21} such that each bin contains 2n(R̃21−R21) codewords; uniformly distribute 2nR̃22 codewords v(l2) into 2 nR22 bins indexed by k2 ∈ {1, . . . , 2nR22} such that each bin contains 2n(R̃22−R22) codewords. The entire codebook is revealed to both senders and receivers. B. Encoding and Transmission We assume that the senders want to transmit a message vector (w1, w21, w22) = (j, k1, k2). Sender 1 simply encodes the message as codeword x1(j) and sends the codeword with n channel uses. Sender 2 will first need to look for a codeword u(l̂1) in bin k1 such that (u(l̂1),w(j),q) ∈ ǫ , and a codeword v(l̂2) in bin k2 such that (v(l̂2),w(j),q) ∈ A(n)ǫ . If sender 2 fails to do so, it will randomly pick a codeword u(l̂1) from bin k1 or a codeword v(l̂2) from bin k2. Sender 2 then transmits codeword x2(l̂1, l̂2, j) through n channel uses. We further assume that the transmissions are perfectly synchronized. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 6 C. Decoding Receiver 1 first looks for all the index pairs (ĵ, l1) such that (w(ĵ),u( l1),y1,q) ∈ A(n)ǫ . If ĵ in all the index pairs found are the same, receiver 1 determines w1 = ĵ, otherwise declares an error. Receiver 2 will first look for all index pairs ( l2) such that (u( l1),v( l2),y2,q) ∈ A(n)ǫ . If ¯̂l1 in all the index pairs found are indices of codewords u( l1) from the same bin with index k̂1, l2 in all the index pairs found are indices of codewords v( l2) from the same bin with index k̂2, then receiver 2 will decode that (w21, w22) = (k̂1, k̂2) were transmitted; otherwise, an error is declared. D. Evaluation of Probability of Error We now derive upper bounds for the probabilities of the respective error events, which may happen during the encoding and decoding process. Due to the symmetry of the codebook generation and encoding processing, the probability of error is not codeword dependent. Without loss of generality, we assume that (w1, w21, w22) = (1, 1, 1) were encoded and transmitted. We next define the following three types of events: Ea,b = (u(a),w(b),q) ∈ A(n)ǫ , Ėa,b = (w(a),u(b),y1,q) ∈ A(n)ǫ , Ëa,b = (u(a),v(b),y2,q) ∈ A(n)ǫ . Let Pe(enc2), Pe(dec1), and Pe(dec2) denote the probabilities of error at the encoder of sender 2, the decoder of receiver 1, and the decoder of receiver 2, respectively. [Evaluation of Pe(enc2).] An error is made if 1) the encoder at sender 2 can not find u(l̂1) in bin 1 such that (u(l̂1),w(1),q) ∈ A(n)ǫ , and/or 2) it can not find v(l̂2) in bin 1 such that (v(l̂2),w(1),q) ∈ A(n)ǫ . Then the probability of error at the encoder of sender 2 is bounded as Pe(enc2) ≤ Pr u(l̂1)∈bin 1 (u(l̂1),w(1),q) /∈ A(n)ǫ  + Pr v(l̂2)∈bin 1 (v(l̂2),w(1),q) /∈ A(n)ǫ u(l̂1)∈bin 1 Pr(Ec l̂1,1 v(l̂2)∈bin 1 Pr(Ec l̂2,1 ≤ (1− Pr(E l̂1,1 n(R̃21−R21) + (1− Pr(E l̂2,1 n(R̃22−R22) ≤ (1− 2−n(I(U ;W |Q)+6ǫ))2n(R̃21−R21) + (1− 2−n(I(V ;W |Q)+6ǫ))2n(R̃22−R22), where (a) follows from the fact that we can obtain Pr(E l̂1,1 ) ≥ 2−n(I(U ;W |Q)+6ǫ) and Pr(E l̂2,1 2−n(I(V ;W |Q)+6ǫ) by setting S′1 = U, S 2 = W , and S 3 = Q, and S 1 = V, S 2 = W , and S 3 = Q in Lemma 1, respectively. Following the same argument in the proof of Lemma 2.1.3 of [10], we conclude that Pe(enc2) → 0 as n → +∞, if R̃21 ≥ R21 + I(U ;W |Q), (10) R̃22 ≥ R22 + I(V ;W |Q), (11) November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 7 are satisfied. We further choose R̃21 = R21 + I(U ;W |Q), (12) R̃22 = R22 + I(V ;W |Q). (13) Note that such a choice still ensures that Pe(enc2) → 0 as n → +∞. [Evaluation of Pe(dec1)] An error is made if 1) Ėc1,l̂1 happens, and/or 2) there exists some ĵ 6= 1 such that Ė happens. Note that l1 is not required to be equal to l̂1, since it is unnecessary for receiver 1 to decode l̂1 correctly. The probability of error at receiver 1 can be upper bounded as Pe(dec1) ≤ Pr(Ėc1,l̂1 ∪ĵ 6=1Ėĵ,ˆ̂l1) ≤ Pr(Ėc 1,l̂1 ĵ 6=1 Pr(Ė = Pr(Ėc 1,l̂1 ĵ 6=1 Pr(Ė ĵ,l̂1 ĵ 6=1, l1 6=l̂1 P (Ė ≤ Pr(Ėc 1,l̂1 ) + 2nR1Pr(Ė2,l̂1) + 2 n(R1+R̃21)Pr(Ė l1 6=l̂1 ). (14) Choosing S′1 = W, S 2 = (Y1,U), and S 3 = Q in Lemma 1, we have Pr(Ė2,l̂1) 2−n(I(W ;Y1U |Q)±6ǫ). Likewise, we have Pr(Ė l1 6=l̂1 = 2−n(I(WU ;Y1|Q)±6ǫ). In addition, it follows from AEP that Pr(Ėc 1,l̂1 ) → 0 as n → +∞. Thus, we infer from (14) that Pe(dec1) → 0 as n → +∞, if R1 ≤ I(W ; Y1U |Q), (15) R1 + R̃21 ≤ I(WU ; Y1|Q), (16) are satisfied. [Evaluation of Pe(dec2)] An error is made if 1) Ëc l̂1,l̂2 happens, and/or 2) there exists some l2) in which either l1 or l2 is not an index of any codeword from the respective bin 1. The probability of the second case is upper bounded by the probability of the event, ˯̂ for some l2) 6= (l̂1, l̂2). Thus, the probability of error at receiver 2 is bounded as Pe(dec2) ≤ Pr(Ëcl̂1,l̂2 l2)6=(l̂1,l̂2) ≤ Pr(Ëc l̂1,l̂2 l2)6=(l̂1,l̂2) P (˯̂ = Pr(Ëc l̂1,l̂2 l1 6=l̂1 Pr(˯̂ l1,l̂2 l2 6=l̂2 Pr(Ë l1 6=l̂1, l2 6=l̂2) Pr(˯̂ ≤ Pr(Ëc l̂1,l̂2 ) + 2nR̃21Pr(˯̂ l1 6=l̂1,l̂2 ) + 2nR̃22Pr(Ë l2 6=l̂2 ) + 2n(R̃21+R̃22)Pr(˯̂ l1 6=l̂1, l2 6=l̂2 November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 8 Applying Lemma 1 to evaluate Pr(˯̂ l1 6=l̂1,l̂2 ), Pr(Ë l2 6=l̂2 ) and Pr(˯̂ l1 6=l̂1, l2 6=l̂2 ) in (17), we conclude that Pe(dec2) → 0 as n → +∞ if the following inequalities, R̃21 ≤ I(U ; Y2V |Q), (18) R̃22 ≤ I(V ; Y2U |Q), (19) R̃21 + R̃22 ≤ I(UV ; Y2|Q), (20) are satisfied. According to (12), (13) and the fact that R2 = R21+R22, we first substitute R̃21 and R̃22 with R21+ I(U ;W |Q) and R22+ I(V ;W |Q) in (15), (16) and (18)–(20), and subsequently substitute R21 with R2 −R22 in the resulting inequalities. After these two substitution steps, we have R1 ≤ I(W ; Y1U |Q), (21) R1 +R2 −R22 ≤ I(WU ; Y1|Q)− I(U ;W |Q), (22) R2 −R22 ≤ I(U ; Y2V |Q)− I(U ;W |Q), (23) R22 ≤ I(V ; Y2U |Q)− I(V ;W |Q), (24) R2 ≤ I(UV ; Y2|Q)− (I(U ;W |Q) + I(V ;W |Q)). (25) Furthermore, applying Fourier-Motzkin elimination [11] to remove R22 from (21)–(25), we have R1 ≤ I(W ; Y1U |Q), (26) R2 ≤ I(UV ; Y2|Q)− (I(U ;W |Q) + I(V ;W |Q)), (27) R2 ≤ I(U ; Y2V |Q)− I(U ;W |Q) + I(V ; Y2U |Q)− I(V ;W |Q), (28) R1 + R2 ≤ I(WU ; Y1|Q)− I(U ;W |Q) + I(V ; Y2U |Q)− I(V ;W |Q). (29) Since I(U ; Y2V |Q)+I(V ; Y2U |Q)−I(UV ; Y2|Q) = I(U ;V |Q)+I(U ;V |Y2Q) ≥ 0, (27) implies (28) and thus (28) is redundant. To ensure that R1, R21 and R22 are non-negative, we enforce four additional constraints (6)–(9). Therefore, the rate region R(p) is achievable for a fixed joint probability distribution p(·) ∈ P , and Theorem 1 follows. Remark 1: The proposed coding scheme exploits three coding methods to achieve any rate pair in the rate region, R. The first method is cooperation that is realized by the superposition relationship between w and x2 through p(x2|ũ2, ṽ2, w, q). The second is collaboration, by which we mean that sender 2 separates its own message into two parts, i.e., w2 = (w21, w22), and encodes w21 at a possibly low rate such that receiver 1 can decode it. By doing so, the effective interference caused by the signals carrying the sender 2’s information may be reduced. The third is Gel’fand-Pinsker coding, which we apply to encode both messages, w21 and w22, from sender 2 by treating the codeword w as known interference. This perhaps allows receiver 2 to be able to decode the messages from sender 2 at the same rate as if the interference caused by sender 1 was not present [12]. IV. RELATING WITH EXISTING RATE REGIONS In this section, we will show that Theorem 1 includes the achievable rate regions in [2], [3]. To demonstrate it, we compromise the advantages of the coding scheme developed in Section III to obtain the following subregions of R. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 9 A. A Subregion of R Let P∗ denote the set of all joint probability distributions p(·) that factors in the form of p(q, w, x1, u, v, ṽ, x2, y1, y2) =p(q)p(x1, w|q)p(u|q)p(v, ṽ|w, q) · p(x2|u, ṽ, w, q)p(y1, y2|x1, x2). (30) Note that the joint distribution (30) differs from (2) in the way that conditioned on Q, U is now independent of any other auxiliary random variables, and Ũ is not present. Let Rsim(p) denote the set of all non-negative rate pairs (R1, R2) such that R1 ≤ I(W ; Y1|UQ), (31) R2 ≤ I(UV ; Y2|Q)− I(V ;W |Q), (32) R1 +R2 ≤ I(WU ; Y1|Q) + I(V ; Y2|UQ)− I(V ;W |Q); (33) 0 ≤ I(V ; Y1|UQ)− I(V ;W |Q), (34) for a joint probability distribution p(·) ∈ P∗. Furthermore, let Rsim = p(·)∈P∗ Rsim(p). Theorem 2: The rate region Rsim is achievable for the discrete memoryless IC-DMS, i.e., Rsim ⊆ R ⊆ C. Proof: The proof can be devised from the proof of Theorem 1 by customizing the original coding scheme for the new joint distribution (30). We change the encoding and decoding method for the message w21 (corresponding to U), i.e., the Gel’fand-Pinsker coding used in the proof of Theorem 1 was replaced by the conventional random coding. Specifically, we generate 2nR21 independent codewords u(k1), k1 ∈ {1, . . . , 2nR21}, according to i=1 p(ui|qi). The encoding and decoding are then adapted to the new codebook accordingly. Evaluating the probability of error in the same way as was done in the proof of Theorem 1, we obtain R̃22 −R22 ≥ I(V ;W |Q); (35) R1 ≤ I(W ; Y1|UQ), (36) R1 +R21 ≤ I(WU ; Y1|Q); (37) R21 ≤ I(U ; Y2|V Q), (38) R̃22 ≤ I(V ; Y2|UQ), (39) R21 + R̃22 ≤ I(UV ; Y2|Q). (40) Again, we choose R̃22 − R22 = I(V ;W |Q) in (35), and then substitute R̃22 with R22 + I(V ; Y2|UQ) as well as R21 with R2 − R22 in the group of (36)–(40). By applying Fourier- Motzkin elimination on the resulting inequalities to remove R22, and adding the constraints that ensure the respective rates R1, R21 and R22 to be non-negative, we obtain (31)–(34). Therefore, the region Rsim(p) is achievable for a given p(·) ∈ P∗, and the theorem follows. Note that simultaneous decoding (simultaneous joint typicality) is applied at both decoders. The advantage of simultaneous decoding over successive decoding is well demonstrated on the IC by Han and Kobayashi in [7]. We next modify the coding scheme by applying successive decoding instead of simultaneous decoding at both decoders to derive a subregion of Rsim. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 10 B. A Subregion of Rsim Let Rsuc(p) denote the set of all achievable rate pairs (R1, R2) such that R1 ≤ I(W ; Y1|UQ), (41) R2 ≤ min{I(U ; Y1|Q), I(U ; Y2|Q)}+ I(V ; Y2|UQ)− I(V ;W |Q); (42) 0 ≤ I(V ; Y1|UQ)− I(V ;W |Q), (43) for a fixed joint probability distribution p(·) ∈ P∗. Define Rsuc = p(·)∈P∗ Rsuc(p). Theorem 3: The rate region Rsuc is achievable for the discrete memoryless IC-DMS, i.e., Rsuc ⊆ Rsim ⊆ R ⊆ C. Proof: The codebook generation, encoding and transmission remain the same as those used to prove Theorem 2, whereas the decoding processes at both decoders are altered. Both decoders decode w21 first, and then decoder 1 decodes w1 and decoder 2 decodes w22 respectively. Then the following can easily be obtained R̃22 −R22 ≥ I(V ;W |Q), (44) R21 ≤ I(U ; Y1|Q), (45) R1 ≤ I(W ; Y1|UQ), (46) R21 ≤ I(U ; Y2|Q), (47) R̃22 ≤ I(V ; Y2|UQ). (48) From (44)–(48), it is straightforward to obtain (41)–(43). Therefore, the region Rsuc(p) is achievable, and the theorem follows immediately. Remark 2: Note that (45) is only necessary when the successive decoding is applied. This is because every decoding step in a successive decoding scheme is expected to have a vanishing probability of error. In what follows, we further specialize the subregion Rsuc to obtain two more achievable rate regions Rsp1 and Rsp2. Let P∗1 denote the set of all joint probability density distributions p(·) that factor in the form of p(q, w, x1, v, ṽ, x2, y1, y2) = p(q)p(x1, w|q)p(v, ṽ|w, q)p(x2|ṽ, w, q)p(y1, y2|x1, x2). (49) Let Rsp1(p) denote the set of all non-negative rate pairs (R1, R2) such that R1 ≤ I(W ; Y1|Q), (50) R2 ≤ I(V ; Y2|Q)− I(V ;W |Q), (51) for a fixed joint distribution p(·) ∈ P∗1 . Define Rsp1 = p(·)∈P∗1 Rsp1(p). Corollary 1: The region Rsp1 is an achievable rate region for the discrete memoryless IC- DMS, i.e., Rsp1 ⊆ Rsuc ⊆ Rsim ⊆ R ⊆ C. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 11 Proof: Fixing the auxiliary random variable U as a constant, we reduce (41) and (42) to (50) and (51), and the corollary follows immediately. Remark 3: We note that the region Rsp1 is similar to the region Rin reported in [3, Theorem 3.1]. It seems that the region Rin is more general than the region Rsp1 in the sense that fixing the auxiliary random variable U in Rin as a constant, one can obtain a region which is the same as Rsp1. Nevertheless, after examining the coding scheme used in [3, Theorem 3.1], one can find that there exists a one-one correspondence between codewords u(w2) and x2(w2), and both codewords are jointly generated and decoded, i.e., p(u, x2) is used to generate two-letter codewords. Thus, one can introduce one auxiliary random variable W such that there exists a one-one mapping between W and U×X2, i.e., f : U×X2 ↔ W , and thus W has the probability mass distribution p(w) = p(f−1(w)) = p(u, x2). Replacing all (X 2 (w2), U n(w2)) by W n(w2) in the proof of [3, Theorem 3.1] will yield the same rate region. Equivalently speaking, for any input distribution achieving a rate region characterized by [3, Theorem 3.1], one can find a corresponding joint distribution in the form of (49) such that Corollary 1 yields exactly the same rate region. Therefore, two rate regions Rsp1 and Rin are identical. Let P∗2 denote the set of all joint probability distributions p(·) that factor in the form of p(q, w, x1, u, x2, y1, y2) = p(q)p(x1, w|q)p(u|q)p(x2|u, w, q)p(y1, y2|x1, x2). (52) Let Rsp2(p) denote the set of all non-negative rate pairs (R1, R2) such that R1 ≤ I(W ; Y1|UQ), (53) R2 ≤ min{I(U ; Y1|Q), I(U ; Y2|Q)}, (54) for a fixed joint distribution p(·) ∈ P∗2 . Define Rsp2 = p(·)∈P∗2 Rsp2(p). Corollary 2: The region Rsp2 is an achievable rate region for the discrete memoryless IC- DMS, i.e., Rsp2 ⊆ Rsuc ⊆ Rsim ⊆ R ⊆ C. Proof: The proof can be devised from the proof of Theorem 3 easily by fixing V as a constant. V. THE GAUSSIAN IC-DMS In the preceding sections, we have derived several achievable rate regions for the discrete memoryless IC-DMS. We now extend these results to obtain corresponding achievable rate regions for the Gaussian IC-DMS (GIC-DMS). A. The Channel Model of the GIC-DMS In general, with no loss of information-theoretic optimality, the GIC-DMS can be converted to the GIC-DMS in the standard form through invertible transformations [2], [6], [11]. We thus only consider the GIC-DMS in the standard form, which is represented as follows Y1 = X1 + c21X2 + Z1, Y2 = X2 + c12X1 + Z2, November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 12 PSfrag replacements x1(w1) x2(w2, w1) Fig. 4. A Gaussian interference channel with degraded message sets. where Zi, i = 1, 2, is the additive white Gaussian noise with zero mean and unit variance, and√ c21 and c12 are the normalized link gains in the GIC-DMS depicted in Fig. 4. Moreover, the transmitted codeword xi = (xi1, . . . , xin), i = 1, 2, is subject to an average power constraint given by ‖xit‖2 ≤ Pi. Since it has been shown in the maximum-entropy theorem in [8] that Gaussian input signals are optimal for Gaussian channels, we will consider Gaussian codewords Xni , i = 1, 2. B. Achievable Rate Regions for the GIC-DMS 1) Gaussian Extension of R: We first extend R to its Gaussian counterpart denoted by G. To obtain the rate region G, we map the random variables involved in the joint distribution (2) to the corresponding Gaussian random variables with the following customary constraints: P1) W , distributed according to N (0, 1), P2) X1 = P1W , P3) Ũ , distributed according to N (0, αβP2), P4) Ṽ , distributed according to N (0, αβ̄P2), P5) U = Ũ + λ1W , P6) V = Ṽ + λ2W , P7) X2 = Ũ + Ṽ + ᾱP2W , where α, β ∈ [0, 1], α + ᾱ = 1, β + β̄ = 1, λ1, λ2 ∈ [0,+∞), and W , Ũ and Ṽ are mutually independent. The input-output relationship of the GIC-DMS can be described by c21ᾱP2 c21Ũ + c21Ṽ + Z1, (55) Y2 = Ũ + Ṽ + ᾱP2 + c12P1 W + Z2. (56) To simplify the derivations, we fix the time-sharing random variable Q as a constant. The issue of how this time-sharing random variable affects the achievable rate region is well addressed November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 13 in [13]. In the Gaussian case, the respective mutual information terms in (3) – (9) need be evaluated with respect to the mappings defined by P1–P7. Since the computation procedure to obtain G and the resulting description of G are fairly lengthy, we relegate them (Theorem 5) to the Appendix. 2) Gaussian Extension of Rsuc: For illustration and comparison purpose, we next show how to obtain the Gaussian counterpart of Rsuc in details. Following the first step in the previous derivation, we also map the random variables involved in (30) to the Gaussian ones with the following constraints: M1) W , distributed according to N (0, 1), M2) X1 = P1W , M3) U , distributed according to N (0, αβP2), M4) Ṽ , distributed according to N (0, αβ̄P2), M5) V = Ṽ + λW , M6) X2 = U + Ṽ + ᾱP2W , where α, β ∈ [0, 1], α + ᾱ = 1, β + β̄ = 1, λ ∈ [0,+∞), and W , U and Ṽ are mutually independent. Using the mappings defined by M1–M6, we express the input-output relationship for the GIC-DMS as: c21ᾱP2 c21U + c21Ṽ + Z1, (57) Y2 = U + Ṽ + ᾱP2 + c12P1 W + Z2. (58) Let Gsuc(α, β) denote the set of all the non-negative rate pairs (R1, R2) such that c21ᾱP2 c21αβ̄P2 + 1 , (59) log2(1 + αβ̄P2) + min c21αβP2 c21ᾱP2 + c21αβ̄P2 + 1 αβ̄P2 + ᾱP2 + c12P1 . (60) Define Gsuc = α,β∈[0,1] Gsuc(α, β). Theorem 4: The region Gsuc is an achievable rate region for the GIC-DMS in the standard form. Proof: It suffices to prove that Gsuc(α, β) is achievable for any given α, β ∈ [0, 1]. Since Gsuc is extended from Rsuc, we need compute the mutual information terms in (41) and (42). The righthand side of (59) can be readily obtained through a straightforward computation of I(W ; Y1|UQ) in (41). Recall that Q is a constant. It is also fairly straightforward to obtain the two terms within the minimum operator in (60) through computing I(U ; Y1|Q) and I(U ; Y2|Q) in (42). We next evaluate the only remaining term I(V ; Y2|UQ)− I(V ;W |Q) for a constant Q. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 14 Defining Ỹ2 = Ṽ + ᾱP2 + c12P1 W + Z2, we have I(V ; Y2|U)− I(V ;W ) = h(Y2|U)− h(Y2|UV )− I(V ;W ) = h(Ỹ2)− h(Ỹ2|V )− I(V ;W ) = h(Ỹ2) + h(V )− h(Ỹ2V )− I(V ;W ). (61) With V = Ṽ + λW , we evaluate (61) as I(V ; Y2|U)− I(V ;W ) αβ̄P2 + ᾱP2 + c12P1 log2(2πe(αβ̄P2 + λ (2πe)2 αβ̄P2 + ᾱP2 + c12P1 (αβ̄P2 + λ αβ̄P2 + λ ᾱP2 + c12P1 αβ̄P2 . (62) It is easy to find that when αβ̄P2 ᾱP2 + c12P1 αβ̄P2 + 1 , (63) the term I(V ; Y |U)− I(V ;W ) is maximized, and the maximum value is max[I(V ; Y2|U)− I(V ;W )] = log2(1 + αβ̄P2). (64) This is in parallel with the result in [12]. Therefore, the rate region Gsuc(α, β) is achievable for any pair α, β ∈ [0, 1], and the theorem follows. In the following, we obtain two corollaries by setting β = 0 and β = 1 in Theorem 4, respectively. Corollary 3: The rate region Gsp1 is an achievable rate region for the GIC-DMS in the standard form with Gsp1 := α∈[0,1] Gsuc(α, 0), i.e., Gsp1 is the union of the sets of non-negative rate pairs (R1, R2) satisfying c21ᾱP2 c21αP2 + 1 log2(1 + αP2), over all α ∈ [0, 1]. Corollary 4: The rate region Gsp2 is an achievable rate region for the GIC-DMS in the standard form with Gsp2 := α∈[0,1] Gsuc(α, 1), i.e., Gsp2 is the union of the sets of non-negative rate pairs (R1, R2) satisfying c21ᾱP2 R2 ≤min c21αP2 c21ᾱP2 ᾱP2 + c12P1 November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 15 0 0.5 1 1.5 2 (bits) (iii) Fig. 5. P1 = P2 = 6, c21 = 0.3, c12 = 0. (i) gives the rate region in Theorem 1 of [1]; (ii) gives the rate region in Corollary 2 of [1]; (iii) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and Theorem 3.5 of [3]). over all α ∈ [0, 1]. Remark 4: Corollaries 3 and 4 correspond the Gaussian extensions of Corollaries 1 and 2 respectively. Particularly, the rate region depicted by Corollary 3 is the same as the rate regions given in [2, Theorem 4.1] and [3, Theorem 3.5]. It has been proven in both [2] and [3] that the rate region Gsp1 is indeed the capacity region for the GIC-DMS in the low-interference-gain regime, i.e., c21 ≤ 1. In addition, the set of achievable rate pairs given in [2, Lemma 4.2] is contained in the region Gsp2 as a subset. C. Numerical Examples We next provide several numerical examples to illustrate improvements of our achievable rate regions over the previously known results in [1]–[3]. Denote the achievable rate regions obtained in [1, Theorem 1] and [1, Corollary2] by Gdmt1 and Gdmt2, respectively. 1) Comparing with Rate Regions in [1]: Fig. 5 compares the rate regions Gdmt1, Gdmt2, and Gsp1 for an extreme case in which receiver 2 does not experience any interference from sender 1, i.e., c12 = 0. As can be seen from Fig. 5, the rate region Gsp1 strictly includes Gdmt1, as well as Gdmt2 obtained through time-sharing between Gdmt1 and a fully-cooperative rate point. The November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 16 coding scheme used to establish Gdmt1 incurs certain rate loss due to the fact that sender 2 does not use its power to help the sender 1’s transmissions even though it has complete and non- causal knowledge about the message being transmitted by sender 1. In contrast, our proposed coding scheme allows sender 2 to use superposition coding to help sender 1, and thus yields an improved rate region. In Fig. 6, we consider another case in which the transmit power of sender 1 is set to zero and c21 ≤ 1. From the figure, we observe that the rate region Gdmt2 is strictly smaller than Gsp1. Note that in this case, the GIC-DMS becomes a Gaussian degraded broadcast channel. According to [8], the optimal coding scheme for this case is: sender 2 uses a portion of its power to transmit the codeword conveying w1, and uses the remaining power to transmit the codeword conveying w2, which is encoded by using the dirty-paper coding [14]. It is easy to verify that this scheme is a special case of the coding scheme developed in Theorem 1. 0 0.2 0.4 0.6 0.8 1 (bits) Fig. 6. P1 = 0, P2 = 6, c21 = 0.5. (i) gives the rate region in Corollary 2 of [1]; (ii) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and Theorem 3.5 of [3]). 2) Comparing with Rate Regions in [2], [3]: As mentioned earlier, the rate region Gsp1, a subregion of G, is the same as the one given in [2, Theorem 4.1] and the one given in [3, Theorem 3.5], which is indeed the capacity region for GIC-DMS in the low-interference-gain regime. In Figs. 7 and 8, we compare G with Gsp1 and Gsp2 in the high-interference-gain regime, i.e., c21 > 1. As can be seen from the figures, the rate region G strictly includes both Gsp1 and Gsp2 in this case. Comparing Fig. 7 with Fig. 8, we observe that the improvement of the rate November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 17 0 0.5 1 1.5 2 2.5 3 (bits) (iii) Fig. 7. P1 = P2 = 6, c21 = 2, c12 = 0.3. (i) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and Theorem 3.5 of [3]); (ii) gives the achievable rate region in Corollary 4; (iii) gives the achievable rate region in Theorem 5. region G over Gsp1 becomes more pronounced as the link gain c21 increases. The improvement is mainly because our coding scheme allows receiver 1 to decode partial information from sender 2, and thus reduces the effective interference experienced by receiver 1. In addition, it can be seen from the figures that in the high-interference-gain regime, Gsp1 is not convex and thus is only suboptimal. VI. CONCLUSIONS In this paper, we have investigated the IC-DMS from an information theoretic perspective. We have developed a coding scheme that combines the advantages of cooperative coding, collaborative coding and Gel’fand-Pinsker coding. With the coding scheme, we have derived a new achievable rate region for such a channel, which not only includes existing results as special cases, but also exceeds them in the high-interference-gain regime. However, we are not able to establish a converse for the derived achievable rate region, because the achievable result is closely related to the achievable results for the interference channel and the broadcast channel, for which there is no converse available in general. APPENDIX November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 18 0 0.5 1 1.5 2 2.5 3 3.5 (bits) (iii) Fig. 8. P1 = P2 = 6, c21 = 6, c12 = 0.3. (i) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and Theorem 3.5 of [3]); (ii) gives the achievable rate region in Corollary 4; (iii) gives the achievable rate region in Theorem 5. AN ACHIEVABLE RATE REGION FOR THE GIC-DMS In this appendix, we show how to extend R, the achievable rate region for the discrete memoryless IC-DMS, to its Gaussian counterpart, G. Note that the mappings M1–M6 of the auxiliary random variables are described in Section V. We first compute the following two covariance matrices: ΣWUY1 = µ11 µ12 µ13 µ21 µ22 µ23 µ31 µ32 µ33 E{W 2} E{WU} E{WY1} E{WU} E{U2} E{UY1} E{WY1} E{UY1} E{Y 21 } P1 λ1P1 η1 λ1P1 αβP2 + λ 1P1 λ1η1 c21αβP2 P1 λ1η1 c21αβP2 η 1 + c21αP2 + 1 November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 19 ΣUV Y2 = ν11 ν12 ν13 ν21 ν22 ν23 ν31 ν32 ν33 E{U2} E{UV } E{UY2} E{UV } E{V 2} E{V Y2} E{UY2} E{V Y2} E{Y 22 } αβP2 + λ 1P1 λ1λ2P1 αβP2 + λ1η2 λ1λ2P1 αβ̄P2 + λ 2P1 αβ̄P2 + λ2η2 αβP2 + λ1η2 P1 αβ̄P2 + λ2η2 P1 αP2 + η 2 + 1 where c21ᾱP2, ᾱP2 + c12P1, and E{·} denotes the expectation of a random variable. Define Γ(x) = log2(x)/2, and ξ = log2(2πe)/2. We express the respective differential entropy terms as: ha = h(W ) = ξ + Γ(µ11), hb = h(UY1) = 2ξ + Γ µ22 µ23 µ32 µ33 hc = h(WUY1) = 3ξ + Γ µ11 µ12 µ13 µ21 µ22 µ23 µ31 µ32 µ33 hd = h(UV ) = 2ξ + Γ ν11 ν12 ν21 ν22 he = h(Y2) = ξ + Γ(ν33), hf = h(UV Y2) = 3ξ + Γ ν11 ν12 ν13 ν21 ν22 ν23 ν31 ν32 ν33 hg = h(WU) = 2ξ + Γ µ11 µ12 µ21 µ22 hh = h(Y1) = ξ + Γ(µ33), hi = h(V ) = ξ + Γ(ν22), hj = h(UY2) = 2ξ + Γ µ11 µ13 µ31 µ33 hk = h(U) = ξ + Γ(ν11), hl = h(V Y2) = 2ξ + Γ µ22 µ23 µ32 µ33 where | · | denotes the determinant of a matrix. November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 20 The mutual information terms in (3)–(9) are then computed as: I1 = ha + hb − hc, I2 = hd + he − hf , I3 = Γ(1 + λ21P1 I4 = Γ(1 + λ22P1 αβ̄P2 I5 = hg + hh − hc, I6 = hi + hj − hf , I7 = hk + hl − hf . Let G(α, β, λ1, λ2) denote the set of all rate pairs (R1, R2) such that the following inequalities are satisfied: R1 ≤ I1, (65) R2 ≤ I2 − I3 − I4, (66) R1 +R2 ≤ I5 + I6 − I3 − I4; (67) 0 ≤ I5 − I3, (68) 0 ≤ I7 − I3, (69) 0 ≤ I6 − I4, (70) 0 ≤ I2 − I3 − I4. (71) for given α, β ∈ [0, 1] and λ1, λ2 ∈ [0,+∞). Note that (65)–(71) are directly extended from (3)–(9). Theorem 5: The rate region G is achievable for the GIC-DMS in the standard form with α,β∈[0,1];λ1,λ2∈[0,+∞) G(α, β, λ1, λ2). November 4, 2018 DRAFT IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 21 REFERENCES [1] N. Devroye, P. Mitran, and V. Tarokh, “Achievable rates in cognitive channels,” IEEE Trans. Inform. Theory, vol. 52, no. 5, pp. 1813–1827, May 2006. [2] A. Jovičić and P. Viswanath, “Cognitive radio: An information-theoretic perspective,” 2006. [Online]. Available: http://arxiv.org/abs/cs/0604107 [3] W. Wu, S. Vishwanath, and A. Arapostathis, “On the capacity of interference channels with degraded message sets,” 2006. [Online]. Available: http://arxiv.org/abs/cs/0605071 [4] I. Maric, R. Yates, and G. Kramer, “The strong interference channel with unidirectional cooperation,” in Proc. UCSD Workshop on Information Theory and its Applications, San Diego, CA, USA, Feb. 2006. [5] S. I. Gel’fand and M. S. Pinsker, “Coding for channel with random parameters,” Problems of Control and Information Theory, vol. 9, no. 1, pp. 19–31, 1980. [6] A. B. Carleial, “Interference channels,” IEEE Trans. Inform. Theory, vol. IT-24, no. 1, pp. 60–70, Jan. 1978. [7] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. Inform. Theory, vol. IT-27, no. 1, pp. 49–60, Jan. 1981. [8] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley & Sons, 1994. [9] T. M. Cover, “An achievable rate region for the broadcast channel,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 399–404, July 1975. [10] T. Berger, “Multiterminal source coding,” in The Information Theory Approach to Communications, ser. CISM Courses and Lectures,G. Longo, Ed. Springer-Verlag, vol. 229, 1978, pp. 171–231. [11] G. Kramer, “Review of rate regions for interference channels,” in Proc. IZS Workshop, Zurich, Feb. 2006. [12] M. H. M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. IT-29, no. 3, pp. 439–441, May 1983. [13] I. Sason, “On achievable rate regions for the Gaussian interference channel,” IEEE Trans. Inform. Theory, vol. 50, no. 6, pp. 1345–1356, June 2004. [14] W. Yu and J. Cioffi, “Trellis precoding for the broadcast channel,” in Proc. IEEE Global Telecommunications Conference (Globecom), San Antonio, TX, Nov. 2001. November 4, 2018 DRAFT http://arxiv.org/abs/cs/0604107 http://arxiv.org/abs/cs/0605071 Introduction The Channel Model An Achievable Rate Region for the Discrete Memoryless IC-DMS Random Codebook Generation Encoding and Transmission Decoding Evaluation of Probability of Error Relating with Existing Rate Regions A Subregion of R A Subregion of Rsim The Gaussian IC-DMS The Channel Model of the GIC-DMS Achievable Rate Regions for the GIC-DMS Gaussian Extension of R Gaussian Extension of Rsuc Numerical Examples Comparing with Rate Regions in Tarokh06:icdmscog Comparing with Rate Regions in jovicic06:cogICDMS,wuwei06icdms Conclusions References

0704.0541

On complete subsets of the cyclic group

On complete subsets of the cyclic group Y. O. Hamidoune A.S. Lladó O. Serra Abstract A subset X of an abelian G is said to be complete if every element of the subgroup generated by X can be expressed as a nonempty sum of distinct elements from X . Let A ⊂ Z be such that all the elements of A are coprime with n. Solving a conjecture of Erdős and Heilbronn, Olson proved that A is complete if n is a prime and if |A| > 2 Recently Vu proved that there is an absolute constant c, such that for an arbitrary large n, A is complete if |A| ≥ c n, and conjectured that 2 is essentially the right value of c. We show that A is complete if |A| > 1 + 2 n− 4, thus proving the last conjecture. 1 Introduction The additive group of integers modulo n will be denoted by Zn. Let G be a finite Abelian group and let X ⊂ G. The subgroup generated by a subset X of G will be denoted 〈X〉. For a positive integer k, we shall write k ∧X = A ⊂ X and |A| = k Following the terminology of [12] we write k ∧X. The set X is said to be complete if SX = 〈X〉. The reader may find the connection between this notion and the corresponding notion for integers in [12]. We shall also write S0X = SX ∪ {0}. Note that S0X = x∈X{0, x}. Université Pierre et Marie Curie, E. Combinatoire, Case 189, 4 Place Jussieu, 75005 Paris, France. [email protected] Universitat Politècnica de Catalunya, Dept. Matemàtica Apl. IV; Jordi Girona, 1, E-08034 Barcelona, Spain. [email protected] Universitat Politècnica de Catalunya, Dept. Matemàtica Apl. IV; Jordi Girona, 1, E-08034 Barcelona, Spain. [email protected] http://arxiv.org/abs/0704.0541v1 Let p denote a prime number and let A ⊂ Zp \ {0}. Erdős and Heilbronn [4] showed that A is complete if |A| ≥ p, and conjectured that 18 can be replaced by 2. This conjecture was proved by Olson[8]. More precisely, Olson’s Theorem states that A is complete if |A| ≥ 4p− 4. This result was sharpened by Dias da Silva and one of the authors [1] by showing that |k∧A| = p, if |A| ≥ 4p − 4, where k = ⌈ p− 1 ⌉. They also showed that |(j ∧ A) ∪ ((j + 1) ∧A)| = p, if |A| ≥ 4p− 8, where j = ⌈ p− 2 ⌉. Let G be a finite abelian group and let A ⊂ G\{0}. Complete sets for general abelian group were investigated by Diderrich and Mann [3]. Diderrich [2] proved that, if |G| = pq is the product of two primes, then A is complete if |A| ≥ p+ q − 1. Let p be the smallest prime dividing |G|. Diderrich conjectured [2] that A is complete, if |G|/p is composite and |A| = p+ |G|/p− 2. This conjecture was finally proved by Gao and one of the authors [5]. More precise results were later proved by Gao and the present authors [6]. Note that the bound of Diderrich is best possible, since one may construct non complete sets of size p+ |G|/p − 3. However the result of Olson was extended recently by Vu [13] to general cyclic groups. Let A ⊂ Zn be such that all the elements of A are coprime with n. Vu proved that there is an absolute constant c such that, for an arbitrary large n, A is complete if |A| ≥ c n. The proof of Vu is rather short and depends on a recent result of Szemerédi and Vu [11]. In the same paper Vu conjectures that the constant is essentially 2. Our main result is the following: Theorem 1.1 Let A be a subset of Zn be such that all the elements of A are coprime with n. If |A| > 1 + 2 n− 4 then A is complete. This result implies the validity of the last conjecture of Vu. We conjecture the following: Conjecture 1.2 Let A ⊂ Zn be such that all the elements of A are coprime with n and |A| ≥√ 4n− 4. Then |k ∧A| = n, where k = ⌈ n− 1 ⌉. 2 Some tools In this section we present known material and some easy applications of it. We give short proofs in order to make the paper self-contained. Recall the following well-known and easy lemma. Lemma 2.1 Let G be a finite group. Let X and Y be subsets of G such that X + Y 6= G. Then |X|+ |Y | ≤ |G|. Proof. Take a ∈ G \ (X + Y ). We have (a− Y ) ∩X = ∅. ✷ We use also the Chowla’s Theorem [7, 10] : Theorem 2.2 (Chowla [7, 10]) Let n be a positive integer and let X and Y be non-empty subsets of Zn. Assume that 0 ∈ Y and that the elements of Y \ {0} are coprime with n. Then |X + Y | ≥ min(n, |X| + |Y | − 1). Proof. The proof is by induction on |Y |, the result being obvious for |Y | = 1. Assume first that Y ⊂ X − x, for all x ∈ X. Then X + Y ⊂ X, and hence X + Y = X. It follows that X + Y = X + nY = Zn. Assume now that Y 6⊂ X − x, for some x ∈ X. Then 0 ∈ Y ∩ (X − x) and |Y ∩ (X − x)| < |Y |. By the induction hypothesis, |X|+ |Y | − 1 ≤ |((X − x) ∪ Y ) + ((X − x) ∩ Y )| ≤ |(X − x) + Y |. Let B ⊂ G and x ∈ G. Following Olson, we write λB(x) = |(B + x) \B|. The following result is implicit in [8]: Lemma 2.3 (Olson, [8]) Let Y be a nonempty subset of G \ {0}, z /∈ Y and y ∈ Y . Put B = S0Y . Then |B| ≥ |S0Y \{y}|+ λB(y), (1) |S0Y ∪{z}| = |S Y |+ λB(z). (2) Proof. Clearly we have B∩(B−y) ⊂ B\S0 Y \{y} and hence λB(y) = |B∩(B−y)| ≤ |B|−|S0Y \{y}|. ¿From S0 Y ∪{z} = B + {0, z} we have |S0 Y ∪{z} | = |B|+ |(B + z) \B|} = |B|+ λB(z). ✷ We need the following helpful result also due to Olson: Lemma 2.4 (Olson [8]) Let B and C be nonempty subsets of an abelian group G such that 0 6∈ C. Then, λB(x) = λB(−x). (3) λB(x+ y) ≤ λB(x) + λB(y). (4) λB(x) ≥ |B|(|C| − |B|+ 1). (5) Proof. For each x ∈ G we have |(B + x) ∩B| = |B + x| − |(B + x) ∩B| = |B − x| − |B ∩ (B − x)| = |B ∩ (B − x)| = λB(−x), proving (3). Let x, y ∈ G. Then, λB(x+ y) = |(B + x+ y) ∩B| = |(B + x) ∩ (B − y)| = |(B + x) ∩B ∩ (B − y)|+ |(B + x) ∩B ∩ (B − y)| ≤ |(B + x) ∩B|+ |B ∩ (B − y)| = λB(x) + λB(y), proving (4). Finally, λB(x) ≥ (|B + x| − |B ∩ (B + x)|) ≥ |C||B| − |B ∩ (B + x)| ≥ |C||B| − x∈G\0 |B ∩ (B + x)| = |B|(|C| − |B|+ 1), proving (5). ✷ 3 The main result The next Lemma is the key tool for our main result. Lemma 3.1 Let A and B be nonempty subsets of Zn. Assume that A ∩ (−A) = ∅ and that each element in A is coprime with n. Put a = |A| and b = |B|. Assume also that a ≥ 3 and 2b ≤ n+ 2. Then λB(x) > a− a(a− 3) . (6) In particular, if 2b ≥ a(a− 3), then λB(x) ≥ a− 1. (7) Proof. Put A∗ = A ∪ (−A) ∪ {0}. Let t < n be a positive integer and set t = 2ma+ r, m ≥ 0, 0 ≤ r ≤ 2a− 1. Let Cj = jA ∗. By Chowla’s theorem, |Cj| ≥ min{n, 2ja + 1} = 2ja + 1, for j ≤ m. Therefore we can choose a set C ⊃ A∗ of cardinality t + 1 which intersects Cj in exactly 2ja elements j = 2, . . . ,m, and intersects Cm+1 in exactly r elements. Let E = C \{0}. Let α = max{λB(x) : x ∈ A}. By (3) we have λB(x) ≤ α, for all x ∈ A∗. For an element x in Cj there are elements x1, · · · , xj ∈ A∗ such that x = x1+ · · ·+xj. In view of (4) we have λB(x) ≤ λ(x1)+ · · ·+λ(xj) ≤ jα. Therefore, λB(x) ≤ α2a+ 2α2a + · · ·+mα2a+ r(m+ 1)α = α(m+ 1)(ma+ r) = α(t− r + 2a)(t+ r) ≤ α(t+ a) By using (5) we have x∈E λB(x) (t+ a)2 ≥ 4ab(t− b+ 1) (t+ a)2 In particular, since 2b ≤ n+ 2, we can set t = 2b− 3 to get, α ≥ 4ab(b− 2) (2b+ a− 3)2 ≥ a(b− 2) (1− a− 3 > a− a(a− 3) where we have used a ≥ 3. In particular, if 2b ≥ a(a − 3), then α > a − 2 so that α ≥ a − 1. This completes the proof. ✷ Lemma 3.1 gives the following estimation for the cardinality of the set of subset sums. Lemma 3.2 Let A ⊂ Zn such that A ∩ (−A) = ∅ and every element of A is coprime with n. Also assume |A| ≥ 2. Then |S0A| ≥ min{ n + 2 , 3 + |A|(|A| − 1) Proof. We shall prove the result by induction on a = |A|, the result being obvious for a = 2. Suppose a > 2. Put B = S0A. We may assume b = |B| ≤ n2 + 1 so that 2b ≤ n + 2. By the induction hypothesis, 2b ≥ 6 + (a− 1)(a− 2) > a(a− 3). By (7) there is an x ∈ A with λB(x) ≥ a− 1. Then, by Lemma 2.3, |B| ≥ |S0A\{x}|+ λB(x) ≥ 3 + (a− 2)(a − 1)/2 + a− 1 = 3 + a(a− 1) as claimed. ✷ We are now ready for the proof of Theorem 1.1. Proof of Theorem 1.1. Suppose A non complete and put |A| = k. Let X,Y be disjoint subsets of A. We clearly have SX + S Y ⊂ SA 6= Zn. Since |SX | ≥ |S0X | − 1, we have |S0X |+ |S0Y | ≤ n+ 1, (8) by Lemma 2.1. Partition A = A1 ∪ A2 into two almost equal parts, i.e. |A1| = ⌈k/2⌉ and |A2| = ⌊k/2⌋, such that Ai ∩ (−Ai) = ∅, i = 1, 2. We must have 3 + ⌊k ⌋ − 1)/2 < (n+ 2)/2, (9) since otherwise, by Lemma 3.2, we have |S0A1 |+ |S | ≥ n+ 2, contradicting (8). Case 1. k even. Then we have by (9) n/2 > 2 + − 1)/2 = 2 + k(k − 2)/8, and hence (k − 1)2 + 16 ≤ 4n, a contradiction. Case 2. k odd. Put a = k−1 . In view of (9), Lemma 3.2 implies |S0A2 | ≥ 3 + a(a− 1)/2. By (7) with B = S0A2 , there is a y ∈ A1 such that λB(y) ≥ a− 1. Put C1 = A1 \ {y} and C2 = A2 ∪ {y}. Then we have, by Lemma 2.3, |S0C2 | ≥ |S |+ λB(y) ≥ 3 + a(a− 1)/2 + a− 1 = 2 + a(a+ 1) On the other hand, from (9) and Lemma 3.2 we get |S0C1 | ≥ 3 + a(a− 1) By (8), n+ 1 ≥ |S0C1 |+ |S | ≥ 3 + a(a− 1)/2 + 2 + a(a+ 1)/2 = 5 + a2. Therefore 4n ≥ 16 + 4a2 = 16 + (k − 1)2, a contradiction. This completes the proof. ✷ References [1] J.A. Dias da Silva and Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc., 26 (1994), 140-146. [2] G.T. Diderrich, An addition theorem for abelian groups of order pq, J. Number Theory 7 (1975), 33-48. [3] G. T. Diderrich and H. B. Mann, Combinatorial problems in finite abelian groups, In: ”A survey of Combinatorial Theory” (J.L. Srivasta et al. Eds.), pp. 95- 100, North- Holland, Amsterdam (1973). [4] P. Erdős and H. Heilbronn, On the Addition of residue classes mod p, Acta Arith. 9 (1964), 149-159. [5] W. Gao and Y.O. Hamidoune, On additive bases, Acta Arith. 88 (1999), 3, 233-237. [6] W. Gao, Y.O. Hamidoune A. S. Lladó and O. Serra, Covering a finite abelian group by subset sums. Combinatorica 23 (2003), no. 4, 599–611. [7] H.B. Mann, Addition Theorems, R.E. Krieger, New York, 1976. [8] J. E. Olson, An addition theorem mod p, J. Comb. Theory 5 (1968), 45-52. [9] S. Chowla, A theorem on the addition of residue classes: applications to the number Γ(k) in Waring’s problem, Proc.Indian Acad. Sc. 2 (1935) 242–243. [10] M. B. Nathanson, Additive Number Theory. Inverse problems and the geometry of sumsets, Grad. Texts in Math. 165, Springer, 1996. [11] E. Szemerédi and V.H. Vu, Long arithmetic progressions in finite and infinite sets, Annals of Math., to appear. [12] T. Tao and V.H. Vu, Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105 (2006), Cambridge Press University. [13] V.H. Vu, Olson Theorem for cyclic groups, Preprint, arXiv:math.NT/0506483 v1, 23 june 2005. http://arxiv.org/abs/math/0506483 Introduction Some tools The main result

0704.0542

Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians

arXiv:0704.0542v2 [math.CO] 23 Apr 2007 Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians K. N. Raghavan Institute of Mathematical Sciences C. I. T. Campus, Taramani Chennai 600 113, INDIA email: [email protected] Shyamashree Upadhyay Chennai Mathematical Institute Plot No. H1, SIPCOT IT Park Padur Post, Siruseri 603 103, Tamilnadu, INDIA email: [email protected] 04 April 2007 Abstract A solution is given to the following problem: how to compute the mul- tiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the problem from geometry to combinatorics. The solution of the resulting combinatorial problem forms the bulk of the pa- per. This approach has been followed earlier to solve the same problem for the Grassmannian and the symplectic Grassmannian. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. Taking the Schubert variety to be of a special kind and the point to be the “identity coset,” our problem specializes to a problem about Pfaffian ideals treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a “generic singularity.” Mathematics Subject Classification 2000: 05E15 (Primary), 13F50, 13P10, 14L35 (Sec- ondary) http://arxiv.org/abs/0704.0542v2 Contents Introduction 4 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 5 Important note added . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I The theorem 7 1 The set up 7 1.1 The statement of the problem . . . . . . . . . . . . . . . . . . . . 7 1.2 Some convenient choices . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Reduction to the case n even . . . . . . . . . . . . . . . . . . . . 8 2 The theorem 9 2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Two fundamental definitions . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Definition of v-chain . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Definition of O-domination . . . . . . . . . . . . . . . . . . 11 2.3 The main theorem and its corollary . . . . . . . . . . . . . . . . . 11 II From geometry to combinatorics 13 3 Reduction to combinatorics 13 3.1 Homogeneous co-ordinate ring of the Schubert variety X(w) . . . 13 3.1.1 The line bundle L on Md(V ) . . . . . . . . . . . . . . . . . 13 3.1.2 The section qθ of L . . . . . . . . . . . . . . . . . . . . . . 13 3.1.3 Standard monomial theory for Md(V ) . . . . . . . . . . . . 14 3.2 Co-ordinate rings of affine patches and tangent cones of X(w) . . 14 3.2.1 Standard monomial theory for affine patches . . . . . . . . . 14 3.2.2 Standard monomial theory for tangent cones . . . . . . . . . 16 4 Further reductions 17 4.1 The main propositions . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 From the main propositions to the main theorem . . . . . . . . . 18 III The proof 21 5 Terminology and notation 21 5.1 Distinguished subsets . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.1.1 Distinguished subsets of N . . . . . . . . . . . . . . . . . . 21 5.1.2 Attaching elements of I(d, 2d) to distinguished subsets of N 21 5.2 The involution # . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2.1 The involution # on I(d, 2d) . . . . . . . . . . . . . . . . . 21 5.2.2 The involution # on N and R . . . . . . . . . . . . . . . . 22 5.3 The subset SC attached to a v-chain C . . . . . . . . . . . . . . 22 5.3.1 Vertical and horizontal projections of an element of ON . . 22 5.3.2 The “connection” relation on elements of a v-chain . . . . . 22 5.3.3 The definition of SC . . . . . . . . . . . . . . . . . . . . . 23 5.3.4 The type of an element α of a v-chain C, and the set SC,α 23 6 O-depth 27 6.1 Definition of O-depth . . . . . . . . . . . . . . . . . . . . . . . . 27 6.2 O-depth and depth . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 O-depth and type . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7 The map Oπ 34 7.1 Description of Oπ . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.2 Illustration by an example . . . . . . . . . . . . . . . . . . . . . . 35 7.3 A proposition about Sj,j+1 . . . . . . . . . . . . . . . . . . . . . 38 7.4 Proof of Proposition 7.1.1 . . . . . . . . . . . . . . . . . . . . . . 40 7.5 More observations . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8 The map Oφ 42 8.1 Description of Oφ . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8.2 Basic facts about Tw,j,j+1 and T w,j,j+1 . . . . . . . . . . . . . . . 45 9 Some Lemmas 46 9.1 Lemmas from the Grassmannian case . . . . . . . . . . . . . . . . 46 9.2 Orthogonal analogues of Lemmas of 9.1 . . . . . . . . . . . . . . 49 9.3 Orthogonal analogues of some lemmas in [7] . . . . . . . . . . . . 51 9.4 More lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 10 The Proof 59 10.1 Proof of Proposition 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 59 10.2 Proof that OφOπ = identity . . . . . . . . . . . . . . . . . . . . 62 10.3 Proof that OπOφ = identity . . . . . . . . . . . . . . . . . . . . 63 10.4 Proof of Proposition 4.1.3 . . . . . . . . . . . . . . . . . . . . . . 64 IV An Application 65 11 Multiplicity counts certain paths 65 11.1 Description and illustration . . . . . . . . . . . . . . . . . . . . . 65 11.2 Justification for the interpretation . . . . . . . . . . . . . . . . . 67 References 71 Index of definitions and notation 73 Introduction In this paper the following problem is solved: given a Schubert variety in an or- thogonal Grassmannian (by which is meant the variety of isotropic subspaces of maximum possible dimension of a finite dimensional vector space with a sym- metric non-degenerate form—see §1 for precise definitions) and an arbitrary point on the Schubert variety, how to compute the multiplicity, or more gener- ally the Hilbert function, of the local ring of germs of functions at that point. In a sense, our solution is but a translation of the problem: we do not give closed form formulas but alternative combinatorial descriptions. The meaning of “alternative” will presently become clear. The same problem for the Grassmannian was treated in [11, 8, 7, 9, 12] and for the symplectic Grassmannian in [4]. The present paper is a sequel to [11, 7, 9, 12, 4] and toes the same line as them. In particular, its strategy is borrowed from them and runs as follows: first translate the problem from geometry to combinatorics, or, more precisely, apply standard monomial theory to obtain an initial combinatorial description of the Hilbert function (the earliest version of the theory capable of handling Schubert varieties in an orthogonal Grassmannian is to be found in [17]); then transform the initial combinatorial description to obtain the desired alternative description. But that is easier said than done. While the problem makes sense for Schubert varieties of any kind and stan- dard monomial theory itself is available in great generality [13, 15], the transla- tion of the problem from geometry to combinatorics has been made—in [14]— only for “minuscule1 generalized Grassmannians.” Orthogonal Grassmannians being minuscule, this translation is available to us and we have an initial combi- natorial description of the Hilbert function. As to the passage from the initial to the alternative description—and this is where the content of the present paper lies—neither the end nor the means is clear at the outset. The first problem then is to find a good alternative description. But how to measure the worth of an alternative description? The interpretation of multi- plicity as the number of certain non-intersecting lattice paths (deduced in §11 from our alternative description) seems to testify to the correctness of our al- ternative description, but we are not sure if there are others that are equally or more correct. The proof of the equivalence of the initial and alternative combinatorial de- scriptions is, unfortunately, a little technically involved. It builds on the details of the proofs of the corresponding equivalences in the cases of the Grassmannian and the symplectic Grassmannian. In [10] it is shown that the equivalence in the case of the Grassmannian is a kind of KRS correspondence, called “bounded KRS.” The proof there is short and elegant and it would be nice to realise the main result of the present paper too in a similar spirit as a kind of KRS corre- spondence. The initial description is in terms of “standard monomials” and the alterna- 1Symplectic Grassmannians are not minuscule but can be treated as if they were. tive description in terms of “monomials in roots.” The equivalence of the two descriptions thus gives a bijective correspondence between standard monomials and monomials in roots. Roughly—but not actually—the correspondence maps each standard monomial to its initial term (with respect to a certain monomial order). Thus it is natural to wonder whether we can compute the initial ideal of the ideal of the tangent cone to the Schubert variety at the given point. We believe that this can be done but that it is far more involved and difficult than the corresponding computation for Grassmannians and symplectic Grassmanni- ans (the natural set of generators of the ideal of the tangent cone do not form a Gröbner basis unlike in those cases). If all goes well, the computation will soon appear [16]. Taking the Schubert variety to be of a special kind and the point to be the “identity coset,” our problem specializes to a problem about Pfaffian ideals con- sidered in [5, 2]. On the other side of the spectrum from the identity coset, so to speak, lie the “generic singularities,” points that are generic in the complement of the open orbit of the stabiliser of the Schubert variety. For these, a geometric solution to the problem appears in [1]. Given that our solution of the problem is but a translation, it makes sense to ask if one can extract more tangible information—closed form formulas for example—from our alternative description. See the papers quoted in the previ- ous paragraph and also [3] for some answers in the special cases they consider. Organization of the paper The table of contents indicates how the paper is organized. There is a brief description at the beginning of every subdivision of the contents therein. An index of definitions and notation is included, for it would otherwise be difficult to find the meanings of certain words and symbols. Important note added The recent article [6] treats some of the questions addressed here and some that could be addressed by using the main result proved here. It includes: • an interpretation of the multiplicity similar to ours. • a closed formula for the multiplicity (as a specialization of a factorial Schur function), thereby answering the question we raised above. • a formula for the restriction to the torus fixed point of the equivariant cohomology class of a Schubert variety. The approach in [6] is quite different from ours. In fact, it is the opposite of ours in that it circumvents the lack of results about initial ideals of tangent cones, while our prime motivation is to remedy the lack. The starting points in the two approaches are also different: [6] takes off from certain results of Kostant-Kumar and Arabia on equivariant cohomology, while our launchpad is standard monomial theory. The appearance of [6] notwithstanding, our approach is worthwhile, for, quite apart from the difference in starting points, there is no way, as far as we can tell, to the Hilbert function via the approach of [6], nor to the initial ideal, both of which are interesting in their own right. Part I The theorem Definitions are recalled, the problem formulated, and the theorem stated. 1 The set up In this section, we state the problem to be addressed after recalling the neces- sary basic definitions, make some choices that are convenient for studying the problem, and see why it is enough to focus on a particular case of the problem. 1.1 The statement of the problem Fix an algebraically closed field of characteristic not equal to 2. Fix a vector space V of finite dimension n over this field and a non-degenerate symmetric bilinear form 〈 , 〉 on V . Let d be the integer such that either n = 2d or n = 2d+ 1. A linear subspace of V is said to be isotropic if the form 〈 , 〉 vanishes identically on it. It is elementary to see that an isotropic subspace of V has dimension at most d and that every isotropic subspace is contained in one of dimension d. Denote by Md(V ) ′ the closed sub-variety of the Grassmannian of d-dimensional subspaces consisting of the points corresponding to isotropic subspaces. The orthogonal group O(V ) of linear automorphisms of V preserving 〈 , 〉 acts transitively on Md(V ) ′, for by Witt’s theorem an isometry between sub- spaces can be lifted to one of the whole vector space. If n is odd the special orthogonal group SO(V ) (consisting of form preserving linear automorphisms with trivial determinant) itself acts transitively on Md(V ) ′. If n is even the spe- cial orthogonal group SO(V ) does not act transitively on Md(V ) ′, and Md(V ) has two connected components. We define the orthogonal Grassmannian Md(V ) to be Md(V ) ′ if n is odd and to be one of the two components of Md(V ) ′ if n is even. The Schubert varieties of Md(V ) are defined to be the B-orbit closures in Md(V ) (with canonical reduced scheme structure), where B is a Borel sub- group of SO(V ). The choice of B is immaterial, for any two of them are con- jugate. The question that is tackled in this paper is this: given a point on a Schubert variety in Md(V ), how to compute the multiplicity (and more gen- erally, the Hilbert function) of the Schubert variety at the given point? The answers are contained in Theorem 2.3.1 and Corollary 2.3.2. But in order to make sense of those statements, we need some preparation. 1.2 Some convenient choices We now make some choices that are convenient for the study of Schubert vari- eties. For k an integer such that 1 ≤ k ≤ n, set k∗ := n + 1 − k. Fix a basis e1, . . . , en of V such that 〈ei, ek〉 = 1 if i = k∗ 0 otherwise The advantage of this choice is: the elements of SO(V ) for which each ek is an eigenvector form a maximal torus, and the elements that are upper triangular with respect to this basis form a Borel subgroup (a linear transformation is upper triangular if for each k, 1 ≤ k ≤ n, the image of ek under the transformation is a linear combination of e1, . . . , ek). We denote this maximal torus and this Borel subgroup by T and B respectively. Our Schubert varieties will be orbit closures of this particular Borel subgroup B. The B-orbits of Md(V ) ′ are naturally indexed by its T -fixed points: each orbit contains one and only one such point. The T -fixed points are evidently of the form 〈ei1 , . . . , eid〉, where 1 ≤ i1 < . . . < id ≤ n and for each k, 1 ≤ k ≤ d, there does not exist j, 1 ≤ j ≤ d, such that i∗k = ij—in other words, for each ℓ, 1 ≤ ℓ ≤ n, such that ℓ 6= ℓ∗, exactly one of ℓ and ℓ∗ appears in {i1, . . . , id}; in addition, if n is odd, then d+ 1 does not appear in {i1, . . . , id}. Denote the set of such d-element subsets {i1 < . . . < id} by I n. We thus have a bijective correspondence between I ′n and the B-orbits of Md(V ) ′. Each B-orbit being irreducible and open in its closure, it follows that B-orbit closures are indexed by the B-orbits. Thus I ′n is an indexing set for B-orbit closures in Md(V ) Suppose that n is even—it will be shown presently that it is enough to consider this case. As already observed, Md(V ) ′ has two connected components on each of which SO(V ) acts transitively. The B-orbits belong to one or the other component accordingly as the parity of the cardinality of the number of entries bigger than d in the corresponding element of I ′n. We take Md(V ) to be the component in which these cardinalities are even. We let In denote the subset of I ′n consisting of elements for which this cardinality is even. Schubert varieties in Md(V ) are thus indexed by elements of In. 1.3 Reduction to the case n even We now argue that it is enough to consider the case n even. Suppose that n is odd. Let ñ := n + 1 and Ṽ be a vector space of dimension ñ with a non-degenerate symmetric form. Let ẽ1, . . . , ẽen be a basis of Ṽ as in 1.2. Put e := ẽd+1 and f := ẽd+2. Take λ to be an element of the field such that λ2 = 1/2. We can take V to be the subspace of Ṽ spanned by the vectors ẽ1, . . . , ẽd, λe + λf, ẽd+3, . . . , ẽen, and a basis of V to be these vectors in that order. There is a natural map from Md+1(Ṽ ) ′ to Md(V ): intersecting with V an isotropic subspace of Ṽ of dimension d + 1 gives an isotropic subspace of V of dimension d. This map is onto, for every isotropic subspace of Ṽ (and hence of V ) is contained in an isotropic subspace of Ṽ of dimension d + 1. It is also elementary to see that the map is two-to-one (essentially because in a two- dimensional space with a non-degenerate symmetric form there are two isotropic lines), and that the two points in any fiber lie one in each component (there is clearly an element in O(Ṽ ) \ SO(Ṽ ) that moves one element of the fiber to the other, and so if there was an element of SO(Ṽ ) that also moved one point to the other, the isotropy at the point would not be contained in SO(Ṽ ), a contradiction). We therefore get a natural isomorphism between Md+1(Ṽ ) and Md(V ). We will now show that the B̃-orbits in Md+1(Ṽ ) correspond under the isomorphism to B-orbits of Md(V ) (we denote by T̃ and B̃ the maximal torus and Borel subgroups of SO(Ṽ ) as in §1.2). It will then follow that Schubert varieties in Md+1(Ṽ ) are isomorphic to those in Md(V ) and the purpose of this subsection will be achieved. The group SO(V ) can be realized as the subgroup of SO(Ṽ ) consisting of the elements that fix e − f . The isomorphism Md+1(Ṽ ) ∼= Md(V ) above is equivariant for SO(V ), and we have T̃ ∩ SO(V ) = T and B̃ ∩ SO(V ) = B. It should now be clear that the preimages in Md+1(Ṽ ) of two elements in the same B-orbit of Md(V ) are in the same B̃-orbit: an element of B that moves one to the other considered as an element of B̃ moves also the preimage of the one to that of the other. On the other hand, the preimages of distinct T -fixed points are distinct T̃ -fixed points, the corresponding map from I ′n to Ien being given as follows: i = {i1 < . . . < id} 7→ {̃i1, . . . , ĩd, d+ 1} if i ∈ In {̃i1, . . . , ĩd, d+ 2} if i ∈ I n \ In where ĩk = ik if ik ≤ d ik + 1 if ik ≥ d+ 2 (Note that d + 1 never occurs as an entry in any element of I ′n and that the elements ĩ1, . . . , ĩd, d + 1 (respectively ĩ1, . . . , ĩd, d + 2) are not in increasing order except in the trivial case i = {1 < . . . < d}.) Given that each B-orbit has a T -fixed point and that distinct T̃ -fixed points belong to distinct B̃-orbits, this implies that the preimages of two elements in distinct B-orbits belong to distinct B̃-orbits, and the proof is over. � 2 The theorem The purpose of this section is to state the main theorem and its corollary. We first set down some basic notation and two fundamental definitions needed in order to state the theorem. 2.1 Basic notation We keep the terminology and notations of §1.1, 1.2. As observed in §1.3, it is enough to consider the case n even. So from now on let n = 2d. Recall that, for an integer k, 1 ≤ k ≤ 2d, k∗ := 2d + 1 − k. As observed in §1.2, Schubert varieties in Md(V ) are indexed by In. Since d now determines n, we will henceforth write I(d) instead of In. In other words, I(d) is the set of d-element of subsets of {1, . . . , 2d} such that • for each k, 1 ≤ k ≤ 2d, the subset contains exactly one of k, k∗, and • the number of elements in the subset that exceed d is even. We write I(d, 2d) for the set of all d-element subsets of {1, . . . , 2d}. There is a natural partial order on I(d, 2d) and so also on I(d): v = (v1 < . . . < vd) ≤ w = (w1 < . . . < wd) if and only if v1 ≤ w1, . . . , vd ≤ wd. Given v ∈ I(d), the corresponding T -fixed point in Md(V ) (namely, the span of ev1 , . . . , evd) is denoted e v. Given w ∈ I(d), the corresponding Schubert variety in Md(V ) (which, by definition, is the closure of the B-orbit of the T - fixed point ew with canonical reduced scheme structure) is denoted X(w). The point ev belongs to X(w) if and only if v ≤ w in the partial order just defined. Since, under the natural action of B on X(w), each point of X(w) is in the B-orbit of a T -fixed point ev for some v such that v ≤ w, it is enough to focus attention on such T -fixed points. For the rest of this section an element v of I(d) will remain fixed. We will be dealing extensively with ordered pairs (r, c), 1 ≤ r, c ≤ 2d, such that r is not and c is an entry of v. Let R denote the set of all such ordered pairs, and set N := {(r, c) ∈ R | r > c} OR := {(r, c) ∈ R | r < c∗} ON := {(r, c) ∈ R | r > c, r < c∗} = OR ∩N d := {(r, c) ∈ R | r = c∗} diagonal boundary (r, c) (c∗, c) (r, r∗) The picture shows a drawing of R. We think of r and c in (r, c) as row index and column index respectively. The columns are indexed from left to right by the entries of v in ascending order, the rows from top to bottom by the entries of {1, . . . , 2d} \ v in ascending order. The points of d are those on the diagonal, the points of OR are those that are (strictly) above the diag- onal, and the points of N are those that are to the South-West of the poly- line captioned “boundary of N”—we draw the boundary so that points on the boundary belong to N. The reader can readily verify that d = 13 and v = (1, 2, 3, 4, 6, 7, 10, 11, 13, 15, 18, 19, 22) for the particular picture drawn. The points of ON indicated by solid circles form a v-chain (see §2.2.1 below). We will be consideringmonomials, also calledmultisets, in some of these sets. A monomial, as usual, is a subset with each member being allowed a multiplicity (taking values in the non-negative integers). The degree of a monomial has also the usual sense: it is the sum of the multiplicities in the monomial over all elements of the set. The intersection of a monomial in a set with a subset of the set has also the natural meaning: it is a monomial in the subset, the multiplicities being those in the original monomial. We will refer to d as the diagonal. 2.2 Two fundamental definitions 2.2.1 Definition of v-chain Given two elements (R,C) and (r, c) in ON, we write (R,C) > (r, c) if R > r and C < c (note that these are strict inequalities). An ordered sequence α, β, . . . of elements of ON is called a v-chain if α > β > . . . . A v-chain α1 > . . . > αℓ has head α1, tail αℓ, and length ℓ. 2.2.2 Definition of O-domination To a v-chain C : α1 > α2 > . . . inON there corresponds, as described in §5.3.3, a subset SC ofN which, as observed in Proposition 5.3.5, is “distinguished” in the sense of §5.1.1. To a distinguished subset of N there corresponds, as described below in §5.1.2, an element of I(d, 2d). Following these correspondences through, we get an element of I(d, 2d) attached to the v-chain C. Let w(C) denote this element—sometimes we write wC . (All this makes sense even when C is empty— w(C) will turn out to be v itself in that case.) Furthermore, as will be obvious from its definition, the monomial SC is “symmetric” in the sense of §5.2.2 and contains evenly many elements of the diagonal d. Thus, by Proposition 5.2.1, the element w(C) of I(d, 2d) belongs to I(d). An element w of I(d) is said toO-dominate C if w ≥ w(C), or, equivalently— and this is important for the proofs—if w dominates in the sense of [7] the mono- mial SC (for the proof of the equivalence, see [7, Lemma 5.5]). An element w of I(d) O-dominates a monomial S of ON (repsectively of OR) if it O-dominates every v-chain in S (respectively in S ∩ON). 2.3 The main theorem and its corollary Theorem 2.3.1 Fix a positive integer d and elements v ≤ w of I(d). Let V be a vector space of dimension 2d with a symmetric non-degenerate bilinear form (over a field of characteristic not 2). Let X(w) be the Schubert variety corresponding to w in the orthogonal Grassmannian Md(V ), and e v the torus fixed point of X(w) corresponding to v. Let Rwv denote the associated graded ring with respect to the unique maximal ideal of the local ring of germs at ev of functions on X(w). Then, for any non-negative integer m, the dimension as a vector space of the homogeneous piece of Rwv of degree m equals the cardinality of the set Sw(v)(m) of monomials of degree m of OR that are O-dominated by w. The proof of this theorem occupies us for most of this paper. It is reduced in §3, by an application of standard monomial theory, to combinatorics. The resulting combinatorial problem is solved in §4–10. For now, let us note the following immediate consequence: Corollary 2.3.2 The multiplicity at the point ev of the Schubert variety X(w) equals the number of monomials in ON of maximal cardinality that are square- free and O-dominated by w. Proof: The proof of Corollary 2.2 of [7] holds verbatim here too. � Part II From geometry to combinatorics The problem is translated from geometry to combinatorics. The main combi- natorial results are formulated. 3 Reduction to combinatorics In this section we translate the problem from geometry to combinatorics. In §3.1 we recall from [17] the theorem that enables the translation. The translation itself is done in 3.2 and follows [14]. 3.1 Homogeneous co-ordinate ring of the Schubert vari- ety X(w) 3.1.1 The line bundle L on Md(V ) Let Md(V ) ⊆ Gd(V ) →֒ P(∧ dV ) be the Plücker embedding (where Gd(V ) denotes the Grassmannian of all d-dimensional subspaces of V ). The pull-back toMd(V ) of the line bundle O(1) on P(∧ dV ) is the square of the ample generator of the Picard group ofMd(V ). Letting L denote the ample generator, we observe that it is very ample and want to describe the homogeneous coordinate rings of Md(V ) and its Schubert subvarieties in the embedding defined by L. 3.1.2 The section qθ of L For θ in I(d, 2d), let pθ denote the corresponding Plücker coordinate. Consider the affine patch A of P(∧dV ) given by pǫ = 1, where ǫ := (1, . . . , d). The intersection A ∩Gd(V ) of this patch with the Grassmannian is an affine space. Indeed the d-plane corresponding to an arbitrary point z of A ∩ Gd(V ) has a basis consisting of column vectors of a matrix of the form where I is the identity matrix and A an arbitrary matrix both of size d × d. The association z 7→ A is bijective. The restriction of a Plücker coordinate pθ to A∩Gd(V ) is given by the determinant of a submatrix of size d× d of M , the entries of θ determining the rows to be chosen from M to form the submatrix. As can be readily verified, a point z of A ∩ Gd(V ) represents an isotropic subspace if and only if the corresponding matrix A = (aij) is skew-symmetric with respect to the anti-diagonal : aij + aj∗i∗ = 0, where the columns and rows of A are numbered 1, . . . , d and d+1, . . . , 2d respectively. For example, if d = 4, then a matrix that is skew-symmetric with respect to the anti-diagonal looks like this:  −d −c −b 0 −g −f 0 b −i 0 f c 0 i g d Since the set of these matrices is connected and contains the point that is spanned by e1, . . . , ed, it follows that A∩Gd(V ) does not intersect the other com- ponent of Md(V ) ′. In other words, pǫ vanishes everywhere on Md(V ) ′ \Md(V ). Now suppose that θ belongs to I(d). Computing pθ/pǫ as a function on the affine patch pǫ 6= 0, we see that it is the determinant of a skew-symmetric matrix of even size, and therefore a square. The square root, which is determined up to sign, is called the Pfaffian. This suggests that pθ itself is a square: more precisely that there exists a section qθ of the line bundle L on Md(V ) such that q2θ = pθ. A weight calculation confirms this to be the case. The qθ are also called Pfaffians. 3.1.3 Standard monomial theory for Md(V ) A standard monomial in I(d) is a totally ordered sequence θ1 ≥ . . . ≥ θt (with repetitions allowed) of elements of I(d). Such a standard monomial is said to be w-dominated for w ∈ I(d) if w ≥ θ1. To a standard monomial θ1 ≥ . . . ≥ θt in I(d) we associate the product qθ1 · · · qθt , where the qθ are the sections defined above of the line bundle L. Such a product is also called a standard monomial and it is said to be dominated by w for w ∈ I(d) if the underlying monomial in I(d) is dominated by w. Standard monomial theory for Md(V ) says: Theorem 3.1.1 (Seshadri [17]) Standard monomials qθ1 · · · qθr of degree r form a basis for the space of forms of degree r in the homogeneous coordinate ring of Md(V ) in the embedding defined by the ample generator L of the Picard group. More generally, for w ∈ I(d), the w-dominated standard monomials of degree r form a basis for the space of forms of degree r in the homogeneous coordinate ring of the Schubert subvariety X(w) of Md(V ). 3.2 Co-ordinate rings of affine patches and tangent cones of X(w) From Theorem 3.1.1 one can deduce rather easily, as we now show, bases for co-ordinate rings of affine patches of the form qv 6= 0 and of tangent cones of Schubert varieties. An element v of I(d) will remain fixed for the rest of this section. To simplify notation we will suppress explicit reference to v. 3.2.1 Standard monomial theory for affine patches Let A denote the affine patch of P(H0(Md(V ), L) ∗) given by qv 6= 0. The origin of the affine space A is identified as the T -fixed point ev. The functions fθ := qθ/qv, v 6= θ ∈ I(d), provide a set of coordinate functions on A. Monomials in these fθ form a k-basis for the polynomial ring k[A] of functions on A, where k denotes the underlying field. Fix w ≥ v in I(d), so that the point ev belongs to the Schubert varietyX(w), and let Y (w) be the affine patch of X(w) defined thus: Y (w) := X(w) ∩ A. The coordinate ring k[Y (w)] of Y (w) is a quotient of the polynomial ring k[A], and the proposition that follows identifies a subset of the monomials in fθ which forms a k-basis for k[Y (w)]. We say that a standard monomial θ1 ≥ . . . ≥ θt in I(d) is v-compatible if for each k, 1 ≤ k ≤ t, either θk v or v θk. Given w in I(d), we denote by SM the set of w-dominated v-compatible standard monomials. Proposition 3.2.1 As θ1 ≥ . . . ≥ θt runs over the set SM w of w-dominated v-compatible standard monomials, the elements fθ1 · · · fθt form a basis for the coordinate ring k[Y (w)] of the affine patch Y (w) = X(w) ∩ A of the Schubert variety X(w). Proof: The proof is similar to the proof of Proposition 3.1 of [7]. First consider a linear dependence relation among the fθ1 · · · fθt . Replacing fθ by qθ and “ho- mogenizing” by qv yields a linear dependence relation among the w-dominated standard monomials qθ1 · · · qθs restricted to X(w), and so the original relation must only have been the trivial one, for by Theorem 3.1.1 the qθ1 · · · qθs are linearly independent on X(w). To prove that fθ1 · · · fθt generate k[Y (w)] as a vector space, we make the following claim: if qµ1 · · · qµr be any monomial in the Pfaffians qθ, and qτ1 · · · qτs a standard monomial that occurs with non-zero co-efficient in the expression for (the restriction to X(w) of) qµ1 · · · qµr as a linear combination of w-dominated standard monomials, then τ1∪· · ·∪τs = µ1∪· · ·∪µr as multisets of {1, . . . , 2d}. To prove the claim, consider the maximal torus T of SO(V ) as in §1.2. The affine patch A is T -stable and there is an action of T on k[Y (w)]. The sections qθ are eigenvectors for T with corresponding characters ǫθ1+· · ·+ǫθd , where ǫk denotes the character of T given by the projection to the diagonal entry on row k. The claim now follows since eigenvectors corresponding to different characters are linearly independent. Let fµ1 · · · fµr be an arbitrary monomial in the fθ. Fix an integer h such that h > r(d − 1) and consider the expression for (the restriction to X(w) of) qµ1 · · · qµr · q v as a linear combination of w-dominated standard monomials. We claim that qv occurs in every standard monomial qτ1 · · · qτr+h in this expression (from which it will follow that the τj are all comparable to v). Suppose that none of τ1, . . . , τr+h equals v. For each τj there is at least one entry of v that does not occur in it. The number of occurrences of entries of v in τ1 ∪ · · · ∪ τr+h is thus at most (r + h)(d − 1). But these entries occur at least hd times in µ1 ∪ · · · ∪ µr ∪ v ∪ · · · ∪ v (where v is repeated h times), a contradiction to the claim proved in the previous paragraph. Hence our claim is proved. Dividing by qr+hv the expression for qµ1 · · · qµr .q v as a linear combination of w-dominated standard monomials provides an expression for fµ1 · · · fµr as a linear combina- tion of fθ1 · · · fθt , as θ1 ≥ . . . ≥ θt varies over SM 3.2.2 Standard monomial theory for tangent cones The affine patch Md(V )∩A of the orthogonal Grassmannian Md(V ) is an affine space whose coordinate ring can be taken to be the polynomial ring in variables of the form X(r,c) with (r, c) ∈ OR, where (as in §2.1) OR = {(r, c) | 1 ≤ r, c ≤ 2d, r 6∈ v, c ∈ v, r < c∗} Taking d = 5 and v = (1, 3, 4, 6, 9) for example, a general element of Md(V )∩A has a basis consisting of column vectors of a matrix of the following form:  1 0 0 0 0 X21 X23 X24 X26 0 0 1 0 0 0 0 0 1 0 0 X51 X53 X54 0 −X26 0 0 0 1 0 X71 X73 0 −X54 −X24 X81 0 −X73 −X53 −X23 0 0 0 0 1 0 −X81 −X71 −X51 −X21  The expression for fθ = qθ/qv in terms of the X(r,c) is a square root of the determinant of the submatrix of a matrix like the one above obtained by choosing the rows given by the entries of θ. Thus fθ is a homogeneous polynomial of degree the v-degree of θ, where the v-degree of θ is defined as one half of the cardinality of v \ θ. Since the ideal of the Schubert variety X(w) in the homogeneous coordinate ring of Md(V ) is generated 2 by the qτ , τ ∈ I(d) such that τ 6≤ w, it follows that the ideal of Y (w) := X(w) ∩ A in Md(V ) ∩ A is generated by the the fτ , τ ∈ I(d) such that τ 6≤ w. We are interested in the tangent cone to X(w) at ev (or, what is the same, the tangent cone to Y (w) at the origin), and since k[Y (w)] is graded, its associated graded ring with respect to the maximal ideal corresponding to the origin is k[Y (w)] itself. Proposition 3.2.1 says that the graded piece of k[Y (w)] of degree m is gener- ated as a k-vector space by elements of degreem of the set SMw of w-dominated v-compatible standard monomials, where the degree of a standard monomial θ1 ≥ . . . ≥ θt is defined to be the sum of the v-degrees of θ1, . . . , θt. To prove Theorem 2.3.1 it therefore suffices to prove the following: 2This is a consequence of Theorem 3.1.1. It is easy to see that the qτ such that τ 6≤ w vanish on X(w). Since all standard monomials form a basis for the homogeneous coordinate ring of Md(V ) in P(H 0(Md(V ), L) ∗), it follows that w-dominated standard monomials span the quotient ring by the ideal generated by such qτ . Since such monomials are linearly independent in the homogeneous coordinate ring of X(w), the desired result follows. Theorem 3.2.2 The set SMw(m) of standard monomials in I(d) of degree m that are w-dominated and v-compatible is in bijection with the set Sw(v)(m) of monomials in OR of degree m that are O-dominated by w. 4 Further reductions In the last section, we reduced the proof of our main theorem (Theorem 2.3.1) to that of Theorem 3.2.2. We now reduce the proof of Theorem 3.2.2 to that of Propositions 4.1.1, 4.1.2 and 4.1.3 below. These propositions will eventually be proved in §10. 4.1 The main propositions Fix once and for all an element v of I(d). The bijection stated in Theorem 3.2.2 will be described by means of two maps Oπ and Oφ whose definitions will be given in §7 and §8 below. We will now state some properties of these maps. In §4.2 we will see how Theorem 3.2.2 follows once these properties are estab- lished. The map Oπ associates to a monomial S in ON a pair (w,S′) consisting of an element w of I(d) and a “smaller” monomial S′ in ON. This map enjoys the following good properties: Proposition 4.1.1 1. w ≥ v. 2. v-degree(w) + degree(S′) = degree(S). 3. w O-dominates S′. 4. w is the least element of I(d) that O-dominates S. The map Oφ, on the other hand, associates a monomial in ON to a pair (w,T) consisting of an element w of I(d) with w ≥ v and a monomial T in ON that is O-dominated by w. Proposition 4.1.2 The maps Oπ and Oφ are inverses of each other. For an integer f , 1 ≤ f ≤ 2d, consider the following conditions, the first on a monomial S in ON, the second on an element w of I(d): (‡) f is not the row index of any element of S and f⋆ is not the column index of any element of S. (‡) f is not an entry of w. (It is convenient to the use the same notation (‡) for both conditions.) Proposition 4.1.3 Assume that v satisfies (‡)—all references to (‡) in this proposition are with respect to a fixed f , 1 ≤ f ≤ 2d. 1. Let w be an element of I(d) with w ≥ v and T a monomial in ON that is O-dominated by w. If w and T both satisfy (‡), then so does Oφ(w,T). 2. If a monomial S in ON satisfies (‡), then so do the “components” w and S′ of its image under Oπ. 4.2 From the main propositions to the main theorem Let us now see how Theorem 3.2.2 follows from the propositions of §4.1. Most of the following argument runs parallel to its counterparts in the case of the Grassmannian and symplectic Grassmannian (Propositions 4.1.1 and 4.1.2 have their counterparts in [7, 4]), but, in the case that d is odd, the part involving the “mirror image” requires additional work. This is where Proposition 4.1.3 comes in. Let S, T , and U , denote respectively the sets of monomials in OR, ON, and OR\ON. Let SMv denote the set of v-compatible standard monomials that are “anti-dominated” by v: a standard monomial θ1 ≥ . . . ≥ θt is anti-dominated by v if θt ≥ v (we can also write θt > v since θt 6= v by v-compatibility). Define the domination map from T to I(d) by sending a monomial in ON to the least element that O-dominates it. Define the domination map from SMv to I(d) by sending θ1 ≥ . . . ≥ θt to θ1. Both these maps take, by definition, the value v on the empty monomial. Notation 4.2.1 In the following, we use subscripts, superscripts, suffixes, and combinations thereof to modify the meanings of S, T , U , SM , and SMv. • superscript: this will be an element w of I(d); when used on T it denotes O-domination (more precisely, Tw denotes the subset of T consisting of those elements that are O-dominated by w); when used on SM or SMv it denotes domination by w. • subscript: denotes anti-domination (applied only to standard monomials). • suffix “(m)”: indicates degree (for example, SMwv (m) denotes the set of v- compatible standard monomials that are anti-dominated by v, dominated by w, and of degree m). Repeated application of Oπ gives a map from T to SMv that commutes with domination (as just defined) and preserves degree. Repeated application of Oφ gives a map from SMv to T . These two maps being inverses of each other (Proposition 4.1.2) and so we have a bijection between SMv and T . In fact, since domination and degree are respected (Proposition 4.1.1), we get a bijection SMwv (m) ∼= Tw(m). As explained below, the “mirror image” of the bijection SMv(m) ∼= T (m) gives a bijection SMv(m) ∼= U(m). Putting these bijections together, we get the desired result: SMw(m) = SMwv (k)× SM v(m− k) Tw(k)× U(m− k) = Sw(m). We now explain how to realize the bijection SMv(m) ∼= U(m) as the “mirror image” of the bijection SMv(m) ∼= T (m). For an element u of I(d), define u∗ := (u∗d, . . . , u 1). In the case d is even, the association u 7→ u ∗ is an order reversing involution, and the argument in [4] for the symplectic Grassmannian holds here too. In the case d is odd, u∗ is not an element of I(d), and so some additional work is required. Recall that a “base element” v of I(d) has been fixed and that our notation does not explicitly indicate this dependence upon v: for example, OR is depen- dent upon v. For a brief while now (until the end of this section) we need to simultaneously handle several base elements of I(d). We will use the following convention: when the base element of I(d) is not v, we will explicitly indicate it by means of a suffix. For instance, SM(v∗) denotes the set of v∗-compatible standard monomials in I(d). Let us first do the case when d is even. We get a bijection SMv ∼= SMv∗(v by associating to θ1 ≥ . . . ≥ θt the element θ t ≥ . . . ≥ θ 1 . The sum of the v-degrees of θ1, . . . , θt equals the sum of the v ∗-degrees of θ∗t , . . . , θ 1 , so that we get a bijection SMv(m) ∼= SMv∗(v ∗)(m). For an element (r, c) of ON(v∗), consider its flip (c, r). Since v belongs to I(d), the complement of v∗ in {1, . . . , 2d} is v, and it follows that (c, r) belongs to OR \ ON. This induces a degree preserving bijection T (v∗) ∼= U . Putting this together with the bijection of the previous paragraph and the one deduced earlier in this section (using Oπ and Oφ), we get what we want: SMv(m) ∼= SMv∗(v ∗)(m) ∼= T (v ∗)(m) ∼= U(m). Now suppose that d is odd. Then the map x 7→ x∗ does not map I(d) to I(d) but to I(d)∗ (defined as the set consisting of those elements u of I(d, 2d) such that, for each k, 1 ≤ k ≤ 2d, exactly one of k, k∗ belongs to u, and the number of entries of u greater than d is odd). We define a map u 7→ ũ from I(d)∗ to I(d + 1) as follows: ũ := {ũ1, . . . , ũd, d + 2} (the elements are not in increasing order except in the trivial case u = (1, . . . , d)), where, for an integer e, 1 ≤ e ≤ 2d, we set ẽ := e if 1 ≤ e ≤ d e+ 2 if d+ 1 ≤ e ≤ 2d This map u 7→ ũ is an order preserving injection. Consider the composition x 7→ x∗ 7→ x̃∗ from I(d) to I(d+1). This is an order reversing injection. The induced map on standard monomials is an injection from SMv to SMfv∗(ṽ ∗). It is readily seen that the image under this map is the subset SMfv∗(ṽ ∗)(‡) consisting of those standard monomials all of whose elements satisfy (‡) with f = d+1. We have already established (using the maps Oπ and Oφ) a bijection SMfv∗(ṽ ∗) ∼= T (ṽ∗). It follows from Proposition 4.1.3 that under this bijection the subset SMfv∗(ṽ ∗)(‡) maps to T (ṽ∗)(‡) (defined as the set of those monomials in ON(ṽ∗) satisfying (‡) with f = d+ 1). Now T (ṽ∗)(‡) is in degree preserving bijection with U : every element of degree 1 of T (ṽ∗)(‡) is uniquely of the form (c̃, r̃) for (r, c) in OR\ON, and the desired bijection is induced from this. Putting all of these together, we finally SMv ∼= SMfv∗(ṽ ∗)(‡) ∼= T (ṽ∗)(‡) ∼= U. Thus, in order to prove our main theorem (Theorem 2.3.1), it suffices to describe the maps Oπ and Oφ and to prove Propositions 4.1.1–4.1.3. Part III The proof The main combinatorial results formulated in §4.1 are proved. An attempt is made to maintain parallelism with the proofs in [7]. 5 Terminology and notation 5.1 Distinguished subsets 5.1.1 Distinguished subsets of N Following [7, §4], we define a multiset S of N to be distinguished , if, first of all, it is a subset in the usual sense (in other words, it is “multiplicity free”), and if, for any two distinct elements (R,C) and (r, c) of S, the following conditions are satisfied: A. R 6= r and C 6= c. B. If R > r, then either r < C or C < c. In terms of pictures, condition A says that (r, c) cannot lie exactly due North or East of (R,C) (or the other way around); so we can assume, interchanging the two points if necessary, that (r, c) lies strictly to the Northeast or Northwest of (R,C); condition B now says that, if (r, c) lies to the Northwest of (R,C), then the point that is simultaneously due North of (R,C) and due East of (r, c) (namely (r, C)) does not belong to N. 5.1.2 Attaching elements of I(d, 2d) to distinguished subsets of N To a distinguished subset S of N there is naturally associated an element w of I(d, 2d) as follows: start with v, remove all members of v which appear as column indices of elements of S, and add row indices of all elements of S. As observed in [7, Proposition 4.3], this association gives a bijection between distinguished subsets of N and elements w ≥ v of I(d, 2d). The unique distinguished subset of N corresponding to an element w ≥ v of I(d, 2d) is denoted Sw. 5.2 The involution # 5.2.1 The involution # on I(d, 2d) There are two natural order reversing involutions on I(d, 2d). First there is w 7→ w∗ induced by the natural order reversing involution j 7→ j∗ on {1, . . . , 2d}: here w∗ has the obvious meaning, namely, it consists of all j∗ such that j belongs to w. Then there is the map taking w to its complement {1, . . . , 2d}\w. These two involutions commute. Composing the two we get an order preserving involution on I(d, 2d) which we denote by w 7→ w#. The elements of the subset I(d) are fixed points under this involution (there are points not in I(d) that are also fixed). 5.2.2 The involution # on N and R For α = (r, c) in N, or more generally in R, define α# = (c∗, r∗). The involution α 7→ α# is just the reflection with respect to the diagonal d. For a subset or even multiset S of N (or R), the symbol S# has the obvious meaning. We call S symmetric if S = S#. Proposition 5.2.1 An element w ≥ v of I(d, 2d) belongs to I(d) if and only if the distinguished subset Sw of N corresponding to it as described in §5.1.2 is symmetric and has evenly many diagonal elements. Proof: That the symmetry of Sw is equivalent to the condition that w = w is proved in [4, Proposition 5.7]. Now suppose that Sw is symmetric. We claim that for an element (r, c) of Sw that is not on the diagonal, either both r and c are bigger than d or both are less than d + 1. It is enough to prove the claim, for w is obtained from v by removing the column indices and adding the row indices of elements of Sw, and it would follow that the number of entries in w that are bigger than d equals the number of such entries in v plus the number of diagonal elements in Sw. We now prove the claim. Since Sw is symmetric, it follows that (c ∗, r∗) also belongs to Sw. Since Sw is distinguished, it follows that in case r < c ∗ (that is, if (r, c) lies above the diagonal), we have r < r∗, and so c < r < r∗; and in case r > c∗, we have c∗ < c, and so c∗ < c < r. Thus the claim is proved. � 5.3 The subset SC attached to a v-chain C 5.3.1 Vertical and horizontal projections of an element of ON For α = (r, c) in ON (or more generally in OR), the elements pv(α) := (c ∗, c) and ph(α) := (r, r ∗) of the diagonal d are called respectively the vertical and horizontal projections of α. In terms of pictures, the vertical projection is the element of the diagonal due South of α; the horizontal projection is the element of the diagonal due East of α. The vertical line joining α to its vertical projection pv(α) and the horizontal line joining α to its horizontal projection ph(α) are called the legs of α. 5.3.2 The “connection” relation on elements of a v-chain Let C : α1 = (r1, c1) > α2 = (r2, c2) > · · · be a v-chain in ON. Two consecutive elements αj and αj+1 of C are said to be connected if the following conditions are both satisfied: • their legs are “intertwined”; equivalently and more precisely, this means that r∗j ≥ cj+1, or, what amounts to the same, rj ≤ c • the point (rj+1, r j ) belongs to N; this just means that rj+1 > r Consider the coarsest equivalence relation on the elements of C generated by the above relation. The equivalence classes of C with respect to this equivalence relation are called the connected components of the v-chain C. This definition has its quirks: The v-chain C : α > β > γ in the pic- ture has {α, β} and {γ} as its connected components; but the “sub” v-chain α > γ of C is connected (as a v-chain in its own right). diagonal boundary 5.3.3 The definition of SC We will define SC as a multiset of N. It is easy to see and in any case stated explicitly as part of Corollary 5.3.5 that it is multiplicity free and so is actually a subset of N. First suppose that C : α1 = (r1, c1) > · · · > αℓ = (rℓ, cℓ) is a connected v-chain in ON. Observe that, if there is at all an integer j, 1 ≤ j ≤ ℓ, such that the horizontal projection ph(αj) does not belong to N, then j = ℓ. Define SC := {pv(α1), . . . , pv(αℓ)} if ℓ is even {pv(α1), . . . , pv(αℓ)} ∪ {ph(αℓ)} if ℓ is odd and ph(αℓ) ∈ N {pv(α1), . . . , pv(αℓ−1)} ∪ {αℓ, α } if ℓ is odd and ph(αℓ) 6∈ N For a v-chain C that is not necessarily connected, let C = C1 ∪ C2 ∪ · · · be the partition of C into its connected components, and set SC := SC1 ∪SC2 ∪ · · · 5.3.4 The type of an element α of a v-chain C, and the set SC,α We introduce some terminology and notation. Their usefulness may not be immediately apparent. Suppose that C : α1 > · · · > αℓ is a connected v-chain. We define the type in C of an element αj , 1 ≤ j ≤ ℓ, of C to be V, H, or S, accordingly as: V: j 6= ℓ, or j = ℓ and ℓ is even. H: j = ℓ, ℓ is odd, and ph(αℓ) ∈ N. S: j = ℓ, ℓ is odd, and ph(αℓ) 6∈ N. The type of an element in a v-chain that is not necessarily connected is defined to be its type in its connected component. The set SC,α of elements of N generated by an element α of C is defined to SC,α := {pv(α)} if α is of type V in C; {pv(α), ph(α)} if α is of type H in C; {α, α#} if α is of type S in C; Observe that, for a v-chain C, the monomial SC defined in §5.3.3 is the union, over all elements α of C, of SC,α. For an element α of a v-chain C, we define qC,α to be pv(α) if α is of type V or H and to be α if it is of type S. If the horizontal projection of an element in a v-chain does not belong to N, then clearly the same is true for every succeeding element. The first such element of a v-chain is called the critical element. Proposition 5.3.1 1. The cardinality is odd of a connected component that has an element of type H or S. Conversely, if the cardinality of a compo- nent is odd, then it has an element of type H or S. 2. An element of type H or S can only be the last element in its connected component. 3. The critical element has type either V or S. No element before it can be of type S and every element after it is of type S. In particular, any element that succeeds an element of type S is of type S. Proof: Clear from definitions. � Proposition 5.3.2 Let α > γ be elements of a v-chain C (we are not assuming that they are consecutive). 1. If α > γ is connected as a v-chain in its own right, then α is connected to its next member in C; that is, α cannot be the last element in its connected component in C. 2. If α > γ is not connected as a v-chain in its own right and the legs of α and γ intertwine, then the connected component of γ in C is the singleton {γ}, and γ has type S in C. Proof: Clear from definitions. � Proposition 5.3.3 Let E : α > . . . > ζ be a v-chain, D and D′ two v-chains with tail α, and C, C′ the concatenations of D, D′ respectively with E. Then 1. The last element in the connected component containing α is the same in C and C′ (and this is the same as in E). Let λ denote this element. 2. The only element among α, . . . , ζ that possibly has different types in C and C′ is λ. Proof: (1): Whether or not two successive elements in a v-chain are connected is independent of other elements in the v-chain. (2): The type of an element in a v-chain is V unless it is the last element in its connected component. And the type of the last element in a component depends on the cardinality of the component. The components of E not contain- ing α are still components in C and C′. In contrast, the component containing α could possibly be larger in C (respectively C′) and hence its cardinality could be different. � For an element α = (r, c) of N, we define α(up) to be α itself if α is either on or above the diagonal d (more precisely, if r ≤ c∗), and to be its “reflection” in the diagonal (more precisely, (c∗, r∗)) if α is below the diagonal (more precisely, if r > c∗). For a monomial S of N, S(up) is defined to be the intersection of S (as a multiset) with the subset ON ∪ d of N. The notations α(down) and S(down) have similar meanings. Caution: It is not true that S(up) = {α(up)|α ∈ S} (in the obvious sense one would make of the right hand side). In particular, for a singleton monomial {α}, it is not always true that {α}(up) = {α(up)}. Proposition 5.3.4 Let α and β be elements of a v-chain C. Let us use α′ and β′ respectively to denote elements of SC,α(up) and SC,β(up). 1. If α > β (these elements are not necessarily consecutive in C), then, given β′, there exists α′ such that α′ > β′. In fact, this is true for every choice of α′ except when (*) α is of type H, and ph(α) 6> β ′ for some β′ ∈ SC,β. In particular, qC,α > β ′ and qC,α > qC,β. 2. Conversely, suppose that α′ > β′ for some choice of α′ and β′. Then α ≥ β; if equality occurs, then α is of type H, α′ = pv(α) and β ′ = ph(α). In particular, if α′ > qC,β (or more specially qC,α > qC,β), then α > β. 3. If (*) holds for α > β in C, then (a) the critical element of C is the one just after α; in particular, α is uniquely determined. (b) all elements of C succeeding α are of type S; in particular, β is of type S and β′ = β. (c) (*) holds for γ in place of β for every γ in C that succeeds α. Proof: (1) If α is of type V or H, we need only take α′ = pv(α), for pv(α) > pv(β), pv(α) > ph(β), and pv(α) > β. Now suppose that α is of type S. Then β too is of type S (Proposition 5.3.1 (3)), so β′ can only be β, and the first part of (1) is proved. It follows from the above that if α′ = pv(α) or if α has type S, then α ′ > β′ independent of the choice of α′. So if α′ 6> β′, then (*) holds and α′ = ph(α). (3) Let λ be the immediate successor of α in C. Then α is not connected to λ (Proposition 5.3.1 (2)). Since ph(α) 6> β ′, it follows that α and β have intertwining legs. Therefore so do α and λ. By Proposition 5.3.2 (2), λ has type S in C. Since α has type H and λ type S, it follows immediately from the definition of the critical element that λ is the critical element. This proves (a). Assertion (b) now follows from Proposition 5.3.1 (3). For (c), write ph(α) = (a, a ∗), λ = (R,C), and γ = (r, c). Then R < a∗, for α and λ have intertwining legs but are not connected. So c < r ≤ R < a∗. This means ph(α) 6> γ. And γ being of type S (by (b)), we can take γ′ = γ. (2) Suppose that α 6≥ β. Then β > α. By the second part of (1) above, β is of type H and β′ = ph(β); by item (b) of (3), α is of type S, so α ′ = α. This leads to the contradiction β > α > ph(β). � Corollary 5.3.5 The multiset SC attached to a v-chain C is a distinguished subset of N in the sense of 5.1.1. Proof: If α in C is of type V or S, then SC,α is a singleton; if it is of type H, then SC,α = {pv(α), ph(β)}. So there can be no violation of conditions A and B of §5.1.1 by elements of SC,α. Suppose α > β. By Proposition 5.3.4 (1), we have α′ > β′ for any choice of α′ ∈ SC,α and β ′ ∈ SC,β except when the condition (*) holds. By (3) of the same proposition, if (*) holds, then β′ = β, and writing β = (r, c), ph(α) = (a, a ∗), we have r < a (since α > β) and c < r < a∗ (see proof of item 3(c) of the proposition). Thus there can be no violation of conditions A and B of §5.1.1. � Corollary 5.3.6 Let S be a v-chain in ON and w an element of I(d). If w O-dominates S, then w dominates in the sense of [7] the monomial S ∪ S# of N. Proof: By [4, Proposition 5.15], it is enough to show that w dominates S. Let C : α1 > . . . > αt be a v-chain in S. Writing αj = (rj , cj) and qC,αj = (Rj , Cj) we have rj ≤ Rj and Cj ≤ cj . By Proposition 5.3.4 (1), we have qC,α1 > . . . > qC,αt . Since w O-dominates S, it in particular dominates qC,α1 > . . . > qC,αt and so also C. � 6 O-depth The concept of O-depth defined in §6.1 below plays a key role in this paper. As the name suggests, it is the orthogonal analogue of the concept of depth of [7]. In §6.2 below, it is observed that the O-depth is no smaller than depth in the sense of [7]. In §6.3, some observations about the relation between O-depths and types of elements in v-chains are recorded. 6.1 Definition of O-depth The O-depth of an element α in a v-chain C in ON is the depth in SC in the sense of [7] of qC,α: in other words, it is the depth in SC of pv(α) in case α is of type V or H, and of α (equivalently of α#) in case α is of type S. It is denoted O-depthC(α). The O-depth of an element α in a monomial S of ON is the maximum, over all v-chains C in S containing α, of the O-depth of α in C. It is denoted O-depth (α). Finally, the O-depth of a monomial S in ON is the maximum of the O-depths in S of all the elements of S. There is a conflict in the above definitions: Is the O-depth of an element of a v-chain C the same as its depth as an element of the monomial C? In other words, could the O-depth of an element in a v-chain be exceeded by its O-depth in a sub-chain? The conflict is resolved by the first item of the following proposition. Proposition 6.1.1 1. For v-chains C ⊆ D, the O-depth in C of an element of C is no more than its O-depth in D. 2. If a v-chain C is an initial segment of a v-chain D, then the O-depths in C and D of an element of C are the same. Proof: (1): By an induction on the difference in the cardinalities of D and C, we may assume that D has one more element than C. Call this extra element δ. Suppose that δ lies between successive elements α and β of C (the modifications needed to cover the extreme cases when it goes at the beginning or the end are being left to the reader). The only elements of C that could possibly undergo changes of type on addition of δ are α and the last element in the connected component of β, which let us call β′. If there are no type changes, then SC ⊆ SD and the assertion is immediate. The only type change that α can undergo is from H to V. The type changes that β′ can undergo are: H to V; V to H; S to V; V to S. An easy enumeration of cases shows that only one of α and β′ can undergo a type change. We need not worry about changes from V to H for in this case SC ⊆ SD. First let us suppose that α undergoes a change of type (from H to V). Then δ is connected to α. It follows from Proposition 5.3.1 (1) that δ has type V in D: the connected component of α in C has odd number of elements, so if δ happens to be the last element in its connected component in D, the number of elements in that component will be even. Replacing an occurrence of ph(α) in a v-chain of SC by pv(δ) would result in a v-chain in SD (by Proposition 5.3.4 (1)), and this case is settled. Now suppose that β′ undergoes a type change. Then δ is connected to β and δ is of type V in D (Proposition 5.3.1 (2)). Replacing by pv(δ) any occurrence in a v-chain in SC of pv(β ′), ph(β ′), β′ accordingly as the type of β′ in C is V, H, or S, (not necessarily in the same place but at an appropriate place) would result in a v-chain in SD (by Proposition 5.3.4 (1)), and we see that the O-depth cannot decrease. (2): It follows from Proposition 5.3.4 (2) that, for an element α of C, con- tributions to SD from elements beyond α (in particular from those not in C) do not affect the depth in SD of qD,α. Looking for the possibility of differences in types in C and D of elements of C, we see that the only element of C that has possibly a different type in D is its last element. And this too can change type only from H to V. The above two observations imply that the calculations of O-depths in C and D of an element α of C are no different: we would be considering the depth in SC and SD respectively of the same element (either pv(α) or α), and the differences in SD and SC have no effect on this consideration. � Corollary 6.1.2 If C ⊆ D are v-chains in ON, then wC ≤ wD (although it is not always true that SC ⊆ SD). Proof: By [7, Lemma 5.5], it is enough to show that every v-chain in SC is dominated by wD. Let β1 = (r1, c1) > · · · > βt = (rt, ct) be an arbitrary v-chain in SC . To show that it is dominated by wD, it is enough, by [7, Lemma 4.5], to show the existence of a v-chain (R1, C1) > · · · > (Rt, Ct) in SD with rj ≤ Rj and Cj ≤ cj for 1 ≤ j ≤ t. Such a v-chain exists by the proof of (1) of Proposi- tion 6.1.1. � Corollary 6.1.3 1. Let S be a monomial in ON and α ∈ S. Then there ex- ists a v-chain C in S with tail α such that O-depth (α) = O-depthC(α). 2. For elements α > γ in a v-chain C (these need not be consecutive), we have O-depthC(α) < O-depthC(γ). 3. For elements α > γ of a monomial S in ON, we have O-depth (α) < O-depth 4. No two elements of the same O-depth in a monomial in ON are compa- rable. Proof: (1) This follows from (2) of the Proposition above and the definition of O-depth. (2) This follows from Proposition 5.3.4 (1) and the definition of O-depth. (3) By (1), there exists a v-chain C with tail α such that O-depth (α) = O-depthC(α). Concatenate C with α > γ and letD denote the resulting v-chain. By (2) of the Proposition above, O-depthC(α) = O-depthD(α). By (2) above, O-depthD(α) < O-depthD(γ). And finally, O-depthD(γ) ≤ O-depthS(γ) by the definition of O-depth (4) Immediate from (3). � Corollary 6.1.4 Let β > γ be elements of a v-chain C of elements of ON. Let E be a v-chain in SC with tail qC,γ and length O-depthC(γ). Then qC,β occurs in E. Proof: It is enough to show that for α′ 6= qC,β in E, either α ′ > qC,β or qC,β > α ′. Let α be in C such that qC,β 6= α ′ ∈ SC,α. If β ≥ α, then qC,β > α by Proposition 5.3.4 (1). If α > β and α′ 6> qC,β , then, by (1) and (3) of the same proposition, α′ 6> qC,γ , a contradiction. � 6.2 O-depth and depth Lemma 6.2.1 The O-depth of an element α in a monomial S of ON is no less than its depth (in the sense of [7]) in S ∪S#. Proof: Let C : α1 > . . . > αt be a v-chain in S∪S # with tail αt = α, where t is the depth of α in S ∪S#. We then have α1(up) > . . . > αt(up), so we may assume C to be in S. By Proposition 5.3.4 (1), qC,α1 > . . . > qC,αt in SC . So depth S∪S#(α) = t ≤ depthSC (qC,αt) ≤ O-depthS(α). � 6.3 O-depth and type We begin by defining some useful terminology. Let (r, c) and (R,C) be two elements of R. To say that (R,C) dominates (r, c) means that r ≤ R and C ≤ c (in terms of pictures, (r, c) lies (not necessarily strictly) to the Northeast of (R,C)). To say that they are comparable means that either (R,C) > (r, c) or (r, c) > (R,C). While this is admittedly strange, there will arise no occasion for confusion. For an integer i, we let i(odd) be the largest odd integer not bigger than i and i(even) the smallest even integer not smaller than i. Lemma 6.3.1 1. For consecutive elements α > β of a v-chain C, O-depthC(β) = O-depthC(α) + 2 if and only if α is of type H and ph(α) > β O-depthC(α) + 1 otherwise 2. For an element of a v-chain C such that either its horizontal projection belongs to N or it is connected to its predecessor, the parity of its O-depth in C is the same as that of its ordinality in its connected component in C. 3. The O-depth in a v-chain of an element of type H is odd. 4. If in a v-chain an element of type V is the last in its connected component, then its O-depth is even. 5. If in a v-chain C there is an element of O-depth d, then (a) for every odd integer d′ not exceeding d, there is in C an element of O-depth d′. (b) if, for an even integer d′ not exceeding d, there is no element in C of O-depth d′, then the element α in C of O-depth d′ − 1 is of type H, and ph(α) > β, where β denotes the immediate successor of α in C. 6. Let C be a v-chain and α an element of type H in C. Then the depth in SC of ph(α) equals O-depthC(α) + 1. In particular, this depth is even. Proof: (1): From items 1 and 3(a) of Proposition 5.3.4, it follows that, for γ in C with γ > α, if γ′ 6> qC,α for some γ ′ in SC,γ , then γ ′ 6> qC,β. Thus O-depthC(β) exceeds O-depthC(α) by the number of elements in SC,α that dominate qC,β . This number is 1 if α is of type V, or of type S, or of type H and ph(α) 6> β; it is 2 if α is of type H and ph(α) > β (note that ph(α) > β if and only if ph(α) > qC,β). (2): Let λ be such an element. Everything preceding λ in C is of type H or V (Proposition 5.3.1 (3)). Let λ belong to the kth connected component, and n1, . . . , nk be respectively the cardinalities of the first, . . . , k th connected components. By (1) above and item 3(b) of Proposition 5.3.4, O-depthC(λ) is n1(even) + · · · + nk−1(even) plus the ordinality of λ in the k th connected component. (3) and (4): These are special cases of 2. (5): This follows easily from (1) and (3). (6): It follows from Proposition 5.3.4 (2) that there is no element γ in SC that lies between pv(α) and ph(α) (meaning pv(α) > γ > ph(α)), so the asser- tion holds. � Corollary 6.3.2 For a v-chain C in ON, if the O-depths of elements in C are bounded by k, then the depths of elements in SC are bounded by k(even). Proof: The depth of qC,α in SC for any α in C is at most k by hypothesis. An element of SC that is not qC,α for any α in C can only be of the form ph(α) for some α. By Proposition 5.3.4, depthSCpv(α) = depthSCph(α) − 1, which implies depth ph(α) ≤ k + 1. If, moreover, k is even, then by (3) of Lemma 6.3.1 depth ph(α) = depthSCpv(α) + 1 ≤ (k − 1) + 1 = k. � Proposition 6.3.3 Given a monomial S in ON and an element α in it, there exists a v-chain C in S with tail α such that O-depthC(β) = O-depthS(β) for every β in C. Proof: Proceed by induction on d := O-depth (α). Choose a v-chain D in S with tail α such that O-depthD(α) = O-depthS(α) (such a v-chain exists by Corollary 6.1.3 (1)). Let α′ be the element in D just before α. It follows from item (3) of Corollary 6.1.3 and item (1) of Lemma 6.3.1 that O-depth (α′) (as also O-depthD(α ′)) is either d − 1 or d − 2. By induction, there exists a v- chain C′ with tail α′ that has the desired property. Let C be the concatenation of C′ with α′ > α. We claim that C has the desired property. The only thing to be proved is that O-depthC(α) = d. By item (1) of Lemma 6.3.1, we have O-depthC(α) ≥ O-depthC′(α ′) + 1. In particular, the claim is proved in case O-depthC′(α is d − 1, so let us assume that O-depthC′(α ′) is d− 2. It now follows from the same item that α′ has type H in D and ph(α ′) > α; it further follows that it is enough to show that α′ has type H in C. Since α′ has type H in D, it follows (from item (2) of Proposition 5.3.1) that α′ > α is not connected and (from item (3) of Lemma 6.3.1) that d− 2 is odd. Now, by item (4) of Lemma 6.3.1, the type in C′ of α′ cannot be V, so it is H, and the claim is proved. � Corollary 6.3.4 Let S be a monomial in ON, β an element of S, and i an integer such that i < O-depth (β). Then (a) If i is odd, there exists an element α in S of O-depth i such that α > β. (b) If i is even and there is no element α in S of O-depth i such that α > β, then there is element α in S of O-depth i− 1 such that ph(α) > β. Proof: Choose a v-chain C in S having tail β and the good property of Propo- sition 6.3.3. Apply Lemma 6.3.1 (5). � Corollary 6.3.5 Let C be a v-chain in ON with tail α such that O-depthC(α) is odd. Let A be a v-chain in ON with head α, and D the concatenation of C with A. Let C′ denote the v-chain C \ {α}. Then 1. The type of an element of A is the same in both A and D. In particular, SA ⊆ SD and qA,β = qD,β for β in A. 2. The type of an element of C′ is the same in both C′ and D. In particular, SC′ ⊆ SD. 3. SD = SC′ ∪ SA (disjoint union); letting j0 := O-depthC(α) we have j0 = SA and (SD)1 ∪ · · · ∪ (SD)j0−1 = SC′ . (For a monomial S, the subset of elements of depth at least i is denoted Si, and the subset of elements of depth exactly i is denoted Si.) Proof: (1) Generally (meaning without the assumption that O-depthC(α) is odd), the only element of A that could possibly have a different type in D is the last one in the first connected component of A; whether or not it changes type depends exactly upon whether or not the parity of the cardinality of its connected component in D is different from that in A. Under our hypothesis, this parity does not change, for, by (4) of Lemma 6.3.1, the type of α in C is H or S, and so the cardinality of the connected component of α in C is odd. (2) Generally (meaning without the assumption that O-depthC(α) is odd), the only element of C′ that could possibly have a different type in D is the last one of C′; it changes type if and only if it is connected to α and the cardinality of its connected component in C′ is odd. Under our hypothesis, this cardinality is even, for the same reason as in (1). (3) That SD = SC′ ∪ SA (disjoint union) is an immediate consequence of (1) and (2). By Lemma 6.3.1 (1), qA,α = qD,α dominates every element of SA, so SA ⊆ (SD) j0 (depth qD,α = O-depthD(α) = O-depthC(α) = j0). It is enough to prove the following claim: every element of SC′ has depth less than j0 in SD. Let γ ′ be an element of SC′ . If γ ′ > qD,α then the claim is clear. If not, then, by Proposition 5.3.4 (1), γ′ = ph(γ). By Lemma 6.3.1 (3), O-depthD(γ) is odd. Since the claim is already true for qD,γ = pv(γ), we have O-depthD(γ) = depthSDpv(γ) ≤ j0 − 2. By (6) of the same lemma, depth γ′ = O-depthD(γ)+1, so depthDγ ′ ≤ j0−1, and the claim is proved.� Proposition 6.3.6 Let S be a monomial in ON and j an odd integer. For β in Sj,j+1(:= {α ∈ S |O-depth (α) ≥ j}), we have O-depth Sj,j+1 (β) = O-depth (β)− j + 1 Proof: Proceed by induction on j. For j = 1, the assertion reduces to a tautology. Suppose that the assertion has been proved upto j. By the induction hypothesis, we have Sj+2,j+3 = (Sj,j+1)3,4, and we are reduced to proving the assertion for j = 3. Let A be a v-chain in S3,4 with tail β and O-depthA(β) = O-depthS3,4(β). Let α be the head of A. We may assume that O-depth (α) = 3 for, if O-depth (α) > 3, we can find, by Lemma 6.3.1 (5), α′ of O-depth 3 in S with α′ > α, and extending A by α′ will not decrease the O-depth in A of β (Proposition 6.1.1 (1)). Let E be a v-chain in SA with tail qA,β and length O-depthA(β). The head of E is then qA,α (see Proposition 5.3.4 (1)). Choose C in S with tail α such that O-depthC(α) = 3. Let D be the concatenation of C with A. By Corollary 6.3.5, E is contained in SD, qD,α = qA,α, and qD,β = qA,β. By Proposition 6.1.1 (2), the O-depth of α is the same in D as in C. Choose a v-chain F in SD with tail qD,α = qA,α. Concatenating F with E we get a v-chain inSD with tail qD,β = qA,β of lengthO-depthS3,4(β)+2. This proves that O-depth (β) ≥ O-depth (β) + 2. To prove the reverse inequality, we need only turn the above proof on its head. Let D be a v-chain in S with tail β such that O-depth (β) = O-depthD(β). Let G be a v-chain in SD with tail qD,β and length O-depthS(β). There exists an element α in D of O-depth 3 in D (by Lemma 6.3.1 (5)). Let C be the part of D upto and including α, and A the part α > . . . > β. By Proposition 6.1.1 (2), O-depthC(α) = 3 and, as above, Corollary 6.3.5 applies. By Corollary 6.1.4, qA,α = qD,α occurs in G. The part F of G upto and in- cluding qA,α is of length at most 3, and the part E : qD,α > . . . > qD,β belongs also to SA (Proposition 5.3.4 (2)). Thus the length of G is at most 2 more than the the length of E which is at most O-depth (β). � Corollary 6.3.7 For odd integers i, j, we have (Si,i+1)j,j+1 = Si+j−1,i+j . � Corollary 6.3.8 Let E : α > . . . > ζ be a v-chain, D and D′ two v-chains with tail α, and C, C′ the concatenations of D, D′ respectively with E. Then 1. O-depthC(ζ)−O-depthC(α) ≤ O-depthC′(ζ) −O-depthC′(α) + 1; 2. equality holds if and only if the type of λ is H in C and V in C′, and ph(λ) > µ, where λ is the last element in the connected component con- taining α of E and µ is the immediate successor in E of λ. Proof: These assertions follow from combining (2) of Proposition 5.3.3 with (1) of Lemma 6.3.1. � Corollary 6.3.9 Let ζ be an element of a monomial S in ON. Let C be a v-chain in S with tail ζ such that O-depthC(ζ) = O-depthS(ζ). Then 1. O-depthC(α) ≥ O-depthS(α) − 1 for any α in C. 2. If O-depthC(α) = O-depthS(α) − 1 for some α in C, then (a) letting λ be the last element in the connected component containing α and µ the element next to λ, the type of λ in C is H and ph(λ) > µ. (b) O-depthC(γ) = O-depthS(γ) − 1 for all γ in C between α and λ (both inclusive). Proof: (1) Let α be in C. Let E denote the part of C beyond (and including) α. Let D′ be a v-chain in S with tail α such that O-depthD′(α) = O-depthS(α). Let C′ be the concatenation of D′ and E. Applying Proposition 6.3.8 (1), we O-depthC(α) ≥ O-depthC(ζ) −O-depthC′(ζ) +O-depthC′(α)− 1. But O-depthC(ζ)−O-depthC′(ζ) = O-depthS(ζ)−O-depthC′(ζ) ≥ 0, and, by the choice of D′ and Proposition 6.1.1 (2), O-depthC′(α) = O-depthD′(α) = O-depth (2) Assertions (a) and (b) follow respectively from the “only if ” and “if” parts of item (2) of Proposition 6.3.8. � 7 The map Oπ The purpose of this section is to describe the map Oπ. The description is given in §7.1. It relies on certain claims which are proved in §§7.3, 7.4. Those proofs in turn refer to results from §9, but there is no circularity—to postpone the definition of Oπ until all the results needed for it have been proved would hurt rather than help readability. The observations in §7.5 are required only in §10. The symbol j will be reserved for an odd positive integer throughout this section. 7.1 Description of Oπ The map Oπ takes as input a monomial S in ON and produces as output a pair (w,S′), where w is an element of I(d) such that w ≥ v and S′ is a “smaller” monomial, possibly empty, in ON. If the input S is empty, no output is produced (by definition). So now suppose that S is non-empty. We first partition S into subsets according to the O-depths of its elements. Let S be the sub-monomial of S consisting of those elements of S that have O-depth k—the superscript “pr” is short for “preliminary”. It follows from Corollary 6.1.3 (4) that there are no comparable elements in S and so we can arrange the elements of S in ascending order of both row and column indices. Let σk be the last element of S in this arrangement. Let now j be an odd integer. We set j,j+1 := S We say that S is truly orthogonal at j if ph(σj) belongs to N (that is, if r > r where σj = (r, c)), Let Sj,j+1 denote the monomial in N defined by Sj,j+1 := j,j+1 \ {σj} j,j+1 \ {σj} ∪ {pv(σj), ph(σj)} if S is truly orthogonal at j j,j+1 ∪ j,j+1 otherwise Here S j,j+1 \ {σj} and other terms on the right are to be understood as mul- tisets. As proved in Corollary 7.3.4 (1) below, Sj,j+1 has depth at most 2. Let Sj (respectively Sj+1 be the subset (as a multiset) of elements of depth 1 (respectively 2) of Sj,j+1. Now, for every integer k, we apply the map of π of [7, §4] to Sk to obtain a pair (w(k),S′k), where w(k) is an element of I(d, 2d) andS k is a monomial in N. Let Sw(k) be the distinguished monomial in N associated to w(k)—see §5.1.2. Proposition 7.1.1 1. Sw(k) and S k are symmetric. And therefore so are ∪kSw(k) and ∪kS 2. ∪kSw(k) is a distinguished subset of N (in particular, the Sw(k) are dis- joint). 3. For j an odd integer, either • both Sw(j) and Sw(j+1) meet the diagonal, or • neither of them meets the diagonal, precisely as whether or not S is truly orthogonal at j. And therefore ∪kSw(k) has evenly many diagonal elements. 4. No S′k intersects the diagonal. And therefore neither does ∪kS The proposition will be proved below in §7.4. Finally we are ready to define the image (w,S′) of S under Oπ. We let w be the element of I(d, 2d) associated to the distinguished subset ∪kSw(k) of N; since ∪kSw(k) is symmetric and has evenly many diagonal elements, it follows from Proposition 5.2.1 that w is in fact an element of I(d). And we take S′ := k ∩ON. Remark 7.1.2 Setting π(Sj,j+1) := (wj,j+1,S j,j+1), S ′ := ∪j oddS j,j+1 ∩ON, and defining w to be the element of I(d, 2d) associated to ∪j oddSwj,j+1 would give an equivalent definition of Oπ. 7.2 Illustration by an example We illustrate the map Oπ by means of an example. Let d = 15, and v = (1, 2, 3, 4, 9, 10, 14, 16, 18, 19, 20, 23, 24, 25, 26). A monomial S in ON is shown in Figure 7.2.1. Solid black dots indicate the elements that occur in S with non- zero multiplicity. Integers written near the solid dots indicate multiplicities. The O-depth of S is 5. The element (21, 9) has O-depth 3 although it has depth 2 in S. Figure 7.2.2 shows the monomials S 1,2, S 3,4, and S 5,6. Solid dots, open dots, and crosses indicate elements of these monomials respectively. The monomial S is truly orthogonal at 1 and 3 but not at 5: σ1 = (28, 2), σ3 = (21, 9), and σ5 = (15, 14). Figure 7.2.3 shows the monomials S1,2, S3,4, and S5,6 of N and also their decomposition into blocks, and Figure 7.2.4 the monomialsS′1,2, S 3,4, and S We have Sw = {(15, 14), (17, 16), (21, 10), (7, 4), (27, 24), (28, 3), (30, 1), (29, 2)} hence w = (7, 9, 15, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30). It is easy to check that w ∈ I(d). The monomial S′ is the intersection with ON of the union of S′1,2, S 3,4, and S 5,6—in other words it is just the monomial lying above d in Figure 7.2.4. diagonal 1 2 3 4 9 10 14 16 18 19 20 23 24 25 26 2 1 3 4 3 2 6 2 2 3 2 3 1 2 3 Figure 7.2.1: The monomial S diagonal 1 2 3 4 9 10 14 16 18 19 20 23 24 25 26 2 1 3 4 3 2 6 2 2 1 3 2 3 1 2 3 Figure 7.2.2: S 1,2, S 3,4, and S diagonal 1 2 3 4 9 10 14 16 18 19 20 23 24 25 26 2 1 3 4 3 2 6 2 2 1 3 2 3 1 2 2 4 3 1 4 2 1 3 2 1 2 1 3 2 1 2 4 6 5 3 1 2 1 4 Figure 7.2.3: S1,2, S3,4, and S5,6 diagonal 1 2 3 4 9 10 14 16 18 19 20 23 24 25 26 2 1 4 3 6 5 3 2 1 3 2 1 1 4 3 1 4 2 1 3 2 Figure 7.2.4: S′1,2, S 3,4, and S 7.3 A proposition about Sj,j+1 The aim of this subsection is to show that Sj,j+1 has depth no more than 2— see item (1b) of Proposition 7.3.3. This basic fact was mentioned above in the description of Oπ and is necessary (psychologically although not logically) to make sense of the definitions of Sj and Sj+1. We prepare the way for Proposition 7.3.3 by way of two preliminary propositions. The first of these is about elements of O-depth j and j + 1 in S, the second about the relation of these elements with σj . Proposition 7.3.1 1. S has no comparable elements. 2. For j an odd integer and β an element of S j+1, there exists α in S j such that α > β. In particular, the row index of σj+1 (if σj+1 exists) is less than the row index of σj. Proof: (1) follows from Corollary 6.1.3 (4); (2) follows from Proposition 6.3.3 and Lemma 6.3.1 (5). � Proposition 7.3.2 Let j be an odd integer and let S be truly orthogonal at j. 1. pv(σj) > ph(σj); if α > pv(σj), then α > σj; if α > σj , then α > ph(σj). 2. No element of S j is comparable to pv(σj) or ph(σj). 3. No element of S j+1 is comparable to ph(σj). 4. The following is not possible: α ∈ S j , β ∈ S j+1, and ph(α) > β. Proof: (1) is trivial. (2) follows immediately from the definition of σj . We now prove (3). First suppose β > ph(σj) for some β in S j+1. By (2) of Proposition 7.3.1, there exists α in S j such that α > β. But then the row index of α exceeds that of σj , a contradiction to the choice of σj . We claim that it is not possible for β ∈ S j+1 to satisfy ph(σj) > β. This being a special case of (4), we need only prove that statement. So suppose that α belongs to S j and that ph(α) > β. Let C be a v-chain in S with tail α such that O-depthC(α) = j (see Proposition 6.1.3 (1)). Concatenate C with α > β and call the resulting v-chain D. Then, by Lemma 6.3.1 (4), α is of type H in D, so that, by Lemma 6.3.1 (1), we have O-depthD(β) = O-depthD(α) + 2. But, by Proposition 6.1.1 (2), O-depthD(α) = O-depthC(α) = j, so that O-depth (β) ≥ j + 2, a contradiction. � Let Sj,j+1(ext) denote the set—not multiset—defined by: Sj,j+1(ext) := Sj,j+1 ∪ {σj , σ j } if S is truly orthogonal at j Sj,j+1 otherwise Here Sj,j+1 on the right stands for the underlying set of the multiset Sj,j+1 defined above. The set Sj,j+1(ext) is the disjoint union of the sets Sj(ext) and Sj+1(ext) defined as follows (here again the terms on the right hand side denote the underlying sets of the corresponding multisets): Sj(ext) := ∪ {pv(σ)} if S is truly orthogonal at j otherwise Sj+1(ext) := j+1 ∪ ∪ {ph(σ)} if S is truly orthogonal at j j+1 ∪ otherwise Proposition 7.3.3 1. Sj(ext) (respectively Sj+1(ext)) is precisely the set of elements of depth 1 (respectively 2) in Sj,j+1(ext). In particular, (a) Neither Sj(ext) nor Sj+1(ext) contains comparable elements. (b) The length of a v-chain in Sj,j+1(ext) is at most 2. (c) There is a v-chain of length 2 in Sj,j+1 unless Sj+1(ext) is empty. 2. Let k be a positive integer, not necessarily odd. If there is in S an element of O-depth at least k, then Sk(ext) is non-empty. The converse also holds except possibly if k is even and S is truly orthogonal at k−1. In particular, if Sk(ext) is non-empty, then there is an element of O-depth at least k−1. Proof: (1): It is enough to show that every element of Sj(ext)(up) (respec- tively Sj+1(ext)(up)) is of depth 1 (respectively 2) in Sj,j+1(ext)(up), for • α > β implies α(up) > β(up) for elements α, β of N. • Sj,j+1(ext) = Sj(ext) ∪Sj+1(ext). • Sj,j+1(ext), Sj(ext), and Sj+1(ext) are symmetric. In turn, it is enough to show the following: (i) Every element of Sj(ext)(up) has depth 1. (ii) Sj+1(ext)(up) has no comparable elements. (iii) Every element of Sj+1(ext)(up) has depth at least 2. Item (i) follows from Proposition 7.3.1 and Proposition 7.3.2 (2); item (ii) from Proposition 7.3.1 (1) and Proposition 7.3.2 (3); item (iii) from Propo- sition 7.3.1 (2) and Proposition 7.3.2 (1). (2): The first assertion follows from Lemma 6.3.1 (5): if k is odd there is an element ofO-depth k inS; if k is even and there is no element ofO-depth k inS, then there is in S an element of O-depth k− 1 and of type H, so S is truly or- thogonal at k−1. The second assertion is clear from the definition of Sk(ext). � Corollary 7.3.4 1. No element of Sj,j+1 has depth more than 2. 2. Sj+1(ext) = Sj+1 and Sj(ext) ∩ Sj,j+1 = Sj (as sets). In particular, Sj+1 = Sj,j+1∩Sj+1(ext) and Sj = Sj,j+1∩Sj(ext) as multisets defined by the intersection of a multiset with a subset. Proof: (1): Since Sj,j+1 ⊆ Sj,j+1(ext) (as sets), this follows immediately from (1b) of the proposition above. (2): Since the union of Sj+1(ext) (which always is contained in Sj,j+1) and Sj(ext) ∩Sj,j+1 is all of Sj,j+1, and since Sj , Sj+1 are disjoint, it it enough to show that Sj+1(ext) ⊆ Sj+1 and Sj(ext) ∩Sj,j+1 ⊆ Sj . Now, since elements of Sj(ext) have depth 1 even in Sj,j+1(ext) (by item (1) of the proposition above), it is immediate that Sj(ext) ∩ Sj,j+1 ⊆ Sj . And it follows from the proof of item (iii) in the proof of item (1) of the proposi- tion above that an element of Sj+1(ext) has depth 2 even in Sj,j+1 (not just in Sj,j+1(ext)), so that Sj+1(ext) ⊆ Sj+1. � 7.4 Proof of Proposition 7.1.1 (1) The monomials Sj,j+1 are clearly symmetric. Observe that α in Sj,j+1 has the same depth as α#, for α1 > α2 implies α(up) > α2(up) and α(down) > α2(down) for α1, α2 in N. Thus the monomials Sk are symmetric. Since the map π of [7] respects #—see Proposition 5.7 of [4]—it follows that Sw(k) and S′k are symmetric. Therefore so are ∪kSw(k) and ∪kS (2) This follows from Corollary 9.3.6. (3) IfS is truly orthogonal at j, then pv(σj) and ph(σj) are diagonal elements respectively in Sj and Sj+1—see Corollary 7.3.4 (2). Thus both Sj and Sj+1 have diagonal blocks in the sense of Proposition 5.10 (A) of [4]. It follows from the result just quoted that both Sw(j) and Sw(j+1) meet the diagonal. It is of course clear that each Sw(k) meets the diagonal at most once since diagonal elements are clearly comparable but elements of Sw(k) are not by Lemma 4.9 of [7]. Suppose that S is not truly orthogonal at j. Then σj and σ j belong to different blocks—this is equivalent to the definition of S being not truly orthog- onal at j. By Proposition 7.3.1 (2), it follows that σj+1 and σ j+1 also belong to different blocks. So neither Sj nor Sj+1 has a diagonal block. (4) If S is not truly orthogonal at j, then neither Sj nor Sj+1 has a diagonal block (as has just been said above), and it follows from Proposition 5.10 (A) of [4] that neither S′j nor S j+1 meets the diagonal. So suppose that S is truly orthogonal at j. Then both Sj and Sj+1 have a diagonal entry each of multiplicity 1, namely pv(σj) and ph(σj) respectively. It is clear from the definition of σj that no element of Sj(up) shares its row index with pv(σj). And it follows from Proposition 7.3.1 (2) that no element of Sj+1(up) shares its row index with ph(σj). It now follows from the proof of Proposition 5.10 (B) of [4]—see the last line of that proof—that neither S′j nor S′j+1 meets the diagonal. � 7.5 More observations Proposition 7.5.1 The length of any v-chain in Sj,j+1∪S j+1 is at most 2. Proof: By Corollary 7.3.4 (1), the length of any v-chain in Sj,j+1 is at most 2. Applying Lemma 9.1.1 to Sj,j+1, we get the desired result. � Proposition 7.5.2 1. For an element α′ = (r, c) of S′k(up), there exists an element α = (r, C) of S with C ≤ c. 2. For an element α′ = (r, c) of S′j+1(up), there exists an element α = (R, c) of Sj+1(up) with r ≤ R. 3. For an element α′ of S′j+1(up), there exists an element α of S j with α > α′. Proof: (1) That there exists α inSk(up) with C ≤ c follows from the definition of S′k(up). Clearly such an α cannot be on the diagonal, so α belongs to S (2) As in the proof of (1), it follows from the definition of S′j+1 that there exists α = (R, c) in Sj+1 with r ≤ R. If α lies strictly below the diagonal, then c > R∗, so that α∗ = (c∗, R∗) > α′ = (r, c), a contradiction to Lemma 9.1.1 (α∗ belongs to Sj+1 by the symmetry of Sj+1). Thus α belongs to Sj+1(up). (3) Writing α′ = (r, c), by (1), we can find an β = (r, C) in S j+1 with C ≤ c. By Proposition 7.3.1 (2), there exists α in S j,j+1 such that α > β. � Corollary 7.5.3 If in S′j(up)∪S j+1(up) there exists an element with horizon- tal projection in N, then S is truly orthogonal at j. Proof: Follows directly from Proposition 7.5.2 (1) and (3). � Proposition 7.5.4 The O-depth of an element in S j ∪ S j+1 is at most 2. More strongly, the O-depth of an element in S j,j+1 ∪S j(up) ∪S j+1(up) is at most 2. Proof: It is enough to show that no element in S′j(up)∪S j+1(up) hasO-depth more than 2, for we may assume by increasing multiplicities that S j ⊆ S j(up) and S j+1 ⊆ S j+1(up) (as sets). It follows from Proposition 7.5.1 that a v-chain in S′j(up) ∪ S j+1(up) has length at most 2. Let α 1 = (r1, c1) > α 2 = (r2, c2) be such a v-chain. It follows from the proof of Corollary 4.14 (2) of [7] that α′1 ∈ S j(up) and α 2 ∈ S j+1(up). By item (1) of Lemma 6.3.1, it is enough to rule out the following possibility: α′1 is of type H in α 1 > α 2 and ph(α 1) > α Suppose that this is the case. By Proposition 7.5.2 (1) and (2), it follows that there exist elements α1 = (r1, C1) ∈ S j,j+1 and α2 = (R2, c2) ∈ Sj+1(up) with C1 ≤ c1 and r2 ≤ R2. Since ph(α 1) > α 2, it follows that α1 > α2. Now, if α2 = ph(σj), then Proposition 7.3.2 (2) is contradicted; if α2 belongs to S Proposition 7.3.2 (4) is contradicted (because ph(α1) > α2). � 8 The map Oφ The purpose of this section is to describe the map Oφ and prove some basic facts about it. Certain proofs here refer to results from §9, but there is no circularity—to postpone the definition of Oφ until all the results needed for it have been proved would hurt rather than help readability. As in §7, the symbol j will be reserved for an odd integer throughout this section. 8.1 Description of Oφ The map Oφ takes as input a pair (w,T), where T is a monomial, possibly empty, in ON and w ≥ v an element of I(d) that O-dominates T, and produces as output a monomial T∗ of ON. To describe Oφ, we first partition T into subsets Tw,j,j+1. As the subscript w in Tw,j,j+1 suggests, this partition depends on w. For an odd integer j, let Sjw (respectively Sw,j,j+1) denote the subset of Sw consisting of those elements that are j-deep (respectively that are j deep but not j + 2 deep, or equivalently of depth j or j + 1) in Sw in the sense of [7, §4]. Since Sw is distinguished, symmetric, and has evenly many elements on the diagonal d, it follows that Sjw and Sw,j,j+1 too have these properties, and that, in fact, the number of diagonal elements of Sw,j,j+1 is either 0 or 2 (in the latter case, the elements have to be distinct since Sw is distinguished and so is multiplicity free). Let us denote by wj and wj,j+1 the elements of I(d) corresponding to Sjw and Sw,j,j+1 by Proposition 5.2.1. Let Tw,j,j+1 denote the subset of T consisting of those elements α such that • every v-chain in T with head α is O-dominated by wj , and • there exists a v-chain in T with head α that is not O-dominated by wj+2. It is evident that the subsets Tw,j,j+1 are disjoint (as j varies over the odd integers) and that their union is all of T (for w = w1 O-dominates all v-chains in T by hypothesis and Sjw is empty for large j and so w j = v). In other words, the Tw,j,j+1 form a partition of T. Lemma 8.1.1 1. The length of a v-chain in Tw,j,j+1∪T w,j,j+1 is at most 2. In fact, the O-depth of any element in Tw,j,j+1 is at most 2. 2. wj,j+1 O-dominates Tw,j,j+1. Proof: The lemma follows rather easily from Corollary 9.2.3 as we now show. Let C be a v-chain in Tw,j,j+1. Let τ be the tail of C. Choose a v-chain D in T with head τ that is not O-dominated by wj+2. Let E be the concate- nation of C with D. Since the head of E belongs to Tw,j,j+1, it follows that E is O-dominated by wj . It follows from (the only if part of) Corollary 9.2.3 (applied with S = E and x = wj) that wj,j+1 O-dominates E 1 ∪ E 2 and wj+2 O-dominates E3,pr. This means τ 6∈ E3,pr, so τ ∈ E 1 ∪ E 2 , and so C ⊆ E 1 ∪ E 2 . This proves (2). By Proposition 6.1.1 (2), the O-depths of elements of C are the same in C and E, so C ⊆ C 1 ∪ C 2 , which proves the second assertion of (1). The first assertion of (1) follows from the second (see Lemma 6.2.1). � Corollary 8.1.2 wj,j+1 dominates Tw,j,j+1 ∪ T w,j,j+1 in the sense of [7]. Proof: This follows from (2) of Lemma 8.1.1 and Corollary 5.3.6 (the latter applied with S = Tw,j,j+1 and w = wj,j+1). � We may therefore apply the map φ of [7, §4] to the pair (wj,j+1,Tw,j,j+1 ∪ w,j,j+1) to obtain a monomial (Tw,j,j+1∪T w,j,j+1) ⋆ in N. In applying φ, there is the partitioning of Tw,j,j+1 ∪ T w,j,j+1 into “pieces”, these being indexed by elements of Swj,j+1 = Sw,j,j+1—observe that the elements of depth 1 (respec- tively 2) of Sw,j,j+1 are precisely those of Sw of depth j (respectively j + 1). We denote by Pβ the piece of Tw,j,j+1∪T w,j,j+1 corresponding to β in Swj,j+1 . We also use the notation P∗β as in [7]. Moreover, we will use the phrase piece of T (with respect to w being implicitly understood) to refer to a piece of Tw,j,j+1 ∪ T w,j,j+1 for some odd integer j. Caution: Thinking of T as a monomial in N and w as an element of I(d, 2d) that dominates it, there is, as in [7], the notion of “piece of T” (with respect to w). The two notions of “piece” are different. Lemma 8.1.3 1. The monomial (Tw,j,j+1∪T w,j,j+1) ⋆ is symmetric and has either none or two distinct diagonal elements depending exactly on whether Swj,j+1 = Sw,j,j+1 has 0 or 2 elements on the diagonal. 2. The depth of (Tw,j,j+1 ∪ T w,j,j+1) ⋆ is 2; and ∪β∈(Sw)jP β, ∪β∈(Sw)j+1P are respectively the elements of depth 1 and 2 in (Tw,j,j+1 ∪ T w,j,j+1) Proof: (1) The symmetry follows by combining Proposition 5.6 of [4], which says that the map π respects the involution #, with Proposition 4.2 of [7], which says that π and φ are are inverses of each other. The assertion about diagonal elements follows by combining item (B) of [4, Proposition 5.10], which is an assertion about the existence and relative mul- tiplicities of diagonal elements in B and B′ where B is a diagonal block of a monomial in N, and Proposition 4.2 of [7]. (2) It follows from Propositions 4.2 of [7] that the map π (described in §4 of that paper) applied to (Tw,j,j+1∪T w,j,j+1) ⋆ results in the pair (wj,j+1,Tw,j,j+1∪ Tw,j,j+1). It now follows from Lemma 4.16 of [7] that the depth of (Tw,j,j+1 ∪ w,j,j+1) ⋆ is exactly 2. The latter assertions again follow from the results of [7]— in fact, the proof that π ◦ φ is identity on pages 47–49 of [7] shows that the P∗β are the blocks in the sense of [7] of the monomial (Tw,j,j+1 ∪ T w,j,j+1) Suppose that (Tw,j,j+1 ∪ T w,j,j+1) ⋆ contains the pair (a, a∗), (b, b∗) of di- agonal elements with a > b. We call the pair (b, a∗), (a, b∗) the “twists,” and set δj := (b, a ∗). In other words, δj is the element of the twisted pair that lies above the diagonal—observe that the twisted elements are reflections of each other. We allow ourselves the following ways of expressing the condition that (Tw,j,j+1 ∪ T w,j,j+1) ⋆ has diagonal elements: δj exists; w is diagonal at j (the latter expression is justified by the lemma above). With notation as above, consider the new monomial defined as (Tw,j,j+1 ∪ T w,j,j+1) ⋆ if w is not diagonal at j( (Tw,j,j+1 ∪ T w,j,j+1) ⋆ \ d ∪ {δj, δ j } if w is diagonal at j This new monomial is symmetric and contains no diagonal elements. Its inter- section with ON is denoted T⋆w,j,j+1. In other words, T w,j,j+1 is the intersection of the new monomial with the subset of N of those elements that lie strictly above the diagonal. The union of T⋆w,j,j+1 over all odd integers j is defined to be T w, the result of Oφ applied to (w,T). This finishes the description of the map Oφ. For β in Sw,j,j+1(up), we define the “orthogonal piece-star” OP β corre- sponding to β as OP∗β := P∗β = P β(up) if β is not on the diagonal P∗β ∩ON if β ∈ (Sw)j+1 is on the diagonal {P∗β ∩ON} ∪ {δj} if β ∈ (Sw)j is on the diagonal (8.1.1) With this, we can say that T⋆w is the union of OP β as β varies over Sw(up). Lemma 8.1.4 Suppose that (Tw,j,j+1 ∪ T w,j,j+1) ⋆ contains the pair (a, a∗), (b, b∗) of diagonal elements with a > b. Let . . . , (r1, c1), (a, a ∗), (c∗1, r 1), . . . ; . . . , (r2, c2), (b, b ∗), (c∗2, r 2), . . . be respectively the elements of depth 1 and 2 of (Tw,j,j+1 ∪ T w,j,j+1) ⋆ arranged in increasing order of row and column indices. Then 1. c1 ≤ a ∗ and r1 ≤ b (assuming (r1, c1) exists); and 2. r2 < b and c2 ≤ b ∗ (assuming (r2, c2) exists). Proof: (1) Suppose that (r1, c1) exists. It is clear that c1 ≤ a ∗. From way the map φ of [7] is defined, it follows that (r1, a ∗) is an element of Tw,j,j+1. Suppose that r1 > b. Then ph(r1, a ∗) = (r1, r 1) belongs to N. We consider two cases. If (r2, c2) exists, then, again from the definition of the map φ, it follows that (r2, b ∗) is an element of Tw,j,j+1. But then ph(r1, a ∗) = (r1, r 1) > (b, b ∗) and (b, b∗) dominates (r2, b ∗), which means that the v-chain (r1, a ∗) > (r2, b ∗) (note that a∗ < b∗ because a > b by hypothesis) in Tw,j,j+1 has O-depth more than 2, a contradiction to Lemma 8.1.1 (1). Now suppose that (r2, c2) does not exist. (Then (b, b ∗) is the diagonal ele- ment in (Sw)j+1.) Consider the singleton v-chain C := {(r1, a ∗)} in Tw,j,j+1. Then SC = {(a, a ∗), (r1, r 1)} which is not dominated by wj,j+1, a contradiction to Lemma 8.1.1 (2). (2) Suppose that (r2, c2) exists. Then there exists, by the definition of the map φ, an element (r2, b ∗) in Tw,j,j+1. Since (r2, b ∗) lies above the diagonal, it follows that r2 < b. That c2 ≤ b ∗ is clear. � 8.2 Basic facts about Tw,j,j+1 and T w,j,j+1 Lemma 8.2.1 1. Let α′ > α be elements of T. Let j and j′ be the odd integers such that α′ ∈ Tw,j′,j′+1 and α ∈ Tw,j,j+1. Then j ′ ≤ j. 2. If, further, either (a) there exists µ in T such that α′ > µ > α, or (b) α′ ∈ Pβ′ for β ′ in (Sw)j′+1, then j′ < j. Proof: (1) By hypothesis, every v-chain with head α′ is O-dominated by wj This implies, by Corollary 6.1.2, that every v-chain with head α is O-dominated by wj . This shows j′ ≤ j. (2a) Suppose that j′ = j. It follows from (1) that α′, µ, and α all belong to Tw,j,j+1. But then α ′ > µ > α is a v-chain of length 3 in Tw,j,j+1, a contradiction to Lemma 8.1.1 (1). (2b) Suppose that j′ = j. Then α′ > α is a v-chain in Tw,j,j+1. Being of length 2, it cannot be dominated by (Sw)j+1, which means, by the definition of Pβ′ , that α ′ cannot belong to Pβ′ , a contradiction. � Proposition 8.2.2 1. The length of a v-chain in T⋆w,j,j+1 is at most 2. 2. The O-depth of T⋆w,j,j+1 is at most 2. 3. ∪β∈(Sw)j(up)OP β is precisely the set of depth 1 elements of T w,j,j+1 (in particular, no two elements there are comparable); if δj exists, then it is the last element of ∪β∈(Sw)j(up)OP β when the elements are arranged in increasing order of row and column indices. 4. ∪β∈(Sw)j+1(up)OP β is precisely the set of depth 2 elements of T w,j,j+1 (in particular, no two elements there are comparable); if δj exists, then its row index exceeds the row index of any element in ∪β∈(Sw)j+1(up)OP Proof: For (1), it is enough, given Lemma 8.1.3 (2), to show that δj is not comparable to any element of depth 1 of (Tw,j,j+1 ∪T w,j,j+1) ⋆, and this follows from Lemma 8.1.4 (1). In fact, the above argument proves also (3). For (4), it is enough, given Lemma 8.1.3 (2), the symmetry of the monomi- als involved in that lemma, and the observation that α > β implies α(up) > β(up) for elements α, β of N, to show the following: if (a, a∗) > γ = (e, f) for γ an element of (Tw,j,j+1 ∪ T w,j,j+1) ⋆ lying (strictly) above the diagonal, then δj > γ. But this follows from Lemma 8.1.4 (2): γ is a depth 2 ele- ment in (Tw,j,j+1 ∪ T w,j,j+1) ⋆, and we have e ≤ r2 < b (and a ∗ < f since (a, a∗) > γ). In fact, the above argument proves also (2): observe that f ≤ b∗ (Lemma 8.1.4 (2)). � 9 Some Lemmas The main combinatorial results of this paper are Propositions 4.1.1 and 4.1.2. They are analogues respectively of Propositions 4.1 and 4.2 of [7]. We have tried to preserve the structure of the proofs in [7] of those propositions. The proofs in [7] rely on certain lemmas and it is natural therefore to first establish the orthogonal analogues of those. The purpose of this section is precisely that. Needless to say that the lemmas (especially those in §9.4) may be unintelligible until one tries to read §10. The division of this section into four subsections is also suggested by the structure of the proofs in [7]. Each subsection has at its beginning a brief description of its contents. 9.1 Lemmas from the Grassmannian case In this subsection, the terminology and notation of [7, §4] are in force. The state- ments here could have been made in [7, §4] and would perhaps have improved the efficiency of the proofs there, but do not appear there explicitly. Let S be a monomial in N. Recall from [7] the notion of depth of an ele- ment α in S: it is the largest possible length of a v-chain in S with tail α and denoted depth α. The depth of S is the maximum of the depths in it of all its elements. We denote by Sk the set of elements of depth k of S (as in [7]) and by Sk the set of elements of depth at least k of S. Caution: For a monomial S of ON, we have introduced in §7.1 the notation Sk. That is different from the Sk we have just defined. Lemma 9.1.1 Let S be a monomial in N, and let π(S) = (w,S′), where π is the map defined in [7, §4]. Then the maximum length of a v-chain in S∪S′ is the same as the maximum length of a v-chain in S. Proof: We use the notation of [7, §4] freely. Let d be the maximum length of a v-chain in S. Suppose α1 > . . . > αℓ is a v-chain in S ∪S ′. Let i1, . . . , iℓ be such that αj belongs to Sij ∪ S (the integers ij are uniquely determined— see Corollary 5.4 of [4]). We claim that i1 < . . . < iℓ. This suffices to prove the lemma, for Sk ∪S k is empty for k > d. To prove the claim, it is enough to show i1 < i2. It follows from Lemma 4.10 of [7] that i1 6= i2. We now assume that i1 > i2 and arrive at a contradiction. First suppose that α1 ∈ Si1 . Then, by the definition ofSi1 , there exists β inSi2 with β > α1. Now β > α2 and both β, α2 belong to Si2 ∪S , a contradiction to [7, Lemma 4.10]. If α1 = (r, c) belongs to S , then, by the definition of S′i1 , there exists (r, a) in Si1 with a ≤ c, and there exists β in Si2 with β > (r, a). This leads to the same contradiction as before. � Lemma 9.1.2 Let B and U be monomials in N. Assume that • the elements of B form a single block (in the sense of [7, Page 38]). • U has depth 1 (equivalently, there are no comparable elements in U). • for every β = (r, c) in B, there exist γ1(β) = (R1, C1), and γ2(β) = (R2, C2) in U such that C1 < c, C2 < R1, r < R2 (this holds, for example, when there exists γ(β) in U such that γ(β) > β: take γ1(β) = γ2(β) = γ(β)). Then there exists a unique block C of U such that w(C) > w(B). Proof: It is useful to isolate the following observation: Lemma 9.1.3 Let (r1, c1) and (r2, c2) be elements of N with c2 < r1 ≤ r2. Let γ11 = (R 1 ), γ 1 = (R 1 ) and γ 2 = (R 2 ), γ 2 = (R 2 ) be elements of N such that 1. C11 ≤ c1, C 1 < R 1, r1 ≤ R 2. C12 ≤ c2, C 2 < R 2, r2 ≤ R 3. No two of γ11 , γ 1 , γ 2 , γ 2 are comparable (they could well be equal and this is important for us—see our definition of comparability). Then the monomial {γ11 , γ 1 , γ 2 , γ 2} consists of a single block. Proof: It follows from assumption (1) that γ11 and γ 1 belong to a single block: • if R11 < R 1, then C 1 < R 1 becomes relevant; • if R21 < R 1, then the other two inequalities in (1) become relevant: C11 ≤ c1 < r1 ≤ R Similarly it follows from assumption (2) that γ12 and γ 2 belong to a single block. We therefore need only consider the cases when, in the arrangement of the elements {γ11 , γ 1 , γ 2 , γ 2} in increasing order of row indices, both γ 1 , γ 1 come before or after γ12 , γ 2 . In the former case, the first sequence of inequalities below shows that γ21 and γ 2 belong to the same block, and we are done; in the latter case, the second sequence of inequalities below shows that γ22 and γ belong to the same block, and we are done: • C12 ≤ c2 < r1 ≤ R • C11 ≤ c1 < r1 ≤ r2 ≤ R Continuing with the proof of Lemma 9.1.2, we first prove the existence part. Arrange the elements of B in non-decreasing order of row numbers as well as column numbers (this is possible since there are no comparable elements in B). If β1 = (r1, c1) and β2 = (r2, c2) are successive elements, then c2 < r1 ≤ r2 (since B is a single block). Apply Lemma 9.1.3 with γ11 = γ 1(β1), γ 1 = γ 2(β1), and γ12 = γ 1(β2), γ 2 = γ 1(β2). We conclude that {γ 1 , γ 1 , γ 2 , γ 2} belongs to a single block, say C, of U. Continuing thus, we conclude that all γ1(β) and γ2(β), as β varies over B, belong to C. Since the row (respectively column) index of w(C) is the maximum (respectively minimum) of all row (respectively column) indices of elements of C (and similarly for B), it follows that w(C) > w(B). To prove uniqueness, let C1 and C2 be two blocks of U with w(C1) > w(B) and w(C2) > w(B). Apply the lemma with (r1, c1) = (r2, c2) = w(B) and γ11 = γ 1 = w(C1) and γ 2 = γ 2 = w(C2); it follows from [7, Lemma 4.9] that w(C1) and w(C2) are not comparable. But, unless C1 = C2, neither is the mono- mial {w(C1), w(C2)} a single block, again by [7, Lemma 4.9]. � Lemma 9.1.4 Let S be a monomial in N and x an element of I(d, n). For x to dominate S it is necessary and sufficient that for every α = (r, c) in S there exist β = (R,C) in Sx with C ≤ c, r ≤ R, and depthSxβ ≥ depthSα. (Here Sx denotes the distinguished monomial in N associated to x as in [7, Proposition 4.3].) Proof: The lemma is a corollary of [7, Lemma 4.5] as we now show. First suppose that x dominatesS. Let α = (r, c) be an element ofS, and C a v-chain in S with tail α and length depth α. Since x dominates C, there exists, by [7, Lemma 4.5], a chain in D in Sx of length depthSα and tail β = (R,C) with C ≤ c and r ≤ R, and we are done with the proof of the necessity. To prove the sufficiency, let C : α1 = (r1, c1) > . . . > αk = (rk, ck) be a v-chain in S. By hypothesis, there exist β1 = (R1, C1), . . . , βk = (Rk, Ck) in Sx with Ci ≤ ci, ri ≤ Ri, and depthSxβi = i for 1 ≤ i ≤ k (observe that replacing the ≥ in the latter condition of the statement by an equality yields an equivalent statement). We claim that β1 > . . . > βk. By [7, Lemma 4.5], it suffices to prove the claim. Since βk has depth k in Sx, there exists a β k−1 = (R k−1, C k−1) of depth k − 1 in Sx such that β k−1 > βk. It follows from the distinguishedness of Sx that that β′k−1 = βk−1: if not, then we have two distinct elements of the same depth (namely k−1) in Sx both dominating αk, a contradiction. So βk−1 > βk, and the claim is proved by continuing in a similar fashion. � Let x be an element of I(d, n). Let Sx denote the distinguished monomial in N associated to x as in [7, Proposition 4.3]. For k a positive integer, let xk denote the element of I(d, n) corresponding to the distinguished subset (Sx)k. For a monomial S ofN, let Sk,k+1 := Sk∪Sk+1. Let xk,k+1 denote the element of I(d, n) corresponding to the distinguished monomial (Sx)k,k+1; let x k denote the element of I(d, n) corresponding to the distinguished subset (Sx) Caution: For a monomial S of ON and an odd integer j, we have introduced in §7.1 the notation Sj,j+1. That is different from the Sk,k+1 just defined. Corollary 9.1.5 x dominates S ⇔ xk dominates Sk ∀ k ⇔ x1,2 dominates S1,2 and x 3 dominates S3. Proof: The first equivalence is a restatement of the lemma: in the statement of the lemma we could equally well have written depth β = depth α. The second follows from the first and the following observations: (S1,2)1 = S1, (S1,2)2 = S2, (S 3)k = Sk+2; and (x1,2)1 = x1, (x1,2)2 = x2, (x 3)k = xk+2. � 9.2 Orthogonal analogues of Lemmas of 9.1 Lemma 9.2.2 below is the orthogonal analogue of Lemma 9.1.4 (more precisely, that of the first assertion of Corollary 9.1.5). The following proposition will be used in its proof. Proposition 9.2.1 Let x be an element of I(d) and S a monomial in ON. Then x O-dominates S 1 ∪ S 2 if and only if it O-dominates every v-chain in S of O-depth at most 2. Proof: The “if” part is immediate from definitions (in any case, see also Proposition 7.5.4). For the “only if” part, let C be a v-chain in S of O-depth at most 2. Our goal is to show that x dominates SC . For this, it is enough, by Corollary 9.1.5, to show that x1 dominates (SC)1 and x2 dominates (SC)2 (by choice of C, (SC)k is empty for k ≥ 3). Let α′ ∈ (SC)1. Choose α in C such that α ′ ∈ SC,α. Choose α0 in S 1 such that α0 dominates α. Since x O-dominates the singleton v-chain {α0}, it follows that x1 dominates q{α0},α0 . We claim that q{α0},α0 dominates α ′. To prove the claim, we need only rule out the possibility that α0 is of type S in {α0} and α of type V in C. Since α′ ∈ (SC)1, it follows from Proposition 5.3.4 (1) that α is the first element of C. In particular, if α is of type V in C, then ph(α) ∈ N, so ph(α0) ∈ N, and α0 is of type H in {α0}. The claim is thus proved. Now consider an element of (SC)2. Observe that the length of C is at most 2 (Lemma 6.2.1). So our element is either the horizontal projection ph(α) of the head α of C, or it is qC,β where β is the tail of C. In the first case, let α0 be as in the previous paragraph, and proceed similarly. It is clear that ph(α0) ∈ N (because ph(α) ∈ N); x2 dominates ph(α0) and so also ph(α). Now we handle the second case. If β ∈ S 2 , then C is contained in S and there is nothing to prove. So assume that O-depth (β) ≥ 3. Choose a v-chain D in S with tail β, O-depthD(β) ≥ 3, and with the good property as in Proposition 6.3.3. There occurs in D an element of O-depth 3, say δ. (Lemma 6.3.1 (5)). Let A denote the part δ > . . . of D and C′ the part up to but not including δ. There clearly is an element—call it µ—of depth 2 in SD that dominates qD,β. This element µ belongs to SC′ (Corollary 6.3.5 (3)). Since D has the good property of Proposition 6.3.3, C′ ⊆ S 2 , so µ is dominated by an element in (Sx)2. In particular, qD,β is dominated by the same element of (Sx)2. We are still not done, for it is possible that qD,β be β and qC,β be pv(β). Suppose that this is the case. Then α > β is connected. So ph(α) ∈ N and the legs of α and β intertwine. As seen above in the third paragraph of the present proof, there is an element of (Sx)2 that dominates ph(α). By the distinguished- ness of Sx, it follows that the element in (Sx)2 dominating β is the same as the one dominating ph(α). By the symmetry of Sx, this element lies on the diagonal and so dominates pv(β), and, finally, we are done with the proof in the second case. � Lemma 9.2.2 Let S be a monomial in ON and x an element of I(d). For x to O-dominate S it is necessary and sufficient that, for every odd integer j, every v-chain in S j+1 is O-dominated by xj,j+1. Proof: First suppose that x dominates S. Let j be an odd integer and let A a v-chain in S j+1. We need to show that xj,j+1 dominates SA. For this, we may assume that A is maximal (by Corollary 6.1.2). By Corollary 6.1.3 (3), the length of A is at most 2. By Lemma 6.3.1 (5) (b), for every β in S j+1 there exists α in S j with α > β. Thus we may assume that the head α of A belongs It is enough to show (see [7, Lemma 4.5]) that for any v-chain E in SA • the length of E is at most 2; • there exists an x-dominated monomial in N containing E and the head of E is an element of depth at least j in that monomial. The first of these conditions holds by Proposition 7.5.4. We now show that the second holds. We may assume that E is maximal in SA. By Proposition 5.3.4 (1), the head of E is qA,α. Let C a v-chain in S with tail α such that O-depthC(α) = j. Let D be the concatenation of C with A. We claim that the monomial SD has the desired properties. That SD is x-dominated is clear (since x O-dominates S). By Corollary 6.3.5, it follows that qD,α = qA,α and SA ⊆ SD (in particular that E ⊆ SD). By Proposition 6.1.1 (2), O-depthD(α) = O-depthC(α) = j, that is, depth qD,α = j. The proof of the necessity is thus complete. To prove the sufficiency, proceed by induction on the largest odd integer J such that S J+1 is non-empty. When J = 1, there is nothing to prove, for S 1 ∪ S 2 = S and x1,2 O-dominates S 1 ∪ S 2 . So suppose that J ≥ 3. We implicitly use Corollary 6.3.7 in what follows. By induction, x3 O- dominates S3,4. Let D be a v-chain in S. Our goal is to show that x dominates SD. Let α be the element of D with O-depthD(α) = 3—such an element exists, by Lemma 6.3.1 (5) (if there exists in D an element of O-depth in D exceed- ing 2); the following proof works also in the case when α does not exist. Let A be the part α > . . . of D, and C′ the part up to but not including α. By Proposition 6.1.1 (2), the O-depth (in C′) of elements of C′ is at most 2. By Proposition 9.2.1, x1,2 dominates SC′ . By Corollary 6.3.5 (3), (SD)1,2 = SC′ and (SD) 3 = SA. Since A ⊆ S 3,4, it follows that x3 dominates SA (induction hypothesis). Finally, by an application of Corollary 9.1.5, we conclude that x dominates SD. � Corollary 9.2.3 Let S be a a monomial in ON and x an element of I(d). For x to O-dominate S it is necessary and sufficient that x1,2 O-dominate S and x3 O-dominate S3,4. Proof: It is easy to see that (x3)j,j+1 = xj+2,j+3; it follows from Proposi- tion 6.3.6 that (S3,4) j ∪ (S j+1 = S j+2 ∪S j+3. The assertion follows from the lemma. � 9.3 Orthogonal analogues of some lemmas in [7] The proofs of Propositions 4.1 and 4.2 of [7] are based on assertion 4.9–4.16 (of that paper). Assertion 4.9 being a statement about a single Sk, it is applicable in the present situation. Since references to it are frequent, we recall it below as Lemma 9.3.1. As to assertions 4.10–4.16 of [7], assertions 9.3.2, 9.3.4–9.3.9 below are their respective analogues. A block of a monomial S in ON means a block of Sj,j+1 in the sense of [7] for some odd integer j. Caution: Considering S as a monomial in N, there is the notion of a “block” of S as in [7], which has in fact been used in §9.1, and which is different from the notion just defined. Both notions are used and it will be clear from the context which is meant. Throughout this section S denotes a monomial in ON and j an integer (not necessarily odd). Lemma 9.3.1 If B1, . . . ,Bl are the blocks in order from left to right of some Sk, and w(B1) = (R1, C1), w(B2) = (R2, C2), . . ., w(Bl) = (Rl, Cl), then C1 < R1 < C2 < R2 < . . . < Rl−1 < Cl < Rl Proof: This is merely a recall Lemma 4.9 of [7]. In any case it follows easily from the definitions. � Lemma 9.3.2 No two elements of Sk(ext) ∪ S k are comparable. More pre- cisely, it is not possible to have elements α > β both belonging to Sk(ext)∪S Proof: It follows from Lemma 9.3.1 that Sk ∪ S k contains no comparable elements. If k is even, then Sk(ext) = Sk (Corollary 7.3.4 (2)); if k is odd, we may assumeSk(ext) = Sk (as sets) by increasing the multiplicity of σk in S Lemma 9.3.3 For integers i ≤ k, there cannot exist γ ∈ S′i(up) and β ∈ S such that β > γ. For integers i < k, there cannot exist γ ∈ S′i(up) and β ∈ S such that β dominates γ. Proof: Let γ ∈ S′i(up) and β ∈ S . If i = k and β > γ, then we get a contradiction immediately to Lemma 9.3.2. Now suppose that i < k and that β dominates γ. Apply Corollary 6.3.4 (the notation of the corollary being sug- gestive of how exactly to apply it). Let α be as in its conclusion. The chain α > γ contradicts Lemma 9.3.2 in case i is odd and either Lemma 9.3.2 or Proposition 7.5.4 in case i is even. � Lemma 9.3.4 For (r, c) in S′, there exists a unique block B of S with (r, c) in B′. Proof: The existence is clear from the definition of S′. For the uniqueness, suppose that B and C are two distinct blocks of S with (r, c) in both B′ and C′. We will show that this leads to a contradiction. Let i and k be such that B ⊆ Si and C ⊆ Sk. From Lemma 4.11 of [7] (of which the present lemma is the orthogonal analogue) it follows that i 6= k, so we can assume without loss of generality that i < k. By applying the involu- tion # if necessary, we may assume that (r, c) ∈ S′i(up). Now there exists an element (r, a) in C with a ≤ c (this follows from the definition of C′). Clearly (r, a) ∈ S . Taking β = (r, a) and γ = (r, c), we get a contradiction to Lemma 9.3.3. � Lemma 9.3.5 Let i < j be positive integers. 1. Given a block B of Sj, there exists a unique block C of Si such that w(C) > w(B). 2. Given an element β in Sj(ext)∪S j, there exists α in Si such that α > β. Proof: (1): The assertion follows by applying Lemma 9.1.2 with B = B and U = Si. We need to make sure however that the lemma can be applied. More precisely, we need to check that for every β = (r, c) in B there exist γ1(β) = (R1, C1) and γ2(β) = (R2, C2) in Si such that C 1 < c, C2 < R1, and r < R2. We may assume β = β(up), for, if β = β(down), then β(up) also belongs to Sj because Sj is symmetric, and we can set γ 1(β) = γ2(β(up))(down), and γ2(β) = γ1(β(up))(down)—note that these two belong to Si since Si is symmetric. We consider three cases: 1. β belongs to S. 2. β = ph(σj−1) (in particular, j is even and S is truly orthogonal at j − 1). 3. β = pv(σj) (in particular, j is odd and S is truly orthogonal at j). Define β′ to be β in case 1, σj−1 in case 2, and σj in case 3. Let C be a v-chain in S with tail β′ and having the good property as in Proposition 6.3.3. First suppose that there exists in C an element of O-depth i and denote it by γ. If ph(γ) 6∈ N (this can happen only in case 1), then set γ 1(β) = γ2(β) = γ. Now suppose ph(γ) ∈ N. Then γ ∈ Si except when γ = σi with i odd and σi has multiplicity 1 in S. If γ ∈ Si, take γ 1(β) = γ and γ2(β) = γ# = γ(down); if γ 6∈ Si, then take γ 1(β) = γ2(β) = pv(γ). Now suppose that C has no element ofO-depth i. Then, by Lemma 6.3.1 (5), i is even and there exists in C an element of O-depth i − 1. This element of C is of type H by Lemma 6.3.1 (1), so S is truly orthogonal at i − 1. Set γ1(β) = γ2(β) = ph(σi−1). (2): This proof parallels the proof of (1) above. As in the above proof, we may assume that β = β(up). Suppose β = (r, c) belongs to S′j. Then there exists (r, a) ∈ Sj with a ≤ c. Since S j does not meet the diagonal, it is clear that (r, a) ∈ ON, and thus it is enough to prove the assertion for β ∈ Sj(ext). So now take β ∈ Sj(ext). Let β ′ and C be in the proof of (1). First suppose that there exists in C an element of O-depth i. Denote it by γ. If γ ∈ Si, then take α = γ. If γ 6∈ Si, then pv(γ) ∈ Si, and we take α = pv(γ). In case there is no element in C of O-depth i, we take α = ph(σi−1) (see the above proof). � Corollary 9.3.6 If B and B1 are blocks of S with w(B) = (r, c) and w(B1) = (r1, c1), then exactly one of the following holds: c < r < c1 < r1, c1 < r1 < c < r, c < c1 < r1 < r, or c1 < c < r < r1. Proof: This is a formal consequence of Lemmas 9.3.1 and 9.3.5, just as Corol- lary 4.13 of [7] is of Lemmas 4.9 and 4.12 of that paper. � Corollary 9.3.7 If w(B) > w(C) for blocks B ⊆ Si and C ⊆ Sj of S, then i < j. Proof: This is a formal consequence of Lemmas 9.3.1 and 9.3.5. It follows from the first lemma that i 6= j. Suppose i > j. Then there exists by the second lemma a block C′ ⊆ Sj such that w(C ′) > w(B). But then w(C′) > w(C), a contradiction of the first lemma. � Corollary 9.3.8 Let (s, t) > (s1, t1) be elements of S ′, and B, B1 be blocks of S such that (s, t) ∈ B′, and (s1, t1) ∈ B 1. Then w(B) > w(B1). Proof: Let w(B) = (r, c) and w(B1) = (r1, c1). By Corollary 9.3.6, we have four possibilities. Since (r, c) dominates (s, t) and (r1, c1) dominates (s1, t1), the possibilities c < r < c1 < r1 and c1 < r1 < c < r are eliminated. It is thus enough to eliminate the possibility c1 < c < r < r1. Suppose that this is the case. Then, by Corollary 9.3.7, j1 < j, where j1 and j are such that B ⊆ Sj and B1 ⊆ Sj1 . Now, by Lemma 9.3.5 (2), there exists α in Sj1 such that α > (s, t) > (s1, t1). But then this contradicts Lemma 9.3.2. � Corollary 9.3.9 For a B ⊆ Si of S, the depth of w(B) in Sw is exactly i. Proof: That the depth is at least i follows from Lemma 9.3.5. That the depth cannot exceed i follows from Corollary 9.3.7. � Corollary 9.3.10 Let α ∈ S′k(up), β ∈ S m(up), and α > β. Then k < m. Proof: Corollary 9.3.8 and Corollary 9.3.9. � 9.4 More lemmas This subsection is a collection of lemmas to be invoked in the later subsec- tions. More specifically, Lemma 9.4.1 and Corollary 9.4.2 are invoked in the proof of Proposition 4.1.1 in §10.1, Lemma 9.4.3 in the proof of the first half of Proposition 4.1.2 in §10.2, and Lemma 9.4.4 in the proof of the second half of Proposition 4.1.2 in §10.3. Throughout this subsection, S denotes a monomial in ON. Lemma 9.4.1 Let C be a v-chain in S′, α an element of C, and α′ ∈ SC,α. Then depth α′ ≤ k(even), for k the integer such that α ∈ S′k(up). Proof: Proceed by induction on k. If k = 1, the assertion follows from Corol- lary 9.3.10, so assume k > 1. Choose a v-chain C′ in SC with tail α ′ and depthC′α ′ = depth (α′). The length of a v-chain in SC,α is clearly at most 2. So, if γ′ is the element two steps before α′ in C′ (if γ′ does not exist then there is clearly nothing to prove), then γ′ ∈ SC,γ with γ > α (see Proposition 5.3.4 (2)). We claim that depth (γ′) ≤ k(odd) − 1. It is enough to prove the claim, for then depth (α′) = depthC′α ′ = depthC′γ ′ + 2 ≤ k(odd)− 1 + 2 = k(even). The claim follows by induction from Corollary 9.3.10 if k is odd or more generally if γ ∈ S′l(up) with l ≤ k(odd) − 1. So assume that k is even and γ ∈ S′k−1(up). By 7.5.4, it is not possible that γ is of type H and ph(γ) > α. So the only possibility is that α′ = ph(α) and γ > α is connected. In particular, γ is of type V and α of type H in C and γ′ = pv(γ). Now let µ be the first element in the connected component of α in C. The cardinality of the part µ > . . . > γ of C is even (by Proposition 5.3.1 (1), it follows that the cardinality of µ > . . . > α is odd), say e. Letting m be such that µ ∈ S′m(up), we have, by Proposition 9.3.10, m ≤ k − 1 − (e − 1) = k − e. If m(even) < k − e, then, since depthC′γ ′ = depthC′pv(µ) + e − 1 (by Proposition 5.3.4 (1), since, by Proposition 5.3.1 (2), µ, . . . , γ all have type V in C) and depthC′pv(µ) ≤ m(even) by induction, it follows that depthC′γ k − e+ e− 1 = k − 1, and we are done. So suppose that m(even) = k − e. Let ν be the element just before µ in C (if such an element does not exist, then depthC′γ ′ = e ≤ k − 2—observe that m(even) ≥ 2—and we are done). Then ν > µ is not connected (by choice of µ). So ph(ν) > µ. By Proposition 7.5.4, this means that j ≤ m(even) − 2 where j is the odd integer defined by ν ∈ S′j(up) ∪ S j+1(up). So, again by induction, depthC′γ ′ = depthC′ph(ν) + e ≤ m(even) − 2 + e = k − 2, and the claim is proved. � Corollary 9.4.2 The O-depth of an element α in S′ is at most k where k is such that α ∈ S′k(up). Proof: Let C′ be a v-chain in SC with tail qC,α. If k is even, then, by the lemma, depthC′qC,α ≤ k. So suppose that k is odd. Let γ ′ be the immediate pre- decessor of qC,α in C ′. By Proposition 5.3.4 (2), γ > α, and so γ ∈ S′l(up) with l ≤ k− 1 (see the observation in the first paragraph of the proof of the lemma). So depthC′γ ′ ≤ k − 1 (by the lemma) and depthC′α ′ = depthC′γ ′ + 1 ≤ k. � Lemma 9.4.3 Let S be a monomial in ON and Oπ(S) = (w,S′). Let i < k be integers, α an element of S′i(up), and δ an element of (Sw)k(up) that dominates α. 1. If k is even, then there exists β ∈ S′k(up) with α > β. 2. If k is odd and wk,k+1 O-dominates the singleton v-chain α, then either there exists β ∈ S′k(up) with α > β or there exists γ ∈ S k+1(up) with ph(α) > γ. Proof: Write α = (r, c) and δ = (A,B). By Corollary 9.3.9, there exists a block B of Sk such that δ = w(B). Let (D,B) be the first element of B (arranged in increasing order of row and column indices). We have the following possibilities: (i) D ≤ A and (D,B) ∈ S (ii) k is odd, S is truly orthogonal at k, (D,B) = (A,B) = pv(σk), and B consists of the single diagonal element (D,B) = (B∗, B). (iii) k is even, S is truly orthogonal at k−1, (D,B) = (A,B) = ph(σk−1), and B consists of the single diagonal element (D,B) = (B∗, B). We claim the following: in case (i), D < r (in particular, D < A); in case (ii), the row index of σk is less than r; and case (iii) is not possible. The first two assertions and also the third in the case i < k − 1 follow readily from Lemma 9.3.3; in case (iii) holds and i = k − 1, then σk−1 > α, a contradiction to Lemma 9.3.2. First suppose that possibility (ii) holds. Write σk = (s,B). Since s < r and ph(σk) ∈ N, it is clear that ph(α) = (r, r ∗) also belongs to N. From the hypothesis that wk,k+1 O-dominates {α}, it follows that there is an element of (Sw)k+1 that dominates ph(α) = (r, r ∗). Such an element must be diagonal (because of the distinguishedness ofSw), and so must be the w(C) for the unique diagonal block C ofSk+1. In particular, this means that there are elements other than (s, s∗) in Sk+1, and so S k+1 is non-empty. In the arrangement of elements of S′k+1(up) in increasing order of row and column numbers, let γ = (e, s ∗) be the last element. Then e < s < r and r∗ < s∗, so ph(α) > γ, and we are done. Now suppose that possibility (i) holds. Let (p, q) be the element of Sk such that p is the largest row index that is less than r, and, among those elements with row index p, the maximum possible column index is q. The arrangement of elements of Sk (in increasing order or row and column indices) looks like this: . . . , (p, q), (s, t), . . . Since p < r ≤ A and w(B) = (A,B), we can be sure that (p, q) is not the last element of B. We first consider the case c < t. Then α = (r, c) > β := (p, t) ∈ S′k. If β ∈ S′k(up), then we are done. It is possible that (p, q) lies on or below the diagonal so that β lies below the diagonal, in which case, α > β(up) and β(up) ∈ S′k(up), and again we are done. Now suppose that t ≤ c. We claim that: • (s, t) belongs to the diagonal; • k is odd and S is truly orthogonal at k; and • σk = (u, t) with u < r. Suppose that (s, t) does not belong to the diagonal. Since r ≤ s (by choice of (p, q)), it follows that (s, t) dominates (r, c). This leads to a contradiction to Lemma 9.3.3, for either (s, t) or its reflection (t∗, s∗) (whichever is above the diagonal) belongs to S and dominates α = (r, c) in S′i(up). This shows that (s, t) belongs to the diagonal. If k is even, then (s, t) = ph(σk−1), which means σk−1 > α, again contradicting Lemma 9.3.3, so k must be odd. It also follows that S is truly orthogonal at k and that (s, t) = pv(σk). Writing σk = (u, t), if r ≤ u, then σk would dominate α, again contradicting Lemma 9.3.3. So u < r, and the claim is proved. To finish the proof of the lemma, now proceed as in the proof when possi- bility (ii) holds. � Lemma 9.4.4 Let T be a monomial in ON and w an element of I(d) that O- dominates T. Let β′ > β be elements Sw(up). Let d−1 and d be their respective depths in Sw. Let α be an element of OP β or more generally an element of ON such that (a) it is dominated by β, (b) it is not comparable to any element of Pβ, and (c) in case d is odd, then {α} ∪ Tw,d,d+1 has O-depth at most 2. 1. there exists α′ ∈ P∗β′(up) with α ′ > α; 2. for α′ as in (1), if α′ is diagonal, then ph(δd−2) > α if d is odd and δd−1 > α if d is even. Proof: Assertion (2) is rather easy to prove. If d is odd, then, in fact, ph(δd−2) = α ′; if d is even, then δd−1 has the same column index as α ′ and, by Proposition 8.2.2 (4), has row index more than that of α, so δd−1 > α. Let us prove (1). Write α = (r, c), β = (R,C), and β′ = (R′, C′). There exists, by the definition of P∗β′ , an element in P β′ with column index C ′. We have C′ < c (for C′ < C ≤ c). Let (r′, c′) be the element of P∗β′ such that c is maximum possible subject to c′ < c and among those elements with column index c′ the maximum possible row index is r′. If r < r′, then we are done (if (r′, c′) is below the diagonal, its mirror image would have the desired properties). It suffices therefore to suppose that r′ ≤ r and arrive at a contradiction. In the arrangement of elements of P∗β′ in non-decreasing order of row and column indices, there is a portion that looks like this: . . . , (r′, c′), (a, b), . . . Since there is in P∗β an element with row index R ′ (and clearly r′ ≤ r < R < R′), it follows that (a, b) exists (that is, (r′, c′) is not the last element in the above arrangement). It follows from the construction of P∗β′ from Pβ′ that (r ′, b) is an element in Pβ′ . By the choice of (r ′, c′), we have c ≤ b. Thus (r, c) dominates (r′, b). The proof now splits into two cases accordingly as d is even or odd. First suppose that d is even. Then, since β dominates (r′, b) and yet (r′, b) does not belong to Pβ, there exists a v-chain in Tw,d−1,d of length 2 and head (r ′, b). The tail of this v-chain then belongs toPβ and is dominated by (r, c), a contradiction to our assumption that α is not comparable to any element of Pβ. Now suppose that d is odd. Choose a v-chain C in T with head (r′, b) that is not O-dominated by wd. Let D be the part of C consisting of elements of O-depth (in C) at most 2. We claim that D is O-dominated by wd,d+1. In fact, we claim the following: Any v-chain F with head (r′, b) and O-depth at most 2 is O-dominated by wd,d+1. To prove the claim, we first prove the following subclaim: (†) If the horizontal projection of (r′, b) belongs to N, then β is on the diagonal and dominates the vertical projection of (r′, b), and the diagonal element β1 of (Sw)d+1 dominates the horizontal projection of (r′, b). Let ph(r ′, b) ∈ N. Then β belongs to the diagonal because Sw is distinguished and symmetric. Once β is on the diagonal, it is clear that it dominates pv(r ′, b) (from our assumptions, β dominates (r, c) and (r, c) dominates (r′, b)). It follows from Proposition 8.2.2 (3) that the row index of β1 exceeds the row index r of (r, c), so β1 dominates ph(r ′, b). This finishes the proof of the subclaim (†). To begin the proof of the claim, observe that F has length at most 2. Suppose first that F consists only of the single element (r′, b). The type of (r′, b) in F is either H or S. If it is S, then since β dominates (r′, b), the claim follows immediately. If it is H, then the claim follows immediately from the subclaim (†). Continuing with the proof of the claim, let now F consist of two elements: (r′, b) > µ. Let γ be the element of Sw such that µ ∈ Pγ , and let e be the depth of γ in Sw. From Lemma 8.2.1 (2b) it follows that e ≥ d. If e = d, then γ = β (by the distinguishedness of Sw), and the comparability of (r, c) and µ contradicts our hypothesis (b). So e ≥ d+1, and there exists δ of depth d+1 in Sw that dominates µ. We have β > δ (again by the distinguishedness of Sw). The possibilities for the types of (r′, b) and µ in F are: S and S, V and V, H and S (in the last case ph(r ′, b) 6> µ by Lemma 6.3.1 (1)). Noting the existence in (Sw)d,d+1 of the v-chain β > δ in the first case and also of β > β1 (where β1 is as in the subclaim) in the last case, the proof of the claim in these cases is over. So suppose that the second possibility holds. The distinguishedness of Sw implies that δ = β1. Since δ is diagonal, it dominates the vertical projection of µ. Noting the existence of the v-chain in β > δ in (Sw)d,d+1, the proof of the claim in this case too is over. We continue with the proof of the lemma. It follows from the claim that D is O-dominated by wd,d+1. From Corollary 9.2.3 it follows that the complement E of D in C is not O-dominated by wd+2,d+3 (in particular, that E is non-empty) and that every v-chain in T with head ǫ (where ǫ denotes the head of E) is O-dominated by wd (given such a v-chain, the concatenation of D with it is O-dominated by wd−2, and ǫ continues to have O-depth 3 in the concatenated v-chain). Thus ǫ belongs to Tw,d,d+1. From (1) and (2b) of Lemma 8.2.1 it follows that the element µ of C in between (r′, b) and ǫ (if it exists at all) also belongs to Tw,d,d+1. Now consider the v-chain obtained as follows: take the part of C up to (and including) ǫ and replace its head (r′, b) by (r, c). This chain has O-depth 3 and lives in {α} ∪ Tw,d,d+1, a contradiction to hypothesis (c). � Corollary 9.4.5 Let T be a monomial in ON and w an element of I(d) that O-dominates T. Let β′ > β be elements of Sw(up), α an element of OP β, and d′ := depth 1. If d′ is odd, there exists α′ ∈ OP∗β′ such that α ′ > α. 2. If there does not exist α′ ∈ OP∗β′ such that α ′ > α then (d′ is even by (1) above and) there exists α′′ ∈ OP∗β′′ such that ph(α ′′) > α, where β′′ is the unique element of (Sw)d′−1 such that β ′′ > β′. Proof: Immediate from the lemma. � Corollary 9.4.6 Let T be a monomial in ON and w an element of I(d) that O-dominates T. Let β, β′ be elements of Sw(up), and α, α ′ elements of OP∗β and OP∗β′ respectively. 1. If α′ > α then β′ > β (in particular, depth β′ < depth 2. If ph(α ′) > α and depth β is even, depth β′ ≤ depth β − 2. Proof: (1) Writing β = (r, c) and β′ = (r′, c′), there are, since both β and β′ dominate α and Sw is distinguished, the following four possibilities: c < r < c1 < r1, c1 < r1 < c < r, c < c1 < r1 < r, c1 < c < r < r1 Since α′ > α, and α, α′ are dominated respectively by β, β′ (this is because α, α′ belong to OP∗β, OP β′ respectively), the possibilities c < r < c1 < r1 and c1 < r1 < c < r are eliminated (by the distinguishedness of Sw). It is thus enough to eliminate the possibility β > β′. Suppose, by way of contradiction, that β > β′. By Corollary 9.4.5, either there exists γ ∈ OP∗β such that γ > α in which case the v-chain γ > α in OP∗β contradicts Proposition 8.2.2 (3) or (4), or d := depth β is even and there exists (with β′′ being the unique element in Sw such that β ′′ > β and depth β′′ = d− 1) an element α′′ ∈ OP∗β′′ with ′′) > α′, in which case the v-chain α′′ > α in T⋆w,d−1,d has O-depth 3 and so contradicts Proposition 8.2.2 (2). (2) Set d := depth β. If depth β′ were d− 1, then the v-chain α′ > α in T⋆w,d−1,d would be ofO-depth 3 and so would contradict Proposition 8.2.2 (2). � 10 The Proof The aim of this section is to prove Propositions 4.1.1 and 4.1.2. The proof of first proposition appears in §10.1 and that of the second in §§10.2, 10.3. In §9.4 some lemmas are established that are used in the proofs. Needless to say that the lemmas maybe unintelligible until one tries to read the proofs in the later subsections. 10.1 Proof of Proposition 4.1.1 (1) By definition, w is the element of I(d) associated to the distinguished mono- mial ∪kSw(k). By the very definition of this association, we have w ≥ v. (2) This follows from the corresponding property of the map π of [7]. More precisely, that property justifies the third equality below. The other equalities are clear from the definitions. v-degree(w) + degree(S′) = degree(Sw) + degree(S′k) degree(Sw(k)) + degree(S degree(Sk) j odd degree(Sj,j+1) j odd degree(S j ) + degree(S = degree(S) (3) We have: w O-dominates S′ ⇔ w ≥ wC ∀ v-chain C in S ⇔ w dominates SC ∀ v-chain C in S ⇔ ∀ v-chain C in S′, ∀ α′ = (r, c) ∈ SC , ∃ β = (R,C) ∈ Sw with C ≤ c, r ≤ R, and depth β ≥ depth The first equivalence above follows from the definition of O-domination, the second from [7, Lemma 4.5], the third from Lemma 9.1.4. Now let C be a v-chain in S′ and α′ = (r, c) in SC . We will show that there exists β in Sw that dominates α and satisfies depthSwβ ≥ depthSCα Let α be the element in C such that α′ ∈ SC,α, let k be such that α ∈ S k(up), and let B be the block of Sk such that α ∈ B ′. Writing α = (r1, c1) and w(B) = (R1, C1), we have C1 ≤ c1 and r1 ≤ R1 straight from the definition of w(B). By Corollary 9.3.9, depth w(B) = k. First suppose that w(B) dominates α′ (meaning C1 ≤ c and r ≤ R1). If k ≥ depth α′, we are clearly done; by Corollary 9.4.2, this is the case when α′ = qC,α. So suppose that α is of type H, α ′ = ph(α), and that k < depth α′. By Lemma 9.4.1, depth α′ ≤ k(even). It follows that k is odd and depth α′ = k + 1. By Corollary 7.5.3, S is truly orthogonal at k, which means that Sk+1 has a diagonal block, say C. Note that w(C) dominates ph(σk) which in turn dominates ph(α). Since depthSww(C) = k+1 by Corollary 9.3.9, we are done. Now suppose that w(B) does not dominate α′. Then B is non-diagonal and α′ = pv(α). Since B is non-diagonal, ph(α) 6∈ N, and α cannot be of type H. So α is of type V in C. It follows easily (see Proposition 5.3.1 (3)) that α is the critical element in C, and and that last element in its connected component in C; by Lemma 6.3.1 (4), O-depthC(α) = depthSC qC,α =: d is even. By Proposition 5.3.1 (1), (2), the cardinality of the connected component of α in C is even. The immediate predecessor γ of α in C is connected to α (this follows from what has been said above). It is of type V in C, ph(γ) belongs to N, and depth pv(γ) = d − 1 (see Lemma 6.3.1 (1)). Let ℓ be such that γ ∈ S ℓ(up). Let C be the block of Sℓ such that γ ∈ C ′. Since ph(γ) ∈ N, C is diagonal. Note that w(C) dominates pv(γ) and that pv(γ) > pv(α). By Corollary 9.3.9, depth w(C) = ℓ. Thus if d ≤ ℓ we are done. On the other hand, d− 1 ≤ ℓ by Corollary 9.4.2. So we may assume that ℓ = d− 1. By Corollary 7.5.3, S is truly orthogonal at d − 1. This implies that Sd has a diagonal block, say D. Note that w(D) dominates ph(σd−1) which in turn dominates ph(γ). Writing γ = (r2, c2), since γ > α is connected, it follows that (r1, r 2) belongs to ON. Now both w(B) and w(D) dominate (r1, r 2). Since Sw is distinguished and symmetric and w(B) is not on the diagonal, it follows that w(D) > w(B). This implies, since w(D) is on the diagonal, w(D) > pv(α). Since depthSww(D) = d by Corollary 9.3.9, we are done. (4) Let x be an element of I(d) that O-dominates S. We will show that x ≥ w. By [7, Lemma 5.5], it is enough to show that x dominates Sw. By Lemma 9.1.4, it is enough to show the following: for every block B of S, there exists β in Sx such that β dominates w(B) and depthSxβ ≥ depthSww(B). Let B be a block of S. By Corollary 9.3.9, depth w(B) = k where B ⊆ Sk. Let S x denote the set of elements of Sx of depth at least k. Our goal is to show that there exists β in Skx that dominates w(B). It follows easily from the distinguishedness of Sx and the fact that B is a block, that it suffices to show the following: given α ∈ B, there exists β in Skx (depending upon α) that dominates α. Moreover, since B and Skx are symmetric, we may assume that α = α(up). So now let α = α(up) belong to B. Then either 1. α belongs to S 2. k is odd, S is truly orthogonal at k, and α = pv(σk), or 3. k is even, S is truly orthogonal at k − 1, and α = ph(σk−1). The proofs in the three cases are similar. In the first case, choose a v-chain C in S with tail α such that O-depthC(α) = k (see Corollary 6.1.3 (1)). Then depth qC,α = k and, clearly, qC,α dominates α. Since x dominates SC , there exists, by Lemma 4.5 of [7], β in Skx that dominates qC,α (and so also α). In the second case, choose a v-chain C in S with tail σk with the property that O-depthC(σk) = k. Then depthSC qC,σk = k. Since ph(σk) belongs to N, σk is of type V or H in C, so qC,σk = α. Since x dominates SC , there exists, by [7, Lemma 4.5], β in Skx that dominates qC,σk = α. In the third case, choose a v-chain C in S with tail σk−1 such that the O-depth in C of σk−1 is k − 1. Then depthSCqC,σk−1 = k − 1. Since ph(σk−1) belongs to N, σk−1 is of type V or H in C, so qC,σk−1 = pv(σk−1). From Lemma 6.3.1 (4), it follows, since k − 1 is odd, that σk−1 is of type H. Since pv(σk−1) > ph(σk−1) = α, it follows that depthSCph(σk−1) ≥ k (in fact equality holds as is easily seen). Since x dominates SC , there exists, by [7, Lemma 4.5], β in Skx that dominates ph(σk−1) = α. � 10.2 Proof that OφOπ = identity LetS be a monomial inON and let Oπ = (w,S′). We need to show thatOφ ap- plied to the pair (w,S′) gets us back toS. We know from (3) of Proposition 4.1.1 that w O-dominates S′, so Oφ can indeed be applied to the pair (w,S′). The main ingredients of the proof are the corresponding assertion in the case of Grassmannian [7, Proposition 4.2] and the following claim which we will presently prove: (S′)w,j,j+1 = S j(up) ∪S j+1(up) for every odd integer j Let us first see how the assertion follows assuming the truth of the claim, by tracing the steps involved in applying Oφ to (w,S′). From the claim it follows that when we partition S′ into pieces (see §8), we get S′j(up) ∪S j+1(up) (for odd integers j). Adding the mirror images will get us to S′j ∪ S j+1. From Corollary 9.3.9 it follows that wj,j+1 is exactly the element of I(d, 2d) obtained by acting π on Sj ∪ Sj+1. Now, since φ ◦ π = identity, it follows that on application of φ to (wj,j+1,S j+1) we obtain Sj ∪Sj+1. By twisting the two diagonal elements in Sj ∪ Sj+1 (if they exist at all) and removing the elements below the diagonal d, we get back S j,j+1. Taking the union of S j,j+1 (over odd integers j), we get back S. Thus we need only prove the claim. Since S′ is the union over all odd integers of the right hand sides (this follows from the definition of S′), and the left hand sides as j varies are mutually disjoint, it is enough to show that the right hand side is contained in the left hand side. Thus we need only prove: for j an odd integer and α an element in S′j(up) ∪S j+1(up), • every v-chain in S′ with head α is O-dominated by wj . • there exists a v-chain in S′ with head α that is not O-dominated by wj+2. To prove the first item, write T = Sj,j+1 := {α ∈ S|O-depth (α) ≥ j} and set Oπ(T) = (x,T′). By Proposition 6.3.6, we have T i ∪ T i+1 = S i+j−1 ∪ i+j for any odd integer i. Thus, by the description of Oπ, we have T ∪k≥jS k(up). By Corollary 9.3.9 and the description of Oπ, we have x = w By Corollary 9.3.10, any v-chain inS′ with head belonging toS′j(up)∪S j+1(up) is contained entirely in ∪k≥jS k(up). Finally, by Proposition 4.1.1 (3) applied to T, the desired conclusion follows. To prove the second item we use Lemma 9.4.3. Proceed by decreasing induc- tion on j. For j sufficiently large the assertion is vacuous, for S′j(up)∪S j+1(up) is empty. To prove the induction step, assume that the assertion holds for j+2. If the v-chain consisting of the single element α is not O-dominated by wj+2, then we are done. So let us assume the contrary. Since the O-depth of the singleton v-chain α is at most 2, it follows from Lemma 9.2.2 that wj+2,j+3 O- dominates the v-chain α. Apply Lemma 9.4.3 with k = j+2. By its conclusion, either there exists β ∈ S′j+2(up) such that α > β or there exists γ ∈ S j+3(up) such that ph(α) > γ. First suppose that a γ as above exists. By induction, there exists a v-chain in S′—call it D—with head γ that is not O-dominated by wj+4. Let C be the concatenation of α > γ and D. Since elements of D haveO-depth at least 3 in C (Lemma 6.3.1 (1)), it follows from Corollary 9.2.3 that C is not O-dominated by wj+2, and we are done. Now suppose that such a γ does not exist. Then a β as above exists. If α > β is not O-dominated by wj+2 we are again done. So assume the contrary. Since the O-depth of β in α > β is at least 2, it follows that there exists an element of (Sw)j+3 that dominates β. Applying Lemma 9.4.3 again, this time with k = j + 3, we find γ′ ∈ S′j+3(up) such that β > γ ′. Arguing as in the previous paragraph with γ′ in place of γ, we are done. � 10.3 Proof that OπOφ = identity Let T be a monomial in ON and w an element of I(d) that O-dominates T. We can apply Oφ to the pair (w,T) to obtain a monomial T⋆w in ON. We need to show that Oπ applied to T⋆w results in (w,T). The main step of the proof is to establish the following: T⋆w,j,j+1 = (T j,j+1 (10.3.1) (for the meaning of the left and right sides of the above equation, see §8 and §7 respectively). Assuming this for the moment let us show that Oπ ◦ Oφ = identity. We trace the steps involved in applying Oπ to T⋆w. From Eq. (10.3.1) it follows that when we break up T⋆w according to the O-depths of its elements as in §7, we get T⋆w,j,j+1 (as j varies over odd integers). The next step in the application of Oπ is the passage from (T⋆w) j,j+1 to (T w)j,j+1. This involves replacing σj by its projections and adding the mirror image of the remaining elements of (T⋆w) j,j+1. It follows from Proposition 8.2.2 (3) that σj = δj and so (T⋆w)j,j+1 = (Tw,j,j+1 ∪ T w,j,j+1) ⋆. The next step is to apply π to (Tw,j,j+1 ∪ w,j,j+1) ⋆. Since π is the inverse of φ (as proved in [7]), we have π((Tw,j,j+1 ∪ w,j,j+1) ⋆) = (wj,j+1,Tw,j,j+1). Since Sw and T are respectively the unions, as j varies over odd integers, of (Sw)j,j+1 and Tw,j,j+1, we see that Oφ applied to T⋆w results in (w,T). Thus it remains only to establish Eq. (10.3.1). It is enough to show that the left hand side is contained in the right hand side, for the union over all odd j of either side is T⋆w and the right hand side is moreover a disjoint union. In other words, we need only show that the O-depth in T⋆w of an element of T w,j,j+1 is either j or j + 1. We will show, more precisely, that, for any element β of Sw, the O-depth in T w of any element of OP β equals the depth in Sw of β. Lemma 9.4.4 will be used for this purpose. Let α be an element of OP∗β and set e := O-depthT⋆w (α). We first show, by induction on d := depth β, that e ≥ d. There is nothing to prove in case d = 1, so we proceed to the induction step. Let β′ be the element of Sw of depth d − 1 such that β′ > β. If there exists α′ in OP⋆β′ with α ′ > α, the desired conclusion follows from Corollary 6.1.3 (3) and induction. Lemma 9.4.4 says that such an α′ exists in case d is even. So suppose that d is odd and such an α′ does not exist. The same lemma now says that ph(δd−2) > α, so the desired conclusion follows from Lemma 6.3.1 (1). We now show, by induction on e, that d ≥ e. There is nothing to prove in case e = 1, so we proceed to the induction step. Let C be a v-chain in T⋆w with tail α and having the good property of Proposition 6.3.3. Let α′ be the immediate predecessor in C of α. Let β′ in Sw be such that α ′ ∈ OP∗β′ (we are not claiming at the moment that β′ is unique although that is true and follows from the assertion that we are proving, the distinguishedness of Sw, and the fact that β′ dominates α′). It follows from Corollary 9.4.6 that β′ > β. Let d′ := depth β′. It follows from Corollary 6.1.3 (3) that e′ < e where e′ := O-depth (α′). We have, d ≥ d′+1 ≥ e′+1 ≥ (e−2)+1 = e−1, the first equality being justified because β′ > β, the second by the induction hypothesis, and the last by Lemma 6.3.1 (1). It suffices to rule out the possibility that d = e − 1. So assume d = e − 1. Then d = d′ + 1 and d′ = e′ = e − 2. It follows from (1) of Lemma 6.3.1 that the v-chain α′ > α has O-depth 3 and from (3) of the same lemma that e′ is odd. But then we get a contradiction to Proposition 8.2.2 (2) (α′ and α belong to T⋆w,d′,d′+1). The proof of Eq. (10.3.1) is thus over. � 10.4 Proof of Proposition 4.1.3 Observe that the condition (‡) makes sense also for a monomial of N. By virtue of belonging to I(d), v has f∗ as an entry. It follows from the description of the bijection w ↔ Sw of §5.1.2 that for an element w of I(d) to satisfy (‡) it is necessary and sufficient that Sw (equivalently all its parts Sw,j,j+1) satisfy (‡). (1) Since T satisfies (‡), so do its parts Tw,j,j+1 and Tw,j,j+1 ∪ T w,j,j+1 (adding the mirror image preserves (‡)). Since Sw,j,j+1 also satisfies (‡), it follows from the description of the map φ of [7] (observe the passage from a piece P to its “star” P∗) that the (Tw,j,j+1 ∪ T w,j,j+1) ⋆ satisfy (‡). Since the “twisting” involved in the passage from (Tw,j,j+1 ∪ T w,j,j+1) ⋆ to T⋆w,j,j+1 involves only a rearrangement of row and column indices, it follows that the T⋆w,j,j+1 satisfy (‡). Finally so also does their union T (2) The parts S j,j+1 of S clearly satisfy (‡). Therefore so do the Sj,j+1, for, first of all, adding the mirror image preserves (‡), and then the removal of σj and addition of its projections involves only a rearrangement of row and column indices. It follows from description of the map π of [7] (observe the passage from a block B to the pair (w(B),B′)) that both Sw,j,j+1 and S j,j+1 satisfy (‡). Finally, Sw and S ′ being the union (respectively) of Sw,j,j+1 and S j,j+1(up), they satisfy (‡). � Part IV An Application As an application of the main theorem (Theorem 2.3.1), an interpretation of the multiplicity is presented. 11 Multiplicity counts certain paths Fix elements v, w in I(d) with v ≤ w. It follows from Corollary 2.3.2 that the multiplicity of the Schubert variety X(w) in Md(V ) at the point e v can be in- terpreted as the cardinality of a certain set of non-intersecting lattice paths. We first illustrate this by means of two examples and then justify the interpreta- tion. 11.1 Description and illustration The points of N can be represented, in a natural way, as the lattice points of a grid. The column indices of the points of the grid are the entries of v and the row indices are the entries of {1, . . . , 2d} \ v. In Figure 11.1.1 the points of ON and those of the diagonal in N are shown (for the specific choice of v in Example 11.1.1). The open circles represent the points of Sw(up), where Sw is the distinguished monomial in N that is associated to w as in §5.1.2. From each point β of Sw(up) we draw a vertical line upwards from β and let β(start) denote the top most point of ON on this line. In case β is not on the diagonal, draw also a horizontal line rightwards from β and let β(finish) denote the right most point of ON on this line. In case β is on the diagonal, then β(finish) is not a fixed point but varies subject to the following constraints: • β(finish) is one step away from the diagonal (that is, it is of the form (r, c), for some entry c of v, where r is the largest integer less than c∗ that is not an entry of v); • the column index of β(finish) is not less than that of β; • if depth β is odd, then the horizontal projection of β(finish) is the same as the vertical projection of γ(finish) where γ is the diagonal element of Sw of depth 1 more than that of β. With v and w as in Example 11.1.1, we have β(start) = (6, 3) and β(finish) = (9, 5) for β = (9, 3); β(start) = β(finish) = (21, 20) for β = (21, 20); β(start) = (15, 11) for the diagonal element β = (36, 11); β(start) = (6, 1) for the diagonal element β = (46, 1). In the particular case (of non-intersecting lattice paths) drawn in Figure 11.1.1, β(finish) = (27, 19) for β = (36, 11) and β(finish) = (28, 14) for β = (46, 1). A lattice path between a pair of such points β(start) and β(finish) is a se- quence α1, . . . , αq of elements of ON with α1 = β(start) and αq = β(finish) such that, for 1 ≤ j ≤ q − 1, if we let αj = (r, c), then αj+1 is either (R, c) or (r, C) where R is the least element of {1, . . . , 2d} \ v that is bigger than r and C the least element of v that is bigger than c. Note that if β(start) = (r, c) and β(finish) = (R,C), then q equals |({1, . . . , 2d} \ v) ∩ {r, r + 1, . . . , R}|+ |v ∩ {c, c+ 1, . . . , C}| − 1, where | · | is used to denote cardinality. Consider the set Pathsw of all tuples (Λβ)β∈Sw(up) of paths where • Λβ is a lattice path between β(start) and β(finish) (if β is on the diagonal, then β(finish) is allowed to vary in the manner described above); • Λβ and Λγ do not intersect for β 6= γ. The number of such p-tuples, where p := |Sw(up)|, is the multiplicity of X(w) at the point ev. Example 11.1.1 Let d = 23, v = (1, 2, 3, 4, 5, 11, 12, 13, 14, 19, 20, 22, 23, 26, 29, 30, 31, 32, 37, 38, 39, 40, 41), w = (4, 5, 9, 10, 14, 17, 18, 21, 23, 25, 27, 28, 31, 32, 34, 35, 36, 39, 40, 41, 44, 45, 46), so that Sw ={(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), (27, 26), (28, 19), (34, 30), (35, 29), (36, 11), (44, 38), (45, 37), (46, 1)} and Sw(up) = {(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), (28, 19), (36, 11), (46, 1)}. A particular element of Pathsw is depicted in Figure 11.1.1. � Example 11.1.2 Figure 11.1.2 shows all the elements of Pathsw in the follow- ing simple case: d = 7, v = (1, 2, 3, 4, 7, 9, 10), and w = (4, 6, 7, 10, 12, 13, 14). We have Sw = {(6, 3), (12, 9), (13, 2), (14, 1)}, Sw(up) = {(6, 3), (13, 2)(14, 1)}. There are 15 elements in Pathsw and thus the multiplicity in this case is 15. � Example 11.1.3 Let d = 10, v = (1, 2, 3, 4, 6, 8, 11, 12, 14, 16), and w = (8, 9, 11, 14, 15, 16, 17, 18, 19, 20). so that Sw = {(20, 1)(19, 2)(18, 3), (17, 4), (9, 6)(15, 12)}. Figure 11.1.3 shows a tuple of paths that is disallowed (meaning one that is not in Pathsw). The elements of ON are represented as usual by a grid. The slanted line represents 1 2 3 4 5 11 12 13 14 19 20 22 23 Figure 11.1.1: An element of Pathsw with v and w as in Example 11.1.1 the diagonal d. The solid dot represents the point of Sw(up) that is not on d, and the crosses on d represent the points of Sw(up) that lie on d. The tuple is disallowed because the horizontal projection of the last point of the path Λβ1 is not the vertical projection of the last point of the path Λβ2 , where β1 = (20, 1) and β2 = (19, 2) are the diagonal elements of Sw of depths 1 and 2 respectively. � 11.2 Justification for the interpretation We now justify the interpretation in the previous subsection of the multiplicity. Corollary 2.3.2 says that the multiplicity is the number of monomials in OR of maximal cardinality that are square-free and O-dominated by w. Any such monomial contains OR \ON, for, by the definition of O-domination, adding or removing elements of OR\ON to or from a monomial does not alter the status of its O-domination. One could therefore equally well consider the number of monomials in ON of maximal cardinality that are square-free and O-dominated ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ Figure 11.1.2: All the 15 non-intersecting lattice paths of Example 11.1.2 diagonal 1 2 3 4 6 8 11 12 14 16 The horizontal projection of the last point of the path associated to (20,1) is not the same as the vertical projection of the last point of the path associated to (19,2). This is the reason that this tuple is disallowed. Figure 11.1.3: A disallowed tuple of lattice paths (see Example 11.1.3) by w. We now establish a bijection between the set Monw of such monomials and the set Pathsw of non-intersecting lattice paths as in §11.1. Each element Λ of Pathsw can be thought of, in the obvious way, as a monomial in ON. We will continue to denote the corresponding monomial by Λ. It is clear that the monomial Λ is square-free and that all such monomials Λ have the same cardinality (in particular, that if Λ1 ⊆ Λ2 for two such monomials then Λ1 = Λ2). In order to establish the bijection it therefore suffices to prove the following proposition. Proposition 11.2.1 1. w is the element of I(d) obtained on application of Oπ to the monomial Λ (in particular (see Proposition 4.1.1), the monomial Λ is O-dominated by w). 2. Given a monomial T of ON that is square-free and O-dominated by w, there exists Λ such that T ⊆ Λ. Proof: (1) Write Λ = (Λβ)β∈Sw(up). From the description of the map Oπ in §7, it follows that it suffices to show that Λ (in the notation of §7) is the union ∪Λβ where β runs over all elements of depth k in Sw(up). In other words, it suffices to show that the O-depth in Λ of any element of Λβ equals the depth in Sw of β. To prove this, we observe the following (these assertions are easily seen to be true thinking in terms of pictures): for fixed β ∈ Sw(up) and α ∈ Λβ , (A) For β′ in Sw(up) such that β ′ > β, there exists α′ ∈ Λβ′ such that α ′ > α. (B) If α′ > α for some α′ in Λβ′ for some β ′ in Sw(up), then β ′ > β. If, furthermore, β and β′ are diagonal, their depths in Sw are 1 apart, and the depth in Sw of β is even, then the following is not possible: ph(α belongs to N and ph(α ′) > α. From (A) it is immediate that the O-depth e in Λ of an element α of Λβ is not less than the depth d in Sw of β. We now show, by induction on e, that e ≤ d. For e = 1 there is nothing to show. Suppose that e ≥ 2. Let C be a v-chain in Λ having tail α and the good property of Proposition 6.3.3, α′ the immediate predecessor in C of α, e′ the O-depth of α′ in Λ, β′ the element of Sw(up) such that α′ ∈ Λβ′ , and d ′ the depth in Sw of β ′. From Corollary 6.1.3 (3) it follows that e′ ≤ e − 1, so we may apply induction. From (B) it follows that d′ ≤ d − 1, so that, by induction, e′ ≤ d − 1. If e′ ≤ d − 2, then we are done by Lemma 6.3.1 (1). So suppose that e′ = d′ = d − 1. If d is odd, then the conclusion e ≤ d follows from (1) and (3) of the same lemma. In case d is even, then it follows from condition (B) and (1) of the same lemma. (2) Let T be a square-free monomial in ON that is O-dominated by w. To construct Λ such that T ⊆ Λ, we construct the “components” Λβ . As in §8, let Pβ denote the piece of T corresponding to β ∈ Sw. From every point belonging to Pβ(up) and also from β(start) carve out the South-West quadrant; if β is not diagonal, then do this also from β(finish). The boundary of the carved out portion (intersected with ON) gives a lattice path starting from β(start). In case β is not diagonal, the path ends in β(finish). In this case as well as in the case when β is diagonal and of even depth in Sw, we take Λβ to be this lattice path. In case β is diagonal and of odd depth in Sw we do the carving out from one more point before taking Λβ to be the boundary of the carved out region, namely from the point that is one step away from the diagonal and whose horizontal projection is the vertical projection of the end point of Λγ where γ is the diagonal element of Sw of depth 1 more than β. We need to justify why carving out from the extra point is still valid, and we do this now by applying Lemma 8.1.4. Let us first choose notation that is consistent with that of that lemma. Let β and γ be diagonal elements in Sw of depths d and d + 1. Assume that d is odd. Let the pieces of T corresponding to β and γ, when their elements are arranged in increasing order of row and column indices, look like this: . . . , (r1, a ∗), (a, r∗1), . . . ; . . . , (r2, b ∗), (b, r∗2), . . . It is easy to see that the conditions on the numbers in the above display that pro- vide the requisite justification are: r1 ≤ b and a ∗ < b∗ (if Pβ is empty then the justification is easy). To prove that a∗ < b∗, observe that the diagonal elements in P∗β and P γ are respectively (a, a ∗) and (b, b∗), and apply Lemma 8.1.3 (2). That r1 ≤ b now follows from Lemma 8.1.4 (1). This finishes the justification. It suffices to prove the following claim: the lattice paths Λβ as β varies are non-intersecting. Suppose that Λβ and Λβ′ intersect for β 6= β ′. Let α be a point of intersection. Clearly β dominates all elements of Λβ and in particular α; for the same reason β′ also dominates α. By the distinguishedness of Sw, we may assume without loss of generality that β′ > β. It is easy to see graphically that if γ in Sw is such that β ′ > γ > β then Λγ intersects either Λβ′ or Λβ : consider the open portion of ON “caught between” the segment of Λβ′ from β ′(start) to α and the segment of Λβ from β(start) to α; the starting point γ(start) of Λγ lives in this region but its ending point does not (points strictly to the Northwest of α can neither be of the form γ(finish) for γ not on the diagonal nor can they be one step away from the diagonal); so Λγ must intersect one of the two lattice path segments. We may therefore assume that the depths of β′ and β differ by 1. We now apply Lemma 9.4.4. From the construction of Λβ it readily fol- lows that α satisfies the hypotheses (a), (b), and (c) of that lemma. By the conclusion of Lemma 9.4.4, there exists α′ ∈ P∗β′(up) such that α ′ > α. On the other hand, it follows from the construction of P∗β′ from Pβ′ , and from the construction of Λβ′ that two elements one from P β′ and another from Λβ′ are not comparable. This is a contradiction to the comparability of α′ and α. � References [1] M. Brion and P. Polo, Generic singularities of certain Schubert varieties , Math. Z., 231, no. 2, 1999, pp. 301–324. [2] E. De Negri, Some results on Hilbert series and a-invariant of Pfaffian ideals , Math. J. Toyama Univ., 24, 2001, pp. 93–106. [3] S. R. Ghorpade and C. Krattenthaler, The Hilbert series of Pfaffian rings , in: Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 337–356. [4] S. R. Ghorpade and K. N. Raghavan, Hilbert functions of points on Schubert varieties in the Symplectic Grassmannian, Trans. Amer. Math. Soc., 358, 2006, pp. 5401–5423. [5] J. Herzog and N. V. Trung, Gröbner bases and multiplicity of determinantal and Pfaffian ideals , Adv. Math., 96, no. 1, 1992, pp. 1–37. [6] T. Ikeda and H. Naruse, Excited Young diagrams and equivariant Schubert calculus , arXiv:math/0703637. [7] V. Kodiyalam and K. N. Raghavan, Hilbert functions of points on Schubert varieties in Grassmannians , J. Algebra, 270, no. 1, 2003, pp. 28–54. [8] C. Krattenthaler, On multiplicities of points on Schubert varieties in Grass- mannians. II , J. Algebraic Combin., 22, no. 3, 2005, pp. 273–288. [9] V. Kreiman, Monomial bases and applications for Schubert and Richardson varieties in ordinary and affine Grassmannians , Ph. D. Thesis, Northeast- ern University, 2003. [10] V. Kreiman, Local Properties of Richardson Varieties in the Grassmannian via a Bounded Robinson-Schensted-Knuth Correspondence, preprint, 2005, URL arXiv:math.AG/0511695. [11] V. Kreiman and V. Lakshmibai, Multiplicities of singular points in Schu- bert varieties of Grassmannians , in: Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, pp. 553– [12] V. Kreiman and V. Lakshmibai, Richardson varieties in the Grassmannian, in: Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 573–597. [13] V. Lakshmibai and C. S. Seshadri, Geometry of G/P . V , J. Algebra, 100, no. 2, 1986, pp. 462–557. [14] V. Lakshmibai and J. Weyman, Multiplicities of points on a Schubert va- riety in a minuscule G/P , Adv. Math., 84, no. 2, 1990, pp. 179–208. [15] P. Littelmann, Contracting modules and standard monomial theory for sym- metrizable Kac-Moody algebras , J. Amer. Math. Soc., 11, no. 3, 1998, pp. 551–567. [16] K. N. Raghavan and S. Upadhyay, Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians . In preparation. [17] C. S. Seshadri, Geometry of G/P . I. Theory of standard monomials for minuscule representations , in: C. P. Ramanujam—a tribute, vol. 8 of Tata Inst. Fund. Res. Studies in Math., Springer, Berlin, 1978, pp. 207–239. Index >, relation on ON . . . . . . . . . . . . . . . . . 11 ≤, partial order on I(d, 2d) . . . . . . . . 10 A, affine patch qv 6= 0 of Md(V ) . . . 14 α(down), for α ∈ N . . . . . . . . . . . . . . . . 25 α# for α in N . . . . . . . . . . . . . . . . . . . . . 22 α(up), for α ∈ N . . . . . . . . . . . . . . . . . . .25 anti-domination . . . . . . . . . . . . . . . . . . . 18 B, a specific Borel subgroup . . . . . . . . 8 β(finish), for β ∈ Sw(up) . . . . . . . . . . 65 β(start), for β ∈ Sw(up) . . . . . . . . . . . 65 block in the sense of [7] . . . . . . . . . . . . .47 of a monomial S in ON . . . . . . . 51 comparability, of elements of R . . . . 29 connected components of a v-chain.23 connectedness of two succcessive ele- ments in a v-chain . . . . . . . . 22 critical element (of a v-chain) . . . . . . 24 d, integral part of n/2 (unfortunately also used otherwise) . . . . . 7, 9 degree, of a monomial . . . . . . . . . . . . . 10 degree, of a standard monomial . . . . 16 δj , for j odd . . . . . . . . . . . . . . . . . . . . . . . 44 depth (of an element α in a monomial S in N) = depth α . . . . . . 46 depth (of a monomial S in N) . . . . . 46 diagonal, d . . . . . . . . . . . . . . . . . . . . . 10, 11 distinguished (a subset of N) . . . . . . 21 domination (among elements of R) 29 domination map . . . . . . . . . . . . . . . . . . . 18 e1, . . . , en, a specific basis of V . . . . . . 8 ev, T -fixed point . . . . . . . . . . . . . . . . . . . 10 〈 , 〉, bilinear form on V . . . . . . . . . . . . 7 fθ := qθ/qv . . . . . . . . . . . . . . . . . . . . . . . . 14 head, of a v-chain. . . . . . . . . . . . . . . . . .11 horizontal projection ph(α) . . . . . . . . 22 I(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 I(d, 2d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 i(even), for an integer i . . . . . . . . . . . . 29 i(odd), for an integer i . . . . . . . . . . . . . 29 intersection (of a monomial in a set with a subset) . . . . . . . . . . . . 11 isotropic subspace . . . . . . . . . . . . . . . . . . 7 In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 I ′n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 k, base field, (characteristic 6= 2) .7, 15 k∗(:= n+ 1− k) . . . . . . . . . . . . . . . . 7, 10 L, line bundle . . . . . . . . . . . . . . . . . . . . . 13 Λβ , for β ∈ Sw(up) . . . . . . . . . . . . . . . . 66 lattice path, from β(start) to β(finish), denoted Λβ . . . . . . . . . . . . . . . 65 legs of α, for α ∈ ON . . . . . . . . . . . . . . 22 legs, intertwining of . . . . . . . . . . . . . . . .22 length, of a v-chain . . . . . . . . . . . . . . . . 11 Md(V ), orthogonal Grassmannian 7, 8 Md(V ) ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 monomial. . . . . . . . . . . . . . . . . . . . . . . . . .10 w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 multiplicity, of X(w) at ev . . . . . . . . . 65 multiset := monomial . . . . . . . . . . . . . . 10 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 n := dimV , (even from §2.1 on) . . 7, 9 O(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 O-depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 OP∗β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Oφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 42 Oπ . . . . . . . . . . . . . . . . . . . . . . . . . 17, 34, 35 OR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 O-domination . . . . . . . . . . . . . . . . . . . . . 11 orthogonal Grassmannian (Md(V )) . 7 Paths w . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Pβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 P∗β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Pfaffian qθ . . . . . . . . . . . . . . . . . . . . . . . . . 14 ph(α), horizontal projection. . . . . . . .22 piece of T (see also caution) . . . . . . . 43 pθ, Plücker coordinate . . . . . . . . . . . . . 13 pv(α), vertical projection . . . . . . . . . . 22 qC,α, for α in a v-chain C . . . . . . . . . . 24 qθ, Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . 14 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 S, fixed monomial in ON in §7, §9.3 . S, set of monomials in OR . . . . . . . . 18 modifications . . see Notation 4.2.1 SC , where C is a v-chain . . . . . . . . . . 23 SC,α, for α in a v-chain C . . . . . . . . . 24 Schubert varieties . . . . . . . . . . . . . . . . 7, 8 S(down), for a monomial S . . . . . . . 25 S#, for monomial S in N or R . . . . 22 σk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Sj,j+1, for S in ON, j odd . . . . . . . . 34 Sj,j+1(ext), for S in ON, j odd . . . 38 j,j+1, for S in ON, j odd . . . . . . . . 34 Sk, for monomial S in N . . . . . . . . . . 46 Sk, for monomial S in ON . . . . . . . . 34 Sk(ext), for monomial S in ON . . . 39 Sk,k+1, for monomial S in N . . . . . . 49 , for monomial S in ON . . . . . . . 34 SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 modifications . . see Notation 4.2.1 SMv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 SMw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 SO(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 S′, for monomial S in ON . . . . . . . . 35 standard monomial . . . . . . . . . . . . . . . . 14 v-compatible . . . . . . . . . . . . . . . . . . 15 w-dominated . . . . . . . . . . . . . . . . . . 14 S(up), for a monomial S . . . . . . . . . . 25 Sj,j+1, for monomial S in ON . . . . 32 Sk, for monomial S in N . . . . . . . . . . 46 Sw(v)(m) . . . . . . . . . . . . . . . . . . . . . . . . . .11 Sw, w in I(d, 2d) or I(d, n) . . . . 21, 48 Sw,j,j+1, w in I(d), j odd . . . . . . . . . 42 Sjw, w in I(d), j odd . . . . . . . . . . . . . . 42 symmetric (monomial of N). . . . . . . .22 T , a specific maximal torus . . . . . . . . . 8 T , set of monomials in ON . . . . . . . . 18 modifications . . see Notation 4.2.1 tail, of a v-chain . . . . . . . . . . . . . . . . . . . 11 T, fixed monomial in ON in §8 . . . . . . . truly orthogonal at j (j odd) . . . . . . 34 Tw,j,j+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (Tw,j,j+1 ∪ T w,j,j+1) ⋆ . . . . . . . . . . . . . . 43 T⋆w,j,j+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 T⋆w(:= Oφ(w,T)) . . . . . . . . . . . . . . . . . . 44 type (V, H, S), of an element in a v- chain. . . . . . . . . . . . . . . . . .23–24 U , set of monomials in OR \ON . . 18 modifications . . see Notation 4.2.1 u∗, for u ∈ I(d) . . . . . . . . . . . . . . . . . . . . 19 V , vector space of dimension n . . . . . .7 v, fixed element of I(d) . . . . . . . . . . . . 10 v-chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 v-degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 vertical projection pv(α) . . . . . . . . . . . 22 w(C) (or wC), where C is v-chain. .11 w#, for w an element of I(d, 2d) . . . 21 w(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 (w,S′)(:= Oπ(S)), for S in ON . . 34, w∗, for w an element of I(d, 2d) . . . .21 wj,j+1, j odd . . . . . . . . . . . . . . . . . . . . . . 42 wj , j odd . . . . . . . . . . . . . . . . . . . . . . . . . . 42 X(w), Schubert variety . . . . . . . . . . . . 10 xk, x k, xk,k+1, for x ∈ I(d, n). . .48–49 Xr,c, variable . . . . . . . . . . . . . . . . . . . . . . 16 Y (w)(:= X(w) ∩ A). . . . . . . . . . . . . . . .15

0704.0543

Swift/XRT observes the fifth outburst of the periodic Supergiant Fast X-ray Transient IGR J11215-5952

Astronomy & Astrophysics manuscript no. romanoigrj112 c© ESO 2018 October 24, 2018 Letter to the Editor Swift/XRT observes the fifth outburst of the periodic Supergiant Fast X–ray Transient IGR J11215–5952 P. Romano1,2, L. Sidoli3, V. Mangano4, S. Mereghetti3, and G. Cusumano4 1 INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC), Italy 2 Università degli Studi di Milano, Bicocca, Piazza delle Scienze 3, I-20126 Milano, Italy 3 INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, Via E. Bassini 15, I-20133 Milano, Italy 4 INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, Via U. La Malfa 153, I-90146 Palermo, Italy Received 1 March 2007/Accepted 4 April ABSTRACT Context. IGR J11215-5952 is a hard X–ray transient source discovered in April 2005 with INTEGRAL and a confirmed member of the new class of High Mass X–ray Binaries, the Supergiant Fast X–ray Transients (SFXTs). Archival INTEGRAL data and RXTE observations showed that the outbursts occur with a periodicity of ∼330 days. Thus, IGR J11215–5952 is the first SFXT displaying periodic outbursts, possibly related to the orbital period. Aims. We performed a Target of Opportunity observation with Swift with the main aim of monitoring the source behaviour around the time of the fifth outburst, expected on 2007 Feb 9. Methods. The source field was observed with Swift twice a day (2ks/day) starting from 4th February, 2007, until the fifth outburst, and then for ∼ 5 ks a day afterwards, during a monitoring campaign that lasted 23 days for a total on-source exposure of ∼ 73 ks. This is the most complete monitoring campaign of an outburst from a SFXT. Results. The spectrum during the brightest flares is well described by an absorbed power law with a photon index of 1 and NH ∼ 1 × 1022 cm−2. A 1–10 keV peak luminosity of ∼1036 erg s−1 was derived (assuming 6.2 kpc, the distance of the optical counterpart). Conclusions. These Swift observations are a unique data-set for an outburst of a SFXT, thanks to the combination of sensitivity and time coverage, and they allowed a study of IGR J11215–5952 from outburst onset to almost quiescence. We find that the accretion phase lasts longer than previously thought on the basis of lower sensitivity instruments observing only the brightest flares. The observed phenomenology is consistent with a smoothly increasing flux triggered at the periastron passage in a wide eccentric orbit with many flares superimposed, possibly due to episodic or inhomogeneous accretion. Key words. X-rays: stars: individual: IGR J11215–5952 1. Introduction The hard X–ray transient IGR J11215–5952 was discovered with the INTEGRAL satellite during an outburst in April 2005 (Lubinski et al., 2005) and was associated with HD 306414 (Negueruela et al., 2005), a B1Ia supergiant located at a dis- tance of 6.2 kpc (Masetti et al., 2006). The short duration of the outburst together with the likely optical counterpart suggested that IGR J11215–5952 could be a new member of the class of Supergiant Fast X-ray Transients (SFXTs; Negueruela et al. 2006). Analysing archival INTEGRAL observations of the source field, Sidoli, Paizis, & Mereghetti (2006, hereafter Paper I) discovered two previously unnoticed outbursts (in July 2003 and in May 2004) which demonstrate the recurrent nature of this transient and suggest a possible periodicity of ∼330 days. This periodicity was confirmed by the detection of the fourth outburst from IGR J11215–5952 with RossiXTE/PCA on 2006 March 16–17, 329 days after the third outburst (Smith et al., 2006b). The RXTE/PCA observations showed strong flux variability and a hard spectrum (power-law photon index of 1.7 ± 0.2 in the range 2.5–15 keV) as well as a possible pulse period of ∼195 s (Smith et al., 2006a). The periodicity was confirmed with RXTE observations of the latest outburst, yielding P = 186.78 ± 0.3 s (Swank et al., 2007). Follow-up observations with Swift/XRT Send offprint requests to: P. Romano, [email protected] refined the source position and confirmed the association with HD 306414 (Steeghs et al., 2006). A hard power-law with a high energy cut-off around 15 keV is a good fit to the spectra ob- served with INTEGRAL (Paper I). For the distance of 6.2 kpc, the peak fluxes of the outbursts correspond to a luminosity of ∼ 3×1036 erg s−1 (5–100 keV). All these findings confirmed IGR J11215–5952 as a member of the class of the SFXTs, and the first object of this class of High Mass X–ray Binaries dis- playing periodic outbursts. Predicting a fifth outburst for 2007 Feb 9, we obtained a Target of Opportunity (ToO) observing campaign with Swift, which commenced on Feb 4. The source started showing re- newed activity on Feb 8 (Romano et al., 2007) and under- went a powerful outburst on Feb 9 (Mangano et al., 2007a,b; Sidoli et al., 2007; Swank et al., 2007). This paper presents our observations of IGR J11215–5952 and it is organized as follows. In Sect. 2 we describe our observations and data reduction; in Sect. 3 we describe our spatial, timing and spectral data anal- ysis. Finally, in Sect. 4 we discuss our findings and draw our conclusions. 2. Observations and Data Reduction Table 1 reports the log of the Swift/XRT observations used for this work. Thanks to Swift’s fast-slewing and flexible observing http://arxiv.org/abs/0704.0543v1 2 P. Romano et al.: The fifth outburst of IGR J11215–5952 observed by Swift 140 145 150 155 MJD (−54,000) (a) 1−10 keV Feb 9 14 19 24 0 2×104 4×104 6×104 8×104 Time (s since 2007−02−09 00:03:05 UT) 1 2 3 (b) 1−10 keV 1−4 keV(c) Flare 1 te 4−10 keV 5200 5400 5600 5800 6000 Time (s) 4−10/1−4 1−4 keV (d) Flare 2 3 4−10 keV 1.05×104 1.1×104 1.15×104 Time (s) 4−10/1−4 Fig. 1. XRT light curves, cor- rected for pile-up, PSF losses, vignetting and background- subtracted. a) 1–10 keV light curve for the whole campaign. Different colours denote dif- ferent observations (Table 1), and points before Feb 6 (MJD 54,137) and after Feb 15 (MJD 54,146) are drawn from the sum of several observations. Filled circles are full detections (S/N>3), triangles marginal detections (2 <S/N< 3), while downward-pointing arrows are 3-σ upper limits. The vertical lines mark our time selections for spectroscopic analysis (observation 6, observations 7–8, observations 9–12, end of observation 18). The hor- izontal line marks the region shown enlarged in panel b). b) Detail of observation 6, with a binning that allowed to achieve a S/N in excess of 6. The numbers mark the 5 flares on which we performed spec- troscopic analysis. c) Detail of flare 1, showing the 1–4 keV, 4–10 keV count rates (top and middle panel) and the hardness ratio 4–10/1–4 (bottom). The data were rebinned in order to have at least 20 counts per bin in both the 1–4 keV and 4–10 keV band. d) Same as c), for flare 2. scheduling, the ToO observations started on 2007 Feb 4 with 2 ks per day evenly spread throughout the day to maximize the chances of detection of the outburst onset, and were increased to 5 ks afterwards for a total of 23 days and a total on-source exposure of ∼ 73 ks. We also retrieved from the Swift Archive the data from a 643 s ToO performed on 2006 Mar 20 during the fourth outburst of this source (Steeghs et al., 2006). The XRT data were processed with standard procedures (xrtpipeline v0.10.6), filtering and screening criteria by us- ing FTOOLS in the Heasoft package (v.6.1.2). Given the low rate of the source during the whole campaign, we only consid- ered photon counting data (PC) and further selected XRT grades 0–12 (Burrows et al. 2005). With the exception of observation 6 (2006 Feb 9) the data show an average count rate of < 0.5 counts s−1 and no pile-up correction was necessary. We there- fore extracted the source events from a circular region with a radius of 11 pixels (1 pixel ∼ 2.37′′). During observation 6, pile- up correction was required and we adopted an annular source extraction region with radii 4 and 30 pixels. To account for the background, we also extracted events within an annular region centered on the source, and with radii 40 and 100 pixels, free from background sources. Ancillary response files were gen- erated with xrtmkarf, and account for different extraction re- gions, vignetting and PSF corrections. We used the latest spec- tral redistribution matrices (v008) in the Calibration Database maintained by HEASARC. For timing analysis, the arrival times of XRT events were converted to the Solar System barycentre with the task barycorr and source events were extracted from the circular region with 30 pixels radius to maximize statistics. BAT always observed IGR J11215–5952 simultaneously with XRT, but only survey data products, in the form of Detector Plane Histograms (DPH) with typical integration time of ∼ 300 s, are available. The BAT data were analysed using the stan- dard BAT analysis software distributed within FTOOLS. DPH data were calibrated with the task baterebin using the proper BAT gain/offset files from the housekeeping data directory, and sky images of each observation were extracted in the 15–25, 25–50, 50–100, 100–150 and 15–150 keV energy bands. The batcelldetect task never detected the source above a signal- to-noise ratio (S/N) threshold of 4. This is consistent with the ex- trapolation at the high energies of the XRT data fit (Sect. 3) with an absorbed power-law with exponential cutoff, with e-folding energy of 15 ± 2 keV drawn from the RXTE fit (Swank et al., 2007). Throughout this paper the uncertainties are given at 90% confidence level for one interesting parameter (i.e., ∆χ2 = 2.71) unless otherwise stated. The spectral indices are parameterized as Fν ∝ ν −α, where Fν (erg cm −2 s−1 Hz−1) is the flux density as a function of frequency ν; we also use Γ = α + 1 as the photon index, N(E) ∝ E−Γ (ph cm−2 s−1 keV−1). P. Romano et al.: The fifth outburst of IGR J11215–5952 observed by Swift 3 0 0.5 1 1.5 Phase Fig. 2. Folded 0.2–10 keV light curve of the combined obser- vations 6 though 11, using a period of 186.78 s (Swank et al., 2007). 3. Analysis and Results A refined position was obtained by summing all data taken in 2007 with the exclusion of observation 6 (affected by pile-up) at RA(J2000) = 11h21m46.s90, Dec(J2000) = −59◦ 51′46.′′9, with an error, drawn from the cross-correlation with the USNO-B1.0 catalogue, of 1.′′1 (90% confidence). This position is 1.′′2 from the optical counterpart HD 306414. We extracted light curves in the 1–10 keV (total), 1–4 keV (soft) and 4–10 keV (hard) bands. The 0.2–1 keV band was not used in our analysis because, given the high absorbing column density, its signal was significantly lower than the one of the other bands. The light curves were corrected for Point-Spread Function (PSF) losses, due to the extraction region geometry, bad/hot pixels and columns falling within this region, and for vignetting, by using the task xrtlccorr (v0.1.9), which gen- erates an orbit-by-orbit correction based on the instrument map. We then subtracted the scaled background rate in each band from their respective source light curves and calculated the 4–10/1–4 hardness ratio. The IGR J11215–5952 light curve (Fig. 1) shows an increase in count rate by a factor of ∼ 10 in less than 1.5 hours, and of a factor of ∼ 65 in 17 hours on 2007 Feb 9. However, no significant variation in the hardness ratio can be ev- idenced (panels c,d of Fig. 1). Indeed, fitting the hardness ratio as a function of time (or as a function of count rate) to a constant model yields a value of 0.49±0.03 and χ2 = 1.13 for 80 degrees of freedom, d.o.f. We folded the data at the period of 186.78 s reported by Swank et al. (2007) based on Feb 9 01:20–03:20 UT RXTE/PCA observations and obtained the 0.2–10 keV light curve shown in Fig. 2. Upon examination of the light curve presented in Fig. 1 and the available counting statistics, we selected different time bins over which we accumulated spectra. These include i) the qui- escent phase before the 2007 Feb 9 outburst, ii) the Feb 9 out- burst (observation 6) and iii) the tail phase of the outburst (ob- servations 7–8, 9–12, 7–12, 7–18). We further selected 5 flaring episodes from observation 6 (see Fig. 1b). A comparison with the data collected during the tail of the 2006 outburst was also performed. The data were rebinned with a minimum of 20 counts per energy bin to allow χ2 fitting. However, for the 2006 observa- tion performed (28 counts), before the onset of the outburst (41 0.5 1 1.5 2 2.5 3 NH (10 22 cm−2) 0.5 1 1.5 2 2.5 3 NH (10 22 cm−2) Fig. 3. XRT time-selected spectroscopy. The ∆χ2 = 2.3, 4.61, 9.21 contour levels for the column density in units of 1022 cm−2 vs. the photon index, with best-fit values indicated by crosses. Orange contours are from observation 6, while the blue ones are from the combined observations 7 though 18. counts), for the late flares, and the late observations, the Cash statistic (Cash, 1979) and spectrally unbinned data were used. The spectra were all fit with XSPEC (v11.3.2) in the 0.5–9 keV energy range, adopting the typical pulsar spectral model, an ab- sorbed power law model. The best fit parameters are reported in Table 2 along with the mean luminosity of each time selection. The spectrum of the brightest part of the outburst (observation 6) could be fit with a single power law, with a photon index Γ = 1.00+0.16 −0.14 and an absorbing column density of NH = (1.04 +0.25 −0.20) × 10 22 cm−2 (χ2red = 1.04/83 d.o.f.), while the combined observations 7–18 yielded a photon index of 2.08+0.41 −0.37 and NH = (2.04 +0.62 −0.50) × 10 cm−2 (χ2red = 1.19/19 d.o.f). We note that the 1–10 keV count rate to unabsorbed 1–10 keV flux conversion factor, obtained from the best fit model for observation 6, is 2.9×10−10 erg cm−2 count−1. We also performed fits of observation 6 with an ab- sorbed black-body. Since the column density assumed a value significantly below that resulting from the interstellar medium, the power law fit was favoured. In all cases the best-fit absorb- ing column density is consistent (within 2-σ) with the Galactic absorption along the line of sight of IGR J11215–5952. This value is significantly lower than the column density measured with RXTE/PCA during the 2006 outburst (Smith et al., 2006b), (11±3)×1022 cm−2, which is likely overestimated because it was derived with RXTE/PCA in the energy range 2.5–15 keV. To investigate spectral variations, we created the contour lev- els for the column density vs. the photon index. The most inter- esting example is shown in Fig. 3. They indicate that the photon index showed significant variations, with the spectrum soften- ing as the outburst progresses, confirming the 2006 observation of the outburst tail; there is also evidence of an increasing ab- sorbing column density. These findings are independent on our choice of an absorbed power-law model. 4. Discussion and Conclusions We have carried out the most complete monitoring campaign of an outburst from a SFXT, thanks to the known periodicity of the 4 P. Romano et al.: The fifth outburst of IGR J11215–5952 observed by Swift outburst activity from IGR J11215–5952 (Paper I). This is re- markable, since the transient and unpredictable nature of the out- bursts from all other SFXTs hampers a similar extensive study, from the almost “quiescent” level up to the “flaring” activity. The entire “outburst event” was monitored for 23 days. The source was under the threshold of detectability in the early days of the campaign, with a luminosity below 3.7×1033 erg s−1. On Feb 9 the source underwent a bright outburst up to ∼ 1.1×1036 erg s−1. The bright part of the outburst (Feb 9; see Fig. 1b) is composed of at least five flares, with variable peak flux, each with a du- ration of ∼ 15 min–2 hours. This bright flaring activity lasted about 1 day, then the source underwent a decline phase, not flat, but composed of other equally short flares, one order of magni- tude fainter. This decline phase lasted about 5 days, and then the source faded to a much fainter level, almost below the thresh- old of detectability. The whole outburst lasted about 15 d, after which the source became fainter than 1.2 ×1033 erg s−1. Thus, IGR J11215–5952 reached the typical luminosity of SFXTs dur- ing outburst (around 1036 erg s−1), showing a dynamic range larger than 103 and a hard X–ray spectrum, proper of this kind of sources. The brightest part of the outburst lasted less than a day, on 9th February, and would have been the only flaring activity seen with less sensitive instruments. Indeed up to now, observa- tions of outbursts from SFXTs have been mostly performed with instruments on-board RXTE and/or INTEGRAL, which could only catch the brightest flares, and missed the complete evo- lution of the phenomenon, from the onset of the outburst, and down again to the level of almost quiescence, which is expected at a level around ∼ 1032 erg s−1 (possible magnetospheric emis- sion plus the contribution from the soft X–ray emission from the OB supergiant). Only during XMM-Newton and Chandra ob- servations of IGR J17544-2619 (González-Riestra et al., 2004; in’t Zand, 2005) outbursts were observed starting from the qui- escent emission, but the “post-flare” phases could not be com- pletely followed and thus the duration of the entire outburst phase could not be measured. For IGR J11215–5952 we could exceptionally observe the whole phenomenon, which for the first time reveals that the “short outbursts” (the “flares” lasting minutes or few hours), are actually part of a much longer “outburst event” (lasting several days), which we believe is triggered at the periastron passage in a wide, highly eccentric orbit. Indeed, the IGR J11215–5952 out- burst recurrence time (329 days) is remarkably stable and reveals an underlying clock, which can be naturally associated with the orbital motion in a non-circular orbit. The short flares most prominent on Feb 9 are probably produced by the episodic accre- tion of clumps from the massive wind (Owocki & Cohen, 2006), or by an inhomogeneous accretion stream near periastron (sim- ilar to what proposed to explain the periodic outbursts from the eccentric X-ray pulsar GX 302-1, e.g. Leahy 1991). Thus, both mechanisms originally proposed to explain the SFXTs outbursts, seem to be at work in IGR J11215–5952, i.e., accretion at perias- tron passage in a wide eccentric orbit (Negueruela et al., 2006), and accretion from clumpy winds, (in’t Zand, 2005). Applying a spherically symmetric homogeneous wind model to a B1 Ia spectral type companion, with a mass of 39 M⊙, 42 solar radii, and a wind mass loss of 3.67×10−6 M⊙ yrs −1 (Vink et al., 2000), the short outburst duration implies an eccentricity larger than 0.9. From the spectroscopy of the single flares there is evidence for only minor variations in the local absorbing column density (which would suggest the clear presence of clumps). This may be partly due to the high column density along the line of sight that absorbs most of the radiation below 1 keV, thus preventing us from detecting comparatively small variations of the intrinsic Table 2. Spectral fit results. Spectruma NH Γ χ 2 (d.o.f.)/ L1−10 keV (1022 cm−2) C-stat(%)c (erg s−1) 001 (2006) 0.88+0.96 −0.62 1.89 +1.07 −0.92 167.7 (65.8%) 2.35 × 10 001–005 2.28+2.21 −1.50 1.34 +1.11 −0.96 225.9 (65.8%) 4.32 × 10 006 1.04+0.25 −0.20 1.00 +0.16 −0.14 1.04 (83) 4.78 006 Flare 1 0.85+0.46 −0.32 0.94 +0.31 −0.28 0.66 (17) 8.68 006 Flare 2 1.11+0.79 −0.49 0.91 +0.42 −0.32 1.16 (25) 8.32 006 Flare 3 0.83+0.62 −0.42 0.82 +0.44 −0.40 0.93 (10) 11.1 006 Flare 4 0.88+0.38 −0.31 1.03 +0.32 −0.31 623.6 (39.0%) 4.86 006 Flare 5 2.02+1.01 −0.79 1.51 +0.58 −0.53 436.0 (54.6%) 2.75 007–008 1.75+1.03 −0.83 1.94 +0.64 −0.60 518.3 (66.0%) 2.22 × 10 009–012 1.04+0.47 −0.36 1.48 +0.39 −0.35 576.9 (53.6%) 8.82 × 10 007–012 1.86+0.68 −0.44 1.92 +0.43 −0.38 1.09 (18) 1.40 × 10 007–018 2.04+0.62 −0.50 2.08 +0.41 −0.37 1.19 (19) 5.96 × 10 a Last three digits of observation numbers, see Table 1, column 1. b Luminosity in the 1–10 keV band in units of 1035 erg s−1 obtained from the spectral fits. c Cash statistics (C-stat) and percentage of Monte Carlo realizations that had statistic < C-stat. We performed 104 simulations. column density. However, XRT data show evidence of softening of the spectrum in the long decay to quiescent state (thus con- firming the 2006 observation of the outburst tail) and a possible evidence of an NH growth connected with the same transition. Acknowledgements. We thank the Swift team for making these observations possible, in particular the duty scientists and science planners M. Chester, S. Hunsberger, J. Kennea, C. Pagani and J. Racusin; we thank N. Gehrels for ap- proving this ToO and D. Burrows for a winning observing strategy. We thank S. Campana, P. D’Avanzo, A. Paizis, P. Persi, V.F. Polcaro, and S. Vercellone for insightful discussions. This research has made use of NASA’s Astrophysics Data System Bibliographic Services as well as the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work was supported by MIUR grant 2005-025417, and contract ASI/INAF I/023/05/0. PR thanks INAF-IASFMi, where most of the work was carried out, for their kind hospitality. References Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, Space Science Reviews, 120, 165 Cash, W. 1979, ApJ, 228, 939 González-Riestra, R., Oosterbroek, T., Kuulkers, E., Orr, A., & Parmar, A. N. 2004, A&A, 420, 589 in’t Zand, J. J. M. 2005, A&A, 441, L1 Leahy, D. A. 1991, MNRAS, 250, 310 Lubinski, P., Bel, M. G., von Kienlin, A., et al. 2005, ATel, 469 Mangano, V., Romano, P., & Sidoli, L. 2007a, ATel, 995 Mangano, V., Romano, P., & Sidoli, L. 2007b, ATel, 996 Masetti, N., Pretorius, M. L., Palazzi, E., et al. 2006, A&A, 449, 1139 Negueruela, I., Smith, D. M., & Chaty, S. 2005, ATel, 470 Negueruela, I., Smith, D. M., Reig, P., Chaty, S., & Torrejón, J. M. 2006, in Proceedings of the “The X-ray Universe 2005”, 26-30 September 2005, El Escorial, Madrid, Spain. ESA SP-604, ed. A. Wilson, 165–170 Owocki, S. P. & Cohen, D. H. 2006, ApJ, 648, 565 Romano, P., Sidoli, L., & Mangano, V. 2007, ATel, 994 Sidoli, L., Mereghetti, S., Vercellone, S., et al. 2007, ATel, 997 Sidoli, L., Paizis, A., & Mereghetti, S. 2006, A&A, 450, L9 Smith, D. M., Bezayiff, N., & Negueruela, I. 2006a, ATel, 773 Smith, D. M., Bezayiff, N., & Negueruela, I. 2006b, ATel, 766 Steeghs, D., Torres, M. A. P., & Jonker, P. G. 2006, ATel, 768 Swank, J., Smith, D., & Markwardt, C. 2007, ATel, 997 Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, A&A, 362, 295 P. Romano et al.: The fifth outburst of IGR J11215–5952 observed by Swift 5 Table 1. Observation log. Sequence Start time (MJD) Start time (UT) End time (UT) Net Exposurea (yyyy-mm-dd hh:mm:ss) (yyyy-mm-dd hh:mm:ss) (s) 00030384001 53814.7853 2006-03-20 18:50:47 2006-03-20 19:01:53 643 00030881001 54135.5060 2007-02-04 12:08:34 2007-02-04 23:38:58 2048 00030881002 54136.5092 2007-02-05 12:13:12 2007-02-05 23:44:57 1865 00030881003 54137.1798 2007-02-06 04:18:55 2007-02-06 17:28:17 1941 00030881004 54138.1244 2007-02-07 02:59:06 2007-02-07 17:13:56 1213 00030881005 54139.0604 2007-02-08 01:27:02 2007-02-08 16:03:57 1403 00030881006 54140.0021 2007-02-09 00:03:05 2007-02-09 23:59:57 4668 00030881007 54141.6747 2007-02-10 16:11:33 2007-02-11 00:16:56 3141 00030881008 54142.0696 2007-02-11 01:40:15 2007-02-12 00:10:11 4232 00030881009 54143.6078 2007-02-12 14:35:17 2007-02-12 19:38:57 3337 00030881010 54144.0085 2007-02-13 00:12:17 2007-02-13 16:30:58 3091 00030881011 54145.0182 2007-02-14 00:26:09 2007-02-14 21:25:56 4521 00030881012 54146.0142 2007-02-15 00:20:26 2007-02-15 09:58:57 4590 00030881013 54147.6084 2007-02-16 14:36:05 2007-02-16 19:44:56 5230 00030881014 54148.6848 2007-02-17 16:26:09 2007-02-17 21:23:57 4295 00030881015 54149.2812 2007-02-18 06:44:59 2007-02-18 12:01:58 4804 00030881016 54150.2187 2007-02-19 05:14:52 2007-02-19 12:02:56 4636 00030881017 54151.6337 2007-02-20 15:12:28 2007-02-20 20:19:56 4847 00030881018 54152.0412 2007-02-21 00:59:16 2007-02-21 17:11:57 5194 00030881019 54153.4431 2007-02-22 10:38:04 2007-02-22 15:39:58 3814 00030881021 54155.1683 2007-02-24 04:02:20 2007-02-24 13:46:57 2963 00030881023 54157.5108 2007-02-26 12:15:37 2007-02-26 18:48:58 786 a The exposure time is spread over several snapshots (single continuous pointings at the target) during each observation. Introduction Observations and Data Reduction Analysis and Results Discussion and Conclusions

0704.0544

Crossover behavior in fluids with Coulomb interactions

Crossover behavior in fluids with Coulomb interactions O.V. Patsahan,1 J.-M. Caillol,2 and I.M. Mryglod1 1Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine 2Laboratoire de Physique Théorique CNRS UMR 8627, Bât. 210 Université de Paris-Sud 91405 Orsay Cedex, France (Dated: October 30, 2018) Abstract According to extensive experimental findings, the Ginzburg temperature tG for ionic fluids differs substantially from that of nonionic fluids [Schröer W., Weigärtner H. 2004 Pure Appl. Chem. 76 19]. A theoretical investigation of this outcome is proposed here by a mean field analysis of the interplay of short and long range interactions on the value of tG. We consider a quite general continuous charge-asymmetric model made of charged hard spheres with additional short-range interactions (without electrostatic interactions the model belongs to the same universality class as the 3D Ising model). The effective Landau-Ginzburg Hamiltonian of the full system near its gas-liquid critical point is derived from which the Ginzburg temperature is calculated as a function of the ionicity. The results obtained in this way for tG are in good qualitative and sufficient quantitative agreement with available experimental data. http://arxiv.org/abs/0704.0544v1 I. INTRODUCTION It is known that electrostatic forces determine the properties of various systems: phys- ical as well as chemical or biological. In particular, the Coulomb interactions are of great importance when dealing with ionic fluids i.e., fluids consisting of dissociated cations and ions. In most cases the Coulomb interaction is the dominant interaction and due to its long-range character can substantially affect the critical properties and the phase behavior of ionic systems. Thus, the investigations concerning these issues are of great fundamental interest and practical importance. Over the last ten years, both the phase diagrams and the critical behavior of ionic solu- tions have been intensively studied using both experimental and theoretical methods. These studies were stimulated by controversial experimental results, demonstrating the three types of the critical behavior in electrolytes solutions: (i) classical (or mean-field) and (ii) Ising-like behavior as well as (iii) crossover between the two [1, 2, 3, 4, 5, 6]. In accordance with these peculiarities, ionic solutions were conventionally divided into two classes, namely: “solvo- phobic” systems with Ising-like critical behavior in which Coulomb forces are not supposed to play a major role (the solvent is generally characterized by high dielectric constant) and “Coulombic” systems in which the phase separation is primarily driven by Coulomb in- teractions (the solvent is characterized by low dielectric constant). Hence the criticality of the Coulombic systems became a challenge for theory and experiment. A theoretical model which demonstrates the phase separation driven exclusively by Coulombic forces is a restricted primitive model (RPM) [7, 8]. In this model the ionic fluid is described as an electroneutral binary mixture of charged hard spheres of equal diameter immersed in a structureless dielectric continuum. Early studies [9, 10, 11] established that the model has a gas-liquid phase transition. A reasonable theoretical description of the critical point in the RPM was accomplished at a mean-field (MF) level using integral equation methods [8, 12] and Debye-Hückel theory [13]. Due to controversial experimental findings, the critical be- havior of the RPM has been under active debates [14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and strong evidence for an Ising universal class has been found by recent simulations [21, 24, 25] and theoretical [26, 27, 28, 29] studies. In spite of significant progress in this field, the criticality of ionic systems are far from being completely understood. The investigation of more complex models is very important in understanding the nature of critical behavior of real ionic fluids demonstrating both the charge and size asymmetry as well as other complexities such as short-range attraction. A description of a crossover region when the critical point is approached is of particular interest for such models. Based on the experimental findings one can suggest that in ionic fluids the temperature interval of crossover regime, characterized by the Ginzburg temperature, is much smaller than observed in nonionic systems [5]. In particular, a sharp crossover was reported for the systems Na−NH3 [30] (see also [31, 32, 33]). The analysis of experimental data for various ionic solutions confirmed that such systems generally exhibit crossover or, at least a tendency to crossover from the Ising behavior asymptotically close to the critical point, to the mean-field behavior upon increasing distance from the critical point [34]. Moreover, the systematic experimental investigations of the ionic systems such as tetra- n-butylammonium picrate, Bu4NPic, (for tetra-n-butylammonium picrate we will follow the notations from [5, 6]) in long chain n-alkanols with dielectric constant ranging from 3.6 for 1-tetradecanol to 16.8 for 2-propanol suggest an increasing tendency for crossover to the mean-field behavior when the Coulomb contribution becomes essential [5, 6, 35]. They also indicate that the ”Coulomb limit” reduced temperature of the RPM Tc ≃ 0.05 is valid for the almost non-polar long chain alkanols [6, 35]. It has been stressed [35] that for solutions of Bu4NPic in 1-alkanols, the upper critical solution points are found to increase linearly with the chain length of the alcohols (that corresponds to the decrease of dielectric constant of the solvent). The experimental data for the critical points and the dielectric permittivities for solutions of Bu4NPic in 1-alkanols are given in Table 1 [35]. Theoretically the crossover behavior in ionic systems was firstly studied for the RPM [14, 15, 16]. The results obtained for the Ginzburg temperature were similar to those found for simple fluids in comparable fashion that is in variance to what is expected from the ex- periments [5, 6]. Nearly at the same time in [36] the crossover behavior of the lattice version of a fluid exhibiting the Ising behavior was studied as additional symmetrical electrostatic interactions were turned on. Based on the microscopic ground, the effective Hamiltonian in terms of the fluctuating field conjugate to the number density was derived in this work. Then, the crossover between the mean-field and Ising-like behavior was estimated using the Ginzburg criterion. The resulting crossover temperature calculated as function of the ionicity I, which defines the strength of the Coulomb interaction relative to the short-range inter- action, indicates its weak dependence but with the trends correlating with those observed TABLE I: The experimental parameters of the critical points (critical temperature Tc, critical mass fraction wc) and the corresponding dielectric constants ǫ for solutions of Bu4NPic in 1-alkanols [35]. Solvent ǫ(Tc) Tc/K wc 1-oktanol 9.5 298.55 0.336 1-nonanol 7.9 308.64 0.325 1-decanol 6.4 318.29 0.3152 1-undecanol 5.4 326.98 0.303 1-dodecanol 4.7 335.91 0.2951 1-tridecanol 4.3 342.35 0.284 1-tetradecanol 3.6 351.09 0.2721 experimentally. In this paper we are also interested in the critical behavior of ionic fluids. In particular, we study the effect of the interplay of short-range and long-range interactions on the crossover behavior in such systems. We consider a continuous version of the charge-asymmetric ionic fluid in which both the long-range Coulomb and short-range van-der Waals-like interactions are included. Following [36] we introduce the ionicity I = 1 |q1q2| kBTǫσ , (1) where qi is the charge on ion i, kB is the Boltzmann constant, T is the temperature, σ is collision diameter and ǫ is the dielectric constant. Then we derive the effective Hamiltonian of the charge-asymmetric model in the vicinity of the gas-liquid critical point. As in [36], the coefficients obtained for the effective Hamiltoninan have the forms of expansions in the ionicity but with new terms that appear in this case. Based on this Hamiltonian we estimate the Ginzburg temperatures as functions of the ionicity. The layout of the paper is as follows. In Section 2 we introduce a continuous charge- asymmetric model with additional short-range attractive interactions included. We derive here the functional representation of the grand partition function of the model in terms of the fluctuating fields ϕSk and ϕ k conjugate to the total density and charge density, respectively. Section 3 is devoted to the derivation of the effective GLW Hamiltonian in the vicinity of the critical point. In Section 4 we calculate the Ginzburg temperature as a function of the ionicity for different values of the range of the attractive potential. We conclude in Section 5. II. BACKGROUND A. Model Let us start with a general case of a classical charge-asymmetric two-component system consisting of N particles among which there are N1 particles of species 1 and N2 particles of species 2. The pair interaction potential is assumed to be of the following form: Uαβ(r) = φ αβ (r) + φ αβ (r) + φ αβ(r), (2) where φHSαβ (r) is the interaction potential between the two additive hard spheres of diameters σα and σβ. We call the two-component hard sphere system a reference system. Thermody- namic and structural properties of the reference system are assumed to be known. φSRαβ (r) is the potential of the short-range (van-der-Waals-like ) attraction. φCαβ(r) is the Coulomb potential: φCαβ(r) = qαqβφ C(r)/ǫ, where φC(r) = 1/r and ǫ is the dielectric constant. The solution is made of both positive and negative ions so that the electroneutrality condition is satisfied,i.e. α=1 qαcα = 0, where cα is the concentration of the species α, cα = Nα/N . The ions of the species α = 1 are characterized by their hard sphere diameter σ1 and their electrostatic charge +q0 and those of species α = 2, characterized by diameter σ2, bear opposite charge −zq0 (q0 is elementary charge and z is the parameter of charge asymmetry). In general, the two-component system of hard spheres interacting via the potential φSRαβ (r) can exhibit both the gas-liquid and demixion critical points which belong to the 3D Ising model universal class. We consider the grand partition function (GPF) of the system which can be written as follows: Ξ[να] = α=1,2 exp(ναNα) (dΓ) exp Uαβ(rij) . (3) Here the following notations are used: να is the dimensionless chemical potential, να = βµα−3 lnΛ, µα is the chemical potential of the αth species, β is the reciprocal temperature, Λ−1 = (2πmαβ −1/h2)1/2 is the inverse de Broglie thermal wavelength; (dΓ) is the element of configurational space of the particles: (dΓ) = α dΓα, dΓα = dr 2 . . .dr Let us introduce the operators ρ̂Sk and ρ̂ ρ̂Sk = ρ̂k,α ρ̂ qαρ̂k,α, which are combinations of the Fourier transforms of the microscopic number density of the species α: ρ̂k,α = i exp(−ikrαi ). In this case the part of the Boltzmann factor entering eq. (3) which does not include hard sphere interactions can be presented as follows: (Uαβ(rij)− φHSαβ (rij)) = exp (φ̃SS(k)ρ̂ +φ̃DD(k)ρ̂ −k + 2φ̃SD(k)ρ̂ −k) + (φ̃SRαα (k) + q C(k)) , (4) where φ̃SS(k) = (1 + z)2 z2φ̃SR11 (k) + 2zφ̃ 12 (k) + φ̃ 22 (k) φ̃DD(k) = (1 + z)2 φ̃SR11 (k)− 2φ̃SR12 (k) + φ̃SR22 (k) + φ̃C(k) φ̃SD(k) = (1 + z)2 zφ̃SR11 (k) + (1− z)φ̃SR12 (k)− φ̃SR22 (k) with φ̃X...αβ (k) being a Fourier transform of the corresponding interaction potential defined by φ̃X...αβ (k) = drφX...αβ (r) exp(−ikr), φX...αβ (r) = φ̃X...αβ (k) exp(ikr). Now we simplify our model assuming that • The hard spheres will all be of the same diameter σα = σ. • φ̃SR++(k) = φ̃SR−−(k) = φ̃SR+−(k) = φ̃SR(k). With these restrictions the uncharged system can only exhibit a gas-liquid critical point and a possible demixion is ruled out. Taking into account the assumptions mentioned above we thus have φ̃SS(k) = φ̃ SR(k) < 0, φ̃DD(k) = φ̃ C(k) > 0, φ̃SD(k) ≡ 0. Finally it will be convenient to introduce the effective range bSR of short-range interactions through the relations φ̃SR(k) = φ̃SR(0) 1− (bSR k)2 +O(k4) . (6) B. Functional representation of the grand partition function of an ionic model Let us take advantage of the properties of Gaussian functional integrals to rewrite w̃S(k)ρ̂ (dϕS) exp [w̃S(k)] ρ̂Skϕ w̃C(k)ρ̂ (dϕD) exp [w̃C(k)] ϕDk ϕ ρ̂Dk ϕ NwS = (dϕS) exp [w̃S(k)] NwC = (dϕD) exp [w̃C(k)] ϕDk ϕ (dϕA) = dϕAk = d(ℜϕAk )d(ℑϕAk ), A = S,D. In the above equations we also introduced the notations w̃S(k) = −βφ̃SS(k)/V and w̃C(k) = φ̃ C(k)/V . As a result, we can rewrite Ξ[να] in the form of a functional integral Ξ[να] = (dϕS)(dϕD) exp −H[να, ϕS, ϕD] , (7) where the action H reads as H[να, ϕS, ϕD] = [w̃S(k)] [w̃C(k)] ϕDk ϕ − ln ΞHS[νS + ϕS, νD + iβ1/2ϕD], (8) 1 + z ν̄1 + 1 + z ν̄2, ν̄D = q0(1 + z) (ν̄1 − ν̄2). (9) where the ”renormalized” chemical potentials να are defined as να = να + −w̃S(k) + βq2αw̃C(k) , α = 1, 2. (10) Let us define ∆νS = νS−ϕS0 and ϕ̃Sk = ∆νS+ϕSk with ϕS0 chosen as the chemical potential of the hard spheres. This leads to the relation νS + ϕ S = ϕS0 + ϕ̃ S. (11) Now we present ln ΞHS[. . .] in the form of a cumulant expansion ln ΞHS[. . .] = k1,...,kn 0 , νD; k1, . . . , kn]ϕ̃ . . . ϕ̃Dkin ϕ̃Skin+1 . . . ϕ̃Sknδk1+...+kn , (12) where M 0 , νD; k1, . . . , kn] is the nth cumulant (or the nth order truncated correlation function) defined by 0 , νD; k1, . . . , kn] = ∂n ln ΞHS[. . .] . . . ∂ϕ̃D kin+1 . . . ∂ϕ̃S . (13) In particular it follows from (13) that 0 = lnΞHS[ϕ 0 , νD]. (14) The expressions for the cumulants of higher order (for in ≤ 4) are given in Appendix A. It should be noted that, contrary to [36], (12) includes all powers (even and odd) of the field ϕSk conjugate to the total number density. It should be clear that the coefficients in the cumulant expansion (12) depend on the chemical potential (or, equivalently, on the density). III. EFFECTIVE HAMILTONIAN IN THE VICINITY OF THE CRITICAL POINT Taking into account (12) we can rewrite (7)-(8) as follows Ξ[να] = (dϕ̃S) exp [w̃S(k)] ϕ̃Skϕ̃ + [w̃S(0)] ∆νSϕ̃S0 + k1,...,kn 0 , νD]ϕ̃ . . . ϕ̃Skn ×δk1+...+kn)V[ϕ̃Sk], (15) where H = 1 [w̃S(0)] (∆νS)2 − ln ΞHS[ϕS0 ], V[ϕ̃Sk] = (dϕ̃D) exp [w̃C(k)] ϕ̃Dk ϕ̃ k1,k2,k3 ϕ̃Dk2ϕ̃ δk1+k2+k3 + k1,...,k4 ϕ̃Dk2ϕ̃ ×ϕ̃Sk4δk1+...+k4 + k1,...,k4 ϕ̃Dk2ϕ̃ ϕ̃Sk4δk1+...+k4 + . . . . (16) It is worth noting here that unlike to the case considered in [36] we obtain in (16) terms proportional to and ϕ̃S . While the former is connected with an absence of a lattice symmetry, the the latter stems from charge asymmetry. Our aim now is to derive the effective Landau-Ginzburg (LG) Hamiltonian. Since we are interested in the gas-liquid critical point, this Hamiltonian should be written in terms of fields ϕ̃Sk conjugate to the fluctuation modes of the total number density. To this end we integrate out ϕ̃Dk in (16) using a Gaussian measure. As a result, we can present V[ϕ̃Sk] as follows: V[δϕ̃Sk] = 1 + 〈A〉G + 〈A2〉G + 〈A3〉G + . . . , (17) where 〈. . .〉G means 〈. . .〉G = (dϕD) . . . exp W̃C(k)ϕ with W̃C(k) given by W̃C(k) = [w̃C(k)] + y2G̃1 (18) and y2 being the ionicity introduced by (1): y2 = I Taking into account (1) and the recurrence formulas of Appendix A A may be written as a formal expansion in terms of y2 A = −y k1,k2,k3 G̃2(k1, k2 + k3)ϕ ϕDk2ϕ δk1+k2+k3 − k1,...,k4 G̃3(k1, k2, k3 + k4) ×ϕDk1ϕ ϕSk3ϕ δk1+...+k4 − (1− z)√ k1,...,k4 G̃2(k1, k2 + k3 + k4) ×ϕDk1ϕ ϕDk3ϕ δk1+...+k4 + . . . . (19) In (18)-(19) the “tilde” over ϕ was omitted for the sake of simplicity. It should be mentioned that the dependence of G̃n(k1, k2, . . . , kn) on the ki is very compli- cated. Since we consider here the behavior of the system near the critical point the limiting case of ki = 0 is of particular interest. Therefore, we substitute in (17) G̃n(k1, k2, . . . , kn) ≡ G̃n(0, . . .) n ≥ 3 G̃2(k) = G̃2(0)(1 + g 2k2), (20) G̃22(0) 2G̃2(0) , G̃22(0) = ∂2G̃2(k) |k=0. (21) Having integrated out eq. (17) Ξ[να] takes the form: Ξ[να] = 1 + y2〈N〉HSw̃C(k) (dϕS) exp −Heff [ϕS] Heff [ϕS] = − k1,...,kn . . . ϕ̃Sknδk1+...+kn, (22) where we have for the coefficients an a0 = −H, (23) a1 = 〈N〉HS + [w̃S(0)]−1∆νS − G̃2(0) ∆̃(q) + 3G̃3(0) + (1− z)2 − 2z ×G̃2(0) ∆̃(q) , (24) a2 = −[w̃S(k)]−1 + G̃2(k)− G̃3(0) ∆̃(q) + [G̃2(0)] ∆̃(q)∆̃(| k + q |), (25) a3 = G̃3(0)− G̃4(0) ∆̃(q) + y4G̃2(0)G̃3(0) ∆̃2(q), (26) a4 = G̃4(0)− G̃5(0) ∆̃(q) + 3[G̃3(0)] 2 + 4G̃2(0)G̃4(0) ∆̃2(q), (27) and the propagator ∆̃(q) is written as ∆̃(q) = ∆̃(q; y2) = [W̃C(q)] w̃C(q) 1 + y2〈N〉HSw̃C(q) . (28) Coefficients (23)-(27) have the form of a formal expansion in terms of the ionicity I = y2. In our study all terms which do not exceed the fourth order of y are kept. The ionicity is small enough for large values of the dielectric constant and increases with its decrease. From this point of view we can consider the expansions in (23)-(27) for large values of y2 only as formal ones. It should be also noted that ∆̃(q) ∼ 1/y2 (see (28)) for large values of y2. Let us introduce rSR = [G̃2(0)w̃S(0)] −1 − 1 = T − Tc,0 , (29) where Tc,0 = Tc(I = 0) is the mean-field critical temperature of the uncharged system. Taking into account (29) we can rewrite −Heff as follows: −Heff [ϕS] = −1 [w̃S(0)] (∆νS)2 + lnΞHS[ϕ r0 + τ k1,k2,k3 ϕSk1ϕ ϕSk3δk1+k2+k3 − k1,k2,k3,k4 ϕSk1ϕ ϕSk3ϕ ×δk1+k2+k3+k4 − h0ϕS0 , (30) where ϕSk = G̃2(0)ϕ k and the following notations were introduced: r0 = rSR + G̃3(0) G̃2(0) ∆̃(q)− y G̃2(0) ∆̃2(q) (31) τ 20 = τ G̃2(0) ∆̃(q)∆̃(2)(q) (32) v0 = − [G̃2]1/2 G̃3(0) G̃2(0) G̃4(0) G̃2(0) ∆̃(q) + G̃3(0) ∆̃2(q) , (33) u0 = − G̃4(0) G̃2(0) G̃5(0) G̃2(0) ∆̃(q) + ∆̃2(q) G̃2(0) 3[G̃3(0)] 2 + 4G̃2(0)G̃4(0) , (34) h0 = −[G̃2(0)]1/2 〈N〉HS + [w̃S(0)]−1∆νS G̃2(0) ∆̃(q) + G̃3(0) G̃2(0) (1− z)2 − 2z ∆̃(q)  , (35) where − τ 2SR = g2 + b̄2SR G̃2(0) [w̃S(0)] with b̄2SR = b SRw̃S(0) and ∆̃ (2)(q) = ∂2∆̃(| k+ q |)/∂k2|k=0. Finally, we present (30) as follows: −Heff [ϕS] = −1 [w̃S(0)] (∆νS)2 + lnΞHS[ϕ r + τ 2k2 〈N〉1/2HS k1,k2,k3 ϕSk1ϕ ϕSk3δk1+k2+k3 − 〈N〉HS k1,...,k4 ϕSk1ϕ ϕSk3ϕ ×δk1+k2+k3+k4 − h〈N〉 0 , (37) r = r0, τ 2 = τ 20 , v = 〈N〉1/2HS u = 〈N〉HS, h = h0〈N〉−1/2HS . At the critical point the following equalities hold r = 0, v = 0, h = 0, which give the equations for the critical parameters i.e., the temperature, the density and the chemical potential at the critical point. Equation (37) gives the effective GLW Hamiltonian of the system (2) in the vicinity of the critical point. We are now in position to extract from eq. (37) the Ginzburg temperature as a function of the ionicity. Now let us specify the short-range attraction, φSR(r), in the form of the square-well potential φSR(r) = 0, 0 ≤ r < σ −ε, σ ≤ r < λσ 0, r ≥ λσ It is worth noting here that the system of hard spheres interacting through the potential φSR(r) with λ = 1.4− 1.7 reasonably models most simple fluids [37]. The Fourier transform of φSR(r) for the case of the Weeks-Chandler-Andersen (WCA) regularization inside the hard core [38] has the form: φ̃SR(k) = φ̃SR(0) (λx)3 [−λx cos(λx) + sin(λx)], (38) where x = kσ and φ̃SR(0) = −εσ3 4π To be consistent we also use the WCA regularization scheme for the Coulomb potential which yields φ̃C(x) = 4π sin(x)/x3. (39) IV. GINZBURG TEMPERATURE Following [36] we can present the Ginzburg temperature by tG[ηc(y), y] ≃ u2(y) [1 + t0(y)]τ 6(y) , (40) but in our case all quantities u, t0 and τ 2 should be estimated at critical density ηc(y): u(y) = u(ηc(y), y), t0(y) = t0(ηc(y), y), τ 2(y) = τ 2(ηc(y), y). The density η enters the expressions for u, t0 and τ 2 through the structure factors S̃n. A well-known criterium by Ginzburg predicts that the mean-field theory is valid only when tG <<| t | where t = T−Tc(y)Tc(y) and Tc(y) are the mean-field reduced temperature and the mean-field critical temperature of the charged system at η = ηc(y), respectively. In (40) t0(y) measures the increase of the mean-field temperature of the charged system in respect to the uncharged system t0(y) = Tc(y) − 1 (41) which, for the model under consideration has the form: t0(y) = − G̃3(0) G̃2(0) ∆̃(q) + G̃2(0) ∆̃2(q). (42) Taking into account that [ G̃2(0)w̃C(0) and equation (41), we can rewrite (36) as follows: − τ 2SR = g2 + w̃C(0) (1 + t0(y)). (43) For the uncharged model the Ginzburg temperature reduces to tG(I = 0)) = S̃42 [τSR(I = 0)]6 where S̃n is given by (45) and τSR(I = 0) = τSR(t0 = 0). First we calculate the critical density from the equation v = 0. To this end we take into account (33), (39) and the formulas of Appendix D. As a result, we obtain the dependence of the dimensionless critical density ηc (η = πρσ 3/6) on the ionicity I = y2 (see Fig. 1). 0 4 8 12 16 20 0.105 0.110 0.115 0.120 0.125 0.130 FIG. 1: Critical density as a function of I. In order to calculate the chemical potential at the critical point we introduce ∆ν = ∆νS − νSMF , where νSMF = −〈N〉HSw̃S(0) is the mean-field value of the chemical potential νS(0). ∆νc is obtained from the condition h = 0; taking into account (35) it yields : ∆νc = ∆̃(q)− y 3S̃3 + (1− z)2 − 2z ∆̃(q) , (44) where S̃n(ηc; 0) = G̃n/〈N〉HS (45) is the nth particle structure factor at the critical density ηc(I) when ki = 0. In Fig. 2 ∆νc is displayed as a function of the ionicity for different values of the parameter z. Now we calculate τ 2, u, t0 and tG at η = ηc using (32), (34), (38)-(39), (42) and formulas from Appendices B-D. The dependence of τ 2 on I at different values of the parameter λ is plotted in Fig. 3. The explicit formula for τ 2SR is given in Appendix C. The coefficient u and the shift in the mean-field critical temperature, t0, as functions of I are plotted in Figs. 4 and 5. As is seen, quantities τ 2, t0 and u are increasing functions of I in the whole region under consideration and their dependencies of I are at variance with those obtained in [36] for the lattice model. 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 2: ∆νc as a function of I calculated from (44) for different values of z (η = ηc). The inset depicts the behavior of ∆νc close to the origin. 0 5 10 15 20 λ=1.3 λ=1.5 λ=1.7 λ=2.0 λ=2.2 λ=2.5 FIG. 3: The dependence of τ2 on the ionicity for different λ (η = ηc). Despite this fact, the behavior of the Ginzburg temperature as a function of I calculated in this work is qualitatively similar to that found in [36] (see Figs. 6-8). Moreover, as in [36], the behavior of tG(I) becomes nonmonotonic starting with some value of the attraction potential range (λ in our case). One can see in Fig. 7 that, for λ = 2, tG first drop off (at very small values of the ionicity) then increases slightly and at I ≃ 1.23 again starts to 0 5 10 15 20 FIG. 4: The dependence of u on the ionicity I (η = ηc). 0 4 8 12 16 20 FIG. 5: The reduced shift of the mean-field critical temperature, t0, as a function of I at η = ηc. decrease. In Fig. 8 the ratio of reduced Ginzburg temperatures, tG(I)/tG(0), is shown at different values of λ. It is worth noting that the non-monotonic behavior of tG(I) becomes more pronounced as λ increases. In Table 2 we compare our results for the ionicity dependence of the Ginzburg temper- ature (at λ = 1.5) with the results obtained in [36] for the lattice model as well as with experimental data for the crossover temperatures t× (data for I and t× are taken from [36]). The systems (b)-(d) correspond to the same ionic species Bu4NPic within solvents of different dielectric constant. As is seen, in this case our results are in good agreement (qual- 0 5 10 15 20 0.00 0.02 0.04 0.06 0.08 0.10 0.0920 0.0925 0.0930 0.0935 0.0940 FIG. 6: The reduced Ginzburg temperature, tG, as a function of I at λ = 1.5 (η = ηc). The inset depicts the behavior of tG(I) close to the origin. 0 5 10 15 20 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.00 0.01 0.02 0.03 0.01070 0.01072 0.01074 FIG. 7: The same as in Fig. 6 but at λ = 2. itative and quantitative) with the experimental findings. The system (d) is Na in NH3 and, of course, might be described by the potential φSR(r) with the different attraction range λ. For instance, for λ = 2 we obtain tG(I = 6.97) = 0.8×10−2 (see Fig. 7) that correlates with the experimental value t× = 0.6× 10−2 0 5 10 15 20 λ=1.3 λ=1.5 λ=1.7 λ=2.0 λ=2.2 λ=2.5 FIG. 8: The ratio tG(I)/tG(0) as a function of the ionicity at different values of λ (η = ηc). TABLE II: Experimentally assessed crossover temperature, t×, taken from [36]: (a) tetra-n- butylammonium picrate (Bu4NPic) in 1-tridecanol; (b) Bu4NPic in 1-dodecanol; (c) Bu4NPic in 75% 1-dodecanol plus 25% 1,4-butanediol; (d) Na in NH3; (e) tetra-n-pentylammonium bromide in water and the reduced Ginzburg temperature, tG, found theoretically in [36] and in this work. System Ionicity,I t× tG ([36]) tG (this work) uncharged fluid 0 O(I) 1 ∼ 0.09 (a) 17.9 ∼ 10−3 ∼ 0.712 2.7 × 10−3 (b) 16.8 ∼ 0.9 × 10−2 ∼ 0.717 0.38 × 10−2 (c) 8.9 ∼ 3× 10−2 ∼ 0.777 2.5 × 10−2 (d) 6.97 ∼ 0.6 × 10−2 ∼ 0.807 3.7 × 10−2 (e) ∼ 1.4 O(I) 1 ∼ 0.09 V. SUMMARY In this paper we study the reduced Ginzburg temperature as a function of the interplay between the short- and long-range interactions. The ionic fluid is modelled as a charge asymmetric continuous system that includes additional short-range attractions. The model without Coulomb interactions exhibits a gas-liquid critical point belonging to the Ising class of criticality. We derive an effective GLW Hamiltonian for the model whose coefficients have the form of an expansion in powers of the ionicity. Using these coefficients we calculate a Ginzburg temperature depending on the ionicity. To this end we introduce a specific model which consists of charged hard spheres of the same diameter interacting through the additional square-well potentials. To study the effect of the interplay between short- and long-range interactions we change, besides the ionicity, the range of the square-well potential. As a result, we obtain the similar tendency for the reduced Ginzburg temperature as in [36] when the region of the short-range attraction increases i.e., its nonmonotonic charac- ter but with different numerical characteristics. However, our results demonstrate a much faster decrease of the Ginzburg temperature when the ionicity increases. We found a good qualitative and sufficient quantitative agreement with the experimental findings for Bu4NPic in n-alkanols. This confirms the experimental observations that an interplay between the solvophobic and Coulomb interactions alters the temperature region of the crossover regime i.e., the increase of the ionicity that can be related to the decrease of dielectric constant leads to the decrease of the crossover region. We suggest that the quantitative discrepancy of the results for tG obtained in [36] and in this work could be due to the fact, besides the difference in the symmetry of the two models, that the chemical potential (or density) dependence of the Hamiltonian coefficients was taken into account explicitly in our case. It should be noted that in the approximation considered in this paper only the critical chemical potential depends explicitly on the charge magnitude. In order to obtain the charge dependence of the other quantities terms of order higher than y2 should be taken into account into the effective Hamiltonian. Finally, we emphasize that the functional representation (7)- (8) allows to consider more complicated models in particular models including charge and size asymmetry. VI. APPENDICES A. Recurrence formulas for the cumulants Fourier space. n (k1, k2, . . . , kn) = G̃n(k1, k2, . . . , kn) n (k1, k2, . . . , kn) = 0 n (k1, k2, . . . , kn) = (i) βq2αcαG̃n−1(k1, k2, . . . , |kn−1 + kn|) n (k1, k2, . . . , kn) = (i) β3/2q3αcαG̃n−2(k1, k2, . . . , |kn−2 + kn−1 + kn|) n (k1, k2, . . . , kn) = (i) q2αcα G̃n−2(k1, k2, . . . , |kn−2 + kn−1 + kn|) q4αcα − 3 q2αcα G̃n−3(k1, k2, . . . , |kn−3 + . . .+ kn|) where G̃n(k1, k2, . . . , kn) is the Fourier transform of the n-particle truncated correlation function [39] of a one-component hard sphere system and summation over repeated indices is meant. B. The nth-particle structure factors of a one component hard sphere systems in the Percus-Yevick approximation S2(0) = (1− η)4 (1 + 2η)2 , (46) S3(0) = (1− η)7(1− 7η − 6η2) (1 + 2η)5 , (47) S4(0) = (1− η)10(1− 30η + 81η2 + 140η3 + 60η4) (1 + 2η)8 , (48) S5(0) = (1− η)13(1− 85η + 957η2 − 1063η3 − 3590η4 − 2940η5 − 840η6) (1 + 2η)11 C. Explicit expression for τ2SR Let us write the Ornstein-Zernike equation in the Fourier space S̃2(k) = 1− ρc̃(k) , (50) where c̃(k) is the Fourier transform of the Ornstein-Zernike direct correlation function [40] We have for c̃(k) in the Percus-Yevick approximation [41] ρc̃(k) = −24η αk3(sin(k)− k cos(k)) + βk2(2k sin(k)− (k2 − 2) cos(k)− 2) ηα((4k3 − 24k) sin(k)− (k4 − 12k2 + 24) cos(k) + 24) /k6, (51) where (1 + 2η)2 (1− η)4 , β = −6 η(1 + 1 (1− η)4 From (50) and (51) we get for g2 G̃22(0) 2G̃2(0) = 0.05η (4η6 − 27η5 + 84η4 − 146η3 + 144η2 − 75η + 16) (1 + 2η)2(1− η)4 . Taking into account (38) we have b2SR/w̃S(0) = 0.1λ 2. As a result, τ 2SR is as follows τ 2SR = −0.05 (4η6 − 27η5 + 84η4 − 146η3 + 144η2 − 75η + 16) (1 + 2η)2(1− η)4 + 2λ 2(1 + t0(y)) , (52) where t0(y) is given by (42). D. Explicit expressions for the integrals used in equations (31)-(35) Using (2π)3 (dk) we can present ∆̃(k) = dxx2∆(x), (53) ∆̃(k) π〈N〉HS , (54) ∆̃(k)∆̃(2)(k) = 32ησ2 π〈N〉HS dxx∆(x) (2f1(x) + xf2(x)) , (55) where the following notations are introduced: ∆(x) = sin (x) x3 + κ∗ sin (x) , (56) f1(x) = x2 (cos (x) x− 3 sin (x)) x6 + 2 κ∗ x3 sin (x) + κ∗ 4 − κ∗4 cos2 (x) f2(x) = −x x5 sin (x) + x2κ∗ x2 cos2 (x) + 6 cos (x) x4 − 6 κ∗2x sin (x) cos (x) −12 x3 sin (x) + 6 κ∗2 − 6 κ∗2 cos2 (x) x9 + 3 κ∗ x6 sin (x) + 3 κ∗ −3 κ∗4x3 cos2 (x) + κ∗6 sin (x)− κ∗6 sin (x) cos2 (x) with x = kσ and κ∗ = κDσ = 24y2η being the reduced Debye number. [1] R.R. Singh and K.S. Pitzer, J. Phys. Chem. 92, 6775 (1990). [2] J.M.H. Levelt Sengers and J. A. Given, Mol. Phys. 80, 899 (1993). [3] K.S. Pitzer, J. Phys. Chem. 99, 13070 (1995). [4] K. 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[17] J.-M. Caillol, D. Levesque and J.-J. Weis, J. Chem. Phys. 107, 1565 (1997). [18] J. Valleau and G. Torrie, J. Chem. Phys. 108, 5169 (1998). [19] P.J. Camp and G.N. Patey, J. Chem. Phys. 114, 399 (2001). [20] E. Luijten, M.E. Fisher and A.Z. Panagiotopoulos, J. Chem. Phys. 114, 5468 (2001). [21] J.-M. Caillol, D. Levesque and J.-J. Weis, J. Chem. Phys. 116, 10794 (2002). [22] O.V. Patsahan, Condens. Matter Phys.7, 35 (2004). [23] O.V. Patsahan, I. Mryglod and J.-M. Caillol, J. Phys.: Condens. Matter 17, L251 (2005). [24] E. Luijten, M.E. Fisher and A.Z. Panagiotopoulos, Phys. Rev. Lett. 88, 185701 (2002). [25] Y.C. Kim and M.E. Fisher, Phys. Rev. Lett. 92, 185703 (2004). [26] A. Ciach and G. Stell, J. Mol. Liq. 87, 253 (2000). [27] A. Ciach and G. Stell, Int. J. Mod. Phys. B 21, 3309 (2005). [28] A. Ciach, Phys. Rev. E 73, 066110 (2006). [29] O. Patsahan and I. Mryglod, J. Phys.: Condens. Matter 16, L235 (2004). [30] P.Chieux, M.J. Sienko, J. Chem. Phys. 53, 566 (1970). [31] T.Narayanan and K.S. Pitzer, J. Phys. Chem. 98, 9170 (1994). [32] T.Narayanan and K.S. Pitzer, J. Chem. Phys. 102, 8118 (1995). [33] M.A. Anisimov, J.Jacob, A. Kumar, V.A. Agayan and J.V. Sengers, Phys. Rev. Lett. 85, 2336 (2000). [34] K. Gutkowskii, M.A. Anisimov and J.V. Sengers, J. Chem. Phys. 114, 3133 (2001). [35] M. Kleemeier, S. Wiegand, W. Schröer and H. Weigärtner, J. Chem. Phys. 110, 3085 (1999). [36] A.G. Moreira, M.M. Telo de Gama and M.E. Fisher, J. Chem. Phys. 110, 10058 (1999). [37] D.A. McQuarrie, Statistical Mechanics (Harper-Collins, New York, 1976). [38] J.D. Weeks, D. Chandler and H.C. Andersen, J. Chem. Phys. 54, 5237 (1971). [39] G. Stell, in Phase Transitionss and Critical Phenomena, 5b, edited by C. Domb and M.S. Green (Academic Press, New York, 1975). [40] J.P. Hansen and I.R. McDonald, Theory of simple liquids (Academic Press, 1986). [41] N.W. Ashcroft and J. Lekner, Phys. Rev. B 83, 5237 (1966). Introduction Background Model Functional representation of the grand partition function of an ionic model Effective Hamiltonian in the vicinity of the critical point Ginzburg temperature Summary Appendices Recurrence formulas for the cumulants Fourier space. The nth-particle structure factors of a one component hard sphere systems in the Percus-Yevick approximation Explicit expression for SR2 Explicit expressions for the integrals used in equations (??)-(??) References

0704.0545

Reply to Comment of Kenzelmann and Harris

Mostovoy Reply: In their Comment [1] Kenzelmann and Harris argue against the conclusion made in [2] that spiral magnets are in general ferroelectric. First of all, I believe, this conclusion was proved experimentally. The systematic search for ferroelectricity in magnets with spi- ral ordering recently led to a discovery of new multifer- roic materials, such as CoCr2O4 [3], MnWO4 [4, 5] and LiCu2O2 [6]. Furthermore, Kenzelmann and Harris argue that the continuum theory outlined in [2] leads to misleading pre- dictions about the magnetically-induced electric polar- ization. To prove their point, they consider two hy- pothetical spin configurations shown in Fig. 1 (c) and (d) of their Comment, and argue that the results of the continuum theory are incompatible with crystal symme- tries. While one cannot deny the importance of symme- try considerations, the arguments Kenzelmann and Har- ris are themselves very misleading. They incorrectly as- sert that for the spin configurations shown in Fig. 1 (c) and (d) ‘the spiral theory’ would predict electric polar- ization along, respectively, the c and a axes. The continuum model of multiferroics [2] is based on assumption that the spin state can be described by a sin- gle magnetization vector. For TbMnO3 (see Fig. 1b), where the wave vector of the magnetic spiral is along the b axis and spins are rotating in the bc plane, it predicts electric polarizationP along the c axis, in agreement with experiment. The magnetic structures (c) and (d) are of a different kind, as they are made of spirals rotating in op- posite directions. Thus in the configuration (c) there are two counter-rotating bc spirals in each ab plane, which is why the net polarization along the c axis is zero. Simi- larly, in the configuration (d) the ab spirals in neighboring bc planes rotate in opposite directions, resulting in zero net Pa. It is not difficult to modify the continuum model con- sidered in [2] to describe these more general magnetic orders. For more than one magnetic ion per unit cell one can introduce several independent magnetic order parameters, which increases the number of possible mag- netoelectric coupling terms. For instance, all three spin configurations shown in Fig. 1 of the Comment can be described by three antiferromagnetic order parameters L1 = S1 + S2 − S3 − S4, L2 = S1 − S2 + S3 − S4, L3 = S1 − S2 − S3 + S4 (the labels of the 4 Mn ions in the unit cell of TbMnO3 are the same as in [7]). The spiral configuration (b) can be described by a single order parameter L1 with nonzero Lb and Lc . As discussed in [2], the magneto- electric coupling linear in the gradient of the magnetic order parameter (Lifshitz invariant) allowed by symme- tries has the form P c , which gives rise to magnetically-induced P c. The configuration (c) is described by two different order parameters, Lb . The term Lc does not transform like any of the components of P, so that the induced polar- ization is zero. Finally, for the configuration (d) with nonzero Lb and La , the only possible coupling term is , allowing for nonzero P c. The point is, however, that the spin configurations (c) and (d) considered by Kenzelmann and Harris, are very artificial, as it is difficult to find a system where interac- tions between spins would favor the simultaneous pres- ence of counter-rotating spirals. The average interaction between counter-rotating spirals is zero, while for spirals with spins rotating in the same direction some interac- tion energy can always be gained by properly adjusting their relative phases. This is the reason why the simple model of Ref. [2] with a single vector order parameter successfully describes thermodynamics and magnetoelec- tric properties of many spiral multiferroics. Maxim Mostovoy Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands [1] M. Kenzelmann and A. B. Harris, Comment arXiv/cond-mat0610471. [2] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). [3] Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204 (2006). [4] K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, and T. Arima, Phys. Rev. Lett. 97, 097203 (2006). [5] O. Heyer et al., J. Phys. Condens. Matter 18, L471 (2006). [6] S. Park, Y. J. Choi, C. L. Zhang and S.-W. Cheong, to be published. [7] A. B. Harris and G. Lawes, arXiv/cond-mat0508617. http://arxiv.org/abs/0704.0545v1 References

0704.0546

Persistent Currents in Superconducting Quantum Interference Devices

Microsoft Word - SQubit.doc PERSISTENT CURRENTS IN SUPERCONDUCTING QUANTUM INTERFERENCE DEVICES F. Romeo Dipartimento di Fisica “E. R. Caianiello”, Università degli Studi di Salerno I-84081 Baronissi (SA), Italy R. De Luca CNR-INFM and DIIMA, Università degli Studi di Salerno I-84084 Fisciano (SA), Italy ABSTRACT Starting from the reduced dynamical model of a two-junction quantum interference device, a quantum analog of the system has been exhibited, in order to extend the well known properties of this device to the quantum regime. By finding eigenvalues of the corresponding Hamiltonian operator, the persistent currents flowing in the ring have been obtained. The resulting quantum analog of the overdamped two-junction quantum interference device can be seen as a supercurrent qubit operating in the limit of negligible capacitance and finite inductance. PACS: 74.50.+r, 85.25.Dq Keywords: Josephson junctions, d. c. SQUID, Qubit I INTRODUCTION The d. c. SQUID (Superconducting QUantum Interference Device) is a well known system, widely investigated in the literature [1-3]. This system, though not confined to atomic scale in its dimensions, has been proposed as the basic unit for quantum computing (qubit) by resorting to a characteristic feature of superconductivity: macroscopic quantum coherence [4]. In general, a qubit can be realized by means of a two-level quantum mechanical system [5]. Therefore, the quantum states of a qubit can be a linear combinations of the orthogonal basis 0 and 1 , so that the Hilbert space generated by this basis is two-dimensional. Alternatively, a qubit state can be represented by elements of an infinite-dimensional Hilbert space. In this case, however, the effective potential of the system must show a double-well potential, in such a way that one of the two stationary states can be defined as state 0 and the other as state 1 . The electrodynamic properties of d. c. SQUIDs can be analyzed by means of two-junction quantum interferometer models, where each Josephson junction is assumed to be in the overdamped regime. The simplest possible analysis of these systems is done by assuming negligible values of the inductance L of a single branch of the device, so that 0 β , where 0Φ is the elementary flux quantum, and 21 JJ = is the mean value of the maximum Josephson currents of the junctions. In this case, the dynamical equation for the superconducting phase differences 1φ and 2φ across the two junctions can be written as a single equation for the average phase variable 21 φφϕ = . This equation is similar to the nonlinear differential equation governing the time evolution of a single overdamped junction, so that it can be defined as an equivalent single junction model, and is written as follows: sincos Bex =+ ϕπψ , (1) where t τ , with 21 RRR = , 1R and 2R being the resistive junction parameters, exψ is the externally applied flux normalized to 0Φ and Bi is the bias current normalized to JI . Following the same type of approach, by means of a perturbation analysis, taking β as the perturbation parameter, it can be shown that, to first order in β , the equivalent single junction model can be written as follows for a symmetric SQUID with identical junctions [6]: ( ) ( ) ( ) 2sinsinsincos1 2 Bexex =Ψ+Ψ−+ ϕππβϕπ , (2) where n is an integer. This model allows, at least for small values of the parameter, to calculate in closed form some electrodynamic quantities, such as, for example, the amplitude of the half-integer Shapiro steps appearing in these systems [7]. It has also been shown that, by extending this model to SQUIDs with non-identical junction, one can obtain an effective classical double-well potential in which the transition from one state to the other can be enhanced by applying an opportune external magnetic flux [8]. However, this classical analysis by itself does not allow to define the quantum states of the system. Nonetheless, the equation of the motion (2) could be assumed to be a classical version of the time evolution of a quantum phase state. Therefore, the aim of the present work is to obtain, starting from the time evolution of the superconducting phase difference ϕ , the quantum mechanical Hamiltonian and to compute, by means of this quantum mechanical system, which in the classical limit reduces to Eq. (2), the persistent currents in the SQUID. It is interesting to notice that the resulting “Hamiltonian” quantum model derives from the overdamped limit of a classical dissipative system in the presence of a double well potential. The present analysis can be seen as an alternative approach to the study of the quantum properties of supercurrent qubits: It allows to study the response of the quantum system in the limit of negligible capacitance and finite values of the inductance, as opposed to the case usually considered in the literature, where negligible inductance and finite capacitance is assumed [9 - 10]. II FROM CLASSICAL TO QUANTUM MECHANICS Let us consider the classical dynamical equation ( )xfx =& for the state variable x, where the dot notation indicates the derivative with respect to the normalized time τ. Making use of the previous equation, taking the time derivative of both sides, we can obtain the equation of the motion of the quantity x& as follows: ( ) ( ) ( )xfxfxxfx xx == &&& , (3) where the notation ( )xf x stands for the partial derivative of ( )xf with respect to x. Given the above equation and following the procedure described by Huang and Lin [11], the Lagrangian associated to this problem is obtained in the form: ( )( )[ ]22 xfxL += & . (4) Starting from the Lagrangian L, the Legendre transformation allows us to get the following classical Hamiltonian: ( )( )[ ]22 xfH −= π , (5) where =π is the canonical momentum conjugated to the variable x, while ( ) ( )( )2 xfxU −= could be considered as an effective potential. We are now interested in the quantization of the classical model described so far. According to the standard procedure, the recipe to transform the classical Hamiltonian in a quantum operator is implemented by making the substitution xi∂−=→ππ ˆ , xxx =→ ˆ (in dimensionless units). From the previous definitions, the commutation rule [ ] ix =π̂,ˆ for the conjugated variables follows directly. Furthermore, the Hamiltonian operator can be written as ( )( )[ ]22 1ˆ xfH x +∂−= . The general procedure described above can be adopted to obtain the quantum model of an overdamped d.c. SQUID in the limit in which the reduced two-junctions interferometer model [6] can be applied. In the framework of this model, the phase dynamics can be written (in the homogeneous case) as in Eq. (2), so that the function ( )ϕff = takes the following form: ( ) ( ) ( )ϕϕγϕ 2sinsin baf −−= , (6) where ( )exa πψcos= , ( )exb πψπβ 2sin= , 2 Bi=γ , having chosen 0=n for simplicity. We notice that this analysis cannot be extended to the similar case, considered by Grønbech-Jensen et al. [12], of junctions with finite capacitance. Therefore, by setting ϕ=x and ϕπ &= in the above general analysis, we notice that the phase and the voltage across the two-junction quantum interferometer are conjugate variables of the system. In the present case, therefore, proceeding as we said, by squaring ( )ϕf and exhibiting the final result of the calculation in terms of the higher harmonics of the phase variable instead of powers of trigonometric functions, the following Hamiltonian operator is obtained: ( ) ( ) ( ) ( ) ( ) ( ) ( )γϕγϕγ ϕϕϕϕϕ ,,2sinsin babaab = , (7) where ( ) 222 ba γ is a flux dependent energy shift which will be important in the following discussion. In order to calculate the relevant physical quantities of the system, we introduce the orthonormal complete basis n = of the infinite-dimensional Hilbert space with the inner product == −∫ , where mn,δ is the Kronecker delta. In this representation the matrix elements of the Hamiltonian operator can be written as follows: ( ) ( ) ( ) ( ) ( ) ( ) ( )2,2,1,1,4,4, 1,1,, +−+−+−+− −+−+++++ ++++− mnmnmnmnmnmnmnmn mnmnmnmnmnmn aabban δδδδδ , (8) where the following useful relations have been used: ( ) ( )lmnlmninlm +− −= ,,2 sin δδϕ , (9a) ( ) ( )lmnlmnnlm +− += ,,2 cos δδϕ . (9b) Once the matrix elements of the Hamiltonian operator are known, we can diagonalize a reduced version of the complete infinite-dimensional matrix by introducing an energy cut-off. Such procedure can be safely carried out when we need to characterize low energy states which are located very far from the cut and when the number of the vectors in the basis of the reduced Hilbert space is able to capture the essential features of the low energy states. For instance, the Hilbert space spanned by the first 20 basis functions can be a very effective choice, if we need to study only the lowest energy states close to the ground state of our system. In fact, in our case we have noted that, by halving the number of the basis elements, no evident difference is present in the lowest energy eigenvalues. III PERSISTENT CURRENTS Following the procedure described above, in the present section we shall derive the behavior of the persistent currents associated to each eigenstate of the Hamiltonian as a consequence of the time reversal symmetry breaking provided by the magnetic flux. Such a current, in units of the Josephson current divided by π2 (i. e., in units of JE , where JE is the Josephson energy), can be defined as follows: −= , (10) where nε and exψ are the eigenvalues of the Hamiltonian and the normalized external magnetic flux, respectively. According to the above relation, the persistent current nI can be computed once the pertinent eigenvalue nε is known. Furthermore, it should be noticed that, in the absence of the off-diagonal terms in the Hamiltonian given in Eq. (8), the state independent persistent current computed by means of Eq. (10) would be given by: ( ) ( )[ ]exexI πψβππψπ 222 sin212sin4 −−= . (11) The solution of the full problem can thus be seen as the state dependent correction to the above relation induced by the off-diagonal terms. Last point can be well understood by analyzing Figs. 1a- b. In these figures, even tough we are in the presence of finite off-diagonal terms, the relation given in Eq. (11) is able to describe quite accurately the behavior of the persistent currents which appears insensitive to the state index due to the small value of β . When the value of β is raised (see Fig. 2a), the persistent currents starts to become weakly state sensitive and some deformation of the original shape occurs. Furthermore, the states of higher energy (see Fig. 2b) induce a behavior of the persistent current which is quite insensitive to the state index. A further raising of β (see Fig. 3a) induces a suppression of the persistent current carried by the first excited state in the vicinity of half integer values of the normalized applied magnetic flux. This implies that, in the low energy regime (i. e., when the quantum state can be written as SSS 1 += , where 0 and represent the ground state and the first excited state, respectively), the average persistent current 1 ISISI += close to an half integer flux is mainly related to the ground state properties of the system, since 10 II >> in the vicinity of 2 =exψ (for 2 ≠exψ ). In Fig. 4a, raising once again β , it can be noticed that, in the vicinity of half integer values of the normalized flux, the ground state and the first excited state carries currents of opposite sign, inducing a competing magnetic behavior. Therefore, the average persistent current, and its magnetic behavior, depend, on both coefficients of the decomposition (i. e., on S0 and S1 ). This last point implies that, by measuring the magnetic momentum of the system in a particular magnetic field configuration, we can obtain constraints on the nature of the quantum superposition. For instance, under these conditions, we could prepare the quantum state in such a way that the average persistent current is negligibly small in the vicinity of half integer values of the normalized applied magnetic flux. Furthermore, we point out that a double well potential can be obtained setting the model parameters as done in Fig.5 ( 2.0=β and 7.0=exψ ), where the potential ( )ϕU is shown. Indeed, we notice that for 0=γ two low-energy degenerate states are present, the degeneracy being removed by means of a small current bias. Such bias can drive the response of the system toward one of the two minima of the potential allowing a complete control of the quantum state which can be exploited for technological applications. Finally, we notice that, even thought the chosen β values in Figs. 2 – 5 are close to the validity limits of the first order approximation of the reduced model in ref. [6], the above characteristic response of the system remain qualitatively valid, since we are here considering the leading order in the value of β . IV CONCLUSION Starting from the reduced dynamical model of the two-junction quantum interference device, the applied flux dependence of persistent currents in this system has been studied in the quantum regime. The extension of the dissipative overdamped classical system, from classical to quantum mechanics, allows to consider the electrodynamical response of a supercurrent quantum bit in the limit of negligible capacitance and finite inductance. For null bias current and for opportune values of the externally applied magnetic flux, the quantum analog of the two-junction interferometer shows effective potential with a degenerate ground state; degeneracy can be removed by applying a control non-null bias current. In the literature, the quantum behavior of the two-junction quantum interference device is studied by considering the charging energy of the junctions as preponderant with respect to the energy of the circulating currents [5, 9, 10]. In the present work it is shown that it is possible to obtain an Hamiltonian quantum analog of d. c. SQUIDs containing overdamped junctions in the limit of null capacitance and finite inductance values. In this framework, a flux qubit can be realized, under quite different conditions than those with high junction capacitance value [5]. Finally, the present analysis can also be considered as a link between classical dissipative systems and their corresponding quantum mechanical models. REFERENCES 1. A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (Wiley, NY, 1982). 2. K. K. Likharev, Dynamics of Josephson Junctions and circuits, Gordon and Breach, Amsterdam, 1986. 3. J. Clarke and A. I. Braginski, Eds., The SQUID Handbook, Vol. I (Wiley-VCH, Weinheim, 2004). 4. M. F. Bocko, A. M. Herr and M. J. Feldman, IEEE Tans. Appl. Supercond. 7, 3638 (1997). 5. J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. Lett. 94, 090501 (2005). 6. F. Romeo, R. De Luca, Phys. Lett. A 328, 330 (2004). 7. C. Vanneste, C. C. Chi, W. J. Gallagher, A. W. Kleinsasser, S. I. Raider, and R. L. Sandstrom, J. Appl. Phys. 64, 242 (1988). 8. R. De Luca, F. Romeo, Phys. Rev. B 73, 214518 (2006). 9. G. Burkard, Phys. Rev B 71, 144511 (2005). 10. T. P. Orlando, J. E. Mooij, Lin Tian, Caspar H. van der Wal, L. S. Levitov, Seth Lloyd, J. J. Mazo, Phys. Rev B 60, 15398 (1999). 11. Y.-S. Huang and C.-L. Lin, Am. J. Phys. 70, 741 (2002). 12. N. Grønbech-Jensen, D. B. Thompson, M. Cirillo, C. Cosmelli, Phys. Rev. B 67, 224505 (2003). FIGURE CAPTIONS Fig. 1 (a) Persistent currents 1I (triangle) and 2I (box) plotted as a function of the applied external flux exψ and by fixing 075.0=β and 0=γ . (b) Persistent currents 3I (star) and 4I (diamond) plotted as a function of the applied external flux exψ and by fixing 075.0=β and 0=γ . Fig. 2 (a) Persistent currents 1I (triangle) and 2I (box) plotted as a function of the applied external flux exψ and by fixing 15.0=β and 0=γ . (b) Persistent currents 3I (star) and 4I (diamond) plotted as a function of the applied external flux exψ and by fixing 15.0=β and 0=γ . Fig. 3 (a) Persistent currents 1I (triangle) and 2I (box) plotted as a function of the applied external flux exψ and by fixing 2.0=β and 0=γ . (b) Persistent currents 3I (star) and 4I (diamond) plotted as a function of the applied external flux exψ and by fixing 2.0=β and 0=γ . Fig. 4 (a) Persistent currents 1I (triangle) and 2I (box) plotted as a function of the applied external flux exψ and by fixing 25.0=β and 0=γ . (b) Persistent currents 3I (star) and 4I (diamond) plotted as a function of the applied external flux exψ and by fixing 25.0=β and 0=γ . Fig. 5 Density plot of the potential ( )ϕU plotted as a function of the phase ϕ and of the normalized bias current γ by setting the remaining parameters as: 2.0=β and 7.0=exψ . Lower energy states are represented by darker regions in the plot. Fig. 1 0 0.2 0.4 0.6 0.8 1 �0.75 �0.25 0 0.2 0.4 0.6 0.8 1 Fig. 2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fig. 3 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fig. 4 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fig. 5 0 Π����� Π 3 Π���������� ����� ���������� 2 2 Π

0704.0547

Mid-Infrared Fine Structure Line Ratios in Active Galactic Nuclei Observed with Spitzer IRS: Evidence for Extinction by the Torus

Draft version August 6, 2018 Preprint typeset using LATEX style emulateapj v. 4/12/04 MID-INFRARED FINE STRUCTURE LINE RATIOS IN ACTIVE GALACTIC NUCLEI OBSERVED WITH SPITZER IRS: EVIDENCE FOR EXTINCTION BY THE TORUS R. P. Dudik , J. C. Weingartner , S. Satyapal , J. Fischer , C. C. Dudley , & B. O’Halloran Draft version August 6, 2018 ABSTRACT We present the first systematic investigation of the [NeV] (14µm/24µm) and [SIII] (18µm/33µm) infrared line flux ratios, traditionally used to estimate the density of the ionized gas, in a sample of 41 Type 1 and Type 2 active galactic nuclei (AGNs) observed with the Infrared Spectrograph on board Spitzer. The majority of galaxies with both [NeV] lines detected have observed [NeV] line flux ratios consistent with or below the theoretical low density limit, based on calculations using currently available collision strengths and ignoring absorption and stimulated emission. We find that Type 2 AGNs have lower line flux ratios than Type 1 AGNs and that all of the galaxies with line flux ratios below the low density limit are Type 2 AGNs. We argue that differential infrared extinction to the [NeV] emitting region due to dust in the obscuring torus is responsible for the ratios below the low density limit and we suggest that the ratio may be a tracer of the inclination angle of the torus to our line of sight. Because the temperature of the gas, the amount of extinction, and the effect of absorption and stimulated emission on the line ratios are all unknown, we are not able to determine the electron densities associated with the [NeV] line flux ratios for the objects in our sample. We also find that the [SIII] emission from the galaxies in our sample is extended and originates primarily in star forming regions. Since the emission from low-ionization species is extended, any analysis using line flux ratios from such species obtained from slits of different sizes is invalid for most nearby galaxies. Subject headings: Galaxies: Active— Galaxies: Starbursts— X-rays: Galaxies — Infrared: Galaxies 1. INTRODUCTION Mid-infrared (mid-IR) emission-line spectroscopy of active galactic nuclei (AGNs) is used to investigate the physical conditions of the dust-enshrouded gas that is in close proximity to the active nucleus. In particular, many spectral lines are emitted in the so-called narrow- line region (NLR) of these objects which typically ex- tends between tens to at most a thousand parsecs from the nucleus (Capetti, et al. 1995, 1997, Schmitt & Kin- ney 1996, Falcke et al. 1998; Ferruit et al. 1999, Schmitt et al. 2003). The NLRs of AGNs have been studied extensively us- ing optical spectroscopic observations. However, there have been very few systematic studies of the NLR using infrared spectroscopic observations. Infrared (IR) fine- structure emission lines have a number of special char- acteristics that have been regarded as distinct advan- tages, particularly in determining the electron density of the ionized gas very close to the central AGN. Infrared spectroscopic observations allow access to fine-structure lines from ions with higher ionization potentials than the most widely used optical diagnostic lines. This is impor- tant in many AGNs, where a significant fraction of the line emission from lower ionization species can originate in gas ionized by star forming regions. In addition, it is generally assumed that the density-sensitive infrared line ratios originate in gas with temperatures around 104 K and are less dependent on electron temperature vari- ations, enabling a more straightforward determination of the electron density in the ionized gas. Finally, it 1 George Mason University, Department of Physics & Astron- omy, MS 3F3, 4400 University Drive, Fairfax, VA 22030 2 Naval Research Laboratory, Remote Sensing Division, 4555 Overlook Ave SW, Washington DC, 20375 has long been assumed that the IR diagnostic line ratios are insensitive to reddening corrections–a serious impedi- ment to optical and ultraviolet observations, particularly in the NLRs of AGNs, where the dust composition and spatial distribution are highly uncertain. For these rea- sons, IR spectroscopic observations, especially since the era of the Infrared Space Observatory (ISO), have pro- vided us with some of the most reliable tools for studying the NLRs in AGNs. However, while there are clear ad- vantages of mid-IR fine-structure diagnostics in studying the physical state of the ionized gas, very little work has been done to investigate their robustness in determin- ing the gas densities of the NLRs in a large sample of AGNs. The Spitzer Space Telescope Infrared Spectrome- ter (IRS), with its extraordinary sensitivity and spectral resolution, offers the opportunity to examine for the first time the physical state of NLR gas in a large sample of AGNs. The focus of most previous comparative studies of the infrared fine-structure lines in AGNs has been on the excitation state of the ionized gas, in an effort to de- termine the existence and energetic importance of po- tentially buried AGNs and to constrain their ionizing radiation fields (Genzel et al. 1998, Lutz et al. 1999, Alexander & Sternberg 1999, Sturm et al. 2002, Satya- pal, Sambruna, & Dudik 2004, Spinoglio et al. 2005). Remarkably, very little work has been done in the in- frared on studying the line flux ratios traditionally used to probe the NLR gas densities in a significant number of AGNs. We present in this paper the first systematic in- frared spectroscopic study of the line flux ratios of [NeV] and [SIII] in order to 1) test the robustness of these line ratios as density diagnostics and 2) if possible, to probe the densities of the NLR gas in a large sample of AGNs. http://arxiv.org/abs/0704.0547v2 2. THE SAMPLE We searched the Spitzer archive for galaxies with an ac- tive nucleus and both high- and low-resolution Infrared Spectrometer (IRS; Houck et al. 2004) observations cur- rently available. Only those galaxies with indisputable optical, X-ray, or radio signatures of active nuclei (such as broad Hα or X-ray or radio point sources) were in- cluded in our sample. The sample includes three AGN subclasses: Seyferts, LINERs, and Quasars. The galax- ies in this sample span a wide range of distances (4 to 400 Mpc; median = 21 Mpc), Hubble types, bolomet- ric luminosities (log (LBOL) ∼ 40 to 46, median = 43), and Eddington Ratios (log(L/LEdd) ∼ -6.5 to 0.3; me- dian= -2.5). The entire sample consists of 41 galaxies. The basic properties of the sample are given in Table 1. The black hole masses listed in Table 1 were derived using resolved stellar kinematics, if available, reverber- ation mapping, or by applying the correlation between optical bulge luminosity and central black hole mass de- termined in nearby galaxies only when the host galaxy was clearly resolved. Bolometric luminosities listed in Table 1 were calculated from the X-ray luminosities for most objects. For Seyferts, the relationship LBOL = 10 × LX was adopted (Elvis 1994). For LINERs 1 we assumed LBOL = 34× LX , as derived from the spectral energy dis- tribution of a sample of nearby LINERs from Ho (1999) (see also Dudik et al. 2005 and Satyapal et al. 2005). The bolometric luminosities and black hole masses for quasars and radio galaxies were taken from Woo & Urry (2002) and Marchesini, Celotti, & Ferrarese (2004), re- spectively. A detailed discussion of our methodology and justification of assumptions for determining black hole masses and bolometric luminosities for the various AGN classes represented in Table 1 can be found in Satyapal et al. (2005) and Dudik et al. (2005). Table 1 also lists the AGN type (1 or 2) for the galaxies in our sample based on the presence or absence of broad (full width at half max (FWHM) exceeding 1000 km s−1) Balmer emission lines in the optical spectrum. We emphasize that the selection basis for the objects in our sample was on the availability of high resolution IRS Spitzer observations. The sample should therefore not be viewed as complete in any sense. 3. DATA ANALYSIS AND RESULTS We extracted archival spectral data obtained us- ing the short-wavelength, low-resolution module (SL2, 3.6”×57”, λ = 5.2-7.7µm) and both the short- wavelength, high-resolution (SH, 4.7”×11.3”, λ = 9.9- 19.6µm) and long-wavelength, high-resolution (LH, 11.1”×22.3”, λ = 18.7-37.2µm) modules of IRS. The data presented here were preprocessed by the IRS pipeline (version 13.0) at the Spitzer Science Center (SSC) prior to download. Preprocessing includes ramp fitting, dark-sky subtraction, droop correction, linear- ity correction, flat-fielding, and flux calibration2. The Spitzer data were further processed using the SMART v. 5.5.7 analysis package (Higdon et al. 2004). The slit for 1 We include all galaxies that are classified as LINERs using either the Heckman (1980) or Veilleux & Osterbrock (1987) diag- nostic diagrams. 2 See Spitzer Observers Manual, Chapter 7, http://ssc.spitzer.caltech.edu/documents/som/irs60.pdf Table 1: Properties of the Sample Galaxy Distance Hubble log log log AGN Name (Mpc) Type (MBH) (LX ) (L/LEdd) Type (1) (2) (3) (4) (5) (6) (7) Seyferts NGC4151 13 SABab 7.13a 42.7b -1.53 1r NGC1365 19 SBb 7.64b 41.3d -3.42 2s NGC1097 15 SBb · · · 40.7e · · · · · · NGC7469 65 SABa 6.84a 44.3a 0.34 1t NGC4945 4 SBcd 7.35b 42.5f -1.97 2u Circinus 4 SAb 7.72b 42.1g -2.74 2v Mrk 231 169 SAc 7.24c 42.2h -2.16 · · · Mrk3 54 S0 8.65a 43.5a -2.21 2w Cen A 3 S0 7.24b 41.8i -2.54 2x Mrk463 201 Merger · · · 43.0j · · · 2y NGC 4826 8 SAab 6.76b · · · · · · · · · NGC 4725 16 SABab 7.40b · · · · · · 2r 1 ZW 1 245 Sa · · · 43.9k · · · · · · NGC 5033 19 SAc 7.39b 41.4l -3.13 1r NGC1566 20 SABbc 6.92a 43.5a -0.57 1t NGC 2841 9 SAb 8.21a 42.7a -2.64 · · · NGC 7213 24 SA0 7.99a 43.30a -1.79 · · · LINERs NGC4579 17 SABb 7.85b 41.0b -3.47 · · · NGC3031 4 SAab 7.79b 40.2b -4.16 · · · NGC6240 98 Merger 9.15b 44.2b -1.52 2z NGC5194 8 SAbc 6.90b 41.0b -2.43 2r MRK266NE 112 Merger · · · 40.9b · · · 2t NGC7552 21 SBab 6.99b · · · · · · · · · NGC 4552 17 · · · 8.57b 39.6b -5.52 · · · NGC 3079 15 SBc 7.58b 40.1m -4.05 · · · NGC 1614 64 SBc 6.94b · · · · · · · · · NGC 3628 10 SAb 7.86b 39.9n -4.58 · · · NGC 2623 74 Pec 6.83b · · · · · · 2aa IRAS23128-5919 178 Merger · · · 41.0b · · · 2bb MRK273 151 Merger 7.74b 44.0o -0.31 2t IRAS20551-4250 171 Merger 7.52c 40.9b -3.23 · · · NGC3627 10 SABb 7.16b 39.4p -4.33 2r UGC05101 158 S · · · 40.9b · · · 1cc NGC4125 18 E6 8.50b 38.6b -6.47 · · · NGC 4594 10 SAa 9.04b 40.1q -5.47 · · · Quasars PG 1351+640 353 · · · 8.48a 44.5a -1.08 · · · PG 1211+143 324 · · · 7.49a 44.8a 0.22 1t PG 1119+120 201 · · · · · · · · · · · · 1y PG 2130+099 252 Sa 7.74a 44.47a -0.37 1y PG 0804+761 400 · · · 8.24a 44.93a -0.41 1dd PG 1501+106 146 E · · · · · · · · · 1y Columns Explanation: Col(1):Common Source Names; Col(2): Dis- tance (for H0= 75 km s −1Mpc−1); Col(3): Morphological Class; Col(4): Mass of central black hole in solar masses; Col(5): Log of the hard X-ray luminosity (2-10keV) in erg s−1. Col(6): log of the Eddington Ratio. (* = We include all galaxies that are classified as LINERs using either the Heckman (1980) or Veilleux & Osterbrock (1987) diagnostic diagrams. Col(6): AGN type based on the presence or absence of broad Balmer emission lines.) References:aWoo & Urry 2002, b Satyapal et al. 2005, c Tacconi et al. 2002, d Risaliti et al. 2005, e Terashima et al. 2002, f Guainazzi et al. 2000, g Smith & Wilson 2001, h Gallagher et al. 2002, i Evans et al. 2004, j Iman- ishi & Terashima et al. 2004, kGallo et al. 2004 , l Terashima et al. 1999, m Cappi et al. 2006, n Roberts, Schurch, & Warwick 2001, o Balestra et al. 2005, pGeorgantopoulos et al 2002, q Dudik et al. 2005, r Ho et al. 1997, s Storchi-Bergmann, Mulchaey, & Wilson, 1992, t Veron-Cetty & Veron 2003, u Marconi et al. 2000, vOliva et al. 1994, w Khachikian & Weedman 1974, x Veron-Cetty & Veron 1986y Dahari & De Robertis 1988, z Andreasian, Khachikian, & Ye, 1987, aa Laine et al. 2003, bb Duc, Mirabel, & Maza 1997, cc Sanders et al. 1988, dd Thompson 1992. http://ssc.spitzer.caltech.edu/documents/som/irs60.pdf the SH and LH modules is too small for background sub- traction to take place and separate SH or LH background observations do not exist for any of the galaxies in this sample. For the SL2 module, background subtraction was done using either a designated background file when available or the interactive source extraction option. In the case of the latter, the exact position of the slit on the host galaxy was first checked using Leopard, the data archive access tool available from the SSC. The source was then carefully defined according to the boundary of the slit and the edge of the host galaxy. The background was defined at the edge of the slit, where no other obvious source was present. In some cases, the slit was enveloped in the host galaxy and background subtraction could not take place. For both high and low resolution spectra, the ends of each order were manually cut from the rest of the spectrum. The 41 observations presented in this work are archived from various programs, including the SINGS Legacy Pro- gram, and therefore contain both mapping and staring observations. All of the staring observations were cen- tered on the nucleus of the galaxy. The SH, LH, SL2 staring observations include data from two slit positions overlapping by one third of a slit. In order to isolate the nuclear region in the mapping observations so that we might compare them to the staring observations, we extracted only those 3 overlapping slit positions coin- ciding with either radio or 2MASS nuclear coordinates. Because the slits in both the mapping and staring obser- vations occupy distinctly different regions of the sky, the slits cannot be averaged unless the emission originates from a compact source that is contained entirely in each slit. Therefore the procedure for flux extraction was the following: 1) If the fluxes measured from the two slits differed by no more than the calibration error of the in- strument, then the fluxes were averaged; otherwise, the slit with the highest measured line flux was chosen. 2) If an emission line was detected in one slit, but not in the other, then the detection was selected. This is true for all of the high and low resolution staring and mapping observations. In Tables 2 and 3 we list the line fluxes and statistical errors from the SH and LH observations for the [NeV] 14.3µm and 24.3µm lines, the [SIII] 18.1µm and 33.5µm lines, as well as the 6.2µm PAH emission feature. For all galaxies with previously published fluxes, we list in Tables 2 and 3 the published flux values. Our values dif- fer by no more than a factor of 1.9, much less in most cases, from the Weedman et al. (2005) or Armus et al. (2004, 2006) published values. These differences can be attributed to differences in the pipeline used for prepro- cessing. In all cases detections were defined when the line flux was at least 3σ. For the absolute photometric flux uncertainty we conservatively adopt 15%, based on the assessed values given by the Spitzer Science Center (SSC) over the lifetime of the mission.3 This error is cal- culated from multiple observations of various standard stars throughout the Spitzer mission by the SSC. The dominant component of the total error arises from the 3 See Spitzer Observers Manual, Chapter 7, (http://ssc.spitzer.caltech.edu/documents/som/som7.1.irs.pdf and IRS Data Handbook (http://ssc.spitzer.caltech.edu/irs/dh/dh20v2.pdf, Chapter 7.2 uncertainty at mid-IR wavelengths in the stellar models used in calibration and is systematic rather than Gaus- sian in nature. We note that the spectral resolution of the SH and LH modules of IRS (λ / ∆λ ∼ 600) is in- sufficient to resolve the velocity structure for most of the lines. There are a few galaxies which do show slightly broadened [NeV] line profiles (FWHM ∼ 200 - 1200 km s−1). These results will be discussed in a future paper. Abundance-independent density estimates can readily be obtained using infrared fine-structure transitions from like ions in the same ionization state with different crit- ical densities. The density diagnostics available in the IRS spectra of our objects are: [NeV] 14.32µm, 24.32 µm (ncrit ∼ 4.9 × 10 4 cm−3, and 2.7 × 104 cm−3, where ncrit = Aul/γul, with Aul the Einstein A coefficient and γul the rate coefficient for collisional de-excitation from the upper to the lower level), [NeIII] 15.55µm, 36.04 µm (ncrit ∼ 3 × 10 5 cm−3, and 5 × 104 cm−3, Giveon et al. 2002), and [SIII]18.71µm, 33.48 µm (ncrit ∼ 1.5 × 10 cm−3, and 4.1 × 103 cm−3). The results are very insen- sitive to the shape of the ionizing continuum. Since the [NeIII] 36µm line was either not detected or was outside the wavelength range of the LH module in virtually all galaxies, we omit any analysis of the [NeIII] line ratio from this work. 4. THE [NEV] LINE FLUX RATIOS In Figure 1 we plot the calculated 14µm/24µm line lu- minosity ratio as a function of electron density ne for gas temperatures T = 104K, 105K, and 106K. We include only the five levels of the ground 2s22p2 configuration and neglect absorption and stimulated emission. The results are nearly identical if only the lowest three levels of the ground term are included. We adopt collision strengths from Griffin & Badnell (2000) and radiative transition probabilities from Galavis, Mendoza, & Zeippen (1997). Fig. 1.— [NeV] 14µm/24µm line flux ratio versus electron den- sity, ne, for gas temperatures T = 10 4 K, 105 K, 106 K In Table 2, we list the observed [NeV] line flux ratios http://ssc.spitzer.caltech.edu/documents/som/som7.1.irs.pdf and their associated calibration uncertainties. In calcu- lating the upper and lower limits on the ratios, RMAX and RMIN , shown in Table 2, we did not propagate the errors in quadrature as would be appropriate for statis- tical uncertainties, but propagated them as follows: RMAX = F [NeV]14 + 0.15(F [NeV]14) F [NeV]24 − 0.15(F [NeV]24) RMIN = F [NeV]14 − 0.15(F [NeV]14) F [NeV]24 + 0.15(F [NeV]24) We note that this is conservative, since some components of the calibration errors should cancel in the ratio. Both line fluxes were measured for 19 galaxies. In what fol- lows we compare the line flux ratios measured in all but one, MKN 266, for reasons that are discussed in detail in Section 5.2. Of these 18 AGNs, 13 have ratios that are consistent with the low density limit to within the uncertainties, while only 2, both Type 1, have ratios sig- nificantly above it. The remaining 3, all Type 2, have ratios significantly below the low-density limit. Inter- estingly, we note that a similar range of ratios was also measured with the ISO SWS (Sturm et al. 2002, Alexan- der et al. 1999). There are several possible explanations for this finding. The observed, unphysically low ratios could result from artifacts introduced by variations in the slit sizes from which the line fluxes are obtained, from calibration uncertainties, or from substantial mid- IR extinction. Alternatively, perhaps important physical processes were neglected in calculating the theoretical ra- tios. In addition, errors in the collisional rate coefficients for the [NeV] transitions associated with the mid-infrared lines may be important. We explore these scenarios in the following sections. Observational Effects: Because the IRS LH slit is larger than the SH slit, if the [NeV] emission is extended, or multiple AGNs are present, the 14/24 µm line ratio will be artificially reduced. However, since the ionization potential of [NeV] is ∼ 97 eV, we expect that the [NeV]- emitting gas is ionized by the AGN radiation field only and is concentrated very close to the central source. Vir- tually all of the [NeV] fluxes presented in this work were obtained from IRS staring observations. Thus it is im- possible to determine whether the emission is extended using Spitzer observations alone. However, a number of galaxies have been observed at 14 and 24 µm by ISO. In Table 2 we list in addition to our Spitzer [NeV] fluxes, all available [NeV] fluxes from ISO. The ISO aperture at 14 and 24 µm (14”×27”) is much larger than either the SH or LH slits. In Figure 2 we plot the ratio of the [NeV] flux measured by ISO to that measured by Spitzer for both the 14 and 24 µm lines. The ranges of the [NeV] line flux ratios are consistent with the instrument uncertain- ties and are similar for all galaxies in the sample. Only the 14µm ratio for Mrk 266 falls outside of the expected range. This strongly suggests that the [NeV] emission is indeed compact and originates in the NLR and that the ratios are not affected by aperture variations, except for Mrk 266 which is discussed in detail in Section 5.2. If the data were affected by aperture variations we would expect to see an overall systematic increase of the 14µm/24µm line ratio with distance(See Figure 3). The Spearman rank correlation coefficient (rS, Kendell & Stuart 1976) corresponding to this plot is -0.069 (with [NeV] 14 micron Ratio 0.0 0.5 1.0 1.5 2.0 2.5 [NeV] (ISO) / F [NeV] (Spitzer) Mrk 266 [NeV] 24 micron Ratio Fig. 2.— Ratio of the ISO to Spitzer [NeV] 14µm and 24µm fluxes for those galaxies with overlapping observations. The range indicated with arrows is that corresponding to the absolute flux calibration for ISO (20%) and Spitzer (15%). Within the calibra- tion uncertainties of the instrument, the [NeV] fluxes are virtually the same for all of the galaxies except Mrk 266 (See Section 6.2). This strongly suggests that the [NeV] emission is compact and orig- inates in the NLR. We note that Sturm et al. 2002 find that the [NeV] 24µm detection for NGC 7469 is questionable. The ISO to Spitzer ratio for this galaxy (0.43) is the lowest shown here. a probability of chance correlation of 0.78), where a co- efficient of 1 or -1 indicates a strong correlation and a coefficient of 0 indicates no correlation. Thus we find that there is no correlation between the [NeV] ratio and distance in our sample. However this does not completely rule out aperture effects, if the size of the [NeV] emitting region increases with the bolometric luminosity of the AGN and the sample displays a significant trend in bolo- metric luminosity with distance. In this case, a correla- tion between the [NeV] ratio and distance would not be apparent since aperture variations would affect all galax- ies in the same way, regardless of distance. However this scenario is unlikely since the size of the [NeV]-emitting region would have to increase proportionately with dis- tance in order to remain extended beyond the slit for all galaxies. Nevertheless, we checked for this possibility, both by examining the [NeV] ratio vs. bolometric lu- minosity and by plotting the ratio vs. distance, binning the galaxies according to their bolometric luminosity. We find neither to be correlated over 5 orders of magnitude in LBOL. Thus, in the case of the [NeV] line flux ratio, we find no indication that ratios below the low density limit are artifacts of aperture effects. We point out that the [NeV] 24µm IRS line fluxes in the small overlapping sample plotted in Figure 2 are system- atically higher than the corresponding ISO-SWS fluxes, despite the smaller IRS slit. This indicates that one or both of the instruments is affected by systematic errors more severe than are indicated by the calibration un- certainty estimates. The SWS band 3D that includes the [NeV] 24µm line was characterized by strong fring- ing effects that when combined with the narrow range of the line scan mode introduced sometimes large un- 0.5 1.0 1.5 2.0 2.5 Mrk 266 = 0.069 log(Distance, Mpc) Seyferts Liners Quasars Fig. 3.— The [NeV] 14µm/24µm ratio as a function of distance. Open symbols signify Type 1 AGNs, Filled symbols signify Type 2 AGNs. The error bars shown here mark the calibration uncertain- ties on the line ratio. If the ratio were indeed affected by aperture variations we would expect a systematic increase of the ratio with distance. As can be seen here, this is not the case, and we find no indication that the low ratio is attribuable to aperture effects. certainties in the baseline fitting, and therefore the line flux measurement accuracy. In contrast, the baseline fit- ting over the entire Spitzer IRS SH and LH full spectra can be much more accurate. Moreover pointing accu- racy and stability are an order of magnitude improved over that obtained by ISO. We therefore assume in the sections that follow that the adopted conservative Spitzer IRS calibration uncertainties are accurate characteriza- tions of the IRS measurements. Importantly, regardless of which instrument is used, [NeV] ratios consistent with the low density limit have been observed in a number of sources with both ISO (e.g. Sturm et al. 2002 NGC 1365, NGC 7582, NGC4151, NGC 5506; Alexander et al. 1999, NGC 4151) and Spitzer (Weedman et al. 2005, Haas et al. 2005, and this work). Extinction: We consider the possibility that mid-IR differential extinction toward the [NeV]-emitting regions is responsible for the low [NeV] line ratios. Adopting the low-density limit (LDL) for the intrinsic value of the ratio ([NeV]14µm/24µm ∼0.83 for ne≤200 cm −3) for galaxies with ratios below the LDL, the observed line ratio gives a lower limit to the extinction, for a given MIR extinction curve. We examined the visual extinctions correspond- ing to the mid-IR differential extinction derived using three separate extinction curves: 1) the Draine (1989) extinction curve amended by the more recent ISO SWS extinction curve toward the Galactic center for 2.5-10µm (Lutz et al. 1996), 2) the Chiar & Tielens (2006) ex- tinction curve for the Galactic Center using 2.38-40µm ISO SWS observations of a bright IR source in the Quin- tuplet cluster (GCS3-I) 3) the Chiar & Tielens (2006) extinction curve for the local ISM using 2.38-40µm ISO SWS observations of a WC-type Wolf-Rayet (WR) star (WR98a). The Draine (1989) and Lutz et al. (1996) extinction curve yields AV ∼ 3 to 99 mag (See Table 2). However, these values result from an extinction law that is unexplored beyond 10µm. The Chiar & Tie- lens (2006) Galactic center extinction curve cannot ex- plain the observed [NeV] ratios since the extinction at 24µm is greater than the extinction at 14µm, so we do not discuss it further. The visual extinction resulting from their local ISM extinction curve is unrealistically high (AV median=500mag). The calculated extinction ob- tained using the Draine (1989), Lutz et al. (1996), and Chiar & Tielens (2006) local ISM extinction curves are given in Table 2. The AV derived from the two extinction curves de- scribed above are extremely high in many cases. Even if the extinction is calculated from the upper limit on the ratio to the LDL for the three galaxies whose upper limits are below the LDL, the corresponding visual ex- tinction is still very high (for the Draine 1989 and Lutz 1996 extinction curve AV = 21, 26, and 30 mag for these three galaxies; for the Chiar and Tielens extinction curve AV = 260, 330, and 370 mag). However, we caution the reader that the actual value for extinction is highly un- certain. Indeed very little is known about the 8-40µm extinction curve in AGNs. Specifically, the 10 and 18 µm silicate features in this band are the source of in- consistency. Even within the AGN class, extinction may vary dramatically from 8-40µm because of variations in the silicate features due to differences in grain size, poros- ity, shape, composition, abundance, and location in each galaxy. Hao et al. (2005) show that in five AGNs (4 of which are in our sample), both silicate features vary considerably in strength and width. Sturm et al. (2005) also show that the standard ISM silicate models do not accurately fit NGC 3998, a LINER with silicate emis- sion. Sturm et al. (2005) suggest that increased grain size and possibly the presence of crystalline silicates such as clino-pyroxenes may improve the fit, but that clearly circumnuclear dust in AGNs has very different proper- ties than dust in the Galactic ISM (see also Maiolino et al. 2001a, 2001b, but Weingartner & Murray 2002 for an alternative view). Chiar & Tielens (2006) even show that the GC observations and the local ISM observations within the Galaxy deviate from each other most dramat- ically in the wavelength region between the two silicate absorption features. In their observations, this is the re- gion between ∼ 12-15µm -directly overlapping with the 14µm values in which we are interested. Because of ir- regularity of the silicate features in the mid-IR, it is very difficult to interpret the true extinction there. Moreover, in addition to the uncertainty in the extinction law, the geometry of the obscuring material is unknown and can vary substantially from galaxy to galaxy. The most that can be said here for the galaxies with ratios below the LDL is that if extinction is responsible for the low ratios, then the extinction must be less at 24µm than at 14µm. Physical Processes: It is possible that important physical processes have been neglected in calculating the [NeV] line luminosity ratio as a function of electron den- sity shown in Figure 1. We consider three physical pro- cesses that may affect the line ratios: (1) A source of gas heating in addition to photoioniza- tion (e.g., shocks, turbulence) that may yield gas temper- atures substantially higher than 104K. As can be seen in Figure 1, higher gas temperatures do not yield sig- nificantly lower line ratios in the low-density limit, but could explain the generally low values of the ratios that lie above the LDL. (2) Pumping from the ground term to the first excited term, e.g., by O III resonance lines. The specific energy density required for this to significantly affect the line ratio exceeds 10−14 erg cm−3 Hz−1, which is implausibly large by orders of magnitude. (3) Absorption and stimulated emission within the ground term, which could be important if, e.g., a large quantity of warm dust yielding copious 24µm continuum emission is located close to the [NeV]-emitting region. Figures 4a through 4e show the line ratio as a func- tion of the specific energy density at 24µm, uν(24µm). We display results for electron density ne = 10 2, 103, 104, 105, and 106 cm−3; gas temperature T = 104, 105, and 106K; and ratio of the specific energy density at 14 and 24µm, uν(14µm)/uν(24µm) = 0.4, 1.0, and 1.8 (val- ues were chosen to reproduce the observed range of the 14µm/24µm continuum flux ratios; see Section 5). For the moment, assume that the NeV is located suf- ficiently far from the source of the 14 and 24µm contin- uum emission to treat the source as a point. If hot dust within or near the inner edge of the torus is responsible for this emission, then this assumption requires that the distance to the NeV, rNe, be large compared with the dust sublimation radius, rsub ∼ 1 pcL bol, 46 (Ferland et al. 2002); Lbol, 46 is the bolometric luminosity in units of 1046 erg s−1. In this case, we can obtain a simple estimate of uν(24µm) at the location of the [NeV]-emitting region from the observed specific flux Fν(24µm), the distance to the galaxy D, and rNe: uν(24µm) ∼ Fν(24µm) . (3) With rNe = 100 pc, uν(24µm) estimated in this way ranges from ≈ 10−24 erg cm−3 Hz−1 to somewhat less than 10−20 erg cm−3 Hz−1 for the galaxies in our sam- ple. These can be compared to the results of Hönig et al. (2006), who modeled the infrared emission from clumpy tori. They presented plots of Fν at a distance of 10Mpc from an AGN with bolometric luminosity Lbol = 4 × 10 45 erg s−1. Extrapolating to a distance of 100 pc, we find uν(24µm) as large as a few times 10−21 erg cm−3 Hz−1, close to the estimate for the most luminous AGN in our sample. From Figures 4a through 4e, we see that the infrared continuum can only reduce the line ratio significantly at rNe ≈ 100 pc if T & 10 5K when ne ∼ 10 2 cm−3 and T & 106K when ne ∼ 10 3 cm−3. However, the NeV, as a high-ionization species, may lie closer to the central source than does the bulk of the narrow line region. If rNe ≈ 10 pc, then uν(24µm) increases by a factor ∼ 100. In this case, the observed low line ratios can be explained by this mechanism with T ∼ 104K, if ne ∼ 10 2 cm−3. Higher values of electron density would require higher gas temperatures. In Section 5.1, we suggest that the [NeV]-emitting re- gion may lie within the torus. In this case, absorp- tion and stimulated emission within the ground term are probably important. For the high-luminosity objects, these may even dominate over collisional excitation and de-excitation. At these central locations, gas tempera- tures T ∼ 106 K may be natural (Ferland et al. 2002). Relatively high densities may also be expected, in which case the infrared continuum may not appreciably depress the line ratio (see Figure 4d). Adopting the Mathews & Ferland (1987) spectrum and T ≈ 106K, the ionization parameter U ≡ nγ/ne ∼ 10 in order for a substantial fraction of the Ne to be NeV; nγ is the number density of H-ionizing photons. For this spectrum, nγ ≈ 1.7×10 3Lbol, 46 r Ne, 100 cm −3, where rNe, 100 = rNe/100 pc. If rNe = 1pc, then either (1) ne ∼ 10 10Lbol,46 cm −3 or (2) the nuclear continuum is filtered through a far-UV/X-ray-absorbing medium be- fore reaching the [NeV]-emitting region. If absorption and stimulated emission are indeed rele- vant processes in [NeV] line production, we might expect a relationship between the [NeV] line flux ratio and the 24 µm continuum luminosity that is consistent with one of the curves shown in Figures 4a through 4e. In Figure 5 we plot this relationship for the [NeV] emitting galax- ies in our sample. As can be seen in Figure 5, we find no relationship between the [NeV] line flux ratio and the 24µm continuum luminosity for our sample of galaxies. The Spearman rank correlation coefficient for this plot is -0.01 (probability of chance correlation = 0.95), indi- cating no correlation. As a result, as can be seen from Figures 4a and 4b, stimulated emission and absorption at low densities can be ruled out as possible scenarios be- cause the scatter plot shown in Figure 5 does not follow the model predictions. We note that the location of the [NeV]-emitting region relative to the source of the 24µm continuum emission is uniform among the galaxies in the sample. Variations in the location might obscure any correlation in these plots. Figures 4c and 4d reveal that, for some values of ne, T, and uν(14µm)/uν(24µm), the line ratio is very insensitive to the value of uν(24µm). In these cases, the line ratio remains above ∼ 0.8. Thus, al- though absorption and stimulated emission may be con- tributing processes to [NeV] production, another mecha- nism is required to explain the low (<0.8) [NeV] line flux ratios in our sample. Computed Quantities: Finally, it is possible that there is significant error in the adopted collisional rate coefficients. The accuracy of collisional strengths of in- frared atomic transitions has been a longstanding ques- tion. We adopt the collisional rate coefficients from the state of the art IRON project (Hummer et al. 1993) which produced the most up-to-date and accurate colli- sion strengths for a large database of atomic transitions. While these calculations have been questioned based on recent ISO observations of nebulae (Clegg et al. 1987, Oliva et al. 1996, Rubin et al. 2002, Rubin 2004), it is likely that the discrepancies between the observational and theoretical values can be explained by inaccuracies in the fluxes employed (van Hoof et al. 2000). Uncer- tainties in the collisional rate coefficients for the [NeV] transitions are unlikely to exceed 30% (van Hoof et al. 2000). It is therefore unlikely that the low critical den- sities implied by our data can be attributed to uncer- tainties in the theoretical values of the [NeV] collisional strengths. 5. EXTINCTION EFFECTS OF THE TORUS AND AGN UNIFICATION Although low electron densities, high gas tempera- tures, and/or high infrared radiation densities may play Fig. 4.— The [NeV] line ratio as a function of the specific energy density at 24µm, uν(24µm), for temperatures, T = 10 4, 105, 106 K, for 14µm/24µm continuum ratios of 0.4, 1.0, and 1.8, and finally for electron densites, ne = 10 2, 103, 104, 105, 106 cm−3. Table 2: NeV Line Fluxes and Derived Extinction Galaxy [NeV] [NeV] [NeV] [NeV] [NeV] AV AV AV AV Source 14.32 14.32 24.32 24.32 Ratio Ratio to Ratio to Ratio to Ratio to SH ISO LH ISO LDL (D&L) HDL (D&L) LDL (C&T) HDL (C&T) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Seyferts NGC4151 7.77a 5.50c 6.77a 5.60c 1.15 +0.41 −0.30 · · · 126 · · · 1686 NGC1365 2.20±0.06 2.50c 5.36±0.06 3.90c 0.41 +0.14 −0.11 45 189 570 2519 NGC1097 <0.05 · · · <0.18 · · · · · · · · · · · · · · · · · · NGC7469 1.16a <1.50c 1.47a 0.63c∗ 0.79 +0.28 −0.21 3 149 41 1990 NGC4945 0.28±0.03 <0.50d <0.75 · · · >0.38 · · · · · · · · · · · · Circinus 23.94±0.61 31.70c 24.00±3.90 21.80c 1.00 +0.35 −0.26 · · · 135 · · · 1799 Mrk 231 <0.44a <1.50e <0.69a · · · · · · · · · · · · · · · · · · Mrk3 6.45a 4.60c 6.75a 3.40c 0.96 +0.34 −0.25 · · · 138 · · · 1835 Cen A 2.32a 2.70c 2.99a 2.00c 0.77 +0.27 −0.20 4 150 56 2005 Mrk463 1.83b 1.40c 2.04b · · · 0.90 +0.32 −0.23 · · · 141 · · · 1886 NGC 4826 · · · · · · · · · · · · · · · · · · · · · · · · · · · NGC 4725 <0.09 · · · 0.09±0.03 · · · <1.04 · · · · · · · · · · · · 1 ZW 1 <0.11a 0.27c <0.10a · · · · · · · · · · · · · · · · · · NGC 5033 0.07±0.02 · · · 0.11±0.02 · · · 0.65+0.23 −0.17 16 161 198 2146 NGC1566 0.16±0.05 · · · 0.22±0.04 · · · 0.74+0.26 −0.19 7 153 92 2041 NGC 2841 <0.04 · · · <0.03 · · · · · · · · · · · · · · · · · · NGC 7213 <0.04 · · · <0.09 · · · · · · · · · · · · · · · · · · LINERs NGC4579 <0.06 · · · <0.03 · · · · · · · · · · · · · · · · · · NGC3031 <0.06 · · · <0.04 · · · · · · · · · · · · · · · · · · NGC6240 0.51b <1.00e <0.39b · · · <1.31 · · · · · · · · · · · · NGC5194 0.41±0.04 <0.20c 0.39±0.09 · · · 1.06 +0.37 −0.28 · · · 131 · · · 1751 MRK266∗∗ 0.21±0.02 0.50f 1.19±0.06 · · · 0.18 +0.06 −0.05 100 240 1254 3203 NGC7552 <0.11 · · · <0.83 · · · · · · · · · · · · · · · · · · NGC 4552 <0.06 · · · <0.07 · · · · · · · · · · · · · · · · · · NGC 3079 <0.07a · · · <0.14a · · · · · · · · · · · · · · · · · · NGC 1614 <0.28 · · · <1.49 · · · · · · · · · · · · · · · · · · NGC 3628 <0.06 · · · <0.34 · · · · · · · · · · · · · · · · · · NGC 2623 0.30±0.04 · · · 0.47±0.07 · · · 0.63 +0.22 −0.14 17 163 218 2167 IRAS23128· · · 0.22±0.02 <0.40e 0.34±0.10 · · · 0.65 +0.23 −0.22 16 161 203 2152 MRK273 1.06±0.05 0.82e 2.74±0.19 · · · 0.39 +0.14 −0.10 49 192 617 2565 IRAS20551· · · <0.06 <0.25e <0.25 · · · · · · · · · · · · · · · · · · NGC3627 0.08±0.01 · · · 0.19±0.05 · · · 0.45+0.16 −0.12 40 184 504 2453 UGC05101 0.52b <1.50e 0.49b · · · 1.06+0.37 −0.28 · · · 131 · · · 1750 NGC4125 <0.03 · · · <0.07 · · · · · · · · · · · · · · · · · · NGC 4594 <0.03 · · · <0.04 · · · · · · · · · · · · · · · · · · Quasars PG1351· · · <0.04 · · · <0.07 · · · · · · · · · · · · · · · · · · PG1211· · · 0.04±0.007 · · · <0.04 · · · >1.01 · · · · · · · · · · · · PG1119· · · 0.30±0.06 · · · 0.22±0.02 · · · 1.39 +0.49 −0.36 · · · 115 · · · 1531 PG2130· · · 0.42±0.03 · · · 0.42±0.05 · · · 1.00 +0.35 −0.26 · · · 135 · · · 1798 PG0804· · · <0.06 · · · <0.07 · · · · · · · · · · · · · · · · · · PG1501· · · 0.78±0.02 · · · 0.83±0.02 · · · 0.94 +0.33 −0.25 · · · 138 · · · 1846 Columns Explanation: Col(1):Common Source Names; Col(2): 14.32 µm [NeV] line flux and statistical error in units of 10−20 W cm−2 from Spitzer; Col(3): 14.32 µm [NeV] line flux and statistical error in units of 10−20 W cm−2 from ISO; Col(4): 24.31 µm [NeV] line flux and statistical error in units of 10−20 W cm−2 from Spitzer; Col(5): 24.32 µm [NeV] line flux and statistical error in units of 10−20 W cm−2 from ISO; Col(6): [NeV] Line Ratio used in plots and calculations; Col(7): Extinction required to bring ratios below the low-density limit (LDL) up to the LDL, calculated using the Draine (1989) extinction curve amended by the more recent ISO SWS extinction curve toward the Galactic center for 2.5-10µm (Lutz et al. 1996); Col(8): Extinction required to bring ratios below the low-density limit (LDL) up to the high-density limit (HDL), calculated using the Draine (1989) extinction curve amended by the more recent ISO SWS extinction curve toward the Galactic center for 2.5-10µm (Lutz et al. 1996), Col(9): Extinction required to bring ratios below the low-density limit (LDL) up to the LDL, calculated using the Chiar & Tielens (2006) extinction curve for the local ISM, Col(10): Extinction required to bring ratios below the low-density limit (LDL) up to the high-density limit (HDL), calculated using the Chiar & Tielens (2006) extinction curve for the local ISM.∗ Sturm et al. 2002 find that the [NeV] 24µm detection for NGC 7469 is a questionable one, ∗∗ As discussed in detail in Section 6.2, Mrk 266 is the only galaxy in our sample where we find that aperture variation may affect the observed [NeV] line flux ratio. For this reason it has been excluded from relevent plots and calculations. References for Table 2: a Weedman et al. 2005, b Armus et al. 2004 & 2006, c Sturm et al. 2002, d Verma et al. 2003, e Genzel et al. 1998, f Prieto & Viegas Table 3: SIII Line Fluxes and Derived Extinction Galaxy [SIII] [SIII] [SIII] [SIII] [SIII] Av Av Av Av PAH6.2 Source 18.71 18.71 33.48 33.48 Ratio Ratio to Ratio to Ratio to Ratio to SL2 SH ISO LH ISO LDL (D&L) HDL (D&L) LDL (C&T) HDL (C&T) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) Seyferts NGC4151 7.50a 5.40c 6.57a 8.10c 1.14 +0.40 −0.30 · · · · · · · · · · · · 168.1 NGC1365 5.73±0.05 13.50c 27.20±0.38 36.10c 0.21 +0.07 −0.05 · · · · · · · · · · · · 132.3 NGC1097 2.18±0.02 · · · 11.40±0.23 · · · 0.19 +0.07 −0.05 · · · · · · · · · · · · 151.7 NGC7469 7.70a 9.20c 9.80a 10.40c 0.79 +0.28 −0.20 · · · · · · · · · · · · 415.0 NGC4945 3.18±0.03 6.30d 38.70±1.80 51.40d 0.08 +0.03 −0.02 · · · · · · · · · · · · 671.7 Circinus 19.10±0.70 35.20c 56.30±3.31 93.20c 0.37 +0.13 −0.10 · · · · · · · · · · · · 1018.4 Mrk 231 <0.47a <3.00e <2.30a <3.00e · · · · · · · · · · · · · · · 175.6 Mrk3 5.55a · · · 5.25a · · · 1.06 +0.37 −0.28 · · · 83 · · · 72 18.4 Cen A 4.54a 6.40c 14.80a 22.30c 0.31 +0.11 −0.08 · · · · · · · · · · · · 220.8 Mrk463 1.50b <0.80c 1.35b 1.20c 1.11 +0.39 −0.29 · · · 81 · · · 70 55.4 NGC 4826 3.39±0.03f · · · 4.61±0.08f · · · 0.74 +0.26 −0.19 · · · 96 · · · 82 66.2 NGC 4725 0.02±0.02f · · · 0.11±0.02f · · · 0.23 +0.08 −0.06 23 135 20 116 3.7 1 ZW 1 <0.11a <0.50c <0.18a <1.00c · · · · · · · · · · · · · · · 22.2 NGC 5033 0.85±0.11 · · · 2.42±0.10 · · · 0.35 +0.12 −0.09 · · · · · · · · · · · · 33.9 NGC1566 0.55±0.05f · · · 0.55±0.06f · · · 1.00 +0.35 −0.26 · · · 85 · · · 73 61.0 NGC 2841 0.22±0.04f · · · 0.29±0.03f · · · 0.75 +0.26 −0.20 · · · 95 · · · 82 10.9 NGC 7213 0.47±0.05 · · · 0.59±0.06 · · · 0.80 +0.28 −0.21 · · · · · · · · · · · · 28.7 LINERs NGC4579 0.32±0.06f <0.78g 0.24±0.03f <1.20g 1.33 +0.47 −0.35 · · · 75 · · · 65 17.3 NGC3031 0.61±0.03 · · · 0.09±0.09 · · · 0.67 +0.24 −0.18 · · · · · · · · · · · · 23.4 NGC6240 1.99b <4.00e 2.63b 4.50e 0.76 +0.27 −0.20 · · · 95 · · · 81 399b NGC5194 1.06±0.05f 1.00d 1.48±0.03f 4.60d 0.72 +0.25 −0.19 · · · 96 · · · 83 26.4 MRK266∗∗ 1.00±0.13 · · · 4.65±0.09 · · · 0.21 +0.08 −0.06 25 138 22 118 23.1 NGC7552 17.11±0.08f 24.60d 13.38±0.41f 41.10d 1.28 +0.45 −0.33 · · · 77 · · · 66 872.0 NGC 4552 0.07±0.03f · · · 0.06±0.02f · · · 1.29 +0.45 −0.34 · · · 76 · · · 66 21.5 NGC 3079 1.25a 6.80g 6.08a 6.60g 0.21 +0.07 −0.05 · · · · · · · · · · · · 620.3 NGC 1614 9.63±0.27 · · · 11.60±0.43 · · · 0.83 +0.29 −0.15 · · · 91 · · · 79 508.5 NGC 3628 2.14±0.03 · · · 15.80±0.33 · · · 0.14 +0.05 −0.04 · · · · · · · · · · · · 430.4 NGC 2623 0.88±0.05 · · · 3.16±0.20 · · · 0.28 +0.10 −0.07 16 129 14 111 128.6 IRAS23128· · · 2.62±0.12 0.89e 2.11±0.18 2.80e 1.24 +0.44 −0.32 · · · 78 · · · 67 90.1 MRK273 1.24±0.07 <0.82e 3.88±0.40 2.30e 0.32 +0.11 −0.08 12 124 10 107 69.2 IRAS20551· · · 0.66±0.06 0.30e 1.18±0.13 1.40e 0.56 +0.20 −0.15 · · · 105 · · · 90 38.3 NGC3627 0.38±0.03f · · · 0.57±0.09f · · · 0.67 +0.24 −0.18 · · · 99 · · · 85 153.8 UGC05101 0.98b <1.40e 1.30b 2.50e 0.75 +0.27 −0.20 · · · 95 · · · 82 190b NGC4125 · · ·f · · · 0.06±0.05f · · · · · · · · · · · · · · · · · · 14.9 NGC 4594 0.39±0.03 · · · 1.24±0.13 · · · 0.32 +0.11 −0.08 · · · · · · · · · · · · 14.2 Quasars PG1351· · · 0.34±0.06 · · · <0.13 · · · >2.70 · · · · · · · · · · · · 29.6 PG1211· · · <0.06 · · · <0.08 · · · · · · · · · · · · · · · · · · 25.8 PG1119· · · <0.13 · · · 0.19±0.06 · · · <0.71 · · · · · · · · · · · · 8.7 PG2130· · · <0.19 · · · 0.34±0.06 · · · <0.55 · · · · · · · · · · · · 26.9 PG0804· · · <0.06 · · · <0.21 · · · · · · · · · · · · · · · · · · 27.4 PG1501· · · 0.67±0.15 · · · 0.41±0.05 · · · 1.64 +0.58 −0.43 · · · 68 · · · 59 19.2 Columns Explanation: Col(1):Common Source Names; Col(2): 18.71 µm [SIII] line flux and statistical error in units of 10−20 W cm−2 from Spitzer; Col(3): 18.71 µm [SIII] line flux and statistical error in units of 10−20 W cm−2 from ISO; Col(4): 33.48 µm [SIII] line flux and statistical error in units of 10−20 W cm−2 from Spitzer; Col(5): 33.48 µm [SIII] line flux and statistical error in units of 10−20 W cm−2 from ISO; Col(6):[SIII] line flux ratio used for plots and calculations; Col(7): Extinction required to bring ratios below the low-density limit (LDL) up to the LDL, calculated using the Draine (1989) extinction curve amended by the more recent ISO SWS extinction curve toward the Galactic center for 2.5-10µ (Lutz et al. 1996) for those galaxies with distances greater than 55 Mpc that are not effected by aperture variations, Col(8): Extinction required to bring ratios below the low-density limit (LDL) up to the high-density limit (HDL), calculated using the Draine (1989) extinction curve amended by the more recent ISO SWS extinction curve toward the Galactic center for 2.5-10µ (Lutz et al. 1996) for those galaxies with distances greater than 55 Mpc that are not effected by aperture variations, Col(9): Extinction required to bring ratios below the low-density limit (LDL) up to the LDL, calculated using the Chiar & Tielens (2006) extinction curve for the Galactic Center for those galaxies with distances greater than 55 Mpc that are not effected by aperture variations, Col(10): Extinction required to bring ratios below the low-density limit (LDL) up to the high-density limit (HDL), calculated using the Chiar & Tielens (2006) extinction curve for the Galactic Center for those galaxies with distances greater than 55 Mpc that are not effected by aperture variations, Col(11): 6.2 µm PAH line flux in units of 10−21 W cm−2 ∗∗ The [SIII] ratio for Mrk 266 is known to be affected by aperture variations(See Section 6.2). For this reason it has been excluded from relevent plots and calculations. References for Table 3: a Weedman et al. 2005, b Armus et al. 2004 & 2006, c Sturm et al. 2002, d Verma et al. 2003, e Genzel et al. 1998, f Dale et al. 2006, g Satyapal et al. 2004. 32 33 34 35 36 37 38 = -0.01 log(24 m Specific Luminosity) (W m-1) 14/24 < 0.50 0.50 < 14/24 < 1.0 14/24 > 1.0 Fig. 5.— The observed [NeV] line ratio as a function of the 24µm specific luminosity for our sample of galaxies. The error bars shown here represent the calibration uncertainties on the [NeV] line flux ratio as in Figure 3. The symbol type indicates the 14µm/24µm continuum ratio. a role in lowering the [NeV] line flux ratio, we argue that differential infrared extinction to the [NeV] emitting re- gion due to dust in the obscuring torus is responsible for the low line ratios in at least some AGN. Clearly, this requires that there is significant extinction at mid-IR wavelengths, and specifically toward the [NeV]-emitting regions. Is this reasonable? If there is significant ex- tinction, it is possible that: 1) the [NeV]-emitting region originates much closer to the central source than previ- ously recognized, close enough to be extinguished by the central torus in some galaxies, 2) the [NeV]-emitting por- tion of the NLR is obscured by dust in the host galaxy or in the NLR itself, or 3) some combination of these scenarios. We explore these possibilities in the following analysis. 5.1. The [NeV] originates in gas interior to the central torus. In the conventional picture of an AGN, the broad line region (BLR) is thought to exist within a small region interior to a dusty molecular torus while the NLR origi- nates further out. This of course is the paradigm invoked to explain the Type 1/Type 2 dichotomy. However there have been multiple optical spectroscopic studies that con- tradict the assumption that the observational properties of the NLR are not dependent on the viewing angle and the inclination of the system, suggesting that some of the narrow emission lines originate in gas interior to the torus. For instance, Shuder and Osterbrock (1981) and Cohen (1983) showed that narrow high ionization forbid- den lines such as [Fe VII] λ 6374 (requiring photons with energies ≥ 100eV to ionize) are stronger relative to the low ionization lines in Seyfert 1 galaxies (including inter- mediate Seyferts, 1.2, 1.5 etc.) than in Seyfert 2 galaxies, suggesting that some of the emission is obscured by the torus. In addition, [FeX] λ 6374 and [NeV] λ 3426 have also been shown to be less luminous in Type 2 objects than in Type 1 objects (Murayama & Taniguchi, 1998a; Schmitt 1998, Nagao et al. 2000, 2001a, 2001b, 2003, Tran et al. 2000, see also Jackson and Browne (1990) for narrow line radio galaxies and quasars.) These findings may imply that the emission lines of species with the highest ionization potentials originate closer to the AGN than those of lower ionization species such as [OII]λ3727, [SII]λλ6716, 6731, [OI] λ6300 etc. and therefore may be partially obscured by the central torus. If there is considerable extinction to the line-emitting regions due to the torus, one may expect the mid-infrared continuum to be similarly obscured. To test this scenario we divided our sample into Type 1 or Type 2 objects based on the presence or absence of broad (full width at half max (FWHM) exceeding 1000 km s−1) Balmer emis- sion lines in the optical spectrum. The spectral classifi- cation for the [NeV]-emitting galaxies is given in Table 1. In Figure 6a, we plot the [NeV] 14µm/24µm line flux ra- tio versus the 14µm/24µm continuum ratio of the [NeV] emitting galaxies in our sample. Assuming there is no correlation between the electron density and the contin- uum shape, a correlation between the line flux and con- tinuum ratios would suggest that the mid-IR extinction associated with the torus (such as that found by Clavel et al. 2000) affects the observed line flux ratios. As can be seen, there is a correlation between the line and contin- uum ratios for galaxies with [NeV] emission. Moreover we note that the 3 nuclei with ratios significantly below the LDL are all Type 2 AGNs, while the 2 that lie sig- nificantly above this limit are Type 1 AGNs, suggestive that the extinction of the [NeV]-emitting region in Type 2 AGNs may be due to the torus. We note that the error bars displayed in Figure 6 are based on a conserva- tive estimate (15%) of the absolute calibration error on the flux (see Section 3). Moreover, we have adopted the most conservative approach in propagating the error (see Section 4) for each line ratio. We further note that two of the three nuclei with ratios below the LDL were also observed in high-accuracy peak-up mode, resulting in a pointing accuracy on the continuum for these galaxies of 0.4”. The third galaxy, NGC 3627, was observed in high resolution mapping mode over 15” X 22”. We extracted the spectra and found that the full map and the single slit fluxes agree to within 10%. Thus pointing errors do not appear to be responsible for the low ratios in these galaxies. Finally we find that the ratios for the all galax- ies except Mrk 266 are not sensitive to the line-fitting or flux extraction methods that we have employed. The Spearman rank correlation coefficient for Figure 6 is 0.60 (with a probability of chance correlation of 0.008), indicating a significant correlation between the [NeV] line flux ratios and the mid-IR continuum ratio. We note that some AGNs are known to contain prominent sili- cate emission features (Hao et al. 2005, Sturm et al. 2006) which have not been disentangled from the under- lying continuum in this study. Because of this, the 14µm or 24µm flux may be overestimated in some cases mak- ing intrinsic value of the continuum at 14µm and 24µm somewhat uncertain. However only one galaxy plotted in Figure 6 is currently known to contain such features (PG1211+143, Hao et al. 2005). Variations in ne and the underlying continuum shape will also add scatter to the correlation, as will differences in extinction to the 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 = 0.60 F (14 m) / F (24 m) Continuum Type 1 Objects Type 2 Objects 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 D < 55 Mpc D > 55 Mpc F (14 m) / F (24 m) Continuum Fig. 6.— The [NeV] line ratio vs. the Fν(14µm)/Fν (24µm) continuum ratio for our sample. In both plots, the error bars mark the calibration uncertainties on the line ratio. There is a correlation between the line and continuum ratios which suggests that extinction affects the observed line flux ratios. 6a) The majority of galaxies with ratios below the LDL are Type 2 objects, implying that the extinction toward the [NeV]-emitting region may be due to the torus. 6b) The correlation shown here is not an artifact of aperture vatiations between the SH and LH slits. The correlation holds when only the most distant galaxies are considered. line- and continuum-producing regions. We should note that the correlation seen in Figure 6 is not an artifact of aperture variations between the LH and SH slit. The correlation holds when only the most distant galaxies (closed symbols in Figure 6b) are considered. Independent of this correlation, our most important finding is that the [NeV] line flux ratio is significantly lower for Type 2 AGNs than it is for Type 1 AGNs. Fig- ure 7 shows the relative [NeV] flux ratios for the Type 1 and Type 2 objects in our sample. The mean ratios are 0.97 and 0.72 for the eight Type 1 and ten Type 2 AGNs, respectively, with uncertainties in the mean of about 0.08 for each. Interestingly, although the sample size is limited, precluding us from drawing firm statisti- cally significant conclusions, there is a similar suggestive trend seen in the sample of AGNs observed by Sturm et al. (2002) with ISO-SWS. That is, in their work, the two galaxies with the lowest [NeV] flux ratios are NGC 1365 and NGC 7582, both Type 2 AGNs. The galaxy with the highest ratio in their work is TOL 0109-383 , a Type 1 AGN. If indeed the torus obscures the IR [NeV] emission in Type 2 objects, one would expect the optical/UV [NeV] emission in these objects to be obscured as well. We searched the literature for optical/UV detections of [NeV] λ3426 for all of the galaxies in our sample and found five galaxies with observations at this wavelength. Four of these galaxies (Mrk 463, Mrk 3, NGC 1566, and NGC 4151) were detected at [NeV] λ3426; the other (NGC 3031) was not detected (see, Kuraszkiewicz et al. 2002, 2004 and Forster et al. 2001 for optical/UV fluxes). Of the four galaxies with optical/UV [NeV] detections, two are Type 1 galaxies (NGC 1566 & NGC 4151) and, surprisingly, two are Type 2 galaxies (Mrk 3 and Mrk 463). If the Type 2 galaxies Mrk 3 and Mrk 463 had [NeV] emitting regions interior to the torus, then the op- Type 1 AGN 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 [NeV]14 m/[NeV]24 m Type 2 AGN Fig. 7.— Histogram of the [NeV] 14µm/24µm line flux ratio as a function of AGN type. The [NeV] line ratios for Type 2 AGNs are consistently lower than those from Type 1 AGNs. tical/UV lines in these objects should not be detected due to severe obscuration. We note that Mrk 3 and Mrk 463 have some of the highest X-ray luminosities (both ∼ 1043 erg s−1 ) in the sample and mid-IR [NeV] ra- tios that are comparable to similarly luminous Type 1 objects–consistent with little or no obscuration in the mid-IR for these Type 2 galaxies. This finding may im- ply that, in the most powerful AGNs, the [NeV] emitting region is pushed beyond the torus because the radiation field is so intense, while lines with higher ionization po- tentials than [NeV] (such as [NeVI], [FeX] etc.) are still concealed by the torus. More data, both from the mid- IR and from the optical/UV, are needed to further test this hypothesis. 5.2. The [NeV]-emitting region is obscured by the host galaxy or dust in the NLR. While the correlation in Figures 6a and 6b are very promising explanations for the observed [NeV] line ra- tios, it is not completely clear why some Type 2 galaxies appear obscured and others may not. Perhaps the [NeV] emission is attenuated by dust in the NLR itself or else- where in the host galaxy. Indeed, it is well-known that dust does exist in the NLR (e.g., Radomski et al. 2003, Tran et al. 2000) and that it can be extended and patchy (e.g Alloin et al. 2000; Galliano et al. 2005; Mason et al. 2006). In addition, dust in the host galaxy could be responsible for the extinction seen here. For complete- ness, we have conducted a detailed archival analysis of all of the galaxies in our sample with [NeV] ratios close to or below the LDL in order to see if there is additional evidence for high extinction either in the host galaxy or within the NLR. We find that the majority of galaxies with low densities do indeed have well-known dust lanes, large X-ray inferred column densities, or other properties indicative of extinction. Cen A: This nearby (D = 3.4Mpc) early type (S0) galaxy at one time devoured a smaller gas-rich spiral galaxy (Israel 1998, Quillen 2006). There is clear evi- dence for substantial obscuration toward the nucleus of Cen A. For example, the central region is veiled by a well known dense dust lane thought to be a warped thin disk (Ebneter & Balick 1983, Bland et al. 1986, 1987, Nichol- son et al. 1992, Sparke 1996, Israel 1998, Quillen et al. 2006). Schreier et al. (1996) find V-band extinction av- eraging 4-5 mag and infrared observations by Alonso & Minniti yield AV values exceeding 30 mag in some re- gions. Thus, it is plausible that there are regions toward the nucleus of Cen A that are obscured even at infrared wavelengths. NGC 1566: The optical nuclear spectrum of this nearby galaxy is known to vary dramatically over a pe- riod of months, changing its optical classification from a Type 2 object to a Type 1 object and back again (Pastor- iza & Gerola 1970, de Vaucouleurs 1973, Penfold 1979, Alloin et al. 1985). The narrow optical lines in this ob- ject also show prominent blue wings and the radio prop- erties of this galaxy are more consistent with a Type 2 object than a Type 1 object (Alloin et al. 1985). HST continuum imagery reveals spiral dust lanes within 1” of the nucleus (Griffiths et al. 1997) which might be re- sponsible for the Type 1/Type 2 variability. Baribaud et al. (1992) find hot dust which lies just outside the broad line region in this galaxy and a large covering factor that might explain the steep continuum of the AGN. Ehle et al. (1996) find NH ∼ 2.5 × 10 20 cm−2 from ROSAT X-ray observations of this galaxy. NGC 2623: This galaxy’s tidal tails are evidence of a merger event, however infrared observations reveal a sin- gle symmetric nucleus, implying that the merging galax- ies have coalesced. Multi-color, near infrared observa- tions reveal strong concentrations of obscuring material in the central 500 pc.(Joy & Harvey 1987; Lipari et al. 2004). Lipari et al. (2004) also find an optically obscured nucleus with V-band extinction ≥ 5 mag. IRAS 23128-5919: This galaxy is also in the late stages of a merger. The nuclei of the two galaxies are 4kpc apart and have not yet coalesced. The northern nucleus is a starburst. The southern nucleus is a known AGN, though its optical classification, Seyfert or LINER, is unclear (Duc, Mirabel, & Maza 1997; Charmandaris et al. 2002; Satyapal et al. 2004). IRAS 23128-5919 is an ultraluminous infrared galaxy (ULIRG), clearly con- sistent with the presence of substantial dust towards the nucleus. Optical spectroscopy of the southern nucleus in- dicates very large (1500 km s−1) blue asymmetries in the Hβ and [OIII] lines. This blue wing could be a signature of extinction toward the far side of an expanding region, where the red wing is preferentially obscured. (Johans- son & Bergvall 1988). Mrk 273: This galaxy is also a ULIRG, so significant dust obscuration toward the nucleus is expected. Near- IR imaging and high resolution radio observations show evidence for a double nucleus in this galaxy separated by less than 1 kpc (Ulvestad & Wilson 1984; Mazzarella et al. 1991; Majewski et al. 1993). However, high res- olution Chandra observations reveal only the northern of the two nuclei, suggesting that this galaxy is hosting only one AGN and that perhaps the other ”nucleus” is in fact a portion of the southern radio jet. The soft X- ray emission from the northern nucleus is obscured by column densities of at least 1023 cm−2 (Xia et al. 2002). Although the X-ray-emitting regions are physically dis- tinct from the NLR and some of the obscuration at X-ray wavelengths likely arises in dust-free gas within the sub- limation radius, the high column density derived may be consistent with high extinction toward the central re- gions of this galaxy. Though Xia et al (2002) find that the X-ray morphology of the AGN in Mrk 273 is con- sistent with a Seyfert, Colina et al. (1999) find that it has a LINER optical spectrum, thus implying that some LINER galaxies are in fact heavily absorbed powerful AGN. The soft diffuse X-ray halo in combination with the radio morphology found by Carilli & Taylor (2000) may suggest a circumnuclear starburst surrounding the northern AGN nucleus, again consistent with substantial obscuration toward the AGN. NGC 3627: This nearby galaxy (D ∼ 10Mpc) is thought to have had tidal interactions with NGC 3628, a neighboring galaxy in the Leo Triplet, some 8 × 108 years ago which caused an intense burst of star forma- tion in the nuclear regions around the same time (Rots 1978, Zhang et al. 1993, Afanasiev & Sil’chenko 2005). Zhang et al. (1993) also discovered an extremely dense molecular bar (mass ≥ 4 × 108 M⊙) and Chemin et al. (2003) uncovered a warped disk using Hα observations, both evidence of the tidal interaction. In their spectral fitting to the BeppoSAX observation of NGC 3627, Geor- gantopoulos et al. (2002) find intrinsic column densities of ∼ 1.5 × 1022 cm−2 which, like Mrk 273, may suggest substantial extinction to other regions near the nucleus. NGC 7469: This is a well-known, extensively-studied galaxy with strong, active star formation surrounding a Seyfert 1 nucleus. Meixner et al. (1990) find dense molecular gas (2 × 1010 M⊙), two orders of magnitude above the Galactic value, within the central 2.5kpc of the nucleus. 3.3µm imaging of the galaxy reveals that 80% of the PAH emission comes from an annulus ∼ 1”- 3”in radius around the central nucleus, indicating that there is an elongated region of material that shelters the PAH from the harsh radiation field of the AGN (Cutri et al. 1984, Mazzarella et al. 1994). [OIII] line asym- metries may corroborate the presence of a dense obscur- ing medium, revealing a blue wing resulting when the redshifted gas is obscured by the star forming ring (Wil- son et al. 1986). In addition, Genzel et al. (1995) find variation in the NIR emission attributable to extinction and estimate the extinction from the CO observations of Meixner et al. (1990) to be AV ∼ 10 mag. NGC 1365: This nearby (D = 18.6 Mpc) AGN is known to be circumscribed by embedded young star clus- ters. The galaxy also contains a prominent bar with a dust lane that penetrates the nuclear region (Phillips et al. 1983, Lindblad et al. 1996 & 1999, Galliano et al. 2005). Like NGC 7469, NGC 1365 shows a peak at 3.5µm implying PAH emission in spite of the harsh AGN radia- tion field (Galliano et al. 2005). The large Hα/Hβ ratio found by Alloin et al. (1981) implies substantial extinc- tion toward the emission line regions, ranging from 3-4 mag. Observations with ASCA and ROSAT imply high intrinsic column densities toward the X-ray emitting re- gions, suggesting possibly high obscuration towards other regions near the nucleus (Iyomoto et al. 1997, Komossa & Schulz (1998), see also Schulz et al. 1999). Komossa & Schulz show that the ratio of Hα to both the mid-IR and X-ray radiation is substantially different in NGC 1365 compared with typical Seyfert 1 galaxies, possibly sug- gesting inhomogenous obscuration (Schultz et al. 1999). In an XMM X-ray study of NGC 1365, Risaliti et al. (2005) also find a heavily absorbed Seyfert nucleus. The blueshifted X-ray spectral lines imply high column den- sities of 1023 cm−2 or more. Mrk 266 (NGC 5256): This luminous infrared galaxy is the only galaxy for which aperture effects most likely account for the low 14µm/24µm ratio. Mrk 266 contains a very complicated structure which includes at least two bright nuclei, a Seyfert and a LINER, that are 10” apart–a signature of a merger in progress. The mor- phology of the northeast LINER nucleus is extremely controversial (Wang et al. 1997; Kollatschny & Kowatsch 1998; Satyapal 2004, 2005; Ishigaki et al. 2000; Davies, Ward, & Sugai 2000). Mazzarella et al. (1988) find three non-thermal radio structures, two that coincide with the nuclei and one between the two nuclei. Mazzarella et al. (1988) suggest that the two nuclear structures are associated with classical AGN and are in the stage of a violent interaction in which the center of gravity of the collision produces a massive burst of star formation with supernovae or shocks which are responsible for the third nonthermal radio source. As can be seen in Figure 8, the SH slit, which provides the 14µm flux, overlaps with this third radio source, while the LH slit, responsible for the 24µm flux, encompasses the southwestern nucleus, the third radio source, and part of the northeastern nu- cleus. In this case the two lines observed originate in physically distinct regions that do not each encompass all potential sources of [NeV] emission, resulting in an unphysical 14µm/24µm ratio. This is not to say that Mrk 266 does not suffer from extinction at all. Indeed the possible presence of a circumnuclear starburst im- plies that there may be substantial extinction (Ishigaki et al. 2000; Davies, Ward, & Sugai 2000). We have ver- ified that this is the only distant galaxy in our sample Fig. 8.— 20 cm image of Mrk 266 taken from NED (http://nedwww.ipac.caltech.edu/). As can be seen here, the SH slit (from which the 14µm line is extracted) overlaps with a third radio source, while the LH slit (from which the 24µm line is ex- tracted) encompasses the southwestern nucleus and part of the northeastern nucleus. with a complicated nuclear structure that will result in aperture effects. 5.3. Can the [NeV] line flux ratio be used as a density diagnostic? Our analysis reveals that extinction towards parts of the NLR in some objects is significant and cannot be ig- nored at mid-IR wavelengths. In fact, it is quite possible that extinction affects the [NeV] line flux ratios of those galaxies with ratios above the low density limit (LDL) and the amount of extinction in all cases is highly un- certain. In addition to extinction, the temperature of the [NeV] emitting gas is unknown. If the [NeV] emis- sion originates within the walls of the obscuring central torus, which may be the source of extinction in many of our galaxies, we might expect the temperature of the gas to reach 106 K (Ferland et al. 2002). If, on the other hand, the [NeV] emission comes from further out in the NLR and is instead attenuated by the intervening mate- rial, we might expect the temperature of the gas to be closer to 104 K. As shown in Figure 1, the electron densi- ties inferred from the [NeV] line flux ratios are sensitive to temperature when such large temperature variations are considered. Based on the calculations shown in Fig- ure 1, the low ratios could indicate that the densities in the [NeV] line emitting gas are typically ≤ 3000 cm−3 for T = 104K. However, if the [NeV] gas is characterized by temperatures as high as T = 105K to 106K, densities as high as 105 cm−3 would be consistent with our mea- surements. We note that the [NeV] line flux ratios for the galaxies in our sample (especially the Type 1 AGNs) http://nedwww.ipac.caltech.edu/ all cluster around a ratio of ≈ 1.0. Two separate con- clusions may be drawn from this finding: 1) That the temperatures of the gas are low (∼ 104K) and that the electron density is relatively constant over many orders of magnitude in X-ray Luminosity and Eddington Ratio for these AGNs, or 2) That the temperature of the gas is high (105K to 106 K) and that the AGNs here sam- ple a wide range of electron densities (from 102 cm−3 to 105 cm−3). Since gas temperature, electron density, mid-IR continuum, and extinction are all unknown for these objects, the electron density cannot be determined here. 6. THE SIII LINE FLUX RATIOS In Figure 9 we plot the 18µm/33µm line ratio as a function of electron density ne. As with [NeV], we only consider the five levels of the ground configuration when computing the line ratio and we plot the relationship for gas temperatures of T = 104K and 105K. We adopt col- lision strengths from Tayal & Gupta (1999) and radiative transition probabilities from Mendoza & Zeippen (1982). Fig. 9.— 18µm/33µm line flux ratio in S III versus electron density ne, for gas temperatures T = 10 4 K and 105 K. In Table 3, we list the observed [SIII] line flux ratios for the galaxies in our sample. As with the [NeV] ratios, the [SIII] ratios in many galaxies listed in Table 3 are well below the theoretically allowed value of 0.45 for a gas temperature of T = 104K (13/33 detections). Again we explore the observational effects and the theoretical uncertainties that could artificially lower these ratios. Aperture Effects: The ionization potential of [SIII] is ∼ 35 eV and therefore the [SIII] emission may arise from gas ionized by either the AGN or young stars. In Table 3 we list, in addition to our Spitzer [SIII] fluxes, all available [SIII] fluxes from ISO. Unlike [NeV], the [SIII] fluxes from ISO are significantly larger than the Spitzer fluxes for most galaxies. In Figure 10 we plot the ISO to Spitzer flux ratios for the 18µm and 33µm the [SIII] lines. As can be seen here, the [SIII] emission extends beyond the Spitzer slit for many galaxies (6 out of 9 for [SIII] 18µm and 11 out of 13 for [SIII] 33µm). Similarly, when we compare the [SIII] flux arising from a single slit centered on the nucleus to the flux arising from a more extended region obtained using mapping observations (Dale et al. 2006), we find that in most cases the fluxes from the extended region are much larger than the nuclear single-slit fluxes. Galaxies with fluxes from Dale et al. (2006) are not included in Figure 10 since the extraction aperture for these galaxies is comparable to the 18µm ISO slit. We point out that the value for this ratio is dependent on the orientation of the Spitzer slit relative to the ISO slit and on the distance of each object. We also note that IRAS20551 and IRAS23128 are point sources with Spitzer 18µm fluxes greater than the ISO fluxes from Genzel et al. (1998), however they fall within the Genzel et al. (1998) quoted errors of 30% and the Spitzer calibration error of 15%. Figure 10 suggests that the [SIII] emission may be produced in the extended, circumnuclear star forming regions associated with many AGNs and that aperture effects need to be considered in our analysis of the [SIII] ratio for nearby objects. [SIII] (ISO) / F [SIII] (Spitzer) [SIII] 18 micron Ratio 0 1 2 3 4 5 6 [SIII] 33 micron Ratio Fig. 10.— The ratios of the [SIII] flux from ISO and Spitzer for the 18µm and 33µm lines. The range indicated with arrows is that corresponding to the absolute flux calibration for ISO (20%) and Spitzer (15%). The [SIII] emission is indeed extended beyond the Spitzer slit for many galaxies, suggesting that the [SIII] emission may be produced in star forming regions. We note that IRAS20551 and IRAS23128 are point sources with Spitzer 18µm fluxes greater than the ISO fluxes from Genzel et al. 1998, however they fall within the Genzel et al. (1998) quoted errors of 30% and the Spitzer calibration error of 15%. Galaxies with fluxes from Dale et al. 2006 are not included in this plot since the extraction aperture for these galaxies is comparable to the 18µm ISO slit. The contribution from star formation to the [SIII] lines can be estimated using the strength of the PAH emis- sion, one of the most widely used indicators of the star formation activity in galaxies (e.g. Luhman et al. 2003; Genzel et al. 1998; Roche et al. 1991; Rigopoulou et al. 1999, Clavel et al. 2000; Peeters, Spoon, & Tielens 2004). We examined the [SIII] 18.71 µm/PAH 6.2 µm and [SIII] 33.48 µm/PAH 6.2 µm line flux ratios in 7 starburst galaxies observed by Spitzer and found them to be comparable to the analogous ratios in our entire sample of AGNs as shown in Figure 11. This suggests that the bulk of the [SIII] emission originates in gas ion- ized by young stars. We note that the apertures of the SH and LH IRS modules are smaller than that of the SL2 module, which may artificially raise the line ratios plotted in Figure 11 for nearby galaxies compared with the more distant ones. However, the fact that the line ratios plotted in Figure 11 span a very narrow range sug- gests that the [SIII] line emission has a similar origin in starbursts and in AGNs. Thus, we assume that the bulk of the [SIII] emission originates in gas ionized by young stars and that the electron densities derived using these lines taken from slits of the same size (such those galaxies coming from Dale et al. 2006 mapping observations) or from the most distant galaxies are representative of the gas density in star forming regions. Extinction: We have shown that aperture effects are the likely explanation for why many of the [SIII] ratios for the galaxies in our sample fall below the LDL. How- ever, there are three galaxies in the sample with ratios below the LDL that are distant enough (D>55 Mpc, cor- responding to projected distances greater than 1.2 by 3 kpc and 3 by 6 kpc for the SH and LH slits, respectively) that aperture effects may not be as important (NGC 2623 & Mrk 273, Mrk 266 has been excluded since it is known to be affected by aperture variations See Section 5.2). Extinction may be the explanation for the low ratios in these galaxies. However, even though the SH and LH slits likely cover the entirety of the NLR at these distances, we note that these three galaxies contain well-known, large circumnuclear starbursts (See Section 5.2 for the indi- vidual galaxy summaries) which may produce extremely extended [SIII] emission. It is therefore still possible that the line ratios in these galaxies are artificially lowered by aperture variations between the SH and LH slits. How- ever, in addition to these three distant galaxies, NGC 4725 from Dale et al. (2006) has a [SIII] ratio below the low density limit. The low [SIII] ratio (<0.45) in this case cannot be attributed to aperture variations since the extraction region is the same for both the 18 and 33µm lines. Thus, for completeness, the extinction de- rived using the extinction curves given in Section 4 from the observed [SIII] line ratio for these four sources are given in Table 3. The Draine (1989) and Lutz et al. (1996) extinction curve calculations yield extinction val- ues that range from ∼ 12 to 25 mag. The Chiar and Tielens (2006) extinction curve for the Galactic Center may also be used since, unlike [NeV], the extinction at the longer wavelength line (33µm) is greater than that at the shorter wavelength line (18µm). The values de- rived from this method are quite similar, ranging from ∼ 10 to 22 mag. The Chiar and Tielens (2006) extinction curve from the local ISM cannot be used here since it only extends to 27.0µm. Computed Quantities: As with NeV, there may be uncertainties in the computed SIII infrared collisional rate coefficients. However, there is generally less contro- versy surrounding the [SIII] coefficients and these values are widely accepted. Our analysis suggests that aperture effects severely in- fluence the [SIII] line flux ratios in most cases and that the observed flux is likely dominated by star forming regions. Figure 12, a plot of the [SIII] line ratio as a function of distance, illustrates the influence of aperture effects on the [SIII] line ratio. Most of the galaxies at dis- tances <55 Mpc with [SIII] fluxes extracted from aper- tures of different sizes (i.e. NOT the Dale et al. (2006) galaxies) are below the LDL. On the other hand, galax- ies at larger distances and galaxies with fluxes from Dale et al. (2006) are generally above the LDL. Thus, for the most distant galaxies in our sample and the galaxies with fluxes from Dale et al. (2006) where the aperture for the 18 and 33 µm lines are equal, aperture effects are not problematic, but extinction, as can be seen from Mrk 273, NGC2623, and NGC 4725 in Figure 12, needs to be considered. As with the [NeV] line ratio, the [SIII] line ratio is NOT a tracer of the electron density in our sam- ple. In conclusion, the ambiguity of the intrinsic [SIII] line ratio is primarily the result of aperture variations. However there is at least one case (NGC 4725) where aperture effects cannot explain the low ratio, implying that, in addition to aperture variations, extinction likely plays a role in lowering the [SIII] line flux ratios. 7. SUMMARY We report in this paper the [NeV] 14µm/24µm and [SIII]18µm/33µm line flux ratios, traditionally used to measure electron densities in ionized gas, in an archival sample of 41 AGNs observed by the Spitzer Space Tele- scope. 1. We find that the [NeV] 14µm/24µm line flux ratios are low: approximately 70% of those measured are consistent with the low density limit to within the calibration uncertainties of the IRS. 2. We find that Type 2 AGNs have lower [NeV] 14µm/24µm line flux ratios than Type 1 AGNs. The mean ratios are 0.97 and 0.72 for the eight Type 1 and ten Type 2 AGNs, respectively, with uncertainties in the mean of about 0.08 for each. 3. For several galaxies, the observed [NeV] line ratios are below the theoretical low density limit. All of these galaxies are Type 2 AGNs. 4. We discuss the physical mechanisms that may play a role in lowering the line ratios: differential mid- IR extinction, low density, high temperature, and high mid-IR radiation density. 5. We argue that the [NeV]-emitting region likely originates interior to the torus in many of these AGNs and that differential infrared extinction due to dust in the obscuring torus may be responsible for the ratios below the low density limit. We sug- gest that the ratio may be a tracer of the torus inclination angle to our line of sight. 6. Our results imply that the extinction curve in these galaxies must be characterized by higher extinction at 14µm than at 24µm, contrary to recent studies of the extinction curve toward the Galactic Center. 7. A comparison between the [NeV] line fluxes ob- tained with Spitzer and ISO reveals that there are systematic discrepancies in calibration between the two instruments. However, our results are indepen- dent of which instrument is used; [NeV] line flux ratios are consistently lower in Type 2 AGNs than in Type 1 and [NeV] line flux ratios below the LDL are observed with both ISO and Spitzer. 0 1 2 3 4 5 [SIII]18/F(PAH6.2) Starbursts 0 1 2 3 4 5 AGN 0 1 2 3 4 5 [SIII](33)/F(PAH6.2) Starbursts 0 1 2 3 4 5 AGN Fig. 11.— Distribution of the [SIII]33µm/PAH 6.2µm and the [SIII]18µm/PAH 6.2µm line flux ratios for our sample of AGNs and a small sample of starburst galaxies observed by Spitzer. It is apparent that the line ratios of the AGNs are comparable to the corresponding ratios in starbursts, suggesting that the bulk of the [SIII] emission originates in star forming regions and not the NLRs in our sample of AGNs. 8. Our work provides strong motivation for investigat- ing the mid-IR spectra of a larger sample of galaxies with Spitzer in order to test our conclusions. 9. Finally, an analysis of the [SIII] emission reveals that it is extended in many or all of the galaxies and likely originates in star forming gas and NOT the NLR. Since there is a variation in the aper- tures between the SH and LH modules of the IRS, we cannot use the [SIII] line flux ratios to derive densities for the majority of galaxies in our sam- We are extremely thankful for all of the invaluable data analysis assistance from Dan Watson and Joel Green, without which this work would not have been possible. We are also very grateful to Davide Donato, Eli Dwek, Frederic Galliano, Paul Martini, Kartik Sheth, Eckhard Sturm, Peter van Hoof, and Dan Watson for their en- lightening and thoughtful comments/expertise that sig- nificantly improved this paper. Carissa Khanna was also very helpful in providing assistance in the preliminary data analysis. We are also grateful for the helpful and constructive comments from the referee. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Labora- tory, California Institute of Technology, under contract with the National Aeronautics and Space Administra- tion. SS gratefully acknowledges financial support from NASA grant NAG5-11432 and NAG03-4134X. JCW gratefully acknowledges support from Spitzer Space Tele- scope Theoretical Research Program. JCW is a Cottrell Scholar of Research Corporation. Research in infrared astronomy at NRL is supported by 6.1 base funding. RPD gratefully acknowledges financial support from the NASA Graduate Student Research Program. REFERENCES Alexander, T. & Sternberg, A., 1999, ApJ, 520, 137 Alexander, T., Sturm, E., Lutz, D., et al. 1999, ApJ, 512, 204 Alloin, D., Edmunds, M. G., Lindblad, P. O., & Pagel, B. E. J., 1981, A&A, 101, 377 Alloin, D., Pelat, D., Phillips, M., & Whittle, M., 1985, ApJ, 288, Alloin, D., Pantin, E, Lagage, P.O., & Granato, G. 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0704.0548

Neutrinos and Non-proliferation in Europe

Hawaii Neutrinos & Non-proliferation in Europe Michel Cribier* APC, Paris CEA/Saclay, DAPNIA/SPP The International Atomic Energy Agency (IAEA) is the United Nations agency in charge of the development of peaceful use of atomic energy. In particular IAEA is the verification authority of the Treaty on the Non-Proliferation of Nuclear Weapons (NPT). To do that jobs inspections of civil nuclear installations and related facilities under safeguards agreements are made in more than 140 states. IAEA uses many different tools for these verifications, like neutron monitor, gamma spectroscopy, but also bookeeping of the isotopic composition at the fuel element level before and after their use in the nuclear power station. In particular it verifie that weapon-origin and other fissile materials that Russia and USA have released from their defense programmes are used for civil application. The existence of an antineutrino signal sensitive to the power and to the isotopic composition of a reactor core, as first proposed by Mikaelian et al. [Mik77] and as demonstrated by the Bugey [Dec95] and Rovno experiments, [Kli94], could provide a means to address certain safeguards applications. Thus the IAEA recently ask members states to make a feasibility study to determine whether antineutrino detection methods might provide practical safeguards tools for selected applications. If this method proves to be useful, IAEA has the power to decide that any new nuclear power plants built has to include an antineutrino monitor. Within the Double Chooz collaboration, an experiment [Las06] mainly devoted to study the fundamental properties of neutrinos, we thought that we were in a good position to evaluate the interest of using antineutrino detection to remotely monitor nuclear power station. This effort in Europe, supplemented by the US effort [Ber06], will constitute the basic answer to IAEA of the neutrino community. * On behalf of a collective work by S. Cormon, M. Fallot, H. Faust, T. Lasserre, A. Letourneau, D. Lhuillier, V. Sinev from DAPNIA, Subatech and ILL. - 2 - Figure 1 : The statistical distribution of the fission products resulting from the fission of the most important fissile nuclei 235U and 239Pu shows two humps, one centered around masses 100 and the other one centered around 135. The low mass hump is at higher mass in 239Pu fission than in 235U, resulting in different nuclei and decays. The high penetration power of antineutrinos and the detection capability might provide a means to make remote, non-intrusive measurements of plutonium content in reactors [Ber02]. The antineutrino flux and energy spectrum depends upon the thermal power and on the fissile isotopic composition of the reactor fuel. Indeed, when a heavy nuclei (Uranium, Plutonium) experience a fission, it produce two unequal fission fragments (and a few free neutrons) ; the statistical distribution of the atomic masses is depicted in figure 1. All these nuclei immediately produced are extremely unstable - they are too rich in neutrons - and thus ß decay toward stable nuclei with an average of 6 ß decays. All these process involving several hundreds of unstable nuclei, with their excited states, makes very difficult to understand details of the physics, moreover, the most energetic antineutrinos, which are detected more easily, are produced in the very first decays, involving nuclei with typical lifetime smaller than a second. 235U 239Pu released energy per fission 201.7 MeV 210.0 MeV Mean energy of ν 2.94 MeV 2.84 MeV ν per fission > 1.8 MeV 1.92 1.45 average inter. cross section ≈ 3.2 10-43 ≈ 2.76 10-43 Based on predicted and observed ß spectra, the number of antineutrinos per fission from 239Pu is known to be less than the number from 235U, and the energy released bigger by 5%. Hence an hypothetical reactor able to use only 235U would induce in a detector an antineutrino signal 60% higher than the same reactor producing the same amount of energy but burning only 239Pu (see table). This offers a means to monitor changes in the relative amount of 235U and 239Pu in the core. If made in conjunction with accurate independent - 3 - measurements of the thermal power (with the temperature and the flow rate of cooling water), antineutrino measurements might provide an estimate of the isotopic composition of the core, in particular its plutonium inventories. The shape of the antineutrino spectrum can provide additional information about core fissile isotopic composition. Because the antineutrino signal from the reactor decreases as the square of the distance from the reactor to the detector a precise "remote" measurement is really only practical at distances of a few tens of meters if one is constrained to "small" detectors of the order of few cubic meter in size. Simulations MAGNITUDES OF SOME EFFECTS In our group, the development of detailed simulations using professional reactor codes started (see below), but it seems wise to use less sophisticated methods in order to evaluate already, with some flexibility, the magnitude of some effects. To do that we started from the set of Bateman equations, as depicted graphicaly in figure 2, which discribed the evolution of fuel elements in a reactor. The gross simplification in such treatment is the use of average cross section, depending only on 3 groups (thermal neutron, resonance region, fast neutrons), and moreover the fact that the neutron flux is imposed and not calculated. Figure 2 : The Bateman equations are the set of differential equations which described all transformations of the nuclei submitted to a given neutron flux : capture of neutrons are responsible to move at Z constant (green arrow), ß-decay are responsible to increase the atomic mass by one unit (dark blue arrow), and fission destroy the heavy nuclei and produce energy (orange arrows). Given this we use for each isotope under consideration, the cross section for capture, fission, and also plug in the parameters of the decays. Then it is rather easy (and fast) to simulate the evolution of a given initial core composition ; in the same way, it is possible to « make a diversion » by manipulation the fuel composition at a choosen moment. As an example, the figure 3 show the evolution of a fresh core composed of Uranium enriched at 3.5 % in 235U : the build up of 239Pu and 241Pu is rather well reproduced. - 4 - Figure 3 : In a new reactor the initial fuel consist of enriched uranium rods, with an 235U content typically at 3.5 %, the rest is 238U. As soon as the reactor is operating, reactions described by Bateman equations produce 239Pu (and 241Pu), which then participate to the energy production, at the expense of 238U. Knowing the amount of fissions at a given time, it is straight forward to translate that in a given antineutrino flux using the parametrisation of [Hub04], and finally using the interaction cross section for inverse ß decay reaction, to produce the recorded signal in a given detector placed at a suitable location from the reactor under examination. - 5 - Figure 4 : Positron spectrum recorded in an typical antineutrino detector (10 tons of target) placed at 150m of a nuclear reactor (1000 MWel). Positrons results from the inverse ß-decay reaction used in the detection of anti-neutrino. The signal is the superposition of several component whose spectrum exhibit small but sizeable differences, especialy at high energy. As an example of this type of computation, we show in figure 5, the effect of the modification of fuel composition after 100 days : here the operator, clever enough, knows that he cannot merely remove Plutonium from the core without changing the thermal power which will be immediatly noticed. Hence he takes the precaution to add 28 kg of 235U at the same time where he remove 20 kg of 239Pu : although the thermal power is kept constant, the imprint on the antineutrino signal, although modest, is such that, after 10 days, there is an increase of more than 1 σ in the number of interactions recorded. Such a diversion is clearly impossible in PWR or BWR, but more easy in Candu-type reactor, and even more in a molten salt reactor. - 6 - Figure 5 : An hypothetical diversion scenario where an exchange of 239Pu with 235U is made such that the power does not change, but the antineutrino signal recorded by the monitor is slightly increased, giving some evidence of an abnormal operation. SIMULATIONS OF DIVERSION SCENARIOS The IAEA recommends the study of specific safeguards scenarios. Among its concerns are the confirmation of the absence of unrecorded production of fissile material in declared reactors and the monitoring of the burn-up of a reactor core. The time required to manufacture an actual weapon estimated by the IAEA (conversion time), for plutonium in partially irradiated or spent fuel, lies between 1 and 3 months. The significant quantity of Pu is 8 kg, to be compared with the 3 tons of 235U contained in a Pressurized Water Reactor (PWR) of power 900MWe enriched to 3%. The small magnitude of the researched signal requires a carefull feasability study. The proliferation scenarios of interest involve different kinds of nuclear power plants such as light water or heavy water reactors (PWR, BWR, Candu...), it has to include isotope production reactors of a few tens of MWth, and future reactors (e.g., PBMRs, Gen IV reactors, accelerator-driven sub-critical assemblies for transmutation, molten salt reactors). To perform these studies, core simulations with dedicated Monte-Carlo codes should be provided, coupled to the simulation of the evolution of the antineutrino flux and spectrum over time. We started a simulation work using the widely used particle transport code MCNPX [Mcn05], coupled with an evolution code solving the Bateman equations for the fission - 7 - products within a package called MURE (MCNP Utility for Reactor Evolution) [Mur05]. This package offers a set of tools, interfaced with MCNP or MCNPX, that allows to define easily the geometry of a reactor core. In the evolution part, it accesses, the set of evaluated nuclear data and cross sections. MURE is perfectly adapted to simulate the evolution with time of the composition of the fuel, taking into account the neutronics of a reactor core. We are adapting the evolution code to simulate the antineutrino spectrum and flux, using simple Fermi decay as starting point. The extended MURE simulation will allows to perform sensitivity studies by varying the Pu content of the core in the relevant scenarios for IAEA. By varying the reactor power, the possibility to use antineutrinos for power monitoring can be evaluated. Preliminary results show that nuclei with half-lives lower than 1s emit about 70% (50%) of the 235U( 239Pu) antineutrino spectrum above 6 MeV. The high energy part of the spectrum is the energy region where Pu and U spectra differ mostly. The influence of the ß decay of these nuclei on the antineutrino spectrum might be preponderant also in scenarios where rapid changes of the core composition are performed, e.g. in reactors such as Candu, refueled on line. The appropriate starting point for this scenario is a representative PWR, like the Chooz reactors. For this reactor type, simulations of the evolution of the antineutrino flux and spectrum over time will be provided and compared to the accurate measurement provided by the near detector of Double Chooz. This should tell the precision on the fuel composition and of an independent thermal power measurements. An interesting point to study is at the time of the partial refuelling of the core, thanks to the fact that reactors like Chooz (N4-type) does not use MOX fuel. Without any extra experimental effort, the near detector of the Double Chooz experiment will provide the most important dataset of anti neutrino detected (5x105 ν per year) by a PWR. The precise neutrino energy spectrum recorded at a given time will be correlated to the fuel composition and to the thermal power provided by EDF. This valuable dataset will constitute an excellent experimental basis for the above feasibility studies of potential monitoring and for bench-marking fuel management codes ; it is expected that individual component due to fissile element (235U, 239Pu) could be extracted with some modest precision and serve as a benchmark of this techniques. To fulfil the goal of non-proliferation additional lab tests and theoretical calculations should also be performed to more precisely estimate the underlying neutrino spectra of plutonium and uranium fission products, especially at high energies. Contributions of decays to excited states of daughter nuclei are mandatory to reconstruct the shape of each spectrum. Following the conclusion of P. Huber and Th. Schwetz [Hub04] to achieve this goal a reduction of the present errors on the anti-neutrino fluxes of about a factor of three is necessary. We will see that such improvement needs an important effort. Experimental effort The precise measurement of β-decay spectra from fission products produced by the irradiation of a fissile target can be performed at the high flux reactor at Institut Laue - 8 - Langevin (ILL) in Grenoble, where similar studies performed in the past [Sch85] are the basis of the actual fluxes of antineutrinos used in these reactor neutrino experiment. The ILL reactor produces the highest neutron flux in the world : the fission rate of a fissile material target placed close to the reactor core is about 1012 per second. It is possible to choose different fissile elements as target in order to maximize the yield of the nucleus of interest. Using the LOHENGRIN recoil mass spectrometer [Loh04], measurement of individual β−spectra from short lived fission products are possible ; in the same irradiation channel, measurements of integral ß-spectrum with the Mini-INCA detectors [Mar06], could be envisaged to perform study on the evolution with time of the antineutrino energy spectrum of a nuclear power plant. EXPERIMENTS WITH LOHENGRIN The LOHENGRIN recoil mass spectrometer offers the possibility to measure β- decays of individual fission products. The fissile target (235U, 239Pu, 241Pu, …) is placed into a thermal neutron flux of 6.1014 n/cm2/s, 50 cm from the fuel element. Recoil fission products are selected with a dipolar magnetic field followed by an electrostatic condenser. At the end the fragments could be implanted in a moving tape, and the measurement of subsequent β and γ-rays are recorded by a β-spectrometer (Si-detector) and Ge-clover detectors, respectively. Coincidences between these two quantities could also be made to reconstruct the decay scheme of the observed fission products or to select one fission product. Fragments with half- lives down to 2 µs can be measured, so that nuclei with large Qß (above 4 MeV) can be measured. The LOHENGRIN experimental objectives are to complete existing β-spectra of individual fission products [Ten89] with new measurements for the main contributors to the detected ν-spectra and to clarify experimental disagreements between previous measurements. This ambitious experimental programme is motivated by the fact - noted by C. Bemporad [Bem02] - that unknown decays contribute as much as 25% of the antineutrinos at energies > 4MeV. Folding the antineutrino energy spectrum over the detection cross-section for inverse beta decay enhances the contribution of the high energy antineutrinos to the total detected flux by a factor of about 10 for Eν > 6 MeV. The focus of these experiments will be on neutron rich nuclei with yields very different in 239Pu and 235U fission. In the list : 86Ge,90-92Se, 94Br, 96-98Kr, 100Rb, 100-102Sr, 108-112Mo, 106-113Tc,113-115Ru…contribute to the high energy part of the spectrum and have never been measured. IRRADIATION TESTS IN SUMMER 2005 A test-experiment has been performed during two weeks last in summer 2005. The isobaric chains A=90 and A=94 were studied where some isotopes possess a high Qß energy, contributing significantly to the high energy part of the antineutrino spectra following 235U and 239Pu fissions and moreover produced with very different fission yields after 235U and 239Pu fission [Eng94]. The well-known nuclei, such as 90Br, will serve as a test of the experimental set-up, while the beta decay of more exotic nuclei such as 94Kr and 94Br will constitute a test case for how far one can reach in the very neutron rich region with this experimental device. The recorded data (figure 6) will validate the simulation described in the - 9 - previous section, in particular the evolution over time of the isobaric chains beta decay spectra. Silicon detector Germanium detector Figures 6 : Beta energy spectrum (6a) recorded with the silicon detector corresponding to ß decay of fission products with mass A=94. The fission products arising from the LOHENGRIN spectrometer were implanted on a mylar tape of adjustable velocity in front of the silicon detector. The highest velocity was selected in order to enhance shorter-lived nuclei such as 94Kr and 94Br. The gamma energy spectrum (6b) obtained with the germanium detector corresponding and to the same runs is displayed also. INTEGRAL ß SPECTRA MEASUREMENTS In complement to individual studies on LOHENGRIN, more integral studies can be envisaged using the so called “Mini-INCA chamber” at ILL [Mar06] in return for adding a β- spectrometer (to be developped). The existing α- and γ-spectroscopy station is connected to the LOHENGRIN channel and offers the possibility to perform irradiations in a quasi thermal neutron flux up to 20 times the nominal value in a PWR. Moreover, the irradiation can be repeated as many time as needed. It offers then the unique possibility to characterize the evolution of the ß spectrum as a function of the irradiation time and the irradiation cooling. The expected modification of the β spectrum as a function of the irradiation time is connected to the transmutation induced by neutron capture of the fissile and fission fragment elements. It is thus related to the “natural” evolution of the spent-fuel in the reactor. The modification of the β spectrum as a function of the cooling time is connected to the decaying chain of the fission products and is then a means to select the emitted fragments by their livetime. This - 10 - information is important because long-lived fission fragments accumulate in the core and after few days mainly contribute to the low energy part of the antineutrino-spectra. Due to the mechanical transfer of the sample from the irradiation location to the measurement station an irreducible delay time of 30 mn is imposed leading to the loss of short-live fragments. PROSPECT TO STUDY FISSION OF 238U The integral beta decay spectrum arising from 238U fission has never been measured. All information relies on theoretical computations [Vog89]. Some experiments could be envisaged using few MeV neutron sources in Europe (Van de Graaf in Geel, SINQ in PSI, ALVARES or SAMES accelerators at Valduc, …). Here the total absence of experimental data on the ß emitted in the fission of 238U change the context of this measurement compared to the other isotopes. Indeed any integral measurements performed could be used to constraint the present theoretical estimations of the antineutrino flux produced in the fission of 238U. In any case it seems rather difficult to fulfil the goal of a determination of the isotopic content from antineutrinos measurements as long as in important part of the energy spectrum is so poorly known. Conclusions After the preliminary studies, some thoughts can already be made. A realistic diversion (≈ 10 kg Pu) has an imprint in the antineutrino signal which is very small. The present knowledge on antineutrino spectrum emitted in fissions is not precise enough to allow a determination of the isotopic content in the core sensitive to such diversion. On the other hand, the thermal power measurement is a less difficult job. Neutrinos sample the whole core, without attenuation, and would bring valuable information on the power with totally different systematics than present methods. Even if its measurement is not dissuasive by itself, the operator cannot hide any stops or change of power, and in most case, such a record made with an external and independent device, virtually impossible to fake, will act as a strong constraint. In spite of the uncertainty mentioned previously, we see that the most energetic part offers the best possibility to disentangle fission from 235U and 239Pu. The comparison between the cumulative numbers of antineutrinos as a function of antineutrino energy detected at low vs. high energy is an efficient observable to distinguish pure 235U and 239Pu. IAEA seeks also monitoring large spent-fuel elements. For this application, the likelihood is that antineutrino detectors could only make measurements on large quantities of beta-emitters, e.g., several cores of spent fuel. In the time of the experiment the discharge of parts of the core will happen and the Double-Chooz experiment will quantify the sensitivity of such monitoring. More generally the techniques developed for the detection of antineutrinos could be applied for the monitoring of nuclear activities at the level of a country. Hence a KamLAND type detector deeply submerged off the coast of the country, would offer the sensitivity to - 11 - detect a new underground reactor located at several hundreds of kilometers. All these common efforts toward more reliable techniques, remotely operated detectors, not to mention undersea techniques will automatically benefit to both fields, safeguard and geo-neutrinos. References [Bem02] Bemporad et al.,Rev. of Mod. Phys., Vol. 74, (2002). [Ber02] A. Bernstein, Y. Wang, G. Gratta, and T. West, J. Appl. Phys. 91, 4672 (2002) [Ber06] A. Bernstein, these proceeding [Dec95] Y. Declais et al., Nucl. Phys. B434, 503 (1995) [Eng94] T.R. England and B.F. Rider, ENDF-349, LA-UR-94-3106. [Hub04] P. Huber, Th. Schwetz, Precision spectroscopy with reactor anti- neutrinos Phys.Rev. D70 (2004) 053011 [Kli94] Klimov et al., Atomic Energy, v.76-2, 123, (1994) [Las06] T. Lasserre, these proceeding [Loh04] ILL Instrument Review, 2004/2005. [Mik77] Mikaelian L.A. Neutrino laboratory in the atomic plant, Proc. Int. Conference Neutrino-77, v. 2, p. 383-387 [Mar06] F. Marie, A. Letourneau et al., Nucl. Instr and Meth A556 (2006) 547. [Mcn05] Monte Carlo N-Particle eXtended, LA-UR-05-2675, J.S.Hendricks et al. [Mur05] MURE : MCNP Utility for Reactor Evolution -Description of the methods, first applications and results. MÃl'plan O., Nuttin A., Laulan O., David S., Michel-Sendis F. et al. In Proceedings of the ENC 2005 (CD-Rom) (2005) 1-7. [Sch85] K. Schreckenbach, G. Colvin, W. Gelletly, F.v. Feilitzch, Phys. Lett. B160 (1985) 325 [Ten89] O. Tengblad et al., Nuclear Physics A 503 (1989) 136-160. [Vog89] P. Vogel and J. Engel, Phys. Rev. D39, 3378 (1989)

0704.0549

Kinks and Particles in Non-integrable Quantum Field Theories

Kinks and Particles in Non-integrable Quantum Field Theories G. Mussardoa,b aInternational School for Advanced Studies Via Beirut 1, 34013 Trieste, Italy bIstituto Nazionale di Fisica Nucleare, Sezione di Trieste Abstract In this talk we discuss an elementary derivation of the semi-classical spec- trum of neutral particles in two field theories with kink excitations. We also show that, in the non-integrable cases, each vacuum state cannot generically support more than two stable particles, since all other neutral exitations are resonances, which will eventually decay. http://arxiv.org/abs/0704.0549v1 1 Introduction Two–dimensional massive Integrable Quantum Field Theories (IQFTs) have proven to be one of the most successful topics of relativistic field theory, with a large variety of applications to statistical mechanical models. The main reason for this success consists of their simplified on–shell dynamics which is encoded into a set of elastic and factorized scattering amplitudes of their massive particles [1, 2]. The two- particle S-matrix has a very simple analytic structure, with only poles in the physical strip, and it can be computed combining the standard requirements of unitarity, crossing and factorization together with specific symmetry properties of the theory. The complete mass spectrum is obtained looking at the pole singularities of the S–matrix elements. Off–mass shell quantities, such as the correlation functions, can be also determined once the elastic S–matrix and the mass spectrum are known. In fact, one can compute the exact matrix elements of the (semi)local fields on the asymptotic states with the Form Factor (FF) approach [3], and use them to write down the spectral representation of the correlators. By following this approach, it has been possible, for instance, to tackle successfully the long-standing problem of spelling out the mass spectrum and the correlation functions of the two dimensional Ising model in a magnetic field [2, 4], as well as many other interesting problems of statistical physics (for a partial list of them see, for instance, [5]). The S-matrix approach can be also constructed for massless IQFTs [6, 7, 8, 9], despite the subtleties in defining a scattering theory between massless particles in (1 + 1) dimensions, and turns out to be useful mainly when conformal symmetry is not present. In this case, massless IQFTs generically describe the Renormaliza- tion Group trajectories connecting two different Conformal Field Theories, which respectively rule the ultraviolet and infrared limits of all physical quantities along the flows. Given the large number of remarkable results obtained by the study of IQFTs, one of the most interesting challenges is to extend the analysis to the non–integrable field theories, at least to those obtained as deformations of the integrable ones and to develop the corresponding perturbation theory. The breaking of integrability is expected to considerably increase the difficulties of the mathematical analysis, since scattering processes are no longer elastic. Non–integrable field theories are in fact generally characterized by particle production amplitudes, resonance states and, correspondingly, decay events. All these features strongly effect the analytic structure of the scattering amplitudes, introducing a rich pattern of branch cut singularities, in addition to the pole structure associated to bound and resonance states. For massive non–integrable field theories, a convenient perturbative scheme was originally proposed in [10] and called Form Factor Perturbation Theory (FFPT), since it is based on the knowledge of the exact Form Factors (FFs) of the original integrable theory. It was shown that, even using just the first order correction of the FFPT, a great deal of information can be obtained, such as the evolution of their particle content, the variation of their masses and the change of the ground state energy. Whenever possible, universal ratios were computed and successfully compared with their value obtained by other means. Recently, for instance, it has been obtained the universal ratios relative to the decay of the particles with higher masses in the Ising model in a magnetic field, once the temperature is displayed away from the critical value [11] (see also the contibution by G. Delfino in this proceedings [12]). For other and important aspects of the Ising model along non-integrable lines see the references [13, 14, 15, 16]. Applied to the double Sine–Gordon model [17], the FFPT has been useful in clarifying the rich dynamics of this non–integrable model. In particular, in relating the confinement of the kinks in the deformed theory to the non–locality properties of the perturbed operator and predicting the existence of a Ising–like phase transition for particular ratios of the two frequencies – results which were later confirmed by a numerical study [18]. The FFPT has been also used to study the spectrum of the O(3) non-linear sigma model with a topological θ term, by varying θ [19, 20]. In this talk I would like to focus the attention on a different approach to tackle some interesting non-integrable models, i.e. those two dimensional field theories with kink topological excitations. Such theories are described by a scalar real field ϕ(x), with a Lagrangian density (∂µϕ) 2 − U(ϕ) , (1.1) where the potential U(ϕ) possesses several degenerate minima at ϕ a (a = 1, 2, . . . , n), as the one shown in Figure 1. These minima correspond to the different vacua | a 〉 of the associate quantum field theory. The basic excitations of this kind of models are kinks and anti-kinks, i.e. topological configurations which interpolate between two neighbouring vacua. Semiclassically they correspond to the static solutions of the equation of motion, i.e. ∂2x ϕ(x) = U ′[ϕ(x)] , (1.2) with boundary conditions ϕ(−∞) = ϕ(0)a and ϕ(+∞) = ϕ(0)b , where b = a ± 1. Denoting by ϕab(x) the solutions of this equation, their classical energy density is (A) (B) Figure 1: Potential U(ϕ) of a quantum field theory with kink excitations (A) and istogram of the masses of the kinks (B). given by ǫab(x) = + U(ϕab(x)) , (1.3) and its integral provides the classical expression of the kink masses Mab = ǫab(x) . (1.4) It is easy to show that the classical masses of the kinks ϕab(x) are simply proportional to the heights of the potential between the two minima ϕ a and ϕ : their istogram provides a caricature of the original ptential (see Figura 1). The classical solutions can be set in motion by a Lorentz transformation, i.e. ϕab(x) → ϕab (x± vt)/ 1− v2 . In the quantum theory, these configurations de- scribe the kink states | Kab(θ) 〉, where a and b are the indices of the initial and final vacuum, respectively. The quantity θ is the rapidity variable which parameterises the relativistic dispersion relation of these excitations, i.e. E = Mab cosh θ , P = Mab sinh θ . (1.5) Conventionally | Ka,a+1(θ) 〉 denotes the kink between the pair of vacua {| a 〉, | a+ 1 〉} while | Ka+1,a 〉 is the corresponding anti-kink. For the kink configurations it may be useful to adopt the simplified graphical form shown in Figure 2. The multi-particle states are given by a string of these excitations, with the adja- cency condition of the consecutive indices for the continuity of the field configuration | Ka1,a2(θ1)Ka2,a3(θ2)Ka3,a4(θ3) . . .〉 , (ai+1 = ai ± 1) (1.6) In addition to the kinks, in the quantum theory there may exist other excitations in the guise of ordinary scalar particles (breathers). These are the neutral excitations a,a+1K K a+1,a | a+1> | 0 > | a > | n > Figure 2: Kink and antikink configurations. | Bc(θ) 〉a (c = 1, 2, . . .) around each of the vacua | a 〉. For a theory based on a Lagrangian of a single real field, these states are all non-degenerate: in fact, there are no extra quantities which commute with the Hamiltonian and that can give rise to a multiplicity of them. The only exact (alias, unbroken) symmetries for a Lagrangian as (1.1) may be the discrete ones, like the parity transformation P , for instance, or the charge conjugation C. However, since they are neutral excitations, they will be either even or odd eigenvectors of C. The neutral particles must be identified as the bound states of the kink-antikink configurations that start and end at the same vacuum | a 〉, i.e. | Kab(θ1)Kba(θ2) 〉, with the “tooth” shapes shown in Figure 3. | 0 > | 0 > | 0 > Figure 3: Kink-antikink configurations which may give rise to a bound state nearby the vacuum | 0 〉a. If such two-kink states have a pole at an imaginary value i ucab within the physical strip 0 < Im θ < π of their rapidity difference θ = θ1 − θ2, then their bound states are defined through the factorization formula which holds in the vicinity of this singularity | Kab(θ1)Kba(θ2) 〉 ≃ i θ − iuc | Bc 〉a . (1.7) In this expression gcab is the on-shell 3-particle coupling between the kinks and the neutral particle. Moreover, the mass of the bound states is simply obtained by sub- stituing the resonance value i ucab within the expression of the Mandelstam variable s of the two-kink channel s = 4M2ab cosh −→ mc = 2Mab cos . (1.8) Concerning the vacua themselves, as well known, in the infinite volume their classical degeneracy is removed by selecting one of them, say | k 〉, out of the n available. This happens through the usual spontaneously symmetry breaking mech- anism, even though – stricly speaking – there may be no internal symmetry to break at all. This is the case, for instance, of the potential shown in Figure 1, which does not have any particular invariance. In the absence of a symmetry which connects the various vacua, the world – as seen by each of them – may appear very different: they can have, indeed, different particle contents. The problem we would like to examine in this talk concerns the neutral excitations around each vacuum, in particular the question of the existence of such particles and of the value of their masses. To this aim, let’s make use of a semiclassical approach. 2 A semiclassical formula The starting point of our analysis is a remarkably simple formula due to Goldstone- Jackiw [23], which is valid in the semiclassical approximation, i.e. when the coupling constant goes to zero and the mass of the kinks becomes correspondingly very large with respect to any other mass scale. In its refined version, given in [24] and redis- covered in [25], it reads as follows1 (Figure 4) (θ) = 〈Kab(θ1) | ϕ(0) | Kab(θ2)〉 ≃ dx eiMab θ x ϕab(x) , (2.9) where θ = θ1 − θ2. 1The matrix element of the field ϕ(y) is easily obtained by using ϕ(y) = e−iPµy ϕ(0) eiPµy and by acting with the conserved energy-momentum operator Pµ on the kink state. Moreover, for the semiclassical matrix element FG (θ) of the operator G[ϕ(0)], one should employ G[ϕab(x)]. For instance, the matrix element of ϕ2(0) are given by the Fourier transform of ϕ2 Figure 4: Matrix element between kink states. Notice that, if we substitute in the above formula θ → iπ − θ, the corresponding expression may be interpreted as the following Form Factor (θ) = f(iπ − θ) = 〈a | ϕ(0) | Kab(θ1)Kba(θ2)〉 . (2.10) In this matrix element, it appears the neutral kink states around the vacuum | a〉 we are interested in. Eq. (2.9) deserves several comments. 1. The appealing aspect of the formula (2.9) stays in the relation between the Fourier transform of the classical configuration of the kink, – i.e. the solu- tion ϕab(x) of the differential equation (1.2) – to the quantum matrix ele- ment of the field ϕ(0) between the vacuum | a 〉 and the 2-particle kink state | Kab(θ1)Kba(θ2) 〉. Once the solution of eq. (1.2) has been found and its Fourier transform has been taken, the poles of Fab(θ) within the physical strip of θ identify the neutral bound states which couple to ϕ. The mass of the neutral particles can be extracted by using eq. (1.8), while the on-shell 3-particle coupling gcab can be obtained from the residue at these poles (Figura 5) θ→i uc (θ − iucab)Fab(θ) = i gcab 〈a | ϕ(0) | Bc 〉 . (2.11) 2. It is important to stress that, for a generic theory, the classical kink config- uration ϕab(x) is not related in a simple way to the anti-kink configuration ϕba(x). It is precisely for this reason that neighbouring vacua may have a different spectrum of neutral excitations, as shown in the examples discussed in the following sections. ab ba ab ba Figure 5: Residue equation for the matrix element on the kink states. 3. It is also worth noting that this procedure for extracting the bound states masses permits in many cases to avoid the semiclassical quantization of the breather solutions [22], making their derivation much simpler. The reason is that, the classical breather configurations depend also on time and have, in general, a more complicated structure than the kink ones. Yet, it can be shown that in non–integrable theories these configurations do not exist as exact solutions of the partial differential equations of the field theory. On the contrary, in order to apply eq. (2.9), one simply needs the solution of the ordinary differential equation (1.2). It is worth notice that, to locate the poles (θ), one only needs to looking at the exponential behavior of the classical solutions at x → ±∞, as discussed below. In the next two sections we will present the analyse a class of theories with only two vacua, which can be either symmetric or asymmetric ones. A complete analysis of other potentials can be found in the original paper [27]. 3 Symmetric wells A prototype example of a potential with two symmetric wells is the ϕ4 theory in its broken phase. The potential is given in this case by U(ϕ) = ϕ2 − m . (3.12) Let us denote with | ±1 〉 the vacua corresponding to the classical minima ϕ(0)± = . By expanding around them, ϕ = ϕ ± + η, we have ± + η) = m 2 η2 ±m λ η3 + η4 . (3.13) Hence, perturbation theory predicts the existence of a neutral particle for each of the two vacua, with a bare mass given by mb = 2m, irrespectively of the value of the coupling λ. Let’s see, instead, what is the result of the semiclassical analysis. The kink solutions are given in this case by ϕ−a,a(x) = a , a = ±1 (3.14) and their classical mass is ǫ(x) dx = . (3.15) The value of the potential at the origin, which gives the height of the barrier between the two vacua, can be expressed as U(0) = M0 , (3.16) and, as noticed in the introduction, is proportional to the classical mass of the kink. If we take into account the contribution of the small oscillations around the classical static configurations, the kink mass gets corrected as [22] +O(λ) . (3.17) It is convenient to define > 0 , and also the adimensional quantities ; ξ = 1− πcg . (3.18) In terms of them, the mass of the kink can be expressed as . (3.19) Since the kink and the anti-kink solutions are equal functions (up to a sign), their Fourier transforms have the same poles. Hence, the spectrum of the neutral particles will be the same on both vacua, in agreement with the Z2 symmetry of the model. We have f−a,a(θ) = dx eiMθ xϕ−a,a(x) = i a By making now the analitical continuation θ → iπ−θ and using the above definitions (3.18), we arrive to F−a,a(θ) = 〈a | ϕ(0) | K−a,a(θ1)Ka,−a(θ2)〉 ∝ (iπ−θ) ) . (3.20) The poles of the above expression are located at θn = iπ (1− ξ n) , n = 0,±1,±2, . . . (3.21) and, if ξ ≥ 1 , (3.22) none of them is in the physical strip 0 < Im θ < π. Consequently, in the range of the coupling constant 1 + πc = 1.02338... (3.23) the theory does not have any neutral bound states, neither on the vacuum to the right nor on the one to the left. Viceversa, if ξ < 1, there are n = neutral bound states, where [x] denote the integer part of the number x. Their semiclassical masses are given by = 2M sin = n mb n2 + ... . (3.24) Note that the leading term is given by multiples of the mass of the elementary boson | B1〉. Therefore the n-th breather may be considered as a loosely bound state of n of it, with the binding energy provided by the remaining terms of the above expansion. But, for the non-integrability of the theory, all particles with mass mn > 2m1 will eventually decay. It is easy to see that, if there are at most two particles in the spectrum, it is always valid the inequality m2 < 2m1. However, if ξ < , for the higher particles one always has mk > 2m1 , for k = 3, 4, . . . n . (3.25) According to the semiclassical analysis, the spectrum of neutral particles of ϕ4 theory is then as follows: (i) if ξ > 1, there are no neutral particles; (ii) if 1 < ξ < 1, there Figure 6: Neutral bound states of ϕ4 theory for g < 1. The lowest two lines are the stable particles whereas the higher lines are the resonances. is one particle; (iii) if 1 < ξ < 1 there are two particles; (iv) if ξ < 1 there are particles, although only the first two are stable, because the others are resonances. Let us now briefly mention some general features of the semiclassical methods, starting from an equivalent way to derive the Fourier transform of the kink solution. To simplify the notation, let’s get rid of all possible constants and consider the Fourier transform of the derivative of the kink solution, expressed as G(k) = dx eikx cosh2 x . (3.26) We split the integral in two terms G(k) = dx eikx cosh2 x dx eikx cosh2 x , (3.27) and we use the following series expansion of the integrand, valid on the entire real axis (except the origin) cosh2 x (−1)n+1n e−2n|x| . (3.28) Substituting this expression into (3.27) and computing each integral, we have G(k) = 4i (−1)n+1n ik + 2n −ik + 2n . (3.29) Obviously it coincides with the exact result, G(k) = πk/ sinh π k, but this derivation permits to easily interpret the physical origin of each pole. In fact, changing k to the original variable in the crossed channel, k → (iπ − θ)/ξ, we see that the poles which determine the bound states at the vacuum | a〉 are only those relative to the exponential behaviour of the kink solution at x → −∞. This is precisely the point where the classical kink solution takes values on the vacuum | a〉. In the case of ϕ4, the kink and the antikink are the same function (up to a minus sign) and therefore they have the same exponential approach at x = −∞ at both vacua | ±1〉. Mathematically speaking, this is the reason for the coincidence of the bound state spectrum on each of them: this does not necessarily happens in other cases, as we will see in the next section, for instance. The second comment concerns the behavior of the kink solution near the minima of the potential. In the case of ϕ4, expressing the kink solution as ϕ(x) = 2x − 1 2x + 1 , (3.30) and expanding around x = −∞, we have ϕ(t) = − m√ 1− 2t+ 2t2 − 2t3 + · · · 2 (−1)ntn · · · , (3.31) where t = exp[ 2x]. Hence, all the sub-leading terms are exponential factors, with exponents which are multiple of the first one. Is this a general feature of the kink solutions of any theory? It can be proved that the answer is indeed positive [27]. The fact that the approach to the minimum of the kink solutions is always through multiples of the same exponential (when the curvature ω at the minimum is different from zero) implies that the Fourier transform of the kink solution has poles regularly spaced by ξa ≡ ωπMab in the variable θ. If the first of them is within the physical strip, the semiclassical mass spectrum derived from the formula (2.9) near the vacuum | a 〉 has therefore the universal form mn = 2Mab sin . (3.32) As we have previously discussed, this means that, according to the value of ξa, we can have only the following situations at the vacuum | a 〉: (a) no bound state if ξa > 1; (b) one particle if < ξa < 1; (c) two particles if < ξa < ; (d) particles if ξa < , although only the first two are stable, the others being resonances. So, semiclassically, each vacuum of the theory cannot have more than two stable particles above it. Viceversa, if ω = 0, there are no poles in the Fourier transform of the kink and therefore there are no neutral particles near the vacuum | a 〉. 4 Asymmetric wells In order to have a polynomial potential with two asymmetric wells, one must nec- essarily employ higher powers than ϕ4. The simplest example of such a potential is obtained with a polynomial of maximum power ϕ6, and this is the example discussed here. Apart from its simplicity, the ϕ6 theory is relevant for the class of universality of the Tricritical Ising Model [28]. As we can see, the information available on this model will turn out to be a nice confirmation of the semiclassical scenario. . A class of potentials which may present two asymmetric wells is given by U(ϕ) = ϕ− b m√ ϕ2 + c , (4.33) with a, b, c all positive numbers. To simplify the notation, it is convenient to use the dimensionless quantities obtained by rescaling the coordinate as xµ → mxµ and the field as ϕ(x) → λ/mϕ(x). In this way the lagrangian of the model becomes L = m (∂ϕ)2 − 1 (ϕ+ a)2(ϕ− b)2(ϕ2 + c) . (4.34) The minima of this potential are localised at ϕ 0 = −a and ϕ 1 = b and the corresponding ground states will be denoted by | 0 〉 and | 1 〉. The curvature of the potential at these points is given by U ′′(−a) ≡ ω20 = (a+ b)2(a2 + c) ; U ′′(b) ≡ ω21 = (a+ b)2(b2 + c) . (4.35) For a 6= b, we have two asymmetric wells, as shown in Figure 7. To be definite, let’s assume that the curvature at the vacuum | 0 〉 is higher than the one at the vacuum | 1 〉, i.e. a > b. The problem we would like to examine is whether the spectrum of the neutral particles | B 〉s (s = 0, 1) may be different at the two vacua, in particular, whether it would be possible that one of them (say | 0〉) has no neutral excitations, whereas the other has just one neutral particle. The ordinary perturbation theory shows that both vacua has neutral excitations, although with different value of their mass: m(0) = (a+ b) 2 (a2 + c) , m(1) = (a+ b) 2 (b2 + c) . (4.36) Let’s see, instead, what is the semiclassical scenario. The kink equation is given in this case by = ±(ϕ + a)(ϕ− b) ϕ2 + c . (4.37) Figure 7: Example of ϕ6 potential with two asymmetric wells and a bound state only on one of them. We will not attempt to solve exactly this equation but we can present nevertheless its main features. The kink solution interpolates between the values −a (at x = −∞) and b (at x = +∞). The anti-kink solution does viceversa, but with an important difference: its behaviour at x = −∞ is different from the one of the kink. As a matter of fact, the behaviour at x = −∞ of the kink is always equal to the behaviour at x = +∞ of the anti-kink (and viceversa), but the two vacua are approached, in this theory, differently. This is explicitly shown in Figure 8 and proved in the following. -4 -2 0 2 4 0.005 0.015 0.025 0.035 Figure 8: Typical shape of , obtained by a numerical solution of eq. (4.37). Let us consider the limit x → −∞ of the kink solution. For these large values of x, we can approximate eq. (4.37) by substituting, in the second and in the third term of the right-hand side, ϕ ≃ −a, with the result ≃ (ϕ+ a)(a+ b) a2 + c , x → −∞ (4.38) This gives rise to the following exponential approach to the vacuum | 0〉 ϕ0,1(x) ≃ −a+ A exp(ω0x) , x → −∞ (4.39) where A > 0 is a arbitrary costant (its actual value can be fixed by properly solving the non-linear differential equation). To extract the behavior at x → −∞ of the anti-kink, we substitute this time ϕ ≃ b into the first and third term of the right hand side of (4.37), so that ≃ (ϕ− b)(a+ b) b2 + c , x → −∞ (4.40) This ends up in the following exponential approach to the vacuum | 1〉 ϕ1,0(x) ≃ b− B exp(ω1x) , x → −∞ (4.41) where B > 0 is another constant. Since ω0 6= ω1, the asymptotic behaviour of the two solutions gives rise to the following poles in their Fourier transform F(ϕ0,1) → ω0 + ik (4.42) F(ϕ1,0) → ω1 + ik In order to locate the pole in θ, we shall reintroduce the correct units. Assuming to have solved the differential equation (4.37), the integral of its energy density gives the common mass of the kink and the anti-kink. In terms of the constants in front of the Lagrangian (4.34), its value is given by α , (4.43) where α is a number (typically of order 1), coming from the integral of the adimen- sional energy density (1.4). Hence, the first pole2 of the Fourier transform of the kink and the antikink solution are localised at θ(0) ≃ iπ 1− ω0 1− ω0 (4.44) θ(1) ≃ iπ 1− ω1 1− ω1 2In order to determine the others, one should look for the subleading exponential terms of the solutions. If we now choose the coupling constant in the range , (4.45) the first pole will be out of the physical sheet whereas the second will still remain inside it! Hence, the theory will have only one neutral bound state, localised at the vacuum | 1 〉. This result may be expressed by saying that the appearance of a bound state depends on the order in which the topological excitations are arranged: an antikink-kink configuration gives rise to a bound state whereas a kink-antikink does not. Finally, notice that the value of the adimensional coupling constant can be chosen so that the mass of the bound state around the vacuum | 1 〉 becomes equal to mass of the kink. This happens when . (4.46) Strange as it may appear, the semiclassical scenario is well confirmed by an explicit example. This is provided by the exact scattering theory of the Tricritical Ising Model perturbed by its sub-leading magnetization. Firstly discovered through a numerical analysis of the spectrum of this model [29], its exact scattering theory has been discussed later in [30]. 5 Conclusions In this paper we have used simple arguments of the semi-classical analysis to in- vestigate the spectrum of neutral particles in quantum field theories with kink ex- citations. We have concentrated our analysis on two cases: the first relative to a potential with symmetric wells, the second concerning with a potential with asym- metric wells. Leaving apart the exact values of the quantities extracted by the semiclassical methods, it is perhaps more important to underline some general fea- tures which have emerged through this analysis. One of them concerns, for instance, the existence of a critical value of the coupling constant, beyond which there are no neutral bound states. Another result is about the maximum number n ≤ 2 of neutral particles living on a generica vacuum of a non-integrable theory. An addi- tional aspect is the role played by the asymmetric vacua, which may have a different number of neutral excitations above them. Acknowledgements I would like to thank G. Delfino and V. Riva for interesting discussions. I am particularly grateful to M. Peyrard for very useful and enjoyable discussions on solitons. This work was done under partial support of the ESF grant INSTANS. References [1] A.B. Zamolodchikov and Al.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. [2] A.B. Zamolodchikov, Adv. Stud. Pure Math. 19 (1989), 641. [3] F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, (World Scientific, Singapore, 1992); M. Karowski and P. Weisz, Nucl. Phys. D 139, (1978), 455. [4] G. Delfino and G. Mussardo, Nucl. Phys. B 455, (1995), 724; G. Delfino and P. Simonetti, Phys. Lett. B 383, (1996), 450. [5] G. Mussardo, Phys. Rept. 218 (1992), 215. [6] Al.B.Zamolodchikov, Nucl.Phys. B 358, (1991), 524. [7] A.B.Zamolodchikov and Al.B.Zamolodchikov, Nucl.Phys. B 379, (1992), 602. [8] P. Fendley, H. Saleur and N.P. Werner, Nucl.Phys. B 430, (1994), 577. [9] G.Delfino, G.Mussardo and P.Simonetti, Phys. Rev. D 51, (1995), 6622. [10] G.Delfino, G.Mussardo and P.Simonetti,Nucl.Phys. B 473, (1996), 469. [11] P. Grinza, G. Delfino and G. Mussardo, hep/th 0507133, Nucl. Phys. B in press. [12] G. Delfino, Particle decay in Ising field theory with magnetic field, Proceedings ICMP 2006. [13] B.M. McCoy and T.T. Wu, Phys. Rev. D 18 (1978), 1259. [14] P. Fonseca and A.B. Zamolodchikov, J.Stat.Phys.110 (2003), 527. [15] S.B. Rutkevich, Phys. Rev. Lett. 95 (2005), 250601. [16] P. Fonseca and A.B. Zamolodchikov, Ising Spectoscopy I: Mesons at T < Tc, hep-th/0612304. http://arxiv.org/abs/hep-th/0612304 [17] G. Delfino and G. Mussardo, Nucl. Phys. B 516, (1998), 675. [18] Z. Bajnok, L. Palla, G. Takacs, F. Wagner, Nucl.Phys. B 601, (2001), 503. [19] D. Controzzi and G. Mussardo, Phys. Rev. Lett. 92, (2004), 021601. [20] D. Controzzi and G. Mussardo, Phys. Lett. B 617, (2005), 133. [21] G. Delfino, P. Grinza and G. Mussardo, Nucl. Phys. B 737 (2006), 291. [22] R.F.Dashen, B.Hasslacher and A.Neveu, Phys. Rev. D 10 (1974) 4130; R.F.Dashen, B.Hasslacher and A.Neveu, Phys. Rev. D 11 (1975) 3424. [23] J. Goldstone and R. Jackiw, Phys.Rev. D 11 (1975) 1486. [24] R. Jackiw and G. Woo, Phys. Rev. D 12 (1975), 1643. [25] G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 670 (2003), 464. [26] G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 699 (2004), 545. G. Mussardo, V. Riva and G. Sotkov, Nucl. Phys. B 705 (2005), 548 [27] G. Mussardo, Neutral bound states in kink-like theories, hep-th/0607025, to appear on Nucl. Phys. B. [28] A.B. Zamolodchikov, Sov.J.Nucl.Phys. 44 (1986), 529. [29] M. Lassig, G. Mussardo and J.L. Cardy, Nucl. Phys. B 348 (1991), 591. [30] F. Colomo, A. Koubek and G. Mussardo, Int. Journ. Mod. Phys. A 7 (1992), 5281. http://arxiv.org/abs/hep-th/0607025 Introduction A semiclassical formula Symmetric wells Asymmetric wells Conclusions

0704.0550

The old open clusters Berkeley 32 and King 11

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 23 November 2021 (MN LATEX style file v2.2) The old open clusters Berkeley 32 and King 11⋆ Monica Tosi1,† Angela Bragaglia1 and Michele Cignoni1,2 1 INAF–Osservatorio Astronomico di Bologna, Via Ranzani 1, I-40127 Bologna (Italy) 2 Dipartimento di Astronomia, Università di Bologna, via Ranzani 1, I-40127 Bologna (Italy) ABSTRACT We have obtained CCD BV I imaging of the old open clusters Berkeley 32 and King 11. Using the synthetic colour-magnitude diagram method with three different sets of stellar evo- lution models of various metallicities, with and without overshooting, we have determined their age, distance, reddening, and indicative metallicity, as well as distance from the Galactic centre and height from the Galactic plane. The best parameters derived for Berkeley 32 are: subsolar metallicity (Z=0.008 represents the best choice, Z=0.006 or 0.01 are more marginally acceptable), age = 5.0–5.5 Gyr (models with overshooting; without overshooting the age is 4.2–4.4 Gyr with poorer agreement), (m −M)0 = 12.4 − 12.6, E(B − V ) = 0.12 − 0.18 (with the lower value being more probable because it corresponds to the best metallicity), RGC ∼ 10.7−11 kpc, and |Z| ∼ 231−254 pc. The best parameters for King 11 are: Z=0.01, age=3.5–4.75 Gyr, (m−M)0 = 11.67−11.75,E(B−V ) = 1.03−1.06,RGC ∼ 9.2−10 kpc, and |Z| ∼ 253− 387 pc. Key words: Galaxy: disc – Hertzsprung-Russell (HR) diagram – open clusters and associa- tions: general – open clusters and associations: individual: Berkeley 32, King 11 1 INTRODUCTION This paper is part of the BOCCE (Bologna Open Cluster Chemical Evolution) project, described in detail by Bragaglia & Tosi (2006). With this project, we intend to derive homogeneous measures of age, distance, reddening and chemical abundance for a large sample of open clusters (OCs), to study the present day properties of the Galactic disc and their evolution with time. As part of this project, we present here a photometric study of the two old OCs King 11 (α2000 = 23 h47m40s, δ2000 = +68◦38′30′′, l = 117.◦2, b = +6.◦5) and Berkeley 32 (α2000 = 06h58m07s , δ2000 = +06 ◦25′43′′, l = 208◦ , b = +4.4◦), lo- cated in the second and third Galactic quadrants, respectively. King 11 has been the subject of a few publications in the past. Kaluzny (1989) obtained a rather shallow colour-magnitude dia- gram (CMD) using the 0.9m KPNO telescope. He found it old (about the same age of M 67) and highly reddened, with a distance modulus (m − M)V ∼ 15.3, derived assuming MV (clump)=0.7 mag. Aparicio et al. (1991) acquired deep UBV R data at the 3.5m telescope in Calar Alto on a small field of view (2.7×4.3 arcmin2); ⋆ Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Ob- servatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. † E-mail: [email protected] (MT), [email protected] (AB), [email protected] (MC) they derived a reddening E(B − V ) =1, a distance modulus (m − M)0 ≃ 11.7, a metallicity about solar (with some uncer- tainty, because different methods produced contrasting answers), and an age of 5±1 Gyr. Phelps, Janes & Montgomery (1994) ob- tained not perfectly calibrated BV I photometry and measured a difference in magnitude between the main sequence turn-off point and the red clump of δV =2.3, that translates, using the so-called MAI (Morphological Age Indicator, see Janes & Phelps 1994) into an age of 6.3 Gyr. From their recalibration of the δV - age re- lation, assuming [Fe/H]=−0.23, Salaris, Weiss & Percival (2004) infer an age of 5.5 Gyr. Note that the BDA1 (Mermilliod 1995) indicates a spurious low age for this cluster (1.1 Gyr), directly taken from the Dias et al. (2002) catalogue, whose source is un- clear. Finally, Scott et al. (1995) obtained low resolution spectra of 16 bright stars, from which an average cluster radial velocity (RV) was computed (〈RV 〉 = −35 ± 16 km s−1). These spectra were later reanalyzed by Friel et al. (2002), finding [Fe/H]=−0.27 (rms=0.15) dex. Be 32 has been photometrically studied by Kaluzny & Mazur (1991), Richtler & Sagar (2001) and Hasegawa et al. (2004). Be 32 seems to be quite old (age about 6 Gyr) and moderately metal poor ([Fe/H] between -0.2 and -0.5). We have recently presented the RVs of about 50 stars in Be 32 and a preliminary analysis of the photo- metric data (D’Orazi et al. 2006, hereafter D06) based on isochrone fitting and the magnitude of the red clump. In D06 we also dis- 1 http://www.univie.ac.at/webda//webda.html c© 0000 RAS http://arxiv.org/abs/0704.0550v1 http://www.univie.ac.at/webda//webda.html 2 Tosi et al. cussed the literature related to Be 32 available at the time, and we will not repeat it here. We now refine our determinations, apply- ing the synthetic CMD method, as done for all the clusters in the BOCCE project. Finally, Sestito et al. (2006) presented an analysis of high resolution FLAMES@VLT spectra of 9 red clump giants in Be 32, finding an average metallicity [Fe/H]= −0.29 dex (rms 0.04 dex), in very good agreement with that found by D06. The paper is organized as follows: observations and reductions are presented in Sect. 2, a description of the resulting CMDs can be found in Sect. 3; the derivation of the cluster parameters using the synthetic CMD technique is discussed in Sect. 4, while conclusions and summary are given in Sect. 5. 2 OBSERVATIONS AND DATA REDUCTION Observations in the BV I Johnson-Cousins filters of Be 32 and King 11 were performed at the Telescopio Nazionale Galileo (TNG) in November 2000 (plus three additional exposures in February 2004 for Be 32). We also acquired associated control fields to check the field stars contamination, as detailed in Table 1 and D’Orazi et al. (2006). We used DOLORES (Device Optimized for the LOw RESolution), with scale of 0.275 arcsec/pix, and a field of view 9.4 × 9.4 arcmin2. Of the two November nights, only the first one resulted photometric. Fig. 1 shows the position of our pointings for King 11 and the associated control field. A description of the data and reduction procedure for Be 32 can be found in D’Orazi (2005) and in D06; we report here briefly the analysis of King 11, which is absolutely equivalent to that of Be 32. The standard IRAF 2 routines were utilized for pre- reduction, and the IRAF version of the DAOPHOT-II package (Stetson 1987, Davis 1994) was used with a quadratically varying point spread function (PSF) to derive positions and magnitudes for the stars. Output catalogues for each frame were aligned in position and magnitude, and final (instrumental) magnitudes were computed as weighted averages of the individual values. Even with the short- est exposure times we did not avoid saturation of the brightest red giants in the I filter; unfortunately, we could not obtain additional exposures as we did for Be 32 (D06), so we will mostly concentrate in the following on the V,B − V CMD. The final catalogs have been created including all the objects identified in at least two filters, after applying a moderate selection in the shape-defining parameter sharpness (|sharpness| 6 2) and on the goodness-of-fit estimator χ2 (χ2 6 10). To the two fi- nal catalogs, one for the cluster and one for the comparison field, we applied the transformation to astrometrize the α and δ coor- dinates, using software written by P. Montegriffo at the Bologna Observatory. After application of a correction to the PSF magnitudes to bring them on the same scale of the aperture magnitudes of the stan- dard stars, we calibrated our catalogues to the standard Johnson- Cousins BV I system. We adopted the calibration equations that can be found in D’Orazi et al. (2006), since King 11 was observed in the photometric night beginning on UT 2000 November 25 when Be 32 was observed too. Finally, we determined our completeness level using exten- sive artificial stars experiments: we iteratively added, one at a time, about 50000 simulated stars to the deepest frames and repeated the 2 IRAF is distributed by the National Optical Astronomical Observatory, which are operated by the Association of Universities for Research in As- tronomy, under contract with the National Science Foundation Figure 1. Approximate positions of our pointings on King 11 and the con- trol field. The map is 15 × 45 arcmin2 , has North to the top and East to the left. reduction procedure, determining the ratio of recovered over added stars (see Tosi et al. 2004 for a more detailed description). The re- sults for Be 32 are given in Table 2 and those for King 11 in Table We checked the quality of the calibration comparing our pho- tometry for both clusters with that presented in previous litera- ture papers, i.e. with Kaluzny & Mazur (1991) for B, V and with Richtler & Sagar (2001) for V I in Be 32, and with Aparicio et al. (1991) for King 11 (only for B, V , since there are no other sources to compare the I photometry with). Fig. 2 shows the differences c© 0000 RAS, MNRAS 000, 000–000 Berkeley 32 and King 11 3 Table 1. Log of observations for the clusters and the control fields; exposure times are in seconds. Field α2000 δ2000 exp.timeB exp.timeV exp.timeI UT Date Berkeley 32 06h58m07s +06◦25′43′′ 600, 40, 5 480, 20, 2 480, 20, 1 26/11/2000, 14/02/2004 Be 32 - ext 06h57m27s +06◦08′26′′ 600, 240, 40 300, 120, 20 300, 120, 20 26/11/2000 King 11 23h47m39s +68◦38′25′′ 300, 1200, 240, 40 120, 600, 120, 20 120, 600, 120, 20 25/11/2000, 26/11/2000 King 11 - ext 23h47m40s +68◦08′18′′ 1200, 300, 40 600, 1280, 20 25/11/2000 Figure 2. Comparison between our photometry and literature data. (a) and (b) are for Be 32 by Kaluzny & Mazur (1991); (c) and (d) are for Be 32 by Richtler & Sagar (2001); (e) and (f) for King 11 by Aparicio et al. (1991). The horizontal lines are on zero; stars used to compute the average differences are indicated by (orange) open squares, while the ones discarded are indicated by crosses. c© 0000 RAS, MNRAS 000, 000–000 4 Tosi et al. Table 2. Completeness level for the central (Cols 2, 3 and 4) and external (Cols 5, 6 and 7) fields of Be 32; mag is the calibrated B, V or I magnitude. mag cB cV cI cB cV cI 16.00 1.00 1.00 1.00 1.00 1.00 1.00 16.50 1.00 0.95 0.92 1.00 0.99 0.95 17.00 0.92 0.94 0.88 0.99 0.98 0.94 17.50 0.91 0.93 0.85 0.97 0.97 0.92 18.00 0.89 0.92 0.78 0.97 0.94 0.87 18.50 0.88 0.91 0.68 0.96 0.93 0.84 19.00 0.86 0.87 0.54 0.93 0.93 0.73 19.50 0.82 0.85 0.37 0.91 0.90 0.52 20.00 0.77 0.80 0.21 0.89 0.86 0.29 20.50 0.66 0.74 0.09 0.85 0.78 0.11 21.00 0.51 0.60 0.03 0.69 0.58 0.04 21.50 0.32 0.39 0.01 0.42 0.32 0.01 22.00 0.16 0.19 0.00 0.22 0.15 0.00 22.50 0.06 0.09 0.00 0.07 0.05 0.00 Table 3. Completeness level for the central (Cols 2 and 3) and external (Cols 4 and 5) fields of King 11; mag is the B or V calibrated magnitude. mag cB cV cB cV 16.5 1.0 1.0 1.0 1.0 17.0 1.0 0.99 1.0 0.99 17.5 1.0 0.97 0.99 0.98 18.0 1.00 0.97 0.98 0.95 18.5 1.00 0.95 0.99 0.94 19.0 0.98 0.94 0.96 0.94 19.5 0.97 0.93 0.94 0.93 20.0 0.97 0.92 0.91 0.90 20.5 0.97 0.87 0.88 0.87 21.0 0.95 0.87 0.81 0.82 21.5 0.93 0.74 0.78 0.70 22.0 0.91 0.56 0.63 0.43 22.5 0.88 0.27 0.38 0.21 23.0 0.74 0.06 0.15 0.04 23.5 0.45 0.00 0.02 0.00 24.0 0.18 0.0 0.00 0.0 24.5 0.02 0.0 0.0 0.0 25.0 0.00 0.0 0.0 0.0 with these photometries for the stars in common; the comparison is particularly favorable with the work by Kaluzny & Mazur (1991), but is good in all cases. 3 THE COLOUR - MAGNITUDE DIAGRAMS The CMDs for Be 32 were described in D06 and the data are al- ready available at the BDA. Fig. 3 shows the V,B−V CMD of the stars at various distances from the centre of Be 32 and of the con- trol field. It is apparent that contamination is quite high, with about half the stars likely to be foreground/background objects even in the central regions. However, in the area with a radius of 3′ from the cluster centre the main-sequence (MS), the turn-off (TO) and the subgiant branch (SGB) are well defined. The MS extends more than 5 magnitudes below the TO. With the additional help of the available RVs (from D06 and Randich et al. in preparation, see next section) to select the most probable cluster members, we can Table 4. Stellar evolution models adopted for the synthetic CMDs. The FST models actually adopted here are an updated version of the published ones (Ventura, private communication). Set metallicity overshooting Reference BBC 0.008 yes Fagotto et al. 1994 BBC 0.004 yes Fagotto et al. 1994 BBC 0.02 yes Bressan et al. 1993 FRA 0.006 no Dominguez et al. 1999 FRA 0.01 no Dominguez et al. 1999 FRA 0.02 no Dominguez et al. 1999 FST 0.006 η=0.00,0.02,0,03 Ventura et al. 1998 FST 0.01 η=0.00,0.02,0,03 Ventura et al. 1998 FST 0.02 η=0.00,0.02,0,03 Ventura et al. 1998 satisfactorily identify the TO (V = 16.3, B − V = 0.52 and V − I = 0.60), the SGB, the red giant branch (RGB), and the red clump (V = 13.7, B − V = 1.07 and V − I = 1.10). For King 11, the final, calibrated sample of cluster stars (which will also be made available through the BDA) consists of 1971 ob- jects, and the external field catalogue comprises 880 stars. The cor- responding CMDs are shown in Fig. 4. In spite of a contamination lower than in Be 32, the location of the foreground/background objects in the CMD makes the definition of the evolutionary se- quences more complicated. We can improve the definition by using the information on membership of a few giant stars from Scott et al. (1995), which perfectly define the red clump position. If we con- sider the CMDs of regions with increasing distance from the cluster centre displayed in Fig. 5, it is apparent that a safe identification of the main evolutionary loci becomes difficult beyond a radius of 2′. Within such radius, the cluster main sequence extends for almost 4 magnitudes and the RGB and red clump are well delineated. The Turn-off point is at V = 18.2, B − V ≃ 1.3, while the red clump is at V = 16.0, B − V ≃ 1.8. In the V, V − I CMD of King 11 we lack the brightest RGB stars, because they were saturated even in the shortest image, and the MS is less well defined. For this reason, we refer to the V,B−V CMD to derive the cluster distance, reddening and age and use the I data only to discriminate in metallicity among degenerate solutions (see next Section). 4 CLUSTER PARAMETERS Age, distance and reddening of King 11 and Be 32 have been de- rived with the same procedure applied to all the clusters of our project (see Bragaglia & Tosi 2006 and references therein), namely the synthetic CMD method originally described by Tosi et al. (1991). The best values of the parameters are found by selecting the cases providing synthetic CMDs with morphology, colours, num- ber of stars in the various evolutionary phases and luminosity func- tions (LFs) in better agreement with the observational ones. As for the other clusters of this series, to estimate the effect on the re- sults of different stellar evolution assumptions, we have adopted three different sets of stellar models, with various assumptions on the metallicity, treatment of convection, opacities and equation of state. The adopted models are listed in Table 4. In addition to the usual synthetic CMD method, the cluster parameters have also been searched by means of statistical tests. c© 0000 RAS, MNRAS 000, 000–000 Berkeley 32 and King 11 5 Figure 3. Radial CMDs for Be 32 (upper panels) and equal areas in the comparison field (lower panels); we plot stars within distances of 1, 2, 3 arcmin from the cluster and field centres. The CMDs contain 133, 444, 903 objects in panels (a), (b), (c) respectively, and 57, 229, 524 in panels (d), (e), (f) respectively. Figure 4. (a) V,B −V CMD for King 11; (b) the same CMD, with stars member (open circles, red in the electronic version) and non member (filled squares, blue in the electronic version) according to the RVs in Scott et al. (1995); (c) V,B − V CMD for the comparison field; (d)V, V − I CMD for King 11 The problem of comparing colour-magnitude diagrams (and two dimensional histograms in general) is still unsolved in astrophysics. However, several approaches have been explored. For instance, in Cignoni et al. (2006) the entire CMD is used: data and model CMDs are binned and a function of residuals is minimized. In Gallart et al. (1999), the number of stars in a few regions (repre- sentative of the most important evolutionary phases) is controlled through a χ2 test. The goal of those papers was to recover a com- plex star formation history. Here, the nature of the problem is in principle simpler (single stellar generation), thus we follow a c© 0000 RAS, MNRAS 000, 000–000 6 Tosi et al. Figure 5. Radial CMDs for King 11 (upper panels) and equal areas of the comparison field (lower panels); we plot stars within distances of 1, 2, 3 arcmin from the cluster and field centres. The CMDs contains 173, 531, 941 objects in panels (a), (b), (c) respectively, and 38, 143, 317 in panels (d), (e), (f) respectively. more classical approach: the luminosity and the colour distribution of each model are independently compared with the data using a Kolmogorov-Smirnov (KS) test (Press et al. 1995). One of the ad- vantages of using also the colour distribution lies in the fact that the major drawback of using the LF alone, i.e, the degeneracy among parameters (distance, reddening, age and metallicity) can be miti- gated. Moreover, the KS does not require to bin the data; therefore, arbitrary parametrizations of the CMD (typical of the χ2) can be avoided. In order to reduce the Poisson noise, that is the dominant uncertainty in our luminosity functions, the model CMDs are built with a large number of stars. Only CMDs yielding a KS probability larger than 5% both for the LF and for the colour distribution are accepted. Unavoidably, poorly populated CMD regions like the core he- lium burning region or the RGB are often under-represented by a similar analysis (washed out by Poisson noise). However, also in these cases, a good KS probability still indicates that the most pop- ulous stellar phases (e.g., MS and TO) are well matched. In other words, the adopted statistical procedure provides a quick tool to exclude those solutions for which the synthetic CMD does not re- produce the properties of MS and TO stars. Then, the remaining parameter space is explored with a traditional analysis: i) exploit- ing the difference in luminosity between the lower envelope of the subgiants and the red clump; ii) fitting the SGB; iii) matching the RGB colour. 4.1 King 11 As already said in Sect. 3, for King 11 we have mainly used the V,B− V CMD because the V, V − I lacks the brighter part of the RGB. To minimize contamination from field stars we have selected as reference field the region within a radius of 2′ from its centre. Since this region contains 531 stars, and the control field of the same area contains 143 stars, we assume the cluster members to be 388. Incompleteness and photometric errors are those inferred from the data and described in Section 2. In order to minimize the Poisson noise of the models, all available field stars (∼ 880) are used: hence the synthetic CMDs are built with 3259 synthetic stars (in order to preserve the ratio cluster members/field stars). Only afterwards we randomly extract from the whole sample of synthetic stars 388 objects, as attributed to the cluster central region. Almost all models have been computed assuming a fraction of binary stars of 20% 3 (following Bragaglia & Tosi 2006 prescrip- tions) and a power law IMF with Salpeter’s exponent. The KS test is applied to the stars brighter than V ≈ 20. The constraint on the KS probability doesn’t guarantee a unique solution, mostly because the statistics is dominated by MS stars fainter than the TO, less af- fected than other evolutionary phases by small parameters varia- tions. We have then decided to validate only models with accept- able KS probabilities and with a predicted clump within 0.05 mag 3 The low number of observed TO stars doesn’t permit to infer the actual fraction. c© 0000 RAS, MNRAS 000, 000–000 Berkeley 32 and King 11 7 Figure 6. The range of statistically acceptable ages for King 11. Results for different sets of tracks are shown. of the observed clump (whose membership is also confirmed by ra- dial velocity estimates). Figure 6 shows the results 4; error bars cor- respond to ages for which an appropriate combination of distance and reddening exists. Considering our findings, one can provision- ally accept a range of ages between 3 and 5 Gyr. Only BBC models for Z=0.004 are rejected by the KS test for all ages (meaning that no solution for age, reddening and distance has been found). Figures 7, 8, 9 show a selection of our best synthetic CMDs. To further pro- ceed in the selection, we have used the morphology of the RGB (a poorly populated region, therefore ignored by our statistical test) to give additional constraints on the parameter space. An exami- nation of this evolutionary phase reveals that: 1) the residual BBC models (Z=0.02 and Z=0.008) are all rejected, because they pre- dict excessively red RGBs (the upper panel of Figure 7 shows the best BBC model: age=4.5 Gyr, Z=0.02, E(B − V )=0.93 and (m- M)0=11.85); 2) the same problem exists with the FRA models: the RGB is systematically too red (the lower panel of Figure 7 shows the best FRA model: age=3 Gyr, Z=0.02, E(B−V )=1.01 and (m- M)0=11.95); 3) the FST models seem in good agreement with the data independently of the adopted metallicity. We thus restrict the next considerations only to the FST models. Figure 8 shows the theoretical FST CMDs that best repro- duce the V,B − V data. The best fit parameters turn out to be: Z=0.02, age 4 Gyr, E(B − V )=0.94 and (m-M)0=11.95 (panel a); Z=0.01, age 4.25 Gyr, E(B − V )=1.04 and (m-M)0=11.75 (panel b); Z=0.006, age 4.75 Gyr, E(B − V )=1.09 and (m-M)0=11.65 (panel c). To solve the degeneracy we have made use of the V, V − I CMD: although not complete in the bright part, it remains use- ful, since only models of the right metallicity can fit the ob- served CMDs in all passbands (see also the case of Be 32). Be- cause of the very large reddening, we adopt the reddening law by Dean, Warren, & Cousins (1978, see Appendix, eq. A1): E(V − I) = 1.25×E(B−V )× [1+0.06(B−V )0+0.014E(B−V )], which takes into account a colour dependence. This relation tends to the usual E(V − I) = 1.25 × E(B − V ) for B − V → 0 and E(B − V ) → 0). In Fig. 9 we show the synthetic cases of Fig. 8 both in the 4 FRANEC models for Z=0.006 and Z=0.01, providing the same age of Z=0.02, are not shown in the figure. 1.0 2.0 BBC−0.02 FRA−0.02 Figure 7. The upper panel shows the best solution for King 11 for BBC models: Z=0.02, E(B − V )=0.93 and (m-M)0=11.85); 2) the lower panel shows the best FRANEC model: age=3 Gyr, Z=0.02, E(B − V )=1.01 and (m-M)0=11.95). Both these models predict RGBs that are too red. V,B − V and V, V − I diagrams and with no photometric error, to allow for a more immediate visualization of the theoretical pre- dictions. We can see from Fig. 9 that the three competing models, indistinguishable in B−V (left panel), do separate in V − I (right panel): the best fit is reached for Z=0.01. The solar composition seems definitely ruled out (the MS is too blue), but the Z=0.006 model lies only slightly too red and cannot be completely excluded. This seems to confirm the findings by Friel et al. (2002), who based the analysis on spectroscopic indices. In contrast, Aparicio et al. (1991) preferred a solar abundance on the basis of their CMDs, but in this case different stellar models have been employed. While we are rather confident on a subsolar metallicity, a definitive answer will require analysis of high resolution spectra. The assumption of different levels of core overshooting (η = 0.2 or 0.3) has a minor effect on the results, as expected: King 11 is a sufficiently old cluster that the upper MS stars have masses with small convective cores, and therefore with small overshoot- ing. Comfortably, the predicted number of stars in RGB and clump phase is close to the observed one, confirming that the evolutionary lifetimes of the theoretical models are correct. Finally, in order to evaluate the contribution of the adopted bi- nary fraction and IMF, we performed several tests. Larger fractions of binaries could help to fit the MS, yielding slightly larger distance moduli (with minor effects on the age). Viceversa, if distance, red- dening and age are fixed, the stellar multiplicity that is consistent with the data is wide (between 10% and 60%). In fact, only frac- tions higher than 60% produce an evident plume over the turn-off region, which is not observed. If the same test (fixing distance, red- dening and age) is performed also for the IMF, the results allow c© 0000 RAS, MNRAS 000, 000–000 8 Tosi et al. 1.0 2.0 1.0 2.0 1.0 2.0 1.0 2.0 b c d Figure 8. Comparison between observational and synthetic CMDs for King 11. Panel a shows the data CMD for the central 2′ radius region. Panels b, c and d show the CMDs of the best fitting cases (FST tracks): (b) age 4 Gyr, E(B − V )=0.94 and (m-M)0=11.95, (c) Z=0.01, age 4.25 Gyr, E(B − V )=1.04 and (m-M)0=11.75, (d) Z=0.006, age 4.75 Gyr, E(B − V )=1.09 and (m-M)0=11.65. to rule out only exponents larger than 3.3, for which the synthetic RGBs appear underpopulated. In conclusion, the best parameters for King 11 can be summa- rized in the following intervals: • Z=0.01; • age between 3.5 to 4.75 Gyr; • distance modulus between 11.67 and 11.75; • reddening 1.03 6 E(B − V ) 6 1.06. 4.2 Berkeley 32 For Be 32, we have chosen as reference CMDs those of the region within 3′ from the cluster centre (top panels in Fig. 10), which con- tains 608 stars with magnitudes measured in all the three B, V, I bands. The same area in the control field contains 332 stars with B, V, I . Taking this contamination into account, as well as the cir- cumstance that 27 of the stars within the central area are shown by the RVs not to belong to Be 32, we assume the cluster members to be 249. The top panel of Fig. 10 shows the CMD of the stars located within 3′ from the cluster centre, with the larger symbols indicating the 48 objects whose RVs indicate most probable membership. To help in the RGB definition, also the two brightest RGB members are shown, although outside the selected 3′ radius. The synthetic CMDs have been generated with 249 objects, the incompleteness of Table 2 and the photometric errors described by D06. We have generated the synthetic CMDs with and without binary systems. As for most of our sample clusters, a fraction of 30% of binaries seems more consistent with the data, for all sets of stellar models. We notice, though, that binaries are not sufficient to cover the whole colour extension of the MS: a differential red- dening of about ∆E(B − V ) = ±0.01 would provide a better reproduction of the MS thickness. The results of our analysis are the following. A solar metallic- ity is out of the question, because the synthetic CMDs show V − I colours definitely too blue for all cases when the B − V colours are correct. Of all the synthetic models, only those with metal- licity Z=0.008 are always able to simultaneously reproduce both the B − V and V − I colours of all the evolutionary phases. For Z<0.008, if B−V is reproduced, V − I tends to be too red, while for Z>0.008, if B−V is fine, V − I tends to be too blue. Unfortu- nately, Z=0.008 is available only for the BBC tracks. For the FRA models, an acceptable colour agreement is achieved for Z=0.006, but when we take into account also the shape of the MS and the TO, Z=0.01 may be better. With the FST models, instead, Z=0.006 seem slightly better than Z=0.01. This ambiguity further suggests that the actual metallicity is in between, i.e, Z=0.008. In order to obtain an in depth exploration of the preferred metallicity Z=0.008, we have also applied our statistical procedure. Although the contamination by field stars is quite high, the turn- off region, also thanks to the partial cleaning from non members by the RVs, appears better defined than in King 11. The KS test is simultaneously applied to the V, B-V and V-I distributions, se- lecting only models giving a KS probability above 5 percent. The c© 0000 RAS, MNRAS 000, 000–000 Berkeley 32 and King 11 9 1.5 1.75 2 2.25 2 2.5 3 Z=0.02 Z=0.01 Z=0.006 2 2.5 Z=0.01 Figure 9. Choice of the metallicity for King 11: the left panel shows the V,B− V data and the three best solutions (at Z=0.006, 0.01, 0.02) that all reproduce the observed CMD of the central zone, while the right panel shows the same models overimposed on the V, V − I data (in this case stars from the whole field are shown). Only the solution at Z=0.01 (for an easier understanding it is isolated in the small panel on the right) can well fit at the same time the two different CMDs. only acceptable models resulted to have age between 5 and 6.1 Gyr, distance moduli (m − M)0 = 12.5 − 12.6 and reddening 0.085 < E(B − V ) < 0.12. Whatever the metallicity, it is not easy to reproduce the shape of all the evolutionary phases covered by the stars in Be 32. The BBC models, in spite of the excellent reproduction of the colours, shape and position of MS, SGB and RGB, do not fit precisely the morphology of the TO and predict a clump slightly too bright. The FRA models are the only ones with a TO hooked enough to fit the bluest supposed member of Fig. 10 (which however is in the tail of the RV distribution and is the least safe member), but not for the ages which better reproduce the other CMD sequences. When the TO morphology is fine, the clump is too bright and vice versa. Moreover, the MS of the FRA models is slightly too red at its faint end. The FST models, independently of the overshooting choice η, have TO not much hooked and excessively vertical RGBs, whose brightest portion is therefore too blue. As usual, models without overshooting (FRA) lead to the youngest age. The FST models with maximum overshooting η=0.03 provide results totally equivalent to those with η=0.02; this has been noted also for King 11 and all OC’s old enough to have stars with small (or no) convective cores. The best compromise for each set of stellar models is: • Z=0.008, age 5.2 Gyr, E(B−V )=0.12, (m-M)0=12.6 (BBC); • Z=0.01, age 4.3 Gyr, E(B − V )=0.14, (m-M)0=12.6 (FRA); • Z=0.006, age 5.2 Gyr, E(B − V )=0.18, (m-M)0=12.4 (FST). The CMDs corresponding to these three best cases are shown in Fig. 10, where in V,B−V we plot only the synthetic stars to allow for a direct comparison of the different models, while in V, V − I we overplot the control field objects on the synthetic stars to facili- tate the comparison between theoretical and observational CMDs. The uncertainties mentioned above obviously affect the iden- tification of the best age; however, all our independent tests con- sistently favour an age between 5.0 and 5.5 Gyr with overshooting models (both BBC and FST, although the BBC ones perform better, possibly because of the more appropriate metallicity Z=0.008). Finally, another useful piece of information can be inferred from the comparison of the pure synthetic CMDs of the bottom panels of Fig. 10 with the observational ones of the top panels. The synthetic MSs don’t reach magnitudes fainter than V ≃21 for BBC and FST and V ≃20 for FRA. This limit corresponds to the min- imum stellar mass available in the adopted sets of models: 0.6M⊙ in the BBC and FST sets and 0.7M⊙ in the FRA ones. In the cen- tral row panels, where the external field CMD is overimposed to the synthetic one, the faintest portions are therefore populated only by foreground/background stars. Yet, the synthetic LFs don’t differ too much from the observational one, suggesting that contamina- tion dominates at that magnitude level. 5 SUMMARY AND DISCUSSION The context of this work is the large BOCCE project (Bragaglia & Tosi 2006), devoted to the systematic study of the Galactic disc through open clusters. Distance, reddening and phys- ical properties of the open clusters King 11 and Be 32 have been explored. To this end, synthetic CMDs have been built and com- pared with data using both morphological and statistical criteria. A morphological analysis exploits all the evolutionary phases, but leads to some level of subjectiveness. On the other hand, a pure statistical treatment can establish the significance for each model c© 0000 RAS, MNRAS 000, 000–000 10 Tosi et al. Figure 10. Comparison between observational and synthetic CMDs for Be 32. Panels a and b show the stars measured in B, V, I in the central 3′ radius region. The larger symbols (red in the electronic version) in panel b indicate the objects with higher membership probability from the RVs (see text for details). Panels f, g and h show the B − V CMDs of the best fit case, mentioned in the text, for each type of stellar models. Panels c, d and e show the corresponding V − I CMDs, overimposed to the CMD of the same area in the control field for a more direct comparison. (reducing the subjectiveness of the comparison), but is truly selec- tive only in case of very well defined TOs. In order to extract the maximum level of information, we have used both approaches: 1) we generate synthetic CMDs to best re- produce the main CMD features, especially the late evolutionary phases (RGB, red clump luminosity, SGB); 2) TO and main se- quence are explored by KS test (LF and colour distribution). The final results come from the intersection of these. During the analysis, King 11 and Be 32 have presented differ- ent problems. For King 11, whose metallicity is unknown, the sta- c© 0000 RAS, MNRAS 000, 000–000 Berkeley 32 and King 11 11 Table 5. Comparison of our results and selected literature data for the two clusters. Authors age (Gyr) Z or [Fe/H] (m −M)0 E(B-V) Notes King 11 This work 3.5-4.75 0.01 11.67–11.75 1.03-1.06 BV I Kaluzny ∼ 5 (m −M)V ∼ 15.3 Shallow BV R, comparison to M67/red clump mag Aparicio et al. 5± 1 0.02 11.7 1.00 BV R, synthetic V,B − V CMD Salaris et al. 5.5 −0.23± 0.15 δV , [Fe/H] from liter., age-metallicity-δV relation Berkeley 32 This work 5.0-5.5 0.008 12.4–12.6 0.12 BV I Kaluzny & Mazur 6 −0.37± 0.05 12.45±0.15 0.16 Morphological Age Ratio/MS fitting D’Orazi et al. 6.3 0.008 12.5–12.6 0.10 BV I , isochrone fitting/red clump mag Richtler & Sagar 6.3 −0.2 12.6±0.15 0.08 V I , isochrone fitting/red clump mag Sestito et al. −0.29± 0.04 0.14 High-res spectra tistical treatment has the advantage to explore very quickly a mul- tidimensional parameter space. Nevertheless, King 11 has a very noisy TO, therefore, a morphological analysis plays a key role in refining the results. On the other hand, Be 32 is characterized by well defined TO and MS (and a well defined metallicity), and the statistical approach has provided an independent estimate of the parameters. For King 11, our analysis has produced the following re- sults: (1) the FST tracks give the best chance to reproduce the LF, the colour distribution and the morphological constraints (the clump luminosity, the bottom of the RGB and the RGB colour); (2) the metallicities Z=0.006, Z=0.01, Z=0.02 all produce synthetic V,B − V CMDs whose goodness of fit are indistinguishable but the use of the I band permits to select the right cluster metallicity, i.e. Z=0.01; (4) the synthetic CMDs generated with the FST tracks are consistent with a reddening 1.03 6 E(B − V ) 6 1.06, a dis- tance modulus between 11.67 and 11.75, a cluster age between 3.5 and 4.75 Gyr (the best fit is obtained with 1.04, 11.75 and 4.25, respectively). Our results confirm that King 11 is among the true “old open cluster”, contradicting the Dias et al. (2002) value, but in line with all past direct determinations. For an immediate comparison, Table 5 shows our results together with literature ones. Our derived ages are consistent with the Aparicio et al. (1991) finding (age 5 ± 1 Gyr). The difference (our estimates are systematically younger) may be easily ascribed to the input physics: Aparicio et al. (1991) adopted the Bressan, Bertelli & Chiosi (1981) tracks, characterized by strong core overshooting: although King 11 masses are only marginally affected by this phenomenon, a conspicuous amount of overshooting goes in the direction of rising the estimated age. A similar age is recovered also by Kaluzny (1989), but that work is based on a very shallow sample. Salaris et al. (2004), adopting [Fe/H]=−0.23, provide an age of about 5.5 Gyr from their recali- bration of the relation between δV , metallicity and age, based on ten clusters. The large reddening we have found is in good agree- ment with literature values, in particular with the E(B−V ) = 0.98 derived by the Schlegel et al. (1998) maps. Our choice of metallic- ity is in good agreement with the one by Friel et al. (2002) and slightly discrepant with the other derivation based on photome- try (Aparicio et al. 1991), which, however, is more uncertain since those authors found discrepant results with different methods. In the case of Be 32 our CMDs constrain fairly well the clus- ter metallicity. The BBC tracks for Z=0.008 reproduce all the stel- lar phases in all bands, while other metallicities have problems to simultaneously best fit both the V,B − V and the V, V − I dia- grams. This is in perfect agreement with the finding by Sestito et al. (2006), based on high resolution spectra ([Fe/H]= −0.29± 0.04). The best estimate of the age ranges between 5.0 and 5.5 Gyr, slightly older than King 11. The age derived by D06 with isochrone fitting was 6.3 Gyr, consistent with what we find here once we consider the coarseness of the isochrone grid. Slightly older ages (6.3 and 6.0 Gyr, respectively) were found also by Richtler & Sagar (2001) and Kaluzny & Mazur (1991), while Hasegawa et al. (2004) reach exactly our same conclusion (5.2 Gyr). In addition, the present data for Be 32 suggests a distance modulus (m − M)0 = 12.4 − 12.6, in fair agreement with past studies, and reddening most likely around 0.12. The latter is con- sistent but slightly larger than the E(B − V ) = 0.10 we deter- mined in D06 assuming an older age, and slightly smaller than the value E(B − V ) = 0.16 quoted by Kaluzny & Mazur (1991). A clearly lower reddening (E(B − V ) = 0.08) was found by Richtler & Sagar (2001), but we recall that their study was based only on two passbands and may be plagued by uncertainties like the ones we found in the case of our analysis of King 11. The com- parison to the Schlegel et al. (1998) maps is too uncertain, given the very low latitude of the cluster. We suggest the possibility of a differential reddening of the order of ∆E(B − V ) ≃0.02. We have computed the distances of the two OCs adopting the preferred distance moduli: King 11 has a distance of about 2.2-3.4 kpc from the Sun and about 9.2-10 kpc from the Galactic centre (assuming the Sun to be at 8 kpc from the centre), with a height above the Galactic plane of 253-387 pc; the corresponding values for Be 32 are 3.0-3.3 kpc, 10.7-11 kpc, and 231-254 pc, respec- tively. Neither cluster is far enough from the Galactic centre to be of relevance in the current debate about the metallicity distribution in the outer disc. However, both contribute to enlarge the still small- ish number of old OCs and their metallicity (specially once that of King 11 is confirmed by dedicated high resolution spectroscopy studies) will be important in defining the (possible) variation of the radial metallicity distribution over the Galactic disc lifetime. ACKNOWLEDGEMENTS The King 11 data reduction was performed by Roberto Gualandi of the Loiano Telescope staff. We are grateful to Sofia Randich for the RVs of Be 32 provided in advance of publication. We gratefully acknowledge the use of software written by P. Montegriffo, and of the BDA database, created by J.C. Mermilliod, and now operated c© 0000 RAS, MNRAS 000, 000–000 12 Tosi et al. at the Institute for Astronomy of the University of Vienna. This project has received partial financial support from the Italian MIUR under PRIN 2003029437. REFERENCES Aparicio A., Bertelli G., Chiosi C., Garcia-Pelayo J. M., 1991, A&AS, 88, 155 Bertelli G., Bressan A., Chiosi C., Fagotto F, Nasi E. 1994, A&A, 106, 275 Bressan A., Bertelli G., Chiosi C., 1981, A&A, 102, 25 Bessell M. S., Castelli F., Plez B., 1998, A&A, 333, 231 Bragaglia A., Tosi M. 2006, AJ, 131, 1544 Bressan A., Fagotto F., Bertelli G., Chiosi C., 1993, A&AS, 100, Cignoni M., Degl’Innocenti S., Prada Moroni P. G., Shore, S. N., 2006, A&A, 459, 783 Davis L. E. 1994, A Reference Guide to the IRAF/DAOPHOT Package, IRAF Programming Group, NOAO, Tucson Dean J. F., Warren P. R., Cousins A. W. J., 1978, MNRAS, 183, Dias W. S., Alessi B. S., Moitinho A., Lépine J. R. D., 2002, A&A, 389, 871 Dominguez I., Chieffi A., Limongi M., Straniero O., 1999, ApJ, 524, 226 D’Orazi V. 2005, Laurea Thesis, Dipartimento di Astronomia, Università di Bologna D’Orazi V., Bragaglia A., Tosi M., Di Fabrizio L., Held E. V., 2006, MNRAS, 368, 471 (D06) Fagotto F., Bressan A., Bertelli G., Chiosi C., 1994, A&AS, 105, Friel E. D., Janes K. A., Tavarez M., Scott J., Katsanis R., Lotz J., Hong L., Miller N., 2002, AJ, 124, 2693 Gallart C., Freedman W. L., Aparicio A., Bertelli G., Chiosi C. 1999, AJ, 118, 2245 Hasegawa T., Malasan H. L., Kawakita H., Obayashi H., Kurabayashi T., Nakai T., Hyakkai M., Arimoto N., 2004, PASJ, 56, 295 Janes K. A., Phelps R. L. 1994, AJ, 108, 1773 Kaluzny J., 1989, AcA, 39, 13 Kaluzny J., Mazur B., 1991, AcA, 41, 167 Landolt A. U. 1992, AJ, 104, 340 Mermilliod J. C. 1995, D. Egret, M. A. Albrecht eds, Informa- tion and On-Line Data in Astronomy, Kluwer Academic Press (Dordrecht), p. 127 Phelps R. L., Janes K. A., Montgomery K .A. 1994, AJ 107, 1079 Press W., Flannery B., Teukolsky S., Vetterling W., 1995, Numer- ical Recipes in C. Cambridge Univ. Press Richtler T., Sagar R., 2001, BASI, 29, 53 Salaris M., Weiss A., Percival S. M., 2004, A&A, 414, 163 Schlegel D. J., Finkbeiner D. P., Davis M. 1998, ApJ, 500, 525 Scott J. E., Friel E. D., Janes K. A., 1995, AJ, 109, 1706 Sestito P., Bragaglia A., Randich S., Carretta E., Prisinzano L., Tosi M., 2006, A&A, 458, 121 Stetson P. B. 1987, PASP 99, 191 Straniero O., Chieffi A., Limongi M., 1997, AJ, 490, 425 Tosi M., Greggio L., Marconi G., Focardi P. 1991, AJ, 102, 951 Tosi M., Di Fabrizio L., Bragaglia A., Carusillo P. A., Marconi G. 2004, MNRAS, 354, 225 Ventura P., Zeppieri A., Mazzitelli I., D’Antona F., 1998, A&A, 334, 953 c© 0000 RAS, MNRAS 000, 000–000 Introduction Observations and data reduction The colour - magnitude diagrams Cluster parameters King 11 Berkeley 32 Summary and discussion

0704.0551

The Genetic Programming Collaboration Network and its Communities

7 The Genetic Programming Collaboration Network and its Communities L. Luthi∗ M. Tomassini∗† M. Giacobini†‡ W. B. Langdon§ Abstract Useful information about scientific collaboration structures and pat- terns can be inferred from computer databases of published papers. The genetic programming bibliography is the most complete reference of papers on GP. In addition to locating publications, it contains coau- thor and coeditor relationships from which a more complete picture of the field emerges. We treat these relationships as undirected small world graphs whose study reveals the community structure of the GP collaborative social network. Automatic analysis discovers new com- munities and highlights new facets of them. The investigation reveals many similarities between GP and coauthorship networks in other sci- entific fields but also some subtle differences such as a smaller central network component and a high clustering. 1 Introduction The genetic programming (GP) bibliography1, created and maintained by one of us (WBL) and by S. Gustafson contains most of the GP papers. As such, it is a rich source of data that implicitly describes many aspects of the structure of the GP community. Searching the bibliography and looking at the images2 provides a lot of useful information about the field and the people working on GP. However, a deeper analysis of the data, that goes beyond the mere pictorial aspect, provides a much more complete view. The coauthorship data is a social network since collaborating in a research ∗Information Systems Department, University of Lausanne, Switzerland †Information Systems Department, University of Lausanne, Switzerland ‡Dpt. of Animal Production Epidemiology and Ecology, University of Torino, Italy §Department of Computer Science, University of Essex, UK 1http://www.cs.bham.ac.uk/∼wbl/biblio/ 2http://www.cs.bham.ac.uk/∼wbl/biblio/gp-coauthors/ http://arxiv.org/abs/0704.0551v1 study usually requires that the coauthors become personally acquainted. Thus, studying those ties, their structure, and their evolution allows a better understanding of the factors that shape scientific collaboration. We present a systematic study of the GP coauthorship data base us- ing methods and tools pertaining to complex networks and social network analysis. Social network analysis (see [?] for a survey), although it is an old discipline, has recently received new impetus and tools from the field of complex networks (see [?] for an excellent review). This is mainly due to the relatively recent availability of large machine-readable databases such as the GP bibliography. Social acquaintances involve psychological and other human aspects that are difficult to quantify. However, as it has been done in other fields [?, ?, ?, ?], we use objective data such as joint published work to stand for social bonds. Since this must ignore subtler aspects of a col- laboration relationship, it is obviously far from perfect as a social indicator, yet it is still a good “proxy” for the network of social relationships and can reveal several interesting facts and trends. A preliminary investigation of the GP coauthorship network appears in [?]. In the first part of this article we update this initial study using the most recent data and adding the study of the influence of excluding co- edited proceedings and books. In the second part we offer a new analysis of the finer community structure of the collaboration network. Similar studies have been performed in the last few years on several other collaboration networks in disciplines such as physics, mathematics, medicine, biology, and computer science [?, ?, ?, ?]. A related investigation concerning the EC collaboration network [?] has appeared recently in popular form, but it does not take into account, for example the community structure of the network. [?] deals with some of the same statistical features for the EC community at large as we describe in detail here for GP. The values reported by [?] are in line with those found here for the GP field. Given that the intersection between the GP researchers and general EC is likely to be rather large, it would be interesting to study how they are related to each other. 2 The GP Collaboration Network We treat the genetic programming social network as a graph where each node is a GP researcher, i.e. someone who has at least one entry in the bib- liography. There is a connection between two people if they have coauthored at least one paper, or if they have coedited one or more book or proceedings. As of the start of 2007, there is a total of N = 2809 connected nodes, i.e. authors that have at least one GP collaborator, and a total of 5853 edges (collaborations) in the GP coauthorship network. There are 367 isolated vertices, which represent authors who have not collaborated with others to the extent of coauthoring a paper. Isolated vertices are ignored in our graph statistics. We have also excluded a single paper with 108 coauthors in a nuclear physics journal. This is because we consider it to be an anomalous entry that is not representative of typical collaborations in our discipline. Due to the youth of GP, the graph is relatively small compared to some studied collaboration networks [?, ?, ?]. (Although some published studies have covered much smaller and more specialised networks, e.g. of only 50 people [?].) The main disadvantage of studying a relatively small database is that, like any statistical study, more data allows deeper and more mean- ingful inferences to be drawn. In particular, studies of the form of the distributions (such as whether they follow exponential or power laws) re- quire a large amount of data. The advantages include that the graph almost fully represents the state of the whole GP community. This allows reliable characterisation of collaboration in the community. Also, the problems of multiple authors with the same name (e.g. A. Smith), outliers and different name spelling that plague the larger data sets, are unlikely and easy to spot in our data. Although in many cases in our field co-editing a book or proceedings vol- ume does reflect personal acquaintance, there are some large coeditorships which are not representative and so may give a slanted view. Therefore in the following figures we present two kinds of statistics: those that include all joint publications and those in which co-edited conference proceedings and co-edited books are excluded (but not their contents, of course). Next we present and discuss some basic measures that characterise the GP col- laboration network. 2.1 Number of Papers per Author The average number of papers per author is 3.16 with co-edited books and proceedings and it is 3.14 without. The five most prolific authors are, in de- creasing order: J. Koza, R. Poli, W. B. Langdon, W. Banzhaf and C. Ryan. If we exclude proceedings’ co-editors the ranking remains unchanged. Nat- urally the distribution of the number of papers per author, P (k), has some scatter, particularly in the tail of the distribution. Thus, we present in Fig- ure 1 the graph of the cumulative distribution P (k ≥ n) which is smoother and allows the same inferences to be made. The curves are rather well fitted by a straight line, and thus the distributions follow a power-law P (k) ∝ k−γ 10000 1 10 100 1000 number of entries with coeditors without coeditors power-law fit Figure 1: Cumulative distribution of the number of entries per author. Log- log scale. The straight line is the best mean-square fit and shows the number of authors is ∝ k−2.5. with a calculated exponent γ of 2.5 for both of them. A power-law distribu- tion with similar exponents has been observed for analogous collaboration networks, e.g. 2.86 for a biological publication database (Medline), 3.41 for a computer science database (NCSTRL), 2.4 for mathematics, and 2.1 for a neuroscience papers database [?, ?]. A smaller exponent (in absolute value) means that the tail of the distribution is more stretched towards high values of degree. 2.2 Number of Collaborators per Author The average number of collaborators per author, i.e. the mean degree 〈k〉 of the coauthorship graph, is 4.17 with proceedings and 3.62 without. This is close to the values reported by studies of computer science, physics (exclud- ing high energy physics) and Mathematics, suggesting GP follows similar collaboration patterns to those disciplines. However it is much less than found in high energy physics and medicine. See Table 1. In order and including co-edited volumes, the five authors that have the largest num- ber of collaborators are: W. Banzhaf, J.A. Foster, P. Nordin, W.B. Lang- don, U.-M. O’Reilly. Without co-edited books the ranking is: P. Nordin, W. Banzhaf, J. Daida, C. Ryan and R. Goodacre. The five “pairs” that have the highest number of coauthored papers are, in decreasing order both with or without co-edited proceedings: J. Koza–M. A. Keane, R. Poli–W.B. Langdon, J. Koza–D. Andre, J. Koza–F. Bennet and F. Bennet–M.A. Keane. This shows that J. Koza’s group has been tightly collaborating for a long time, a conclusion that is confirmed in the community study of section 4. It is also evident that the W.B. Langdon–R. Poli association has been an extremely productive one. 10000 1 10 100 number of collaborators with coeditors without coeditors Figure 2: Cumulative distribution of the number of authors with a given number of collaborators. Logarithmic scale on both axes. Figure 2 shows the cumulative distributions of the number of collabora- tors. One sees that the distributions are not pure power-laws, otherwise the points would approximately lie on a straight line. Rather, the distributions shows a power-law regime in the first part followed by an exponential decay in the tail. That is, the whole network cannot be fitted by a power-law. This is quite common. In fact, several measured social networks do not follow a power-law degree distribution [?, ?] and are best fitted either by an exponential degree distribution P (k) ≈ e−k/〈k〉 or by an exponentially truncated power-law of the type P (k) ≈ k−γe−k/kc , where kc represents a critical connectivity and 〈k〉 is the average degree. 2.3 Number of Authors per Paper Figure 3 shows the cumulative distribution of the number of papers written by a given number of coauthors. Here the distribution also has a tail that is longer than that of a Gaussian or exponential distribution, however it does not follow a power-law. The average number of authors per paper is 2.25 (2.22 without co-editors). From Table 1 we can see that these figures are close to the equivalent ones for computer science (NCSTRL) and physics, while Mathematics has a lower number of co-authors per paper. On the other hand, nuclear physics stands out with an unusually high number of coauthors per paper. 10000 1 10 100 number of coauthors with coeditors without coeditors Figure 3: Cumulative distribution of the number of papers with a given number of coauthors on log-log scales. From Figures 2 and 3 one can see that the tails of the distribution with co-editors are longer than without them. Thus, taking co-editorship into account seems to rather artificially inflate the number of publications with many co-authors and, by consequence, the number of collaborators that a person has. 2.4 Connected Components In the theory of Poisson random graphs there is a critical value of average de- gree 〈k〉 = 1 above which there is a sudden appearance of a giant component. This is so-called since most vertices belong to it. The other components are smaller and have an exponentially decreasing size distribution [?]. Although collaboration graphs are not random, a similar phenomenon appears. In- cluding coeditors there are 1025 GP authors in the giant component. This is 36.5% of the total graph. If we exclude coediting proceedings etc. the size is 743, representing the 26.9% of the total. In the giant component the average number of collaborators per author is 5.83 with co-editors and 4.39 without them. The cumulative size distribution of the connected components with and without co-editors are depicted in Figure 4. Figure 4 shows that the proba- bility density functions are well approximated by a power law with exponent of 2.9 (excluding co-editors) and 2.6 (total). Since the other authors did not provide the analogous data for their databases, we do not know how our figures would compare with those for other coauthorship databases. The existence of a big connected component has a social meaning. It suggests 36.5% of GP researchers are members of a single community, since those researchers are either directly connected via a collaboration or they are close to each other in a way that will be made clear in section 3. The size of the giant component is notably smaller in the GP graph with respect to other measured coauthorship networks (see Table 1). This may be due to the comprehensive nature of the GP bibliography. It captures work done by smaller groups which does not get into major journals, whereas, perhaps, the other databases concentrate upon higher impact outlets where work is heavily cited but at the expense of ignoring less regarded authors. This may artificially inflate the fraction of authors within their giant component. Alternatively it may be due to the youth of the GP field, with many semi- isolated individuals and groups starting research independently. One should also consider that all collaboration networks are in a non- equilibrium state as they are continuously evolving [?]. Accordingly, as time goes by, one should observe small components progressively connecting themselves to the large one. For example, in less than one year the size of the giant component including co-editors has grown from 942 to 1025 nodes. This is due in part to a number of newcomers collaborating with people already belonging to the giant component. The other part comes from the absorption of a few disconnected small components into the giant one thanks to one or more new collaborations. This suggests that the size of the giant component has not yet reached its “steady-state” value and it will continue to grow in relative size. Since we possess all the time-stamped data, it is possible to study the evolution of this component, as well as several other indicators from the beginning and up to the present days. This investigation is currently under way. 2.5 Social GP Clusters The clustering coefficient of a node in a graph is the proportion of its neigh- bouring nodes which are also neighbours of each other. The average clus- tering coefficient 〈C〉 is calculated across all nodes in the graph [?]. In other words, 〈C〉 is a simple statistical measure of the amount of local structure 1 10 100 1000 component size with coeditors without coeditors power-law fit power-law fit Figure 4: Cumulative distributions of the number of connected components in the collaboration graph by number of people. Log-log scale. that is present in a graph. Most real-world networks, e.g. the world wide web, roads, electrical power transmission and including the social networks that have been studied to date, have a much larger clustering coefficient than would be expected of a random graph with the same number of ver- tices and edges. Social networks are particularly clustered. For example, the average clustering coefficient is 〈C〉 = 0.665 for the GP collaboration graph including book co-editors, and it is 0.660 without. (We would expect 0.0015 and 0.0013 for the corresponding random graphs). In terms of scien- tific collaborations, a high clustering coefficient means that people tend to collaborate in groups of three or more. This agrees with what we know of the GP field. It may mean that two researchers that collaborate indepen- dently with a third one may, in time, become acquainted and so collaborate themselves. Alternatively it might be due to collaborators coming from the same institution. In all cases, a high value of 〈C〉 for a social network is an indication that collaborations are not made at random at all, and that social forces and processes are at work in the network structure formation. Table 1 summarises the results of this section and compares them with those for some other collaboration networks. Some of the entries in the table will be discussed in the following section. Most GP statistics are similar to those of the larger databases. However one notable difference, as we have already remarked, is the relative smallness of the largest component. The clustering is rather high, which shows that GP researchers know each other quite well within the large component, and the community is rather homoge- Table 1: Basic statistics for some scientific collaboration networks. GP1 is the GP bibliography at the start of 2007, including coedited books and pro- ceedings. GP2 is the same but without coeditors. SPIRES is a data set of papers in high-energy physics. Medline is a database of articles on biomed- ical research. Mathematics comprises articles from Mathematical Reviews. NCSTRL is a database of preprints in computer science. Physics has been assembled from papers posted on the Physics E-print Archive. Details about these databases can be found in [?, ?, ?]. GP1 GP2 SPIRES Medline Mathematics NCSTRL Physics Total number of papers 4564 4504 66652 2163923 1600000 13169 98502 Total number of authors 2809 2765 56627 1520251 253339 11994 52909 Average papers per author 3.16 3.14 11.6 6.4 7 2.55 5.1 Average authors per paper 2.25 2.22 8.96 3.754 1.5 2.22 2.53 Average collaborators per author 4.17 3.62 173 18.1 2.94 3.59 9.7 Size of the giant component (%) 36.5 26.9 88.7 92.6 82.0 57.2 85.0 Clustering coefficient 0.665 0.660 0.726 0.066 0.15 0.496 0.43 Average path length 4.74 5.2 4.0 4.6 7.73 9.7 5.9 neous. In contrast, in biology and medicine or mathematics, where scientist from different sub-disciplines seldom collaborate, the clustering coefficient is lower. Note also the high number of authors per paper, and especially the strikingly high number of collaborators per author in the nuclear physics community (SPIRES). Clearly, nobody can maintain an average of 173 sci- entific partners on a first-hand acquaintance basis and thus this figure does not seem to be socially meaningful. 3 Distances and Centrality A social network can be characterised by a number of measures that give an idea of “how far” people are from each other, or how “central” they are with respect to the whole community. These measures are well known in social network analysis. Here we shall concentrate on average path length and on betweenness centrality. 3.1 Average Path Length The average path length L of a graph is the average value of the shortest paths between all of its pairs of vertices. In random graphs and many real networks, such as the Internet, the World Wide Web and social networks, the average path distance scales as a logarithmic function O(logN) of the number of vertices N . Such networks, if they also have a high clustering coefficient, are known as small worlds networks [?]. Since, even for very large graphs, any two nodes in a small world network are only a few steps apart. In contrast in regular lattices, two nodes are O(N D ) apart. (Where D is the lattice’s dimensionality. For example, for a square lattice L ≤ 2 2 ). The average path length of the giant component of the GP collaboration graph including coeditors is 4.74 (it is 5.2 without coeditors). The longest among all the shortest paths (known as the diameter) is 12 (14 without coeditors). Thus, unsurprisingly, the GP community, as far as its “core” component is concerned, is indeed a small world and is characterised by values that are typical of these kinds of network (see Table 1). Being a small world means that information may circulate quickly and collaborations are easier to set up. These are clearly advantageous for a research community. The connected components following the largest one are themselves small worlds. We expect over time some of them will merge with the largest component. (For this to happen, only a single new collaboration between two scientists each belonging to one of the components is needed.) 3.2 Betweenness The betweenness b(v) of a vertex v is the total number of shortest paths be- tween all possible pairs of vertices that pass through this vertex. Nodes that have a high betweenness potentially have more influence, i.e. they are more central in the network, in that there is more “traffic” that goes through them. The first five authors in terms of betweenness in the network (in- cluding co-editors and in decreasing order) are: W. Banzhaf, H. Iba, U.-M. O’Reilly, H. de Garis and W. B. Langdon. W. Banzhaf is also the researcher that has the highest number of different collaborators. Without co-editors the ranking is: W. B. Langdon, U.-M. O’Reilly, W. Banzhaf, M. Tomassini and P. Nordin. People who have a large value of betweenness play the role of intermediaries or “brokers” in a social sense. 3.3 Non-random collaborations between directly connected authors Most technological and biological networks are disassortive in that they have negative correlation, meaning that high-degree vertices are preferentially connected to low-degree vertices. However most measured social networks are assortative, meaning highly connected nodes tend to be connected with other highly connected nodes [?]. The GP collaboration network confirms this general observation with a correlation coefficient of +0.15 for the gi- Powered by yFiles Lanz Pizz Figure 5: One of the communities belonging to the main network compo- nent. The thickness of the links gives an indication of the number of co- authored papers. The largest thickness indicates more than 16 coauthored works. The thinnest link (light gray) stands for a single collaboration. The different symbols and colours represent sub-communities of the illustrated community. ant component, and +0.30 for the whole graph (including coeditors and excluding the single physicist’s paper). These are close to the coefficients observed for other social networks (specifically 0.127 for Medline and 0.120 for Mathematics [?]). 4 Communities in the Giant Component All the researchers belonging to the largest component of the network can be said to form part of the GP community at large, in the sense that they are all only a few steps away from any other member of the community. However, we know from direct experience that some groups of GPers are more closely connected between themselves than with other people. In other words, they belong to what one might call a group or a tighter community within the global one. It is not easy to give a rigorous quantitative definition of a community within a network. For our purposes a community can be seen as a set of highly connected vertices having few connections with vertices belonging to other communities. In the analysis of social networks, several algorithms that attempt to split a network into communities have been proposed. We used Newman’s method [?], which is based on a measure of the fraction of edges that fall within communities minus the expected value of the same quantity if edges fall at random without regard for the community structure. Since the GP bibliography contains the number of papers that any two collaborators have published together, it is possible to go a step further than just saying that two people have coauthored at least a paper, and give a measure of the intensity of the collaboration. We use the number of papers that two given authors have in common as a measure of the strength of their collaboration. Newman [?] has proposed a more refined measure which takes into account the actual number of coauthors of each paper. However this is more complicated than we need, instead we ignore the total number of coauthors for each paper. Our measure of collaboration strength is used in our communities algorithm to highlight groups of researchers that have collaborated strongly with the aim of uncovering the stability of the scientific relationship. We have also excluded coedited proceedings, books, etc., as we have already seen that these might sometimes represent spurious collaboration relationships. The results of running the algorithm on the subgraph represented by the largest connected component are qualitatively surprisingly close to what one would expect, given our knowledge of the GP field. The advantage is that the analysis makes them explicit and uncovers a number of other relationships that would be difficult to infer without an explicit study of the raw data. As an example of the about 25 communities that the algorithm discovers, Figure 5 shows the structure of the groups around one of us (“Toma”). If we now consider this community as an isolated subgraph and apply again New- man’s algorithm to it, we obtain the groups highlighted by different symbols Powered by yFiles Chio BuxtHoll Figure 6: Another community belonging to the main network component. The thickness of the links gives an indication of the number of co-authored papers. The largest thickness indicates more than 16 coauthored works. The thinnest link (light gray) stands for a single collaboration. The different symbols and colours represent sub-communities. and colours in the figure. Thus, the groups correspond to sub-communities within the main community. The thickness of the links represents the inten- sity of the relationship. It is easy to recognise a “hard core” of collaborating researchers strongly connected to “Toma” forming triads and higher poly- gons of order four and five. The strong triangle (“Foli”, “Pizz”, “Spez”) is relatively loosely connected to the rest, showing that these researchers be- long to the community but often collaborate between themselves. It is also possible to discern institutional and geographical components in the com- munity. For example, most of the upper right part of the figure through the node “Chop” comprises researchers essentially belonging to the University of Geneva, which is close to the University of Lausanne, to which “Toma” belongs. However, geographical closeness is not the key factor in the other groups which belong to Universities in France, Italy, Spain, and the US. We might conjecture that many collaborations start locally at the same or at close institutions and then they spread through people being introduced to others via a common acquaintance, or through people physically moving or visiting other institutions. This is the case in the figure, where “Vann”, ”Chop”, and “Vega” among others have played the role of “bridges” between different institutions and across countries. As a second illustration, let us look at Figure 6 which is the community that revolves around one of us (“Lang”) and “Poli”. In contrast to the pre- vious case, one can see that the graph structure is more “star-like”, with two large directly connected big hubs (“Lang” and “Poli”) who have about 70 co- authored papers, and three other highly connected nodes (“Buxt”,“McPh”, “Rowe”) which are strongly connected to one of the main hubs but not to both. It is interesting to observe the role of “McPh” who, like “Vega” in the previous community (cf. Figure 5), plays a bridging role, this time between some UK and some American researchers. We can also recognise a strong ”theory-oriented” group, which is almost a clique in the graph, formed by (“McPh”,“Poli”,“Rowe”, “Steph”, “Wrig”). There is also another bridge formed by “Cagn” from UK to Italy, again due to a long-standing collabo- ration and friendship. The small cliques or almost cliques at the periphery of the figure essentially represent people that have worked at the same in- stitution in either Italy or the US. The discussion above, motivated by our belonging to the mentioned com- munities, and thus by our direct human knowledge about them, should be enough to get an impression of the many useful observations that one can make on the communities that interlock in the main network component. There are of course several other large well known and interesting commu- nities in the network but unfortunately we cannot describe them here for reasons of space. 5 Conclusions In sections 2 and 3 we characterised the genetic programming (GP) coau- thorship graph using a number of local and global statistics. We extended and updated the findings presented in [?] by studying the influence of coedited volumes and by using the latest data available. Section 3 showed the GP field to be highly clustering and that the GP coauthorship network has a small mean path length. Together these suggest that, at least for the core, GP is indeed a “small world”. We also found, compared with other published collaboration networks, that the fraction of GP authors connected by coauthorship is a relatively small fraction of all GP authors. Section 4 is a study of the community structure of GP. It uses a more pre- cise definition of collaboration, which takes into account the intensity of the relationship. This uncovers many groups of tightly interacting researchers. From the detailed study of two of the communities we have drawn inferences about the pivotal role of some researchers or groups of researchers in pro- moting collaborations within and between academic institutions. Adding our human knowledge about geographical location and personal acquain- tance, allows some conjectures to be drawn about the way in which different continents and countries collaborate on research projects. It should be obvious that the present data driven approach to social network analysis can only provide some answers but not all of them. Algo- rithms and data cannot take into account human aspects such as friendship in scientific collaboration. While these may be buried in the sea of numbers they will never appear explicitly from such analyses. Nevertheless, we feel that our results are interesting and useful in the way that they characterise our community. There is another aspect of the collaboration graph that would be re- vealing: the analysis of its development over the years. Indeed, since each paper has a date of publication, we possess all the data that are needed for such an investigation. This would allow the detailed study of how the network has grown to its present size and structure from the beginning and might give hints as to its future progress. This extension is currently under investigation. Introduction The GP Collaboration Network Number of Papers per Author Number of Collaborators per Author Number of Authors per Paper Connected Components Social GP Clusters Distances and Centrality Average Path Length Betweenness Non-random collaborations between directly connected authors Communities in the Giant Component Conclusions

0704.0552

The Expanding Photosphere Method: Progress and Problems

arXiv:0704.0552v1 [astro-ph] 4 Apr 2007 The Expanding Photosphere Method: Progress and Problems József Vinkó and Katalin Takáts Department of Optics & Quantum Electronics, University of Szeged, Hungary Abstract. Distances to well-observed Type II-P SNe are determined from an updated version of the Expanding Photosphere Method (EPM), based on recent theoretical models. The new EPM distances show good agreement with other independent distances to the host galaxies without any significant systematic bias, contrary to earlier results in the literature. The accuracy of the method is comparable with that of the distance measurements for Type Ia SNe. Keywords: supernovae; core-collapse; distances PACS: 97.10.Vm, 97.60.Bw INTRODUCTION Distance is one of the most fundamental quantities in astrophysics, and it is especially true for supernovae. Type Ia SNe are thought to be the most reliable distance indicators, even up to z ∼ 1.5 redshift, and they play major role in determining the expansion of the Universe as well as the cosmic equation of state. On the other hand, accurate distances to SNe are crucial in understanding not only their physical properties, but also revealing their progenitor objects and the possible explosion mechanisms. The Expanding Photosphere Method (EPM) is a tool for measuring distances to SNe that have large amount of ejected material [1]. The concept of EPM is based on a few assumptions about the general physics of the expanding ejecta. These are the followings: 1. The expansion of the ejected material is spherically symmetric. 2. The ejecta is expanding homologously, i.e. R(t) = v(R) · (t − te), where R(t) is the time-dependent radius of a particular layer in the ejecta, v(R) is the (constant) expansion velocity of this layer and t − te is the time elapsed since the moment of explosion (te). 3. The ejecta is optically thick, i.e. there exists a layer where the optical depth τλ ∼ 1. This layer is the „photosphere” (Rphot ). Because of the expansion, the location of the photosphere moves inward the ejecta, so its velocity (vphot ) is decreasing with time. 4. The photosphere radiates as a blackbody, so the shape of the emergent flux spec- trum is Planckian with a well-defined effective temperature Te f f . However, the ab- solute flux value differs from that of the blackbody due to the dominance of scatter- ing opacity over true absorption in the ejecta. The deviation from the blackbody can be described with a simple scaling, i.e. Fλ = ζ 2πBλ (T ). where Fλ is the surface flux, Bλ (T ) is the Planck function and ζ is the correction (or “dilution”) factor. http://arxiv.org/abs/0704.0552v1 These assumptions are most likely to be valid in Type II-P SNe. These eject a massive, hydrogen-rich envelope that remains optically thick for ∼ 100 days after explosion, and the emergent spectrum is indeed close to be Planckian. Thus, EPM is expected to work best for such SNe. Based on the assumptions, the instantaneous radius of the photosphere can be ex- pressed as Rphot = vphot(t) · (t − te) (the radius of the progenitor is usually neglected). Meantime, the observed flux is fλ = θ 2 · ζ 2πBλ (T ), where θ = Rphot/D is the angular radius of the photosphere from distance D. Combining these two equations, one gets the basic equation of EPM [2, 3]: t = te+D · vphot . (1) Since θ and vphot can be determined from observations, te and D are the only unknowns in Eq.1. These can be derived via least-squares fitting to the observed quantities. If the SNe under study are at high redshifts, the equations should be slightly modified [4]. The definition of the angular radius is connected with the angular distance DA, while in the expression of the observed flux the luminosity distance DL enters. At high z DL = (1+ z) 2DA, so the angular radius of the photosphere can be inferred from fλ (1+ z) πBλ ′(T ) , (2) where λ ′ = λ/(1+ z). One particular advantage of EPM is that it does not require initial calibration, i.e. a sample of objects with a priori known distances. However, the computation of the ζ correction factors needs detailed model atmospheres, which makes the method essen- tially model-dependent. Currently, there are two independent sets of model atmospheres of Type II-P SNe in the literature, which were used to compute correction factors as a function of Te f f [5, 6]. The former one was used in detailed studies of SN 1999em (the most extensively studied SN II-P so far) that resulted in DEPM ≈ 8±1 Mpc [2, 3, 7] . This is in significant disagreement with the Cepheid distance to the host galaxy NGC 1637 being DCep = 11.7±1 Mpc [8]. This problem has been solved in [9] by using a new set of correction factors based on the NLTE radiative transfer code CMFGEN which gave DEPM = 11.5±1.0 Mpc for SN 1999em. NEW EPM DISTANCES TO SNE II-P The method outlined above has been implemented in a new code that needs observed BVRI light curves, radial velocities (determined from the absorption minima of certain spectral features, see below) and reddening information (typically E(B−V )) as input. As in any method based on photometry, the magnitudes must be dereddened, but fortu- nately the results of EPM are quite insensitive to reddening errors, compared with other methods [5]. 3000 4000 5000 6000 7000 8000 9000 Wavelength (Å) SN 1999em (+8 d) Tbb = 16862 K 4 6 8 10 12 14 16 18 20 T (103 K) FIGURE 1. Left panel: Result of fitting a blackbody (solid line) to broadband BVI fluxes (filled symbols) of SN 1999em [3]. The R−band flux is also in good agreement with the fitted blackbody. The flux-calibrated spectrum obtained simultaneously (dotted line) is shown for comparison. Right panel: The correction factor as a function of Te f f from [6] (filled circles) and [5] (open circles). At each epoch, the angular radius has been computed by a simultaneous fitting to the dereddened B, V and I fluxes, as described in [2]. The corresponding effective temper- ature has been derived by fitting a blackbody curve to the broadband fluxes converted from the dereddened magnitudes. Our experience shows that the best results can be achieved by considering all optical+NIR (i.e. BVRI) fluxes simultaneously. Earlier stud- ies were sometimes limited to the usage of B and V bands only, which may result in increased systematic errors due to the large deviation of the B-band fluxes from the blackbody curve at later phases. The left panel of Fig.1 illustrates the optimum fitting of a blackbody to either photometric, or precisely calibrated spectroscopic fluxes. From the resulting Te f f , the correction factor ζ has been computed from the ζBV I(T ) function of Dessart & Hillier [6] for data obtained less than 40 days after explosion. For data measured between 40 - 60 days after explosion, the function given by Eastman et al. [5] was applied. As noted above, the usage of the function of Dessart & Hillier produces better distances, but their models are valid only during the first month after explosion, before the hydrogen starts to recombine. The ζBV I(T ) functions are plotted in the right panel of Fig.1. Beside the correction factors, the other important quantity is the photospheric velocity vphot , because the resulting distance is very sensitive to the velocities that appear in the denominator in Eq.1. Thus, the problem of finding an optimum method to infer vphot from Type II-P SNe spectra has been addressed in several studies (see [2, 3, 6]). We have studied this problem by computing model spectra with the parametrized spectral synthesis code SYNOW [10]. SYNOW computes the emergent spectrum in a homologously expanding atmosphere assuming LTE and pure scattering line formation. The input parameters are the velocity and the blackbody temperature at the photosphere (vphot and Te f f ), the exponent of the atmospheric structure, the list of ions contributing to the spectral features, and the optical depth of one strong line for each ion. Four sets of spectra have been defined corresponding to phases +10, +15, +50 and +95 days after explosion, respectively. The list of ions contained H, He I, Na I, Fe II, Sc 3000 4000 5000 6000 7000 8000 Wavelength (Å) +10 d +15 d +50 d +95 d 0.96 0.98 1.02 1.04 1.06 1.08 2 4 6 8 10 12 14 16 vobs (10 3 km/s) HeI 5876 FeII 4924 FeII 5018 FeII 5169 ScII 5526 FIGURE 2. Left panel: SYNOW model spectra of Type II-P SNe. The phase of each spectrum (ex- pressed in days after explosion) is indicated. Right panel: The ratio of the true photospheric velocity (an input parameter of a SYNOW model) to the „observed” velocity (derived from the absorption minimum of P Cygni lines) as a function of the „observed” velocity. Different symbols mean different photospheric lines indicated on the righ-hand side. II, Ti II and Ba II, because these ions are thought to be responsible for the strongest lines in the optical [3]. The input parameters except vphot were tuned to match real Type II-P SNe spectra. Then, several model spectra were synthesized with different input vphot for each phase. The left panel of Fig.2 shows representative spectra of all phases. The synthesized spectra were used to compute “observed” radial velocities by mea- suring the Doppler-shift of the absorption minima of selected lines. For P Cygni line profiles, this should give exactly vphot if the line is isolated and optically thin. However, in reality the features in a SN spectrum are all blends and may not be optically thin. Therefore, vobs will differ from vphot . In the right panel of Fig.2 the ratio of vphot/vobs is plotted as a function of vobs for the features shown. It is seen that in almost all cases vphot is slightly underestimated. The explanation of such a phenomenon is discussed in [6] for the Hα line. However, the relative difference is below 5 %, thus, these lines are expected to represent vphot with 2 - 4 % accuracy. Motivated by these results, we have selected the He I λ5876 and the Fe II λ5169 features to infer vphot from early-phase (< +20 days) and late-phase spectra of real SNe, respectively. In order to apply the method to real SNe, we have collected the available data of Type II-P SNe from the literature (details and references will be published in a forthcoming paper). Eq.1 was fitted to the observed data via least squares using either t or θ/vphot as the independent variable. The two results for D and te were averaged to obtain their final value. Whenever possible, the fit was restricted to data obtained between +5 – +40 days after explosion, and the angular radii were calculated using the Dessart & Hillier correction factors (see above). In a few cases only late-phase (t ∼ 40− 60 days) data were available. The Eastman et al. correction factors were applied for those SNe. The EPM distances are plotted against the “reference” distances to their host galaxies (mostly Tully-Fisher or SBF-distances for the nearby ones and Hubble-flow distances for the more distant ones) in the left panel of Fig.3. As a comparison, the distances coming 1000 1 10 100 1000 Dref (Mpc) 26 28 30 32 34 36 38 40 FIGURE 3. Left panel: the comparison of EPM (filled circles) and SCM (open triangles) distances with the reference distances of the host galaxies. Right panel: residuals of the distance moduli of Type II-P SNe from EPM (filled symbols) and Type Ia SNe (see text). from the „Standard Candle Method” (SCM) [11] for nearly the same observational sample are also shown. The scattering is very similar for both EPM and SCM. It is concluded that these two methods provide distances to Type II-P SNe with ∼ 15−20 % accuracy. The accuracy of the new EPM distances is also similar to that of individual SNe Ia distances. This is illustrated in the right panel of Fig.3, where the difference between the distance moduli of Type II-P SNe (from this paper) and the low-redshift subsample of Type Ia SNe (from [12]) are plotted against the reference distance moduli (adopting Dre f = cz/H0 for Type Ia SNe). Again, the scattering of the data is similar for the two samples. Thus, the concept of EPM combined with the present knowledge of Type II-P SNe atmospheres may provide consistent and reliable distances, which may be extended toward higher redshifts in the future. This could be a very important, independent test of the Type Ia SNe distance scale. This work was supported by Hungarian OTKA Grants No. T 042509 and TS 049872. REFERENCES 1. Kirshner R.P., Kwan J., ApJ 193, 27 (1974) 2. Hamuy M. et al., ApJ, 558, 615 (2001) 3. Leonard D.C. et al., PASP 114, 35 (2002) 4. Schmidt, B.P. et al., AJ 107, 1444 (1994) 5. Eastman, R.G., Schmidt, B.P., & Kirshner, R., ApJ 466, 911 (1996) 6. Dessart, L. and Hillier, D. J., Astronomy & Astrophysics 439, 671 (2005) 7. Elmhamdi, A. et al. MNRAS 338, 939 (2003) 8. Leonard, D.C., Kanbur, S.M., Ngeow, C.C., Tanvir, N.R. ApJ 594, 247 (2003) 9. Dessart, L. and Hillier, D. J., Astronomy & Astrophysics 447, 691 (2006) 10. Baron E. et al., ApJ 545, 444 (2000) 11. Hamuy, M., in Cosmic Explosions - IAU Colloquium 192, edited by J. M. Marcaide and K. W. Weiler, Springer Proceedings in Physics 99, Springer-Verlag, Berlin, Heidelberg, 2005, pp. 535–541. 12. http://braeburn.pha.jhu.edu/~ariess/R06/Davis07_R07_WV07.dat

0704.0553

Spontaneous Lorentz Violation: Non-Abelian Gauge Fields as Pseudo-Goldstone Vector Bosons

Non-Abelian Gauge Fields as Pseudo-Goldstone Vector Bosons J.L. Chkareuli and J.G. Jejelava E. Andronikashvili Institute of Physics and I. Chavchavadze State University, 0177 Tbilisi, Georgia Abstract We argue that non-Abelian gauge fields can be treated as the pseudo-Goldstone vector bosons caused by spontaneous Lorentz invariance violation (SLIV). To this end, the SLIV which evolves in a general Yang-Mills type theory with the nonlinear vector field constraint Tr(AµA µ) = ±M2 (M is a proposed SLIV scale) imposed is considered in detail. With an internal symmetry group G having D generators not only the pure Lorentz symmetry SO(1, 3), but the larger accidental symmetry SO(D, 3D) of the SLIV constraint in itself appears to be spontaneously broken as well. As a result, while the pure Lorentz violation still generates only one genuine Goldstone vector boson, the accompanying pseudo-Goldstone vector bosons related to the SO(D, 3D) breaking also come into play in the final arrangement of the entire Goldstone vector field multiplet. Remarkably, they remain strictly massless, being protected by gauge invariance of the Yang-Mills theory involved. We show that, although this theory contains a plethora of Lorentz and CPT violating couplings, they do not lead to physical SLIV effects which turn out to be strictly cancelled in all the lowest order processes considered. However, the physical Lorentz violation could appear if the internal gauge invariance were slightly broken at very small distances influenced by gravity. For the SLIV scale comparable with the Planck one the Lorentz violation could become directly observable at low energies. http://arxiv.org/abs/0704.0553v5 1 Introduction The old idea[1] that spontaneous Lorentz invariance violation (SLIV) may lead to an alternative theory of quantum electrodynamics still remains extremely attractive in numerous theoretical contexts[2] (for some later developments, see the papers[3]). The SLIV could generally cause the appearance of massless vector Nambu-Goldstone modes which are identified with photons and other gauge fields underlying the mod- ern particle physics framework like as Standard Model and Grand Unified Theory. At the same time, the Lorentz violation by itself has attracted a considerable at- tention in recent years as an interesting phenomenological possibility appearing in various quantum field and string theories[4-9]. Early models realizing the SLIV conjecture were based on the four fermion (current-current) interaction, where the proposed gauge field may appear as a fermion- antifermion pair composite state[1], in a complete analogy with a massless composite scalar field in the original Nambu-Jona-Lazinio model[10]. Unfortunately, owing to the lack of a starting gauge invariance in such models and composite nature of Goldstone modes appeared it is hard to explicitly demonstrate that these modes really form together a massless vector boson being a gauge field candidate. Actu- ally, one must make a precise tuning of parameters, including a cancellation between terms of different orders in the 1/N expansion (where N is the number of fermion species involved), in order to achieve the massless photon case[11]. Rather, there are in general three separate massless Goldstone modes, two of which may mimic the transverse photons polarizations, while the third one must properly be suppressed. In this connection, the more instructive laboratory for SLIV consideration proves to be some simple class of the QED type models having from the outset a gauge invariant form, whereas the Lorentz violation is realized through the nonlinear dy- namical constraint imposed on the starting vector field Aµ A2µ = n 2 (1) where nµ is an properly oriented unit Lorentz vector, while M is a proposed SLIV scale. This constraint means in essence that the vector field Aµ develops the vacuum expectation value 〈Aµ(x)〉 = nµM and Lorentz symmetry SO(1, 3) breaks down to SO(3) or SO(1, 2) depending on the time-like (n2µ = +1) or space-like (n µ = −1) SLIV. Such QED model was first studied by Nambu a long time ago[12], but only for the time-like SLIV case and in the tree approximation. For this purpose he applied the technique of nonlinear symmetry realizations which appeared successful in the handling of the spontaneous breakdown of chiral symmetry in the nonlinear σ model[13] and beyond1. 1Actually, the simplest possible way to obtain the above supplementary condition (1) could be an inclusion the “standard” quartic vector field potential V (A) = − A2µ + (A2µ) 2 into the QED type Lagrangian, as can be motivated to some extent[14] from the superstring theory. This potential inevitably causes the spontaneous violation of Lorentz symmetry in a standard way, much as an internal symmetry violation is caused in a linear σ model for pions[13]. As a result, one has a In the present paper, we mainly address ourselves to the Yang-Mills gauge fields as the possible vector Goldstone modes (Sec.3) once some basic ingredients of the Goldstonic QED model are established in a general SLIV case (Sec.2). This prob- lem has been discussed many times in the literature within quite different contexts, such as the Yang-Mills gauge fields as the Goldstone modes for the spontaneous breaking of general covariance in a higher-dimensional space[17] or for the nonlinear realization of some special infinite parameter gauge group[18]. However, all these considerations look rather speculative and optional. Specifically, they do not give a correlation between the SLIV induced photon case, from the one hand, and the Yang-Mills gauge field case, from the other. In contrast, our approach is solely based on the spontaneous Lorentz violation thus properly generalizing the Nambu’s QED model[12] to the non-Abelian internal symmetry case. Just in this approach evolved the interrelation between both of cases appears most transparent. We will see that in the Yang-Mills theory case with an internal symmetry group G having D generators not only the pure Lorentz symmetry part SO(1, 3) in the symmetry SO(1, 3) ×G of the Lagrangian, but the larger accidental symmetry SO(D, 3D) of the SLIV constraint Tr(AµA µ) = ±M2 in itself is spontaneously broken as well. Because the starting non-Abelian theory proves to be expanded about the vacuum which violates the much higher accidental symmetry appeared, many extra mass- less modes, the pseudo-Goldstone vector bosons (PGB), have to arise. Actually, while the spontaneous Lorentz violation on its own still generates only one genuine Goldstone vector boson, the accompanying vector PGBs related to the SO(D, 3D) breaking also come into play in the final arrangement of the entire Goldstone vec- tor field multiplet. Remarkably, in contrast to the familiar scalar PGB case[13] the vector PGBs remain strictly massless being protected by the non-Abelian gauge in- variance of the Yang-Mills theory involved. Then in Sec.4 we show by some examples of the lowest order SLIV processes that, while the Goldstonic non-Abelian theory evolved contains a rich variety of Lorentz and CPT violating couplings, it proves to be physically indistinguishable from a conventional Yang-Mills theory. Actually, one of the goals of the present work is to explicitly demonstrate that a conventional Yang-Mills theory (as well as QED) is in fact the spontaneously broken theory. The Lorentz violation, due to the quadratic field constraint of the type (1), renders this theory highly nonlinear in the Goldstone vector modes, while physically equivalent to the usual one. So, as well as in the pure QED case, the SLIV only means the noncovariant gauge choice in the otherwise gauge invariant and Lorentz invariant Yang-Mills theory. However, even a tiny breaking of the starting gauge invariance at massive Higgs mode (with mass 2mA) together with a massless Goldstone mode associated with photon. Furthermore, just as in the pion model one can go from the linear model for the SLIV to the non-linear one taking a limit λA → ∞, m A → ∞ (while keeping the ratio m A/λA to be finite). This immediately leads to the constraint (1) for vector potential Aµ with n 2 = m2A/λA, as it appears from a validity of its equation of motion. Another motivation for the nonlinear vector field constraint (1) might be an attempt to avoid the infinite self-energy of the electron in a classical electrodynamics, as was originally indicated by Dirac[15] and extended later to various vector field theory cases[16]. very small distances influenced by gravity would render the SLIV physically signifi- cant. For the SLIV scale comparable with the Planck one the spontaneous Lorentz violation could become directly observable at low energies. We summarize the results obtained in the final Sec.5. 2 Goldstonic quantum electrodynamics The simplest SLIV model is given by a conventional QED Lagrangian for the charged fermion field ψ L(A,ψ) = − µν + ψ(iγ · ∂ −m)ψ − eAµψγ µψ (2) where the nonlinear vector field constraint (1) is imposed[12]. For the resulting Lorentz violation, one can rewrite the Lagrangian L(A,ψ) in terms of the standard parametrization for the vector potential Aµ Aµ = aµ + (n ·A) (n2 ≡ n2µ) (3) where the aµ is pure Goldstonic mode n · a = 0 (4) while the effective Higgs mode (or the Aµ component in the vacuum direction) is given according to the above nonlinear constraint (1) by n ·A = (M2 − n2a2ν) 2 =M − n2a2ν +O(1/M2) (5) where, for definiteness, the positive sign for the above square root was taken when expanding it in powers of a2ν/M 2. Putting the parametrization (3) with the SLIV constraint (1, 5) into our basic gauge invariant Lagrangian (2) one comes to the truly Goldstonic model for QED. This model might look unacceptable due to the inappropriately large Lorentz violating fermion bilinear eMψ(γ ·n)ψ stemming from the vector-fermion current interaction eAµψγ µψ in the Lagrangian L (2) when the expansion (5) is taken. However, thanks to a local invariance of the Lagrangian L this term can be gauged away by a suitable redefinition of the fermion field ψ → eieM(n·x)ψ (6) after which the above fermion bilinear is exactly cancelled by an analogous term stemming from the fermion kinetic term. So, one eventually comes to the essentially nonlinear SLIV Lagrangian for the Goldstonic aµ field of the type (taken in the first approximation in a2ν/M L(a, ψ) = − δ(n · a)2 − n2a2ρ + (7) +ψ(iγ · ∂ +m)ψ − eaµψγ en2a2ρ ψ(γ · n)ψ We denoted its strength tensor by fµν = ∂µaν − ∂νaµ, while h µν = nµ∂ν −nν∂µ is a new SLIV oriented differential tensor. This tensor hµν acts on the infinite series in a2ρ coming from the expansion of the effective Higgs mode (5) from which the first order term −n2a2ν/2M was only taken in this expansion throughout the Lagrangian L(a, ψ). Also, we explicitly included the orthogonality condition n · a = 0 into Lagrangian through the term which can be treated as the gauge fixing term (taking the limit δ → ∞) and retained the former notation for the fermion ψ. The Lagrangian (7) completes the Goldstonic QED construction for the charged fermion field ψ. The model, as one can see, contains the massless Goldstone modes given by the tree broken generators of the Lorentz group, while keeping the massive Higgs mode frozen. These modes, lumped together, constitute a single Goldstone vector boson associated with photon2. In the limit M → ∞ the model is indistin- guishable from a conventional QED taken in the general axial (temporal or pure axial) gauge. So, for this part of the Lagrangian L(a, ψ) given by the zero-order terms in 1/M the spontaneous Lorentz violation only means the noncovariant gauge choice in otherwise the gauge invariant (and Lorentz invariant) theory. Remarkably, furthermore, also all the other (first and higher order in 1/M) terms in the L(a, ψ) (7), though being by themselves the Lorentz and CPT violating ones, do not lead to the physical SLIV effects which turn out to be strictly cancelled in all the physical processes involved. So, the nonlinear constraint (1) imposed on the standard QED Lagrangian (2) appears, in fact, as a possible gauge choice, while the S-matrix re- mains unaltered under such a gauge convention. This conclusion was first reached at tree level[12] and recently extended to the one-loop approximation[19]. All the one-loop contributions to the photon-photon, photon-fermion and fermion-fermion interactions violating the physical Lorentz invariance were shown to be exactly can- celled as well. This means that the vector field constraint A2µ = n 2 which has been treated as the nonlinear gauge choice at a tree (classical) level, remains just as a pure gauge condition when quantum effects are also taken into account. Re- markably, this conclusion appears to work also for a general Abelian theory case[20], particularly, when the internal U(1) charge symmetry is spontaneously broken hand in hand with the Lorentz one. As a result, the massless photon being first generated by the Lorentz violation become then massive due to the standard Higgs mechanism, while the SLIV condition in itself remains to be a gauge choice3. 2Strictly speaking one can no longer use the standard definition of photon as a state being the spin-1 representation of the (now spontaneously broken) Poincare group. However, due to gauge symmetry of the starting QED Lagrangian (2) the separate SLIV Goldstone modes appear combined in such a way that a standard photon (taken in an axial gauge (4)) emerges. 3Note in this connection that there was discussed[12] a possibility of an explicit construction of the gauge function corresponding to the nonlinear gauge constraint (1) that would eliminate the need for all the kinds of checks of gauge invariance mentioned above. Remarkably, the equation for this gauge function appears to be mathematically equivalent to the classical Hamilton-Jacobi equation of motion for a charged particle. Thus, this gauge function should in principle exist because there is a solution to the classical problem. However, this formal analogy only works for the time-like SLIV (n2µ = +1) in the pure QED leaving aside a general Abelian theory when the gauge invariance can spontaneously be broken. Apart from that, it does not generally extend to 3 Goldstonic Yang-Mills theory In this section, we extend our discussion to the non-Abelian internal symmetry case given by a general group G with generators ti([ti, tj ] = icijktk and Tr(titj) = δij where cijk are structure constants and i, j, k = 0, 1, ...,D − 1). The corresponding vector fields which transform according to its adjoint representation are given in the proper matrix form Aµ = A i, while the matter fields (fermions, for definiteness) are presented in the fundamental representation column ψr (r = 0, 1, ..., d− 1) of G. By analogy with the above Goldstonic QED case we take for them a conventional Yang-Mills type Lagrangian L(A, ψ) = − Tr(F µνF µν) + ψ(iγ · ∂ −m)ψ + gψAµγ µψ (8) (where F µν = ∂µAν − ∂νAµ − ig[Aµ,Aν ] and g stands for the universal coupling constant in the theory) with the nonlinear SLIV constraint Tr(AµA µ) = n2µM 2, n2µ = ±1 (9) imposed4. One can easily see that, although we propose only the SO(1, 3) × G invariance in the theory, the SLIV constraint taken (9) possesses, in fact, the much higher accidental symmetry SO(D, 3D) determined by the dimensionality D of the G group adjoint representation to which the vector fields Aiµ are belonged. This symmetry is indeed spontaneously broken at a scale M Aiµ(x) = niµM (10) with the vacuum direction given now by the ‘unit’ rectangular matrix niµ which describes both of the generalized SLIV cases at once, time-like (SO(D, 3D) → SO(D−1, 3D)) or space-like (SO(D, 3D) → SO(D, 3D−1)), respectively, depending on the sign of the n2µ ≡ n µ,i = ±1. This matrix has only one non-zero element for both of cases determined by the proper SO(D, 3D) rotation. They are, particularly, 0 or n 3 provided that the vacuum expectation value (10) is developed along the i = 0 direction in the internal space and along the µ = 0 or µ = 3 direction, respec- tively, in the Minkowskian space-time. In response to each of these two breakings, side by side with one true vector Goldstone boson and the D − 1 scalar Goldstone bosons corresponding to the spontaneous violation of actual SO(1, 3) ⊗ G symme- try of the total Lagrangian L, the D − 1 vector pseudo-Goldstone bosons related to breaking of the accidental SO(D, 3D) symmetry of the SLIV constraint taken (9) are also produced. Remarkably, in contrast to the familiar scalar PGB case[13] the non-Abelian case (see next Section). 4As in the Abelian case, the existence of such a constraint could be related with some non- linear σ type SLIV model proposed for the vector field multiplet Aiµ in the Yang-Mills theory (8). Note in this connection that, due to its generic antisymmetry, the familiar quadrilinear terms g2Tr([Aµ, Aν ]) 2 in the Lagrangian (8) do not contribute into the SLIV since they identically vanish for any single-valued vacuum configuration the vector PGBs remain strictly massless being protected by the non-Abelian gauge invariance of the starting Lagrangian (8). Together with the aforementioned true vector Goldstone boson they complete the entire Goldstonic vector field multiplet of the internal symmetry group G. As in the Abelian case, upon an explicit use of the corresponding SLIV constraint (9) being so far the only supplementary condition for vector field multiplet Aiµ, one comes to the pure Goldstone field modes aiµ identified in a similar way Aiµ = a (n · A) , n · a ≡ niµa µ,i = 0 (n2 ≡ n2µ) , (11) At the same time, an effective Higgs mode (i.e., the Aiµ component in the vacuum direction niµ) is given by the product n · A ≡ n µ,i determined by the SLIV con- straint n · A = M2 − n2(aiν) 2 =M − 2(aiν) +O(1/M2) (12) where, as earlier in the Abelian case, we took the positive sign for the square root when expanding it in powers of (aiν) 2/M2. Note that the general Goldstonic modes aiµ, apart from pure vector fields, contain the D − 1 scalar ones, a 0 and a (i′ = 1...D − 1), for the time-like (niµ = n 0gµ0δ i0) and space-like (niµ = n 3gµ3δ SLIV, respectively. They can be eliminated from the theory if one puts the proper supplementary conditions on the aiµ fields which were still the constraint free. Using their overall orthogonality (11) to the physical vacuum direction niµ one can formu- late these supplementary conditions in terms of a general axial gauge for the entire aiµ multiplet n · ai ≡ nµa µ,i = 0, i = 0...D − 1 (13) where nµ is the unit Lorentz vector introduced in the Abelian case which is now oriented in Minkowskian space-time so as to be parallel to the vacuum matrix niµ. For such a choice the simple equation holds µ = s inµ (s n · ni ) (14) which shows that the rectangular vacuum matrix niµ has the factorized ”two-vector” form. As a result, apart from the Higgs mode excluded earlier by the orthogonality condition (11), all the scalar fields also appear eliminated, and only pure vector fields, (µ′ = 1, 2, 3) or ai (µ′′ = 0, 1, 2) for time-like or space-like SLIV, respectively, are only left in the theory. We now show that the such constrained Goldstone vector fields aiµ (with the supplementary conditions (13) taken) appear truly massless when the starting non- Abelian Lagrangian L (8) is rewritten in the form determined by the SLIV. Actually, putting the parametrization (11) with the SLIV constraint (12) into the Lagrangian (8) one is led to the highly nonlinear Yang-Mills theory in terms of the pure Gold- stonic gauge field modes aiµ. However, as in the above Abelian case, one should first gauge away (using the local invariance of the Lagrangian L) the enormously large, while false, Lorentz violating terms appearing in the theory in the form of the fermion and vector field bilinears. As one can readily see, they stem from the couplings gψAµγ µψ and −1 g2Tr([Aµ, Aν ]) 2, respectively, when the effective Higgs mode expansion (12) is taken in the Lagrangian (8). Making the appropriate redef- initions of the fermion (ψ ) and vector (aµ ≡ a i) field multiplets ψ → U(ω)ψ , aµ → U(ω)aµU(ω) †, U(ω) = eigM(n i·x)ti (15) and using the evident equalities for the linear (in coordinate) transformations U(ω) with the single-valued vacuum matrix niµ (n 0 or n 3 for the particular SLIV cases) ∂µU(ω) = ign iU(ω) = igU(ω)niµt i (16) one can confirm that the abovementioned Lorentz violating terms are exactly can- celled with the analogous bilinears stemming from their kinetic terms. So, the final Lagrangian for the Goldstonic Yang-Mills theory takes the form (in the first approx- imation in (aiν) 2/M2) L(a,ψ) = − Tr(fµνf δ(n · ai)2 + Tr(fµνh 2(aiν) +ψ(iγ · ∂ −m)ψ + gψaµγ gn2(aiν) ψ(γ · nk)tkψ (17) where the tensor fµν is, as usual, fµν = ∂µaν − ∂νaµ − ig[aµ,aν ], while hµν is a new SLIV oriented tensor of the type hµν = nµ∂ν − nν∂µ + ig([nµ,aν ]− [nν ,aµ]), nµ ≡ n k (18) This tensor hµν acts on the infinite series in (a 2 coming from the expansion of the effective Higgs mode (12) from which only the first order term −n2(aiν) 2/2M was taken throughout the Lagrangian L(a,ψ). We also retained the former notations for the fermion and vector field multiplets after transformations (15), and explicitly in- cluded the (axial) gauge fixing term into Lagrangian according to the supplementary conditions taken (13). The theory derived gives a proper generalization of the nonlinear QED model[12] for the non-Abelian case. It contains the massless vector boson multiplet aiµ (con- sisting of one Goldstone and D − 1 pseudo-Goldstone vector states) which gauges the starting internal symmetry G. In the limit M → ∞ it is indistinguishable from a conventional Yang-Mills theory taken in the general axial gauge. So, for this part of the Lagrangian L(a,ψ) given by the zero-order in 1/M terms the spontaneous Lorentz violation only means the noncovariant gauge choice in the otherwise gauge invariant (and Lorentz invariant) theory. However, one may expect that, just as it appears in the nonlinear QED model, also all the first and higher order in 1/M terms in the L (17), though being by themselves the Lorentz and CPT violating ones, do not lead to the physical SLIV effects due to the mutual cancellation of their contributions into all the physical processes appeared. 4 The lowest order SLIV processes Let us now show that the simple tree level calculations related to the Lagrangian L(a,ψ) confirms in essence this proposition. As an illustration, we consider SLIV processes in the lowest order in g and 1/M being the fundamental parameters of the Lagrangian (17). They are, as one can readily see, the vector-fermion and vector- vector elastic scattering going in the order g/M , which we turn to once the Feynman rules in the Goldstonic Yang-Mills theory are established. 4.1 Feynman rules The corresponding Feynman rules, apart from the ordinary Yang-Mills theory rules (i) the vector-fermion vertex − ig γµ t i (19) (ii) the vector field propagator (taken in a general axial gauge nµaiµ = 0) Dijµν (k) = − gµν − nµkν + kµnν n · k n2kµkν (n · k)2 which automatically satisfies the orthogonality condition nµD µν(k) = 0 and on-shell transversality kµD µν(k) = 0 (k 2 = 0); the latter means that free vector fields with polarization vector ǫiµ(k, k 2 = 0) are always appeared transverse kµǫiµ(k) = 0; (iii) the 3-vector vertex (with vector field 4-momenta k1, k2 and k3; all 4-momenta in vertexes are taken ingoing throughout) gcijk[(k1 − k2)γgαβ + (k2 − k3)αgβγ + (k3 − k1)βgαγ ] (21) include the new ones, violating Lorentz and CPT invariance, for (iv) the contact 2-vector-fermion vertex (γ · nk)τkgµν δ ij (22) (v) another 3-vector vertex (k1 · n i)k1,αgβγδ jk + (k2 · n j)k2,βgαγδ ki + (k3 · n k)k3,γgαβδ where the second index in the vector field 4-momenta k1, k2 and k3 denotes their Lorentz components; (vi) the extra 4-vector vertex (with the vector field 4-momenta k1,2,3,4 and their proper differences k12 ≡ k1 − k2 etc.) [cijpδklgαβgγδ(n p · k12) + c klpδijgαβgγδ(n p · k34) + +cikpδjlgαγgβδ(n p · k13) + c jlpδikgαγgβδ(n p · k24) + (24) +cilpδjkgαδgβγ(n p · k14) + c jkpδilgαδgβγ(n p · k23)] where only the terms which can not lead to contractions of the rectangular vacuum matrix n µ with vector field polarization vectors ǫ µ(k) are presented. These contrac- tions are in fact vanished due to the gauge taken (13), np · ǫi = sp(n · ǫi) = 0 (with a factorized two-vector form for the matrix n µ (14) used). Just the rules (i-vi) are needed to calculate the lowest order amplitudes of the processes we have mentioned in the above. 4.2 Vector boson scattering on fermion This process is directly related to two SLIV diagrams one of which is given by the contact a2-fermion vertex (22), while another corresponds to the pole diagram with the longitudinal a-boson exchange between Lorentz violating a3 vertex (23) and ordinary a-boson-fermion one (19). Since ingoing and outgoing a-bosons appear transverse (k1 · ǫ i(k1) = 0, k2 · ǫ j(k2) = 0) only the third term in this a 3 coupling (23) contributes to the pole diagram so that one comes to a simple matrix element iM for both of diagrams iM = i ū(p2)τ (γ · nl) + i(k · nl)γµkνDµν(k) u(p1)[ǫ(k1) · ǫ(k2)] (25) where the spinors u(p1,2) and polarization vectors ǫ µ(k1) and ǫ µ(k2) stand for the in- going and outgoing fermions and a-bosons, respectively, while k is the 4-momentum transfer k = p2 − p1 = k1 − k2. Upon the further simplifications in the square bracket related to the explicit form of the a boson propagator Dµν(k) (20) and ma- trix niµ (14), and using the fermion current conservation ū(p2)(p̂2 − p̂1)u(p1) = 0, one is finally led to the total cancellation of the Lorentz violating contributions to the a-boson-fermion scattering in the g/M approximation. Note, however, that such a result may be in some sense expected since from the SLIV point of view the lowest order a-boson-fermion scattering discussed here is hardly distinct from the photon-fermion scattering considered in the nonlinear QED case[12]. Actually, the fermion current conservation which happens to be crucial for the above cancellation works in both of cases, whereas the couplings being peculiar to the Yang-Mills theory have not yet touched on. In this connection the next example seems to be more instructive. 4.3 Vector-vector scattering The matrix element for this process in the lowest order g/M is given by the contact SLIV a4 vertex (24) and the pole diagrams with the longitudinal a-boson exchange between the ordinary a3 vertex (21) and Lorentz violating a3 one (23), and vice versa. There are six pole diagrams in total describing the elastic a − a scattering in the s- and t-channels, respectively, including also those with an interchange of identical a-bosons. Remarkably, the contribution of each of them is exactly canceled with one of six terms appeared in the contact vertex (24). Actually, writing down the matrix element for one of the pole diagrams with ingoing a-bosons (with momenta k1 and k2) interacting through the vertex (21) and outgoing a-bosons (with momenta k3 and k4) interacting through the vertex (23) one has cijpδkl[(k1 − k2)µgαβ + (k2 − k)αgβµ + (k − k1)βgαµ] · ·Dpqµν(k)gγδkν(n q · k)[ǫi,α(k1)ǫ j,β(k2)ǫ k,γ(k3)ǫ l,δ(k4)] (26) where polarization vectors ǫi,α(k1), ǫ j,β(k2), ǫ k,γ(k3) and ǫ l,δ(k4) belong, respectively, to ingoing and outgoing a-bosons, while k = −(k1 + k2) = k3 + k4 according to the momentum running in the diagrams taken above. Again, as in the previous case of vector-fermion scattering, due to the fact that outgoing a-bosons appear transverse (k3 · ǫ k(k3) = 0 and k4 · ǫ l(k4) = 0), only the third term in the Lorentz violating a coupling (23) contributes to this pole diagram. Upon evident simplifications related to the a-boson propagator Dµν(k) (20) and matrix n µ (14) one comes to the expres- sion which is exactly cancelled with the first term in the contact SLIV vertex (24) when it is properly contracted with a-boson polarization vectors. Likewise, other terms in this vertex provide the further one-to-one cancellation with the remaining pole matrix elements iM (2−6) . So, again, the Lorentz violating contribution to the vector-vector scattering is absent in Goldstonic Yang-Mills theory in the lowest g/M approximation. 4.4 Other processes Other tree level Lorentz violating processes, related to a bosons and fermions, appear in higher orders in the basic SLIV parameter 1/M . They come from the subsequent expansion of the effective Higgs mode (12) in the Lagrangian (17). Again, their amplitudes are essentially determined by an interrelation between the longitudinal a-boson exchange diagrams and the corresponding contact a-boson interaction dia- grams which appear to cancel each other thus eliminating physical Lorentz violation in theory. Most likely, the same conclusion can be derived for SLIV loop contributions as well. Actually, as in the massless QED case considered earlier [19], the corre- sponding one-loop matrix elements in Goldstonic Yang-Mills theory either vanish by themselves or amount to the differences between pairs of the similar integrals whose integration variables are shifted relative to each other by some constants (be- ing in general arbitrary functions of external four-momenta of the particles involved) that in the framework of dimensional regularization leads to their total cancellation. So, the Goldstonic vector field theory (17) for a non-Abelian charge-carrying matter is likely to be physically indistinguishable from a conventional Yang-Mills theory. 5 Conclusion The spontaneous Lorentz violation in 4-dimensonal flat Minkowskian space-time was shown to generate vector Goldstone bosons both in Abelian and non-Abelian theo- ries with the corresponding nonlinear vector field constraint (1) or (9) imposed. In the Abelian case such a massless vector boson is naturally associated with photon. In non-Abelian case, although the pure Lorentz violation still generates only one genuine Goldstone vector boson, the accompanying vector PGBs related to a vio- lation of the larger accidental symmetry SO(D, 3D) of the SLIV constraint (9) in itself come also into play in the final arrangement of the entire Goldstone vector field multiplet of the internal symmetry group G. Remarkably, they remain strictly massless being protected by the gauge invariance of the Yang-Mills theory involved. These theories, both Abelian and non-Abelian, while being essentially nonlinear in the Goldstone vector modes, are physically indistinguishable from conventional QED and Yang-Mills theory. One could actually see that just the gauge invariance not only provides these theories to be free from the unreasonably large Lorentz violation stemming from the fermion and vector field bilinears (see Sections 2 and 3), but also render all the other physical SLIV effects (including those which are suppressed by the Lorentz violation scale M) non-observable (Section 4). As a result, Abelian and non-Abelian SLIV theory appear, respectively, as standard QED and Yang-Mills theory taken in the nonlinear gauge (to which the vector field constraints (1) and (9) are virtually reduced), while the S-matrix remains unaltered under such a gauge convention. So, while at present the Goldstonic nature of gauge fields, both Abelian and non- Abelian, seems to be highly plausible, the most fundamental question of physical Lorentz violation in itself, that only could uniquely point toward such a possibility, is still an open question. Note, that here we are not dealing with direct (and quite arbitrary in essence) Lorentz non-invariant extensions of QED or Standard Model which were intensively discussed on their own in recent years [6-8]. Rather, the case in point is a construction of genuine SLIV models which would generate gauge fields as the proper vector Goldstone bosons, from one hand, and could lead to observed Lorentz violating effects, from the other. In this connection, somewhat natural framework for physical Lorentz violation to occur would be a model where the internal gauge invariance were slightly broken at very small distances through some high-order operators stemming from the gravity-influenced area. Such physical SLIV effects would be seen in terms of powers of ratio M/MP l (where MP l is the Planck mass). So, for the SLIV scale comparable with the Planck one they would become directly observable. Remarkably enough, if one has such internal gauge symmetry breaking in an ordinary Lorentz invariant theory this breaking appears vanishingly small at laboratory being properly suppressed by the Planck scale. However, the spontaneous Lorentz violation would render it physically significant: the higher Lorentz scale, the greater SLIV effects observed. If true, it would be of particular interest to have a better understanding of the internal gauge symmetry breaking mechanism that brings out the spontaneous Lorentz violation at low energies. We return to this basic question elsewhere. Acknowledgments We would like to thank Colin Froggatt, Rabi Mohapatra and Holger Nielsen for useful discussions and comments. One of us (J.L.C.) is grateful for the warm hospitality shown to him during a visit to Center for Particle and String Theory at University of Maryland where part of this work was carried out. References [1] W. Heisenberg, Rev. Mod. Phys. 29 (1957) 269; J.D. Bjorken, Ann. Phys. (N.Y.) 24 (1963) 174; I. Bialynicki-Birula, Phys. Rev. 130 (1963) 465 ; G. Guralnik, Phys. Rev. 136 (1964) B1404; T. Eguchi, Phys.Rev. D 14 (1976) 2755; H. Terazava, Y. Chikashige and K. Akama, Phys. Rev. D 15 (1977) 480 . [2] C.D. Froggatt and H.B. Nielsen, Origin of Symmetries (World Scientific, Sin- gapore, 1991). [3] J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Phys. Rev. Lett. 87 (2001) 091601; J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen Nucl. Phys. B 609 (2001) 46; J.D. Bjorken, hep-th/0111196; Per Kraus and E.T. Tomboulis, Phys. Rev. D 66 (2002) 045015; A. Jenkins, Phys. Rev. D 69 (2004) 105007; J.L. Chkareuli, C.D. Froggatt, R.N. Mohapatra and H.B. Nielsen, hep-th/0412225; J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, hep-th/0610186. [4] D. Colladay and V.A. Kostelecky, Phys. Rev. D58 (1998) 116002 ; V.A. Kostelecky, Phys. Rev. D69 (2004) 105009 ; R. Bluhm and V.A. Kostelecky, Phys. Rev. D 71(2005) 065008; CPT and Lorentz Symmetry, ed. A. Kostelecky (World Scientific, Singapore, 1999, 2002, 2005). [5] S.M. Carroll, G.B. Field and R. Jackiw, Phys. Rev. D 41 (1990) 1231; R. Jackiw and V.A. Kostelecky, Phys. Rev. Lett. 82 (1999) 3572. [6] S. Coleman and S.L. Glashow, Phys. Rev. D 59 (1999) 116008. http://arxiv.org/abs/hep-th/0111196 http://arxiv.org/abs/hep-th/0412225 http://arxiv.org/abs/hep-th/0610186 [7] J. W. Moffat, Int. J. Mod.Phys. D2 (1993) 351; J.W. Moffat, Int. J. Mod.Phys. D12 (2003) 1279. [8] O. Bertolami and D.F. Mota, Phys. Lett. B 455 (1999) 96. [9] T. Jacobson, S. Liberati and D. Mattingly, Ann. Phys. (N.Y.) 321 (2006) 150. [10] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [11] M. Suzuki, Phys. Rev. D 37 (1988) 210 . [12] Y. Nambu, Progr. Theor. Phys. Suppl. Extra 190 (1968). [13] S. Weinberg, The Quantum Theory of Fields, v.2, Cambridge University Press, 2000. [14] V.A. Kostelecky and S. Samuel, Phys. Rev. D 39 (1989) 683; V.A. Kostelecky and R. Potting, Nucl. Phys. B 359 (1991) 545. [15] P.A.M. Dirac, Proc. Roy. Soc. 209A (1951) 292; P.A.M. Dirac, Proc. Roy. Soc. 212A (1952) 330. [16] R. Righi and G. Venturi, Lett. Nuovo Cim. 19 (1977) 633; R. Righi, G. Venturi and V. Zamiralov, Nuovo Cim. A47 (1978) 518. [17] Y.M. Cho and P.G.O. Freund, Phys. Rev. D 12 (1975) 1711. [18] E. A. Ivanov and V.I. Ogievetsky, Lett. Math. Phys. 1 (1976) 309 . [19] A.T. Azatov and J.L. Chkareuli, Phys. Rev. D 73 (2006) 065026. [20] J.L. Chkareuli and Z.R. Kepuladze, Phys. Lett. B 644 (2007) 212; J.L. Chkareuli and Z.R. Kepuladze, Proc. of XIV Int. Seminar “Quarks-2006”, eds. S.V. Demidov at al (Moscow, INR, 2006); hep-th/0610227. http://arxiv.org/abs/hep-th/0610227 Introduction Goldstonic quantum electrodynamics Goldstonic Yang-Mills theory The lowest order SLIV processes Feynman rules Vector boson scattering on fermion Vector-vector scattering Other processes Conclusion

0704.0554

In-medium effects on particle production in heavy ion collisions

arXiv:0704.0554v1 [nucl-th] 4 Apr 2007 In-Medium Effects on Particle Production in Heavy Ion Collisions V. Prassa a G.Ferini b T. Gaitanos c H.H. Wolter d G.A. Lalazissis a M. Di Toro b aDepartment of Theoretical Physics, Aristotle University of Thessaloniki, Thessaloniki Gr-54124,Greece bLaboratori Nazionali del Sud INFN, I-95123 Catania, Italy cInstitut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany dSektion Physik, Universität München, D-85748 Garching, Germany email: [email protected] The effect of possible in-medium modifications of nucleon-nucleon (NN) cross sections on particle production is investigated in heavy ion collisions (HIC) at intermediate energies. In particular, using a fully covariant relativistic transport approach, we see that the den- sity dependence of the inelastic cross sections appreciably affects the pion and kaon yields and their rapidity distributions. How- ever, the (π−/π+)- and (K0/K+)-ratios depend only moderately on the in-medium behavior of the inelastic cross sections. This is particularly true for kaon yield ratios, since kaons are more uni- formly produced in high density regions. Kaon potentials are also suitably evaluated in two schemes, a chiral perturbative approach and an effective meson-quark coupling method, with consistent re- sults showing a similar repulsive contribution for K+ and K0. As a consequence we expect rather reduced effects on the yield ratios. We conclude that particle ratios appear to be robust observables for probing the nuclear equation of state (EoS) at high baryon density and, particularly, its isovector sector. Key words: Asymmetric nuclear matter, symmetry energy, relativistic heavy ion collisions, particle production. PACS numbers: 25.75.-q, 21.65.+f, 21.30.Fe, 25.75.Dw. Preprint submitted to Elsevier Preprint 4 November 2018 http://arxiv.org/abs/0704.0554v1 1 Introduction The knowledge of the properties of highly compressed and heated hadronic matter is an important issue for the understanding of astrophysical processes, such as the mechanism of supernovae explosions and the physics of neutron stars [1,2]. Heavy ion collisions provide the unique opportunity to explore highly excited hadronic matter, i.e. the high density behavior of the nuclear EoS, under controlled conditions (high baryon energy densities and tempera- tures) in the laboratory [3]. Of particular recent interest is also the still poorly known density dependence of the isovector channel of the EoS. Suggested observables have been the nucleon collective flows [3,4] and the distributions of produced particles such as pions and, in particular, particles with strangeness (kaons) [5,6]. Because of the rather high energy threshold (Elab = 1.56 GeV for Nucleon-Nucleon collisions), kaon production in HICs at energies in the range 0.8− 1.8 AGeV is mainly due to secondary processes involving ∆ resonances and pions (π). On the other hand, secondary processes require high baryon density. This explains why the kaon production around threshold is intimately connected to the high density stage of the nucleus- nucleus collision. Furthermore, the relatively large mean free path of positive charged (K+) and neutral (K0) kaons inside the hadronic environment causes hadronic matter to be transparent for kaons [7]. Therefore kaon yields and generally strangeness ratios have been proposed as important signals for the investigation of the high density behavior of the nuclear EoS. This idea, as firstly suggested by Aichelin and Ko [8], has been recently applied in HIC at intermediate energies in terms of strangeness ratios, e.g. the ratio of the kaon yields in Au+Au and C+C collisions [5,9]. In these studies it was found that this ratio is very sensitive to the stiffness of the nuclear EoS. Indeed comparisons with KaoS data [10] favored a soft behavior of the high density nuclear EoS, a statement which is particularly consistent with elliptic flow data of the FOPI collaboration [11]. The idea of studying particle ratios in HICs around the kinematical threshold has been recently applied in the determination of the isovector channel of the nuclear EoS, i.e. the high density dependence of the symmetry energy Esym. It has turned out that particle ratios, such as (π−/π+) [12] or (K0/K+) [13–15], are sensitive to the stiffness of the symmetry energy and, in particular to the strength of the vector isovector field. However in medium effects on the kaon propagation have been neglected so far. Here we will test the robustness of the yield ratio against the inclusion and the variation of the corresponding kaon potentials. At the same time in Ref. [16] the role of the in-medium modifica- tions of NN cross sections has been studied in terms of baryon and strangeness dynamics. It was found that the pion and kaon yields are sensitively influenced by the reduced effective NN cross sections for inelastic processes. Here we will see that the kaon yield ratio appears robust even with respect to the density dependence of the in-medium inelastic NN cross sections, while at variance the pion ratio seems to be more sensitive. The collision dynamics is rather complex and involves the nuclear mean field (EoS) and binary 2-body collisions. In the presence of a nuclear medium the treatment of binary collisions represents a non-trivial problem. The NN cross sections for elastic and inelastic processes, which are the crucial physical pa- rameters here, are experimentally accessible only in free space and not for 2-body scattering at finite baryon density. Recent microscopic studies, based on the G-matrix approach, have shown a strong decrease of the elastic NN cross section [17,18] in a hadronic medium. These in-medium effects of the elastic NN cross section considerably influence the hadronic reaction dynam- ics [19]. Obviously the question arises whether similar in-medium effects of the inelastic NN cross sections may affect the reaction dynamics and, in particular, the production of particles (pions and kaons). Furthermore, the strangeness propagation inside the nuclear medium is even more complex and involves the additional consideration of kaon mean field potentials in the dynamical description. This is an important issue when com- paring with experimental kaon data [10]. In a Chiral Perturbation approach at the lowest order (ChPT Potentials), the kaon (antikaon) potential has an attractive scalar and a repulsive (attractive) vector part [20]. This leads to weakly repulsive (strongly attractive) potentials for kaons (antikaons) with corresponding scalar and vector kaon-nucleon coupling constants depending on the parametrization [20,21] accounted for. Similar results can be obtained in an effective meson-coupling model (OBE Potentials, in the RMF spirit), where the K-meson couplings are simply related to the nucleon-meson ones, in the spirit of ref. [22]. The latter approach has the advantage of being fully consistent with the covariant transport equations used to simulate the reaction dynamics [14,15]. We remind that the high density dependence of the kaon self energies is still an object of current debate, e.g. see Refs. [23,7] in which the role of the kaon potential has been investigated in terms of kaon in-plane and out-of-plane flows. Moreover for studies aimed to the determination of the symmetry energy from strangeness production one has to consider with particular care the isospin dependence of the kaon mean field potential. The main focus of the present work is on a detailed study of the robustness of the pionic (π−/π+) and, in particular, the strangeness ratio (K0/K+) with re- spect to the in-medium modifications of the imaginary part of the nucleon self energy, i.e. the NN cross sections, and to the in-medium variations of the kaon self energy, i.e. the density dependence of the kaon potential. This analysis, which goes beyond our previous investigations of [14,15], is also motivated by new measurements of the FOPI collaboration [24] by means of the strangeness ratios. The paper is organized as follows: The next Section describes the theoret- ical treatment of the reaction dynamics within the Relativistic Boltzmann- Uheling-Uhlenbeck (RBUU) transport equation. A detailed discussion on the in-medium modifications of the inelastic NN cross sections is presented. In Sec- tion 3 we discuss the kaon mean field potentials (in both ChPT and OBE/RMF schemes) and their expected isospin dependence. Section 4 is devoted to a short introduction to the dynamical calculations. Results are then shown in Section 5, mostly for central 197Au+197 Au collisions at 1AGeV , in terms of pion and kaon yields. The initial presentation of the absolute yields is relevant for a de- tailed discussion as well as for a comparison with theoretical results of other groups and with experimental data of the KaoS and FOPI collaborations. All together this intermediate step is important for testing the reliability of the calculations, since ratios do not do it. Finally we present the pion and strangeness ratios and discuss their dependence on the in-medium modifica- tions of the cross NN cross sections and of the kaon potentials, including the isospin effects. In Section 6 we conclude with a summary and some general comments and perspectives. 2 Theoretical description of the collision dynamics In this chapter we briefly discuss the transport equation focusing on the treat- ment of two features important for kaon dynamics: (a) the collision integral by means of the cross sections; (b) the kaon mean field potential and its isospin dependence. 2.1 The RBUU equation The theoretical description of HICs is based on the semiclassical kinetic the- ory of statistical mechanics, i.e. the Boltzmann Equation with the Uehling- Uhlenbeck modification of the collision integral [25]. The relativistic analog of this equation is the Relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) equa- tion [26] k∗µ∂xµ + (k µν +M∗∂µxM ∗) ∂k f(x, k∗) = 2(2π)9 W (kk2|k3k4) f3f4f̃ f̃2 − ff2f̃3f̃4 , (1) where f(x, k∗) is the single particle distribution function. In the collision term the short-hand notations fi ≡ f(x, k∗i ) for the particle and f̃i ≡ (1− f(x, k∗i )) 0 50 100 150 200 250 300 350 [MeV] =1.1 fm =1.34 fm =1.7 fm pn-data Fig. 1. Elastic in-medium neutron-proton cross section σel at various Fermi momenta kF as a function of the laboratory energy Elab. The free cross section (kF = 0) is compared to the experimental total np cross section [17]. for the hole distributions are used, with E∗ M∗2 + k2. The collision in- tegral explicitly exhibits the final state Pauli-blocking while the in-medium scattering amplitude includes the Pauli-blocking of intermediate states. The dynamics of the drift term, i.e. the lhs of eq.(1), is determined by the mean field. Here the attractive scalar field Σs enters via the effective mass M∗ = M − Σs and the repulsive vector field Σµ via the kinetic momenta k∗µ = kµ − Σµ and via the field tensor F µν = ∂µΣν − ∂νΣµ. The dynamical description according to Eq.(1) involves the strangeness propagation in the nuclear medium. This topic will be discussed in more detail at the end of this section. 2.2 In-medium effects on NN cross sections The in-medium cross sections for 2-body processes (see below) enter in the collision integral via the transition amplitude W = (2π)4δ4 (k + k2 − k3 − k4) (M∗)4|T |2 (2) with T the in-medium scattering matrix element. In the kinetic equation (1) both physical input quantities, the mean field (EoS) and the collision integral (cross sections) should be derived from the same underlying effective two-body interaction in the medium, i.e. the in-medium T-matrix; Σ ∼ ℜTρB, σ ∼ ℑT , respectively. However, in most practical applications phenomenological mean fields and cross sections have been used. In such approach the strategy is to adjust to the known bulk properties of nuclear matter around the saturation point, and to try to constrain the models at supra-normal densities with the help of heavy ion reactions [27,28]. Medium modifications of the NN cross sections are usually not taken into account. In spite of that for several observables the comparison to experimental data appears to work astonishingly well [27–30]. However, in particular kinematical regimes a sensitivity to the elastic NN cross sections of dynamical observables, such as collective flows and stopping [19,31] or transverse energy transfer [32], has been observed. Microscopic Dirac-Brueckner-Hartree-Fock (DBHF) studies for nuclear matter above the Fermi energy regime show a strong density dependence of the elastic [17] and inelastic [18,33] NN cross sections. In such studies one starts from the bare NN-interaction in the spirit of the One-Boson-Exchange (OBE) model by fitting the parameters to empirical nucleon-nucleus scattering and solves then the equations of the nuclear matter many body problem in the T -matrix or ladder approximation. It is not the aim of the present work to go into further details on this topic. An important feature of such microscopic calculations is the inclusion of the Pauli-blocking effect in the intermediate scattering states of the T -matrix elements and their in-medium modifications, i.e. the density dependence of the nucleon mass and momenta. Here of particular interest are the in-medium modifications of the inelastic NN cross sections since they di- rectly influence the production mechanism of resonances and thus the creation of pions and kaons according to the channels listed later (see Sect.3). DBHF studies on inelastic NN cross sections are rare and in limited regions of density and momentum [18]. For this reason we will first discuss in the following the in-medium dependence of the elastic NN cross sections, which will be then used as a starting basis for a detailed analysis of the density dependence of the inelastic NN cross sections. The microscopic in-medium dependence of the elastic cross sections can be seen in Fig. 1, where the energy dependence of the in-medium neutron-proton (np) cross section at Fermi momenta kF = 0.0, 1.1, 1.34, 1.7fm −1, correspond- ing to ρB ∼ 0, 0.5, 1, 2ρ0 (ρ0 = 0.16fm−3 is the nuclear matter saturation den- sity) is shown. These results are obtained from relativistic Dirac-Brueckner calculations [17]. The presence of the medium leads to a substantial suppres- sion of the cross section which is most pronounced at low laboratory energy Elab and high densities where the Pauli-blocking of intermediate states is most efficient. At larger Elab asymptotic values of 15-20 mb are reached. Also the angular distributions are affected by the presence of the medium. E.g. the ini- tially strongly forward-backward peaked np cross sections become much more isotropic at finite densities, mainly due to the Pauli suppression of intermedi- ate soft modes (π-exchange) [17]. As a consequence a larger transverse energy transfer can be expected. The case of the inelastic NN cross sections is similar, but more complicated. The presence of the medium influences not only the matrix elements, but also the threshold energy Etr, which is an important quantity at beam energies be- low or near the threshold of particle production. In free space it is calculated from the invariant quantity s = (p 1 + p 2 )(p1µ + p2µ) with p i , (i = 1, 2) the 4-momenta of the two particles in the ingoing collision channel, e.g. NN −→ N∆. This quantity is conserved in binary collisions in free space, from which one determines the modulus of the momenta of the particles in the outgoing channel. The threshold condition reads Etr ≡ s ≥ M1 +M2. Cross sections in free space are usually parametrized in terms of s or the corresponding mo- mentum in the laboratory system plab within the One-Boson-Exchange (OBE) model, see e.g. [34] for details. At finite density, however, particles carry kinetic momenta and effective masses and obey a dispersion relation p∗µp ∗µ = m∗2 modified with respect to the free case. These in-medium effects shift the threshold energy in the free space according to s∗ = (p 2 )(p 2µ) and the threshold condition for inelastic processes inside the medium reads now E∗ s∗ ≥ m∗ . The requirement of energy-momentum conservation can be carried out in terms of the quantity s∗ or s, only as long as the in-medium mean fields or the corresponding self energies do not change between ingoing and outgoing channels. The application of free parametrizations of cross sections for inelastic processes in dynamical situations of HICs at finite density leads thus to an inconsistency, since the threshold condition is performed in terms of effective quantities, but the matrix elements are carried out in free space, e.g. by fitting their parameters to free empirical NN scattering. This effect can be seen in Fig. 2 (left panel) where the free inelastic NN −→ N∆ cross section σinel as a function of the laboratory energy Elab is displayed, at various baryon densities ρB. The threshold energy in the free space is Etr = s = 2.014 GeV (for M = 0.939 GeV and Mmin = 1.076 for the nucleon and the lower limit mass of the ∆ resonance). The corresponding threshold value of the laboratory energy Elab = (E − 4M2)/2M is 0.32 GeV. However, at finite density the threshold is shifted towards lower energies, i.e. the free cross section increases, due to the reduction of the free masses of the outgoing particles in the threshold condition E∗ . Obviously at higher energies far from threshold the free cross section does not depend on the density. A more consistent approach is the determination of the inelastic cross section under the consideration of in-medium effects, i.e. the Pauli-blocking of inter- mediate scattering states and in-medium modified spinors in the determina- tion of the matrix elements within the OBE model. A simultaneous treatment of the transport equation and the structure equations of DBHF for actual anisotropic momentum configurations is not possible, due to its high com- plexity. For this reason we have applied the same method as for the case 0 0.5 1 1.5 2 (GeV) 0 0.5 1 1.5 2 (GeV) =0.5ρ0 free effective Fig. 2. Inelastic NN −→ N∆ cross section σinel at various baryon densities ρB (in units of the saturation density ρ0 = 0.16 fm −3) as a function of the laboratory energy Elab using the free parametrizations (left) and the in-medium modified ones from DBHF [18] (right). of elastic binary processes, i.e. in-medium parametrizations of the inelastic cross sections of the type NN −→ N∆ within the same underlying DBHF approach as already used for the elastic processes. Haar and Malfliet [18] investigated this topic for infinite nuclear matter with the result of a strong in-medium modification of the inelastic cross sections due to the reasons given above. However, these studies were performed at various densities but only in a limited region of momenta. For a practical application in HICs we have thus extended these DBHF calculations using an extrapolation technique. We have imposed an exponential decay law of the form ae−bplab on the values of the in-medium cross sections of the channel NN → N∆ given in ref. [18]. The parameter a normalizes to the last value of the extrapolated cross sec- tion and b is defined by fitting the slope of the free cross section, since it does not change with density. For the density dependence we have enforced a correction of the form f(ρB) = 1 + a0(ρB/ρ0) + a1(ρB/ρ0) 2 + a2(ρB/ρ0) where a0 = −0.601, a1 = 0223, a2 = −0.0035, with ρ0 saturation density, are extracted from the results of ref.[18]. The same modification is imposed on the cross sections of all the inelastic channels, in a form of the type σeff = σfree(Elab)f(ρB), with σfree taken from the standard free parametriza- tions of Ref. [34]. Such a procedure is well appropriate at low energies but at higher momenta can be less accurate. This, however, should not be a problem at the reaction energies below the kaon production threshold considered in this work. Fig. 2 (right panel) shows the energy dependence of the inelastic NN cross section at various densities as obtained from DBHF calculations [17] for sym- metric nuclear matter. As in the case of elastic processes (see Fig. 1), the inelastic one drops with increasing baryon density ρB mainly due to the Pauli blocking of intermediate scattering states and the in-medium modification of the effective Dirac mass [17]. There are also phenomenological studies [16,33] which give similar medium effects on the inelastic cross sections, within the limitation to isospin symmetric nuclear matter. More suitable results would come from a DBHF approach to isospin asymmetric nuclear matter. Only re- cently such studies have been started [35], however, limiting to low momenta regions, below the threshold energy of inelastic channels. 3 Kaon Potentials Before starting with the presentation of the results, it is important to analyse the in-medium kaon potential, since it could be relevant when theoretical re- sults will be compared with experiments. In fact it has been widely discussed whether the kaon potential plays a crucial role in describing kaon production and their dynamics [23,7,9]. Kaplan and Nelson [20] found that the explicit chi- ral symmetry breaking is not so small forK mesons and this leads to significant corrections to the free kaon mass at finite baryon density. There are different models for the description of kaon properties in the nuclear medium. Here we will briefly discuss two main approaches, one based on Chiral Perturbation Theory (ChPT ) and a second on effective meson couplings (OBE/RMF ), more consistent with the general frame of our covariant reaction dynamics. The results are in good agreement and this is not surprising on the basis of a simple physics argument. It is well established [7] that kaons (K0,+) feel a weak repulsive potential in nuclear matter, of the order of 20 − 30 MeV at normal density. This can be described as the net result of the cancellation of an attractive scalar and a repulsive vector interaction terms. Such a mechanism can be reproduced in the ChPT approach through the competition between an attractive scalar Kaplan-Nelson term [20] and a repulsive vector Weinberg- Tomozawa [36] term. The same effect can be obtained in an effective meson field scheme just via a coupling to the attractive σ-scalar and to the repulsive ω-vector fields. In this paper antikaons K− and their strong attractive potential will be not discussed, since for the higher threshold they have been not considered in the energy range of interest here. Finally, for studies aimed to the determination of the symmetry energy from strangeness production one has to treat with particular care the isospin de- pendence of the kaon mean field potential. 3.1 Chiral Perturbative Results Starting from an effective chiral Lagrangian for the K mesons one obtains a density and isospin dependence for the effective kaon (K0,+) masses [7]. In isospin asymmetric matter we finally get m∗K = m2K − ρs3 + VµV µ (upper sign, K +), (3) where ρs, ρs3 are total and isospin scalar densities, with mK = 494MeV the free kaon mass, fπ = 93MeV the pion decay constant, and ΣKN the kaon- nucleon sigma term (attractive scalar), here chosen as 450 MeV. The vector potential is given by: 8f ∗2π 8f ∗2π jµ3 (upper sign, K +), (4) with jµ, jµ3 baryon and isospin currents. The f π is an in-medium reduced pion decay constant. It is expected to scale with density in a way similar to the chiral condensate [37]. This leads to a reduction around normal density f ∗2π ≃ 0.6f 2π . Such a reduction is compensated in one-loop ChPT by other contributions in the scalar attractive term so we will use f ∗π only for the vector potential, with an enhanced repulsive effect [7]. The constant C has been fixed from the Gell-Mann-Okubo mass formula (i.e. in free space) to a value of 33.5MeV [22]. In Eqs. (3-4) upper signs hold for K+ and lower signs for K0. As can be seen, the vector term, which dominates over the scalar one at high density, is more repulsive for K0 than for K+. This leads to a higher (lower) K0 (K+) kaon in-medium energy given by the dispersion relation EK(k) = k0 = k2 +m∗2K + V0 (5) The density dependence, evaluated in the chiral approach, of the quantity EK(k)k=0 = m K + V0 for K 0,+, that directly influences the in-medium pro- duction thresholds is shown by the upper curves in Fig. 3 (left panel). In particular, it can be noted that that K0 and K+ in medium-energy differs by ≈ 5% at ρB = 2ρ0 (with EK0 > EK+), at a fixed isospin asymmetry around 0.2. Therefore, the inclusion of isovector terms favors K+ over K0 production, with a consequent reduction of the K0/K+ strangeness ratio. 0 1 2 3 0 1 2 3 0 - E Fig. 3. Density dependence (ρ0 is the saturation density) of in medium kaon energy (left panel) in unit of the free kaon mass (mK = 0.494GeV ). Upper curves refer to ChPT model calculations: the central line corresponds to symmetric matter, the other two give the isospin effect (up K0, down K+). Bottom curves are obtained in the OBE/RMF approach, the solid one is for symmetric matter. The isospin splitting is given by the dashed (NLρ) and dotted (NLρδ) lines, again up K0, down K+. Right panel: relative weight of the isospin splitting, see text. All the curves are obtained considering an asymmetry parameter α = 0.2. 3.2 Relativistic Mean Field Results Kaon potentials can be also derived within an effective meson field OBE ap- proach, fully consistent with the RMF transport scheme used to simulate the reaction dynamics, see Eq.(1). We will use a simple constituent quark-counting prescription to relate the kaon-meson couplings to the nucleon-meson cou- plings, i.e. just a factor 3 reduction. Following the chiral argument discussed before, only for the scalar vector case we have further increased the kaon coupling to gωK ≃ 1.4/3gωN . This will ensure the required repulsion around normal densities for K+s. Consistently the isospin dependence will be directly derived from the coupling between the kaon fields and the ρ and δ isovector mesons [22]. The in-medium energy carried by kaons will have the same form as in Eq.(5) but with effective masses and vector potentials given by m∗K = m2K −mK(gσKσ ± gδKgδN m2K −mK(gσKσ ± fδρs3) (6) gωKgωN gρKgρN ∗ρB ± fρρB3) (7) where upper signs are forK+s. The fi ≡ g2iN/m2i , i = σ, ω, ρ, δ are the nucleon- meson coupling constants used in our RMF Lagrangians and f ∗ω = 1.4fω due to the enhanced kaon-scalar/vector coupling. σ represents the solution of the non linear equation for the scalar/isoscalar field which gives the reduction of the nucleon mass in symmetric matter, therefore we can directly evaluate the kaon-σ coupling using gσKσ = (M −M∗) where M∗ is the nucleon effective mass at the fixed baryon density. In this RMF approach we can derive an almost analytical expression for the isospin effects on the kaon in-medium energy Eq.(5) at k = 0. Using the approximate form ρs ≃ M∗/E∗FρB for the scalar density, we get a rel- ative weight of the isospin splitting of the kaon potentials ∆EK(k)k=0 ≡ EK0(k)k=0 −EK+(k)k=0 given by 2α(fρ − M f ∗ω + (mK − 16(M −M∗)) with α ≡ ρB3/ρB the asymmetry parameter. We can now easily estimate the isospin splitting of K0 vs. K+ for the two isovector mean field Lagrangians used here, NLρ and NLρδ. The effect will be clearly larger when the δ coupling is included since we have to increase the ρ-coupling fρ, see [14,15], but still the expected weight is relatively small, going from about 1.5% (NLρ) to about 3.0% (NLρδ) at ρB = 2ρ0, for a fixed isospin asymmetry around 0.2. The complete results are also shown in Fig. 3 (right panel). The agreement with the ChPT estimations is rather good, but in the RMF scheme we see an overall reduced repulsion and a smaller isospin splitting. Both effects are of interest for our discussion, the first affecting the K0,+ absolute yields, the second important for the K0/K+ yield ratios. 4 Numerical realization and notations The Vlasov term of the RBUU equation (1) is treated within the Relativistic Landau-Vlasov method, in which the phase space distribution function f(x, p∗) is represented by covariant Gaussians in coordinate and momentum space [38]. For the nuclear mean field or the corresponding EoS in symmetric matter the fσ (fm 2) fω (fm 2) fρ (fm 2) fδ (fm 2) A (fm−1) B NLρ 9.3 3.6 1.22 0.0 0.015 -0.004 NLρδ 9.3 3.6 3.4 2.4 0.015 -0.004 Table 1 Coupling parameters in terms of fi ≡ ( gimi ) 2 for i = σ, ω, ρ, δ, A ≡ a and B ≡ b for the non-linear NL models [14] using the ρ (NLρ) and both, the ρ and δ mesons (NLρδ) for the description of the isovector mean field. NL2 parametrization [26] of the non-linear Walecka model [39] is adopted with a compression modulus of 200 MeV and a Dirac effective mass of m∗ = 0.82 M (M is the bare nucleon mass) at saturation. The momentum dependence enters via the relativistic treatment in terms of the vector component of the baryon self energy. The isovector components in the mean fields are introduced in the NLρ,NLρδ Lagrangians as in the recent Refs. [14,15]. In Table 1 we report all the coupling constants and the coefficients of the non-linear σ-terms. The collision integral is treated within the standard parallel ensemble algo- rithm imposing energy-momentum conservation. For the elastic NN cross sec- tions the DBHF calculations of Ref. [17] have been used throughout this work. At intermediate relativistic energies up to the threshold of kaon (K0,+) pro- duction, i.e. Elab = 1.56 GeV, the major inelastic channels are (B, Y,K stand for a baryon (nucleons N or a ∆-resonance), hyperon and kaon, respectively) – NN ←→ N∆ (∆-production and absorption) – ∆←→ πN (π-production and absorption) – BB −→ BYK, Bπ −→ Y K (K-production from BB and Bπ-channels) The produced resonances propagate in the same mean field as the nucleons, and their decay is characterized by the energy dependent lifetime Γ which is taken from Ref. [34]. The produced pions propagate under the influence of the Coulomb interaction with the charged hadrons. Kaon production is treated hereby perturbatively due to the low cross sections, taken from Refs. [40]. Kaons undergo elastic scattering and their phase space trajectories are deter- mined by relativistic equations of motion, if the kaon potential is accounted In the next section the results of transport calculations in terms of pion and kaon yields and their rapidity distributions will be presented. The following cases for the inelastic NN cross sections σinel and the kaon potential ΣK (scalar and vector) will be particularly discussed: – free σinel, without ΣK (w/o K-pot σfree) – free σinel, with ΣK (w K-pot σfree) – free σinel, with isospin dependent ΣK (w ID K-pot σfree) – effective σinel, without ΣK (w/o K-pot σeff ) – effective σinel, with ΣK (w K-pot σeff ) – effective σinel, with isospin dependent ΣK (w ID K-pot σeff ) For pions only the different cases of σinel will be labelled, since they do not ex- perience any potential, apart coulomb. One should note that in all calculations only inelastic processes including the lowest mass resonance ∆(1232MeV ) have been considered, without accounting for the N∗(1440) resonance. This will have not appreciable consequences for pions yields, but it slightly reduces the kaon multiplicities. 5 Results As mentioned in the introduction, the main topic of the present work is to study the sensitivity of particle ratios to physical parameters such as in- medium effects of cross sections and the isospin dependence of the kaon po- tential. This is an important issue to clarify since there is some evidence sug- gesting the yield ratios as good observables in determining the high density behavior of the symmetry energy. In a near future these data will be exper- imentally accessible with the help of reactions with radioactive ion beams. However, a comparison of absolute values with experimental data, although it is not the aim of this work, is essential and it has to be included in order to show the consistency of our approach. Thus we will start the presentation of the results first in terms of absolute yields, and comparison with data, before passing to the main section on the particle ratios. Most calculations refer to central 197Au+197 Au collisions at 1 AGeV . 5.1 Effects of in-medium inelastic NN cross sections on particle yields 5.1.1 Resonance and Pion Production Here we study the role of the density dependence of the effective inelastic NN cross sections on particle yields (pions and kaons). We start with the temporal evolution of the ∆ resonances and the produced pions, as shown in Fig. 4. The maximum of the multiplicity of produced ∆-resonances occurs around 15 fm/c which corresponds to the time of maximum compression. Due to their finite lifetimes these resonances decay into pions (and nucleons) as ∆ −→ πN . Some of these pions are re-absorbed in the inverse process, i.e. πN −→ ∆ but chemical equilibrium is never reached, as pointed out in [15]. This 0 10 20 30 40 50 60 time (fm/c) 0 10 20 30 40 50 60 time (fm/c) Fig. 4. Time evolution of the ∆-resonances (left panel) and total pion yield (right panel) for a central (b = 0 fm) Au+Au reaction at 1 AGeV incident energy. Cal- culations with free (solid lines) and effective (DBHF, dashed lines) σinel are shown. mechanism continues until all resonances have decayed leading to a saturation of the pion yield for times t ≥ 50 fm/c (the so-called freeze-out time). The resonance production takes place during the high density phase, where the in-medium effects of the effective cross sections are expected to dominate. In fact, the transport results with the in-medium modified σinel reduce the multiplicity of inelastic processes, and thus the yields of ∆ resonances and pions. However, the in-medium effect is not so pronounced here with respect to similar phenomenological studies of Ref. [16,33], which should come from the moderate density dependence of the effective cross sections, see also again Fig. 2. Fig. 5 shows the centrality dependence of the charged pion yields for Au+Au collisions at 1.0 AGeV incident energy. The degree of centrality is characterized by the observable Apart, which gives the number of participant nucleons and can be calculated within a geometrical picture using smooth density profiles for the nucleus [41]. Obviously Apart increases with decreasing impact parameter b and its value approaches the total mass number of the two colliding nuclei in the limiting case of b = 0 fm. As can be seen in Fig. 5, the charged pion yields are enhanced with increasing Apart, particularly in a non-linear Apart- dependence. As pointed out in [41], the charged pion multiplicities show a similar non-linear increase also in the data. However, by directly comparing the theoretical charged pion yields with the experiments [41] we observe that our calculations overpredict the data, even when the in-medium reductions in σinel are accounted for. This discrepancy is a general feature of the transport models and may lie on the role of the rescattering processes that take place in the spectator region, 0.0 100.0 200.0 300.0 400.0 0 0 5 5 10 10 15 15 20 20 (FOPI) (FOPI) E=1.0 AGeV Au+Au Fig. 5. Centrality dependence (in terms of Apart) of the negative (π −) and positive (π+) charged pions for Au+Au collisions at 1 AGeV incident energy. Calculations with free (solid lines, filled circles) and effective (dashed lines, filled squares) cross sections are shown as indicated. Experimental data, taken from FOPI collaboration [41], are also displayed for comparison. where nuclear surface effects can play a crucial role. In order to check this point we have performed a selection on pions produced at central rapidity, where data are also available [41]. In Fig. 6 we present the inclusive (all centralities) pion rapidity distributions vs. the FOPI data for charged pions. We see that the agreement is rather good at mid-rapidity while we see a definite overcounting in the spectator sources. Such a good evaluation of the pion production ad mid-rapidity is confirmed by the results shown in Fig. 7, where we present the inclusive (all centralities) pion transverse spectrum at midrapidity (−0.2 < y0 < 0.2). We first note that this is also not much affected by the inclusion of the in-medium inelastic cross sections. Moreover we see again that our results are in good agreement with the experimental values from the FOPI collaboration [41], in the same rapidity selection. The overestimation of the pion yields shown in Fig. 5 probably results from other rapidity regions where the role of the spectator sources is more evident. We have also to say that we are not imposing any experimental filter to our results. The point is rather delicate since the main discrepancies appear in high rapidity regions. In any case such a fine agreement at mid- -2 -1 0 1 2 -2 -1 0 1 2 Fig. 6. Inclusive (all centralities) pion rapidity distributions for a Au+Au reaction at Ebeam = 1 AGeV incident energy. Comparison with the experimental values given by FOPI collaboration [41]; as in the data we have used a transverse momentum cut to pt > 0.1GeV/c. 0 200 400 600 800 (MeV) FOPI FOPIπ 0.1*π Fig. 7. Inclusive transverse spectrum at midrapidity of π−, π+ for a Au+Au reaction at Ebeam = 1 AGeV incident energy. Comparison with the experimental values given by FOPI collaboration [41]. The cross sections are normalized to a rapidity interval dy = 1. rapidity is very important for the reliability of our results on kaon production, mostly produced in that rapidity range via secondary πN,∆N channels, see [15]. The pion reaction dynamics is furthermore not sensitively affected by the in-medium inelastic cross sections. We restrict here the analysis to central Au+Au collisions at 1 AGeV. In Fig. 8 we show cross section effects on the -2 -1 0 1 2 -2 -1 0 1 2 Fig. 8. Rapidity distributions of negative and positive charged pions (left and right panels, respectively) for a central (b = 0 fm) Au+Au reaction at Ebeam = 1 AGeV incident energy. rapidity distributions (normalized to the projectile rapidity in the cm sys- tem) for π±, an observable which characterizes the degree of stopping or the transparency of the colliding system. This is due to the fact that the global dynamics is mainly governed by the total NN cross sections, in which its elas- tic contribution is the same for all the cases. In previous studies [19,31] the in-medium effects of the elastic NN cross sections gave important contribu- tions to the degree of transparency or stopping. It was found that a reduction of the effective NN cross section particularly at high densities is essential in describing the experimental data [19], as confirmed by various other analy- ses [31]. The density effects on the inelastic NN cross section influence only those nucleons associated with resonance production, and therefore they do not affect the global baryon dynamics significantly. 5.1.2 Kaon Production The situation is different for kaon production, see Fig. 9. The influence of the in-medium dependence of σinel is important, and reduces the kaon abundancies by a factor of ≈ 30%. This is due to the fact that the leading channels for kaon production are N∆ −→ BY K and Nπ −→ ΛK. Thus kaon production is essentially a twostep process and the medium-modified inelastic cross sections enter twice, leading to an increased sensitivity. Fig. 10 shows the rapidity distributions of kaons, where the in-medium effect is more visible with respect to the corresponding pion rapidity distributions (see Fig. 8). These results seem to show that kaon production could be used to determine the in-medium dependence of the NN cross section for inelas- 0 10 20 30 40 50 60 time (fm/c) 0 10 20 30 40 50 60 time (fm/c) w/o K-pot σ w/o K-pot σ Fig. 9. Time evolution of the K0 (left panel) and K+ (right panel) multiplicities, for the same reaction and models as in Fig. 4, with free and in-medium inelastic cross sections, without the inclusion of the kaon potentials. -2 -1 0 1 2 w/o K-pot σ w/o K-pot σ -2 -1 0 1 2 Fig. 10. Same as in Fig. 9, but for the normalized rapidity distributions. tic processes. Similar phenomenological studies based on the BUU approach [16,33] strongly support in-medium modifications of the free cross sections. It is of great interest to perform an extensive comparison with experimental data on kaon production, in order to have a more clear image of the effect of the in-medium cross sections on their production. The point is that kaon absolute yields are also largely affected by the kaon potentials, see the following, as expected from the general discussion of the previous section. However since kaons are mainly produced in more uniform high density regions the effects of the medium on cross sections tend to disappear in the yield ratios. In the next section we will show that the same holds true for the K0, K+ potentials. Our conclusion is that the kaon yield ratios might finally be a rather robust 0 10 20 30 40 50 60 time (fm/c) 0 10 20 30 40 50 60 time (fm/c) w/o K-pot w K-pot w ID K-pot Fig. 11. Time evolution of the K0 (left panel) and K+ (right panel) multiplicities for the same reaction as in Fig. 4. Calculations without (w/o K-pot, solid), with (w K-pot, dashed) and with the isospin dependent (w ID K-pot,dotted-dashed) kaon potential are shown. In all the cases the free choice for σinel is adopted. observable to probe the nuclear EoS at high baryon densities. 5.2 The role of the kaon potential As discussed in the previous sections, the important quantity which influences the kaon production threshold is the in-medium energy at zero momentum [7]. This quantity rises with increasing baryon density and in the general case of isospin asymmetric matter shows a splitting between K0 and K+, see Fig. 3. We are presenting here several K-production results in ab initio collision sim- ulations using the Chiral determination of the K-potentials, ChPT . Fig. 11 shows the time dependence of the two isospin states of the kaon with respect to the role of the kaon potential and its isospin dependence. First of all, the repulsive kaon potential considerably reduces the kaon yields, at least in this ChPT evaluation. The inclusion of the isospin dependent part of the kaon potential slightly modifies the kaon yields, towards a larger K+ production in neutron-excess matter. However by comparing to the corresponding isospin dependence of the in-medium kaon energy, see Fig. 3, the effect is less pronounced in the dynamical situation. This is due to the fact that in heavy ion collisions the local asymmetry in the interacting region varies with time, see [15]. In particular, it decreases with respect to the initial asymmetry because of partial isospin equilibration due to stopping and inelastic processes with associated isospin exchange. This is reflected also in the kaon rapidity distributions, see Fig. 12, -2 -1 0 1 2 -2 -1 0 1 2 w/o K-pot w K-pot w ID K-pot Fig. 12. Same as in Fig. 11, but for the rapidity distributions. -1 0 1 2 normalized rapidity y -1 0 1 2 normalized rapidity y -1 0 1 2 normalized rapidity y w/o K-pot w K-pot w ID K-pot Fig. 13. K+ rapidity distributions for semi-central(b < 4 fm) Ni+Ni reactions at 1.93 AGeV. Theoretical calculations (as indicated) are compared with the exper- imental data of FOPI (open triangles) and KaoS (open diamonds) collaborations [42,43]. where the role of the kaon potential is crucial, but not its isospin dependence. As we have already seen even in-medium modifications of inelastic cross sec- tions are affecting the kaon absolute yields, so it appears of interest to look at the combined effects. For that purpose we have performed calculations for a semi-central (b < 4fm) Ni+Ni system at 1.93AGeV , where data are existing from the FOPI [42] and KaoS [43] collaborations. The results for K+ rapidity distributions, compared to experimental data, are shown in Fig.13. We observe that although the kaon yields are reduced when using the in- medium inelastic cross section, we are still rather far away from the data, left panel of Fig. 13. We note that the reduction due to the density dependence of the effective inelastic cross sections is rather moderate here with respect to that of the heavier Au-system (see Fig. 10). for kaons). This is due to the less compression achieved for the lighter Ni-systems. The inclusion of the kaon potential, without (central panel) and with (right panel) isospin dependence, is further suppressing the K+ yield, towards a better agreement with data, as expected for the repulsive behavior at high density. In fact the results obtained with kaon potentials and effective cross sections seem to underestimate the data. This could be an indication that the ChPT K-potentials are too repulsive at densities around 2ρ0 where kaons are pro- duced, see [15]. We like to remind that the parameters of ChPT potentials are essentially derived from free space considerations. When we follow a more consistent RMF approach, directly linked to the effective Lagrangians used to describe bulk properties of the nuclear matter as well as the relativistic trans- port dynamics, we see less repulsion, bottom curves in Fig. 3 (left panel). This is valid also for the isospin dependent part of the K-potentials, that more di- rectly will affect the K0/K+ yield ratio. We see from the same Fig. 3 (right panel) that in the RMF frame this splitting is reduced to a few percent for all the different isovector interactions. The conclusion is that when kaon po- tentials are evaluated within a consistent effective field approach we have a better agreement with data for absolute yields, with a very similar reduction of the K0 and K+ rates. This is important for the yield ratio, that then should be not much sensitive to the in-medium effects on kaon propagation. A similar conclusion on K-potential effects, obtained within the ChPT ap- proach, can be drawn from the centrality dependence of the K+ yields shown in Fig. 14 in the case of Au+Au collisions at 1 AGeV beam energy and com- pared with KaoS data [44]. The trend in centrality can be reproduced by all theoretical calculations (with different cross sections), however, all of them seem to underestimate the experimental yields. In fact we have to mention that another possible source of the discrepancy with data can be that in all our simulations only the lowest mass resonance ∆(1232MeV ) has been dynamically included. Transport calculations from other groups, that take care also of the N∗(1440MeV ) resonance, are getting an enhancement of the K+ yield for Au+Au collisions at 1 AGeV incident energy [7]. This significant dependence of the kaon yields on the N∗ resonance comes from the 2-pionic N∗-decay channel, i.e. N∗ −→ ππN . Therefore, since the most important channels of kaon production are the pionic ones, we can expect some underestimation of the absolute yields in our calculations. Just to confirm this point, in Fig. 14 we report also transport results from the Tübingen group, in which all resonances are accounted for [7]. We finally re- 0,0 0,2 0,4 0,6 0,8 5,0×10 1,0×10 1,5×10 2,0×10 0,0 0,2 0,4 0,6 0,8 1,0 w K-pot w ID K-pot Fig. 14. K+ centrality dependence in Au+Au reactions at 1 AGeV incident energy. Our theoretical calculations (as indicated) are compared with KaoS data from [44] (open diamonds) and with results of the Tübingen group (open squares). mark that the inclusion of other nucleon resonances in neutron-rich matter will further contribute to increase the K0 yield through a larger intermedi- ate π− production. This can contribute to compensate the opposite effect of isospin dependent part of the K-potentials on the K0/K+ yield ratios. 5.3 Pionic and Strangeness Ratios A crucial question is whether particle yield ratios are influenced by in-medium effects both on inelastic cross section and kaon potentials. This point is of ma- jor importance particularly for kaons, since ratios of particles with strangeness have been widely used in determining the nuclear EoS at supra-normal density. Relative ratios of kaons between different colliding systems have been utilized in determining the isoscalar sector of the nuclear EoS [5]. More recently, the (π−/π+)- and (K0/K+)-ratios have been proposed in order to explore the high density behavior of the symmetry energy, i.e. the isovector part of the nuclear mean field [14,15,13,12]. Fig. 15 shows the incident energy dependence of the pionic (π−/π+, left panel) and strangeness (K0/K+, right panel) ratios for the different choices of in- elastic cross sections and kaon potentials, as widely discussed in the previous sections. First of all, a rapid decrease of the pionic ratio with increasing beam energy 0.8 1 1.2 1.4 1.6 1.8 2 (AGeV) 0.8 1 1.2 1.4 1.6 1.8 2 (AGeV) free w/o K-pot eff w/o K-pot 0.8 1 1.2 1.4 1.6 1.8 2 (AGeV) free w K-pot eff w K-pot 0.8 1 1.2 1.4 1.6 1.8 2 (AGeV) tio σ free w ID K-pot eff w ID K-pot Fig. 15. Energy dependence of the π−/π+ (left panel) and K0/K+ (right panel) ratios for central (b = 0 fm) Au+Au reactions. is observed, related to the opening of secondary rescattering processes (reab- sorption/recreation of pions with associated isospin exchange) channels. The corresponding strangeness ratio depends only moderately on beam energy due to the absence of secondary interactions with the hadronic environment. The pionic ratio is partially affected by the in-medium effects of σinel, as it can be seen in the left panel of Fig.15. Its slope is slightly changing with respect to beam energy. The situation is similar for the strangeness ratio, which actually appears even more robust vs. in-medium modifications, even with the kaon potentials. This can be seen in the right panel of Fig.15, where for all the considered beam energies the ratio remains almost unchanged. Such a result is consistent with those of the previous sections, where it was found that the absolute kaon yields decrease in the same way when the effective σinel are applied and when the K-potentials are included. The different sensitivity to variations in the inelastic cross sections of pionic vs. strangeness ratios can be easily understood. For the large rescattering and lower masses pions can be produced at different times during the collision, in different density regions. At variance kaons are mainly produced at early times in a rather well definite compression stage, i.e. in a source with a more uniform high density, and so the density dependence of the inelastic cross sections will affect in the same way neutral and charged kaon yields, leaving the ratio unchanged. At this level of investigation one could argue that the strangeness ratio is a very promising observable in determining the nuclear EoS and particularly its isospin dependent part. This has been also the main conclusion in Ref. [15]. However, a strong isospin dependence of the kaon potentials could directly NL NLρ NLρδ K0/K+ (w/o K-pot) 1.24 (± 0.02) 1.35 (± 0.01) 1.43 (± 0.02) K0/K+ (w ID K-pot) 1.02 (± 0.03) 1.22 (± 0.04) 1.34 (± 0.05) Table 2 Sensitivity of the strangeness ratio K0/K+ to the isospin dependent kaon potential and to the isovector mean field (NL, no isovector fields, NLρ and NLρδ). The considered reaction is a central (b = 0 fm) Au+Au collision at 1 AGeV incident energy. affect the ratio, since the K0 and K+ rates will be modified in opposite ways. This is shown by the two triangle points at 1 AGeV in the right panel of Fig. 15. As already discussed this large isospin dependence of the kaon potential, clearly present in the ChPT evaluation, is greatly reduced in a consistent mean field approach, see Fig. 3 and the arguments presented in Section 3. In any case this point deserves more detailed studies. We plan to perform ab initio kaon-production simulations within the OBE/RMF evaluation of kaon potentials, with an isospin part fully consistent with the isovector fields of the Hadronic Lagrangians used for the reaction dynamics. An interesting final comment is that the sensitivity of the strangeness ratio to the isovector part of the nuclear EoS remains even when strong isospin dependence of the kaon potentials is inserted, as in the ChPT case. In order to check this, we have repeated for Au+Au at 1 AGeV incident energy the calculations by varying the isovector part of the nuclear mean field. As in Refs. [14,15], three options for the isovector mean field have been applied: the NL (no isovector fields), NLρ and NLρδ parametrizations, but now in- cluding the isospin effect in the kaon potential in the ChPT evaluation. The options of the symmetry energy differ from each other in the high density stiff- ness. NL gives a relatively soft Esym, NLρδ a relatively stiff one, and NLρ lies in the middle between the other limiting cases [14]. Table 2 shows the strangeness ratio as function of these different cases for the isovector mean field, keeping now constant the other parameters (free σinel, isospin dependent kaon potential). The ratio, indeed, strongly decreases when the isospin part in the kaon potential is accounted for. The more interesting result is, however, that the relative difference between the different choices of the symmetry en- ergy remains stable. This can be understood from the fact that in the kaon self energies the isospin sector contains only the isospin densities and currents without additional parameters such meson-nucleon coupling constants. Since the local asymmetry does not strongly vary from one case to the other (NL, NLρ, NLρδ), one would expect a robustness of the EoS dependence. Thus we conclude that the strangeness ratio appears to be well suited in determin- ing the isovector EoS, however, a fully consistent mean field approach is still missing. 6 Conclusions We have investigated the role of the in-medium modifications of the inelastic cross section and of the kaon mean field potentials on particle production in intermediate energy heavy ion collisions within a covariant transport equation of a Boltzmann type. We have used for both, the elastic and inelastic NN cross sections the same DBHF approach which provide in a parameter free manner the in-medium modifications of the imaginary part of the self energy in nu- clear matter. The kaon potential has been evaluated in two ways, following a Chiral Perturbative approach and an Effective Field scheme, considering va- lence quark-meson couplings. We have applied these modifications of the cross sections and kaon potentials to the collision integral of the transport equation and analyzed Au+Au and Ni+Ni collisions at intermediate relativistic energies around the kaon threshold energy. Our studies have shown a good sensitivity of the particle multiplicities and rapidity distributions of pions and kaons. In particular, a moderate reduction for pions has been seen when the in-medium effects in the inelastic cross section are accounted for. The pion yields are still overestimating the inclusive data while we have a very nice agreement with the pion spectra and multiplicities at mid-rapidity. The latter point is important for trusting the kaon production, mainly due to secondary pion collisions at mid-rapidity. At variance the kaon (K0,+) yields show a larger sensitivity to the reduction of the inelastic cross sections, with a decrease of about 30 %. However we see that the introduction of a repulsive kaon potential is essential in order to reproduce even inclusive data. We have then focused our attention on π−/π+ and K0/K+ yield ratios, re- cently suggested as good probes of the isovector part of the EoS at high den- sities. The pionic ratios, due to their strong secondary interaction processes with the hadronic environment, show a dependence on the density behavior of the inelastic cross sections. A further selection of the production source, i.e. a transverse momentum discrimination, could be required in order to have a more reliable probe of the nuclear EoS. The situation appears more favorable for the kaon ratios. In fact we find that the multiplicities of K0 and K+ are influenced in such a way that their ratio is not affected by the density dependence of the inelastic cross sections. This is due to the long mean free path of the K0,+ that are produced only in the compression stage of the collision [15]. The effects of the in medium kaon potentials are also largely compensating in the K0/K+ yield ratio, due to the similar repulsive field seen by K0 and K+ mesons. Such a result can be modified by the isospin dependence of the kaon potentials which is expected to act in opposite directions for neutral and charged kaons rates. Actually this is a rather stimulating open problem. In our analysis with two completely different approaches, ChPT vs. RMF , we get a good agreement for the isoscalar kaon potential but a rather different prediction for the isovector part. However, a study in terms of the different choices of the isovector kaon field has shown that the relative dependence of the strangeness ratios on the stiff- ness of the isovector nuclear EoS remains a well robust observable. This is an important issue in determining the high density behavior of the symmetry energy in more systematic analyses in the future, when more experimental data will be available. Acknowledgments. This work is supported by BMBF, grant 06LM189 and the State Scholarships Foundation (I.K.Y.). It is also co-funded by European Union Social Fund and National funded Pythagoras II - EPEAEK II, under project 80861. One of the authors (V.P.) would like to thank H.H. Wolter and M. Di Toro for the warm hospitality during her short stays at their institutes. References [1] J. M. Lattimer, M.Prakash, Nucl. Phys A777, (2006) 479 and refs. therein; B. Liu, H. Guo, V. Greco, U. Lombardo, M. Di Toro and Cai-Dian Lue, Eur. Phys. J. A22, (2004) 337. [2] T.Klähn et al., Phys. Rev. C74, (2006) 035802. [3] P. Danielewicz, R. Lacey, W.G. Lynch, Science 298, (2002) 1592 [4] N. Hermann, J.P. Wessels, T. Wienold, Annu. Rev. Nucl. Part. Sci. 49, (1999) [5] C. Fuchs et al., Phys. Rev. Lett. 86, (2001) 1974. [6] A.B. Larionov, U. Mosel, Phys. Rev. C72, (2005) 014901. [7] C. Fuchs, Prog. Part. Nucl. Phys. 56, (2006) 1. [8] J. Aichelin, C.M. Ko, Phys. Rev. Lett. 55, (1985) 2661. [9] C. Hartnack, H. Oeschler, J. Aichelin, Phys. Rev. Lett. 90, (2003) 102302. [10] C. Sturm et al. (KaoS Collaboration), Phys. Rev. Lett. 86, (2001) 39. [11] A. Andronic et al. (FOPI Collaboration), Phys. Lett. B612, (2005) 173. [12] Bao-An Li, Phys.Rev.C71, (2005) 014608. [13] Qing-feng Li, Zhu-xia Li, En-guang Zhao, Raj K. Gupta, Phys. Rev. C71, (2005) 054907. [14] G. Ferini, M. Colonna, T. Gaitanos, M. Di Toro, Nucl. Phys. A762, (2005) 147-166. [15] G. Ferini, T. Gaitanos, M. Colonna, M. Di Toro, H.H. Wolter, Phys. Rev. Lett. 97, (2006) 202301.. [16] A.B. Larionov, W. Cassing, S. Leupold, U. Mosel, Nucl.Phys. A696, (2001) [17] C. Fuchs et al., Phys. Rev. C 64, (2001) 024003. [18] B. Ter Haar, R. Malfliet, Phys. Rev. C36, (1987) 1611. [19] T. Gaitanos, C. Fuchs, H.H. Wolter, Phys. Lett. B609, (2005) 241; E. Santini, T. Gaitanos, M. Colonna, M. Di Toro, Nucl. Phys. A756, (2005) T. Gaitanos, C. Fuchs, H.H. Wolter, Prog. Part. Nucl. Phys. 53, (2004) 45. [20] D.B. Kaplan, A.E. Nelson, Phys. Lett. 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0704.0555

The Graham conjecture implies the Erdos-Turan conjecture

arXiv:0704.0555v1 [math.NT] 4 Apr 2007 THE GRAHAM CONJECTURE IMPLIES THE ERDÖS-TURÁN CONJECTURE LIANGPAN LI Abstract. Erdös and Turán once conjectured that any set A ⊂ N with 1/a = ∞ should contain infinitely many progressions of arbitrary length k ≥ 3. For the two-dimensional case Graham conjectured that if B ⊂ N × N satisfies (x,y)∈B x2 + y2 then for any s ≥ 2, B contains an s × s axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some s ≥ 2, then the Erdös-Turán conjecture is true for k = 2s− 1. 1. Introduction One famous conjecture of Erdös and Turán [2] asserts that any set A ⊂ N with∑ a∈A 1/a = ∞ should contain infinitely many progressions of arbitrary length k ≥ 3. There are two important progresses towards this direction due to Szemerédi [7] and Green and Tao [5] respectively, which assert that if A has positive upper density or A is the set of the prime numbers, then A contains infinitely many progressions of arbitrary length. If one considers the similar question in the two-dimensional plane, Graham [4] conjectured that if B ⊂ N× N satisfies (x,y)∈B x2 + y2 then B contains the four vertices of an axes-parallel square. More generally, for any s ≥ 2 it should be true that B contains an s× s axes-parallel grid. Furstenberg and Katznelson [3] proved the two-dimensional Szemerédi theorem, that is, any set B ⊂ N × N with positive upper density contains an s × s axes-parallel grid. In another words, such a set B contains any finite pattern. The purpose of this paper is to show that if the Graham conjecture is true, then the Erdös-Turán conjecture is also true. 2. The Graham conjecture implies the Erdös-Turán conjecture Suppose that the Erdös-Turán conjecture is false for k = 3. Then there exists a A = {a1 < a2 < a3 < · · · } ⊂ N Date: April 4, 2007. 2000 Mathematics Subject Classification. 11B25. http://arxiv.org/abs/0704.0555v1 2 LIANGPAN LI n∈N 1/an = ∞ such that A contains no arithmetic progression of length 3. Define a set B ⊂ N× N by (an +m,m) : n ∈ N,m ∈ N (x,y)∈B x2 + y2 (an +m)2 +m2 (an +m)2 +m2 (an + an)2 + a2n In the sequel we indicate that B contains no square and argue it by contradiction. This would mean that the Graham conjecture is false for s = 2. Suppose that for some n,m, l ∈ N, B contains a square of the following form: (an +m,m+ l), (an +m+ l,m+ l), (an +m,m), (an +m+ l,m). It follows easily from the construction of B that an − l, an, an + l ∈ A, which yields a contradiction since A contains no arithmetic progression of length 3 according to the initial assumption. Similarly, if the Graham conjecture is true for some s ≥ 2, then the Erdös-Turán conjecture is true for k = 2s− 1. The interested reader can easily provide a proof. 3. Concluding Remarks Let r(k,N) be the maximal cardinality of a subset A of {1, 2, . . . , N} which is free of k-term arithmetic progressions. Behrend [1] and Rankin [6] had shown that r(k,N) ≥ N · exp(−c(logN)1/(k−1)). Similarly, let r̃(s,N) be the maximal cardinality of a subset B of {1, 2, . . . , N}2 which is free of s× s axes-parallel grids. For any set A ⊂ {1, 2, . . . , N}, define Θ(A) = {(a+m,m) : a ∈ A,m = 1, 2, . . . , N} ⊂ {1, 2, . . . , 2N}2. Following the discussion in Section 2, one can easily deduce that if A is free of 2s−1 term of arithmetic progression, then Θ(A) is free of s× s axes-parallel grid. Hence r̃(s, 2N) ≥ r(2s− 1, N) ·N ≥ N2 exp(−c(logN)1/(2s−2)). We end this paper with a question. Does the Erdös-Turán conjecture imply the Graham conjecture? THE GRAHAM CONJECTURE IMPLIES THE ERDÖS-TURÁN CONJECTURE 3 References [1] F.A. Behrend, On sets of integers which contain no three terms in arithmetic progression, Proc. Nat. Aca. Sci. 32 (1946), 331–332. [2] P. Erdös and P.Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264. [3] H. Furstenberg and Y. Katznelson, An ergodic Szemeredi theorem for commuting transfor- mation, J. d’Analyse Math. 34 (1979), 275–291. [4] R. Graham, Conjecture 8.4.6 in Discrete and Computational Geometry (J.E. Goodman and J. O’Rourke, eds), CRC Press, Boca Raton, NY, p.11. [5] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, to appear in Ann. of Math. [6] R.A. Rankin, Sets of integers containing not more than a given number of terms in arithmetic progression, Proc. Roy. Soc. Edinburgh Sect A. 65 (1960/61), 332–344. [7] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 299–345. Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, Peo- ple’s Republic of China E-mail address: [email protected]

0704.0556

Effective conservation of energy and momentum algorithm using switching potentials suitable for molecular dynamics simulation of thermodynamical systems

Effective conservation of energy and momentum algorithm using switching potentials suitable for molecular dynamics simulation of thermodynamical systems Christopher G. Jesudason ∗ Laboratory of Physics and Helsinki Institute of Physics, P.O.Box 1100, FIN-02015 HUT, Finland. Email: [email protected], [email protected] 4 April, 2007 Abstract During a crossover via a switching mechanism from one 2-body poten- tial to another as might be applied in modeling (chemical) reactions in the vicinity of bond formation, energy violations would occur due to finite step size which determines the trajectory of the particles relative to the potential interactions of the unbonded state by numerical (e.g. Verlet) integration. This problem is overcome by an algorithm which preserves the coordinates of the system for each move, but corrects for energy dis- crepancies by ensuring both energy and momentum conservation in the dynamics. The algorithm is tested for a hysteresis loop reaction model with an without the implementation of the algorithm. The tests involve checking the rate of energy flow out of the MD simulation box; in the equilibrium state, no net rate of flows within experimental error should be observed. The temperature and pressure of the box should also be invariant within the range of fluctuation of these quantities. It is demon- strated that the algorithm satisfies these criteria AMS (MSC2000) Subject Classification. 00A71-2, 70H05, 80A20 1 PRELIMINARIES The dimeric particle reaction simulated may be written A2 (1.1) ∗on leave from Chemistry Department, University of Malaya, 50603 Kuala Lumpur, Malaysia. http://arxiv.org/abs/0704.0556v1 where k1 is the forward rate constant and k−1 is the backward rate constant. The reaction simulation was conducted at extremely high temperatures which are off-scale and not used in ordinary simulations of LJ (Lennard-Jones) fluids where normally [1] the reduced temperatures T ∗ ranges ∼ 0.3 − 1.2, whereas here, T ∗ ∼ 8.0 − 16.0, well above the supercritical regime of the LJ fluid At these temperatures, the normal choices for time step increments do not obtain without also taking into account energy-momentum conservation algorithms in regions where there are abrupt changes of gradient [1, 2, 3]. The global literature does not seem to cover such extreme conditions of simulation with discrete time steps using the Verlet velocity algorithm. The units used here are reduced LJ ones [1]. The simulation was at density ρ = 0.70 with 4096 atomic particls which could react. The potentials used are as given in Fig. (1) where rb = 1.20 for the vicinity where the bond of the dimer is broken and where 2 free particles emerge, and rf = 0.85 is the point along the hysteresis potential curve where the dimer is defined to exist for two previously free particles which collide. The reaction proceeds as follows; all particles interact with the splined LJ pair potential uLJ except for the dimeric pair (i, j) formed from particles i and j which interact with a harmonic-like intermolecular potential modified by a switch u(r) given u(r) = uvib(r)s(r) + uLJ [1− s(r)] (1.2) where uvib(r) is the vibrational potential given by eq.(1.3) below uvib(r) = u0 + k(r − r0) 2 (1.3) The switching function s(r) is defined as s(r) = )n (1.4) where s(r) → 1 if r < rsw s(r) → 0 for r > rsw The switching function becomes effective when the distance between the atoms approach the value rsw (see Fig. (1) ). Some of the other parameters used in the equations that follow include: u0 = −10, r0 = 1.0, k ∼ 2446 (exact value is determined by the other input pa- rameters), n = 100, rf = 0.85, rb = 1.20, and rsw = 1.11. Particles i and j above also interact with all other particles not bonded to it via uLJ . Full simulation details are given elsewhere [2]; suffice to say the activation energy at rf is ex- tremely high at approximately 17.5. At rf , the molecular potential is turned on where at this point there is actually a crossing of the potential curves although the gradients of the molecular and free uLJ potentials are ”‘very close”’. On the other hand, at rb , the switch forces the two curves to coalesce, but detailed examination shows that there is an energy gap of about the same magnitude as the cut-off point in a normal non-splined LJ potential (∼ 0.04 energy units), meaning there is no crossing of the potentials. The current algorithm is applied for both these cross-over regions with their different mechanisms of cross-over. The MD cell is rectangular, with unit distance along the axis ( x direction) of the cell length, whereas the breadth and height was both 1/16, implying a thin pencil-like system where the thermostats were placed at the ends of the MD cell, and the energy supplied per unit time step δt at both ends of the cell (orthogonal to the x axis) in the vicinity of x = 0 and x = 1 maintained at temperatures Th and Tl could be monitored, where this energy per unit step time is respectively ǫh and ǫl. At equilibrium, (when Th = Tl), the net energy supplied within statistical error (meaning 1-3 units of the standard error of the ǫ distributions ) is zero, i.e. ǫl ≈ ǫh ≈ 0. The cell is divided up uniformly into 64 rectangular regions along the x axis and its thermodynamical properties of temperature and pressure are probed. The resulting values of the ǫ’s and the relative invariance of the pressure and temperature profiles would be a measure of the accuracy of the algorithm from a thermodynamical point of view at the steady state. For systems with a large number of particles such thermodynam- ical criteria are appropriate. The synthetic thermostats now frequently used in conjunction with ”‘non-Hamiltonian”’ MD [3] cannot be employed for this type of study, where actual energy increments are sampled. The runs were for 4 million time steps, with averages taken over 100 dumps, where each variable is sampled every 20 time steps. The final averages were over the 20-100 dump values of averaged quantities. 0.8 1 1.2 1.4 1.6 1.8 r/LJ distance units Potentials for simulation model intermolecular potential s(r) switching function atomic LJ potential Figure 1: Potentials used for this work. The temperature T and pressure p are computed by the equipartition and 0 20 40 60 80 x layer number Figure 2: Temperature profile across the cell for different set conditions a−e for temperature T ∗ and step time δt pairs (T ∗, δt) where a = (8.0, 2.0 ep− 3), b = (8.0, 5.0 ep− 4), c = (8.0, 5.0 ep− 5), d = (12.0, 5.0 ep− 5), e = (16.0, 5.0 ep− 5). The curves {l1, l3, t1, t2, t3} results with the application of the algorithm at rb and rf with associated conditions l1 ⇔ a, l3 ⇔ b, t1 ⇔ c, t2 ⇔ d, t3 ⇔ e whilst the curves {l2, l4, l5, l6, l7} are for the cases without implementing the algorithm with the associated conditions l2 ⇔ a, l4 ⇔ b, l5 ⇔ c, l6 ⇔ d, l7 ⇔ e, where ep x ≡ 10x. 0 20 40 60 80 x layer number Figure 3: Pressure profile across the cell for different runs.The conditions of the runs and the labeling of the curves are exactly as in Fig. (2). Curve ǫh ǫl Mean Temperature l1 -.2274E+00 ±0.19E-02 -.2295E+00 ±0.21E-02 0.9063E+01 ±0.62E-02 l2 -.5602E+00 ±0.22E-02 -.5596E+00 ±0.22E-02 0.1032E+02 ±0.63E-02 l3 -.4161E-01 ±0.14E-02 -.4089E-01 ±0.14E-02 0.8774E+01 ±0.79E-02 l4 -.5201E-01 ±0.16E-02 -.5103E-01 ±0.17E-02 0.8980E+01 ±0.98E-02 t1 -.5312E-03 ±0.92E-03 -.3334E-03 ±0.76E-03 0.8082E+01 ±0.49E-02 l5 0.1311E-02 ±0.82E-03 0.1147E-02 ±0.84E-03 0.7731E+01 ±0.97E-02 t2 -.6823E-03 ±0.12E-02 -.1507E-02 ±0.13E-02 0.1216E+02 ±0.17E-01 l6 0.7291E-02 ±0.12E-02 0.6343E-02 ±0.14E-02 0.1088E+02 ±0.15E-01 t3 -.9348E-03 ±0.18E-02 -.3379E-02 ±0.17E-02 0.1622E+02 ±0.18E-01 l7 0.1918E-01 ±0.14E-02 0.1938E-01 ±0.16E-02 0.1329E+02 ±0.20E-01 Table 1: Table with values for the mean heat supply per unit step and temper- ature. The error is one unit of standard error for the quantities. Virial expression given respectively by pi.pi/mi = 3NkBT andP = ρkBT +W/V where W = − 1 w(rij) and the intermolecular pair Virial w(r) is given by w(r) = r dv(r) with v being the potential. 2 ALGORITHMAND AND ANALYSIS OF NU- MERICAL RESULTS The velocity Verlet algorithm [4, p. 81]used here [1] and allied types generate a trajectory at time nδt from that at (n− 1)δt with step increment δt through a mapping Tm where (v(nδt), r(nδt)) = Tm(v((n − 1)δt), r((n − 1)δt)) which does not scale linearly with δt. For a Hamiltonian H whose potential V is dependent only on position r having momentum components pi, the system without external perturbation has constant energy E, and the normal assump- tion in MD (NAMD)is that for the nth step, ∆En = |H(nδt)− E| ≤ ǫ and also i=1 ∆Ei ≤ ǫ s for the specified ǫ′s. In the simulation under NAMD, the force fields are constant and do not change for any one time step. In these cases, the energy is a constant of the motion for any time interval δtT when no external perturbations (e.g. due to thermostat interference) are impressed. When there is a crossing of potentials at such a time interval interval from φb to φa at an inter particle distance(icd ) rc (such as points rf and rb of Fig. (1)) of general particle 1 and 2 (the (1, 2) particle pair) due to a reactive process (such as oc- curs in either direction of (1.1)) a bifurcation occurs where the MD program computes the next step coordinates as for the unreacted system (potential φb), which needs to be corrected. Let the icd at time step i be ri (with φb potential) and at step i + 1 after interval δt be rf = ri+1 where rf < rc < ri. Due to this crossover, a different Hamiltonian H ′ is operative after point rc is crossed, where under NAMD, the other coordinates not undergoing crossover are not affected. For what follows, subscripts refer to the particle concerned. Let the interparticle potential at rf be Ea = Ef = φa(rf ) and at rf be Eb = φb(rf ), where ∆ = Eb − Ea. Then if rf be the final coordinate due to the φb potential and force field, two questions may be asked: (i) Can the velocities of (1,2) be scaled, so that there is no energy or momentum violation during the crossover based on the φb trajectory calculation and (ii) Can a pseudo stochastic potential be imposed from coordinates rc (at virtual time tc) to rf such that (i) above is true? For (ii) we have Theorem 2.1 A virtual potential which scales velocities to preserve momentum and energy can be constructed about region rc. proof The external work done δW on particles 1 and 2 over the time step is proportional to the distance traveled since these forces are constant and so for each of these particles i, Fext,i.∆ri = δWi where ∆ri is the distance increment during at least part of the time step from rc to rf . For the non-reacting trajec- tory over time λδt (λ ≤ 1) (virtual because it is not the correct path due to the crossover at rc), δW2 + δW1 − (φb(rf )− φb(rc)) = ∆ (K.E.) (2.1) where ∆ (K.E.) is the change of kinetic energy for the (1, 2) pair from the First Law between the end points rf , rc. Now over time interval tc to tf , for the reactive trajectory, we introduce a ”‘virtual potential”’ V vir that will lead to the same positional coordinates for the pair at the end of the time step with different velocities than for the non-reactive transition leading to the transition δW2 + δW1 − (V vir(rf )− V vir(rc)) = ∆ ′(K.E.) (2.2) where ∆ ′(K.E.) is the change of kinetic energy for the pair with V vir turned on and along this trajectory, the change of potential for V vir is equated to the change in the K.E. of the pair as given in the results of theorem (2.2) for all three orthogonal coordinates, i.e. δV vir(r) − δφb(r) = δ (K.E.x,y,z)−∆ ′(K.E.)x,y,z with momentum conservation, that is δV vir(ri) = δφa(ri) for the variation along the ri coordinate, but δφa(ri) = −δK.E. along internuclear coordinate ri whereas δV vir = −K.E. (scaled about all three axes). Continuity of potential implies φa(rf ) = V vir(rf );φa(rc) = V vir(rc);φb(rc) = V vir(rc); (2.3) Subtracting (2.1) from (2.2) and applying b.c.’s (2.6) leads to ∆ = φb(rf )− V vir(rf ) = φb(rf )− φa(rf ) = Eb − Ea (2.4) ′(K.E.)−∆ (K.E.) (2.5) The above shows that a conservative virtual potential could be said to be oper- ating in the vicinity of the transition (from tc to ta) .• Question (i) above leads to: Theorem 2.2 Relative to the velocities at any rf due to the φb potential, the rescaled velocities v ′ due to the potential difference ∆ leading to these final velocities due to the virtual potential can have a form given by ′ = (1 + α)vi + β (2.6) (where i = 1, 2) for a vector β. proof From the v velocities at rf due to φb we compute the v ′ velocities at rf due to the virtual potential. Since net change of momentum is due to the external forces only, which is invariant for the (1, 2) pair, conservation of total momentum relating v′ and v in (2.6) yields a definition of β ( summation from 1 to 2 for what follows, where the mass of particle i is mi) β = −α mivi/ mi (2.7) Defining for any vector s2 = s.s,β2 = α2Q, where (2.8) then the rescaled velocities become from (2.6) = (1 + α)2vi 2 + 2(1 + α)vi.β + β 2. (2.9) With ∆ = Eb − Ea, Energy conservation implies 2 = ∆ (2.10) The coupling of (2.9-2.10) leads, after several steps of algebra to α2m1m2 2(m1 +m2) 2 + v2 2 − 2v1.v2 (2.11) 2αm1m2 2(m1 +m2) 2 + v2 2 − 2v1.v2 Defining a = (v1 − v2) 2, q = m2m1/[2(m1 +m2)], (q > 0, a ≥ 0), then the above is equivalent to the quadratic equation α2qa+ 2qaα−∆ = 0 (2.12) and in simulations, only α is unknown and can be determined from (2.12) where real solutions exist for ∆/qa ≥ −1. • The above Inequality leads to a certain asymmetry concerning forward and backward reactions, even for reversible re- actions where the region of formation and breakdown of molecules are located in the same region with the reversal of the sign of approximate ∆. For this simulation, a reaction in either direction (formation or breakdown of dimer ) proceeds if (??) is true; if not then the trajectory follows the one for the initial trajectory without any reaction (i.e. no potential crossover). Interpretation of results. Fig. (1) shows a rapidly changing potential curve with several inflexion points used in the simulation at very high temperature (as far as I know such ranges have not been reported in the literature for non- synthetic methods) warranting smaller time steps; larger ones would introduce errors due to the rapidly changing potential and high K.E.; thus, even with the application of the algorithm between cordinates rf and rb, curves l1 and l2 have too large a δt value to achieve equilibrium - meaning flat or invariant - temperature (see Fig. (2) ) or pressure (see Fig. (3))or unit step thermostat heat supply (see Table 1)(ǫh and ǫl) profiles where for these curves, the (ǫh, ǫl) values show net heat absorption; the curve at t1 (with δt = 5.0 ep− 5 show flat profiles (within statistical fluctuations and 2 standard errors of variation) for temperature, pressure and net zero heat supply; and this choice of time step interval was found adequate for runs at much higher temperatures (T = 12 and T = 16) which was used to determine thermodynamical properties [2]. For this δt value and all others, no reasonable stationary equilibrium conditions could be obtained without the application of the algorithm (curves l2,l4,l5,l6 and l7). The algorithm is seen to be effective over a wide temperature range for this complex dimer reaction simulated under extreme values of thermodynamical variables and the results here do not vary for longer runs and greater sampling statistics (e.g. 6 or 10 million time steps). The thin, pencil-like geometry of the rectangular cell with thermostats located at the ends would highlight the energy non-conservation leading to a non-flat temperature distribution, as observed and which was used to determine the regime of validity of the algorithm. References [1] J. M. Haile,Molecular Dynamics Simulation,JohnWiley & Sons,Inc.,New York, 1992. [2] C. G. Jesudason, Model hysteresis dimer molecule. I. Equilibrium prop- erties. J. Math. Chem. JOMC, Accepted 2006. [3] D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications, Vol(1) of Computational Science Series, Aca- demic Press, San Diego, Second Ed., 2002. [4] M.P. Allen and D. J. Tildesley, Computer Simulation of Liquids,Oxford Univ. Press, Oxford, 1992 PRELIMINARIES ALGORITHM AND AND ANALYSIS OF NUMERICAL RESULTS

0704.0557

Mixed chemistry phenomenon during late stages of stellar evolution

Baltic Astronomy, vol. 16, xxx–xxx, 2007. MIXED CHEMISTRY PHENOMENON DURING LATE STAGES OF STELLAR EVOLUTION R. Szczerba, M.R. Schmidt, M. Pulecka1 1 Nicolaus Copernicus Astronomical Center, ul. Rabiańska 8, 87-100 Toruń, Poland Received 2006 October 15; revised — Abstract. We discuss phenomenon of simultaneous presence of O- and C- based material in surroundings of evolutionary advanced stars. We concentrate on silicate carbon stars and present observations that directly confirm the binary model scenario for them. We discuss also class of C-stars with OH emission detected, to which some [WR] planetary nebulae do belong. Key words: stars: Asymptotic Giant Branch, carbon stars, chemical com- position, planetary nebulae, stars: individual (V778 Cyg, IRAS 04496−6859, IRAS 06238+0904, M 2−43) 1. INTRODUCTION During Asymptotic Giant Branch (AGB) phase of evolution stars with ini- tial masses between 0.8 and 8M⊙ lose a significant amount of their initial mass by ejecting the matter into interstellar space with rates between 10−7 and 10−4 M⊙ yr −1. The chemistry in the formed circumstellar envelopes is determined by the photospheric C/O ratio and is O-based for n(O)>n(C) (usually less evolution- ary advanced stars) and C-based when carbon abundance exceeds that of oxygen (evolved stars which experienced thermal pulses and dredged-up carbon to the surface). This dichotomy is a consequence of CO molecule (very stable) formation which is so efficient that less abundant element (C or O) is mostly used. There- fore, the detection of co-existence of O-rich and C-rich material in surroundings of evolved stars was (and still is) surprising and attracts a significant attention. Hereafter, we call this phenomenon a mixed chemistry phenomenon. Already, due to the IRAS observations it was realized that there is a group of carbon stars which show typical for O-rich environment the 9.7 and 18µm amorphous silicate features (Little-Marenin 1986, Willems & de Jong 1986). The Infrared Space Observatory (ISO) observations (Yamamura et al., 2000) showed that 9.7µm feature in one of such objects (V778 Cyg) is very stable and did not change during the last 15 years (the time spanned between IRAS and ISO obser- vations). This put a very strong constraint on a model and evolutionary status of this class of objects with most likely explanation being a long-lived reservoir of O-rich material located inside or around a binary system. In this review we discuss MERLIN interferometer observations of V778 Cyg which proved existence of such reservoir (disk) around companion of C-rich star. We note that the recent Spitzer Space Telescope (SST) data showed that the first extra-galactic silicate http://arxiv.org/abs/0704.0557v1 2 R. Szczerba, M.R. Schmidt, M. Pulecka carbon star (IRAS 04496−6859, Trams et al. 1999) is in fact a normal carbon star and do not show the 9.7µm dust emission (see Speck et al. 2006). There is another group of carbon stars suspected to have mixed chemistry. Namely, carbon stars with OH maser emission. Lewis (1992) listed a group of stars with SiC emission seen in the IRAS Low Resolution Spectra (LRS) and OH maser emission detected. While most of these sources appeared to have wrong LRS clas- sification the 3 C-stars with OH maser emission remained and Chen et al. (2001) added 6 more sources to this class. However, this class of mixed chemistry sources did not attract a significant attention (except of [WR] planetary nebulae – see below), since OH emission is not well spatially resolved and this group of sources may be result of spatial coincidence between OH maser emission from interstellar medium and location of C-star. For example, Szczerba et al. (2002) presented observational evidence that IRAS 05373−0810 (C-star with OH maser emission) is a genuine carbon star and that OH maser and SiO thermal emission detected toward this star is not coming from its envelope, but from molecular clouds. Here we discuss a case of another C-star with OH maser emission (IRAS 06238+0904) toward which we have detected, using IRAM radiotelescope, the SiO thermal emis- sion coming from its envelope. Here, we present arguments that shock and Photon Dominated Region (PDR) chemistry allow to form a significant amount of SiO in C-rich environment. One of the most important achievements of the ISO mission was detection of crystalline silicates. Surprisingly, crystalline silicates were detected also in [WR] planetary nebulae, which have H-poor and C-rich central stars of WR-type1. [WR] planetary nebulae show at the same time presence of Polycyclic Aromatic Hydro- carbons (PAHs) and crystalline silicates (Waters et al. 1998, Cohen et al. 1999). Scenarios proposed to explain simultaneous presence of PAHs and crystalline sil- icates include: destruction of fossil comets orbiting the star, ejection of matter before star become C-rich, formation of stable O-rich disk or torus around com- panion or system at some point of binary evolution. Hajduk, Szczerba & Gesicki (this Proceedings) present an attempt to determine spatial location of PAHs and crystalline silicates inside the [WR] planetary nebula M 2-43, by means of the radiative transfer modelling of ISO spectrum. They concluded that crystalline silicates have to be located at significant distance from the central star to avoid their emission at about 10 µm. We note also an attempt to find precursors of [WR] planetary nebulae (C-rich stars with C- and O-rich material in their circumstellar shells) among proto-planetary nebulae (Szczerba et al. 2003). The authors have argued that formation of crystalline silicates is necessary before proto-planetary nebula phase, while post-AGB star may be still O-rich and change to C-rich one during the fatal thermal pulse. They indicated five proto-planetary nebulae as a possible precursors of [WR] planetary nebulae, including famous Red Rectangle, other C-rich source with crystalline silicates (IRAS 16279-4757), as well as three O-rich sources which show presence of crystalline silicates in their ISO spectra: AC Her, IRAS 18095+2704 and IRAS 19244+1115. In this review we will not cover such cases as: NGC 6302 – O-rich planetary nebula which show presence of crystalline silicates as well as PAHs (e.g. Kemper et al. 2002); HD 233517 – an evolved O-rich red giant with orbiting polycyclic 1Note, that Zijlstra et al. (1991) detected OH maser emission from [WR] planetary nebula IRAS 07027−7934. Therefore, at least this [WR] planetary nebula belongs also to the discussed above class of C-stars with OH maser emission. Mixed chemistry phenomenon 3 aromatic hydrocarbons (Jura et al. 2006); IRAS 09425−6040 – a carbon-rich AGB star with the highest abundance of crystalline silicates detected up to now (Molster et al. 2001); IRC +10216 – a well known C-rich AGB star with water and OH maser lines detected (Melnick et al. 2001, Ford et al. 2003); and possibly some other spectacular sources which we, not intentionally, have overlooked. 2. V778 CYG A SILICATE CARBON STAR To test the hypotheses related to the mixed chemistry phenomenon observed in silicate carbon stars, we observed water masers towards V778Cyg at high angular resolution. Details of our observations and data analysis are presented by Szczerba et al. (2006), so here we repeat only some of the most important points and findings. The observations were taken on 2001 October 12/13 under good weather con- ditions, using five telescopes of MERLIN. The longest MERLIN baseline of 217 km gave a fringe spacing of 12mas at 22.235080GHz. The bandwidth was 2MHz with 256 spectral channels per baseline providing a channel separation of 0.105km s−1. The continuum calibrator sources were observed in 16MHz band with 16 channels. The data were obtained in left and right circular polarisation and the velocities were measured with respect to the local standard of rest. We used the phase referencing method; 4min scans on V778Cyg were inter- leaved by 2min scans on the source 2021+614 (at 3.◦8 from the target) over 11.5 h. The flux density of 2021+614 at K band of 1.48 Jy was derived from observation of 4C39.25. At the epoch of observation the flux density of 3C39.25 was 7.5±0.3Jy (Terasranta 2002, private communication). This source was also used for bandpass calibration. After initial calibration with the MERLIN software the data were processed using the AIPS package. In order to derive phase and amplitude corrections for atmospheric and instrumental effects the phase reference source was mapped and self-calibrated. These corrections were applied to the target visibility data. The absolute position of the brightest feature at −15.1 km s−1 was determined. The phase solutions for this feature were obtained with self-calibration method and were then applied for the all channels. The target was mapped and cleaned using a 12mas circular restoring beam. The map noise of ∼27mJy beam−1 for I Stokes parameter in a line-free channel was close to the predicted thermal noise level. In order to determine the position and the brightness of the maser components two dimensional Gaussian components were fitted to the emission in channel maps. The position uncertainty of this fitting depends on the signal to noise ratio in the channel map and is lower than 1mas for about 80% of the maser components to- wards V778Cyg. The absolute position of the phase reference source is known with an accuracy of ∼3mas. The highest uncertainties in the absolute position of maser spots are due to tropospheric effects and errors in the telescope positions. The first effects, estimated by observing the phase rate on the point source 3C39.25, introduce the position error of ∼9mas, whereas uncertainties in telescope positions of 1−2 cm cause an error of spot positions of ∼10mas. In order to check the posi- tion accuracy of maser spots we applied a reverse phase referencing scheme. The emission of 15 channels around the reference feature at −15.1 km s−1 was averaged and mapped. The map obtained was used as a model to self-calibrate the raw tar- get data then these target solutions were applied to the raw data of 2021+614. 4 R. Szczerba, M.R. Schmidt, M. Pulecka The position of the reference source was shifted by ∼2mas with respect to the catalogue position. This indicates excellent phase connection when referencing 2021+614 to the set of the brightest maser spots. The above discussed factors im- ply the absolute position accuracy of the maser source to be of order of ∼25mas. Fig. 1. The absolute positions of the H2O 22GHz maser components towards V778Cyg relative to the reference feature at −15.1 km s−1 (RA(J2000) = 20h 36m07.s3833, DE(J2000) = 60◦05′26.′′024). The symbols indicate the ranges of component velocities in km s−1. The size of each symbol is proportional to the logarithm of peak brightness of the corresponding component. The maser emission brigther than 150mJ beam−1 (∼ 5σ) was found in 51 spectral channels. In these channels single and unresolved component only was detected. The overall structure of the H2O maser is shown in Fig. 1. The maser com- ponents form a distorted ”S” like shape structure along a direction of position angle of about −10◦. There is a clear velocity gradient along this structure with weak south components blueshifted with respect to the brightest north compo- nents. The angular extend of maser emisson is 18.5mas. The axis of alonga- tion of the maser structure is fairly perpendicular to the line towards the op- tical position of V778Cyg measured by Tycho2 (see Fig. 2). Angular separa- tion between the optical star and the maser reference component is 0.192±0.′′048. Mixed chemistry phenomenon 5 200 150 100 50 0 -50 Relative RA [mas] Tycho-2 MERLIN Fig. 2. Comparison of th optical position of V778 Cyg as determined in the Tycho-2 catalogue with the radio position of the H2O 22 GHz maser components as obtained from the MERLIN measurements. The epochs of optical and radio observations differ by about 10 yrs. Szczerba et al. (2006) have argued that such separation cannot be explained by proper motion and instead provide direct observational evidence for the binary system model of Yamamura et al. (2000). They suggested that the observed wa- ter maser components can be interpreted as an almost edge-on warped Keplerian disk located around a companion object and tilted by no more 20◦ relative to the orbital plane. More detailed model of disk around companion in V778 Cyg system is presented by Babkovskaia et al. (2006). Finally, note that recently Ohnaka et al. (2006) reported indirect detection of disk around another silicate carbon star (IRAS 08002−38003). They argued that oxygen-rich material is stored in circumbinary disk surrounding the carbon-rich primary star and its putative low- luminosity companion. These two findings may suggest that there are two different kinds of silicate carbon stars: with circumbinary disk and disk around companion only. 3. IRAS 06238+0904 - AN OH MASER C-STAR OR GENUINE CARBON STAR? Genuine carbon stars are formed during evolution on AGB. The star on that stage posses extended circumstellar envelope (CSE). In its inner part (near the photosphere) physical conditions (T∼2500 K, ρ ∼ 1014 cm−3) make the material mainly molecular, with composition determined by the local thermodynamic equi- librium (LTE). In carbon CSE (C/O>1) after CO formation there is almost no oxygen left. However silicon monoxide (SiO) is observed in carbon stars. Recent observations (Schöier et al. 2006) show relatively high SiO fractional abundances (1× 10−7− 5× 10−5), while LTE models give ∼ 5× 10−8 (Millar 2004). Therefore 6 R. Szczerba, M.R. Schmidt, M. Pulecka the non-equilibrium processes should be considered in modelling of circumstellar chemistry. In this review we focus on IRAS 06238+0904 – an OH maser C-star (see Chen et al. 2001). We first built model of carbon circumstellar envelope (CSE) and then computed radiative transfer in molecular rotational lines of HCN J=1-0, CS J=3-2, CS J=5-4 and SiO J=3-2, detected by us with the IRAM radiotelescope. Spectral energy distribution (SED) for IRAS 06238+0904 was modelled by means of the code and method described in Szczerba et al. (1997). The best fit (see Fig. 3) is obtained for the star’s effective temperature T∗=2500 [K], luminosity to distance ratio L/d2=6500 [L⊙kpc −2], mass loss rate Ṁ = 2×10−5 [M⊙ yr −1], dust temperature at the inner boundary Tdust(R in )=900 [K], amorphous carbon (AC) and silicon carbide (SiC) to gas ratios: ρ(AC)/ρgas=0.001, ρ(SiC)/ρgas=0.00019. Fig. 3. Spectral energy distribution for IRAS 06238+0904. See text for details concerning assumed and estimated parameters. The chemical model is computed with the network based on RATE99 database Le Teuff et al. (2000) composed of 343 species made of 10 elements. The gas tem- perature profile is approximated by the power law function r−1.8 established by iterations from the best fits to CS lines. We assume solar gas composition with modifications of carbon (C/O=1.5) and sulfur ǫ(S)=6.71 abundances. As initial concentrations we put LTE values of 23 important species, where SiO number den- sity is equal 1×10−8 [cm−3] The effect of dust formation is included by reduction of Si and C by amount locked up in SiC and amorphous carbon grains according to dusty model. This results in decrease of silicon abundance to ǫ(Si)=7.39 and decrease of C to O ratio to 1.3. Mixed chemistry phenomenon 7 The radiative transfer is computed in Sobolev approximation with molecular data taken from the Leiden database (Schöier et al. 2005). Only interstellar radiation is taken into account as an important source of UV photons. Level populations of investigated molecules were computed for the assumed temperature and molecular densities resulting from chemical model. The half-width main beam (HPBW) for SiO rotational transition J=3-2 (v=130 GHz) is equal to 18.9′′, 16.7′′ for CS(3-2), 10.0′′ for CS(5-4) and 28.9′′ for HCN(1-0) transition, in case of the IRAM telescope observations. The synthetic profile was computed for assumed distance to IRAS 06238+0904 being equal 2.3 kpc. The observed and obtained molecular line profiles of SiO(3-2), CS(3-2) and CS(5-4) are shown in 3 panels of Fig. 4. During line profiles modelling we included simple treatment of CO self-shielding based on Mamon et al. (1988). This process has considerable influence on all molecules, and is especially important for SiO. As one can see in Fig. 4, when self-shielding is not included (solid line) we can explain observed spectrum solely by the PDR chemistry. Around 1×1016 [cm] we observed considerable reproduction of SiO. On the other hand, inclusion of CO self-shielding prevents formation of SiO (dashed line). Partial reproduction in PDR is still present. In both cases the exchange reaction OH + Si → SiO + H is a main process responsible for formation of silicon monoxide. Exchanges between atomic oxygen and SiH, SiC, and HCSi molecules (O + SiH → SiO + H, O + HCSi → SiO + CH, and O + SiC → SiO + C) are also important. Simulation of the shock passage (see Willacy & Cherchneff 1998) enlarge initial abundance of SiO, in comparison to the LTE value, for about one order. Profile obtain with abundance of this molecule increased by factor of 10 is shown as dashed-dotted line in Fig. 4. Fig. 4. Observed and modelled molecular rotational lines without (solid line) and with (dashed line) CO self-shielding. Dashed-dotted line in the left panel show results when the intial LTE abundance of SiO is increased ten times due to the shock passage. Therefore, we can conclude that IRAS 06238+0904 is a genuine C-star and no assumption of mixed chemistry is necessary. Chemical reactions considered in network can reproduce O-bearing SiO molecule in C-rich environment if no CO self-shielding is considered. In presence of CO self-shielding the computed SiO 8 R. Szczerba, M.R. Schmidt, M. Pulecka emission is too low. This may be improved, however, if we consider the effect of shock passage which can increase the initial SiO abundance by order of magnitude as predicted by Willacy & Cherchneff (1998). ACKNOWLEDGMENTS. This work has been supported by grants 2.P03D.017.25 and 1.P03D.010.29 of the Polish State Committee for Scientific Research. REFERENCES Babkovskaia N., Poutanen J., Richards A. M. S., Szczerba R. 2006, MNRAS, 370, Chen P. S., Szczerba R., Kwok S, Volk K. 2001, AA, 368, 1006 Cohen M., Barlow M. J., Sylvester R. J. et al. 1999, ApJ, 513, L135 Ford K. E. S., Neufeld D. A., Goldsmith P. F., Melnick G. J. 2003, ApJ, 589, 430 Jura M., Bohac C. J., Sargent B. et al. 2006, ApJ, 637, L45 Lewis B. M., 1992, ApJ, 396, 251 Le Teuff Y. H., Millar T. 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Salama, ESA SP-511, 149 Szczerba R., Szymczak M., Babkovskaia N. et al. 2006, AA, 452, 561 Trams N. R., van Loon J. Th., Zijlstra A. A. et al. 1999, AA, 344, L17 Waters L. B. F. M., Beintema D. A., Zijlstra A. A. 1998, AA, 331, L61 Willacy K., Cherchneff I., 1998, AA, 330, 676 Willems F. J., de Jong T. 1986, ApJ, 309, L39 Yamamura I., Dominik C., de Jong T. 2000, AA, 363, 629 Zijlstra A. A., Gaylard M. J., Te Lintel Hekkert P. et al. 1991, AA, 243, L9

0704.0558

M-regularity of the Fano surface

M-REGULARITY OF THE FANO SURFACE ANDREAS HÖRING Abstract. In this note we show that the Fano surface in the intermediate Jacobian of a smooth cubic threefold is M -regular in the sense of Pareschi and Popa. 1. Introduction Let X3 ⊂ P4 be a smooth cubic threefold, then its intermediate Jacobian J(X) := H2,1(X,C)∗/H3(X,Z) is a principally polarised abelian variety (J(X),Θ) of dimension five that is not a Ja- cobian of a curve [4, Thm.0.12]. The Fano scheme F parametrising lines contained in X is a smooth surface, and the Abel-Jacobi map F → J(X) is an embedding that induces an isomorphism Alb(F )) ≃ J(X) [4, Thm.0.6,0.9]. Furthermore the cohomology class of F ⊂ J(X) is minimal, that is [F ] = There is only one other known family of examples of principally polarised abelian varieties (A,Θ) of dimension n such that for 1 ≤ d ≤ n− 2, a minimal cohomology class Θ (n−d)! can be represented by an effective cycle of dimension d: the Jacobians of curves J(C) where the suvarietiesWd(C) ⊂ J(C) have minmal cohomology class. O. Debarre has shown that on a Jacobian these are the only subvarieties having minimal class [5, Thm.5.1], furthermore by a theorem of Z. Ran [11, Thm.5], the only principally polarised abelian fourfolds with a subvariety of minimal class are (products of) Jacobians of curves. In higher dimension few things are known about subvarieties having minimal class. In [9], G. Pareschi and M. Popa introduce a new approach to the characterisation of these subvarieties: they consider the (probably more tractable) cohomological properties of the twisted structure sheaf of the subvariety. More precisely we have the following conjecture. 1.1. Conjecture. [5],[9] Let (A,Θ) be an irreducible principally polarised abelian varieties of dimension n, and let Y be a nondegenerate subvariety (cf. [11, p.464]) of A of dimension d ≤ n− 2. The following statements are equivalent. 1.) The variety Y has minimal cohomology class, i.e. [Y ] = Θ (n−d)! 2.) The twisted structure sheaf OY (Θ) is M -regular (cf. definition 1.4 below), and h0(Y,OY (Θ)⊗ Pξ) = 1 for Pξ ∈ Pic 0(A) general. Date: 4th April, 2007. http://arxiv.org/abs/0704.0558v2 3.) Either (A,Θ) is the Jacobian of a curve of genus n and Y is a translate of Wd(C) or −Wd(C), or n = 5, d = 2 and (A,Θ) is the intermediate Jacobian of a smooth cubic threefold and Y is a translate of F or −F . The implication 2) ⇒ 1) is the object of [9, Thm.B]. The implication 3) ⇒ 2) has been shown for Jacobians of curves in [8, Prop.4.4]. We complete the proof of this implication by treating the case of the intermediate Jacobian. 1.2. Theorem. Let X3 ⊂ P4 be a smooth cubic threefold, and let (J(X),Θ) be its intermediate Jacobian. Let F ⊂ J(X) be an Abel-Jacobi embedded copy of the Fano variety of lines in X. Then OF (Θ) is M -regular and h 0(F,OF (Θ)⊗ Pξ) = 1 for Pξ ∈ Pic 0 J(X) general. Since the properties considered are invariant under isomorphisms, the theorem implies the same statement for −F . The study of the remaining open implications of conjecture 1.1 is a much harder task than the proof of theorem 1.2. In an upcoming paper we will start to investi- gate this problem under the additional hypothesis that (A,Θ) is the intermediate Jacobian of a generic smooth cubic threefold. In this case we can show the following statement. 1.3. Theorem. [6] Let X3 ⊂ P4 be a general smooth cubic threefold. Let (J(X),Θ) be its intermediate Jacobian, and let F ⊂ J(X) be an Abel-Jacobi embedded copy of the Fano variety of lines in X. Let S ⊂ J(X) be a surface that has minimal cohomology class, i.e. [S] = Θ . Then S is a translate of F or −F . Notation and basic facts. We work over an algebraically closed field of characteristic different from 2. We will denote by D ≡ D′ the linear equivalence of divisors, and by D ≡num D ′ the numerical equivalence. For (A,Θ) a principally polarised abelian variety (ppav), we identify A with  = Pic0(A) via the morphism induced by Θ. If ξ ∈ A is a point, we denote by Pξ the corresponding point in  = Pic0(A) which we consider as a numerically trivial line bundle on A. 1.4. Definition. [10] Let (A,Θ) be a ppav of dimension n, and let F be a coherent sheaf on A. For all n ≥ i > 0, we denote by V iF := {ξ ∈ A | h i(A,F ⊗ Pξ) > 0} the i-th cohomological support locus of F . We say that F is M -regular if codimV i for all i ∈ {1, . . . , n}. If l ⊂ X is a line, we will denote by [l] the corresponding point of the Fano surface F and by Dl ⊂ F the incidence curve of l, that is, Dl parametrises lines in X that meet l. Furthermore we have by [4, §10], [12, §6] and Riemann-Roch that OF (Θ) ≡num 2Dl,(1.5) KF ≡num 3Dl,(1.6) Dl ·Dl = 5,(1.7) χ(F,OF (Θ)) = 1.(1.8) 2. Prym construction of the Fano surface We recall the construction of the Fano surface as a special subvariety of a Prym variety [3, 2]: let C̃ := Dl0 ⊂ F be the incidence curve of a general line l0 ⊂ X . Let X ′ be the blow-up of X in l0. Then the projection from l0 induces a conic bundle structure X ′ → P2 with branch locus C ⊂ P2 a smooth quintic. This conic bundle induces a natural connected étale covering of degree two π : C̃ → C (cf. [1, Ch.I] for details), and we denote by σ : C̃ → C̃ the involution induced by π. The kernel of the normmorphism Nm : JC̃ → JC has two connected components which we will denote by P and P1. The zero component P is called the Prym variety associated to π, and it is isomorphic as a ppav to J(X) [1, Thm.2.1]. Let H ⊂ C be an effective divisor given by a hyperplane section in P2. Then H has degree five and h0(C,OC(H)) = 3, so the complete linear system g 5 corresponds to a P2 ⊂ C(5). We choose a divisor H̃ ∈ C̃(5) such that π(5)([H̃ ]) = [H ], where π(5) : C̃(5) → C(5) is the morphism induced by π on the symmetric products. Let φH : C (5) → JC and φ : C̃(5) → JC̃ be the Abel-Jacobi maps given by H and H̃. We have a commutative diagram C̃(5) The fibre of φ (C̃(5)) → φH(C (5)) over the point 0 (and thus the intersection of (C̃(5)) with kerNm) has two connected components F0 ⊂ P and F1 ⊂ P1. If we identify P and P1 via H̃ − σ(H̃), we obtain an identification F1 = −F0 [3, p.360]. The (non-canonical) isomorphism of ppavs P ≃ J(X) transforms F0 into a translate of the Fano surface F [3, Thm.4]. From now on we will identify P (resp. F0) and J(X) (resp. some Abel-Jacobi emdedded copy of the Fano surface F ). We will now prove two technical lemmata on certain linear systems on C̃. The first is merely a reformulation of [2, §2,ii)]. 2.9. Lemma. The line bundle O (C̃) is a base-point free pencil of degree five such that any divisor D ∈ |O (C̃)| satisfies π∗D ≡ H. Proof. We define a morphism µ : C̃ = Dl0 → l0 ≃ P 1 by sending [l] ∈ C̃ to l∩l0. Since l0 is general and through a general point of l0 there are five lines distinct from l0, the morphism µ has degree 5. If [l] ∈ F , then Dl · Dl0 = 5 by formula (1.7), so for [l] 6= [l0] the divisor Dl0 ∩ Dl ∈ |OC̃(Dl)| is effective. Furthermore π∗Dl ≡ H , since π∗Dl is the intersection of C ⊂ P 2 with the image of l under the projection X ′ → P2. By specialisation the linear system |O (C̃)| is not empty and a general divisor D in it corresponds to the five lines distinct from l0 passing through a general point of l0. Hence OC̃(C̃) ≃ µ ∗OP1(1) and π∗D ≡ H . � 2.10. Lemma. The sets V ′0 := {ξ ∈ P | h 0(C̃,O (C̃)⊗ Pξ) > 0} V ′1 := {ξ ∈ P | h 0(C̃,O (2C̃)⊗ Pξ) > 1} are contained in translates of F ∪ (−F ). Proof. 1) Let D ∈ |O (C̃) ⊗ Pξ| be an effective divisor. Then π∗C̃ ≡ π∗D ≡ H . It follows that D ∈ (φ (C̃(5)) ∩ kerNm), so D is in F or −F . 2) We follow the argument in [2, §3]. By [2, §2,iv)] we have h0(C̃,O (C̃ + σ(C̃))) = 4, so h0(C̃,O (2C̃)) is odd. It follows from the deformation invariance of the parity [7, p.186f] that V ′1 = {ξ ∈ P | h 0(C̃,O (2C̃)⊗ Pξ) ≥ 3}. Fix ξ ∈ P such that h0(C̃,O (2C̃)⊗ Pξ) ≥ 3 and D ∈ |OC̃(2C̃)⊗ Pξ|. Let s and t be two sections of O (C̃) such that the associated divisors have disjoint supports, then we have an exact sequence 0 → O (D − C̃) (t,−s) (D)⊕2 (s,t) (D + C̃) → 0. This implies h0(C̃,O (D − C̃)) + h0(C̃,O (D + C̃)) ≥ 2h0(C̃,O (D)) = 6, furthermore by Riemann-Roch h0(C̃,O (D + C̃)) = h0(C̃,O −D − C̃)) + 5. Now K −D ≡ σ(D) and h0(C̃,O (σ(D) − C̃)) = h0(C̃,O (D − σ(C̃))) imply h0(C̃,O (D − C̃)) + h0(C̃,O (D − σ(C̃))) ≥ 1. Hence D ≡ C̃ + D′ or D ≡ σ(C̃) + D′ where D′ is an effective divisor such that ′ ≡ H . We see as in the first part of the proof that the effective divisors D′ such that π∗D ′ ≡ H are parametrised by a set that is contained in a translate of F ∪ (−F ). � 3. Proof of theorem 1.2. Since OF (Θ) ≡num OF (2C̃) by formula (1.5), it is equivalent to verify the stated properties for the sheaf OF (2C̃). Step 1. The second cohomological support locus is contained in a translate of F ∪ (−F ). By formula (1.6), we have KF ≡ OF (3C̃)⊗Pξ0 for some ξ0 ∈ P . Hence by Serre duality h2(F,OF (2C̃)⊗Pξ) = h 0(F,OF (C̃)⊗P ξ ⊗Pξ0), so it is equivalent to consider the non-vanishing locus V0 := {ξ ∈ P | h 0(F,OF (C̃)⊗ Pξ) > 0}. If l ∈ F is a line on X , the corresponding incidence curve Dl ⊂ F is an effective divisor numerically equivalent to C̃, so it is clear that ±F is (up to translation) a subset of V0. In order to show that we have an equality, consider the exact sequence 0 → OF ⊗ Pξ → OF (C̃)⊗ Pξ → OC̃(C̃)⊗ Pξ → 0. Clearly h0(F,OF ⊗ Pξ) = 0 for ξ 6= 0, so h 0(F,OF (C̃)⊗ Pξ) ≤ h 0(C̃,O (C̃)⊗ Pξ) for ξ 6= 0. Since a divisor D ∈ |O (C̃)| satisfies π∗D ≡ H , we conclude with Lemma 2.10. Step 2. The first cohomological support locus is is contained in a union of trans- late of F ∪ (−F ). Since χ(F,OF (2C̃)) = χ(F,OF (Θ)) = 1 (formula (1.8)), we h1(F,OF (2C̃)⊗ Pξ) = h 0(F,OF (2C̃)⊗ Pξ) + h 0(F,OF (C̃)⊗ P ξ ⊗ Pξ0)− 1. Since h0(F,OF (2C̃)⊗ Pξ) = h 0(F,OF (Θ)⊗ Pξ) ≥ 1 for all ξ ∈ P , the first cohomological support locus is contained in the locus where h0(F,OF (C̃)⊗ P ξ ⊗Pξ0) > 0 or h 0(F,OF (2C̃)⊗ Pξ) > 1. By step 1 the statement follows if we show the following claim: the set V1 := {ξ ∈ P | h 0(F,OF (2C̃)⊗ Pξ) > 1} is contained in a union of translates of F ∪ (−F ). Step 3. Proof of the claim and conclusion. Consider the exact sequence 0 → OF (C̃)⊗ Pξ → OF (2C̃)⊗ Pξ → OC̃(2C̃)⊗ Pξ → 0. By the first step we know that h0(F,OF (C̃)⊗ Pξ) = 0 for ξ in the complement of a translate of F ∪ (−F ), so h0(F,OF (2C̃)⊗ Pξ) ≤ h 0(C̃,O (2C̃)⊗ Pξ) for ξ in the complement of a translate of F ∪ (−F ). The claim is then immediate from Lemma 2.10. By the same lemma h0(C̃,O (2C̃)⊗Pξ) = 1 for ξ ∈ P general, so h0(F,OF (2C̃)⊗ Pξ) = h 0(F,OF (Θ)⊗ Pξ) = 1 for ξ ∈ P general. � Remark. It is possible to strengthen a posteriori the statements in the proof: since Theorem 1.2 holds, we can use the Fourier-Mukai techniques from [9] to see that the cohomological support loci are supported exactly on the theta-dual of F (ibid, Definition 4.2), which in our case is just F . Acknowledgements. I would like to thank Mihnea Popa for suggesting to me to work on this question. Olivier Debarre has shown much patience at explaining to me the geometry of abelian varieties. For this and many discussions on minimal cohomology classes I would like to express my deep gratitude. References [1] A. Beauville. Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. École Norm. Sup. (4), 10(3):309–391, 1977. [2] A. Beauville. Les singularités du diviseur Θ de la jacobienne intermédiaire de l’hypersurface cubique dans P4. In Lect. Notes Math. 947., pages 190–208. Springer, Berlin, 1982. [3] A. Beauville. Sous-variétés spéciales des variétés de Prym. Comp. Math., 45(3):357–383, 1982. [4] C. H. Clemens and P. A. Griffiths. The intermediate Jacobian of the cubic threefold. Ann. of Math. (2), 95:281–356, 1972. [5] O. Debarre. Minimal cohomology classes and Jacobians. J. Alg. Geom., 4(2):321–335, 1995. [6] A. Höring. Paper in preparation. Soon on this server, 2007. [7] D. Mumford. Theta characteristics of an algebraic curve. Ann. Sci. École Norm. Sup. (4), 4:181–192, 1971. [8] G. Pareschi and M. Popa. Regularity on abelian varieties. I. J. Amer. Math. Soc., 16(2):285– 302, 2003. [9] G. Pareschi and M. Popa. Generic vanishing and minimal cohomology classes on abelian varieties. arXiv:math.AG/0610166, 2006. [10] G. Pareschi and M. Popa. GV-sheaves, Fourier-Mukai transform, and Generic Vanishing. arXiv:math.AG/0608127, 2006. [11] Z. Ran. On subvarieties of abelian varieties. Inventiones Math., 62:459–479, 1981. [12] G. E. Welters. Abel-Jacobi isogenies for certain types of Fano threefolds, volume 141 of Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam, 1981. Andreas Höring, IRMA, Université Louis Pasteur, 7 rue René Descartes, 67084 Stras- bourg, France E-mail address: [email protected] http://arxiv.org/abs/math/0610166 http://arxiv.org/abs/math/0608127 1. Introduction 2. Prym construction of the Fano surface 3. Proof of theorem ??. References

0704.0559

Signal for space-time noncommutativity: the Z -> gamma gamma decay in the renormalizable gauge sector of the theta-expanded NCSM

arXiv:0704.0559v1 [hep-ph] 4 Apr 2007 Signal for space-time noncommutativity: the Z → γγ decay in the renormalizable gauge sector of the θ-expanded NCSM ∗ Josip Trampetić† Rudjer Bošković Institute, Zagreb, Croatia Abstract We propose the Z → γγ decay, a process strictly forbidden in the standard model, as a signal suitable for the search of noncommutativity of coordinates at very short distances. We compute the Z → γγ partial widthin the framework of the recently proposed renormalizable gauge sector of the noncommutative standard model. The one-loop renormalizability is obtained for the model containing the usual six representations of matter fields of the first generation. Even more, the noncommutative part is finite or free of divergences, showing that perhaps new interaction symmetry exists in the noncommutative gauge sector of the model. Discovery of such symmetry would be of tremendous importance in further search for the violation of the Lorentz invariance at very high energies. Experimental possibilities of Z → γγ decay are analyzed and a firm bound to the scale of the noncommutativity parameter is set around 1 TeV. ∗ Based on presentation given at the IV Summer School in Modern Mathematical Physics, Belgrad, Serbia, September 3-14, 2006 and LHC Days in Split, Croatia, October 2-7, 2006. Work supported by the Croatian Ministry of Science, Education and Sport project 098-0982930-2900. † e-mail address: [email protected] http://arxiv.org/abs/0704.0559v1 The title 2 Gauge theories can be extended to a noncommutative (NC) setting in different ways. In our model, the classical action is obtained via a two-step procedure. First, the noncommutative Yang-Mills (NCYM) is equipped with a star product carrying information about the underlying noncommu- tative manifold, and, second, the ⋆-product and noncommutative fields are expanded in the noncommutative parameter θ using the Seiberg-Witten (SW) map [1]. In this approach, noncommutativity is treated perturba- tively. The major advantage is that models with any gauge group and any particle content can be constructed [2, 3, 4, 5, 6, 7], so we can construct the standard model (SM). Commutative gauge symmetry is the underlying symmetry of the theory and is present in each order of the θ-expansion. Noncommutative (NC) symmetry, on the other hand, exists only in the full theory, i.e. after summation. There are a number of versions of the noncommutative standard model (NCSM) in the θ-expanded approach, [3, 4, 5, 6]. The action is gauge in- variant; furthermore, it has been proved that the action is anomaly free whenever its commutative counterpart is also anomaly free [8]. The ar- gument of renormalizability was previously included in the construction of field theories on noncommutative Minkowski space producing not only the one-loop renormalizable model [9], but the model containing one-loop quantum corrections free of divergences [10], contrary to previous results [11, 12]. In [10] we analyzed the gauge theory based on the U(1)Y × SU(2)L × SU(3)C group: we succeeded in constructing a model which had the renor- malizable gauge sector to θ-linear order. The condition of the gauge sector renormalizability determines the additional θ-linear interactions between gauge bosons. Experimental evidence for noncommutativity coming from the gauge sector should be searched for in the process of the Z → γγ decay, kinemati- cally allowed for on-shell particles [10, 7]. As it is forbidden in the SM by an- gular momentum conservation and Bose statistics (Landau-Pomeranchuk- Yang Theorem), it would serve as a clear signal for the existence of space- time noncommutativity. Signatures of noncommutativity were discussed previously within particle physics in [7, 13, 14]. The noncommutative space which we consider is the flat Minkowski space, generated by four hermitian coordinates x̂µ which satisfy the com- mutation rule [x̂µ, x̂ν ] = iθµν = const. (1) The algebra of the functions φ̂(x̂), χ̂(x̂) on this space can be represented by the algebra of the functions φ̂(x), χ̂(x) on the commutative R4 with the Moyal-Weyl multiplication: φ̂(x) ⋆ χ̂(x) = e θµν ∂ ∂yν φ̂(x)χ̂(y)|y→x . (2) It is possible to represent the action of an arbitrary Lie group G (with the generators denoted by T a) on noncommutative space. In analogy to the ordinary case, one introduces the gauge parameter Λ̂(x) and the vector The title 3 potential V̂µ(x). The main difference is that the noncommutative Λ̂ and V̂µ cannot take values in the Lie algebra G of the group G: they are enveloping algebra-valued. The noncommutative gauge field strength F̂µν is F̂µν = ∂µV̂ν − ∂ν V̂µ − i(V̂µ ⋆ V̂ν − V̂ν ⋆ V̂µ). (3) There is, however, a relation between the noncommutative gauge symmetry and the commutative one: it is given by the Seiberg-Witten (SW) mapping [1]. Namely, the matter fields φ̂, the gauge fields V̂µ, F̂µν and the gauge parameter Λ̂ can be expanded in the noncommutative θµν and in the com- mutative Vµ and Fµν . This expansion coincides with the expansion in the generators of the enveloping algebra of G, {T a, : T aT b :, : T aT bT c :}; here : : denotes the symmetrized product. The SW map is obtained as a solution to the gauge-closing condition of infinitesimal (noncommutative) transformations. The expansions of the NC vector potential and of the field strength, up to first order in θ, read V̂ρ(x) = Vρ(x)− θµν {Vµ(x), ∂νVρ(x) + Fνρ(x)}+ . . . , (4) F̂ρσ = Fρσ + 2{Fµρ, Fνσ} − {Vµ, (∂ν +Dν)Fρσ} + . . . , (5) where Dν = ∂ν − i[Vν , ] is the commutative covariant derivative. The solution for the SW map given above is not unique and along with (5) all expressions V̂ ′µ, F̂ µν of the form V̂ ′µ = V̂µ +Xµ, F̂ µν = F̂µν +DµXν −DνXµ (6) are solutions to the closing condition to linear order, if Xµ is a gauge covariant expression linear in θ, otherwise arbitrary. One can think of this transformation as of a redefinition of the fields Vµ and Fµν . Taking the action of the noncommutative gauge theory, analogous to that of the ordinary Yang-Mills theory with the commutative field strengths replaced by the noncommutative ones, S = − d4x F̂µν ⋆ F̂ µν , (7) and expanding the fields as in (4-5) and the ⋆-product in θ, we obtain the expression S = − d4xFµνF µν + θµν Tr FµνFρσ − FµρFνσ F ρσ , (8) which is the starting point for the analysis of θ-expanded noncommutative gauge models. Due to the renormalizability condition, we add term, includ- ing NC freedom parameter 1 (a− 1), to the original Lagrangian, producing the following general form of the noncommutative gauge field action: S = − d4xFµνF µν + θµν Tr d4x ( FµνFρσ − FµρFνσ)F ρσ. (9) The title 4 The most general form of the NC action, invariant under the NC gauge transformation, is given in [3, 5, 6, 4], Sgauge = − R(F̂µν) ⋆R(F̂ . (10) The sum in (10) is, in principle, taken over all irreducible representations R of GSM with arbitrary weights CR. Obviously, gauge models are rep- resentation dependent in the NC case: the choice of representations has a strong influence on the theory, on both the form of interactions and the renormalizability properties. Expanding the NC gauge action (10) to first order in the noncommuta- tivity parameter θ, we obtain Sgauge = − d4xR(Fµν)R(F µν) (11) R(Fµν)R(Fρσ)−R(Fµρ)R(Fνσ) R(F ρσ). The arbitrariness in the gauge action, introduced through the coefficient a, reflects in part also the nonuniqueness of the SW map. As we have already mentioned, renormalizability points out the value a = 3 as physical; however, we keep the value of a arbitrary in calculations and use a = 3 at the end. Note that by generalizing the expression (5) to equivalent form F̂µν(a) = Fµν + 2{Fµρ, Fντ} − a{Vρ, (∂τ +Dτ )Fµν} , (12) one could also obtain the actions (9,11) directly from (7,10).1 The im- portant question, if the freedom parameter a is eventually comming from different class of SW maps and/or some other new interaction symmetry extends the purpose of this presentation and, consequentlly, shall be dis- cussed elsewhere. The noncommutative correction, that is the θ-linear part of the La- grangian, reads Lθi = g ′3κ1θ fµνfρσf ρσ − fµρfνσf + g3κ BiµνB ρσk −BiµρB + g3Sκ GaµνG ρσc −GaµρG 1This is in part due to the properties of the integral over the two-function ⋆-product, i.e. the Stokes theorem. The title 5 + g′g2κ2θ ρσi − fµρB ρσi + c.p. + g′g2Sκ3θ ρσa − fµρG ρσa + c.p. , (13) where the c.p. in (13) denotes the addition of the terms obtained by a cyclic permutation of fields without changing the positions of indices. Here, fµν , B µν , and G µν are the physical field strengths which correspond to U(1)Y, SU(2)L, and SU(3)C, respectively. The couplings κi, (i = 1, ..., 5), as functions of the weights CR, that is of the Ci(= 1/g i ), i = 1, ..., 6, are parameters of the model. The couplings in (13) are defined as follows: CRd(R2)d(R3)R1(Y ) 3, (14) CRd(R3)R1(Y )Tr (R2(T L)R2(T L)), (15) CRd(R2)R1(Y )Tr (R3(T S )R3(T S)), (16) CRd(R3)Tr ({R2(T L),R2(T L)}R2(T L)), (17) κabc5 = CRd(R2)Tr ({R3(T S ),R3(T S)}R3(T S)). (18) The κ1, . . . , κ5 depend on the representations of matter fields through the dependence on the coefficients CR. For the first generation of the standard model there are six such representations, summarized in Table 1 of [4]; they produce six independent constants CR 2. However, one can immediately verify that κ 4 = 0. This follows from the fact that the symmetric coeffi- cients dijk of SU(2) vanish for all irreducible representations. In addition, we take that κabc5 = 0. The argument for this assumption is related to the invariance of the color sector of the SM under charge conjugation. Although apparently in Table 1 from [4] one has only the fundamental representa- tion 3 of SU(3)C, there are in fact both 3 and 3̄ representations with the same weights, C3 = C3̄. In the Lagrangian this corresponds to writing each minimally-coupled quark term as a half of the sum of the original and the charge-conjugated terms. Since the symmetric coefficients for the 3 and 3̄ representations satisfy dabc = −dabc , we obtain κabc5 = C3d = 0. (19) 2We assume that CR > 0; therefore the six CR’s were denoted by , i = 1, ..., 6, in [3, 6]. The title 6 -0.3 -0.2 -0.1 0 0.1 ΓΓΓ -0.2 -0.3 -0.2 -0.1 0 0.1 Figure 1: (a) The three-dimensional simplex that bounds possi- ble values for the coupling constants Kγγγ , KZγγ and KZgg at the MZ scale. The vertices of the simplex are: (−0.184,−0.333, 0.054), (−0.027,−0.340,−0.108), (0.129,−0.254, 0.217), (−0.576, 0.010,−0.108), (−0.497,−0.133, 0.054), and (−0.419, 0.095, 0.217). (b) The allowed region for KZγγ and Kγγγ at theMZ scale, projected from the simplex given in Fig (a). The vertices of the polygon are: (−0.333, −0.184), (−0.340, −0.027), (−0.254, 0.129), (0.095, −0.419), (0.0095, −0.576), and (−0.133, −0.497). We are left only with three nonvanishing couplings, κ1, κ2, and κ3, depend- ing on six constants C1, . . . , C6: κ1 = −C1 − κ2 = − C6 ; κ3 = C5 . (20) There are three relations among Ci’s: = 2C1 + C2 + C5 + C6 , = C2 + 3C5 + C6 ; = C3 + C4 + 2C5 , (21) in effect representing three consistency conditions imposed on (8) in a way to match the SM action at zeroth order in θ. See detailes in [6]. Fig.(1) shows the three-dimensional simplex that bounds allowed values for the dimensionless coupling constants Kγγγ , KZγγ and KZgg. For any choosen point within the simplex in Fig.(1) the remaining coupling con- stants KZZγ, KZZZ, KWWγ, KWWZ and Kγgg are uniquely fixed by the NCSM [6, 4]. This is true for any combination of three coupling constants. The title 7 Our total classical action reads Scl = SSM + Sθi = g ′3κ1θ fµνfρσf ρσ − fµρfνσf + g′g2κ2θ ρσi − fµρB ρσi + c.p. + g′g2Sκ3θ ρσa − fµρG ρσa + c.p. . (22) The term Sθ1 in (22) is one-loop renormalizable to linear order in θ [9] since the one-loop correction to the Sθ1 is of the second order in θ. We need to investigate only the renormalizability of the remaining Sθ2 and S 3 parts of the action (22). To realize the one-loop renormalization of the gauge part action (22), we apply, as before [9, 10], the background field method [15, 16]. As we have already explained the details of the method in [12], here we only discuss the points needed for this computation. The main contribution to the func- tional integral is given by the Gaussian integral. However, technically, this is achieved by splitting the vector potential into the classical-background plus the quantum-fluctuation parts, that is, φV → φV +ΦV , and by comput- ing the terms quadratic in the quantum fields. In this way we determine the second functional derivative of the classical action, which is possible since our interactions (22) are of the polynomial type. The quantization is performed by the functional integration over the quantum vector field ΦV in the saddle-point approximation around the classical (background) configuration φV . First, an advantage of the background field method is the guarantee of covariance, because by doing the path integral the local symmetry of the quantum field ΦV is fixed, while the gauge symmetry of the background field φV is manifestly preserved. Since we are dealing with gauge symmetry, our Lagrangian (22) is sin- gular owing to its invariance under the gauge group. Therefore, a proper quantization of (22) requires the presence of the gauge fixing term Sgf [φ], i.e. the Feynman-Fadeev-Popov ghost appears in the effective action Γ[φ] = Scl[φ] + Sgf [φ] + Γ (1)[φ], Sgf [φ] = − d4x(DµΦ )2 . (23) The one-loop effective part Γ(1)[φ] is given by Γ(1)[φ] = log detS(2)[φ] = Tr logS(2)[φ]. (24) In (24), the S(2)[φ] is the 2nd-functional derivative of the classical action, with the following structure: S2 = ✷+N1 +N2 + T2 + T3 + T4 . (25) The title 8 Here N1, N2 are commutative vertices, while T2, T3, T4 are noncommutative ones. The indices denote the number of classical fields. The one-loop effective action computed by using the background field method is θ,2 = Tr log I +✷−1(N1 +N2 + T2 + T3 + T4) (−1)n+1 −1N1 +✷ −1N2 +✷ −1T2 +✷ −1T3 +✷ As the conventions and the notation are the same as in [10], we only en- counter and discuss the final results. The divergent one-loop vertex correction to (22) as a function of the SW freedom parameter a is [10] Γdiv = 3(4π)2ǫ BiµνB µνi + GaµνG 3(4π)2ǫ g′g2κ2(3− a)θ ρσ − fµρB 3(4π)2ǫ g′g2Sκ3(3− a)θ ρσ − fµρG ρσa . From (27) it is clear that the expanded gauge action (22) is renormalizable only for the value a = 3 and, its noncommutative part is finite or free of di- vergencies, so the noncommutativity parameter θ need not be renormalized. The results for the bare fields and couplings, are given in [10]. Note that we have also analized the renormalizability properties of the pure NC SU(N) gauge sector, for vector fields in the adjoint representation [17]. We have found that this model is also renormalizable for a = 3. However, to obtain renormalizability, we had to pay a price by necessity for the renormalization of the noncommutative deformation parameter h. In this way the parameter h and/or the scale of noncommutativity ΛNC become running quantities, dependent on energy [17]. In addition, it was shown that the one-loop contributions to the U(1) gauge-field part of the noncommutative gauge theories in the enveloping- algebra formalism are renormalizable at first order in θ even if the scalar matter, with and without spontaneous symmetry breaking, contributions are taken into account [18]. There is reasonable hope that the same con- clusion should hold for SU(N), but the computations are expected to be extremely involving. Nevertheless, the results [18] further strengthen the philosophy which is embraced in our latest papers [10, 17]. From the action (22) we extract the triple-gauge boson terms which are not present in the commutative SM Lagrangian. In terms of the physical fields A, W±, Z, and G they are Lθγγγ = sin 2θW Kγγγθ ρτAµν (aAµνAρτ − 4AµρAντ ) , Kγγγ = gg′(κ1 + 3κ2); (28) The title 9 LθZγγ = sin 2θW KZγγ θ × [2Zµν (2AµρAντ − aAµνAρτ ) + 8ZµρA µνAντ − aZρτAµνA µν ] , KZγγ = − 2g2 ; (29) where Aµν ≡ ∂µAν − ∂νAµ, etc. The structure of the other interactions such as ZZγ, WWZ, ZZZ, Zgg, and γgg is given in [4, 6]. Next we focus on the branching ratio of the Z → γγ decay in the renor- malizable model. Note that each term from the θ-expanded action (22), (28) and (29) is manifestly invariant under the ordinary gauge transforma- tions. The gauge-invariant amplitude AθZ→γγ for the Z(k1) → γ(k2) γ(k3) decay in the momentum space reads AθZ→γγ = −2e sin 2θWKZγγΘ 3 (a; k1,−k2,−k3)ǫµ(k1)ǫν(k2)ǫρ(k3). (30) The tensor Θ 3 (a; k1, k2, k3) is given by 3 (a; k1, k2, k3) = − (k1θk2) (31) × [(k1 − k2) ρgµν + (k2 − k3) µgνρ + (k3 − k1) νgρµ] − θµν [k 1 (k2k3)− k 2 (k1k3)] − θνρ [k 2 (k3k1)− k 3 (k2k1)] − θρµ [kν3 (k1k2)− k 1 (k3k2)] + (θk2) gνρ k23 − k + (θk3) gνρ k22 − k + (θk3) gµρ k21 − k + (θk1) gµρ k23 − k + (θk1) gµν k22 − k + (θk2) gµν k21 − k + θµα(ak1 + k2 + k3)α [g νρ (k3k2)− k + θνα(k1 + ak2 + k3)α [g µρ (k3k1)− k + θρα(k1 + k2 + ak3)α [g µν (k2k1)− k 1 ] , where the 4-momenta k1, k2, k3 are taken to be incoming, satisfying the momentum conservation (k1+ k2+ k3 = 0). In (31) the freedom parameter a appears symmetric in physical gauge bosons which enter the interaction point, as one would expect. The amplitude (30), for a = 3, with the Z boson at rest gives the total rate for the Z → γγ decay: ΓZ→γγ = sin2 2θWK ~E2θ + ~B2θ ), (32) where ~Eθ = {θ 01, θ02, θ03} and ~Bθ = {θ 23, θ31, θ12} are dimensionless coef- ficients of order one, representing the time-space and space-space noncom- mutativity, respectively. For the Z boson at rest, polarized in the direction The title 10 of the third axis, we obtain the following polarized partial width: Γ3Z→γγ = sin2 2θWK ~E2θ + ~B2θ + 42 (θ03)2 + (θ12)2 . (33) In order to estimate the scale of noncommutativity ΛNC from ΓZ→γγ,we consider new experimental possibilities at LHC. According to the CMS Physics Technical Design Report [19], around 107 Z → e+e− events are expected to be recorded with 10 fb−1 of the data. From this one can estimate the expected number of Z → γγ events per 10 fb−1. Assuming that BR(Z → γγ) ∼ 10−8 and using BR(Z → e+e−) = 3 × 10−2, we may expect to have ∼ 3 events of Z → γγ with 10 fb−1. Now the question is: What would be the background from Z → e+e− when the electron radiates a very high-energy bremsstrahlung photon in the beam pipe or in the first layer(s) of the Pixel Detector and is thus lost for the tracker reconstruction? In that case, the electron would not be reconstructed and would be misidentified as a photon. The probability of such an event should be evaluated from the full detector simulation. According to the CMS note [20] which studies the Z → e+e− background for Higgs → γγ, the probability to misidentify the electron as a photon is huge (see Fig. 3 in [20]) but the situation can be improved by applying more stringent selections to the photon candidate when searching for Z → γγ events [21]. However, the irreducible di-photon background (Fig. 3 in [20]) might also kill the signal. In that case, one can only set the upper limits to the scale of noncommutativity from the Z → γγ rate. In accord with the analysis of the LHC experimental expectations [19, 20, 21] it is bona fide reasonable to assume that the lower bound for the branching ratio is BR(Z → γγ) ∼ 10−8. Next, choosing the lower central value of |KZγγ | = 0.05, from the figures and the Table in [6], we find that the upper bound to the scale of noncommutativity is ΛNC ∼ 1.0 TeV for ~E2θ + ~B2θ ≃ 1. The obtained bound is strongly supported in [18]. Clearly, the measurement of the Z → γγ decay branching ratio would fix the quantity |KZγγ/Λ NC|, while the inclusion of other triple gauge boson interactions through 2 → 2 scattering experiments [14] would sufficiently reduce the available parameter space of our model by more precisely de- termining the relations among the couplings Kγγγ , KZγγ , KZZγ, KZZZ , KWWγ, and KWWZ. Next, we summarize our results and compare with those obtained previously. The first Z → γγ calculation [22] was performed within a different model which has different symmetries in comparison with ours and, because of the absence of the SW map, the model does not possess the commutative gauge invariance. Also, the Z → γγ rate obtained in [22] by imposing the unitarity of the theory in the usual manner, θ0i = 0, [23, 24], vanishes 3. The partial width for the same process was obtained in [6] in the frame- work of similar theories, which, however, were not renormalizable. The 3The condition of unitarity can be covariantly generalized to θµνθ µν = 2( ~B2θ − ~E2θ) > 0 [25]. The title 11 present results for the partial widths ΓZ→γγ and Γ Z→γγ are about three times larger than those in [6] and consistently symmetric with respect to time-space and space-space noncommutativity. In the polarized rate (33) the third components ((θ03)2 + (θ12)2) are enhanced relative to the other two components by a large factor, as expected. Also, the rate (33) is en- hanced by a factor of ∼ 3 with respect to the total rate (32). The upper limit to the scale of noncommutativity ΛNC ∼ 1 TeV is significantly higher than in [6]. This bound is now firmer owing to the regular behavior of the triple gauge boson interactions (28-29) with respect to the one-loop renormalizability of the NCSM gauge sector. After 10 years of the LHC running the integrated luminosity is expected to reach ∼ 1000 fb−1, [20]. This means that for the assumed BR(Z → γγ) ∼ 10−8 we should have ∼ 300 events of Z → γγ, that is we should be well above the background. On the other hand, this result can also be understood as ∼ 3 events with the BR(Z → γγ) ∼ 10−10, which lifts the scale of noncommutativity up by a factor of ∼ 3. Therefore, with a more stringent selection of photon candidates and if the irreducible di-photon contamination becomes controllable, the Z → γγ decay will become a clean signature of space-time noncommutativity in LHC experiments. 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0704.0560

Laser spectroscopy of hyperfine structure in highly-charged ions: a test of QED at high fields

Laser spectroscopy of hyperfine structure in highly-charged ions: a test of QED at high fields D.F.A. Winters ∗, M. Vogel GSI mbH, Planckstrasse 1, Darmstadt D-64291, Germany D.M. Segal, R.C. Thompson Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW, United Kingdom W. Nörtershäuser Universität Mainz, Fritz-Strassmann-Weg 2, Mainz D-55099, Germany GSI mbH, Planckstrasse 1, Darmstadt D-64291, Germany Abstract An overview is presented of laser spectroscopy experiments with cold, trapped, highly-charged ions, which will be performed at the HITRAP facility at GSI in Darmstadt (Germany). These high-resolution measurements of ground state hyperfine splittings will be three orders of magnitude more precise than previous measurements. Moreover, from a comparison of measurements of the hyperfine splittings in hydrogen- and lithium-like ions of the same isotope, QED effects at high electromagnetic fields can be determined within a few percent. Several candidate ions suited for these laser spectroscopy studies are presented. Key words: QED, highly-charged ions, hyperfine structure, laser spectroscopy, trapping and cooling PACS: 12.20.Fv, 21.10.Ky, 32.10.Fn, 32.30.-r 1. Introduction Quantum electrodynamics (QED) was the first quantum field theory to be formulated and has suc- cessfully passed every experimental test at low and intermediate fields. A well-known example of QED effects at low fields (∼ 109 V/cm) is the Lamb shift in hydrogen [1]. At low fields, the QED effects (self-energy and vacuum polarisation) can still be ∗ Corresponding author. Email address: [email protected] (D.F.A. Winters). treated as a perturbation, only taking into account lower order terms [2]. However, up to now QED calculations have never been tested at high fields (∼ 1015 V/cm) because such fields cannot be pro- duced in a laboratory, nor by the strongest lasers available. At high fields, perturbative QED is no longer valid and higher order terms become impor- tant as well [2]. Experiments carried out at high fields therefore test different aspects of QED cal- culations and are complementary to high-precision tests of the lower order terms. Heavy atoms that have been stripped of almost Preprint submitted to Canadian Journal of Physics 4 November 2018 http://arxiv.org/abs/0704.0560v2 all their electrons, the so-called highly-charged ions (HCI), are ideal ‘laboratories’ for tests of QED at high fields. These ions have, for example, electric field strengths of the order of 1015 V/cm close to the nucleus [2] and can be produced at high veloc- ities at the Gesellschaft für Schwerionenforschung (GSI) in Darmstadt, Germany. At the HITRAP facility, which is currently be- ing built at GSI, ions coming from the experimen- tal storage ring (ESR) with MeV energies will be slowed down by linear and radiofrequency stages to keV kinetic energies, trapped and cooled down to sub-eV energies, and finally made available for experiments. Within the HITRAP project, instru- mentation is being developed for high-precision measurements of atomic and nuclear properties, mass and g-factor measurements and ion-atom and ion-surface interaction studies [3,4,5]. 2. Hydrogen- and lithium-like ions Hydrogen- and lithium-like ions are the best can- didates for our studies, since they have s-electrons which are very close to the nucleus. The (higher order) QED effects are most pronounced at the high fields close to the nucleus, therefore the best measurable quantity is the ground state hyperfine splitting (HFS). Due to the simple electronic struc- ture of H- and Li-like species, accurate (higher or- der) calculations of ground state HFS can be done, which will then be compared with accurate exper- imental results. As a first approximation, good within about 4%, the energy of the (1s) 2S1/2 ground state HFS of hydrogen-like ions is given by [2,6]: EHFS = α(Zα) 2(2I + 1) 2As(1− δ)(1) where α is the fine structure constant, gI = µ/(µNI) is the nuclear g-factor (with µ the nuclear magnetic moment and µN the nuclear magneton), I the nuclear spin, me and mp are the electron and proton mass, respectively, and c is the speed of light. Equation (1) represents the normal ground state HFS multiplied by a correction As for the relativistic energy of the s-electron, and by a fac- tor (1− δ), which takes the ‘Breit-Schawlow’ (BS) effect into account. The BS effect is due to the spa- tial distribution of the nuclear charge. It corrects for the fact that we cannot assume a homogeneous charge distribution over a spherical nucleus. The values for δ were taken from [6], those for gI and I from [7]. In principle eq.(1) should also contain a correction for the finite nuclear mass, but since this correction is very small it can be neglected [2]. The energy of the (1s22s) 2S1/2 ground state HFS of lithium-like ions only differs from eq.(1) by a factor 1/n3 = 1/8 and by the As-value [2]. However, eq.(1) requires two further important corrections, the one of most interest to us being that which corrects for the QED effects. The other correction takes the ‘Bohr-Weisskopf’ (BW) effect into account [8]. The BW effect is due to the spa- tial distribution of the nuclear magnetisation and is only known with an accuracy of 20-30 %, which is mainly due to the single-particle model used for its calculation [9]. Unfortunately, the QED effects are of the same order of magnitude as the uncer- tainty in the BW effect [10]. Thus, from a HFS measurement of a single species (i.e. H- or Li-like) the QED effects cannot be determined accurately. Equation (1) can also be written as E1sHFS = C1s +E1sQED, where the constant C 1s includes ev- erything except the QED effects. Since the equa- tions for the (1s) and (2s) states are so similar, it is possible to write the difference between the two HFS as ∆EHFS = E HFS−ξE HFS = Enon−QED+ EQED [10]. The factor ξ only contains non-QED terms and can be calculated to a high precision [10]. From the difference between the HFS measure- ments of H- and Li-like ions of the same isotope, the QED effects can thus be determined within a few percent. However, this requires measurements of transitionwavelengthswith an experimental res- olution of the order of 10−6. The transition lifetime t is defined as t = A−1 (see e.g. [11]). The transition probability A, for an M1 transition from the excited to the lower hyperfine state, is given by [2] 4α(2πν)3~2I (2κ+ 1) 27m2ec 4 (2I + 1) where ~ is Planck’s constant divided by 2π and κ Table 1 Calculated HFS transition wavelengths (λ) and lifetimes (t) of the most interesting ion species for systematic studies. Also shown are the nuclear spin (I) and magnetic moment (µ), taken from [7]. The half-lives of these species are longer than 10 minutes. (The values listed are truncated and the QED and BW effects are not included.) element ion type λ (nm) t (ms) I µ (µN ) lead 207Pb81+ H-like 973 45 1/2 0.59 bismuth 209Bi82+ H-like 239 0.38 9/2 4.11 209Bi80+ Li-like 1469 87 protactinium 231Pa90+ H-like 262 0.64 3/2 2.01 231Pa88+ Li-like 1511 123 lead [12] 207Pb+ P3/2 - P1/2 710 41 1/2 0.59 chlorine [13] 35Cl+ 3P2 - 1D2 858 - 3/2 0.82 3P1 - 1D2 913 - argon [14] 37Ar2+ 3P2 - 1D2 714 - 7/2 1.3 3P1 - 1D2 775 - is related to the electron’s angular momentum [2]. From eq.(2) it can be seen that A scales with the transition frequency as ν3, whereas ν is propor- tional to Z3, see eq.(1). Therefore, the transition lifetime scales with Z as t ∝ Z−9 and is roughly of the order of milliseconds for Z > 70. In table 1 the calculated transition wavelengths (λ) and lifetimes (t), together with their corre- sponding I and µ values, of the most interesting species for our laser spectroscopy studies are listed. (The QED and BW effects are not taken into ac- count.) The half-lives of these species exceed 10 minutes, which corresponds to the minimum time required for a measurement. Although the wave- lengths span a broad range, roughly from 200 to 1600 nm, these transitions are still accessible with standard laser systems. The three species (Pb [12], Cl [13] and Ar [14]) at the bottom of the table are considered as candidates for pilot experiments. They are singly charged ions, which are easily pro- duced, have M1 transitions at convenient wave- lengths, and can be used to test the laser spec- troscopy part of the experiment. A measurement of the HFS in 207Pb+ is of special interest, be- cause it will be possible to extract the value of µ. Currently two different values exist, which unfor- tunately leads to a 2% difference in the HFS cal- culations [15]. In principle, similar experiments could be car- ried out with metastable hafnium (180Hf, level energy 1141 keV, half-life 5.5 h [7]). For H-like hafnium, the transition values are λ = 217 nm and t = 0.25 ms. For the Li-like ion, λ = 1434 nm and t = 72 ms are obtained. The difficulty with this isotope is that its nucleus is in an excited state, which is difficult to produce. Figure 1 shows the calculated transition wave- lengths of all H-like lead, and all H- and Li-like bismuth isotopes with half-lives exceeding 10 min- utes. (The QED and BW effects are not included.) The isotopes are labelled by their corresponding atomic mass units (in u) and the stable isotopes (207Pb and 209Bi) are indicated by the small ar- rows. For Pb, only the H-like isotopes are acces- sible with standard lasers, because the transition wavelengths of Li-like isotopes are much longer than 1600 nm. For Bi, many isotopes of both ion species are accessible, although their transition wavelengths differ considerably. From Fig.1 it is clear that both elements offer many candidates for laser spectroscopy measure- ments of ground state HFS and that bismuth, in particular, allows for a systematic study of the (higher order) QED effects at high fields. Further- more, a systematic study of different isotopes of the same species, for example a study of the H-like Pb isotopes, will make it possible to study trends in nuclear properties across a range of isotopes. There already exist two previous measurements of the 2s ground state HFS in 209Bi. A direct mea- surement [16] was carried out using the ESR at GSI (Darmstadt), but unfortunately no resonance was found at the predicted value of ≈ 1554 nm [10]. An indirect measurement [17] was performed in an electron beam ion trap (EBIT) and yielded a value of ≈ 1512 nm, but the error in the mea- surement was rather large (≈ 50 nm). In the ESR the ions have relativistic velocities (≈ 200MeV/u), H-like Li-like H-like Fig. 1. Calculated transition wavelengths for H-like Pb and Bi isotopes (full circles), and Li-like Bi isotopes (open circles). Only isotopes with half-lives exceeding 10 minutes are shown. The small arrows indicate the stable isotopes, the numbers are the masses in u. (The QED and BW effects are not included.) which are used to shift the transition wavelength to a lower value (≈ 532 nm), and the transitions are Doppler-broadened (≈ 40 GHz). In the EBIT the ions have temperatures of several hundreds of eV (∼ 106 K), which lead to considerable Doppler broadening (≈ 10 GHz). The resolution obtained in previous measurements at the ESR is of the or- der of 10−4, whereas that of the EBIT measure- ment is of the order of 10−2. 3. Experiment overview A detailed description of the proposed experi- ments, as well as a treatment of the techniques used, can be found elsewhere [18,19]. Briefly, an externally produced bunch of roughly 105 HCI at an energy of a few keV is loaded into a cylindrical open-endcap Penning trap [20] on axis, i.e. along the magnetic field lines. Electron capture (neutral- isation) by collisions is strongly reduced by operat- ing the trap at cryogenic temperatures under UHV conditions. The HCI are captured in flight, con- fined, cooled by ‘resistive cooling’ [21] and radially compressed by a ‘rotating wall’ [22] technique. Af- ter these steps a cold and dense ion cloud is ob- tained. The spectroscopy laser enters the trap ax- ially through an open-endcap and will fully irradi- ate the ion cloud. The fluorescence from the excited HCI is detected perpendicular to the cooled axial motion (trap axis) through segmented ring elec- trodes, which are covered by a highly-transparent copper mesh. (The ring is segmented for the rotat- ing wall technique.) The above mentioned transition lifetimes imply that, for a detection efficiency of ∼ 10−3, accept- able fluorescence rates, up to a few thousand counts per second, from M1 transitions can be expected from a (∼ 3mm diameter) cloud of 105 ions [18,19]. Confining the HCI in a trap, and cooling and com- pressing the cloud, will thus enable fluorescence detection and ensure long interrogation times by the laser. However, due to the high density of HCI in the cloud, space charge effects will play a role and will lead to shifts of the motional frequencies of the trapped ions. We have studied this effect in detail and understand the corresponding frequency shifts well [23]. Since these shifts are fairly small, the (fre- quency dependent) cooling and compression tech- niques can still be applied. The HCI also need to be strongly cooled to re- duce Doppler broadening of the transitions. This will be achieved by resistive cooling of the (axial) ion motion in the trap. For example, for the F = 1 → F = 0 transition in 207Pb81+ at ν ≈ 3× 1014 Hz, the Doppler broadened linewidth at a temper- ature of 4 K is ∆νD ≈ 3× 10 7 Hz. The anticipated resolution is therefore of the order of 107/1014 = 10−7. This is three orders of magnitude better than any previous measurement, see e.g. [15,24], and good enough to measure the QED effects within a few percent. 4. Acknowledgments This work is supported by the European Com- mission within the framework of the HITRAP project (HPRI-CT-2001-50036). W.N. acknowl- edges funding by the Helmholtz Association (VH- NG-148). References [1] W.E. Lamb Jr. and R.C. Rhetherford, Phys. Rev. 72, (1974) 241. [2] T. Beier, Phys. Rep. 339, (2000) 79. [3] W. Quint J. Dilling, S. Djekic, H. Häffner, N. Hermanspahn, H.-J. Kluge, G. Marx, R. Moore, D. Rodriguez, J. Schönfelder, G. Sikler, T. Valenzuela, J. Verdú, C. Weber and G. Werth, Hyp. Int. 132, (2001) [4] T. Beier, L. Dahl, H.-J. Kluge, C. Kozhuharov, W. Quint and the HITRAP collaboration, Nucl. Instr. Meth. Phys. Res. B 235, (2005) 473. [5] H.-J. Kluge, T. Beier, K. Blaum, M. Block, L. Dahl, S. Eliseev, F. Herfurth, S. Heinz, O. Kester, C. Kozhuharov, T. Kühl, G. Maero, W. Nörtershäuser, T. Stöhlker, W. Quint, G. Vorobjev, G. Werth, and the HITRAP Collaboration, Proceedings of the Memorial Symposium for Gerhard Soff, Topics in Heavy Ion Physics (Eds Walter Greiner and Joachim Reinhardt), pages 89-101 (2005), EP Systema (Budapest). [6] V.M. Shabaev, J. Phys. B 27, (1994) 5825. [7] R.B. Firestone and V.S. Shairley, Table of Isotopes (Appendix E), Wiley (1998). [8] A. Bohr and V.F. Weisskopf, Phys. Rev. 77, 94 (1950). [9] V.M. Shabaev, M. Tomaselli, T. Kühl, A.N. Artemyev and V.A. Yerokhin, Phys. Rev. A 56, (1997) 252. [10] V.M. Shabaev, A.N. Artemyev, V.A. Yerokhin, O.M. Zherebtsov and G. Soff, Phys. Rev. Lett. 86, (2001) 3959. [11] W. Demtröder, Laser Spectroscopy, Springer, New York (1996). [12] A. Roth, Ch. Gerz, D. Wilsdorf and G. Werth, Z. Phys. D 11, (1989) 283. [13] I.S. Bowen, Astrophys. J. 132, (1960) 1. [14] M.H. Prior, Phys. Rev. A 30, (1984) 3051. [15] P. Seelig, S. Borneis, A. Dax, T. Engel, S. Faber, M. Gerlach, C. Holbrow, G. Huber, T. Kühl, D. Marx, K. Meier, P. Merz, W. Quint, F. Schmitt, M. Tomaselli, L. Völker, H. Winter, M. Würtz, K. Beckert, B. Franzke, F. Nolden, H. Reich, M. Steck and T. Winkler, Phys. Rev. Lett. 81, (1998) 4824. [16] S. Borneis, A. Dax, T. Engel, C. Holbrow, G. Huber, T. Kühl, D. Marx, P. Merz, W. Quint, F. Schmitt, P. Seelig, M. Tomaselli, H. Winter, K. Beckert, B. Franzke, F. Nolden, H. Reich and M. Steck, Hyp. Int. 127, (2000) 305. [17] P. Beiersdorfer, A.L. Osterheld, J.H. Scofield, J.R. Crespo López-Urrutia and K. Widmann, Phys. Rev. Lett. 80, (1998) 3022. [18] D.F.A. Winters, A.M. Abdulla, J.R. Castrejón Pita, A. de Lange, D.M. Segal and R.C. Thompson, Nucl. Instr. Meth. Phys. Res. B 235, (2005) 201. [19] M. Vogel, D.F.A. Winters, D.M. Segal and R.C. Thompson, Rev. Sci. Instrum. 76, (2005) 103102. [20] G. Gabrielse, L. Haarsma and S.L. Rolston, Int. J. Mass Spectr. Ion Proc. 88, (1989) 319. [21] D.J. Wineland and H.G. Dehmelt, J. Appl. Phys. 46, (1975) 919. [22] W.M. Itano, J.J. Bollinger, J.N. Tan, B. Jelenković, X.-P. Huang and D.J. Wineland, Science 279, (1998) [23] D.F.A. Winters, M. Vogel, D.M. Segal and R.C. Thompson, J. Phys. B: At. Mol. Opt. Phys. 39, (2006) 3131. [24] I. Klaft, S. Borneis, T. Engel, B. Fricke, R. Grieser, G. Huber, T. Kühl, D. Marx, R. Neumann, S. Schröder, P. Seelig and L. Völker, Phys. Rev. Lett. 73, (1994) 2425. Introduction Hydrogen- and lithium-like ions Experiment overview Acknowledgments References

0704.0561

Confinement into a state with persistent current by thermal quenching of loop of Josephson junctions

Confinement into a state with persistent current by thermal quenching of loop of Josephson junctions Jorge Berger Physics Department, Ort Braude College, P. O. Box 78, 21982 Karmiel, Israel∗ Abstract We study a loop of Josephson junctions that is quenched through its critical temperature. For three or more junctions, symmetry breaking states can be achieved without thermal activation, in spite of the fact that the relaxation time is practically constant when the critical temperature is approached from above. The probability for these states decreases with quenching time, but the dependence is not allometric. For large number of junctions, cooling does not have to be fast. For this case, we evaluate the standard deviation of the induced flux. Our results are consistent with the available experimental data. PACS numbers: 74.40.+k, 74.81.Fa, 05.70.Fh, 11.15.Ex http://arxiv.org/abs/0704.0561v1 We consider a process in which a superconducting loop that contains n identical Joseph- son junctions is cooled through its critical temperature, in the absence of applied fields, and monitor the spontaneous generation of metastable states with persistent current. From the theoretical point of view, this process is enlightening, because it provides and additional example of a phase transition that is dominated by the time evolution of the system param- eters, rather than by thermal equilibrium; this is a subject that is still far from being closed, and is thought to be relevant both to condensed matter physics and to cosmology. From the practical point of view, this process has significant importance, since flux trapping is a major obstacle for reproducible functioning of large scale ultra-high-speed superconductiv- ity digital applications [1] and we would like to comprehend how it depends on the system parameters. The best known theory for the description of dynamic phase transitions is the Kibble– Zurek mechanism (KZM) [2, 3, 4]. It states that in these transitions the equilibrium critical scalings predict various aspects of the nonequilibrium dynamics of symmetry breaking, in- cluding the density of residual topological defects. Several numeric simulations have tested the predictions of the KZM, particularly the dependence of the density of defects on the quenching time. Some of these simulations lead to refinements of the KZM [5] and others disagree with it [6]. The KZM has also been tested in several experiments; among them we will mainly be interested on those performed in superconducting loops [7, 8, 9]. The study of a loop of Josephson junctions is appealing, because the “rules of the game” are particularly simple [10] and the system parameters can be tailored practically at will. This system may be neater and qualitatively different from other systems, because it can be trully divided into n identical subsystems, whereas in other systems the division depends on a continuously varying coherence length. Moreover, experimental results are already avilable The supercurrent through Josephson junction i is given by IJi = Ic(t) sin γi , (1) where γi is the gauge-invariant phase difference across the junction. Ic vanishes above the critical temperature and increases when the temperature is lowered. We will consider a quenching process in which Ic(t) grows from 0 to Ic0. This supercurrent may be interpreted as arising from a potential energy term − i Ic cos γi. This potential energy gives rise to metastable asymmetric states in which the system can be trapped. There are two typical processes for the formation of topological defects in the KZM. In one case defects become confined when the order parameter becomes unable to follow the change of the parameters of the system. Another scenario is activation due to thermal fluctuations close to the Ginzburg temperature. We shall see that the present system does not fit in either of these cases; the relaxation time does not diverge at the critical temperature and no activation energy is required in order to enter a metastable state. More precisely, let R be the resistance of each junction, L the self-inductance of the loop, let us assume that the capacitance is small and the resistance of the loop itself (above the critical temperature Tc) is much smaller than that of the junctions. Then there are two relaxation times in the problem: one of them is ~/2eRIc, which is infinite regardless of the temperature above Tc, and the other is L/nR, which remains constant and refers to a process that does not attempt to align the order parameters of neighboring segments into the same phase. In a sense, our problem is similar to that of decompression of He4 from the λ-line [11]. Let Ic0 be the maximal superconducting current through the junctions at low temperature and let us take R, Ic0, ~ and 2e as units. Accordingly, the units of voltage, energy, inductance and time will be RIc0, 2eRIc0, ~/2eIc0 and ~/2eRIc0. The state of the loop will be described by the set of values {γi}. The sum of these values can be interpreted as minus the magnetic flux enclosed by the loop. Since we assume that there is no applied magnetic flux, γi = −LI , (2) where I is the current around the loop. Our goal is to find the probabilities for the possible values of i=1 γi after the loop has been cooled. The rules for the evolution of {γi} are stated in several textbooks [10]. The ac Josephson relation is dγi/dt = Vi , (3) where Vi is the voltage across junction i. The total current through junction i is I = IJi + Vi + CdVi/dt+ INi , (4) where IJi is the supercurrent, given by Eq. (1), C is the capacitance of each junction, and INi is the Johnson current. We will assume that during a period of time τ1 the system is kept above the critical temperature and Ic = 0, then, during a period of time τ2 the system is quenched and Ic grows up to Ic0 and finally, during a period of time τ3, Ic = Ic0. In most of our calculations we will assume that the capacitance is negligible. In this case, from Eqs. (2), (3) and (4), = −IJi − INi − γi . (5) The case C 6= 0 will be discussed below. We integrate Eq. (5) by Euler iterations. For this purpose we divide the process into short periods of time of duration ∆t. The Johnson current averaged over a single period is given by INi = ηgi/ ∆t , (6) where gi is a random number with zero average and variance 1 and η = (2kBT/R) 1/2, with kB Boltzmann’s constant and T the temperature. We assume that the temperature remains close to the critical temperature during the entire process; accordingly, η will be taken as constant. We first consider the case in which the loop is cooled instantaneously, i.e., τ2 = 0. If we ignore the Johnson current, Eq. (5) is equivalent to viscosity-dominated motion of a particle in n-dimensional space that feels a potential energy (1/2L)( i=1 γi) 2 − Ic i=1 cos γi. In this situation, {γi} evolves towards a local minimum of the potential energy. During the first stage, Ic(t) = 0 and the only local minimum is the plane i=1 γi = 0. During the last stage, Ic(t) = 1. At this stage the absolute minimum is located at the origin, γi = 0, but for sufficiently large values of L additional local minima may also exist. We ask whether the values {γi} could wander in the plane i=1 γi = 0 and then, when the temperature is lowered, flow into a local minimum different from γi = 0. For our present purpose, two states such that their respective values of γi differ by integer multiples of 2π and i=1 γi is the same for both, will be considered equivalent. Figure 1 shows that this situation is possible for the case n = 3. The figure shows five evolution curves that start at the plane i=1 γi = 0 and, in the absence of thermal fluctuations, flow to a local minimum of the potential energy. Note that although the temperature is assumed to change instantaneously, i=1 γi builds up during a lapse of time of the order of ~/eRIc0. FIG. 1: Several evolution lines for the case of three junctions, for L = 4 and in the absence of thermal fluctuations. All these curves start at the plane γ1+γ2+γ3 = 0 and converge to the point γ1 = γ2 = −0.98, γ3 = 2π − 0.98. We studied the evolution including thermal fluctuations, for n = 3 and n = 4, for several values of η. In all cases, we started from values of {γi} randomly located in the interval −π < γi < π. In order to achieve initially an equilibrium distribution, evolution was followed during a period τ1 with Ic = 0. After that, evolution was followed during a period τ3 with Ic = Ic0. In order to decide what is the “final” state, we should average over an additional period of time, in order to filter out thermal fluctuations. We found it easier to turn off at this stage the fluctuations and let the state converge to the nearest local minimum. For each set of values, this process was reapeated 1000 times and the probability for confinement in a given state was evaluated as the number of times this state was obtained, divided by 1000. In most cases the final state was the ground state γi = 0, but the first excited state was also reached. Due to the symmetry of the problem, these states are degenerate, i.e., the γi’s can be permuted and all the signs can be inverted. For the parameters we considered, we did not find cases with higher excited states. The results are shown in Fig. 2. We avoided values of η that might be too small to enable thermalization during the period τ1. The probabilities shown in the graph correspond to the 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 FIG. 2: Probability for a current-carrying metastable state as a function of the size of thermal fluctuations. Parameters used: τ1 = 40000, τ2 = 0, τ3 = 10000, ∆t = 0.1, C = 0. ♦ n = 3, L = 4; ⋆ n = 4, L = 4; � n = 4, L = 2. total probability of reaching any of the (degenerate) excited states. As a general trend, we see that the probability of ending at an excited state increases with the number of junctions and with the normalized self-inductance. We also see that this probability is fairly independent of the size of thermal fluctuations, until a sufficiently large value of η is reached. Beyond this value, there is a fast decrease of this probability. It is reasonable to expect that the probability for the metastable state will decrease when the thermal energy kBT becomes comparable to the energy barrier that confines this state, i.e., the difference between the energy at the saddle-point and the energy at the local minimum. For n = 3, L = 4, a local minimum is at γ1 = γ2 = −0.98, γ3 = 2π − 0.98, the corresponding saddle point is at γ1 = γ2 = −0.67, γ3 = π + 0.67 and the energy difference is 0.25; for n = 4, L = 4, a local minimum is at γ1 = γ2 = γ3 = −0.83, γ4 = 2π − 0.83, the saddle point is at γ1 = γ2 = γ3 = −0.54, γ4 = π + 0.54 and the energy difference is 0.42; similarly, for n = 4, L = 2, the energy barrier is 0.13. In all cases we find that the probability for the metastable state decreases to about half its maximum value when the thermal energy is about one eighth of the barrier energy. Clearly, the precise value depends on τ3; in principle, for τ3 → ∞, the metastable state should always decay. If the thermal energy becomes of the order of the energy difference between the excited and the ground state, then the probability for the excited state will increase with temperature (equilibrium probability), but we are not interested in this regime. Let us now study the influence of the cooling time on the probability for a metastable 0 10 20 30 40 50 60 FIG. 3: Probability for a metastable state as a function of the cooling time for loops with 3 or 4 junctions. The empty (filled) symbols correspond to Ic(t) proportional to t (to t 2) and are fitted by dashed (continuous) lines. For visibility, the line for n = 3, L = 4 and Ic(t) ∝ t2 has been lowered by 0.025. Parameters used: τ1 = 40000, τ2 + τ3 = 10000, ∆t = 0.1, C = 0. Unless stated otherwise, η = 0.1. ♦ n = 3, L = 4; � n = 4, L = 2; △ n = 4, L = 4, η = 0.2. 0 500 1000 1500 2000 0 5000 FIG. 4: Like Fig. 3, for n = 5 and n = 6. The inset shows P (τ2) in the range 0 ≤ τ2 ≤ 104. For visibility, the line for n = 5, L = 2 and Ic(t) ∝ t has been raised by 0.025.♦ n = 5, L = 2; △ n = 5, L = 1; ◦ n = 6, L = 2. state. At the moment that Ic(t) becomes different from zero, there will be an incentive for leaving this state; on the other hand, as long as Ic(t) is small, the confining barrier will also be small and the way out will be easy. We therefore expect that the trapping probability will decrease with τ2. In most of our calculations we assumed that Ic(t) is proportional to the temperature below Tc and therefore increases linearly with t, but we also considered the case Ic(t) ∝ t2, which is more realistic for strong coupling [9]. For the present purpose, simulations were repeated 104 times. Figures 3 and 4 show our results for several values of n, L and η. The topological charge for a given state may be defined as the sum of the topological charges of all junctions, where the topological charge of junction i is the closest integer to γi/2π. For all the cases considered in Figs. 3–4, the topological charge was 0 or ±1. The probabilities shown in Figs. 3–4 are also the expectations of the absolute value of the topological charge. It is therefore tempting to identify this probability with the density of defects, and anticipate that it will decrease as a power of τ2. However, the arguments that lead to the time dependence of the defect density in [4] seem to be irrelevant in the present case; there is no obvious way to associate the presence of topological charge to some primordial coherence length and, indeed, our results cannot be fitted by a power dependence. Denoting the probability by P , most of the curves in our results (typically for small n and L) can be fitted by the empiric form P (τ2) ∝ exp[−(τ2/τ0) 2/n] in the case Ic(t) ∝ t and by P (τ2) ∝ exp[−(τ2/τ0) 2/(n−1)] in the case Ic(t) ∝ t2. The characteristic time τ0 depends very strongly on the number of junctions and only weakly on the size of the energy barrier or on the temperature. For n = 3, 16 . τ0 . 17; for n = 4, 26 . τ0 . 39; for n = 5, 350 . τ0 . 540. Our empirical fits suggest that for n ≫ 1 the probabilities for metastable states decrease very slowly with the cooling time. Indeed, in the experiment that involved 214 junctions [7], the distribution of permanent currents was found to be independent of the cooling time (up to the order of a minute). Part of the probabilities shown in Figs. 3–4 do not decrease at a uniform rate. Instead, they seem to decay in two stages. A possible explanation might be that the region in phase space that in the absence of thermal fluctuations would flow into a metastable state can be divided into two subregions, such that escape from one subregion is much easier than escape from the other. Let us now consider large values of n and L, as were encountered in the experiment. In this case many different final metastable states are possible, and the most significant experimental quantity will be the variance of the induced flux. Our results are shown in Fig. 5. As in the case of small values of n and L, the general trend is increase of the typical flux with increase of either n or L. However, these individual increases appear to saturate. For instance, for L . 200, the standard deviation of the flux actually decreases with n in 50 100 200 500 2eLIc0�Ñ 50 100 200 FIG. 5: Standard deviation of the induced flux, Φ/Φ0 = − i γi, as a function of the self-inductance and the number of junctions. Two curves are for fixed n, and L is shown in the lower abscissa; the other two are for fixed L, and n is shown in the upper abscissa. The abscissas are in logarithmic scale. The symbols have been joined for visibility. The parameters are as in Fig. 2 and η = 0.1. Each simulation was repeated 400 times. � n = 100; △ n = 214; ◦ L = 100; ⋆ L = 600. the range 100 . n . 200. In the experiment [7], n = 214 and 〈Φ2〉1/2/Φ0 = 7.4 ± 0.7, where Φ is the induced flux and Φ0 the quantum of flux. Comparison with Fig. 5 indicates that 2eIc0L/~ should be in the range between ∼ 100 and ∼ 150. The estimates of Ref. [7] suggest that 2eIc0L/~ ∼ 600. Since the experimental estimate was not based on a measurement, but rather on a plausibility argument for the size of Ic0, and the junctions are not really all identical, the agreement is reasonable. Let us finally consider the case C 6= 0. In this case we integrated Eqs. (3) and (4) as a system of differential equations. In digital applications, a preferred value is C = ~/2eIc0R which provides for fast switching without oscillations. We have repeated our calculations for this case and for several representative values of the other parameters. We found that a capacity of this size has no qualitative effect. In summary, we have performed simulations that describe the formation of symmetry breaking states when a loop of Josephson junctions is quenched. Among the typical systems in which symmetry breaking occurs in a dynamics-dominated process, the present system constitutes a class of its own. Our results agree with the experiment in the case of large n and raise predictions for the case of small n. This work has been supported by the Israel Science Foundation under grant 4/03-11.7. I am grateful to Alan Kadin for useful comments. ∗ Electronic address: [email protected] [1] M. Jeffery, T. Van Duzer, J. R. Kirtley, and M. B. Ketchen, Appl. Phys. Lett. 67, 1769 (1995). [2] T.W. B. Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep. 67, 183 (1980). [3] W. H. Zurek, Nature (London) 317, 505 (1985); Acta Phys. Pol. B 24, 1301 (1993). [4] W. H. Zurek, Phys. Rep. 276, 177 (1996). [5] P. Laguna and W. H. Zurek, Phys. Rev. Lett. 78, 2519 (1997); A. Yates and W. H. Zurek, Phys. Rev. Lett. 80, 5477 (1998). [6] M. Hindmarsh and A. Rajantie, Phys. Rev. Lett. 85, 4660 (2000). [7] R. Carmi, E. Polturak, and G. Koren, Phys. Rev. Lett. 84, 4966 (2000). [8] J. R. Kirtley, C.C. Tsuei, and F. Tafuri, Phys. Rev. Lett. 90, 257001 (2003). [9] R. Monaco, J. Mygind, M. Aaroe, R. J. Rivers, and V. P. Koshelets, Phys. Rev. Lett. 96, 180604 (2006). [10] A. M. Kadin, Introduction to Superconducting Circuits (Wiley, New York, 1999); M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York, 1996); K. K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon and Breach, New York, 1986); A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (Wiley, New York, 1982). [11] P.C. Hendry, N.S. Lawson, R.A.M. Lee, P.V.E. McClintock, and C.H.D. Williams, in: Forma- tion and Interactions of Topological Defects, ed. A.C. Davis and R.N. Brandenberger (Plenum, New York,1995). mailto:[email protected] References

0704.0562

Frequency modulation Fourier transform spectroscopy

Title Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 Frequency modulation Fourier transform spectroscopy Julien Mandon, Guy Guelachvili, Nathalie Picqué Laboratoire de Photophysique Moléculaire, CNRS; Univ. Paris-Sud, Bâtiment 350, 91405 Orsay, France Corresponding author: Dr. Nathalie Picqué, Laboratoire de Photophysique Moléculaire Unité Propre du CNRS, Université Paris Sud, Bâtiment 350 91405 Orsay Cedex, France Phone number: +33 1 69156649 Fax number: +33 1 69157530 Email: [email protected] Web: http://www.laser-fts.org Abstract: A new method, FM-FTS, combining Frequency Modulation heterodyne laser spectroscopy and Fourier Transform Spectroscopy is presented. It provides simultaneous sensitive measurement of absorption and dispersion profiles with broadband spectral coverage capabilities. Experimental demonstration is made on the overtone spectrum of C2H2 in the 1.5 µm region. OCIS codes: 120.6200, 300.6300, 300.6380, 300.6360, 300.6310, 300.6390, 120.5060 120.6200 Spectrometers and spectroscopic instrumentation, 300.6300 Spectroscopy, Fourier transforms, 300.6380 Spectroscopy, modulation, 300.6360 Spectroscopy, laser, 300.6310 Spectroscopy, heterodyne, 300.6390 Spectroscopy, molecular, 120.5060 Phase modulation mailto:[email protected] http://www.laser-fts.org/ Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 Improving sensitivity is presently one of the major concern of spectroscopists. This may be obtained both from the enhancement of the intrinsic signal, and from the reduction of the background noise. In this latter case, modulation has been one of the most effective approach. In particular, Frequency Modulation (FM) absorption spectroscopy [1] has reached detection sensitivity near to the fundamental quantum noise limit, by shifting the frequency modulation of the measurements to a frequency range where the 1/f noise becomes negligible. Moreover, FM spectroscopy benefits from high-speed detection and simultaneous measurement of absorption and dispersion signals. Since Bjorklund’s first demonstrations [1,2] of the efficiency of FM spectroscopy with a single-mode continuous-wave dye laser, the technique has been widely used as a tunable laser spectroscopic method in fields such as laser stabilization [3], two-photon spectroscopy [4], optical heterodyne saturation spectroscopy [5], trace gas detection [6]. In most schemes, the laser wavelength is scanned across the atomic/molecular resonance to retrieve the line shape. More rarely, the modulation frequency is tuned. However in both cases, the measurements are limited to narrow spectral ranges. This letter reports the first results in FM broadband spectroscopy. This work is motivated by our ongoing effort of implementing a new spectroscopic approach simultaneously delivering sensitivity, resolution, accuracy, broad spectral coverage and rapid acquisition. The basic idea, named FM-FTS, is to associate the advantages of FM spectroscopy and high-resolution Fourier transform spectroscopy (FTS). FTS is able to record at once extended ranges, with no spectral restriction. In particular it gives easy access to the infrared domain. In this letter, a new way of modulating the interferogram is implemented. The key concept is that a radio frequency (RF) modulation is performed. The beat signal at the output port of the Fourier transform spectrometer is modulated at constant RF, which is about 104 times greater than the audio frequency generally delivered by the interferometer optical conversion. Together with the advantage, over classical FTS, of measurements performed at much higher frequency, our approach benefits from the synchronous detection ability and from the simultaneous acquisition of both the absorption and the dispersion of the recorded profiles. The experimental principle is presented in Fig. 1. The light emitted by the broadband source is first passing through the interferometer. The output beam is then phase-modulated by an electro-optic modulator (EOM) before entering the absorption cell and falling on the fast detector. The synchronous detection of the detector signal is realized by the lock-in amplifier at the EOM driver reference frequency fm. Recorded data are finally stored on the computer disk with their corresponding path difference position ∆. Their Fourier transform is the spectrum. In more details, the electric field E at the output of the interferometer may be written as: 0 ( )( , ) 1 exp exp( )d c.c. (1) c c ct i i tc ω ω ω ⎡ ⎤∆⎛ ⎞ ∆ = + − +⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ where E0 is the electric field amplitude of the source at ωc optical pulsation, c is the velocity of light and c.c the conjugate complex of the preceding expression in Eq. 1. The EOM effect on the beam is assumed to have a low modulation index M. As a consequence, each carrier wave of pulsation ωc, has two weak sidebands located at ± ωm = ± 2π fm. Equation (1) becomes: ( ) ( ) } 0 ( )( , ) 1 exp exp M exp M exp d c.c. (2) c m c m c t i i t i t i t ω ω ω ω ω ⎡ ⎤∆⎛ ⎞ ∆ = + −⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ ⎡ ⎤ ⎡ ⎤+ + − − +⎣ ⎦ ⎣ ⎦ When interacting with the gas, the carrier and the sidebands experience attenuation and phase- shift due to absorption and dispersion. Following the notations introduced in [1], this interaction may be written as exp(-δ(ω)- i φ(ω)) where δ is the amplitude attenuation and φ is Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 the phase shift. The following convention is adopted: δn and φn denotes for n = 0, ±1 the respective components at ωc and ωc± ωm. Then Eq. 2 may be written: ( ) ( ){ ( ) ( ) ( ) ( ) } 1 1 1 1 ( , ) 1 exp exp exp M exp exp M exp exp d c.c. (3) c m c m c t i i i t i i t i i t ω δ φ ω δ φ ω ω δ φ ω ω ω+ + − − ⎡ ⎤∆⎛ ⎞ ∆ = + − − −⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ ⎡ ⎤ ⎡ ⎤+ − − + − − − − +⎣ ⎦ ⎣ ⎦ The intensity I detected by the fast photodetector is proportional to : *I( , ) ( , ) ( , ). (4)t t t∆ ∝ ∆ ∆E E ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 I( , ) exp 2 exp 2 exp 2 1 cos 2M cos exp cos exp cos 1 cos 2M sin exp sin exp sin 1 cos δ δ δ ω ω δ δ φ φ δ δ φ φ ω ω δ δ φ φ δ δ φ φ ω + + − − − − + + ⎛ ⎡ ⎤∆⎛ ⎞⎡ ⎤∆ ∝ − + − + − +⎜ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦⎝ ⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − − − − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ ∆⎛ ⎞⎡ ⎤+ − − − − − − − + ⎜⎣ ⎦ ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 2M cos 2 exp sin 1 cos 2M sin 2 exp cos 1 cos d . (5) m c c ω δ δ φ φ ω ω δ δ φ φ ω ω + − − + + − − + ⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − + ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎞⎡ ⎤∆⎛ ⎞⎡ ⎤+ − − − + ⎟⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎠ After synchronous detection at fm frequency and with the assumption that |δ0-δj |<<1 and |φ0- φj|<<1 (with j = ±1), the in-phase Icos(∆) and the in-quadrature Isin(∆) parts of the electric signal are given by ( )( )cos 0 1 1I ( ) M 1 cos exp 2 d . (6)c cc ω δ δ δ ω− + ⎡ ⎤∆⎛ ⎞ ∆ ∝ + − −⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ ( )( )sin 0 1 1 0I ( ) M 1 cos exp 2 2 d . (7)c cc ω δ φ φ φ ω+ − ⎡ ⎤∆⎛ ⎞ ∆ ∝ + − + −⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦ Summarizing, two interferograms are simultaneously measured, allowing to obtain broadband FM spectra. The in-phase interferogram provides spectrally resolved information on the difference of absorption experienced by each group of two sidebands. The in- quadrature interferogram gives the difference between the average of the dispersions experienced by the sidebands and the dispersion undergone by each carrier. For this first experimental demonstration, a narrow-band emission source covering 0.25 cm-1 (7.5 GHz) has been implemented as a test source. It is made of a fiber-coupled distributed feedback laser diode emitting around 1530 nm with an output power of a few mW. The current of the laser diode is modulated at about 20 Hz by a ramp generator. At each path difference step, while the interferometer is recording one interferogram sample, the laser frequency excursion is equal to 7.5 GHz, corresponding to one period of the triangular ramp. Consequently, for the interferometer, the laser diode behaves as a continuous emission source emitting over 0.25 cm-1. The interferometer output light is phase-modulated at fm = 150 MHz by the EOM and passes through an 80-cm cell filled at 10 hPa with acetylene in natural abundance. The light is next focused on an InGaAs nanosecond infrared photodetector, which according to Eq.5 delivers a signal proportional to the intensity of the beam containing a beat signal at the RF modulation frequency. The amplified detector signal is mixed with the reference signal at fm, down to d.c., using a commercial high frequency dual-phase lock-in amplifier. The reference may be phase-shifted with respect the signal used to drive the EOM. The two channels detected in-phase and in-quadrature are measured simultaneously. Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 Figure 2 shows a typical in-phase interferogram of C2H2. Its shape is characteristic of an interferogram of first-derivative type line-shapes. The 3 cm period amplitude modulation is due to the beat between the two strongest acetylene lines in the explored spectral domain. Figure 3 shows the two narrow-band spectra, Fourier transform of the in-phase (absorption) and in-quadrature (dispersion) interferograms. The spectral domain extension is limited by the tuning capabilities of the diode laser, which was used as a test source. This does not restrict the generality of the present demonstration. The lines belong to the ν1+ν3 and ν1+ν3+ν51-ν51 overtone bands of 12C2H2. The unapodised spectral instrumental resolution: 12.5 10-3 cm-1 (0.375 GHz) is narrower than the Doppler width of the lines. Signal to noise ratio is of the order of 1200. The total recording time of the order of 15 minutes is due to the need of adapting the interferometer recording mode procedure to the rather low laser diode frequency excursion period. The present validation of FM-FTS with a narrow band light source made the experience much simpler. Indeed, in wideband FTS, processing the signal of the interferogram needs special dynamic range solutions. Thanks to the only 0.25 cm-1-wide spectrum analysed in this experiment, a sophisticate RF detection chain, presently under development, was not necessary. The design of our Connes-type interferometer allows a balanced detection of the signals recorded at the two output ports. This will be helpful to remove the part of the interferogram which is not modulated by path difference and to consequently improve the dynamic range of the measurements. Similar solutions have already been successfully practiced for time-resolved FTS [7]. In FM-FTS, they are formally even easier to implement since the signal may be band-pass filtered around the modulation radio-frequency. In the present experimental set-up, the light should sequentially reach the equipment parts as shown in Fig.1. Briefly, to have a broadband equivalent of FM tunable laser spectroscopy, the sidebands generated by the EOM must not be resolved by the spectrometer. Also, since each carrier and its sidebands have to experience different attenuation and phase- shift, the EOM must be placed before the cell containing the gas of interest. This matter will be discussed in more detail elsewhere. This first FM-FTS experiment demonstrates the feasibility of coupling broadband laser sources, Fourier spectrometers and RF detection. This opens new perspectives in high sensitivity multiplex spectroscopy. FM-FTS may be coupled to a large variety of high brightness sources. This includes broadband cw lasers, supercontinua sources, mode-locked lasers as demonstrated recently [8], and Amplified Spontaneous Emission sources. Frequency nonlinear conversion may also be used when no laser source is available in the spectral range of interest. FM-FTS may be practiced with any kind of Fourier transform spectrometers, including commercially available instruments, at the expense of reasonable modifications in the signal detection scheme. The approach is also suitable at low spectral resolution. In such case, modulation frequencies lying in the GHz domain may be used. Moreover, FM-FTS induces new practices in Fourier transform spectroscopy. The modulation frequency is very high. The optical fringes generated by the interferometer can then be scanned at a much higher frequency than what is usually practiced nowadays. Path difference variation of the order of 1 m/s, is easily affordable. It corresponds to acquisition times expressed in second when presently the most efficient existing high resolution interferometers need 1 to 10 hours to record interferograms. Additionally, due to the low étendue of the analysed laser beams in our method, miniaturized instruments may be implemented. In addition to the radio-frequency detection scheme, sensitivity may be further enhanced by using an external optical resonator, thus increasing the effective absorption length. With FM-FTS, both the absorption and the dispersion associated with each spectral features are measured simultaneously. Despite its recognized interest for lineshape parameters retrieval, traditional dispersion spectroscopy has been poorly developed, only at low spectral Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 resolution, mostly due to its experimental complexity. FM-FTS should represent an easy manner of getting this information over extended spectral domains, which may induce new interest to the experimental investigation of dispersion profiles. References [1] G.C. Bjorklund, Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions, Optics Letters 5, 15-17 (1980). [2] G.C. Bjorklund, M.D. Levenson, W. Lenth, C. Ortiz, Frequency-modulation (FM) spectroscopy. Theory of lineshapes and signal-to-noise analysis, Applied Physics B 32, 145- 152 (1983). [3] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley and H. Ward, Laser phase and frequency stabilization using an optical resonator, Applied Physics B 31, 97-105 (1983). [4] W. Zapka, M. D. Levenson, F. M. Schellenberg, A. C. Tam, G. C. Bjorklund, Continuous- wave Doppler-free two-photon frequency-modulation spectroscopy in Rb vapor Optics Letters 8, 27-29 (1983) [5] J.L. Hall, L. Hollberg, T. Baer, H.G. Robinson, Optical heterodyne saturation spectroscopy, Applied Physics Letters 39, 680-682 (1981). [6] P. Maddaloni, P. Malara, G. Gagliardi, P. De Natale, Two-tone frequency modulation spectroscopy for ambient-air trace gas detection using a portable difference-frequency source around 3 µm, Applied-Physics-B-Lasers-and-Optics B85, 219-22 (2006). [7] N. Picqué, G. Guelachvili, High-information time-resolved Fourier transform spectroscopy at work, Applied Optics 39, 3984-3990 (2000). [8] J. Mandon, G. Guelachvili, N. Picqué, Frequency Comb Spectrometry with Frequency Modulation, in preparation, 2007. Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 Figure captions Fig. 1. Schematic of the experimental setup. Fig. 2. Absorption interferogram using in-phase RF detection with FM-FTS. Maximum path difference is 40 cm corresponding to 12.5 10-3 cm-1 unapodized resolution. Fig. 3. FM-FTS dispersion and absorption spectra of the acetylene molecule at 1528.6 nm. The middle plot represents the line relative intensities taken from the HITRAN database. Frequency modulation Fourier transform spectroscopy Mandon, Guelachvili, Picqué, 2007 Broadband source FTS EOM Sample Cell Driver Lock-In Detector Fig. 1. Schematic of the experimental setup. 0 10 20 30 40 Path difference (cm) FM-FTS Interferogram (absorption channel) Fig. 2. Absorption interferogram using in-phase RF detection with FM-FTS. Maximum path difference is 40 cm corresponding to 12.5 10-3 cm-1 unapodized resolution. 6541.50 6541.75 6542.00 Dispersion Absorption (2) R ν1+ν3+ν51-ν51 ν1+ν3 (cm-1) Fig. 3. FM-FTS dispersion and absorption spectra of the acetylene molecule at 1528.6 nm. The middle plot represents the line relative intensities taken from the HITRAN database. Frequency modulation Fourier transform spectroscopy

0704.0563

Universe Without Singularities. A Group Approach to De Sitter Cosmology

Microsoft Word - DeSitterEJTPnew.doc UNIVERSE WITHOUT SINGULARITIES A GROUP APPROACH TO DE SITTER COSMOLOGY Ignazio Licata Isem, Institute for Scientific Methodology, Pa, Italy IxtuCyber for Complex Systems, via Favorita 9, 91025, Marsala (TP), Italy [email protected] Abstract: In the last years the traditional scenario of “Big Bang” has been deeply modified by the study of the quantum features of the Universe evolution, proposing again the problem of using “local” physical laws on cosmic scale, with particular regard to the cosmological constant role. The “group extention” method shows that the De Sitter group univocally generalizes the Poincarè group, formally justifies the cosmological constant use and suggests a new interpretation for Hartle- Hawking boundary conditions in Quantum Cosmology. Key-words: Group Methods in Theoretical Physics; Projective Relativity; De Sitter Universe;Quantum Cosmology. 1. Introduction There are strong theoretical coherence reasons which impose to critically reconsider the approach to cosmological problem on the whole. The Quantum Cosmology’s main problem is to individuate the proper boundary conditions for the Universe’s wave function in the Wheeler-DeWitt equation. These conditions have to be such to allow the confrontation between a probability distribution of states and the observed Universe. In particular, it is expected to select a path in the configuration space able to solve the still open problems of the Big-Bang traditional scenario: flat space, global homogeneity (horizon problem) and the “ruggedness” necessary to explain the tiny initial dishomogeneities which have led to the formation of the galactic structures. The inflationary cosmology ideas has partly supplied with a solution to the standard model wants by introducing the symmetry breaking and phase transition notions which are at the core of Quantum Cosmology. The last one also finds its motivation in the necessity to provide with a satisfactory physical meaning to the initial singularity problem, unavoidable in GR under the condition of the Hawking-Penrose theorem (Hawking & Ellis, 1973). The Hartle-Hawking “no-boundary” condition seems to provide a very powerful constraint for the Quantum Cosmology main requirements, but appears as an “ad hoc” solution which could be deduced by a fundamental approach. Particularly, the mix of topologies used to conciliate the without boundary Universe symmetry with the Big-Bang evolutionary scenario is unsatisfactory. We realize that most part of the Quantum Cosmology problems inherit the uncertainties of the Fridman model in GR, so they derive from the euristic use of the local laws on cosmic scale. A possible way-out is the Fantappié-Arcidiacono group approach which allows to individuate a Universe model without recourse to arbitrary extrapolations of the symmetry groups valid in physics. The group extension theory naturally finds again the Hartle-Hawking condition on the Universe wave function and allows to firmly founding theoretically the Quantum Cosmology. The price to pay is a subtle methodological question on using the GR in cosmology. In fact, in 1952 Fantappié pointed out that the problem of the use of local laws to define the cosmological boundary conditions is due to the fact that GR describes matter in terms of local curvature, but leaves the question of space-time global structure indeterminate. It happens because, differently from RR, GR has not be built on group base, which thing should be central in building any theory up, especially when it aims to express universally valid statements on physical world, the class of the superb theories, how Roger Penrose called them. We are going to examine here the foundations of the group extension method (par. 2) and the relativity in the De Sitter Universe (par. 3, 4), we introduce the conditions to define matter-fields (par.5).In (par.6) we analyze the physical significance of the observers in an istantonic Universe at imaginary time, and in (par.7) investigate the physical meaning of an Hartle-Hawking condition in an hyper-spherical universe. 2. An Erlangen Program for Cosmology In 1872 Felix Klein (1849-1925) presented the so-called Erlangen program for geometry, centred upon the symmetry transformations group. From 1952, Fantappié, basing on a similar idea and in perfect consonance with Relativity spirit, proposed an Erlangen program for physics, where a Universe is univocally individuated by a symmetry group which let its physical laws invariant (Fantappié,1954, 1959). It has to be underlined that in the theory Universe means any physical system characterized by a symmetry group. The space-time isotropy and homogeneity principle with respect to physical laws tells us that the physical law concept itself is based upon symmetry. So the essential idea is to individuate physical laws starting from the transformations group which let them invariant. We observe here that there are infinite possible transformations group which individuate an isotropic and homogeneous space-time. In order to build the next improvements in physics using the group extension method, we can follow the path indicated by the two groups we know to be two valid description levels of the physical world: the Galilei group and the Lorentz-Poincarè one. It is useful to remember that the Galilei group is a particular case of the Lorentz one when ∞→c ,i.e. when it is not made use of the field notion and the interactions velocity is considered to be infinite. Staying within a quadrimensional space-time and consequently considering only groups at 10 parameters and continuous transformations, Fantappié showed that the Poincaré group can be considered a limit case of a broader group depending with continuity on c and another parameter r: the Fantappié group; moreover this group cannot be further extended under the condition to stay within a group at 10 parameters. So we have the sequence: 1031 31 +++ →→ FLG Where G is the Galilei group, L the Lorentz one and F the Fantappié final one, from which with ∞→R , we get the L group. It is shown that such sequence of universes is univocal. The Lorentz group can be mathematically interpreted as the group of roto-translations such to let that particular object that is the Minkowski space-time invariant. Similarly, the Fantappié group is the one of the pentadimensional rotations of a new space-time: the hyper-spherical and at constant curvature De Sitter universe (maximally symmetric). We point out we have obtained the De Sitter model without referring to the gravitational interaction, differently from the GR where the De Sitter universe is one of the possible solutions of the Einstein equations with cosmological constant. From a formal viewpoint we make recourse to pentadimensional rotations because in the De Sitter universe there appears a new constant r, which can be interpreted as the Universe radius. The group extension mechanism individuates an univocal sequence of symmetry groups; for each symmetry group we have a corresponding level of physical world description and a new universal constant, so providing the most general boundary conditions and constraining the form of the possibile physical laws. The Fantappié group fixes the c and r constants and defines a new relativity for the inertial observers in De Sitter Universe. In this sense, the Theory of Universes- based on group extension method- is actually a version of what is sought for in the Holographic Principle: the possibility to describe laws and boundaries in a compact and unitary way. In 1956 G. Arcidiacono proposed to study the De Sitter S4 absolute universe by means of the tangent relative spaces where observers localize and describe the physical events by using the Beltrami-Castelnuovo 4P projective representation in the Projective Special Relativity, PSR (Arcidiacono,1956; 1976; 1984). We note that we pass from hyper-spherical S4 to its real representation as hyperboloid by means of an inverse Wick rotation, rotating τ→it and associating the great circles on the hyper-sphere with a family of geodesics on the hyperboloid. In this way, we get a realization of the Weyl principle for defining a Universe model, because it fixes a set of privileged observers (Ellis & Williams, 1988). So, the choice of 4P Beltrami-Castelnuovo is equivalent to study a relativity in 4S . 3. The Fantappié Group Transformations To study the De Sitter 4S universe according to Beltrami-Castelnuovo representation we have to set the projectivities which let the Cayley-Klein interval invariant: (1.3) 0222222 =+−++ rtczyx . The (1.3) meets the time axis in the two 0tt ±= “singularities”, where crt =0 is the time it takes light to run the Universe r radius. In this case the singularities’ meaning is purely geometrical, not physical, and they represent the hyperboloid rims (1.3), since the De Sitter universe is lacking in “structural” singularities. The 4S invariant transformations are the 5-dimensional space rotations which lead on the 4P observer’s space the projectivities that let the (1.3) unchanged. Let’s introduce the five homogeneous projective coordinates (Weierstrass condition): (2.3) 2rxx aa = , with .4,3,2,1,0=a The ix space-time coordinates, with i = 1,2,3,4 are: (3.3) xx =1 , yx =2 , zx =3 , ictx =4 . The connection between the (2.3) and (3.3) is given by the relation: (4.3) 0xxrx ii = from which, owing to (2.3), we get the inverse relation: (5.3) arx =0 , axx ii = , where 2222 11 γα −+=+= rxxa ii , with rx =α and 0tt=γ . The searched transformation between the two 'O and O observers consequently has the form: (6.3) ax = bab xα with abα orthogonal matrix. Limiting ourselves, just for simplicity reasons, to the 410 ,, xxx variables and following the standard method, also used in RR, we get 3 families of transformations: A) the space translations along the x axis, given by the ( 10 , xx ) rotation: (7.3) ϑϑ sincos 01 1 xxx += +−= ϑsin1 0 xx ϑcos0x 4 4 xx = . Using the (4.3) and putting αϑ == r Ttg , we get the space-time transformations with T parameter: (8.3) ' , 1 2' . The (8.3) for r indeterminate, i.e. ∞→r , are reduced to the well-known space translations of the classical and relativistic cases, connected by the T parameter. B) the T0 parameter time translation, given by the ( 40 , xx ) rotation: (9.3) 0004 4 sincos ϑϑ xxx += +−= 04 0 sinϑxx 00 cosϑx 1 1 xx = . Putting γϑ itTitg == 000 we obtain: (10.3) = , Also the (10.3), when ∞→r are reduced to the known cases of classical and relativistic physics. C) the V parameter inertial transformations, given by the ( 41 , xx ) rotation: (11.3) 04011 sincos ϕϕ xxx += +−= 01 4 sinϕxx 04 cosϕx 0 0 xx = . Putting βϕ icVitg == , here we find again the Lorentz transformations: (12.3) x , The (A), (B) and (C) transformations form the Fantappié projective group which for two variables (x,t) and three parameters (T,T0,V), with T translations and V velocity along x, can be written: (13.3) ( )[ ] ( ) ( ) 0 ttrxab bTctax αβγβγα γβγαβ ( )[ ] ( ) ( ) 0 ttrxab bTtcxa αβγβγα αβγαβ where we have put 221 γα −+=a and 22 )(1 βγαβ −+−=b , with rx=α , cV=β and 0tt=γ . For ∞→r we get a = 1 and 21 β−=b , and from (13.3) we obtain the Poincaré group with three parameters (T, T0,V). The Fantappié group can be synthesized by a very clear geometrical viewpoint, saying that the De Sitter universe at 21 r constant curvature shows an elliptic geometry in its hyper spatial global aspect (Gauss-Riemann) and an hyperbolic geometry in its space-time sections (Lobacevskij). Making the “natural” r unit of this two geometries tend towards infinity we obtain the parabolic geometry of Minkowski flat space. 4. The Projective Relativity in De Sitter Universe The Projective Special Relativity (PSR) widens and contextualizes the relativistic results in De Sitter geometry.Just like in any physics there exists a wll-defined connection between mechanics and geometry. Therefore the PSR makes use of the notion of observer’s private space, redifining it on the basis of a constant curvature. In PSR it is introduced a space temporal double scale which connects a ( τχ , ) point of S4 with a (x,t) one of P4 by means the (1.3) projective invariant. Given a AB straight line and put as R and S the intersections with (1.3), the projective distance is given by the logarithm of the (ABRS) bi-ratio: (1.4) ( ) ( ) ( ) ( ) ( )ASBRBSARtABRStAB ⋅⋅== log2log2 00 . From the (1.4) we obtain: (2.4) rarctg=χ and 00 log From the (2.4) second one, similar to the Milne’s formula, we can see that the “formal” singularities are related to the projective description which depicts a universe with infinite space and finite time, whereas the De Sitter one is with finite space and infinite time. It is important to underline that such equivalence between an “evolutionary” model and a “stationary” one, differently from what is often stated, is purely geometrical and has nothing to do with the physical processes, but it deals with the cosmological observer definition.We will speak again about such fundamental point further. The addition of durations’ new law: (3.4) 1 tdd it is obtained by the (10.3) formulae and finds its physical meaning in the appearing of the new crt =0 , interpretable as the “universe age” for any 4P observer family. Let us consider a uniform motion with U velocity, given by '' Utx = , by means of Fantappié transformations we have a uniform motion with W velocity given by: (4.4) ( ) ( ) ( ) cUVcVU −+−++ For the visible universe of the O observer, inside the light-cone, it is valid the condition γα ±= and a=1 , and the (4.4) can be simplified as: (5.4) ( )( ) cVcUcVU For V = c then W=c, according to RR, while for U=c we have: (6.4) ( ) ( ) ccVcVccW ≠+−±= 112 2α . The (6.4) expresses the possibility of observing hyper-c velocity in PSR. The outcome is less strange than it can seem at first sight, because now the space-time of an observer is defined not only by the c constant but also by r, and the light-cone is at variable aperture. In straighter physical terms it means that when we observe a far universe region of the crt =0 order, the cosmic objects’ velocity appears to be superior to c value, even if the region belongs to the light-cone of the observer’s past. For b=0 we obtain the angular coefficients of the tangents to the (1.3) Cayley-Klein invariant starting from a P point of the Beltrami-Castelnuovo projection, which represent the two light-cone’s straight lines. Differently from RR, here the light-cone’s angle is not constant and depends on the P point according to the formula: (7.4) ( )222 γαϑ += atg . From the (7.4) derives the C variation of the light velocity with time: (8.4) 21 γ− C , with from which follows that ∞→C in the two 0t± singularities which fix the limit duration according to the addition of durations’ new law (3.4). Another remarkable consequence of the projective group is the expansion-collapse law, that is the connection between the two singularities. Differentiating the (10.3) and dividing them we obtain the velocities’ variation law for a translation in time: (9.4) ( ) 002' 11 txttVV γγγ −+=− . For 1=γ and 00 tT = we have the law of projective expansion valid for 00 <<− tt : (10.4) = , or also If ( )00 == tγ , we can write (11.4) HxtxV == 0 , ( )αβ = , where 01 trcH == is the well-known Hubble constant. The analogous procedure will be followed for the law of projective collapse valid for 00 tt << , with 1−=γ and 00 tT −= : (12.4) = , or We note that in singularities the expansion-collapse velocity becomes infinite. In PSR such process, differently from GR, is not connected to gravitation, but derives from Beltrami- Castelnuovo geometry. From the Fantappié group it also follows a new formula for the Doppler effect: (13.4) ( ) ( ) 2' 11 αββωω ++−= , where ω is the frequency. For 1=β , which is V=c, we get nothing but the traditional proportionality between distance and frequency, αωω =' . For V=0 there follows a Doppler effect depending on distance: (14.4) 2' 1 αωω += . The z red-shift is defined by '1 ωω=+ z and the (13.4) becomes: (15.4) ( ) ( ) ( ) 21111 αββ ++−=+ z , which was historically introduced- in a 1930 Accademia dei Lincei famous memoir- by Castelnuovo to explain the “new” Hubble observations on galactic red-shift. If we are placed on the observer’s light-cone where the (12.4) becomes )1( ααβ −= , the (15.4) will be: (16.4) ( )α−=+ 111 z . The red-shift tends towards infinity for x = r, and hyper- c velocities are possible if z > 1. As everybody would naturally expect, modifying geometry implies, as well as in RR, a deep redefinition of mechanics. In PSR, the m mass of a body varies with velocity and distance according (17.4) bamm 20= . From the (17.4) it follows that for a = 0, in singularities, the mass is null, while on the light- cone, for b = 0, ∞→m . The mass of a body at rest varies with t according to: (18.4) ( )20 1 γ−= mm , from which we deduce that at the initial and final instant, 1±=γ , the mass vanishes. Another greatly important outcome (Arcidiacono, 1977) is the relation between m mass and the J polar inertia momentum of a body: (19.4) 2mrJ = A remarkable consequence is that the universe M mass varies with t: (20.4) ( ) ( ) MtM +−= γ , where M0 is the mass for 0=t , and J the polar momentum with respect to the observer. So the overall picture for an inertial observer in a De Sitter Universe is that of a universe coming into existence in a singularity at –t0 time, expanding and collapsing at t0 time and where c light velocity is only locally constant. In the initial and final instants the light velocity is infinite and the global mass is zero while in the expansion-collapse time it varies according to (20.4). In the projective scenario the space flatness is linked to the observer geometry in a universe at constant curvature. All this is linked to the fact that in PSR the translations and rotations are indivisible. In the singularities there is no “breakdown” of the physical laws because the global space-time structure is univocally individuated by the group which is independent of the matter-energy distribution. In this case, the singularities in 4P are – more properly- an horizon of events with a natural “cosmic censure” fixed by observers’ geometry. 5. The Projective Gravitation The connection between the metric approach to Einstein gravitation and Fantappié-Arcidiacono group one is the aim of Projective General Relativity(PGR), which describes a universe globally at constant curvature and locally at variable curvature. It can be done by following the Cartan idea, where any 4V Riemann manifold is associated with an infinite family of Euclidean, pseudo- Euclidean, non-Euclidean spaces tangent to it in each of its P points. Those spaces’ geometry is individuated by a holonomy group. The Cartan connection law links the tangent spaces so as to obtain both the 4V local characteristics (curvature and torsion) and the global ones (holonomy group). The GR holonomy group is the one at four dimension rotations, i.e. the Lorentz group. So we get a general method which builds a bridgeway up between differential geometry and group theory (Pessa, 1973; Arcidiacono, 1986) To make a PGR it is introduced the 5V Riemann manifold which allows as holonomy group the De Sitter-Fantappié one, isomorphic to the 5S five-dimensional rotations’ group. The 5V geometry is successively written in terms of Beltrami projective inducted metric for a anholomonous 4V manifold at variable curvature. The Veblen projective connection: (1.5) { }ABCABC =π = ( )BCSCSBBSCAS gggg ∂−∂+∂2 defines a projective translation law which let the field of the Q quadrics invariant in the tangent spaces, in each 4V point, 0== BAAB xxgQ ,where ABg are the coefficients of the five-dimensional metric, the Kx are the homogeneous projective coordinates, and (ABC)=0,1,..,4.From the (1.5) we build the projective torsion-curvature tensor: (2.5) SBC BCDR ππππππ −+∂−∂= . So the gravitation equations of Projective General Relativity are: (3.5) ABABAB TRgR χ=− 2 with ABT energy-momentum tensor, and χ Einstein gravitational constant. The (2.5) tensor is projectively flat, i.e. when it vanishes we get the De Sitter space at constant curvature. The deep link between rotations and translations in 4S naturally leads the (3.5) to include the torsion, showing an interesting formal analogy with Einstein-Cartan- Sciama-Kibble spin-fluids theory. The construction is analogous to the GR one, but in lieu of the relation between Riemann curvature and Minkowski s-t, we get here a curvature-torsion connected to the De Sitter-Fantappié holonomy group. It has to be noted that, in concordance with the equivalence principle, the PGR gives a metric description of the local gravity, valid for single( i.e., non cosmological) systems. It is here proposed again the problem of the relations between local physics and its extension on cosmic scale. In fact, if we take the starting expression of standard cosmology based upon GR, i.e. let us consider the whole matter of Universe, and transfer it within the ambit of PGR, we can ask ourselves if the torsion role, associated to the rotation one, could get a feed-back on the background metric, modifying it deeply. Generally, the syntax of a purely group-based theory does not get the tools to give an answer, because it is independent from gravity and the hypotheses on ABT . For example, Snyder (Snyder, 1947) showed that in a De Sitter space it is introduced an uncertainty relation linked to a curvature of the kind: 21 rxx ki ≈∆∆ . Only a third quantization formalism, able to take into account the dynamical two-way inter-relations between local and global, will succeed in giving an answer. The essential point we have to underline here is that the introduction of a cosmological constant, both as additional hypothesis on Einstein equations or via group, is a radical alternative to the “machian philosophy” of the GR. So, for a Universe without metter-fields we assume the constant curvature as a sort of “pre- matter” which describes in topological terms the most general conditions for the quantum vacuum. Therefore the Einstein equations in the following form are valid: (4.5) ABAB gG Λ= and ( ) ABAB gRR Λ−= 2 , with their essentially physical content, i.e. the deep connection among curvature, radius and matter- energy’s density vacρ by means of the cosmological constant: (5.5) vac π 6. De Sitter Observers, Singularities and Wick Rotations From a quantum viewpoint the 4S interesting aspect is that it is at imaginary cyclic time and without singularities. It means that it is impossible to define on De Sitter a global temporal coordinate. So it has an istanton feature, individuated by its Euler topological number which is 2 (Rajaraman,1982). This leads to a series of formal analogies both with black holes’ quantum physics and the theoretical proposals for the “cure” for singularities. Let us consider the De Sitter-Castelnuovo metric in real time: (1.6) 222 2 11 Ω+�� drdrr ds , where 2222 sin ϕϑϑ ddd +=Ω in polar coordinates. As we have seen in PSR, the singularity in Hcr = becomes an horizon of events for any observer when it passes to the Euclidean metric with it−→τ : (2.6) ( )222 22 sincos Ω++= rddrH dds ττ , with a close analogy with the Schwarzschild solution’s case. The τ period is Hπβ 2= ; for the observers in De Sitter it implies the possibility to define a temperature, an entropy and an area of the horizon, respectively given by: (3.6) 1 −== β Tb ; π From the (3.6) we get the following fundamental outcome: (4.6) AS which is the well-known expression of the t’Hooft-Susskind-Bekenstein Holographic Principle(Susskind,1995). The (4.6) connects the non-existence of a global temporal coordinate with the information accessible to any observer in the De Sitter model. In this way we obtain a deep physical explanation for applying the Weyl Principle in the De Sitter Universe, and sum up that in cosmology, as well as in QM, a physical system cannot be fully specified without defining an observer. G. Arcidiacono stated that the hyper-spherical Universe is like a book written with seven seals ( Apocalypse, 6-11), and consequently two operations are necessary to investigate its physics: 1) inverse Wick rotation and 2) Beltrami-Castelnuovo representation. That’s the way we can completely define a relativity in De Sitter. The association of imaginary time with temperature gets a remarkable physical significance which implies some considerations on the statistical partition function (Hawking, 1975). For our aims it will be sufficient to say that such temperature is linked to the (4.6) relation, i.e. to the information that an observer spent within his area of events. Which thing has patent implications from the dynamical viewpoint, because it is the same as to state that, as well as in Schwarzschild black hole’ s case, the De Sitter space and the quantum field defined on it behave as if they were immersed in background fluctuations. The transition amplitude from a configuration of a φ generic field in dttt =− 12 time will be given by the iHdte− matrix element which acts as a ( )1U group transformation of the ( ) ( )timespace UU 11 ⇔ . It means that a transition amplitude on 4S will appear to an observer as the ( )tR scale factor’s variation with H variation rate. It makes possible to link the hyper-spherical description with the Big-Bang evolutionary scenario and to get rid of the thermodinamic ambiguities which characterize its “beginning” and “ending” notions. The last ones have to be re-interpretated as purely quantum dynamics of the matter-fields on the hyper-sphere free of singularities. 7. Physical Considerations for Further Developments Such considerations suggest a research program we are going here to shortly delineate ; it furthermore develops the analogy between black holes, istantons and De Sitter Universes (see – for example – Frolov, Markov, Mukhanov, 1989;Strominger, 1992). It is known that the Hartle- Hawking proposal of “no-boundary” condition removes the initial singularity and allows to calculate the Universe wave function (Hartle-Hawking, 1989). In fact, it is possible – as in the usual QFT- to calculate the path integrals by using a Wick rotation as “Euclidization” procedure. In such way also the essential characteristics of the inflationary hypotheses are englobed (A. Borde, A. Guth and A. Vilenkin, 2003). The derived formalism is similar to that used in the ordinary QM for the tunnel effect, an analogy which should explain the physics at its bottom (Vilenkin, 1982; S.W. Hawking and I.G. Moss, 1982). The group extension method provides this procedure with a solid foundation, because the De Sitter space, maximally symmetric and simply connected, is univocally individuated by the group structure, and consequently is directly linked to the space-time homogeneity and isotropy principle with respect to physical laws. The original Hartle-Hawking formulation operates a mix of topologies hardly justified both on the formal level and the conceptual one. The “no-boundary” condition is only valid if we works with imaginary time, and the theory does not contain a strict logical procedure to explain the passage to real time. This corresponds to a quite vague attempt to conciliate an hyper-spherical description at imaginary time with an evolutive one at real time according to the traditional Big-Bang scenario.In fact, it has been observed that the Hartle-Hawking condition is the same as to substitute a singularity with a “nebulosity”. The spontaneous proposal, at this point, is considering the Hartle-Hawking conditions on primordial space-time as a consequence of a global charaterization of the hyper-sphere and directly developing quantum physics on 4S .Which thing does not contradict the quantum mechanics formulation and its fundamental spirit, which is to say the Feynman path integrals. In other words, quantum mechanics has not to be applied to cosmology for the Universe smallness at its beginning, but because each physical system – without exception- gets quantum histories with amplitude interferences. We point out that such view is in perfect consonance with the so-called quantum mechanics Many Worlds Interpretation ( Halliwell, 1994). The “by nothing creation” means that we cannot “look inside” an istanton (hyper-spherical space), but we have to recourse to an “evolutionary” description which separates space from time. The projective methods tell us how to do it. An analogous problem– to some extent – is that of the Weyl Tensor Hypothesis. Recently, Roger Penrose has suggested a condition on the initial singularity that, within the GR, ties entropy and gravity and makes a time arrow emerge (Penrose,1989). It is known that the ABCDW Weyl conformal tensor describes the freedom degrees of the gravitational field. The Penrose Hypothesis is that 0→ABCDW in the Big-Bang, while ∞→ABCDW in the Big-Crunch. The physical reason is that in the Universe’s initial state we have an highly uniform matter distribution at low entropy ( entalpic order), while in Big-Crunch, just like a black hole, we have an high entropy situation. This differentiates the two singularities and provides a time arrow. In an hyper-spherical Universe there is no “beginning” and “ending”, but only quantum transitions.Consequently, the Penrose Hypothesis can only be implemented in terms of projective representation within the ambit of PGR. Finally, we can take into consideration the possibility to build a Quantum Field Theory on 4S . A QFT, for T tending towards zero, is a limit case of a theory describing some physical fields interacting with an external environment at T temperature. Without this external environment we could not speak of dechoerence , could not introduce concepts such as like dissipation, chaos, noise and, obviously, the possibility to describe phase transitions would vanish too. Therefore, it is of paramount importance to write a QFT on De Sitter background metric and then studying it in projective representation. If we admit decoherence processes on 4S , it is possible to interpret the Weyl Principle as a form of Anthropic Principle: the “classical” and observable Universes are the ones where it can be operated a description at real time. In conclusion, it is possible to delineate an alternative, but not incompatible with traditional cosmology scenario.The Universe is the quantum configuration of the quantum fields on 4S .Thus developing a Quantum Cosmology coincides with developing a Quantum Field Theory on a space free of singularities.The Big-Bang is a by vacuum nucleation in an hyper-spherical background at imaginary time, and so the concepts of “beginning”, “expansion” and “ending” belong to the space- time foreground and gain their meaning only by means of a suitable representation which defines a family of cosmological observers. Acknowledgements: I owe my knowledge of the group extension method to the regretted Prof. G. Arcidiacono (1927 – 1998), during our intense discussions while strolling throughout Rome. Special thanks to my friends E. Pessa and L. Chiatti for the rich exchange of viewpoints and e- mails. References Arcidiacono, G.(1956), Rend. Accad. Lincei,20, 4 Arcidiacono, G. (1976), Gen. Rel. And Grav., 7, 885 Arcidiacono, G.(1977), Gen. Rel. And Grav.,7, 865 Arcidiacono, G.(1984), in De Sabbata,V. & T.M.Karade (eds), Relativistic Astrophysics and Cosmology, World Scientific, Singapore Arcidiacono, G. (1986), Projective Relativity,Cosmology and Gravitation, Hadr.Press,Cambridge,USA Borde, A. , Guth,A., Vilenkin, A. (2003),Phys. Rev. Lett. 90 Ellis, G.F.R. & Wiliams,R. (1988), Flat and Curved Space-Times, Clarendon Press Fantappié, L. (1954), Rend. Accad. Lincei,17,5 Fantappié, L. (1959), Collectanea Mathematica,XI, 77 Frolov, V.P., Markov,M.A., Mukhanov,V.F.(1989), Phys.Lett.,B216,272 Halliwell,J.J. (1994), in Greenberger,D. & Zeilinger,A.(eds), Fundamental Problems in Quantum Theory, New York Academy of Sciences,NY Hartle, J.B. & Hawking,S.W. (1983), Phys.Rev.D,28,12 Hawking,S.W. & Ellis, G.F.R.,(1973) The Large Scale Structure of Space-Time,Cambridge Univ.Press Hawking,S.W. (1975), Commun.Math.Phys.,43 Hawking, S.W. & Moss, I.G. (1982), Phys.Lett.,B110,35 Pessa,E. (1973), Collectanea Mathematica,XXIV,2 Rajaraman,R.(1982), Solitons and Istantons,North-Holland Publ.,NY Snyder, H.S. (1947), Phys.Rev., 51,38 Strominger, A. (1992), Phys.Rev.D,46,10 Susskind, L. (1995), Jour.Math.Phys.,36 Vilenkin, A. (1982), Phys.Lett.,117B,1.

0704.0564

Spectral action on noncommutative torus

CENTRE DE PHYSIQUE THÉORIQUE 1 CNRS–Luminy, Case 907 13288 Marseille Cedex 9 FRANCE Spectral action on noncommutative torus D. Essouabri2, B. Iochum1,3, C. Levy1,3 and A. Sitarz4,5 Dedicated to Alain Connes on the occasion of his 60th birthday Abstract The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophan- tine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context. February 2007 PACS numbers: 11.10.Nx, 02.30.Sa, 11.15.Kc MSC–2000 classes: 46H35, 46L52, 58B34 CPT-P06-2007 1 UMR 6207 – Unité Mixte de Recherche du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’Université du Sud Toulon-Var – Laboratoire affilié à la FRUMAM – FR 2291 2 Université de Caen (Campus II), Laboratoire de Math. Nicolas Oresme (CNRS UMR 6139), B.P. 5186, 14032 Caen, France, [email protected] 3 Also at Université de Provence, [email protected], [email protected] 4 Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland [email protected] 5 Partially supported by MNII Grant 115/E-343/SPB/6.PR UE/DIE 50/2005–2008 http://arxiv.org/abs/0704.0564v2 1 Introduction The spectral action introduced by Chamseddine–Connes plays an important role [3] in noncom- mutative geometry. More precisely, given a spectral triple (A,H,D) where A is an algebra acting on the Hilbert space H and D is a Dirac-like operator (see [8, 23]), they proposed a physical action depending only on the spectrum of the covariant Dirac operator DA := D +A+ ǫ JAJ−1 (1) where A is a one-form represented on H, so has the decomposition ai[D, bi], (2) with ai, bi ∈ A, J is a real structure on the triple corresponding to charge conjugation and ǫ ∈ { 1,−1 } depending on the dimension of this triple and comes from the commutation relation JD = ǫDJ. (3) This action is defined by S(DA,Φ,Λ) := Tr Φ(DA/Λ) where Φ is any even positive cut-off function which could be replaced by a step function up to some mathematical difficulties investigated in [16]. This means that S counts the spectral values of |DA| less than the mass scale Λ (note that the resolvent of DA is compact since, by assumption, the same is true for D, see Lemma 3.1 below). In [18], the spectral action on NC-tori has been computed only for operators of the form D+A and computed for DA in [20]. It appears that the implementation of the real structure via J , does change the spectral action, up to a coefficient when the torus has dimension 4. Here we prove that this can be also directly obtained from the Chamseddine–Connes analysis of [4] that we follow quite closely. Actually, S(DA,Φ,Λ) = 0<k∈Sd+ − |DA|−k +Φ(0) ζDA(0) +O(Λ−1) (5) where DA = DA + PA, PA the projection on KerDA, Φk = 12 Φ(t) tk/2−1 dt and Sd+ is the strictly positive part of the dimension spectrum of (A,H,D). As we will see, Sd+ = { 1, 2, · · · , n } |DA|−n = |D|−n. Moreover, the coefficient ζDA(0) related to the constant term in (5) can be computed from the unperturbed spectral action since it has been proved in [4] (with an invertible Dirac operator and a 1-form A such that D +A is also invertible) that ζD+A(0) − ζD(0) = (−1)q −(AD−1)q, (6) using ζX(s) = Tr(|X|−s). We will see how this formula can be extended to the case a noninvert- ible Dirac operator and noninvertible perturbation of the form D+ à where à := A+ εJAJ−1. All this results on spectral action are quite important in physics, especially in quantum field theory and particle physics, where one adds to the effective action some counterterms explicitly given by (6), see for instance [2–5,17,18,20,22,28,35–38]. Since the computation of zeta functions is crucial here, we investigate in section 2 residues of series and integrals. This section contains independent interesting results on the holomorphy of series of holomorphic functions. In particular, the necessity of a Diophantine constraint is naturally emphasized. In section 3, we revisit the notions of pseudodifferential operators and their associated zeta functions and of dimension spectrum. The reality operator J is incorporated and we pay a particular attention to kernels of operators which can play a role in the constant term of (5). This section concerns general spectral triple with simple dimension spectrum. Section 4 is devoted to the example of the noncommutative torus. It is shown that it has a vanishing tadpole. In section 5, all previous technical points are then widely used for the computation of terms in (5) or (6). Finally, the spectral action (6) is obtained in section 6 and we conjecture that the noncom- mutative spectral action of DA has terms proportional to the spectral action of D + A on the commutative torus. 2 Residues of series and integral, holomorphic continuation, etc Notations: In the following, the prime in means that we omit terms with division by zero in the summand. Bn (resp. Sn−1) is the closed ball (resp. the sphere) of Rn with center 0 and radius 1 and the Lebesgue measure on Sn−1 will be noted dS. For any x = (x1, . . . , xn) ∈ Rn we denote by |x| = x21 + · · ·+ x2n the euclidean norm and |x|1 := |x1|+ · · · + |xn|. N = {1, 2, . . . } is the set of positive integers and N0 = N ∪ {0} the set of non negative integers. By f(x, y) ≪y g(x) uniformly in x, we mean that |f(x, y)| ≤ a(y) |g(x)| for all x and y for some a(y) > 0. 2.1 Residues of series and integral In order to be able to compute later the residues of certain series, we prove here the following Theorem 2.1. Let P (X) = j=0 Pj(X) ∈ C[X1, · · · ,Xn] be a polynomial function where Pj is the homogeneous part of P of degree j. The function ζP (s) := P (k) , s ∈ C has a meromorphic continuation to the whole complex plane C. Moreover ζP (s) is not entire if and only if PP := {j : u∈Sn−1 Pj(u) dS(u) 6= 0} 6= ∅. In that case, ζP has only simple poles at the points j + n, j ∈ PP , with s=j+n ζP (s) = u∈Sn−1 Pj(u) dS(u). The proof of this theorem is based on the following lemmas. Lemma 2.2. For any polynomial P ∈ C[X1, . . . ,Xn] of total degree δ(P ) := i=1 degXiP and any α ∈ Nn0 , we have P (x)|x|−s ≪P,α,n (1 + |s|)|α|1 |x|−σ−|α|1+δ(P ) uniformly in x ∈ Rn verifying |x| ≥ 1, where σ = ℜ(s). Proof. By linearity, we may assume without loss of generality that P (X) = Xγ is a monomial. It is easy to prove (for example by induction on |α|1) that for all α ∈ Nn0 and x ∈ Rn \ {0}: |x|−s β,µ∈Nn0 β+2µ=α |β|1+|µ|1 ) (|β|1+|µ|1)! β! µ! |x|σ+2(|β|1+|µ|1) It follows that for all α ∈ Nn0 , we have uniformly in x ∈ Rn verifying |x| ≥ 1: |x|−s ≪α,n (1 + |s|)|α|1 |x|−σ−|α|1 . (7) By Leibniz formula and (7), we have uniformly in x ∈ Rn verifying |x| ≥ 1: xγ |x|−s ∂β(xγ) ∂α−β |x|−s ≪γ,α,n β≤α;β≤γ xγ−β (1 + |s|)|α|1−|β|1 |x|−σ−|α|1+|β|1 ≪γ,α,n (1 + |s|)|α|1 |x|−σ−|α|1+|γ|1 . Lemma 2.3. Let P ∈ C[X1, . . . ,Xn] be a polynomial of degree d. Then, the difference ∆P (s) := P (k) Rn\Bn P (x) which is defined for ℜ(s) > d+ n, extends holomorphically on the whole complex plane C. Proof. We fix in the sequel a function ψ ∈ C∞(Rn,R) verifying for all x ∈ Rn 0 ≤ ψ(x) ≤ 1, ψ(x) = 1 if |x| ≥ 1 and ψ(x) = 0 if |x| ≤ 1/2. The function f(x, s) := ψ(x) P (x) |x|−s, x ∈ Rn and s ∈ C, is in C∞(Rn × C) and depends holomorphically on s. Lemma 2.2 above shows that f is a “gauged symbol” in the terminology of [24, p. 4]. Thus [24, Theorem 2.1] implies that ∆P (s) extends holomorphically on the whole complex plane C. However, to be complete, we will give here a short proof of Lemma 2.3: It follows from the classical Euler–Maclaurin formula that for any function h : R → C of class CN+1 verifying lim|t|→+∞ h(k)(t) = 0 and |h(k)(t)| dt < +∞ for any k = 0 . . . , N + 1, that we have ∑ h(k) = h(t) + (−1)N (N+1)! BN+1(t) h (N+1)(t) dt where BN+1 is the Bernoulli function of order N + 1 (it is a bounded periodic function.) Fix m′ ∈ Zn−1 and s ∈ C. Applying this to the function h(t) := ψ(m′, t) P (m′, t) |(m′, t)|−s (we use Lemma 2.2 to verify hypothesis), we obtain that for any N ∈ N0: ψ(m′,mn) P (m ′,mn) |(m′,mn)|−s = ψ(m′, t) P (m′, t) |(m′, t)|−s dt+RN (m′; s) (8) where RN (m′; s) := (−1) (N+1)! BN+1(t) N+1 (ψ(m ′, t) P (m′, t) |(m′, t)|−s) dt. By Lemma 2.2, ∣∣∣BN+1(t) ∂ ψ(m′, t) P (m′, t) |(m′, t)|−s ) ∣∣∣ dt ≪P,n,N (1 + |s|)N+1 (|m′|+ 1)−σ−N+δ(P ). m′∈Zn−1 RN (m′; s) converges absolutely and define a holomorphic function in the half plane {σ = ℜ(s) > δ(P ) + n−N}. Since N is an arbitrary integer, by letting N → ∞ and using (8) above, we conclude that: (m′,mn)∈Zn−1×Z ψ(m′,mn) P (m ′,mn) |(m′,mn)|−s− m′∈Zn−1 ψ(m′, t) P (m′, t) |(m′, t)|−s dt has a holomorphic continuation to the whole complex plane C. After n iterations, we obtain that ψ(m) P (m) |m|−s − ψ(x) P (x) |x|−s dx has a holomorphic continuation to the whole C. To finish the proof of Lemma 2.3, it is enough to notice that: • ψ(0) = 0 and ψ(m) = 1, ∀m ∈ Zn \ {0}; • s 7→ ψ(x) P (x) |x|−s dx = {x∈Rn:1/2≤|x|≤1} ψ(x) P (x) |x|−s dx is a holomorphic function on C. Proof of Theorem 2.1. Using the polar decomposition of the volume form dx = ρn−1 dρ dS in Rn, we get for ℜ(s) > d+ n, Rn\Bn Pj(x) ρj+n−1 Pj(u) dS(u) = j+n−s Pj(u) dS(u). Lemma 2.3 now gives the result. 2.2 Holomorphy of certain series Before stating the main result of this section, we give first in the following some preliminaries from Diophantine approximation theory: Definition 2.4. (i) Let δ > 0. A vector a ∈ Rn is said to be δ−diophantine if there exists c > 0 such that |q.a−m| ≥ c |q|−δ, ∀q ∈ Zn \ { 0 } and ∀m ∈ Z. We note BV(δ) the set of δ−diophantine vectors and BV := ∪δ>0BV(δ) the set of diophantine vectors. (ii) A matrix Θ ∈ Mn(R) (real n × n matrices) will be said to be diophantine if there exists u ∈ Zn such that tΘ(u) is a diophantine vector of Rn. Remark. A classical result from Diophantine approximation asserts that for all δ > n, the Lebesgue measure of Rn \ BV(δ) is zero (i.e almost any element of Rn is δ−diophantine.) Let Θ ∈ Mn(R). If its row of index i is a diophantine vector of Rn (i.e. if Li ∈ BV) then tΘ(ei) ∈ BV and thus Θ is a diophantine matrix. It follows that almost any matrix of Mn(R) ≈ is diophantine. The goal of this section is to show the following Theorem 2.5. Let P ∈ C[X1, · · · ,Xn] be a homogeneous polynomial of degree d and let b be in S(Zn × · · · × Zn) (q times, q ∈ N). Then, (i) Let a ∈ Rn. We define fa(s) := P (k) e2πik.a. 1. If a ∈ Zn, then fa has a meromorphic continuation to the whole complex plane C. Moreover if S is the unit sphere and dS its Lebesgue measure, then fa is not entire if and only u∈Sn−1 P (u) dS(u) 6= 0. In that case, fa has only a simple pole at the point d + n, with s=d+n fa(s) = u∈Sn−1 P (u) dS(u). 2. If a ∈ Rn \ Zn, then fa(s) extends holomorphically to the whole complex plane C. (ii) Suppose that Θ ∈ Mn(R) is diophantine. For any (εi)i ∈ {−1, 0, 1}q , the function g(s) := l∈(Zn)q b(l) fΘ i εili extends meromorphically to the whole complex plane C with only one possible pole on s = d+n. Moreover, if we set Z := {l ∈ (Zn)q : i=1 εili = 0} and V := l∈Z b(l), then 1. If V P (u) dS(u) 6= 0, then s = d+ n is a simple pole of g(s) and s=d+n g(s) = V u∈Sn−1 P (u) dS(u). 2. If V P (u) dS(u) = 0, then g(s) extends holomorphically to the whole complex plane C. (iii) Suppose that Θ ∈ Mn(R) is diophantine. For any (εi)i ∈ {−1, 0, 1}q , the function g0(s) := l∈(Zn)q\Z b(l) fΘ i=1 εili where Z := {l ∈ (Zn)q : i=1 εili = 0} extends holomorphically to the whole complex plane C. Proof of Theorem 2.5: First we remark that If a ∈ Zn then fa(s) = P (k) . So, the point (i.1) follows from Theorem 2.1; g(s) := l∈(Zn)q\Z b(l) fΘ i εili (s) + l∈Z b(l) P (k) . Thus, the point (ii) rises easily from (iii) and Theorem 2.1. So, to complete the proof, it remains to prove the items (i.2) and (iii). The direct proof of (i.2) is easy but is not sufficient to deduce (iii) of which the proof is more delicate and requires a more precise (i.e. more effective) version of (i.2). The next lemma gives such crucial version, but before, let us give some notations: F := { P (X) (X21+···+X r/2 : P (X) ∈ C[X1, . . . ,Xn] and r ∈ N0}. We set g =deg(G) =deg(P )− r ∈ Z, the degree of G = P (X) (X21+···+X r/2 ∈ F . By convention we set deg(0) = −∞. Lemma 2.6. Let a ∈ Rn. We assume that d (a.u,Z) := infm∈Z |a.u−m| > 0 for some u ∈ Zn. For all G ∈ F , we define formally, F0(G; a; s) := e2πi k.a and F1(G; a; s) := (|k|2+1)s/2 e2πi k.a. Then for all N ∈ N, all G ∈ F and all i ∈ {0, 1}, there exist positive constants Ci := Ci(G,N, u), Bi := Bi(G,N, u) and Ai := Ai(G,N, u) such that s 7→ Fi(G;α; s) extends holomorphically to the half-plane {ℜ(s) > −N} and verifies in it: Fi(G; a; s) ≤ Ci(1 + |s|)Bi d (a.u,Z) Remark 2.7. The important point here is that we obtain an explicit bound of Fi(G;α; s) in {ℜ(s) > −N} which depends on the vector a only through d(a.u,Z), so depends on u and indirectly on a (in the sequel, a will vary.) In particular the constants Ci := Ci(G,N, u), Bi = Bi(G,N) and Ai := Ai(G,N) do not depend on the vector a but only on u. This is crucial for the proof of items (ii) and (iii) of Theorem 2.5! 2.2.1 Proof of Lemma 2.6 for i = 1: Let N ∈ N0 be a fixed integer, and set g0 := n+N + 1. We will prove Lemma 2.6 by induction on g =deg(G) ∈ Z. More precisely, in order to prove case i = 1, it suffices to prove that: Lemma 2.6 is true for all G ∈ F verifying deg(G) ≤ −g0. Let g ∈ Z with g ≥ −g0+1. If Lemma 2.6 is true for all G ∈ F such that deg(G) ≤ g−1, then it is also true for all G ∈ F satisfying deg(G) = g. • Step 1: Checking Lemma 2.6 for deg(G) ≤ −g0 := −(n+N + 1). Let G(X) = P (X) (X21+···+X r/2 ∈ F verifying deg(G) ≤ −g0. It is easy to see that we have uniformly in s = σ + iτ ∈ C and in k ∈ Zn: |G(k) e2πi k.a| (|k|2+1)σ/2 |P (k)| (|k|2+1)(r+σ)/2 ≪G 1(|k|2+1)(r+σ−deg(P ))/2 ≪G (|k|2+1)(σ−deg(G))/2 ≪G 1(|k|2+1)(σ+g0)/2 . It follows that F1(G; a; s) = (|k|2+1)s/2 e2πi k.a converges absolutely and defines a holo- morphic function in the half plane {σ > −N}. Therefore, we have for any s ∈ {ℜ(s) > −N}: |F1(G; a; s)| ≪G (|k|2+1)(−N+g0)/2 (|k|2+1)(n+1)/2 ≪G 1. Thus, Lemma 2.6 is true when deg(G) ≤ −g0. • Step 2: Induction. Now let g ∈ Z satisfying g ≥ −g0+1 and suppose that Lemma 2.6 is valid for all G ∈ F verifying deg(G) ≤ g − 1. Let G ∈ F with deg(G) = g. We will prove that G also verifies conclusions of Lemma 2.6: There exist P ∈ C[X1, . . . ,Xn] of degree d ≥ 0 and r ∈ N0 such that G(X) = P (X)(X21+···+X2n+1)r/2 and g =deg(G) = d− r. Since G(k) ≪ (|k|2 +1)g/2 uniformly in k ∈ Zn, we deduce that F1(G; a; s) converges absolutely in {σ = ℜ(s) > n+ g}. Since k 7→ k + u is a bijection from Zn into Zn, it follows that we also have for ℜ(s) > n+ g F1(G; a; s) = P (k) (|k|2+1)(s+r)/2 e2πi k.a = P (k+u) (|k+u|2+1)(s+r)/2 e2πi (k+u).a = e2πi u.a P (k+u) (|k|2+2k.u+|u|2+1)(s+r)/2 e2πi k.a = e2πi u.a α∈Nn0 ;|α|1=α1+···+αn≤d ∂αP (k) (|k|2+2k.u+|u|2+1)(s+r)/2 e2πi k.a = e2πi u.a |α|1≤d ∂αP (k) (|k|2+1)(s+r)/2 2k.u+|u|2 (|k|2+1) )−(s+r)/2 e2πi k.a. Let M := sup(N + n+ g, 0) ∈ N0. We have uniformly in k ∈ Zn 2k.u+|u|2 (|k|2+1) )−(s+r)/2 −(s+r)/2 )(2k.u+|u|2)j (|k|2+1)j +OM,u ( (1+|s|)M+1 (|k|2+1)(M+1)/2 Thus, for σ = ℜ(s) > n+ d, F1(G; a; s) = e 2πi u.a |α|1≤d ∂αP (k) (|k|2+1)(s+r)/2 2k.u+|u|2 (|k|2+1) )−(s+r)/2 e2πi k.a = e2πi u.a |α|1≤d −(s+r)/2 ∂αP (k)(2k.u+|u|2) (|k|2+1)(s+r+2j)/2 e2πi k.a +OG,M,u (1 + |s|)M+1 (|k|2+1)(σ+M+1−g)/2 . (9) Set I := {(α, j) ∈ Nn0 × {0, . . . ,M} | |α|1 ≤ d} and I∗ := I \ { (0, 0) }. Set also G(α,j);u(X) := ∂αP (X)(2X.u+|u|2) (|X|2+1)(r+2j)/2 ∈ F for all (α, j) ∈ I∗. Since M ≥ N + n+ g, it follows from (9) that (1 − e2πi u.a) F1(G; a; s) = e2πi u.a (α,j)∈I∗ −(s+r)/2 G(α,j);u;α; s +RN (G; a;u; s) (10) where s 7→ RN (G; a;u; s) is a holomorphic function in the half plane {σ = ℜ(s) > −N}, in which it satisfies the bound RN (G; a;u; s) ≪G,N,u 1. Moreover it is easy to see that, for any (α, j) ∈ I∗, G(α,j);u = deg(∂αP ) + j − (r + 2j) ≤ d− |α|1 + j − (r + 2j) = g − |α|1 − j ≤ g − 1. Relation (10) and the induction hypothesis imply then that (1− e2πi u.a) F1(G; a; s) verifies the conclusions of Lemma 2.6. (11) Since |1− e2πi u.a| = 2| sin(πu.a)| ≥ d (u.a,Z), then (11) implies that F1(G; a; s) satisfies conclu- sions of Lemma 2.6. This completes the induction and the proof for i = 1. 2.2.2 Proof of Lemma 2.6 for i = 0: Let N ∈ N be a fixed integer. Let G(X) = P (X) (X21+···+X r/2 ∈ F and g = deg(G) = d− r where d ≥ 0 is the degree of the polynomial P . Set also M := sup(N + g + n, 0) ∈ N0. Since P (k) ≪ |k|d for k ∈ Zn\{ 0 }, it follows that F0(G; a; s) and F1(G; a; s) converge absolutely in the half plane {σ = ℜ(s) > n+ g}. Moreover, we have for s = σ + iτ ∈ C verifying σ > n+ g: F0(G; a; s) = k∈Zn\{ 0 } (|k|2+1−1)s/2 e2πi k.a = ′ G(k) (|k|2+1)s/2 |k|2+1 )−s/2 e2πi k.a (−1)j G(k) (|k|2+1)(s+2j)/2 e2πi k.a (1 + |s|)M+1 ′ |G(k)| (|k|2+1)(σ+2M+2)/2 (−1)jF1(G; a; s + 2j) (1 + |s|)M+1 ′ |G(k)| (|k|2+1)(σ+2M+2)/2 . (12) In addition we have uniformly in s = σ + iτ ∈ C verifying σ > −N , ′ |G(k)| (|k|2+1)(σ+2M+2)/2 ′ |k|g (|k|2+1)(−N+2M+2)/2 |k|n+1 < +∞. So (12) and Lemma 2.6 for i = 1 imply that Lemma 2.6 is also true for i = 0. This completes the proof of Lemma 2.6. 2.2.3 Proof of item (i.2) of Theorem 2.5: Since a ∈ Rn \ Zn, there exists i0 ∈ {1, . . . , n} such that ai0 6∈ Z. In particular d(a.ei0 ,Z) = d(ai0 ,Z) > 0. Therefore, a satisfies the assumption of Lemma 2.6 with u = ei0 . Thus, for all N ∈ N, s 7→ fa(s) = F0(P ; a; s) has a holomorphic continuation to the half-plane {ℜ(s) > −N}. It follows, by letting N → ∞, that s 7→ fa(s) has a holomorphic continuation to the whole complex plane C. 2.2.4 Proof of item (iii) of Theorem 2.5: Let Θ ∈ Mn(R), (εi)i ∈ {−1, 0, 1}q and b ∈ S(Zn × Zn). We assume that Θ is a diophantine matrix. Set Z := { l = (l1, . . . , lq) ∈ (Zn)q : i εili = 0 } and P ∈ C[X1, . . . ,Xn] of degree d ≥ 0. It is easy to see that for σ > n+ d: l∈(Zn)q\Z |b(l)| ′ |P (k)| |e2πi k.Θ i εili | ≪P l∈(Zn)q\Z |b(l)| |k|σ−d l∈(Zn)q\Z |b(l)| < +∞. g0(s) := l∈(Zn)q\Z b(l) fΘ i εili (s) = l∈(Zn)q\Z ′ P (k) e2πi k.Θ i εili converges absolutely in the half plane {ℜ(s) > n+ d}. Moreover with the notations of Lemma 2.6, we have for all s = σ + iτ ∈ C verifying σ > n+ d: g0(s) = l∈(Zn)q\Z b(l)fΘ i εili (s) = l∈(Zn)q\Z b(l)F0(P ; Θ εili; s) (13) But Θ is diophantine, so there exists u ∈ Zn and δ, c > 0 such |q. tΘu−m| ≥ c (1 + |q|)−δ , ∀q ∈ Zn \ { 0 }, ∀m ∈ Z. We deduce that ∀l ∈ (Zn)q \ Z, .u−m| = | .tΘu−m| ≥ c 1 + | εili| )−δ ≥ c (1 + |l|)−δ. It follows that there exists u ∈ Zn, δ > 0 and c > 0 such that ∀l ∈ (Zn)q \ Z, d εili).u;Z ≥ c (1 + |l|)−δ . (14) Therefore, for any l ∈ (Zn)q \Z, the vector a = Θ i εili verifies the assumption of Lemma 2.6 with the same u. Moreover δ and c in (14) are also independent on l. We fix now N ∈ N. Lemma 2.6 implies that there exist positive constants C0 := C0(P,N, u), B0 := Bi(P,N, u) and A0 := A0(P,N, u) such that for all l ∈ (Zn)q \ Z, s 7→ F0(P ; Θ i εili; s) extends holomorphically to the half plane {ℜ(s) > −N} and verifies in it the bound F0(P ; Θ εili; s) ≤ C0 (1 + |s|)B0 d εili).u;Z This and (14) imply that for any compact set K included in the half plane {ℜ(s) > −N}, there exist two constants C := C(P,N, c, δ, u,K) and D := D(P,N, c, δ, u) (independent on l ∈ (Zn)q \ Z) such that ∀s ∈ K and ∀l ∈ (Zn)q \ Z, F0(P ; Θ εili; s) ≤ C (1 + |l|)D . (15) It follows that s 7→ l∈(Zn)q\Z b(l)F0(P ; Θ iεili; s) has a holomorphic continuation to the half plane {ℜ(s) > −N}. This and ( 13) imply that s 7→ g0(s) = l∈(Zn)q\Z b(l)fΘ i εili (s) has a holomorphic contin- uation to {ℜ(s) > −N}. Since N is an arbitrary integer, by letting N → ∞, it follows that s 7→ g0(s) has a holomorphic continuation to the whole complex plane C which completes the proof of the theorem. Remark 2.8. By equation (11), we see that a Diophantine condition is sufficient to get Lemma 2.6. Our Diophantine condition appears also (in equivalent form) in Connes [7, Prop. 49] (see Remark 4.2 below). The following heuristic argument shows that our condition seems to be necessary in order to get the result of Theorem 2.5: For simplicity we assume n = 1 (but the argument extends easily to any n). Let θ ∈ R \Q. We know (see this reflection formula in [15, p. 6]) that for any l ∈ Z \ {0}, gθl(s) := e2πiθlk s−1/2 ) hθl(1− s) where hθl(s) := |θl+k|s So, for any (al) ∈ S(Z), the existence of meromorphic continuation of g0(s) := l∈Z al gθl(s) is equivalent to the existence of meromorphic continuation of h0(s) := al hθl(s) = |θl+k|s So, for at least one σ0 ∈ R, we must have |al||θl+k|σ0 = O(1) uniformly in k, l ∈ Z It follows that for any (al) ∈ S(Z), |θl + k| ≫ |al|1/σ0 uniformly in k, l ∈ Z∗. Therefore, our Diophantine condition seems to be necessary. 2.2.5 Commutation between sum and residue Let p ∈ N. Recall that S((Zn)p) is the set of the Schwartz sequences on (Zn)p. In other words, b ∈ S((Zn)p) if and only if for all r ∈ N0, (1 + |l1|2 + · · · |lp|2)r |b(l1, · · · , lp)|2 is bounded on (Zn)p. We note that if Q ∈ R[X1, · · · ,Xnp] is a polynomial, (aj) ∈ S(Zn)p, b ∈ S(Zn) and φ a real-valued function, then l := (l1, · · · , lp) 7→ ã(l) b(−l̂p)Q(l) eiφ(l) is a Schwartz sequence on (Zn)p, where ã(l) := a1(l1) · · · ap(lp), l̂i := l1 + . . .+ li. In the following, we will use several times the fact that for any (k, l) ∈ (Zn)2 such that k 6= 0 and k 6= −l, we have |k + l|2 = |k|2 − 2k.l + |l|2 |k|2|k + l|2 . (16) Lemma 2.9. There exists a polynomial P ∈ R[X1, · · · ,Xp] of degree 4p and with positive coefficients such that for any k ∈ Zn, and l := (l1, · · · , lp) ∈ (Zn)p such that k 6= 0 and k 6= −l̂i for all 1 ≤ i ≤ p, the following holds: |k + l̂1|2 . . . |k + l̂p|2 ≤ 1|k|2p P (|l1|, · · · , |lp|). Proof. Let’s fix i such that 1 ≤ i ≤ p. Using two times (16), Cauchy–Schwarz inequality and the fact that |k + l̂i|2 ≥ 1, we get |k+bli|2 2|k||bli|+|bli| (2|k||bli|+|bli| |k|4|k+bli|2 |l̂i|+ |l̂i|2 + 4|k|3 |l̂i| 3 + 1 |l̂i|4. Since |k| ≥ 1, and |l̂i|j ≤ |l̂i|4 if 1 ≤ j ≤ 4, we find |k+bli|2 |l̂i|j ≤ 5|k|2 1 + 4|l̂i|4 1 + 4( |lj |)4 |k+bl1|2...|k+blp|2 |k|2p 1 + 4( |lj |)4 Taking P (X1, · · · ,Xp) := 5p 1 + 4( j=1Xj) now gives the result. Lemma 2.10. Let b ∈ S((Zn)p), p ∈ N, Pj ∈ R[X1, · · · ,Xn] be a homogeneous polynomial function of degree j, k ∈ Zn, l := (l1, · · · , lp) ∈ (Zn)p, r ∈ N0, φ be a real-valued function on Zn × (Zn)p and h(s, k, l) := b(l)Pj(k) e iφ(k,l) |k|s+r|k + l̂1|2 · · · |k + l̂p|2 with h(s, k, l) := 0 if, for k 6= 0, one of the denominators is zero. For all s ∈ C such that ℜ(s) > n+ j − r − 2p, the series H(s) := (k,l)∈(Zn)p+1 h(s, k, l) is absolutely summable. In particular, l∈(Zn)p h(s, k, l) = l∈(Zn)p h(s, k, l) . Proof. Let s = σ + iτ ∈ C such that σ > n+ j − r − 2p. By Lemma 2.9 we get, for k 6= 0, |h(s, k, l)| ≤ |b(l)Pj(k)| |k|−r−σ−2p P (l), where P (l) := P (|l1|, · · · , |lp|) and P is a polynomial of degree 4p with positive coefficients. Thus, |h(s, k, l)| ≤ F (l)G(k) where F (l) := |b(l)|P (l) and G(k) := |Pj(k)||k|−r−σ−2p. The summability of l∈(Zn)p F (l) is implied by the fact that b ∈ S((Zn)p). The summability of∑′ k∈ZnG(k) is a consequence of the fact that σ > n + j − r − 2p. Finally, as a product of two summable series, k,lF (l)G(k) is a summable series, which proves that k,lh(s, k, l) is also absolutely summable. Definition 2.11. Let f be a function on D× (Zn)p where D is an open neighborhood of 0 in C. We say that f satisfies (H1) if and only if there exists ρ > 0 such that (i) for any l, s 7→ f(s, l) extends as a holomorphic function on Uρ, where Uρ is the open disk of center 0 and radius ρ, (ii) the series l∈(Zn)p ‖H(·, l)‖∞,ρ is summable,where ‖H(·, l)‖∞,ρ := sups∈Uρ |H(s, l)|. We say that f satisfies (H2) if and only if there exists ρ > 0 such that (i) for any l, s 7→ f(s, l) extends as a holomorphic function on Uρ − {0}, (ii) for any δ such that 0 < δ < ρ, the series l∈(Zn)p ‖H(·, l)‖∞,δ,ρ is summable, where ‖H(·, l)‖∞,δ,ρ := supδ<|s|<ρ |H(s, l)|. Remark 2.12. Note that (H1) implies (H2). Moreover, if f satisfies (H1) (resp. (H2) for ρ > 0, then it is straightforward to check that f : s 7→ l∈(Zn)p f(s, l) extends as an holomorphic function on Uρ (resp. on Uρ \ { 0 }). Corollary 2.13. With the same notations of Lemma 2.10, suppose that r + 2p − j > n, then, the function H(s, l) := k∈Znh(s, k, l) satisfies (H1). Proof. (i) Let’s fix ρ > 0 such that ρ < r + 2p − j − n. Since r + 2p − j > n, Uρ is inside the half-plane of absolute convergence of the series defined by H(s, l). Thus, s 7→ H(s, l) is holomorphic on Uρ. (ii) Since ∣∣|k|−s ∣∣ ≤ |k|ρ for all s ∈ Uρ and k ∈ Zn \ { 0 }, we get as in the above proof |h(s, k, l)| ≤ |b(l)Pj(k)| |k|−r+ρ−2p P (|l1|, · · · , |lp|). Since ρ < r + 2p − j − n, the series k∈Zn |Pj(k)||k|−r+ρ−2p is summable. Thus, ‖H(·, l)‖∞,ρ ≤ K F (l) where K := ′|Pj(k)||k|−r+ρ−2p <∞. We have already seen that the series l F (l) is summable, so we get the result. We note that if f and g both satisfy (H1) (or (H2)), then so does f + g. In the following, we will use the equivalence relation f ∼ g ⇐⇒ f − g satisfies (H1). Lemma 2.14. Let f and g be two functions on D × (Zn)p where D is an open neighborhood of 0 in C, such that f ∼ g and such that g satisfies (H2). Then l∈(Zn)p f(s, l) = l∈(Zn)p g(s, l) . Proof. Since f ∼ g, f satisfies (H2) for a certain ρ > 0. Let’s fix η such that 0 < η < ρ and define Cη as the circle of center 0 and radius η. We have g(s, l) = Res f(s, l) = 1 f(s, l) ds = u(t, l)dt . where I = [0, 2π] and u(t, l) := 1 ηeitf(η eit, l). The fact that f satisfies (H2) entails that the series l∈(Zn)p ‖f(·, l)‖∞,Cη is summable. Thus, since ‖u(·, l)‖∞ = η ‖f(·, l)‖∞,Cη , the series∑ l∈(Zn)p ‖u(·, l)‖∞ is summable, so, as a consequence, l∈(Zn)p u(t, l)dt = l∈(Zn)p u(t, l)dt which gives the result. 2.3 Computation of residues of zeta functions Since, we will have to compute residues of series, let us introduce the following Definition 2.15. ζ(s) := Zn(s) := |k|−s, ζp1,...,pn(s) := 1 · · · k |k|s , for pi ∈ N, where ζ(s) is the Riemann zeta function (see [25] or [14]). By the symmetry k → −k, it is clear that these functions ζp1,...,pn all vanish for odd values of pi. Let us now compute ζ0,··· ,0,1i,0··· ,0,1j ,0··· ,0(s) in terms of Zn(s): Since ζ0,··· ,0,1i,0··· ,0,1j ,0··· ,0(s) = Ai(s) δij , exchanging the components ki and kj , we get ζ0,··· ,0,1i,0··· ,0,1j ,0··· ,0(s) = Zn(s− 2). Similarly, |k|s+8 n(n−1) Zn(s+ 4)− 1n−1 |k|s+8 but it is difficult to write explicitly ζp1,...,pn(s) in terms of Zn(s− 4) and other Zn(s−m) when at least four indices pi are non zero. When all pi are even, ζp1,...,pn(s) is a nonzero series of fractions P (k) where P is a homogeneous polynomial of degree p1 + · · ·+ pn. Theorem 2.1 now gives us the following Proposition 2.16. ζp1,...,pn has a meromorphic extension to the whole plane with a unique pole at n+ p1 + · · ·+ pn. This pole is simple and the residue at this pole is s=n+p1+···+pn ζp1,...,pn(s) = 2 )···Γ( n+p1+···+pn when all pi are even or this residue is zero otherwise. In particular, for n = 2, ′ kikj |k|s+4 = δij π , (18) and for n = 4, ′ kikj |k|s+6 = δij ′ kikjklkm |k|s+8 = (δijδlm + δilδjm + δimδjl) . (19) Proof. Equation (17) follows from Theorem (2.1) s=n+p1+···+pn ζp1,...,pn(s) = k∈Sn−1 1 · · · kpnn dS(k) and standard formulae (see for instance [32, VIII,1;22]). Equation (18) is a straightforward consequence of Equation (17). Equation (19) can be checked for the cases i = j 6= l = m and i = j = l = m. Note that Zn(s) is an Epstein zeta function associated to the quadratic form q(x) := x 1+...+x so Zn satisfies the following functional equation Zn(s) = π s−n/2Γ(n/2− s/2)Γ(s/2)−1 Zn(n− s). Since πs−n/2Γ(n/2−s/2) Γ(s/2)−1 = 0 for any negative even integer n and Zn(s) is meromorphic on C with only one pole at s = n with residue 2πn/2Γ(n/2)−1 according to previous proposition, so we get Zn(0) = −1. We have proved that Zn(s+ n) = 2π n/2 Γ(n/2)−1, (20) Zn(0) = −1. (21) 2.4 Meromorphic continuation of a class of zeta functions Let n, q ∈ N, q ≥ 2, and p = (p1, . . . , pq−1) ∈ Nq−10 . Set I := {i | pi 6= 0} and assume that I 6= ∅ and I := {α = (αi)i∈I | ∀i ∈ I αi = (αi,1, . . . , αi,pi) ∈ N 0 } = We will use in the sequel also the following notations: - for x = (x1, . . . , xt) ∈ Rt recall that |x|1 = |x1|+ · · ·+ |xt| and |x| = x21 + · · ·+ x2t ; - for all α = (αi)i∈I ∈ I = i∈I N |α|1 = |αi|1 = |αi,j| and 2.4.1 A family of polynomials In this paragraph we define a family of polynomials which plays an important role later. Consider first the variables: - for X1, . . . ,Xn we set X = (X1, . . . ,Xn); - for any i = 1, . . . , 2q, we consider the variables Yi,1, . . . , Yi,n and set Yi := (Yi,1, . . . , Yi,n) and Y := (Y1, . . . , Y2q); - for Y = (Y1, . . . , Y2q), we set for any 1 ≤ j ≤ q, Ỹj := Y1 + · · ·+ Yj + Yq+1 + · · ·+ Yq+j. We define for all α = (αi)i∈I ∈ I = i∈I N 0 the polynomial Pα(X,Y ) := (2〈X, Ỹi〉+ |Ỹi|2)αi,j . (22) It is clear that Pα(X,Y ) ∈ Z[X,Y ], degXPα ≤ |α|1 and degY Pα ≤ 2|α|1. Let us fix a polynomial Q ∈ R[X1, · · · ,Xn] and note d := degQ. For α ∈ I, we want to expand Pα(X,Y )Q(X) in homogeneous polynomials in X and Y so defining L(α) := {β ∈ N(2q+1)n0 : |β|1 − dβ ≤ 2|α|1 and dβ ≤ |α|1 + d } where dβ := 1 βi, we set Pα(X,Y )Q(X) =: β∈L(α) cα,βX where cα,β ∈ R, Xβ := Xβ11 · · ·X n and Y β := Y 1,1 · · ·Y β(2q+1)n 2q,n . By definition, X β is a homogeneous polynomial of degree in X equals to dβ . We note Mα,β(Y ) := cα,β Y 2.4.2 Residues of a class of zeta functions In this section we will prove the following result, used in Proposition 5.4 for the computation of the spectrum dimension of the noncommutative torus: Theorem 2.17. (i) Let 1 Θ be a diophantine matrix, and ã ∈ S (Zn)2q . Then s 7→ f(s) := l∈[(Zn)q]2 |k + l̃i|pi |k|−sQ(k) eik.Θ has a meromorphic continuation to the whole complex plane C with at most simple possible poles at the points s = n+ d+ |p|1 −m where m ∈ N0. (ii) Let m ∈ N0 and set I(m) := { (α, β) ∈ I × N(2q+1)n0 : β ∈ L(α) and m = 2|α|1 − dβ + d }. Then I(m) is a finite set and s = n+ d+ |p|1 −m is a pole of f if and only if C(f,m) := (α,β)∈I(m) Mα,β(l) u∈Sn−1 uβ dS(u) 6= 0, with Z := {l : 1 lj = 0} and the convention ∅ = 0. In that case s = n + d + |p|1 −m is a simple pole of residue Res s=n+d+|p|1−m f(s) = C(f,m). In order to prove the theorem above we need the following Lemma 2.18. For all N ∈ N we have |k + l̃i|pi = α=(αi)i∈I∈ i∈I{0,...,N} ) Pα(k,l) |k|2|α|1−|p|1 +ON (|k||p|1−(N+1)/2) uniformly in k ∈ Zn and l ∈ (Zn)2q verifying |k| > U(l) := 36 ( ∑2q−1 i=1, i 6=q |li|)4. Proof. For i = 1, . . . , q − 1, we have uniformly in k ∈ Zn and l ∈ (Zn)2q verifying |k| > U(l), ∣∣2〈k,eli〉+|eli|2 . (23) In that case, |k + l̃i| = |k|2 + 2〈k, l̃i〉+ |l̃i|2 = |k| 2〈k,eli〉+|eli| |k|2u−1 P iu(k, l) where for all i = 1, . . . , q − 1 and for all u ∈ N0, P iu(k, l) := 2〈k, l̃i〉+ |l̃i|2 with the convention P i0(k, l) := 1. In particular P iu(k, l) ∈ Z[k, l], degk P iu ≤ u and degl P iu ≤ 2u. Inequality (23) implies that for all i = 1, . . . , q − 1 and for all u ∈ N, |k|2u |P iu(k, l)| ≤ uniformly in k ∈ Zn and l ∈ (Zn)2q verifying |k| > U(l). Let N ∈ N. We deduce from the previous that for any k ∈ Zn and l ∈ (Zn)2q verifying |k| > U(l) and for all i = 1, . . . , q − 1, we have |k + l̃i| = |k|2u−1 P iu(k, l) +O |k| | |k|)−u |k|2u−1 P iu(k, l) +ON |k|(N−1)/2 It follows that for any N ∈ N, we have uniformly in k ∈ Zn and l ∈ (Zn)2q verifying |k| > U(l) and for all i ∈ I, |k + l̃i|pi = αi∈{0,...,N} |k|2|αi|1−pi P iαi(k, l) +ON |k|(N+1)/2−pi where P iαi(k, l) = j=1 P (k, l) for all αi = (αi,1, . . . , αi,pi) ∈ {0, . . . , N}pi and |k + l̃i|pi = α=(αi)∈ i∈I{0,...,N} |k|2|α|1−|p|1 Pα(k, l) +ON |k|(N+1)/2−|p|1 where Pα(k, l) = i∈I P (k, l) = j=1 P (k, l). Proof of Theorem 2.17. (i) All n, q, p = (p1, . . . , pq−1) and ã ∈ S (Zn)2q are fixed as above and we define formally for any l ∈ (Zn)2q F (l, s) := |k + l̃i|pi Q(k) eik.Θ 1 lj |k|−s. (24) Thus, still formally, f(s) := l∈(Zn)2q ãl F (l, s). (25) It is clear that F (l, s) converges absolutely in the half plane {σ = ℜ(s) > n + d + |p|1} where d = degQ. Let N ∈ N. Lemma 2.18 implies that for any l ∈ (Zn)2q and for s ∈ C such that σ > n+ |p|1+d, F (l, s) = |k|≤U(l) |k + l̃i|pi Q(k) eik.Θ 1 lj |k|−s α=(αi)i∈I∈ i∈I{0,...,N} |k|>U(l) |k|s+2|α|1−|p|1 Pα(k, l)Q(k) e 1 lj +GN (l, s). where s 7→ GN (l, s) is a holomorphic function in the half-plane DN := {σ > n+ d+ |p|1 − N+12 } and verifies in it the bound GN (l, s) ≪N,σ 1 uniformly in l. It follows that F (l, s) = α=(αi)i∈I∈ i∈I{0,...,N} Hα(l, s) +RN (l, s), (26) where Hα(l, s) := ′ (1/2 |k|s+2|α|1−|p|1 Pα(k, l)Q(k) e 1 lj , RN (l, s) := |k|≤U(l) |k + l̃i|pi Q(k) eik.Θ 1 lj |k|−s |k|≤U(l) α=(αi)i∈I∈ i∈I{0,...,N} ) Pα(k,l) |k|s+2|α|1−|p|1 Q(k) eik.Θ 1 lj +GN (l, s). In particular there exists A(N) > 0 such that s 7→ RN (l, s) extends holomorphically to the half-plane DN and verifies in it the bound RN (l, s) ≪N,σ 1 + |l|A(N) uniformly in l. Let us note formally hα(s) := ãlHα(l, s). Equation (26) and RN (l, s) ≪N,σ 1 + |l|A(N) imply that f(s) ∼N α=(αi)i∈I∈ i∈I{0,...,N} hα(s), (27) where ∼N means modulo a holomorphic function in DN . Recall the decomposition Pα(k, l)Q(k) = β∈L(α)Mα,β(l) k β and we decompose similarly hα(s) = β∈L(α) hα,β(s). Theorem 2.5 now implies that for all α = (αi)i∈I ∈ i∈I{0, . . . , N}pi and β ∈ L(α), - the map s 7→ hα,β(s) has a meromorphic continuation to the whole complex plane C with only one simple possible pole at s = n+ |p|1 − 2|α|1 + dβ , - the residue at this point is equal to s=n+|p|1−2|α|1+dβ hα,β(s) = ãlMα,β(l) u∈Sn−1 uβdS(u) (28) where Z := {l ∈ (Z)n)2q : 1 lj = 0}. If the right hand side is zero, hα,β(s) is holomorphic on By (27), we deduce therefore that f(s) has a meromorphic continuation on the halfplane DN , with only simple possible poles in the set {n+ |p|1+k : −2N |p|1 ≤ k ≤ d }. Taking now N → ∞ yields the result. (ii) Let m ∈ N0 and set I(m) := { (α, β) ∈ I × N(2q+1)n0 : β ∈ L(α) and m = 2|α|1 − dβ + d }. If (α, β) ∈ I(m), then |α|1 ≤ m and |β|1 ≤ 3m+ d, so I(m) is finite. With a chosen N such that 2N |p|1 + d > m, we get by (27) and (28) s=n+d+|p|1−m f(s) = (α,β)∈I(m) Mα,β(l) u∈Sn−1 uβ dS(u) = C(f,m) with the convention ∅ = 0. Thus, n+d+ |p|1−m is a pole of f if and only if C(f,m) 6= 0. 3 Noncommutative integration on a simple spectral triple In this section, we revisit the notion of noncommutative integral pioneered by Alain Connes, pay- ing particular attention to the reality (Tomita–Takesaki) operator J and to kernels of perturbed Dirac operators by symmetrized one-forms. 3.1 Kernel dimension We will have to compare here the kernels of D and DA which are both finite dimensional: Lemma 3.1. Let (A,H,D) be a spectral triple with a reality operator J and chirality χ. If A ∈ Ω1D is a one-form, the fluctuated Dirac operator DA := D +A+ ǫJAJ−1 (where DJ = ǫ JD, ǫ = ±1) is an operator with compact resolvent, and in particular its kernel KerDA is a finite dimensional space. This space is invariant by J and χ. Proof. Let T be a bounded operator and let z be in the resolvent of D + T and z′ be in the resolvent of D. Then (D + T − z)−1 = (D − z′)−1 [1− (T + z′ − z)(D + T − z)−1]. Since (D− z′)−1 is compact by hypothesis and since the term in bracket is bounded, D+ T has a compact resolvent. Applying this to T = A+ ǫJAJ−1, DA has a finite dimensional kernel (see for instance [27, Theorem 6.29]). Since according to the dimension, J2 = ±1, J commutes or anticommutes with χ, χ commutes with the elements in the algebraA andDχ = −χD (see [10] or [23, p. 405]), we get DAχ = −χDA and DAJ = ±JDA which gives the result. 3.2 Pseudodifferential operators Let (A,D,H) be a given real regular spectral triple of dimension n. We note P0 the projection on KerD , PA the projection on KerDA , D := D + P0 ,DA := DA + PA . P0 and PA are thus finite-rank selfadjoint bounded operators. We remark that D and DA are selfadjoint invertible operators with compact inverses. Remark 3.2. Since we only need to compute the residues and the value at 0 of the ζD, ζDA functions, it is not necessary to define the operators D−1 or D−1A and the associated zeta func- tions. However, we can remark that all the work presented here could be done using the process of Higson in [26] which proves that we can add any smoothing operator to D or DA such that the result is invertible without changing anything to the computation of residues. Define for any α ∈ R OP 0 := {T : t 7→ Ft(T ) ∈ C∞ R,B(H) OPα := {T : T |D|−α ∈ OP 0 }. where Ft(T ) := e it|D| T e−it|D| = eit|D| T e−it|D| since |D| = |D|+ P0. Define δ(T ) := [|D|, T ], ∇(T ) := [D2, T ], σs(T ) := |D|sT |D|−s, s ∈ C. It has been shown in [13] that OP 0 = p≥0Dom(δ p). In particular, OP 0 is a subalgebra of B(H) (while elements of OPα are not necessarily bounded for α > 0) and A ⊆ OP 0, JAJ−1 ⊆ OP 0, [D,A] ⊆ OP 0. Note that P0 ∈ OP−∞ and δ(OP 0) ⊆ OP 0. For any t > 0, Dt and and |D|t are in OP t and for any α ∈ R,Dα and |D|α are in OPα. By hypothesis, |D|−n ∈ L(1,∞)(H) so for any α > n, OP−α ⊆ L1(H). Lemma 3.3. [13] (i) For any T ∈ OP 0 and s ∈ C, σs(T ) ∈ OP 0. (ii) For any α, β ∈ R, OPαOP β ⊆ OPα+β . (iii) If α ≤ β, OPα ⊆ OP β. (iv) For any α, δ(OPα) ⊆ OPα. (v) For any α and T ∈ OPα, ∇(T ) ∈ OPα+1. Proof. See the appendix. Remark 3.4. Any operator in OPα, where α ∈ R, extends as a continuous linear operator from Dom |D|α+1 to Dom |D| where the Dom |D|α spaces have their natural norms (see [13,26]). We now introduce a definition of pseudodifferential operators in a slightly different way than in [9,13,26] which in particular pays attention to the reality operator J and the kernel of D and allows D and |D|−1 to be a pseudodifferential operators. It is more in the spirit of [4]. Definition 3.5. Let us define D(A) as the polynomial algebra generated by A, JAJ−1, D and A pseudodifferential operator is an operator T such that there exists d ∈ Z such that for any N ∈ N, there exist p ∈ N0, P ∈ D(A) and R ∈ OP−N (p, P and R may depend on N) such that P D−2p ∈ OP d and T = P D−2p +R . Define Ψ(A) as the set of pseudodifferential operators and Ψ(A)k := Ψ(A) ∩OP k. Note that if A is a 1-form, A and JAJ−1 are in D(A) and moreover D(A) ⊆ ∪p∈N0OP p. Since |D| ∈ D(A) by construction and P0 is a pseudodifferential operator, for any p ∈ Z, |D|p is a pseudodifferential operator (in OP p.) Let us remark also that D(A) ⊆ Ψ(A) ⊆ ∪k∈ZOP k. Lemma 3.6. [9, 13] The set of all pseudodifferential operators Ψ(A) is an algebra. Moreover, if T ∈ Ψ(A)d and T ∈ Ψ(A)d′, then TT ′ ∈ Ψ(A)d+d′ . Proof. See the appendix. Due to the little difference of behavior between scalar and nonscalar pseudodifferential operators (i.e. when coefficients like [D, a], a ∈ A appears in P of Definition 3.5), it is convenient to also introduce Definition 3.7. Let D1(A) be the algebra generated by A, JAJ−1 and D, and Ψ1(A) be the set of pseudodifferential operators constructed as before with D1(A) instead of D(A). Note that Ψ1(A) is subalgebra of Ψ(A). Remark that Ψ1(A) does not necessarily contain operators such as |D|k where k ∈ Z is odd. This algebra is similar to the one defined in [4]. 3.3 Zeta functions and dimension spectrum For any operator B and if X is either D or DA, we define ζBX(s) := Tr B|X|−s ζX(s) := Tr |X|−s The dimension spectrum Sd(A,H,D) of a spectral triple has been defined in [9,13]. It is extended here to pay attention to the operator J and to our definition of pseudodifferential operator. Definition 3.8. The spectrum dimension of the spectral triple is the subset Sd(A,H,D) of all poles of the functions ζPD := s 7→ Tr P |D|−s where P is any pseudodifferential operator in OP 0. The spectral triple (A,H,D) is simple when these poles are all simple. Remark 3.9. If Sp(A,H,D) denotes the set of all poles of the functions s 7→ Tr P |D|−s where P is any pseudodifferential operator, then, Sd(A,H,D) ⊆ Sp(A,H,D). When Sp(A,H,D) = Z, Sd(A,H,D) = {n − k : k ∈ N0 }: indeed, if P is a pseudodifferential operator in OP 0, and q ∈ N is such that q > n, P |D|−s is in OP−ℜ(s) so is trace-class for s in a neighborhood of q; as a consequence, q cannot be a pole of s 7→ Tr P |D|−s Remark 3.10. Sp(A,H,D) is also the set of poles of functions s 7→ Tr B|D|−s−2p where p ∈ N0 and B ∈ D(A). 3.4 The noncommutative integral We already defined the one parameter group σz(T ) := |D|zT |D|−z, z ∈ C. Introducing the notation (recall that ∇(T ) = [D2, T ]) for an operator T , ε(T ) := ∇(T )D−2, we get from [4, (2.44)] the following expansion for T ∈ OP q σz(T ) ∼ g(z, r) εr(T ) mod OP−N−1+q (29) where g(z, r) := 1 ) · · · (z − (r − 1)) = with the convention g(z, 0) := 1. We define the noncommutative integral by − T := Res ζTD(s) = Res T |D|−s Proposition 3.11. [13] If the spectral triple is simple, is a trace on Ψ(A). Proof. See the appendix. 4 Residues of ζDA for a spectral triple with simple dimension spectrum We fix a regular spectral triple (A,H,D) of dimension n and a self-adjoint 1-form A. Recall that DA := D + à where à := A+ εJAJ−1, DA := DA + PA where PA is the projection on KerDA. Remark that à ∈ D(A) ∩OP 0 and DA ∈ D(A) ∩OP 1. We note VA := PA − P0. As the following lemma shows, VA is a smoothing operator: Lemma 4.1. (i) k≥1Dom(DA)k ⊆ k≥1Dom |D|k. (ii) KerDA ⊆ k≥1Dom |D|k. (iii) For any α, β ∈ R, |D|βPA|D|α is bounded. (iv) PA ∈ OP−∞. Proof. (i) Let us define for any p ∈ N, Rp := (DA)p−Dp, so Rp ∈ OP p−1 and Rp Dom |D|p Dom |D| (see Remark 3.4). Let us fix k ∈ N, k ≥ 2. Since DomDA = DomD = Dom |D|, we have Dom(DA)k = {φ ∈ Dom |D| : (Dj +Rj)φ ∈ Dom |D| , ∀j 1 ≤ j ≤ k − 1 }. Let φ ∈ Dom(DA)k. We prove by recurrence that for any j ∈ { 1, · · · , k − 1 }, φ ∈ Dom |D|j+1: We have φ ∈ Dom |D| and (D +R1)φ ∈ Dom |D|. Thus, since R1 φ ∈ Dom |D|, Dφ ∈ Dom |D|, which proves that φ ∈ Dom |D|2. Hence, case j = 1 is done. Suppose now that φ ∈ Dom |D|j+1 for a j ∈ { 1, · · · , k − 2 }. Since (Dj+1 +Rj+1)φ ∈ Dom |D|, and Rj+1 φ ∈ Dom |D|, we get Dj+1 φ ∈ Dom |D|, which proves that φ ∈ Dom |D|j+2. Finally, if we set j = k − 1, we get φ ∈ Dom |D|k, so Dom(DA)k ⊆ Dom |D|k. (ii) follows from KerDA ⊆ k≥1Dom(DA)k and (i). (iii) Let us first check that |D|αPA is bounded. We define D0 as the operator with domain DomD0 = ImPA ∩Dom |D|α and such that D0 φ = |D|α φ. Since DomD0 is finite dimensional, D0 extends as a bounded operator on H with finite rank. We have φ∈Dom |D|αPA, ‖φ‖≤1 ‖|D|αPA φ‖ ≤ sup φ∈DomD0, ‖φ‖≤1 ‖|D|α φ‖ = ‖D0‖ <∞ so |D|αPA is bounded. We can remark that by (ii), DomD0 = ImPA and Dom |D|αPA = H. Let us prove now that PA|D|α is bounded: Let φ ∈ DomPA|D|α = Dom |D|α. By (ii), we have ImPA ⊆ Dom |D|α so we get ‖PA|D|α φ‖ ≤ sup ψ∈ImPA, ‖ψ‖≤1 | < ψ, |D|α φ > | ≤ sup ψ∈ImPA, ‖ψ‖≤1 | < |D|αψ, φ > | ≤ sup ψ∈ImPA, ‖ψ‖≤1 ‖|D|αψ‖ ‖φ‖ = ‖D0‖ ‖φ‖ . (iv) For any k ∈ N0 and t ∈ R, δk(PA)|D|t is a linear combination of terms of the form |D|βPA|D|α, so the result follows from (iii). Remark 4.2. We will see later on the noncommutative torus example how important is the difference between DA and D + A. In particular, the inclusion KerD ⊆ KerD + A is not satisfied since A does not preserve KerD contrarily to Ã. The coefficient of the nonconstant term Λk (k > 0) in the expansion (5) of the spectral action S(DA,Φ,Λ) is equal to the residue of ζDA(s) at k. We will see in this section how we can compute these residues in term of noncommutative integral of certain operators. Define for any operator T , p ∈ N, s ∈ C, Kp(T, s) := (− s2) 0≤t1≤···≤tp≤1 σ−st1(T ) · · · σ−stp(T ) dt with dt := dt1 · · · dtp. Remark that if T ∈ OPα, then σz(T ) ∈ OPα for z ∈ C and Kp(T, s) ∈ OPαp. Let us define X := D2A −D2 = ÃD +DÃ+ Ã2, XV := X + VA, thus X ∈ D1(A) ∩OP 1 and by Lemma 4.1, XV ∼ X mod OP−∞. (30) We will use Y := log(D2A)− log(D2) which makes sense since D2A = D2A + PA is invertible for any A. By definition of XV , we get Y = log(D2 +XV )− log(D2). Lemma 4.3. [4] (i) Y is a pseudodifferential operator in OP−1 with the following expansion for any N ∈ N k1,··· ,kp=0 (−1)|k|1+p+1 |k|1+p ∇kp(X∇kp−1(· · ·X∇k1(X) · · · ))D−2(|k|1+p) mod OP−N−1. (ii) For any N ∈ N and s ∈ C, |DA|−s ∼ |D|−s + Kp(Y, s)|D|−s mod OP−N−1−ℜ(s). (31) Proof. (i) We follow [4, Lemma 2.2]. By functional calculus, Y = I(λ) dλ, where I(λ) ∼ (−1)p+1 (D2 + λ)−1XV (D2 + λ)−1 mod OP−N−3. By (30), (D2 + λ)−1XV (D2 + λ)−1X mod OP−∞ and we get I(λ) ∼ (−1)p+1 (D2 + λ)−1X (D2 + λ)−1 mod OP−N−3. We set Ap(X) := (D2 + λ)−1X (D2 + λ)−1 and L := (D2 + λ)−1 ∈ OP−2 for a fixed λ. Since [D2 + λ,X] ∼ ∇(X) mod OP−∞, a recurrence proves that if T is an operator in OP r, then, for q ∈ N0, A1(T ) = LTL ∼ (−1)k∇k(T )Lk+2 mod OP r−q−5. With Ap(X) = LXAp−1(X), another recurrence gives, for any q ∈ N0, Ap(X) ∼ k1,··· ,kp=0 (−1)|k|1∇kp(X∇kp−1(· · ·X∇k1(X) · · · ))L|k|1+p+1 mod OP−q−p−3, which entails that I(λ) ∼ (−1)p+1 k1,··· ,kp=0 (−1)|k|1∇kp(X∇kp−1(· · ·X∇k1(X) · · · ))L|k|1+p+1 mod OP−N−3. (D2 + λ)−(|k|1+p+1)dλ = 1 |k|1+p D−2(|k|1+p), we get the result provided we control the remainders. Such a control is given in [4, (2.27)]. (ii) We have |DA|−s = eB−(s/2)Y e−B |D|−s where B := (−s/2) log(D2). Following [4, Theorem 2.4], we get |DA|−s = |D|−s + Kp(Y, s)|D|−s . (32) and each Kp(Y, s) is in OP Corollary 4.4. For any p ∈ N and r1, · · · , rp ∈ N0, εr1(Y ) · · · εrp(Y ) ∈ Ψ1(A). Proof. If for any q ∈ N and k = (k1, · · · , kq) ∈ Nq0, Γkq(X) := (−1)|k|1+q+1 |k|1+q ∇kq(X∇kq−1(· · ·X∇k1(X) · · · )), then, Γkq (X) ∈ OP |k|1+q. For any N ∈ N, k1,··· ,kq=0 Γkq(X)D −2(|k|1+q) mod OP−N−1. (33) Note that the Γkq(X) are in D1(A), which, with (33) proves that Y and thus εr(Y ) = ∇r(Y )D−2r, are also in Ψ1(A). We remark, as in [11], that the fluctuations leave invariant the first term of the spectral action (5). This is a generalization of the fact that in the commutative case, the noncommutative integral depends only on the principal symbol of the Dirac operator D and this symbol is stable by adding a gauge potential like in D+A. Note however that the symmetrized gauge potential A+ ǫJAJ−1 is always zero in this case for any selfadjoint one-form A. Lemma 4.5. If the spectral triple is simple, formula (6) can be extended as ζDA(0)− ζD(0) = (−1)q −(ÃD−1)q. (34) Proof. Since the spectral triple is simple, equation (32) entails that ζDA(0)− ζD(0) = Tr(K1(Y, s)|D|−s)|s=0 . Thus, with (29), we get ζDA(0) − ζD(0) = −12 Y . Replacing A by Ã, the same proof as in [4] gives − Y = (−1)q −(ÃD−1)q. Lemma 4.6. For any k ∈ N0, s=n−k ζDA(s) = Res s=n−k ζD(s) + r1,··· ,rp=0 s=n−k h(s, r, p) Tr εr1(Y ) · · · εrp(Y )|D|−s where h(s, r, p) := (−s/2)p 0≤t1≤···≤tp≤1 g(−st1, r1) · · · g(−stp, rp) dt . Proof. By Lemma 4.3 (ii), |DA|−s ∼ |D|−s + p=1Kp(Y, s)|D|−s mod OP−(k+1)−ℜ(s), where the convention ∅ = 0 is used. Thus, we get for s in a neighborhood of n− k, |DA|−s − |D|−s − Kp(Y, s)|D|−s ∈ OP−(k+1)−ℜ(s) ⊆ L1(H) which gives s=n−k ζDA(s) = Res s=n−k ζD(s) + s=n−k Kp(Y, s)|D|−s . (35) Let us fix 1 ≤ p ≤ k and N ∈ N. By (29) we get Kp(Y, s) ∼ (− s2) 0≤t1≤···tp≤1 r1,··· ,rp=0 g(−st1, r1) · · · g(−stp, rp) εr1(Y ) · · · εrp(Y ) dt mod OP−N−p−1. (36) If we now take N = k − p, we get for s in a neighborhood of n− k Kp(Y, s)|D|−s − r1,··· ,rp=0 h(s, r, p) εr1(Y ) · · · εrp(Y )|D|−s ∈ OP−k−1−ℜ(s) ⊆ L1(H) so (35) gives the result. Our operators |DA|k are pseudodifferential operators: Lemma 4.7. For any k ∈ Z, |DA|k ∈ Ψk(A). Proof. Using (36), we see that Kp(Y, s) is a pseudodifferential operator in OP −p, so (31) proves that |DA|k is a pseudodifferential operator in OP k. The following result is quite important since it shows that one can use for D or DA: Proposition 4.8. If the spectral triple is simple, Res P |DA|−s P for any pseudodiffer- ential operator P . In particular, for any k ∈ N0 − |DA|−(n−k) = Res s=n−k ζDA(s). Proof. Suppose P ∈ OP k with k ∈ Z and let us fix p ≥ 1. With (36), we see that for any N ∈ N, PKp(Y, s)|D|−s ∼ r1,··· ,rp=0 h(s, r, p)Pεr1(Y ) · · · εrp(Y )|D|−s mod OP−N−p−1+k−ℜ(s). Thus if we take N = n− p+ k, we get PKp(Y, s)|D|−s n−p+k∑ r1,··· ,rp=0 h(s, r, p) Tr Pεr1(Y ) · · · εrp(Y )|D|−s Since s = 0 is a zero of the analytic function s 7→ h(s, r, p) and s 7→ TrPεr1(Y ) · · · εrp(Y )|D|−s has only simple poles by hypothesis, we see that Res h(s, r, p) Tr Pεr1(Y ) · · · εrp(Y )|D|−s PKp(Y, s)|D|−s = 0. (37) Using (31), P |DA|−s ∼ P |D|−s + p=1 PKp(Y, s)|D|−s mod OP−n−1−ℜ(s) and thus, Tr(P |DA|−s) = − P + PKp(Y, s)|D|−s . (38) The result now follows from (37) and (38). To get the last equality, one uses the pseudodiffer- ential operator |DA|−(n−k). Proposition 4.9. If the spectral triple is simple, then − |DA|−n = − |D|−n. (39) Proof. Lemma 4.6 and previous proposition for k = 0. Lemma 4.10. If the spectral triple is simple, − |DA|−(n−1) = − |D|−(n−1) − (n−1 − X|D|−n−1. − |DA|−(n−2) = − |D|−(n−2) + n−2 − X|D|−n + n − X2|D|−2−n Proof. (i) By (31), s=n−1 ζDA(s)− ζD(s) = Res s=n−1 (−s/2)Tr Y |D|−s = −n−1 Y |D|−(n−1)|D|−s where for the last equality we use the simple dimension spectrum hypothesis. Lemma 4.3 (i) yields Y ∼ XD−2 mod OP−2 and Y |D|−(n−1) ∼ X|D|−n−1 mod OP−n−1 ⊆ L1(H). Thus, Y |D|−(n−1)|D|−s = Res X|D|−n−1|D|−s − X|D|−n−1. (ii) Lemma 4.6 (ii) gives s=n−2 ζDA(s) = Res s=n−2 ζD(s) + Res s=n−2 h(s, r, 1) Tr εr(Y )|D|−s + h(s, 0, 2) Tr Y 2|D|−s We have h(s, 0, 1) = − s , h(s, 1, 1) = 1 )2 and h(s, 0, 2) = 1 )2. Using again Lemma 4.3 (i), Y ∼ XD−2 − 1 ∇(X)D−4 − 1 X2D−4 mod OP−3. Thus, s=n−2 Y |D|−s − X|D|−n − 1 −(∇(X) +X2)|D|−2−n. Moreover, using ∇(X)|D|−k = 0 for any k ≥ 0 since is a trace, s=n−2 ε(Y )|D|−s = Res s=n−2 ∇(X)D−4|D|−s − ∇(X)|D|−2−n = 0. Similarly, since Y ∼ XD−2 mod OP−2 and Y 2 ∼ X2D−4 mod OP−3, we get s=n−2 Y 2|D|−s = Res s=n−2 X2D−4|D|−s − X2|D|−2−n. Thus, s=n−2 ζDA(s) = Res s=n−2 ζD(s)+(−n−22 )( − X|D|−n − 1 −(∇(X) +X2)|D|−2−n) − ∇(X)|D|−2−n + 1 − X2|D|−2−n. Finally, s=n−2 ζDA(s) = Res s=n−2 ζD(s) + (−n−22 ) − X|D|−n − 1 − X2|D|−2−n − X2|D|−2−n and the result follows from Proposition 4.8. Corollary 4.11. If the spectral triple is simple and satisfies |D|−(n−2) = ÃD|D|−n =∫ DÃ|D|−n = 0, then − |DA|−(n−2) = n(n−2)4 − ÃDÃD|D|−n−2 + n−2 − Ã2|D|−n Proof. By previous lemma, s=n−2 ζDA(s) = − Ã2|D|−n + n −( ÃDÃD +DÃDÃ+ ÃD2Ã+DÃ2D )|D|−n−2 Since ∇(Ã) ∈ OP 1, the trace property of yields the result. 5 The noncommutative torus 5.1 Notations Let C∞(TnΘ) be the smooth noncommutative n-torus associated to a non-zero skew-symmetric deformation matrix Θ ∈Mn(R) (see [6], [30]). This means that C∞(TnΘ) is the algebra generated by n unitaries ui, i = 1, . . . , n subject to the relations ui uj = e iΘij uj ui, (40) and with Schwartz coefficients: an element a ∈ C∞(TnΘ) can be written as a = k∈Zn ak Uk, where {ak} ∈ S(Zn) with the Weyl elements defined by Uk := e− k.χk u 1 · · · uknn , k ∈ Zn, relation (40) reads UkUq = e k.Θq Uk+q, and UkUq = e −ik.Θq UqUk (41) where χ is the matrix restriction of Θ to its upper triangular part. Thus unitary operators Uk satisfy U∗k = U−k and [Uk, Ul] = −2i sin( k.Θl)Uk+l. Let τ be the trace on C∞(TnΘ) defined by τ k∈Zn ak Uk := a0 and Hτ be the GNS Hilbert space obtained by completion of C∞(TnΘ) with respect of the norm induced by the scalar product 〈a, b〉 := τ(a∗b). On Hτ = { k∈Zn ak Uk : {ak}k ∈ l2(Zn) }, we consider the left and right regular representations of C∞(TnΘ) by bounded operators, that we denote respectively by L(.) and R(.). Let also δµ, µ ∈ { 1, . . . , n }, be the n (pairwise commuting) canonical derivations, defined by δµ(Uk) := ikµUk. (42) We need to fix notations: let AΘ := C∞(TnΘ) acting on H := Hτ ⊗ C2 with n = 2m or n = 2m+ 1 (i.e., m = ⌊n ⌋ is the integer part of n ), the square integrable sections of the trivial spin bundle over Tn. Each element of AΘ is represented on H as L(a) ⊗ 12m where L (resp. R) is the left (resp. right) multiplication. The Tomita conjugation J0(a) := a ∗ satisfies [J0, δµ] = 0 and we define J := J0 ⊗ C0 where C0 is an operator on C2 . The Dirac operator is given by D := −i δµ ⊗ γµ, (43) where we use hermitian Dirac matrices γ. It is defined and symmetric on the dense subset of H given by C∞(TnΘ)⊗ C2 . We still note D its selfadjoint extension. This implies α = −εγαC0, (44) D Uk ⊗ ei = kµUk ⊗ γµei, where (ei) is the canonical basis of C 2m . Moreover, C20 = ±12m depending on the parity of m. Finally, one introduces the chirality (which in the even case is χ := id⊗ (−i)mγ1 · · · γn) and this yields that (AΘ,H,D, J, χ) satisfies all axioms of a spectral triple, see [8, 23]. The perturbed Dirac operator VuDV ∗u by the unitary Vu := L(u)⊗ 12m L(u)⊗ 12m defined for every unitary u ∈ A, uu∗ = u∗u = U0, must satisfy condition (3) (which is equivalent toH being endowed with a structure ofAΘ-bimodule). This yields the necessity of a symmetrized covariant Dirac operator: DA := D +A+ ǫJ AJ−1 since VuDV ∗u = DL(u)⊗12m [D,L(u∗)⊗12m ]: in fact, for a ∈ AΘ, using J0L(a)J 0 = R(a ∗), we get L(a)⊗ γα J−1 = −R(a∗)⊗ γα and that the representation L and the anti-representation R are C-linear, commute and satisfy [δα, L(a)] = L(δαa), [δα, R(a)] = R(δαa). This induces some covariance property for the Dirac operator: one checks that for all k ∈ Zn, L(Uk)⊗ 12m [D, L(U∗k )⊗ 12m ] = 1⊗ (−kµγµ), (45) so with (44), we get Uk[D, U∗k ] + ǫJUk[D, U∗k ]J−1 = 0 and VUk D V = D = DL(Uk)⊗12m [D,L(U∗k )⊗12m ]. (46) Moreover, we get the gauge transformation: VuDAV ∗u = Dγu(A) (47) where the gauged transform one-form of A is γu(A) := u[D, u∗] + uAu∗, (48) with the shorthand L(u)⊗ 12m −→ u. As a consequence, the spectral action is gauge invariant: S(DA,Φ,Λ) = S(Dγu(A),Φ,Λ). An arbitrary selfadjoint one-form A, can be written as A = L(−iAα)⊗ γα, Aα = −A∗α ∈ AΘ, (49) DA = −i δα + L(Aα)−R(Aα) ⊗ γα. (50) Defining Ãα := L(Aα)−R(Aα), we get D2A = −gα1α2(δα1 + Ãα1)(δα2 + Ãα2)⊗ 12m − 12Ωα1α2 ⊗ γ α1α2 where γα1α2 := 1 (γα1γα2 − γα2γα1), Ωα1α2 := [δα1 + Ãα1 , δα2 + Ãα2 ] = L(Fα1α2)−R(Fα1α2) Fα1α2 := δα1(Aα2)− δα2(Aα1) + [Aα1 , Aα2 ]. (51) In summary, D2A = −δα1α2 δα1 + L(Aα1)−R(Aα1) δα2 + L(Aα2)−R(Aα2) ⊗ 12m L(Fα1α2)−R(Fα1α2) ⊗ γα1α2 . (52) 5.2 Kernels and dimension spectrum We now compute the kernel of the perturbed Dirac operator: Proposition 5.1. (i) KerD = U0 ⊗C2 , so dimKerD = 2m. (ii) For any selfadjoint one-form A, KerD ⊆ KerDA. (iii) For any unitary u ∈ A, KerDγu(A) = Vu KerDA. Proof. (i) Let ψ = k,j ck,j Uk ⊗ ej ∈ KerD. Thus, 0 = D2ψ = k,i ck,j|k|2 Uk ⊗ ej which entails that ck,j|k|2 = 0 for any k ∈ Zn and 1 ≤ j ≤ 2m. The result follows. (ii) Let ψ ∈ KerD. So, ψ = U0 ⊗ v with v ∈ C2 and from (50), we get DAψ = Dψ + (A+ ǫJAJ−1)ψ = (A+ ǫJAJ−1)ψ = −i[Aα, U0]⊗ γαv = 0 since U0 is the unit of the algebra, which proves that ψ ∈ KerDA. (iii) This is a direct consequence of (47). Corollary 5.2. Let A be a selfadjoint one-form. Then KerDA = KerD in the following cases: (i) Au := L(u)⊗ 12m [D, L(u∗)⊗ 12m ] when u is a unitary in A. (ii) ||A|| < 1 (iii) The matrix 1 Θ has only integral coefficients. Proof. (i) This follows from previous result because Vu(U0 ⊗ v) = U0 ⊗ v for any v ∈ C2 (ii) Let ψ = k,j ck,j Uk ⊗ ej be in KerDA (so k,j |ck,j|2 <∞) and φ := j c0,j U0 ⊗ ej . Thus ψ′ := ψ − φ ∈ Ker DA since φ ∈ KerD ⊆ KerDA and 06=k∈Zn, j ck,j kα Uk ⊗ γαej ||2 = ||Dψ′||2 = || − (A+ ǫJAJ−1)ψ′||2 ≤ 4||A||2||ψ′||2 < ||ψ′||2. Defining Xk := α kαγα, X α |kα|2 12m is invertible and the vectors {Uk ⊗Xkej }06=k∈Zn, j are orthogonal in H, so 06=k∈Zn, j |kα|2 |ck,j|2 < 06=k∈Zn, j |ck,j|2 which is possible only if ck,j = 0, ∀k, j that is ψ′ = 0 et ψ = φ ∈ Ker D. (iii) This is a consequence of the fact that the algebra is commutative, thus A+ǫJAJ−1 = 0. Note that if Ãu := Au + ǫJAuJ −1, then by (45), ÃUk = 0 for all k ∈ Zn and ‖AUk‖ = |k|, but for an arbitrary unitary u ∈ A, Ãu 6= 0 so DAu 6= D. Naturally the above result is also a direct consequence of the fact that the eigenspace of an iso- lated eigenvalue of an operator is not modified by small perturbations. However, it is interesting to compute the last result directly to emphasize the difficulty of the general case: Let ψ = l∈Zn,1≤j≤2m cl,j Ul⊗ ej ∈ KerDA, so l∈Zn,1≤j≤2m |cl,j |2 <∞. We have to show that ψ ∈ Ker D that is cl,j = 0 when l 6= 0. Taking the scalar product of 〈Uk ⊗ ei| with 0 = DAψ = l, α, j cl, j(l αUl − i[Aα, Ul])⊗ γαej , we obtain l, α, j cl, j lαδk,l − i〈Uk, [Aα, Ul]〉 〈ei, γαej〉. If Aα = α,l aα,l Ul ⊗ γα with { aα,l }l ∈ S(Zn), note that [Ul, Um] = −2i sin(12 l.Θm)Ul+m and 〈Uk, [Aα, Ul]〉 = l′∈Zn aα,l′(−2i sin(12 l ′.Θl)〈Uk, Ul′+l〉 = −2i aα,k−l sin(12k.Θl). cl, j lαδk,l − 2aα,k−l sin(12k.Θl) 〈ei, γαej〉, ∀k ∈ Zn, ∀i, 1 ≤ i ≤ 2m. (53) We conjecture that KerD = KerDA at least for generic Θ’s: the constraints (53) should imply cl,j = 0 for all j and all l 6= 0 meaning ψ ∈ KerD. When 12πΘ has only integer coefficients, the sin part of these constraints disappears giving the result. Lemma 5.3. If 1 Θ is diophantine, Sp C∞(TnΘ),H,D = Z and all these poles are simple. Proof. Let B ∈ D(A) and p ∈ N0. Suppose that B is of the form B = arbrDqr−1|D|pr−1ar−1br−1 · · · Dq1 |D|p1a1b1 where r ∈ N, ai ∈ A, bi ∈ JAJ−1, qi, pi ∈ N0. We note ai =: l ai,l Ul and bi =: i bi,l Ul. With the shorthand kµ1,µqi := kµ1 · · · kµqi and γ µ1,µqi = γµ1 · · · γµqi , we get Dq1 |D|p1a1b1 Uk ⊗ ej = l1, l a1,l1b1,l′1 Ul1UkUl′1 |k + l1 + l′1|p1 (k + l1 + l′1)µ1,µq1 ⊗ γ µ1,µq1 ej which gives, after r iterations, BUk⊗ej = ãlb̃lUlr · · ·Ul1UkUl′1 · · ·Ul′r |k+ l̂i+ l̂′i|pi(k+ l̂i+ l̂′i)µi1,µiqi ⊗γ qr−1 · · · γµ q1 ej where ãl := a1,l1 · · · ar,lr and b̃l′ := b1,l′1 · · · br,l′r . Let us note Fµ(k, l, l ′) := i=1 |k + l̂i + l̂′i|pi(k + l̂i + l̂′i)µi1,µiqi and γ µ := γ µr−11 ,µ qr−1 · · · γµ Thus, with the shortcut ∼c meaning modulo a constant function towards the variable s, B|D|−2p−s ãlb̃l′ τ U−kUlr · · ·Ul1UkUl′1 · · ·Ul′r )Fµ(k,l,l′) |k|s+2p Tr(γµ) . Since Ulr · · ·Ul1Uk = UkUlr · · ·Ul1e−i 1 li.Θk we get U−kUlr · · ·Ul1UkUl′1 · · ·Ul′r = δPr 1 li+l eiφ(l,l ′) e−i 1 li.Θk where φ is a real valued function. Thus, B|D|−2p−s eiφ(l,l ′) δPr 1 li+l ãlb̃l′ Fµ(k,l,l ′) e−i 1 li.Θk |k|s+2p Tr(γµ) ∼c fµ(s)Tr(γµ). The function fµ(s) can be decomposed has a linear combination of zeta function of type described in Theorem 2.17 (or, if r = 1 or all the pi are zero, in Theorem 2.5). Thus, s 7→ Tr B|D|−2p−s has only poles in Z and each pole is simple. Finally, by linearity, we get the result. The dimension spectrum of the noncommutative torus is simple: Proposition 5.4. (i) If 1 Θ is diophantine, the spectrum dimension of C∞(TnΘ),H,D equal to the set {n− k : k ∈ N0 } and all these poles are simple. (ii) ζD(0) = 0. Proof. (i) Lemma 5.3 and Remark 3.9. (ii) ζD(s) = 1≤j≤2m〈Uk ⊗ ej , |D|−sUk ⊗ ej〉 = 2m( + 1) = 2m(Zn(s) + 1). By (21), we get the result. We have computed ζD(0) relatively easily but the main difficulty of the present work is precisely to calculate ζDA(0). 5.3 Noncommutative integral computations We fix a self-adjoint 1-form A on the noncommutative torus of dimension n. Proposition 5.5. If 1 Θ is diophantine, then the first elements of the expansion (5) are given − |DA|−n = − |D|−n = 2m+1πn/2 Γ(n )−1. (54) − |DA|n−k = 0 for k odd. − |DA|n−2 = 0. We need few technical lemmas: Lemma 5.6. On the noncommutative torus, for any t ∈ R, − ÃD|D|−t = − DÃ|D|−t = 0. Proof. Using notations of (49), we have Tr(ÃD|D|−s) ∼c 〈Uk ⊗ ej,−ikµ|k|−s[Aα, Uk]⊗ γαγµej〉 ∼c −iTr(γαγµ) kµ|k|−s〈Uk, [Aα, Uk]〉 = 0 since 〈Uk, [Aα, Uk]〉 = 0. Similarly Tr(DÃ|D|−s) ∼c 〈Uk ⊗ ej , |k|−s aα,l 2 sin (l + k)µUl+k ⊗ γµγαej〉 ∼c 2Tr(γµγα) aα,l sin (l + k)µ |k|−s〈Uk, Ul+k〉 = 0. Any element h in the algebra generated by A and [D,A] can be written as a linear combination of terms of the form a1 p1 · · · anpr where ai are elements of A or [D,A]. Such a term can be written as a series b := a1,α1,l1 · · · aq,αq,lqUl1 · · ·Ulq ⊗ γα1 · · · γαq where ai,αi are Schwartz sequences and when ai =: l alUl ∈ A, we set ai,α,l = ai,l with γα = 1. We define L(b) := τ a1,α1,l1 · · · aq,αq,lqUl1 · · ·Ulq Tr(γα1 · · · γαq ). By linearity, L is defined as a linear form on the whole algebra generated by A and [D,A]. Lemma 5.7. If h is an element of the algebra generated by A and [D,A], h|D|−s ∼c L(h)Zn(s). In particular, Tr h|D|−s has at most one pole at s = n. Proof. We get with b of the form a1,α1,l1 · · · aq,αq,lqUl1 · · ·Ulq ⊗ γα1 · · · γαq , b|D|−s a1,α1,l1 · · · aq,αq,lqUl1 · · ·UlqUk〉 Tr(γα1 · · · γαq )|k|−s ∼c τ( a1,α1,l1 · · · aq,αq,lqUl1 · · ·Ulq)Tr(γα1 · · · γαq )Zn(s) = L(b)Zn(s). The results follows now from linearity of the trace. Lemma 5.8. If 1 Θ is diophantine, the function s 7→ Tr εJAJ−1A|D|−s extends meromor- phically on the whole plane with only one possible pole at s = n. Moreover, this pole is simple εJAJ−1A|D|−s = aα,0 a m+1πn/2 Γ(n/2)−1. Proof. With A = L(−iAα) ⊗ γα, we get ǫJAJ−1 = R(iAα) ⊗ γα, and by multiplication εJAJ−1A = R(Aβ)L(Aα)⊗ γβγα. Thus, εJAJ−1A|D|−s 〈Uk, AαUkAβ〉 |k|−s Tr(γβγα) aα,l aβ,−l e ik.Θl |k|−sTr(γβγα) ∼c 2m aα,l a ik.Θl |k|−s. Theorem 2.5 (ii) entails that l aα,l a ik.Θl |k|−s extends meromorphically on the whole plane C with only one possible pole at s = n. Moreover, this pole is simple and we aα,l a ik.Θl |k|−s = aα,0 aα0 Res Zn(s). Equation (20) now gives the result. Lemma 5.9. If 1 Θ is diophantine, then for any t ∈ R, − X|D|−t = δt,n 2m+1 aα,l a −l + aα,0 a 2πn/2 Γ(n/2)−1. where X = ÃD +DÃ+ Ã2 and A =: −i l aα,l Ul ⊗ γα. Proof. By Lemma 5.6, we get X|D|−t = Ress=0Tr(Ã2|D|−s−t). Since A and εJAJ−1 commute, we have Ã2 = A2 + JA2J−1 + 2εJAJ−1A. Thus, Tr(Ã2|D|−s−t) = Tr(A2|D|−s−t) + Tr(JA2J−1|D|−s−t) + 2Tr(εJAJ−1A|D|−s−t). Since |D| and J commute, we have with Lemma 5.7, Ã2|D|−s−t ∼c 2L(A2)Zn(s+ t) + 2Tr εJAJ−1A|D|−s−t Thus Lemma 5.8 entails that Tr(Ã2|D|−s−t) is holomorphic at 0 if t 6= n. When t = n, Ã2|D|−s−t = 2m+1 aα,l a −l + aα,0 a 2πn/2 Γ(n/2)−1, (55) which gives the result. Lemma 5.10. If 1 Θ is diophantine, then − ÃDÃD|D|−2−n = −n−2 − Ã2|D|−n. Proof. With DJ = εJD, we get − ÃDÃD|D|−2−n = 2 − ADAD|D|−2−n + 2 − εJAJ−1DAD|D|−2−n. Let us first compute ADAD|D|−2−n. We have, with A =: −iL(Aα)⊗ γα =: −i l aα,lUl ⊗ γα, ADAD|D|−s−2−n l1,l2 aα2,l2 aα1,l1 τ(U−kUl2Ul1Uk) kµ1(k+l1)µ2 |k|s+2+n Tr(γα,µ) where γα,µ := γα2γµ2γα1γµ1 . Thus, − ADAD|D|−2−n = − aα2,−l aα1,lRes ′ kµ1kµ2 |k|s+2+n Tr(γα,µ). We have also, with εJAJ−1 = iR(Aα)⊗ γa, εJAJ−1DAD|D|−s−2−n l1,l2 aα2,l2aα1,l1τ(U−kUl1UkUl2) kµ1 (k+l1)µ2 |k|s+2+n Tr(γα,µ). which gives − εJAJ−1DAD|D|−2−n = aα2,0aα1,0Res ′ kµ1kµ2 |k|s+2+n Tr(γα,µ). Thus, − ÃDÃD|D|−2−n = aα2,0aα1,0 − aα2,−laα1,l Ress=0 ′ kµ1kµ2 |k|s+2+n Tr(γα,µ). kµ1kµ2 |k|s+2+n δµ1µ2 Zn(s+ n) and Cn := Ress=0 Zn(s+ n) = 2π n/2Γ(n/2)−1 we obtain − ÃDÃD|D|−2−n = aα2,0aα1,0 − aα2,−laα1,l Tr(γα2γµγα1γµ). Since Tr(γα2γµγα1γµ) = 2 m(2− n)δα2,α1 , we get − ÃDÃD|D|−2−n = 2m − aα,0 aα0 + aα,−l a )Cn(n−2) Equation (55) now proves the lemma. Lemma 5.11. If 1 Θ is diophantine, then for any P ∈ Ψ1(A) and q ∈ N, q odd, − P |D|−(n−q) = 0. Proof. There exist B ∈ D1(A) and p ∈ N0 such that P = BD−2p + R where R is in OP−q−1. As a consequence, P |D|−(n−q) = B|D|−n−2p+q. Assume B = arbrDqr−1ar−1br−1 · · · Dq1a1b1 where r ∈ N, ai ∈ A, bi ∈ JAJ−1, qi ∈ N. If we prove that B|D|−n−2p+q = 0, then the general case will follow by linearity. We note ai =: l ai,l Ul and bi =: l bi,l Ul. With the shorthand kµ1,µqi := kµ1 · · · kµqi and γ µ1,µqi = γµ1 · · · γµqi , we get Dq1a1b1Uk ⊗ ej = a1,l1 b1,l′1 Ul1UkUl (k + l1 + l 1)µ1,µq1 ⊗ γ µ1,µq1ej which gives, after iteration, B Uk ⊗ ej = ãlb̃lUlr · · ·Ul1UkUl′1 · · ·Ul′r (k + l̂i + l̂ i)µi1,µiqi qr−1 · · · γµ where ãl := a1,l1 · · · ar,lr and b̃l′ := b1,l′1 · · · br,l′r . Let’s note Qµ(k, l, l ′) := i=1 (k + l̂i + l̂ i)µi1,µiqi and γµ := γ qr−1 · · · γµ q1 . Thus, − B |D|−n−2p+q = Res ãl b̃l′ τ U−kUlr · · ·Ul1UkUl′1 · · ·Ul′r ) Qµ(k,l,l′) |k|s+2p+n−q Tr(γµ) . Since Ulr · · ·Ul1Uk = UkUlr · · ·Ul1e−i 1 li.Θk, we get U−kUlr · · ·Ul1UkUl′1 · · ·Ul′r = δPr 1 li+l eiφ(l,l ′) e−i 1 li.Θk where φ is a real valued function. Thus, − B |D|−n−2p+q = Res eiφ(l,l ′) δPr 1 li+l ãl b̃l′ Qµ(k,l,l ′)e−i 1 li.Θk |k|s+2p+n−q Tr(γµ) =: Res fµ(s)Tr(γ We decompose Qµ(k, l, l ′) as a sum h=0Mh,µ(l, l ′)Qh,µ(k) where Qh,µ is a homogeneous poly- nomial in (k1, · · · , kn) and Mh,µ(l, l′) is a polynomial in (l1)1, · · · , (lr)n, (l′1)1, · · · , (l′r)n Similarly, we decompose fµ(s) as h=0 fh,µ(s). Theorem 2.5 (ii) entails that fh,µ(s) extends meromorphically to the whole complex plane C with only one possible pole for s+2p+n−q = n+d where d := deg Qh,µ. In other words, if d+ q− 2p 6= 0, fh,µ(s) is holomorphic at s = 0. Suppose now d+ q− 2p = 0 (note that this implies that d is odd, since q is odd by hypothesis), then, by Theorem 2.5 (ii) fh,µ(s) = V u∈Sn−1 Qh,µ(u) dS(u) where V := l,l′∈ZMh,µ(l, l ′) eiφ(l,l ′) δPr 1 li+l ,0 ãl b̃l′ and Z := { l, l′ : i=1 li = 0 }. Since d is odd, Qh,µ(−u) = −Qh,µ(u) and u∈Sn−1 Qh,µ(u) dS(u) = 0. Thus, Res fh,µ(s) = 0 in any case, which gives the result. As we have seen, the crucial point of the preceding lemma is the decomposition of the numer- ator of the series fµ(s) as polynomials in k. This has been possible because we restricted our pseudodifferential operators to Ψ1(A). Proof of Proposition 5.5. The top element follows from Proposition 4.9 and according to (20), − |D|−n = Res |D|−s−n = 2mRes Zn(s + n) = 2m+1πn/2 Γ(n/2) For the second equality, we get from Lemmas 5.7 and 4.6 s=n−k ζDA(s) = r1,··· ,rp=0 h(n− k, r, p) − εr1(Y ) · · · εrp(Y )|D|−(n−k). Corollary 4.4 and Lemma 5.11 imply that εr1(Y ) · · · εrp(Y )|D|−(n−k) = 0, which gives the result. Last equality follows from Lemma 5.10 and Corollary 4.11. 6 The spectral action Here is the main result of this section. Theorem 6.1. Consider the n-NC-torus C∞(TnΘ),H,D where n ∈ N and 1 Θ is a real n×n skew-symmetric real diophantine matrix, and a selfadjoint one-form A = L(−iAα)⊗ γα. Then, the full spectral action of DA = D +A+ ǫJAJ−1 is (i) for n = 2, S(DA,Φ,Λ) = 4πΦ2Λ2 +O(Λ−2), (ii) for n = 4, S(DA,Φ,Λ) = 8π2 Φ4Λ4 − 4π Φ(0) τ(FµνF µν) +O(Λ−2), (iii) More generally, in S(DA,Φ,Λ) = Φn−k cn−k(A)Λ n−k +O(Λ−1), cn−2(A) = 0, cn−k(A) = 0 for k odd. In particular, c0(A) = 0 when n is odd. This result (for n = 4) has also been obtained in [20] using the heat kernel method. It is however interesting to get the result via direct computations of (5) since it shows how this formula is efficient. As we will see, the computation of all the noncommutative integrals require a lot of technical steps. One of the main points, namely to isolate where the Diophantine condition on Θ is assumed, is outlined here. Remark 6.2. Note that all terms must be gauge invariants, namely, according to (48), invariant by Aα −→ γu(Aα) = uAαu∗ + uδα(u∗). A particular case is u = Uk where Ukδα(U∗k ) = −ikαU0. In the same way, note that there is no contradiction with the commutative case where, for any selfadjoint one-form A, DA = D (so A is equivalent to 0!), since we assume in Theorem 6.1 that Θ is diophantine, so A cannot be commutative. Conjecture 6.3. The constant term of the spectral action of DA on the noncommutative n-torus is proportional to the constant term of the spectral action of D+A on the commutative n-torus. Remark 6.4. The appearance of a Diophantine condition for Θ has been characterized in di- mension 2 by Connes [7, Prop. 49] where in this case, Θ = θ with θ ∈ R. In fact, the Hochschild cohomology H(AΘ,AΘ∗) satisfies dim Hj(AΘ,AΘ∗) = 2 (or 1) for j = 1 (or j = 2) if and only if the irrational number θ satisfies a Diophantine condition like |1−ei2πnθ|−1 = O(nk) for some k. Recall that when the matrix Θ is quite irrational (see [23, Cor. 2.12]), then the C∗-algebra generated by AΘ is simple. Remark 6.5. It is possible to generalize above theorem to the case D = −i gµν δµ ⊗ γν instead of (43) when g is a positive definite constant matrix. The formulae in Theorem 6.1 are still valid. 6.1 Computations of In order to get this theorem, let us prove a few technical lemmas. We suppose from now on that Θ is a skew-symmetric matrix in Mn(R). No other hypothesis is assumed for Θ, except when it is explicitly stated. When A is a selfadjoint one-form, we define for n ∈ N , q ∈ N, 2 ≤ q ≤ n and σ ∈ {−,+}q + := ADD−2, − := ǫJAJ−1DD−2, σ := Aσq · · ·Aσ1 . Lemma 6.6. We have for any q ∈ N, −(ÃD−1)q = −(ÃDD−2)q = σ∈{+,−}q − Aσ. Proof. Since P0 ∈ OP−∞, D−1 = DD−2 mod OP−∞ and (ÃD−1)q = (ÃDD−2)q. Lemma 6.7. Let A be a selfadjoint one-form, n ∈ N and q ∈ N with 2 ≤ q ≤ n and σ ∈ {−,+}q. Then ∫ − Aσ = − A−σ. Proof. Let us first check that JP0 = P0J . Since DJ = εJD, we get DJP0 = 0 so JP0 = P0JP0. Since J is an antiunitary operator, we get P0J = P0JP0 and finally, P0J = JP0. As a consequence, we get JD2 = D2J , JDD−2 = εDD−2J , JA+J−1 = A− and JA−J−1 = A+. In summary, JAσiJ−1 = A−σi . The trace property of now gives − Aσ = − Aσq · · ·Aσ1 = − JAσqJ−1 · · · JAσ1J−1 − A−σq · · ·A−σ1 = − A−σ. Definition 6.8. In [4] has been introduced the vanishing tadpole hypothesis: − AD−1 = 0, for all A ∈ Ω1D(A). (56) By the following lemma, this condition is satisfied for the noncommutative torus, a fact more or less already known within the noncommutative community [34]. Lemma 6.9. Let n ∈ N, A = L(−iAα)⊗γα = −i l∈Zn aα,l Ul⊗γα, Aα ∈ AΘ, { aα,l }l ∈ S(Zn), be a hermitian one-form. Then, ApD−q = (ǫJAJ−1)pD−q = 0 for p ≥ 0 and 1 ≤ q < n (case p = q = 1 is tadpole hypothesis.) (ii) If 1 Θ is diophantine, then BD−q = 0 for 1 ≤ q < n and any B in the algebra generated by A, [D,A], JAJ−1 and J [D,A]J−1. Proof. (i) Let us compute ∫ − Ap(ǫJAJ−1)p′D−q. With A = L(−iAα)⊗ γα and ǫJAJ−1 = R(iAα)⊗ γα, we get Ap = L(−iAα1) · · ·L(−iAαp)⊗ γα1 · · · γαp (ǫJAJ−1)p = R(iAα′1 ) · · ·R(iAα′ )⊗ γα′1 · · · γα We note ãα,l := aα1,l1 · · · aαp,lp . Since L(−iAα1) · · ·L(−iAαp)R(iAα′1) · · ·R(iAα′p′ )Uk = (−i) ãα,l ãα′,l′ Ul1 · · ·UlpUkUl′ · · ·Ul′1 , Ul1 · · ·UlpUk = UkUl1 · · ·Ulp e−i( i li).Θk, we get, with Ul,l′ := Ul1 · · ·UlpUl′ · · ·Ul′1 , gµ,α,α′(s, k, l, l ′) := eik.Θ kµ1 ...kµq |k|s+2q ãα,l ãα′,l′ , ′,µ := γα1 · · · γαpγα′1 · · · γα p′γµ1 · · · γµq , Ap(ǫJAJ−1)p D−q|D|−sUk ⊗ ei ∼c (−i)p ip gµ,α,α′(s, k, l, l ′)UkUl,l′ ⊗ γα,α ′,µei. Thus, Ap(ǫJAJ−1)p D−q = Res f(s) where f(s) : = Tr Ap(ǫJAJ−1)p D−q|D|−s ∼c (−i)p ip 〈Uk ⊗ ei, gµ,α,α′(s, k, l, l ′)UkUl,l′ ⊗ γα,α ′,µei〉 ∼c (−i)p ip gµ,α,α′(s, k, l, l ′)Ul,l′ Tr(γµ,α,α ∼c (−i)p ip gµ,α,α′(s, k, l, l Ul,l′ Tr(γµ,α,α It is straightforward to check that the series k,l,l′gµ,α,α′(s, k, l, l Ul,l′ is absolutely summable if ℜ(s) > R for a R > 0. Thus, we can exchange the summation on k and l, l′, which gives f(s) ∼c (−i)p ip gµ,α,α′(s, k, l, l Ul,l′ Tr(γµ,α,α If we suppose now that p′ = 0, we see that, f(s) ∼c (−i)p ′ kµ1 ...kµq |k|s+2q ãα,l δ Tr(γµ,α,α which is, by Proposition 2.16, analytic at 0. In particular, for p = q = 1, we see that AD−1 = 0, i.e. the vanishing tadpole hypothesis is satisfied. Similarly, if we suppose p = 0, we get f(s) ∼c (−i)p ′ kµ1 ...kµq |k|s+2q ãα,l′ δPp′ i=1 l Tr(γµ,α,α which is holomorphic at 0. (ii) Adapting the proof of Lemma 5.11 to our setting (taking qi = 0, and adding gamma matrices components), we see that − BD−q = Res eiφ(l,l ′) δPr 1 li+l ãα,l b̃β,l′ kµ1 ···kµq e 1 li.Θk |k|s+2q Tr(γ(µ,α,β)) where γ(µ,α,β) is a complicated product of gamma matrices. By Theorem 2.5 (ii), since we suppose here that 1 Θ is diophantine, this residue is 0. 6.1.1 Even dimensional case Corollary 6.10. Same hypothesis as in Lemma 6.9. (i) Case n = 2: − AqD−q = −δq,2 4π τ (ii) Case n = 4: with the shorthand δµ1,...,µ4 := δµ1µ2δµ3µ4 + δµ1µ3δµ2µ4 + δµ1µ4δµ2µ3 , − AqD−q = δq,4 π Aα1 · · ·Aα4 Tr(γα1 · · · γα4γµ1 · · · γµ4)δµ1,...,µ4 . Proof. (i, ii) The same computation as in Lemma 6.9 (i) (with p′ = 0, p = q = n) gives − AnD−n = Res (−i)n ′ kµ1 ...kµn |k|s+2n l∈(Zn)n ãα,lUl1 · · ·Uln Tr(γα1 · · · γαnγµ1 · · · γµn) and the result follows from Proposition 2.16. We will use few notations: If n ∈ N, q ≥ 2, l := (l1, · · · , lq−1) ∈ (Zn)q−1, α := (α1, · · · , αq) ∈ {1, · · · , n}q, k ∈ Zn\{0}, σ ∈ {−,+}q, (ai)1≤i≤n ∈ (S(Zn))n, lq := − 1≤j≤q−1 lj , λσ := (−i)q j=1...q σj , ãα,l := aα1,l1 . . . aαq ,lq , φσ(k, l) := 1≤j≤q−1 (σj − σq) k.Θlj + 2≤j≤q−1 σj (l1 + . . .+ lj−1).Θlj , gµ(s, k, l) := kµ1 (k+l1)µ2 ...(k+l1+...+lq−1)µq |k|s+2|k+l1|2...|k+l1+...+lq−1|2 with the convention 2≤j≤q−1 = 0 when q = 2, and gµ(s, k, l) = 0 whenever l̂i = −k for a 1 ≤ i ≤ q − 1. Lemma 6.11. Let A = L(−iAα) ⊗ γα = −i l∈Zn aα,l Ul ⊗ γα where Aα = −A∗α ∈ AΘ and { aα,l }l ∈ S(Zn), with n ∈ N, be a hermitian one-form, and let 2 ≤ q ≤ n, σ ∈ {−,+}q. Then, Aσ = Res f(s) where f(s) := l∈(Zn)q−1 φσ(k,l) gµ(s, k, l) ãα,l Tr(γ αqγµq · · · γα1γµ1). Proof. By definition, Aσ = Res f(s) where Tr(Aσq · · ·Aσ1 |D|−s) ∼c 〈Uk ⊗ ei, |k|−s Aσq · · ·Aσ1Uk ⊗ ei〉 =: f(s). Let r ∈ Zn and v ∈ C2m . Since A = L(−iAα)⊗ γα, and ǫJAJ−1 = R(iAα)⊗ γα, we get +Ur ⊗ v = ADD−2Ur ⊗ v = A rµ|r|2+δr,0Ur ⊗ γ µv = −i rµ |r|2+δr,0 AαUr ⊗ γαγµv , −Ur ⊗ v = ǫJAJ−1DD−2Ur ⊗ v = ǫJAJ−1 rµ|r|2+δr,0Ur ⊗ γ µv = i |r|2+δr,0 UrAα ⊗ γαγµv. With UlUr = e Ur+l and UrUl = e Ur+l, we obtain, for any 1 ≤ j ≤ q, σjUr ⊗ v = (−σj) i eσj r.Θl rµ |r|2+δr,0 aα,l Ur+l ⊗ γαγµv. We now apply q times this formula to get |k|−sAσq · · ·Aσ1Uk ⊗ ei = λσ l∈(Zn)q φσ(k,l) gµ(s, k, l) ãα,l Uk+ ⊗ γαqγµq · · · γα1γµ1ei φσ(k, l) := σ1 k.Θl1 + σ2 (k + l1).Θl2 + . . .+ σq (k + l1 + . . . + lq−1).Θlq. Thus, f(s) = l∈(Zn)q φσ(k,l) gµ(s, k, l) ãα,l U Tr(γαqγµq · · · γα1γµ1) l∈(Zn)q φσ(k,l) gµ(s, k, l) ãα,l δ( lj)Tr(γ αqγµq · · · γα1γµ1) l∈(Zn)q−1 φσ(k,l) gµ(s, k, l) ãα,l Tr(γ αqγµq · · · γα1γµ1) where in the last sum lq is fixed to − 1≤j≤q−1 lj and thus, φσ(k, l) = 1≤j≤q−1 (σj − σq) k.Θlj + 2≤j≤q−1 σj (l1 + . . .+ lj−1).Θlj . By Lemma 2.10, there exists a R > 0 such that for any s ∈ C with ℜ(s) > R, the family φσ(k,l) gµ(s, k, l) ãα,l (k,l)∈(Zn\{ 0 })×(Zn)q−1 is absolutely summable as a linear combination of families of the type considered in that lemma. As a consequence, we can exchange the summations on k and l, which gives the result. In the following, we will use the shorthand c := 4π Lemma 6.12. Suppose n = 4. Then, with the same hypothesis of Lemma 6.11, (i) 1 −(A+)2 = 1 −(A−)2 = c aα1,l aα2,−l lα1 lα2 − δα1α2 |l|2 (ii) − 1 −(A+)3 = −1 −(A−)3 = 4c li∈Z4 aα3,−l1−l2 a aα1,l1 sin l1.Θl2 (iii) 1 −(A+)4 = 1 −(A−)4 = 2c li∈Z4 aα1,−l1−l2−l3 aα2,l3 a aα2l1 sin l1.Θ(l2+l3) sin l2.Θl3 (iv) Suppose 1 Θ diophantine. Then the crossed terms in (A+ + A−)q vanish: if C is the set of all σ ∈ {−,+}q with 2 ≤ q ≤ 4, such that there exist i, j satisfying σi 6= σj, we have∑ Aσ = 0. Proof. (i) Lemma 6.11 entails that A++ = Res l∈Zn −f(s, l) where f(s, l) := ′ kµ1 (k+l)µ2 |k|s+2|k+l|2 ãα,l Tr(γ α2γµ2γα1γµ1) and ãα,l := aα1,l aα2,−l . We will now reduce the computation of the residue of an expression involving terms like |k+l|2 in the denominator to the computation of residues of zeta functions. To proceed, we use (16) into an expression like the one appearing in f(s, l). We see that the last term on the righthandside yields a Zn(s) while the first one is less divergent by one power of k. If this is not enough, we repeat this operation for the new factor of |k + l|2 in the denominator. For f(s, l), which is quadratically divergent at s = 0, we have to repeat this operation three times before ending with a convergent result. All the remaining terms are expressible in terms of Zn functions. We get, using three times (16), |k+l|2 − 2k.l+|l| (2k.l+|l|2)2 − (2k.l+|l| |k|6|k+l|2 . (57) Let us define fα,µ(s, l) := ′ kµ1 (k+l)µ2 |k|s+2|k+l|2 ãα,l so that f(s, l) = fα,µ(s, l)Tr(γ α2γµ2γα1γµ1). Equation (57) gives fα,µ(s, l) = f1(s, l)− f2(s, l) + f3(s, l)− r(s, l) with obvious identifications. Note that the function r(s, l) = ′ kµ1 (k+l)µ2 (2kl+|l| |k|s+8|k+l|2 ãα,l is a linear combination of functions of the typeH(s, l) satisfying the hypothesis of Corollary 2.13. Thus, r(s, l) satisfies (H1) and with the previously seen equivalence relation modulo functions satisfying this hypothesis we get fα,µ(s, l) ∼ f1(s, l)− f2(s, l) + f3(s, l). Let’s now compute f1(s, l). f1(s, l) = ′ kµ1 (k+l)µ2 |k|s+4 ãα,l = ãα,l ′ kµ1kµ2 |k|s+4 Proposition 2.1 entails that s 7→ ′ kµ1kµ2 |k|s+4 is holomorphic at 0. Thus, f1(s, l) satisfies (H1), and fα,µ(s, l) ∼ −f2(s, l) + f3(s, l). Let’s now compute f2(s, l) modulo (H1). We get, using several times Proposition 2.1, f2(s, l) = ′ kµ1 (k+l)µ2 (2kl+|l| |k|s+6 ãα,l = ′ (2kl)kµ1kµ2+(2kl)kµ1 lµ2+|l| 2kµ1kµ2+lµ2 |l| |k|s+6 ãα,l ∼ 0 + ′ (2kl)kµ1 lµ2 |k|s+6 ãα,l + ′ |l|2kµ1kµ2 |k|s+6 ãα,l + 0 . Recall that |k|s+6 Zn(s + 4). Thus, f2(s, l) ∼ 2lilµ2 ãα,l Zn(s+ 4) + |l|2 ãα,l δµ1µ2 Zn(s+ 4). Finally, let us compute f3(s, l) modulo (H1) following the same principles: f3(s, l) = ′ kµ1 (k+l)µ2 (2kl+|l| |k|s+8 ãα,l ′ (2kl)2kµ1kµ2+(2kl) 2kµ1 lµ2+|l| 4kµ1kµ2+|l| 4kµ1 lµ2+(4kl)|l| 2kµ1kµ2+(4kl)|l| 2kµ1 lµ2 |k|s+8 ãα,l ∼ 4lilj ′ kikjkµ1kµ2 |k|s+8 ãα,l + 0. In conclusion, fα,µ(s, l) ∼ −14(2lµ1 lµ2 + |l| 2 δµ1µ2)ãα,lZn(s+ 4) + 4l ilj ãα,l ′ kikjkµ1kµ2 |k|s+8 =: gα,µ(s, l). Proposition (2.1) entails that Zn(s+ 4) and s 7→ ′ kikjkµ1kµ2 |k|s+8 extend holomorphically in a punctured open disk centered at 0. Thus, gα,µ(s, l) satisfies (H2) and we can apply Lemma 2.14 to get −(A+)2 = Res f(s, l) = gα,µ(s, l)Tr(γ α2γµ2γα1γµ1) =: g(s, l). The problem is now reduced to the computation of Res g(s, l). Recall that Ress=0 Z4(s+4) = 2π by (20) or (17), and Ress=0 ′ kikjklkm |k|s+8 = (δijδlm + δilδjm + δimδjl) Thus, gα,µ(s, l) = −π ãα,l (lµ1 lµ2 + |l|2δµ1µ2). We will use Tr(γµ1 · · · γµ2j ) = Tr(1) all pairings of { 1···2j } s(P ) δµP1µP2 δµP3µP4 · · · δµP2j−1µP2j (58) where s(P ) is the signature of the permutation P when P2m−1 < P2m for 1 ≤ m ≤ n. This gives Tr(γα2γµ2γα1γµ1) = 2m(δα2µ2δα1µ1 − δα1α2δµ2µ1 + δα2µ1δµ2α1). (59) Thus, g(s, l) = −c ãα,l (lµ1 lµ2 + 12 |l| 2δµ1µ2)(δ α2µ2δα1µ1 − δα1α2δµ2µ1 + δα2µ1δµ2α1) = −2c ãα,l (lα1 lα2 − δα1α2 |l|2). Finally, −(A+)2 = 1 −(A−)2 = c aα1,l aα2,−l lα1 lα2 − δα1α2 |l|2 (ii) Lemma 6.11 entails that A+++ = Res (l1,l2)∈(Zn)2 f(s, l) where f(s, l) := l1Θl2 kµ1 (k+l1)µ2 (k+ bl2)µ3 |k|s+2|k+l1|2|k+bl2|2 ãα,l Tr(γ α3γµ3γα2γµ2γα1γµ1) =: fα,µ(s, l)Tr(γ α3γµ3γα2γµ2γα1γµ1), and ãα,l := aα1,l1 aα2,l2 aα3,−bl2 with l̂2 := l1 + l2. We use the same technique as in (i): |k+l1|2 − 2k.l1+|l1| (2k.l1+|l1| |k|4|k+l1|2 |k+bl2|2 |k|2 − 2k.bl2+|bl2| (2k.bl2+|bl2| |k|4|k+bl2|2 and thus, |k+l1|2|k+bl2|2 − 2k.l1 − 2k.bl2 +R(k, l) (60) where the remain R(k, l) is a term of order at most −6 in k. Equation (60) gives fα,µ(s, l) = f1(s, l) + r(s, l) where r(s, l) corresponds to R(k, l). Note that the function r(s, l) = l1Θl2 kµ1(k+l)µ2 (k+ bl2)µ3R(k,l) |k|s+2 ãα,l is a linear combination of functions of the type H(s, l) satisfying the hypothesis of Corollary (2.13). Thus, r(s, l) satisfies (H1) and fα,µ(s, l) ∼ f1(s, l). Let us compute f1(s, l) modulo (H1) f1(s, l) = l1Θl2 kµ1 (k+l1)µ2 (k+ bl2)µ3 |k|s+6 ãα,l − l1Θl2 kµ1 (k+l1)µ2 (k+ bl2)µ3 (2k.l1+2k. |k|s+8 ãα,l l1Θl2 kµ1kµ2 bl2µ3+kµ1kµ3 l1µ2 |k|s+6 ãα,l − l1Θl2 kµ1kµ2kµ3 (2k.l1+2k. |k|s+8 ãα,l = i e l1Θl2 ãα,l (l1µ2δµ1µ3 + l̂2µ3δµ1µ2) Z4(s+ 4)− 2(li1 + l̂i2) ′ kµ1kµ2kµ3ki |k|s+8 =: gα,µ(s, l). Since gα,µ(s, l) satisfies (H2), we can apply Lemma 2.14 to get −(A+)3 = Res (l1,l2)∈(Zn)2 f(s, l) (l1,l2)∈(Zn)2 gα,µ(s, l)Tr(γ α3γµ3γα2γµ2γα1γµ1) =: Recall that l3 := −l1 − l2 = −l̂2. By (17) and (19), gα,µ(s, l)i e l1Θl2 ãα,l 2(−li1 + li3)π (δµ1µ2δµ3i + δµ1µ3δµ2i + δµ1iδµ2µ3) + (l1µ2δµ1µ3 − l3µ3δµ1µ2)π We decompose Xl in five terms: Xl = 2 l1Θl2 ãα,l (T1 + T2 + T3 + T4 + T5) where T0 := (−li1 + li3)(δµνδρi + δµρδνi + δµiδνρ) + l1νδµρ − l3ρδµν , T1 := (δ α3ρδα2νδα1µ − δα3ρδα2α1δµν + δα3ρδα2µδα1ν)T0, T2 := (−δα2α3δρνδα1µ + δα2α3δα1ρδµν − δα2α3δρµδα1ν)T0, T3 := (δ α3νδα2ρδα1µ − δα3νδα1ρδα2µ + δα3νδρµδα1α2)T0, T4 := (−δα1α3δα2ρδµν + δα1α3δρνδα2µ − δα1α3δρµδα2ν)T0, T5 := (δ α3µδα2ρδα1ν − δα3µδρνδα1α2 + δα3µδα1ρδα2ν)T0. With the shorthand p := −l1 − 2l3, q := 2l1 + l3, r := −p− q = −l1 + l3, we compute each Ti, and find 3T1 = δ α1α2(2− 2m)pα3 + δα3α1qα2 − δα2α1qα3 + δα3α2qα1 + δα3α2rα1 − δα2α1rα3 + δα3α1rα2 , 3T2 = (2 m − 2)δα2α3pα1 − 2mδα2α3qα1 − 2mδα2α3rα1 , 3T3 = δ α1α3pα2 − δα2α3pα1 + δα1α2pα3 + 2mδα2α1qα3 + δα3α2rα1 − δα3α1rα2 + δα1α2rα3 , 3T4 = −δα1α32mpα2 − δα1α32mqα2 + δα1α3(2m − 2)rα2 , 3T5 = δ α1α3pα2 − δα1α2pα3 + δα3α2pα1 + δα3α2qα1 − δα1α2qα3 + δα3α1qα2 + (2− 2m)δα1α2rα3 . Thus, Xl = 2 m 2π2 l1.Θl2 ãα,l (q α3δα1α2 + rα2δα1α3 + pα1δα2α3) (61) −(A+)3 = i 2c (S1 + S2 + S3), where S1, S2 and S3 correspond to respectively q α3δα1α2 , rα2δα1α3 and pα1δα2α3 . In S1, we permute the li variables the following way: l1 7→ l3, l2 7→ l1, l3 7→ l2. Therefore, l3.Θ l1 7→ l3.Θ l1 and q 7→ r. With a similar permutation of the αi, we see that S1 = S2. We apply the same principles to prove that S1 = S3 (using permutation l1 7→ l2, l2 7→ l3, l3 7→ l1). Thus, −(A+)3 = i 2c ãα,l e l1.Θl2 (l1 − l2)α3δα1α2 = S4 − S5, where S4 correspond to l1 and S5 to l2. We permute the li variables in S5 the following way: l1 7→ l2, l2 7→ l1, l3 7→ l3, with a similar permutation on the αi. Since l1.Θ l2 7→ −l1.Θ l2, we finally get −(A+)3 = −4c aα1,l1 aα2,l2 aα3,−l1−l2 sin l1.Θl2 lα31 δ α1α2 . (iii) Lemma 6.11 entails that A++++ = Res (l1,l2,l3)∈(Zn)3 fµ,α(s, l)Tr γ µ,α where θ := l1.Θl2 + l1.Θl3 + l2.Θl3, Tr γµ,α := Tr(γα4γµ4γα3γµ3γα2γµ2γα1γµ1), fµ,α(s, l) := θ kµ1 (k+l1)µ2 (k+ bl2)µ3 (k+ bl3)µ4 |k|s+2|k+l1|2|k+bl2|2|k+bl3|2 ãα,l, ãα,l := aα1,l1 aα2,l2 aα3,l3 aα4,−l1−l2−l3 . Using (16) and Corollary 2.13 successively, we find fµ,α(s, l) ∼ θ kµ1kµ2kµ3kµ4 |k|s+2|k+l1|2|k+l1+l2|2|k+l1+l2+l3|2 ãα,l ∼ θ kµ1kµ2kµ3kµ4 |k|s+8 ãα,l. Since the function θ kµ1kµ2kµ3kµ4 |k|s+8 ãα,l satisfies (H2), Lemma 2.14 entails that −(A+)4 = (l1,l2,l3)∈(Zn)3 ãα,l Res ′ kµ1kµ2kµ3kµ4 |k|s+8 Tr γµ,α =: Therefore, with (19), we get Xl = ãα,l e (A+B + C), where A := Tr(γα4γµ4γα3γµ4γ α2γµ2γα1γµ2), B := Tr(γα4γµ4γα3γµ2γα2γµ4γ α1γµ2), C := Tr(γα4γµ4γα3γµ2γ α2γµ2γα1γµ4). Using successively {γµ, γν} = 2δµν and γµγµ = 2m 12m , we see that A = C = 4 Tr(γα4γα3γα2γα1), B = −4 Tr(γα4γα3γα1γα2) + Tr(γα4γα2γα3γα1) Thus, A+B +C = 8 2m δα4α3δα2α1 + δα4α1δα3α2 − 2δα4α2δα3α1 , and ãα,l δα4α3δα2α1 + δα4α1δα3α2 − 2δα4α2δα3α1 . (62) By (62), we get ∫ −(A+)4 = 2c (−2T1 + T2 + T3), where T1 := l1,...,l4 aα4,l4 aα3,l3 aα2,l2 aα1,l1 e δα4α2 δα3α1 , T2 := l1,...,l4 aα4,l4 aα3,l3 aα2,l2 aα1,l1 e δα4α3 δα2α1 , T3 := l1,...,l4 aα4,l4 aα3,l3 aα2,l2 aα1,l1 e δα4α1 δα3α2 . We now proceed to the following permutations of the li variables in the T1 term : l1 7→ l2, l2 7→ l1, l3 7→ l4, l4 7→ l3. While i li is invariant, θ is modified : θ 7→ l2.Θl1 + l2.Θl4 + l1.Θl4. With δ0, in factor, we can let l4 be −l1 − l2 − l3, so that θ 7→ −θ. We also permute the αi in the same way. Thus, l1,...,l4 aα3,l3 aα4,l4 aα1,l1 aα2,l2 e δα3α1 δα4α2 . Therefore, 2T1 = 2 l1,...,l4 aα4,l4 aα3,l3 aα2,l2 aα1,l1 cos δα4α2 δα3α1 . (63) The same principles are applied to T2 and T3. Namely, the permutation l1 7→ l1, l2 7→ l3, l3 7→ l2, l4 7→ l4 in T2 and the permutation l1 7→ l2, l2 7→ l3, l3 7→ l1, l4 7→ l4 in T3 (the αi variables are permuted the same way) give l1,...,l4 aα4,l4aα3,l3aα2,l2 aα1,l1 e δα4α2 δα3α1 , l1,...,l4 aα4,l4 aα3,l3aα2,l2 aα1,l1 e δα4α2 δα3α1 where φ := l1.Θ l2 + l1.Θ l3 − l2.Θ l3. Finally, we get −(A+)4 = 4c l1,...,l4 aα1,l4 aα2,l3 a − cos θ l1,...,l3 aα1,−l1−l2−l3 aα2,l3 a l1.Θ(l2+l3) sin l2.Θl3 . (64) (iv) Suppose q = 2. By Lemma 6.11, we get − Aσ = Res λσfα,µ(s, l)Tr(γ α2γµ2γα1γµ1) where fα,µ(s, l) := ′ kµ1 (k+l)µ2 |k|s+2|k+l|2 eiη k.Θl ãα,l and η := 1 (σ1 − σ2) ∈ {−1, 1}. As in the proof of (i), since the presence of the phase does not change the fact that r(s, l) satisfies (H1), we get fα,µ(s, l) ∼ f1(s, l)− f2(s, l) + f3(s, l) where f1(s, l) = ′ kµ1(k+l)µ2 |k|s+4 eiη k.Θl ãα,l, f2(s, l) = ′ kµ1(k+l)µ2 (2k.l+|l| |k|s+6 eiη k.Θl ãα,l, f3(s, l) = ′ kµ1(k+l)µ2 (2k.l+|l| |k|s+8 eiη k.Θl ãα,l. Suppose that l = 0. Then f2(s, 0) = f3(s, 0) = 0 and Proposition 2.1 entails that f1(s, 0) = kµ1kµ2 |k|s+4 ãα,0 is holomorphic at 0 and so is fα,µ(s, 0). Since 1 Θ is diophantine, Theorem 2.5 3 gives us the result. Suppose q = 3. Then Lemma 6.11 implies that − Aσ = Res l∈(Zn)2 fµ,α(s, l) Tr(γ µ3γα3 · · · γµ1γα1) where fµ,α(s, l) := ik.Θ(ε1l1+ε2l2)e σ2l1.Θl2 kµ1 (k+l1)µ2 (k+l1+l2)µ3 |k|s+2|k+l1|2|k+l1+l2|2 ãα,l, and εi := (σi − σ3) ∈ {−1, 0, 1}. By hypothesis (ε1, ε2) 6= (0, 0). There are six possibilities for the values of (ε1, ε2), corresponding to the six possibilities for the values of σ: (−,−,+), (−,+,+), (+,−,+), (+,+,−), (−,+,−), and (+,−,−). As in (ii), we see that fµ,α(s, l) ∼ ′ eik.Θ(ε1l1+ε2l2)kµ1 (k+l1)µ2 (k+ bl2)µ3 |k|s+6 ′ eik.Θ(ε1l1+ε2l2)kµ1 (k+l1)µ2 (k+ bl2)µ3 (2k.l1+2k. |k|s+8 λσ ãα,l e σ2l1.Θl2 . With Z := {(l1, l2) : ε1l1 + ε2l2 = 0}, Theorem 2.5 (iii) entails that l∈(Zn)2\Z fµ,α(s, l) is holomorphic at 0. To conclude we need to prove that g(σ) := fµ,α(s, l) Tr(γ µ3γα3 · · · γµ1γα1) is holomorphic at 0. By definition, λσ = iσ1σ2σ3 and as a consequence, we check that g(−,−,+) = −g(+,+,−), g(+,−,+) = −g(+,−,−), g(−,+,+) = −g(−,+,−), which implies that σ g(σ) = 0. The result follows. Suppose finally that q = 4. Again, Lemma 6.11 implies that − Aσ = Res l∈(Zn)3 fµ,α(s, l) Tr(γ µ4γα4 · · · γµ1γα1) where fµ,α(s, l) := i=1 εili e (σ2l1.Θl2+σ3(l1+l2).Θl3) kµ1 (k+l1)µ2 (k+l1+l2)µ3 (k+l1+l2+l3)µ4 |k|s+2|k+l1|2|k+l1+l2|2|k+l1+l2+l3|2 ãα,l and εi := (σi − σ4) ∈ {−1, 0, 1}. By hypothesis (ε1, ε2, ε3) 6= (0, 0, 0). There are fourteen pos- sibilities for the values of (ε1, ε2, ε3), corresponding to the fourteen possibilities for the values of σ: (−,−,−,+), (−,−,+,+), (−,+,−,+), (+,−,−,+), (−,+,+,+), (+,−,+,+), (+,+,−,+), (+,+,+,−), (−,−,+,−), (−,+,−,−), (+,−,−,−), (−,+,+,−), (+,−,+,−) and (+,+,−,−). As in (ii), we see that, with the shorthand θσ := σ2l1.Θl2 + σ3(l1 + l2).Θl3, fµ,α(s, l) ∼ i=1 εili e θσ kµ1kµ2kµ3kµ4 |k|s+8 ãα,l =: gµ,α(s, l) . With Zσ := {(l1, l2, l3) : i=1 εili = 0}, Theorem 2.5 (iii), the series l∈(Zn)3\Zσ fµ,α(s, l) is holomorphic at 0. To conclude, we need to prove that g(σ) := gµ,α(s, l) Tr(γ µ4γα4 · · · γµ1γα1) = 0. Let C be the set of the fourteen values of σ and C7 be the set of the seven first values of σ given above. Lemma 6.7 implies ∑ g(σ) = 2 g(σ). Thus, in the following, we restrict to these seven values. Let us note Fµ(s) := kµ1kµ2kµ3kµ4 |k|s+8 so that g(σ) = Res Fµ(s)λσ θσ ãα,l Tr(γ µ4γα4 · · · γµ1γα1). Recall from (62) that Fµ(s)Tr(γ µ4γα4 · · · γµ1γα1) = 2c δα4α3δα2α1 + δα4α1δα3α2 − 2δα4α2δα3α1 As a consequence, we get, with ãα,l := aα1,l1 · · · aα4,l4 , g(σ) = 2cλσ l∈(Zn)4 θσ ãα,l δP4 i=1 li,0 i=1 εili,0 δα4α3δα2α1 + δα4α1δα3α2 − 2δα4α2δα3α1 =: 2cλσ(T1 + T2 − 2T3). We proceed to the following change of variable in T1: l1 7→ l1, l2 7→ l3, l3 7→ l2, l4 7→ l4. Thus, we get θσ 7→ ψσ := σ2l1.Θl3 + σ3(l1 + l3).Θl2, and i=1 εili 7→ ε1l1 + ε3l2 + ε2l3 =: uσ(l). With a similar permutation on the αi, we get l∈(Zn)4 ψσ ãα,l δP4 i=1 li,0 δε1l1+ε3l2+ε2l3,0 δ α4α2δα3α1 . We proceed to the following change of variable in T2: l1 7→ l2, l2 7→ l3, l3 7→ l1, l4 7→ l4. Thus, we get θσ 7→ φσ := σ2l2.Θl3 + σ3(l2 + l3).Θl1, and i=1 εili 7→ ε3l1 + ε1l2 + ε2l3 =: vσ(l). After a similar permutation on the αi, we get l∈(Zn)4 φσ ãα,l δP4 i=1 li,0 δε3l1+ε1l2+ε2l3,0 δ α4α2δα3α1 . Finally, we proceed to the following change of variable in T3: l1 7→ l2, l2 7→ l1, l3 7→ l4, l4 7→ l3. Thus, we get θσ 7→ −θσ, and i=1 εili 7→ (ε2− ε3)l1+(ε1− ε3)l2− ε3l3 =: wσ(l). With a similar permutation on the αi, we get l∈(Zn)4 θσ ãα,l δP4 i=1 li,0 δ(ε2−ε3)l1+(ε1−ε3)l2−ε3l3,0δ α4α2δα3α1 . As a consequence, we get g(σ) = 2c l∈(Zn)4 Kσ(l1, l2, l3) ãα,l δP4 i=1 li,0 δα4α2δα3α1 , where Kσ(l1, l2, l3) = λσ ψσ δuσ(l),0 + e φσ δvσ(l),0 − e θσ δP3 i=1 εili,0 θσ δwσ(l),0 The computation of Kσ(l1, l2, l3) for the seven values of σ yields K−−++(l1, l2, l3) = δl1+l3,0 + δl2+l3,0 − δl1+l2,0 − δl1+l2,0, K−+−+(l1, l2, l3) = δl1+l2,0 + δl1+l2,0 − δl1+l3,0 − δl1+l3,0, K−−++(l1, l2, l3) = δl2+l3,0 + δl1+l3,0 − δl2+l3,0 − δl2+l3,0, K−−−+(l1, l2, l3) = − l1.Θl2δP3 i=1 li,0 l2.Θl1δP3 i=1 li,0 l2.Θl1δP3 i=1 li,0 l1.Θl2δl3,0 K−+++(l1, l2, l3) = − l3.Θl2δl1,0 + e l3.Θl1δl2,0 − e l2.Θl3δl1,0 − e l3.Θl1δl2,0 K+−++(l1, l2, l3) = − l1.Θl2δl3,0 + e l2.Θl1δl3,0 − e l1.Θl3δl2,0 − e l3.Θl2δl1,0 K++−+(l1, l2, l3) = − l1.Θl3δl2,0 + e l2.Θl3δl1,0 − e l1.Θl2δl3,0 − e l2.Θl1δP3 i=1 li,0 Thus, ∑ Kσ(l1, l2, l3) = 2i(δl3,0 − δP3 i=1 li,0 ) sin l1.Θl2 and ∑ g(σ) = i4c l∈(Zn)4 (δl3,0 − δP3 i=1 li,0 ) sin l1.Θl2 ãα,l δP4 i=1 li,0 δα4α2δα3α1 . The following change of variables: l1 7→ l2, l1 7→ l2, l3 7→ l4, l4 7→ l3 gives l∈(Zn)4 1 li,0 sin l1.Θl2 ãα,l δP4 1 li,0 δα4α2δα3α1 = − l∈(Zn)4 δl3,0 sin l1.Θl2 ãα,l δP4 1 li,0 δα4α2δα3α1 g(σ) = i8c l∈(Zn)4 δl3,0 sin l1.Θl2 ãα,l δP4 1 li,0 δα4α2δα3α1 . Finally, the change of variables: l2 7→ l4, l4 7→ l2 gives l∈(Zn)4 δl3,0 sin l1.Θl2 ãα,l δP4 1 li,0 δα4α2δα3α1 = − l∈(Zn)4 δl3,0 sin l1.Θl2 ãα,l δP4 1 li,0 δα4α2δα3α1 which entails that g(σ) = 0. Lemma 6.13. Suppose n = 4 and 1 Θ diophantine. For any self-adjoint one-form A, ζDA(0) − ζD(0) = −c τ(Fα1,α2Fα1α2). Proof. By (34) and Lemma 6.6 we get ζDA(0) − ζD(0) = (−1)q σ∈{+,−}q − Aσ. By Lemma 6.12 (iv), we see that the crossed terms all vanish. Thus, with Lemma 6.7, we get ζDA(0)− ζD(0) = 2 (−1)q −(A+)q. (65) By definition, Fα1α2 = i aα2,k kα1 − aα1,k kα2 aα1,k aα2,l [Uk, Ul] (aα2,k kα1 − aα1,k kα2)− 2 aα1,k−l aα2,l sin( τ(Fα1α2F α1α2) = α1, α2=1 (aα2,k kα1 − aα1,k kα2)− 2 l′∈Z4 aα1,k−l′ aα2,l′ sin( k.Θl′ (aα2,−k kα1 − aα1,−k kα2)− 2 l”∈Z4 aα1,−k−l” aα2,l” sin( k.Θl” One checks that the term in aq of τ(Fα1α2F α1α2) corresponds to the term (A+)q given by Lemma 6.12. For q = 2, this is l∈Z4, α1, α2 aα1,l aα2,−l lα1 lα2 − δα1α2 |l|2 For q = 3, we compute the crossed terms: k,k′,l (aα2,k kα1 − aα1,k kα2) a Uk[Uk′ , l] + [Uk′ , Ul]Uk which gives the following a3-term in τ(Fα1α2F α1α2) aα3,−l1−l2 a aα1,l1 sin l1.Θl2 For q = 4, this is aα1,−l1−l2−l3 aα2,l3 a l1.Θ(l2+l3) sin l2.Θl3 which corresponds to the term (A+)4. We get finally, (−1)q −(A+)q = − c τ(Fα1,α2F α1α2). (66) Equations (65) and (66) yield the result. Lemma 6.14. Suppose n = 2. Then, with the same hypothesis as in Lemma 6.11, −(A+)2 = −(A−)2 = 0. (ii) Suppose 1 Θ diophantine. Then − A+A− = − A−A+ = 0. Proof. (i) Lemma 6.11 entails that A++ = Res l∈Z2 −f(s, l) where f(s, l) := kµ1 (k+l)µ2 |k|s+2|k+l|2 ãα,l Tr(γ α2γµ2γα1γµ1) =: fµ,α(s, l)Tr(γ α2γµ2γα1γµ1) and ãα,l := aα1,l aα2,−l. This time, since n = 2, it is enough to apply just once (16) to obtain an absolutely convergent series. Indeed, we get with (16) fµ,α(s, l) = ′ kµ1 (k+l)µ2 |k|s+4 ãα,l − ′ kµ1 (k+l)µ2 (2k.l+|l| |k|s+4|k+l|2 ãα,l. and the function r(s, l) := kµ1 (k+l)µ2 (2k.l+|l| |k|s+4|k+l|2 ãα,l is a linear combination of functions of the type H(s, l) satisfying the hypothesis of Corollary 2.13. As a consequence, r(s, l) satisfies (H1) and fµ,α(s, l) ∼ ′ kµ1 (k+l)µ2 |k|s+4 ãα,l ∼ ′ kµ1kµ2 |k|s+4 ãα,l Note that the function (s, l) 7→ hµ,α(s, l) := kµ1kµ2 |k|s+4 ãα,l satisfies (H2). Thus, Lemma 2.14 yields f(s, l) = hµ,α(s, l)Tr(γ α2γµ2γα1γµ1). By Proposition 2.16, we get Res hµ,α(s, l) = δµ1µ2 π ãα,l. Therefore, − A++ = −π ãα,l Tr(γ α2γµγα1γµ) = 0 according to (59). (ii) By Lemma 6.11, we obtain that A−+ = Res l∈Z2 λσfα,µ(s, l)Tr(γ α2γµ2γα1γµ1) where λσ = −(−i)2 = 1 and fα,µ(s, l) := ′ kµ1 (k+l)µ2 |k|s+2|k+l|2 eiη k.Θl ãα,l and η := 1 (σ1 − σ2) = −1. As in the proof of (i), since the presence of the phase does not change the fact that r(s, l) satisfies (H1), we get fα,µ(s, l) ∼ ′ kµ1 (k+l)µ2 |k|s+4 eiη k.Θl ãα,l := gα,µ(s, l) . Since 1 Θ is diophantine, the functions s 7→ l∈Z2\{0} gα,µ(s, l) are holomorphic at s = 0 by Theorem 2.5 3. As a consequence, − A−+ = Res gα,µ(s, 0)Tr(γ α2γµ2γα1γµ1) = Res ′ kµ1kµ2 |k|s+4 ãα,0 Tr(γ α2γµ2γα1γµ1). Recall from Proposition 2.1 that Ress=0 |k|s+4 = δij π. Thus, again with (59), − A−+ = ãα,0 π Tr(γα2γµγα1γµ) = 0. Lemma 6.15. Suppose n = 2 and 1 Θ diophantine. For any self-adjoint one-form A, ζDA(0) − ζD(0) = 0. Proof. As in Lemma 6.13, we use (34) and Lemma 6.6 so the result follows from Lemma 6.14. 6.1.2 Odd dimensional case Lemma 6.16. Suppose n odd and 1 Θ diophantine. Then for any self-adjoint 1-form A and σ ∈ {−,+}q with 2 ≤ q ≤ n, ∫ − Aσ = 0 . Proof. Since Aσ ∈ Ψ1(A), Lemma 5.11 with k = n gives the result. Corollary 6.17. With the same hypothesis of Lemma 6.16, for any self-adjoint one-form A, ζDA(0)− ζD(0) = 0. Proof. As in Lemma 6.13, we use (34) and Lemma 6.6 so the result follows from Lemma 6.16. 6.2 Proof of the main result Proof of Theorem 6.1. (i) By (5) and Proposition 5.5, we get S(DA,Φ,Λ) = 4πΦ2Λ2 +Φ(0) ζDA(0) +O(Λ−2), where Φ2 = Φ(t) dt. By Lemma 6.15, ζDA(0) − ζD(0) = 0 and from Proposition 5.4, ζD(0) = 0, so we get the result. (ii) Similarly, S(DA,Φ,Λ) = 8π2 Φ4Λ4+Φ(0) ζDA(0)+O(Λ−2) with Φ4 = 12 Φ(t) t dt. Lemma 6.13 implies that ζDA(0)−ζD(0) = −c τ(FµνFµν) and by Proposition 5.4, ζDA(0) = −c τ(FµνFµν) leading to the result. (iii) is a direct consequence of (5), Propositions 5.4, 5.5, and Corollary 6.17. A Appendix A.1 Proof of Lemma 3.3 (i) We have |D|T |D|−1 = T + δ(T )|D|−1 and |D|−1T |D| = T − |D|−1δ(T ). A recurrence proves that for any k ∈ N, |D|kT |D|−k = δq(T )|D|−q and we get |D|−kT |D|k = q=0(−1)q |D|−qδq(T ). As a consequence, since T , |D|−q and δq(T ) are in OP 0 for any q ∈ N, for any k ∈ Z, |D|kT |D|−k ∈ OP 0. Let us fix p ∈ N0 and define Fp(s) := δp(|D|sT |D|−s) for s ∈ C. Since for k ∈ Z, Fp(k) is bounded, a complex interpolation proves that Fp(s) is bounded, which gives |D|sT |D|−s ∈ OP 0. (ii) Let T ∈ OPα and T ′ ∈ OP β. Thus, T |D|−α, T ′|D|−β are in OP 0. By (i) we get |D|βT |D|−α|D|−β ∈ OP 0, so T ′|D|−β|D|βT |D|−β−α ∈ OP 0. Thus, T ′T |D|−(α+β) ∈ OP 0. (iii) For T ∈ OPα, |D|α−β and T |D|−α are in OP 0, thus T |D|−β = T |D|−α|D|α−β ∈ OP 0. (iv) follows from δ(OP 0) ⊆ OP 0. (v) Since ∇(T ) = δ(T )|D|+ |D|δ(T )− [P0 , T ], the result follows from (ii), (iv) and the fact that P0 is in OP A.2 Proof of Lemma 3.6 The non-trivial part of the proof is the stability under the product of operators. Let T, T ′ ∈ Ψ(A). There exist d, d′ ∈ Z such that for any N ∈ N, N > |d|+ |d′|, there exist P,P ′ in D(A), p, p′ ∈ N0, R ∈ OP−N−d , R′ ∈ OP−N−d such that T = PD−2p + R, T ′ = P ′D−2p′ + R′, PD−2p ∈ OP d and P ′D−2p′ ∈ OP d′ . Thus, TT ′ = PD−2pP ′D−2p +RP ′D−2p + PD−2pR′ +RR′. We also have RP ′D−2p ′ ∈ OP−N−d′+d′ = OP−N and similarly, PD−2pR′ ∈ OP−N . Since RR′ ∈ OP−2N , we get TT ′ ∼ PD−2pP ′D−2p′ mod OP−N . If p = 0, then TT ′ ∼ QD−2p′ mod OP−N where Q = PP ′ ∈ D(A) and QD−2p′ ∈ OP d+d′ . Suppose p 6= 0. A recurrence proves that for any q ∈ N0, D−2P ′ ∼ (−1)k∇k(P ′)D−2k−2 + (−1)q+1D−2∇q+1(P ′)D−2q−2 mod OP−∞ . By Lemma 3.3 (v), the remainder is in OP d ′+2p′−q−3, since P ′ ∈ OP d′+2p′ . Another recurrence gives for any q ∈ N0, D−2pP ′ ∼ k1,··· ,kp=0 (−1)|k|1∇|k|1(P ′)D−2|k|1−2p mod OP d′+2p′−q−1−2p. Thus, with qN = N + d+ d ′ − 1, TT ′ ∼ k1,··· ,kp=0 (−1)|k|1P∇|k|1(P ′)D−2|k|1−2(p+p′) mod OP−N . The last sum can be written QND −2rN where rN := p qN + (p + p ′). Since QN ∈ D(A) and −2rN ∈ OP d+d′ , the result follows. A.3 Proof of Proposition 3.11 Let P ∈ OP k1 , Q ∈ OP k2 ∈ Ψ(A). With [Q, |D|−s] = Q− σ−s(Q) |D|−s and the equivalence Q− σ−s(Q) ∼ − r=1 g(−s, r) εr(Q) mod OP−N−1+k2 , we get P [Q, |D|−s] ∼ − g(−s, r)Pεr(Q)|D|−s mod OP−N−1+k1+k2−ℜ(s) which gives, if we choose N = n+ k1 + k2, P [Q, |D|−s] n+k1+k2∑ g(−s, r)Tr Pεr(Q)|D|−s By hypothesis s 7→ Tr Pεr(Q)|D|−s has only simple poles. Thus, since s = 0 is a zero of the analytic function s 7→ g(−s, r) for any r ≥ 1, we have Res g(−s, r) Tr Pεr(Q)|D|−s = 0, which entails that Res P [Q, |D|−s] = 0 and thus − PQ = Res P |D|−sQ When s ∈ C with ℜ(s) > 2max(k1 + n + 1, k2), the operator P |D|−s/2 is trace-class while |D|−s/2Q is bounded, so Tr P |D|−sQ |D|−s/2QP |D|−s/2 σ−s/2(QP )|D|−s Thus, using (29) again, P |D|−sQ − QP + n+k1+k2∑ g(−s/2, r)Tr εr(QP )|D|−s As before, for any r ≥ 1, Res g(−s/2, r)Tr εr(QP )|D|−s = 0 since g(0, r) = 0 and the spectral triple is simple. Finally, P |D|−sQ − QP. Acknowledgments We thank Pierre Duclos, Emilio Elizalde, Victor Gayral, Thomas Krajewski, Sylvie Paycha, Joe Varilly, Dmitri Vassilevich and Antony Wassermann for helpful discussions and Stéphane Louboutin for his help with Proposition 2.16. A. Sitarz would like to thank the CPT-Marseilles for its hospitality and the Université de Provence for its financial support and acknowledge the support of Alexander von Humboldt Foundation through the Humboldt Fellowship. References [1] A. L. Carey, J. Phillips, A. Rennie and F. A. Sukochev, “The local index formula in semifi- nite von Neumann algebras I: Spectral flow”, Advances in Math. 202 (2006), 415–516. [2] L. Carminati, B. Iochum and T. Schücker, “Noncommutative Yang-Mills and noncommu- tative relativity: a bridge over troubled water, Eur. Phys. J. C 8 (1999) 697–709. [3] A. Chamseddine and A. Connes, “The spectral action principle”, Commun. Math. Phys. 186 (1997), 731–750. [4] A. Chamseddine and A. 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Vassilevich, “Non-commutative heat kernel”, Lett. Math. Phys. 67 (2004), 185–194. http://arxiv.org/abs/math/0601171 [37] D. V. Vassilevich, “Heat kernel, effective action and anomalies in noncommutative theories”, JHEP 0508 (2005), 085. [38] D. V. Vassilevich, “Induced Chern–Simons action on noncommutative torus”, [arXiv:hep-th/0701017]. http://arxiv.org/abs/hep-th/0701017 Introduction Residues of series and integral, holomorphic continuation, etc Residues of series and integral Holomorphy of certain series Proof of Lemma ?? for i=1: Proof of Lemma ?? for i=0: Proof of item (i.2) of Theorem ??: Proof of item (iii) of Theorem ??: Commutation between sum and residue Computation of residues of zeta functions Meromorphic continuation of a class of zeta functions A family of polynomials Residues of a class of zeta functions Noncommutative integration on a simple spectral triple Kernel dimension Pseudodifferential operators Zeta functions and dimension spectrum The noncommutative integral Residues of DA for a spectral triple with simple dimension spectrum The noncommutative torus Notations Kernels and dimension spectrum Noncommutative integral computations The spectral action Computations of Even dimensional case Odd dimensional case Proof of the main result Appendix Proof of Lemma ?? Proof of Lemma ?? Proof of Proposition ??

0704.0565

The Lifshitz-Slyozov-Wagner equation for reaction-controlled kinetics

THE LIFSHITZ–SLYOZOV–WAGNER EQUATION FOR REACTION-CONTROLLED KINETICS APOSTOLOS DAMIALIS Abstract. We rigorously derive a weak form of the Lifshitz–Slyozov–Wagner equation as the homogenization limit of a Stefan-type problem describing reaction-controlled coarsening of a large number of small spherical particles. Moreover, we deduce that the effective mean-field description holds true in the particular limit of vanishing surface-area density of particles. 1. Introduction The late-stage behavior of a material undergoing a first-order phase transition (due to changes in temperature and/or pressure for example) is characterized by thermodynamic instability resolved through phase separation and consequent coars- ening of the emerging phase. In the case of the new phase occupying much smaller volume fraction, and thus appearing as well-separated particles, this coarsening pro- cess (known as Ostwald ripening) is driven by the minimization of surface energy at the interface via diffusional mass exchange between particles while the total mass or volume of each phase is conserved. The result of this kind of mass diffusion from regions of high to regions of low interfacial curvature is the growth of large parti- cles and the shrinkage and final extinction of smaller ones. For a review of some aspects of Ostwald ripening, mainly from the physical and modeling viewpoint, see the survey by Voorhees [21] or the book by Ratke and Voorhees [18]. In this coarsening scenario the mass-diffusion process can be controlled by two different mechanisms: either by the diffusion of atoms away from the particles and into the bulk, or by the reaction-rate of attachment of atoms at the phase interface. In the former case (diffusion control), the random exchange of atoms between the particles and the bulk is sufficiently rapid and the surrounding of each particle is in thermal equilibrium with the atoms in it; in the latter (interface-reaction control), detachment and attachment are slow compared to diffusion and the surrounding bulk can be out of equilibrium with the particle interface. We refer to the physics literature for more details, for example, Slezov and Sagalovich [19], Bartelt, Theis, and Tromp [3]; for a related mathematical treatment see Dai and Pego [5]. The classical theory for Ostwald ripening was developed by Lifshitz and Slyozov [9] and Wagner [22] in the case of supersaturated solid solutions in three dimensions. The Lifshitz–Slyozov–Wagner theory statistically characterizes the evolution by the particle-radius density n(t, R), where n(t, R) dR is defined to be the number of particles with radii between R and dR at time t per unit volume. In the late stages of the phase transition nucleation and coalesence of particles can be neglected since new nuclei dissolve immediately and since particles cannot merge because of the large distances between them. Thus, the particle-radius density satisfies the This work was supported by the DFG through the Graduiertenkolleg RTG-1128 “Analysis, Numerics, and Optimization of Multiphase Problems” at the Humboldt-Universität zu Berlin. http://arxiv.org/abs/0704.0565v4 2 APOSTOLOS DAMIALIS continuity equation (see [18, §5.1]) n(t, R) + v(t, R)n(t, R) where v(t, R) denotes the growth rate of particles of radius R at time t. Using a mean-field ansatz (cf. Section 3), Lifshitz, Slyozov, and Wagner formally calculate n(t, R) + (Rū− 1)n(t, R) ū(t) = n(t, R) dR Rn(t, R) dR, in the diffusion-controlled case, and n(t, R) + n(t, R) ū(t) = Rn(t, R) dR R2n(t, R) dR, in the reaction-controlled one, both results valid in the limit of vanishing mass or volume fraction of particles. In [11] and [12] Niethammer rigorously derived the effective equations in the dif- fusion-controlled case, starting from a quasi-static one-phase Stefan problem with surface tension and kinetic undercooling, −∆u = 0 in Ω \G, V = ∇u · n on ∂G, u = H + βV on ∂G, (1.1) and restricting it to spherical particles. The same was also done in [11] for the full time-dependent parabolic problem but without the kinetic-drag term βV . Here, u is a chemical potential, n is the outer normal to the particle phase G, V is the nor- mal velocity of the phase interface ∂G, and H is its mean curvature. The domain Ω ⊂ R3 is considered bounded and β is a parameter that comes from the nondi- mensionalization and scales like diffusivity over mobility. The second boundary condition is the Gibbs–Thomson law, coupling the curvature of the interface with the chemical potential, modified by accounting for kinetic drag. Note that while under diffusion control the parameter β is small and the kinetic drag can even be neglected (thus yielding the well-known Mullins–Sekerka model [10]), in the reaction-controlled case the values of β are large and, therefore, the kinetic-drag term is necessary. For a derivation of such sharp-interface free-boundary problems from continuum mechanics and thermodynamics see the book of Gurtin [8]. The goal in the following is to use the techniques developed in [11] and [12] to derive the effective equations in the reaction-controlled case. This involves passing over to a different time scale incorporating the parameter β tending to infinity (see Section 2) and, as a result, some extra manipulations in the proofs. Except for the scaling, in Section 2 we also give short proofs of some useful preliminaries and discuss the validity of the mean-field description while in Section 3 we prove pointwise estimates for approximate solutions and for the growth rates of particles. Finally, using these estimates, in Section 4 we pass to the homogenization limit of infinitely-many particles and obtain a weak form of the Lifshitz–Slyozov–Wagner equation. THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 3 In comparison with the results in the diffusion-controlled case, we make precise that the crucial quantity that has to vanish in order to neglect direct interactions between particles and justify the expected mean-field law is the surface-area density of the particles in contrast to their capacity in the other case (see [11] and [12]). This difference is of interest since the asymptotic limits of vanishing surface area and capacity have different physical interpretations and further refine the näıve general limit of vanishing mass or volume. For the reaction-controlled case though, the result is in some sense to be expected since the limit of vanishing surface- area density corresponds to the physics of the interface-reaction-controlled scenario, where there is an obvious dependence on the area of the interface. 2. Formulation, scaling, and preliminary estimates We start with problem (1.1) where the quasi-static approximation to the para- bolic diffusion equation is justified by the small interfacial velocities present during late-stage coarsening. (See the discussion in Mullins and Sekerka [10].) We further suppose that the solid phase consists of spherical particles with cen- ters fixed in space, a simplification that can be justified by the work of Alikakos and Fusco [1], [2], and Velázquez [20]. Denoting these particles as Bi, where each Bi is the closed ball B(xi, Ri(t)), the particle phase is then the union ∪Bi and its isotropic evolution can be modeled by averaging the flux in the Stefan condition, i.e., V = Ṙi(t) := − ∇u · n , where the average integral is defined as for a function f on some domain D, and where the overdot denotes a derivative with respect to time; the Gibbs–Thomson law becomes then + βṘi, since in the case of spheres the mean curvature is the inverse radius. To have many small particles in a bounded domain, for a system with size of order O(1), say the unit cube [0, 1]3, let δ be the typical particle distance with 0 < δ ≪ 1. For the distribution of particle centers in space, we assume, for simplicity, that they are situated on a three-dimensional lattice of spacing δ. Then, the initial number density of particles Ni(δ) will be bounded by 1/δ 3, and for the particles to be small let the typical particle size be δα for α > 1. For times t ∈ [0, T ] we choose a δ small enough so that adjacent particles of size δα will not collide during the evolution up to a maximal time T . Concerning the assumption on the spatial distribution of particles, a more general assumption like infi6=j |xi−xj| > cδ, for a constant c > 0, would still be enough for our purposes in this work. These considerations will also be used in the proof of Lemma 3.2 where we approximate a certain sum over all particles by an integral. For an approach using more sophisticated deterministic and stochastic assumptions on the distribution of particles with respect to homogenization we refer to Niethammer and Velázquez [16], [17], where also further refinements of the theory are made. To have particle sizes of order O(1) as well, we rescale Rδi := 4 APOSTOLOS DAMIALIS and motivated by the scaling invariance of problem (1.1) (cf. [5]), uδ := δαu, tδ := Notice that this rescaling is another way of addressing the reaction-controlled regime. Instead of rescaling time by β and then letting β tend to infinity, we keep β fixed and positive, and specially rescale as above letting δ tend to zero. Since now β plays no significant role, we will set it to unity in what follows. In addition, one easily sees that the transformations Rδi , u δ, and tδ preserve the form of the equations. From hereon we also drop the superscript δ from the notation for time and to denote the dependence on the new scale we write Bδi := B(xi, δ αRδi ). Finally, note that under diffusion control the relevant scale for time would be δ3α instead of δ2α. This difference is key to all that follows, leading to different consid- erations on the validity of the mean-field model. (Cf. the remarks following Lemma 2.1.) As initial data, for every particle-center xi we associate a corresponding bounded initial radius Rδi (0) with the assumption that supi∈NiR i (0) ≤ R0, uniformly for some constant R0. To consider a closed system, we impose a no-flux Neumann boundary condition on the outer boundary of Ω, i.e., ∇uδ · n = 0 on ∂Ω. In case the ith particle vanishes at time ti := sup{t | Rδi (t) > 0}, for times later than ti we define R i to be zero, reduce the number N(t) := {j | Rδj(t) > 0} of active particles by one, and neglect the boundary ∂Bδi in the boundary conditions. In the following, all sums, unions, and suprema will run over the set N(t), with N(0) ≡ Ni, and any further reference to the particle-number density will mean the active particle-number density N unless otherwise noted. Summarizing, the restricted and rescaled problem for the particle radii can be considered as a nonlocal, N -dimensional system of ordinary differential equations Ṙδi (t) = 4πδ2αRδi (t) ∇uδ · n on ∂Bδi (t), (2.1) for times t ∈ (0, ti), ti < T , and with bounded initial data Rδi (0) for every i, while the chemical potential is determined by −∆uδ(t, x) = 0 in Ω \ ∪Bδi (t), (2.2) uδ(t, x) = Rδi (t) + Ṙδi (t) on ∂B i (t), (2.3) and the Neumann condition on the outer boundary. Global existence and uniqueness of continuous, piecewise-smooth solutions for a similar restricted Stefan problem was proved in [12] by an application of the Picard–Lindelöf theorem, the only difference being the different time scale. These solutions are not globally smooth due to the singularities arising from the extinction of particles; however, they are smooth in the intervals between the extinction times ti. In the following, when we mention solutions of the problem we will mean such continuous, piecewise-smooth solutions that exist up to any given time T . THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 5 It is easy to see that equations (2.1), (2.2), (2.3), along with the outer boundary condition conserve the volume and decrease the interfacial area of the particle phase. Indeed, differentiating the total volume of particles with respect to time gives Rδi (t) 3 = 3 Rδi (t) 2Ṙδi (t) = 3 Rδi (t) 4πδ2αRδi (t) ∇uδ · n where the last sum vanishes due to the divergence theorem, equation (2.2), and the no-flux condition on ∂Ω. The decrease of total surface area follows from the next a priori estimate. Lemma 2.1. For any time t ∈ (0, T ), the solutions of the problem satisfy the following energy equality. (Rδi ) 2|Ṙδi |2 + Rδi (t) 4πδ2α Ω\∪Bδ |∇uδ|2 = 1 Rδi (0) Proof. Multiplying −∆uδ = 0 with uδ, integrating over Ω \ ∪Bδi , and integrating by parts gives Ω\∪Bδ |∇uδ|2 + (∇uδ · n)uδ − (∇uδ · n)uδ = 0, where the last term vanishes due to the Neumann condition on the outer boundary. Thus, using equations (2.3) and (2.1) we get Ω\∪Bδ |∇uδ|2 = + Ṙδi ∇uδ · n + Ṙδi 4πδ2α(Rδi ) 2Ṙδi = 4πδ2α Rδi Ṙ i + (R 2|Ṙδi |2 and after rearranging, (Rδi ) 2|Ṙδi |2 + Rδi Ṙ 4πδ2α Ω\∪Bδ |∇uδ|2 = 0. (2.4) The result follows from an integration over time. � After normalization with respect to the initial particle-number density Ni, this energy equality can yield useful information on the validity of the mean-field ap- proach. In fact, we have (Rδi ) 2|Ṙδi |2+ Rδi (t) 4πNiδ2α Ω\∪Bδ |∇uδ|2 = 1 Rδi (0) where the right-hand side is uniformly bounded by the assumption on the initial radii. For the left-hand side to stay bounded as well, if the quantity Niδ 2α tends to zero, the same must hold for |∇uδ| and it is exactly this limit of vanishing surface- area density of particles that results in a mean field that is constant in space since, in particular, ∇uδ → 0 in L2 0, T ;H1(Ω) Here and in the following, to obtain global estimates that are uniform in δ we extend uδ to the interior of particles, and thus to the whole of Ω, by its boundary values. It is important to note that in our scaling setup, for the surface area to vanish as δ 6 APOSTOLOS DAMIALIS tends to zero, the exponent α must be strictly larger than 3/2 since Ni is O(1/δ These facts will be made precise in Corollary 3.3 where we give an estimate of the mean-field effect. Note also that we do not address here the critical case α = 3/2 that corresponds to finite surface area. For that one would have to use the different methods developed by Niethammer and Otto in [13]. Finally, note that for similar considerations under diffusion control, the corre- sponding quantity would be the capacity Niδ α due to the different time scale. In three dimensions, this capacity effect fits to general homogenization results as in the work of Cioranescu and Murat [4]; to our knowledge though, the surface-area effect has not been explicitly discussed in the relevant literature. 3. Approximation and growth-rate estimates As in the mean-field ansatz of Lifshitz, Slyozov, and Wagner, we suppose that the system is dilute enough so that particles behave as if they were isolated and we base our approximation on the solution of a single-particle problem. Consider problem (2.1), (2.2), (2.3) for a single spherical particle centered at the origin and with initially unscaled radius r that we rescale as rδ := r/δα, along with the corresponding reaction-controlled rescalings for a chemical potential uδr and time, as in Section 2. For this rescaled particle Bδr we consider the following problem in the whole space: ṙδ(t) = 4πδ2αrδ(t)2 ∇uδr · n on ∂Bδr , where the chemical potential uδr(t, x) satisfies −∆uδr(t, x) = 0 in R3 \Bδr , uδr(t, x) = rδ(t) + ṙδ(t) for x ∈ ∂Bδr , and the mean-field assumption is posed as a condition at infinity, i.e., |x|→∞ uδr(t, x) = ū r(t). This problem can be explicitly solved to give uδr(t, x) = ū r(t) + δαrδ(t) 1 + δαrδ(t) 1− ūδr(t)rδ(t) ṙδ(t) = 1 + δαrδ(t) ūδr(t)− rδ(t) Note that in the formal limit of δ tending to zero, the expected effective equations take the general form u(t, x) = ū(t) and ṙ = ū− 1 as in the reaction-controlled Lifshitz–Slyozov–Wagner theory. Going now back to the many-particle problem, a calculation using the single- particle growth rate above along with the requirement that the volume is conserved gives the following expression for the mean field ūδ = 1 + δαRδi (Rδi ) 1 + δαRδi . (3.1) THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 7 The effect of this mean field plus a sum of single-particle solutions will be the monopole approximation to the solution uδ supposing that there are no direct interactions between particles. To this end, let us define the approximate solution ζδ(t, x) := ūδ(t) + δαRδi (t) 1 + δαRδi (t) 1− ūδ(t)Rδi (t) |x− xi| (3.2) for x ∈ Ω \ ∪Bδi (t). Below is a maximum principle tailored to our setting that will be used to compare the approximation and the solution in the lemma next. Its proof can be found in [12]. Lemma 3.1. Let Ω be a Lipschitz domain and let ∪Bi ⊂ Ω be a finite collection of disjoint closed balls. Then, a function v which is constant on each of the boundaries ∂Bi and satisfies −∆v = 0 in Ω \ ∪Bi, v − ci ∇v · n ≥ 0 on ∂Bi, ∇v · n ≥ 0 on ∂Ω, where ci ≥ 0 for all i, also satisfies v ≥ 0 in Ω \ ∪Bi. Lemma 3.2. For any time t ∈ (0, T ) and small positive ε, the chemical potential and its approximation satisfy ‖uδ − ζδ‖L∞(Ω\∪Bδ )(t) ≤ Cδ2α−3−ε supRδi (t) 1 + ūδ(t) supRδi (t) Proof. Since the difference uδ − ζδ is already harmonic in Ω \ ∪Bδi as ζδ is a su- perposition of fundamental solutions, we would like to estimate to what extent it satisfies the maximum principle’s boundary conditions. For the condition on the particle boundaries, we use equations (2.1), (2.3), and the definition of ζδ to calculate for x on the boundary ∂Bδi of the ith particle, ζδ(t, x)−− ∇ζδ · n uδ(t, x)−− ∇uδ · n ζδ − 1 ∇ζδ · n ūδ − 1 δαRδj 1 + δαRδj (1− ūδRδj) |x− xj | |x− xj | and since by the divergence theorem there holds for j 6= i, |x− xj | · n = 0, while for j = i, |x− xi| · n = − 1 δα(Rδi ) 8 APOSTOLOS DAMIALIS we continue the calculation to get ūδ − 1 1− ūδRδi Rδi (1 + δ αRδi ) δαRδj 1 + δαRδj (1− ūδRδj) |x− xj | j 6=i δαRδj 1 + δαRδj (1− ūδRδj) |x− xj | ≤ δ2α−3 supRδj (1 + ūδ supRδj ) j 6=i |x− xj | ≤ Cδ2α−3 supRδj(1 + ūδ supRδj). (3.3) In the last step, keeping in mind the assumptions on the spatial distribution of particle centers, the sum is bounded for j 6= i since it is considered as a Riemann- sum approximation to the integral |x− y| which in turn is bounded using radial symmetry around the singularity and where the factor δ3 in the sum compensates for the scaling in space. To further fulfil the maximum principle’s outer boundary condition on ∂Ω, we consider the comparison function ζδ+zδ, where the auxiliary function zδ solves the problem −∆zδ = ∇ζδ · n in Ω, ∇zδ · n = −∇ζδ · n on ∂Ω, zδ = 0, (3.4) such that the comparison function ζδ + zδ has zero normal derivative on ∂Ω. To work with the maximum principle, zδ also needs to be harmonic in Ω and for that we need that the integral ∇ζδ · n vanishes. But, ∇ζδ · n = δα δαRδi 1 + δαRδi (1−Rδi ūδ) |x− xi| · n , where the last integral equals −4π, independent of i. Thus, zδ is harmonic if and only if ūδ = 1 + δαRδi (Rδi ) 1 + δαRδi which is exactly the mean field (3.1) as dictated by the single-particle ansatz in the beginning of the section. Moreover, since now zδ is harmonic, the divergence theorem further gives ∇zδ · n = 0. A construction as in Lemma 3 of [11] and elliptic regularity theory (see Gilbarg and Trudinger [7]) give the estimate ‖zδ‖L∞(Ω) ≤ Cεδ2α−3−ε supRδi (1 + ūδ supRδi ), where ε is a small positive number. THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 9 Let us now apply the maximum principle to the function f+ := u δ − ζδ − zδ + Cδ2α−3−ε supRδi (1 + ūδ supRδi ). For a large enough constant C, say 2Cε, the following hold for f+: it is harmonic, there holds ∇f+ · n = 0 on ∂Ω by the construction of zδ, and for the constants ci = 1/4πδ 2α(Rδi ) 2, estimate (3.3) gives uδ − ζδ − zδ + Cδ2α−3−ε supRδi (1 + ūδ supRδi )− ci ∇(uδ − ζδ − zδ) · n ≥ 0. Thus, f+ satisfies the maximum principle’s conditions and therefore, f+ ≥ 0 in Ω \ ∪Bδi , i.e., uδ − ζδ − zδ ≥ −Cδ2α−3−ε supRδi (1 + ūδ supRδi ). Using the maximum principle with −v instead of v, the function f− := u δ − ζδ − zδ − Cδ2α−3−ε supRδi (1 + ūδ supRδi ), again satisfies the corresponding conditions and, as above, yields f− ≤ 0 in Ω\∪Bδi , i.e., uδ − ζδ − zδ ≤ Cδ2α−3−ε supRδi (1 + ūδ supRδi ). Combining the last two inequalities, we get ‖uδ − ζδ − zδ‖L∞(Ω\∪Bδ ) ≤ Cδ2α−3−ε supRδi (1 + ūδ supRδi ) and the lemma follows by the triangle inequality using the regularity of zδ. � In the previous lemma it is clear that our approach excludes the critical case α = 3/2. In the following we introduce, for technical reasons, a new exponent γ > 0 with the property δγ := max {δα, δ2α−3, δ2α−3−ε} for each α greater than 3/2 + ε. As a corollary to the previous lemma we can now estimate the effect of the mean field. Corollary 3.3. For any time t ∈ (0, T ) and γ > 0, the chemical potential and the mean field satisfy ‖uδ − ūδ‖L∞(Ω\∪Bδ )(t) ≤ Cδγ 1 + 2 supRδi (t) 1 + ūδ(t) supRδi (t) Proof. By the triangle inequality and Lemma 3.2 there holds ‖uδ − ūδ‖L∞(Ω\∪Bδ ) ≤ ‖ζδ − ūδ‖L∞(Ω\∪Bδ ) + Cδ 2α−3−ε supRδj(1 + ū δ supRδj). To estimate ‖ζδ− ūδ‖L∞(Ω\∪Bδ ), by the definition of ζ δ there holds for x ∈ Ω\∪Bδj , ζδ(t, x)− ūδ(t) δαRδj 1 + δαRδj (1− ūδRδj) |x− xj | δαRδi 1 + δαRδi (1 + ūδRδi ) |x− xi| j 6=i δαRδj 1 + δαRδj (1− ūδRδj) |x− xj | and since |x− xi| ≥ δαRδi in Ω \ ∪Bδj , arguing as in estimate (3.3) gives α(1 + ūδRδi ) 1 + δαRδi + Cδ2α−3 supRδj(1 + ū δ supRδj) ≤ C(δα + δ2α−3 supRδj )(1 + ūδ supRδj), 10 APOSTOLOS DAMIALIS thus, ‖ζδ − ūδ‖L∞(Ω\∪Bδ ) ≤ C(δα + δ2α−3 supRδj )(1 + ūδ supRδj), and finally, ‖uδ − ūδ‖L∞(Ω\∪Bδ ) ≤ C(δα + δ2α−3 supRδj + δ2α−3−ε supRδj )(1 + ūδ supRδj). Using the exponent γ, we get ‖uδ − ūδ‖L∞(Ω\∪Bδ ) ≤ Cδγ(1 + 2 supRδi )(1 + ūδ supRδi ). � The following lemma gives an estimate for the growth rate of particles in accor- dance with the reaction-controlled Lifshitz–Slyozov–Wagner theory. Lemma 3.4. For any time t ∈ (0, T ) and γ > 0, for the growth rates of particles holds Ṙδi − ūδ − 1 ≤ Cδγ(1 + ūδ supRδi ) 1 + (1 + δγ supRδi )(1 + 2 supR Proof. Let wδi be the capacity potential of the ball B i with respect to a larger ball Bλδi := B(xi, λδ αRδi ) for λ > 1, i.e., let wi solve −∆wδi = 0 in Bλδi \Bδi , wδi = 0 on ∂B wδi = 1 in B (3.5) An explicit calculation gives wδi = 1− λδ |x− xi| (3.6) and also ∇wδi · n = ∇wδi · n = 4π 1− λδ αRδi . (3.7) Using equations (2.1), (2.2), (2.3), and the Neumann boundary condition, along with the above properties of wδi , and integrating by parts, gives 4πδ2α(Rδi ) 2Ṙδi = ∇uδ · n wδi∇uδ · n ∇wδi∇uδ uδ∇wδi · n − uδ∇wδi · n + Ṙδi ∇wδi · n − uδ∇wδi · n δαRδi + Ṙδi − ūδ (uδ − ūδ)∇wδi · n , THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 11 where in the last equation we used (3.7) and added and subtracted ūδ. After rearranging, we have Ṙδi − ūδ − 1 δαRδi Ṙ 4πλδαRδi (uδ − ūδ)∇wδi · n ≤ λ− 1 δαRδi |Ṙδi |+ 4πλδαRδi (uδ − ūδ)∇wδi · n ≤ δαRδi |Ṙδi |+ ‖uδ − ūδ‖L∞(Ω\∪Bδ ), (3.8) where in the last step we again used equation (3.7). But by using equation (2.3) for uδ on ∂Bδi we have Rδi |Ṙδi | ≤ 1 +Rδi |uδ| ≤ 1 +Rδi (‖uδ − ūδ‖L∞(Ω\∪Bδ ) + ū δ). (3.9) Substituting back in (3.8) and using Corollary 3.3 gives the final estimate. � The next lemma ensures that the bounds in the approximation and the growth- rate estimates are indeed uniform. Lemma 3.5. For any time t ∈ (0, T ), the mean field and the radii of the particles are uniformly bounded, i.e., ūδ(t) ≤ C and supRδi (t) ≤ C. Proof. For the mean field (3.1) holds ūδ = 1 + δαRδi (Rδi ) 1 + δαRδi ≤ supRδi (1 + δα supRδi ) (Rδi ) and since by Hölder’s inequality (Rδi ) (Rδi ) δ3(Rδi ) conservation of the total volume of particles gives ūδ ≤ C supRδi (1 + δα supRδi ). or, using the exponent γ, ūδ ≤ C supRδi (1 + δγ supRδi ). (3.10) Consider now the set t | supRδi (t) ≤ then, for times t ∈ A, plugging (3.10) in estimate (3.9) and using Corollary 3.3 gives (Rδi ) 2 ≤ C sup(Rδi )2 + C. Integrating over the time interval (0, T ), Gronwall’s inequality implies that supi supt∈A∩[0,T ] (R 2 ≤ C(T ), therefore, [0, T ] ⊂ A, i.e., the radii are bounded up to time T as is the mean field by estimate (3.10). � Finally, the following lemma gives control over the growth rates of vanishing particles and will prove useful for some regularity considerations in the next section. 12 APOSTOLOS DAMIALIS Lemma 3.6. For any time t ∈ (0, T ) such that Rδi (t) ≤ 1/4 supt,δ ūδ(t) and for sufficiently small δ, there holds ≤ Ṙδi ≤ − and √ ti − t ≤ Rδi ≤ 2 ti − t. Proof. For δαRδi 1 + δαRδi |x− xi| it can be verified that the function uδ−g satisfies the assumptions of the maximum principle in Lemma 3.1 for the constants ci = 1/4πδ 2α(Rδi ) 2, thus yielding uδ ≥ g in Ω \ ∪Bδi . But since uδ = g on the boundary ∂Bδi , monotonicity implies that ∇uδ · n ≥ ∇g · n on ∂Bδi and taking the average integrals over ∂Bδi we have Ṙδi ≥ − Rδi (1 + δ αRδi ) ≥ − 2 Moreover, Lemma 3.4 gives Ṙδi ≤ ūδ − + Cδγ(1 + ūδ supRδi ) 1 + (1 + δγ supRδi )(1 + 2 supR Using now the assumption that Rδi ≤ 1/4 supt,δ ūδ and since from Lemma 3.5 it follows that for sufficiently small δ the O(δγ) term is uniformly bounded by 1/4Rδi , we get Ṙδi ≤ ≤ − 1 Let now y1 := ti − t, y2 := 2 ti − t be sub- and supersolutions that respectively solve ẏ1 = − , ẏ2 = − By comparison, we get the lemma’s second assertion, i.e., y1 ≤ Rδi ≤ y2. � 4. Homogenization In order to pass to the homogenization limit of infinitely-many particles, we need first describe the particle-radius density in the limit. To that end, define at any time t ∈ (0, T ) the empirical measure νδt as 〈φ, νδt 〉 = t, Rδi (t) dνδt := t, Rδi (t) for φ ∈ Cc, i.e., for functions φ(t, R) continuous and compactly supported in the radius variable. Using now the estimates from the previous section, we can prove the following Lemma 4.1. For a subsequence δ → 0 and for a function ū ∈ W 1, p(0, T ), for p < 2, holds ūδ → ū in L2(0, T ), uδ → ū in L2 0, T ;H1(Ω) THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 13 Furthermore, the measures νδt converge to a family νt of probability measures such φdνδt → φa(t) dνt uniformly in t, where a(t) denotes the percentage of active particles in the limit. Proof. As a consequence of Lemma 3.6, we have sup ‖Ṙδi ‖Lp(0,T ) ≤ C(p) for p < 2, thus, conservation of volume and boundedness of the radii give for p < 2, Lp(0,T ) ≤ C sup ‖Ṙδi ‖Lp(0,T ) ≤ C. Therefore, ūδ ∈ W 1, p(0, T ), for p < 2, and the compactness following from the Rellich–Kondrachov theorem gives that ūδ converges to a limit ū in L2. Taking into consideration that uδ is extended to the whole of Ω and using the lemmas in the previous section, ζδ converges to ū in L2(Ω) and uδ − ζδ converges uniformly to 0 as δ → 0, therefore, uδ converges to ū in L2(Ω). By the energy equality in Lemma 2.1 we have further control over ‖∇uδ‖L2 and thus, we have strong convergence in L2(0, T ;H1(Ω)). For the measures νδt holds ‖νδt ‖ := sup‖φ‖Cc≤1|〈φ, ν t 〉| ≤ 1 in the norm of (Cc) ∗, so for a subsequence δ → 0 there holds that νδt converges weakly-* to νt. Furthermore, for positive functions φ the limit measure νt is non- negative and from this it follows that νt becomes zero if there are no particles left in the system. Choosing now a function ψ(t) that depends only on time, we calculate ψ(t) dνδt = ψ(t). The ratio N/Ni is the percentage of active particles at time t. This ratio is bounded by 1 and decreasing, therefore it is uniformly bounded in the space BV (0, T ) and by the compact embedding of BV (0, T )∩L∞(0, T ) in L2(0, T ), it converges in L2, for a subsequence δ → 0, to a limit a ∈ BV (0, T ). If we project now the measure νt to the interval [0, T ], we get that the projection satisfies proj[0, T ] νt = a(t) dt and according to [6, Ch. 1, Thm. 10], the decomposition and convergence to νt follow from the slicing of measures. � We conclude with the following theorem which states that the limit measure νt satisfies the Lifshitz–Slyozov–Wagner equation in a weak sense. Note that in the theorem’s statement, the initial condition is defined as t, Rδi (t) dνδ0 := 0, Rδi (0) Theorem 4.2. The measure νt satisfies the Lifshitz–Slyozov–Wagner equation in the sense that φ(t, R) + ū− 1 φ(t, R) a(t) dνt + φ(0, R) dν0 = 0, (4.1) 14 APOSTOLOS DAMIALIS for all smooth and compactly supported functions φ ∈ C∞c ([0, T )× R+), where the mean field ū is given by R dνt R2 dνt. Proof. We begin by computing the mean-field limit ū(t). For a continuous function φ(t) there holds, by the definition of ūδ, 1 + δαR dνδt = ∑ R2i 1 + δαRi 1 + δαRi 1 + δαR dνδt . Taking the limit δ → 0 on both sides, Lemma 4.1 gives R dνt R2 dνt. Consider now a smooth and compactly supported function φ as in the theorem’s statement. Then, the fundamental theorem of calculus and Lemma 3.4 give t, Rδi (t) dνδt + 0, Rδi (0) t, Rδi (t) + Ṙδi (t) t, Rδi (t) dνδt + 0, Rδi (0) t, Rδi (t) ūδ − 1 t, Rδi (t) dνδt +O(δ 0, Rδi (0) dνδ0 . The result follows by taking the limit for a subsequence δ → 0 and using the strong convergence of ūδ. � As a concluding remark, we note that the well-posedness (existence, uniqueness, and continuous dependence on initial data) of the weak formulation (4.1) can be treated by the methods developed by Niethammer and Pego in [14] and [15]. Acknowledgments Thanks are due to Barbara Niethammer for her substantial help and to Nick Alikakos and Bob Pego for helpful discussions. Thanks are also due to the anony- mous referee for a careful reading of the manuscript. References [1] N. D. Alikakos and G. Fusco. The equations of Ostwald ripening for dilute systems. J. Stat. Phys. 95 No. 5/6 (1999), pp. 851–866. [2] N. D. Alikakos and G. Fusco. Ostwald ripening for dilute systems under quasistationary dynamics. Comm. Math. Phys. 238 No. 3 (2003), pp. 429–479. [3] N. C. Bartelt, W. Theis, and R. M. Tromp. Ostwald ripening of two-dimensional islands on Si(001). Phys. Rev. B 54 (1996), pp. 11741–11751. [4] D. Cioranescu and F. Murat. A strange term coming from nowhere. In Topics in the math- ematical modelling of composite materials, A. Cherkaev, R. Kohn eds. Birkhäuser, Boston, MA, 1997, pp. 45–94. THE LSW EQUATION FOR REACTION-CONTROLLED KINETICS 15 [5] S. Dai and R. L. Pego. Universal bounds on coarsening rates for mean-field models of phase transitions. SIAM J. Math. Anal. 37 No. 2 (2005), pp. 347–371. [6] L. C. Evans. Weak convergence methods for nonlinear partial differential equations. American Mathematical Society, Providence, RI, 1990. [7] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Springer-Verlag, Berlin, second edition, 1983. [8] M. E. Gurtin. Thermomechanics of evolving phase boundaries in the plane. The Clarendon Press, Oxford, 1993. [9] I. M. Lifshitz and V. V. Slyozov. The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19 (1961), pp. 35–50. [10] W. W. Mullins and R. F. Sekerka. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34 No. 2 (1963), pp. 323–329. [11] B. Niethammer. Derivation of the LSW-theory for Ostwald ripening by homogenization meth- ods. Arch. Rat. Mech. Anal. 147 (1999), pp. 119–178. [12] B. Niethammer. The LSW model for Ostwald ripening with kinetic undercooling. Proc. R. Soc. Edinburgh 130A No. 6 (2000), pp. 1337–1361. [13] B. Niethammer and F. Otto. Ostwald ripening: The screening length revisited. Calc. Var. 13 No. 1 (2001), pp. 33–68. [14] B. Niethammer and R. L. Pego. On the initial-value problem in the Lifshitz–Slyozov–Wagner theory of Ostwald ripening. SIAM J. Math. Anal. 31 No. 3 (2000), pp. 467–485. [15] B. Niethammer and R. L. Pego. Well-posedness for measure transport in a family of nonlocal domain coarsening models. Indiana Univ. Math. J. 54 No. 2 (2005), pp. 499–530. [16] B. Niethammer and J. J. L. Velázquez. Homogenization in coarsening systems I: Deterministic case. Math. Mod. Meth. Appl. Sci. 14 No. 8 (2004), pp. 1211–1233. [17] B. Niethammer and J. J. L. Velázquez. Homogenization in coarsening systems II: Stochastic case. Math. Mod. Meth. Appl. Sci. 14 No. 9 (2004), pp. 1–24. [18] L. Ratke and P. W. Voorhees. Growth and coarsening: Ostwald ripening in material process- ing. Springer-Verlag, Berlin, 2002. [19] V. V. Slezov and V. V. Sagalovich. Diffusive decomposition of solid solutions. Sov. Phys. Usp. 30 No. 1 (1987), pp. 23–45. [20] J. J. L. Velázquez. On the effect of stochastic fluctuations in the dynamics of the Lifshitz– Slyozov–Wagner model. J. Stat. Phys. 99 No. 1/2 (2000), pp. 231–252. [21] P. W. Voorhees. The theory of Ostwald ripening. J. Stat. Phys. 38 No. 1/2 (1985), pp. 231– [22] C. Wagner. Theorie der Alterung von Niederschlägen durch Umlösen. Z. Elektrochem. 65 No. 7/8 (1961), pp. 581–591. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany E-mail address: [email protected] Current address: Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece mailto:[email protected] 1. Introduction 2. Formulation, scaling, and preliminary estimates 3. Approximation and growth-rate estimates 4. Homogenization Acknowledgments References

0704.0566

Canonical singular hermitian metrics on relative canonical bundles

Canonical singular hermitian metrics on relative canonical bundles Hajime TSUJI November 5, 2007 Abstract We introduce a new class of canonical AZD’s (called the supercanonical AZD’s) on the canonical bundles of smooth projective varieties with pseu- doeffective canonical classes. We study the variation of the supercanonical AZD ĥcan under projective deformations and give a new proof of the in- variance of plurigenera. This paper is a continuation of [T5]. MSC: 14J15,14J40, 32J18 Contents 1 Introduction 1 1.1 Canonical AZD hcan . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Supercanonical AZD ĥcan . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Variation of the supercanonical AZD ĥcan . . . . . . . . . . . . . 5 2 Proof of Theorem 1.7 6 2.1 Upper estimate of K̂Am . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Lower estimate of K̂Am . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Independence of ĥcan,A from hA . . . . . . . . . . . . . . . . . . 9 2.4 Completion of the proof of Theorem 1.7 . . . . . . . . . . . . . . 10 2.5 Comparison of hcan and ĥcan . . . . . . . . . . . . . . . . . . . . 10 3 Variation of ĥcan under projective deformations 11 3.1 Construction of ĥcan on a family . . . . . . . . . . . . . . . . . . 12 3.2 Semipositivity of the curvature current of ĥm,A . . . . . . . . . . 13 3.3 Uniqueness of ĥcan,A for singular hA’s . . . . . . . . . . . . . . . 16 3.4 Case dimS > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Completion of the proof of Theorem 1.10 . . . . . . . . . . . . . 17 4 Appendix 20 1 Introduction Let X be a smooth projective variety and let KX be the canonical bundle of X . In algebraic geometry, the canonical ring R(X,KX) := ⊕∞m=0Γ(X,OX(mKX)) is one of the main object to study. http://arxiv.org/abs/0704.0566v5 Let X be a smooth projective variety such that KX is pseudoeffective. The purposes of this article are twofold. The first purpose is to construct a singular hermitian metric ĥcan on KX such that 1. ĥcan is uniquely determined by X . ĥcan is semipositive in the sense of current. 3. H0(X,OX(mKX)⊗I(ĥmcan)) ≃ H0(X,OX(mKX)) holds for every m ≧ 0, where I(ĥmcan) denotes the multiplier ideal sheaf of ĥmcan as is defined in [N]. And the second purpose is to study the behavior of ĥcan on projective families. We may summerize the 2nd and the 3rd conditions by introducing the following notion. Definition 1.1 (AZD)([T1, T2]) Let M be a compact complex manifold and let L be a holomorphic line bundle on M . A singular hermitian metric h on L is said to be an analytic Zariski decomposition (AZD in short), if the followings hold. 1. Θh is a closed positive current. 2. For every m ≥ 0, the natural inclusion H0(M,OM (mL)⊗ I(hm)) → H0(M,OM (mL)) is an isomorphim. Remark 1.2 A line bundle L on a projective manifold X admits an AZD, if and only if L is pseudoeffective ([D-P-S, Theorem 1.5]). � In this sense, the first purpose of this article is to construct an AZD on KX depending only on X , when KX is pseudoeffective (by Remark 1.2 this is the minimum requirement for the existence of an AZD). The main motivation to construct such a singular hermitian metric is to study the canonical ring in terms of it. This is indeed possible. For example, we obtain the invariance of plurigenera under smooth projective deformations (cf. Corollary 1.12). In fact the hermitian metric constructed here is useful in many other con- texts. Other applications and a generalization to subKLT pairs will be treated in the forthcoming papers ([T6]). I would like to express thanks to Professor Bo Berndtsson who pointed out an error in the previous version. 1.1 Canonical AZD hcan If we assume the stronger assumption that X has nonnegative Kodaira dimen- sion, we have already konwn how to construct a canonical AZD for KX . Let us review the construction in [T5]. Theorem 1.3 ([T5]) Let X be a smooth projective variety with nonnegative Kodaira dimension. We set for every point x ∈ X Km(x) := sup{| σ | m (x);σ ∈ Γ(X,OX(mKX)), | (σ ∧ σ̄) 1m |= 1} K∞(x) := lim sup Km(x). hcan := the lower envelope of K is an AZD on KX. � Remark 1.4 By the ring structure of R(X,KX), we see that lim sup Km(x) = sup Km(x) holds. � Remark 1.5 Since h∞ depends only on X, the volume h−1can is an invariant of X. � Apparently this construction is very canonical, i.e., hcan depends only on the complex structure of X . We call hcan the canonical AZD of KX . But this construction works only if we know that the Kodaira dimension of X is nonneg- ative apriori. This is the main defect of hcan. For example, hcan is useless to solve the abundance conjecture. 1.2 Supercanonical AZD ĥcan To avoid the defect of hcan we introduce the new AZD ĥcan. Let us use the following terminology. Definition 1.6 Let (L, hL) be a singular hermitian line bundle on a complex manifold X. (L, hL) is said to be pseudoeffective, if the curvature current of hL is semipositive. � Let X be a smooth projective n-fold such that the canonical bundle KX is pseudoeffective. Let A be a sufficiently ample line bundle such that for every pseudoeffective singular hermitian line bundle (L, hL) on X , OX(A+L)⊗I(hL) and OX(KX+A+L)⊗I(hL) are globally generated. Such an ample line bundle A extists by L2-estimates. Let hA be a a C ∞ hermitian metric on A with strictly positive curvature 1. Let us fix a C∞ volume form dV on X . By the L2-extension theorem ([O]) we may and do assume that A is sufficiently ample 1Later we shall also consider the case that hA is any C ∞ hermitian metric (without posi- tivity of curvature) or a singular hermitian metric on A. so that for every x ∈ X and for every pseudoeffective singular hermitian line bundle (L, hL), there exists a bounded interpolation operator Ix : A 2(x, (A+ L)x, hA · hL, δx) → A2(X,A+ L, hA · hL, dV ) such that the operator norm of Ix is bounded by a positive constant independent of x and (L, hL), where A 2(X,A + L, hA · hL, dV ) denotes the Hilbert space defined by A2(X,A+L, hA·hL, dV ) := {σ ∈ Γ(X,OX(A+L)⊗I(hL)) | | σ |2 ·hA·hL·dV < +∞} with the L2 inner product (σ, σ′) := σ · σ̄′ · hA · hL · dV and A2(x, (A+L)x, hA ·hL, δx) is defined similarly, where δx is the Dirac measure supported at x. We note that if hL(x) = +∞, then A2(x, (A+L)x, hA ·hL, δx) = 0. For every x ∈ X we set K̂Am(x) := sup{| σ | m (x) | σ ∈ Γ(X,OX(A+mKX)), | A · (σ ∧ σ̄) m |= 1}. Here | σ | 2m is not a function on X , but the supremum is takan as a section of the real line bundle |A | 2m ⊗ |KX |2 in the obvious manner2. Then h A · K̂Am is a continuous semipositive (n, n) form on X . Under the above notations, we have the following theorem. Theorem 1.7 We set K̂A∞ := lim sup A · K̂ ĥcan,A := the lower envelope of K̂ Then ĥcan,A is an AZD of KX . And we define ĥcan := the lower envelope of inf ĥcan,A, where inf means the pointwise infimum and A runs all the ample line bundles on X. Then ĥcan is a well defined AZD 3 depending only on X. � Definition 1.8 (Supercanonical AZD) We call ĥcan in Theorem 1.7 the supercanonical AZD of KX . And we call the semipositive (n, n) form ĥ can the supercanonical volume form on X. � Remark 1.9 Here “super” means that corresponding volume form ĥ−1can satisfies the inequality : ĥ−1can ≧ h if X has nonnegative Kodaria dimension (cf. Theorem 2.9). � In the statement of Theorem 1.7, one may think that ĥcan,A may dependent of the choice of the metric hA. But later we prove that ĥcan,A is independent of the choice of hA (cf. Theorem 2.7). 2We have abused the notations |A|, |KX| here. These notations are similar to the notations of corresponding linear systems. But I think there is no fear of confusion. 3I believe that ĥcan,A is already independent of the sufficiently ample line bundle A. 1.3 Variation of the supercanonical AZD ĥcan Let f : X −→ S be an algebraic fiber space, i.e., X,S are smooth projective varieties and f is a projective morphism with connected fibers. Suppose that for a general fiber Xs := f −1(s), KXs is pseudoeffective 4. In this case we may define a singular hermitian metric ĥcan on KX/S similarly as above. Then ĥcan have a nice properties on f : X −→ S as follows. Theorem 1.10 Let f : X −→ S be an algebraic fiber space such that for a general fiber Xs, KXs is pseudoeffective. We set S ◦ be the maximal nonempty Zariski open subset of S such that f is smooth over S◦ and X◦ = f−1(S◦). Then there exists a unique singular hermitian metric ĥcan on KX/S such that 1. ĥcan has semipositive curvature in the sense of current. 2. ĥcan |Xs is an AZD of KXs for every s ∈ S◦. 3. There exists the union F of at most countable union of proper subvarieties of S such that for every s ∈ S \F , ĥcan|Xs ≦ ĥcan,s holds, where ĥcan,s denotes the supercanonical AZD of KXs . 4. There exists a subset G of measure 0 in S◦, such that for every s ∈ S◦ \G, ĥcan |Xs = ĥcan,s holds. Remark 1.11 Even for s ∈ G, ĥcan|Xs is an AZD of KXs by 2. I do not know whether F or G really exists in some cases. � By Theorem 1.10 and the L2-extension theorem ([O-T, p.200, Theorem]), we obtain the following corollary immediately. Corollary 1.12 ([S1, S2, T3]) Let f : X −→ S be a smooth projective family over a complex manifold S. Then plurigenera Pm(Xs) := dimH 0(Xs,OXs(mKXs)) is a locally constant function on S � The following corollary is immediate consequence of Theorem 1.10, since the supercanonical AZD is always has minimal singularities (cf. Definition 2.2 and Remark 2.8). Corollary 1.13 Let f : X −→ Y be an algebraic fiber space. Suppose that KX and KY are pseudoeffective. Let ĥcan be the canonical singular hermitian metric on KX/Y constructed as in Theorem 1.10. Let ĥcan,X , ĥcan,Y be the supercanonical AZD’s of KX and KY respectively. Then there exists a positive constant C such that ĥcan,X ≦ C · ĥcan · f∗ĥcan,Y holds on X. � 4This condition is equivalent to the one that for some regular fiber Xs, KXs is pseudoef- fective. This is well known. For the proof, see Lemma 3.7 below for example. Cororally 1.13 is very close to Iitaka’s conjecture which asserts that Kod(X) ≧ Kod(Y ) + Kod(F ) holds for any algebraic fiber space f : X −→ Y , where F is a general fiber of f : X −→ Y and Kod(M) denotes the Kodaira dimension of a compact complex manifold M . In this paper all the varieties are defined over C. And we frequently use the classical result that the supremum of a family of plurisubharmonic functions locally uniformly bounded from above is again plurisubharmonic, if we take the uppersemicontinuous envelope of the supremum ([L, p.26, Theorem 5]). For simpliciy, we denote the upper(resp. lower)semicontinuous envelope simply by the upper(resp. lower) envelope. We note that this adjustment occurs only on the set of measure 0. In this paper all the singular hermitian metrics are supposed to be lowersemicontinuous. There are other applications of the supercanonical AZD. Also it is imme- diate to generalize it to the log category and another generalization involving hermitian line bundles with semipositive curvature is also possible. These will be discussed in the forthcoming papers. 2 Proof of Theorem 1.7 In this section we shall prove Theorem 1.7. We shall use the same notations as in Section 1.2. The upper estimate of K̂Am is almost the same as in [T5], but the lower estimate of K̂Am requires the L 2 extension theorem ([O-T, O]). 2.1 Upper estimate of K̂Am Let X be as in Theorem 1.7 and let n denote dimX and let x ∈ X be an arbitrary point. Let (U, z1, · · · , zn) be a coordinate neighbourhood ofX which is biholomorphic to the unit open polydisk ∆n such that z1(x) = · · · = zn(x) = 0. Let σ ∈ Γ(X,OX(A+mKX)). Taking U sufficiently small, we may assume that (z1, · · · , zn) is a holomorphic local coodinate on a neighbourhood of the closure of U and there exists a local holomorphic frame eA of A on a neighbour- hood of the clousure of U . Then there exists a bounded holomorphic function fU on U such that σ = fU · eA · (dz1 ∧ · · · ∧ dzn)⊗m holds. Suppose that A · (σ ∧ σ̄) m |= 1 holds. Then we see that | fU (z) | m dµ(z) ≦ (inf hA(eA, eA)) hA(eA eA) m | fU |2 dµ(z) ≦ (inf hA(eA, eA)) hold, where dµ(z) denotes the standard Lebesgue measure on the coordinate. Hence by the submeanvalue property of plurisubharmonic functions, A · | σ | m (x) ≦ { hA(eA, eA)(x) infU hA(eA, eA) m · π−n· |dz1 ∧ · · · ∧ dzn |2 (x) holds. Let us fix a C∞ volume form dV on X . Since X is compact and every line bundle on a contractible Stein manifold is trivial, we have the following lemma. Lemma 2.1 There exists a positive constant C independent of the line bundle A and the C∞ metric hA such that lim sup A · K̂ m ≦ C · dV holds on X. � 2.2 Lower estimate of K̂Am Let hX be any C ∞ hermitian metric on KX . Let h0 be an AZD of KX defined by the lower envelope of : inf{h(x) | h is a singular hermitian metric on KX with Θh ≧ 0,h ≧ hX}. Then by the classical theorem of Lelong ([L, p.26, Theorem 5]) it is easy to verify that h0 is an AZD of KX (cf. [D-P-S, Theorem 1.5]). h0 is of minimal singularities in the following sense. Definition 2.2 Let L be a pseudoeffective line bundle on a smooth projective variety X. An AZD h on L is said to be an AZD of minimal singularities, if for any AZD h′ on L, there exists a positive constant C such that h ≦ C · h′ holds. � Let us compare h0 and ĥcan. By the L2-extension theorem ([O]), we have the following lemma. Lemma 2.3 There exists a positive constant C independent of m such that K(A+mKX , hA · hm−10 ) ≧ C · (hA · hm0 )−1 holds, where K(A+mKX , hA · hm−10 ) is the (diagonal part of) Bergman kernel of A+mKX with respect to the L 2-inner product: (σ, σ′) := ( σ ∧ σ̄′ · hA · hm−10 , where we have considered σ, σ′ as A+ (m− 1)KX valued canonical forms. � Proof of Lemma 2.3. By the extremal property of the Bergman kernel (see for example [Kr, p.46, Proposition 1.4.16]) we have that K(A+mKX , hA·hm−10 )(x) = sup{|σ(x) | 2| σ ∈ Γ(X,OX(A+mKX)⊗I(hm−10 )), ‖σ‖= 1}, holds for every x ∈ X , where ‖σ‖= (σ, σ) 12 . Let x be a point such that h0 is not +∞ at x. Let dV be an arbitrary C∞ volume form on X as in Section 1.2. Then by the L2-extension theorem ([O, O-T]) and the sufficiently ampleness of A (see Section 1.2), we may extend any τx ∈ (A+mKX)x with hA·hm−10 ·dV −1(τx, τx) = 1 to a global section τ ∈ Γ(X,OX(A+mKX)⊗ I(hm−10 )) such that ‖τ ‖≦ C0, where C0 is a positive constant independent of x and m. Let C1 be a positive constant such that h0 ≧ C1 · dV −1 holds on X . By (1), we obtain the lemma by taking C = C−10 · C1. � Let σ ∈ Γ(X,OX(A+mKX)⊗ I(hm−10 )) such that σ ∧ σ̄ · hA · hm−10 = 1 | σ |2 (x) = K(A+mKX , hA · hm−10 )(x) hold, i.e., σ is a peak section at x. Then by the Hölder inequality we have that A · (σ ∧ σ̄) m | ≦ ( hA · hm0 · | σ |2 ·h−10 ) m · ( h−10 ) h−10 ) hold. Hence we have the inequality: K̂Am(x) ≧ K(A+mKX , hA · hm−10 )(x) m · ( h−10 ) m (2) holds. Now we shall consider the limit lim sup A ·K(A+mKX , hA · h Let us recall the following result. Lemma 2.4 ([D, p.376, Proposition 3.1]) lim sup A ·K(A+mKX , hA · h m = h−10 holds. � Remark 2.5 In ([D, p.376, Proposition 3.1], Demailly only considered the local version of Lemma 2.4. But the same proof works in our case by the sufficiently ampleness of A. This kind of localization principle for Bergman kernels is quite standard. � In fact the L2-extension theorem ([O-T, O]) implies the inequality lim sup A ·K(A+mKX , hA · h m ≧ h−10 and the converse inequality is elementary. See [D] for details and applications. Hence letting m tend to infinity in (2), by Lemma 2.4, we have the following lemma. Lemma 2.6 lim sup A · K̂ m ≧ ( h−10 ) −1 · h−10 holds. � By Lemmas 2.1 and 2.6, we see that K̂A∞ := lim sup A · K̂ exists as a bounded semipositive (n, n) form on X . We set ĥcan,A := the lower envelope of (K 2.3 Independence of ĥcan,A from hA In the above construction, ĥcan,A depends on the choice of the C ∞ hermitian metric hA apriori. But actually ĥcan,A is independent of the choice of hA. Let h′A be another C ∞-hermitian metric on A. We define (K̂Am) ′ := sup{| σ | 2m ; σ ∈ Γ(X,OX(A+mKX)), | (h′A) m · (σ ∧ σ̄) 1m |= 1}. We note that the ratio hA/h A is a positive C ∞-function on X and m = 1 uniformly on X . Since the definitions of K̂Am and (K̂ ′ use the extremal prop- erties, we see easily that for every positive number ε, there exists a positive integer N such that for every m ≧ N (1− ε)(K̂Am)′ ≦ K̂Am ≦ (1 + ε)(K̂Am)′ holds on X . Hence we obtain the following uniqueness theorem. Theorem 2.7 K̂A∞ = lim supm→∞ h A · K̂Am is independent of the choice of the C∞ hermitian metric hA. Hence hcan,A is independent of the choice of the C hermitian metric hA. � 2.4 Completion of the proof of Theorem 1.7 Let h0 be an AZD of KX constructed as in Section 2.1. Then by Lemma 2.6 we see that ĥcan,A ≦ ( h−10 ) · h0 holds. Hence we see I(ĥmcan,A) ⊇ I(hm0 ) holds for every m ≧ 1. This implies that H0(X,OX(mKX)⊗I(hm0 )) ⊆ H0(X,OX(mKX)⊗I(ĥmcan,A)) ⊆ H0(X,OX(mKX)) hold, hence H0(X,OX(mKX)⊗ I(ĥmcan,A)) ≃ H0(X,OX(mKX)) holds for every m ≧ 1. And by the construction and the classical theorem of Lelong ([L, p.26, Theorem 5]) stated in Section 1.3, ĥcan,A has semipositive curvature in the sense of current. Hence ĥcan,A is an AZD of KX and depends only on X and A by Lemma 2.7. Let us consider K̂∞ := sup K̂∞,A where sup means the pointwise supremum and A runs all the sufficiently am- ple line bundle on X . Then Lemma 2.1, we see that K̂∞ is a well defined semipositive (n, n) form on X . We set ĥcan := the lower envelope of K̂ Then by the construction, ĥcan ≦ ĥcan,A for every ample line bundle A. Since ĥA is an AZD ofKX , ĥcan is also an AZD ofKX indeed (again by [L, p.26, Theorem 5]). Since ĥcan,A depends only on X and A, ĥcan is uniquely determined by X . This completes the proof of Theorem 1.7. � Remark 2.8 As one see Section 2.2, we see that ĥcan is an AZD of KX of minimal singularities (cf. Definition 2.2). � 2.5 Comparison of hcan and ĥcan Suppose that X has nonnegative Kodaira dimension. Then by Theorem 1.3, we can define the canonical AZD hcan on KX . We shall compare hcan and ĥcan. Theorem 2.9 ĥcan,A ≦ hcan holds on X. In particular ĥcan ≦ hcan holds on X � Proof of Theorem 2.9. If X has negative Kodaira dimension, then the right hand side is infinity. Hence the ineuqality is trivial. Suppose thatX has nonnegative Kodaira dimension. Let σ ∈ Γ(X,OX(mKX)) be an element such that (σ ∧ σ̄) m |= 1 Let x ∈ X be an arbitrary point on X . Since OX(A) is globally generated by the definition of A, there exists an element τ ∈ Γ(X,OX(A)) such that τ(x) 6= 0 and hA(τ, τ) ≦ 1 on X . Then we see that hA(τ, τ) m · (σ ∧ σ̄) 1m ≦ 1 holds. This implies that K̂Am(x) ≧|τ(x) | m ·Km(x) holds at x. Noting τ(x) 6= 0,letting m tend to infinity, we see that K̂A∞(x) ≧ K∞(x) holds. Since x is arbitrary, this completes the proof of Theorem 2.9. � Remark 2.10 The equality hcan = ĥcan implies the abundance of KX . � By the same proof we obtain the following comparison theorem (without assuming X has nonnegative Kodaira dimension). Theorem 2.11 Let A,B a sufficiently ample line bundle on X. Suppose that B −A is globally generated, then ĥcan,B ≦ ĥcan,A holds. � Remark 2.12 Theorem 2.11 implies that ĥcan = lim ĥcan,ℓA holds for any ample line bundle A on X. � 3 Variation of ĥcan under projective deforma- tions In this section we shall prove Theorem 1.10. The main ingredient of the proof is the variation of Hodge structure. 3.1 Construction of ĥcan on a family Let f : X −→ S be an algebraic fiber space as in Theorem 1.10. The construction of ĥcan can be performed simultaeneously on the family as follows. The same construction works for flat projective family with only canonical singularities. But for simplicity we shall work on smooth category. Let S◦ be the maximal nonempty Zariski open subset of S such that f is smooth over S◦ and let us set X◦ := f−1(S◦). Hereafter we shall assume that dimS = 1. The general case of Theorem 1.10 easily follows from just by cutting down S to curves. Let A be a sufficiently ample line bundle on X such that for every pseudoeffective singular hermitian line bundle (L, hL), OX(A+L)⊗I(hL) and OX(KX+A+L)⊗I(hL) are globally generated and OXs(A+L |Xs)⊗I(hL |Xs) and OXs(KXs+A+L |Xs)⊗I(hL |Xs) are globally generated for every s ∈ S◦ as long as hL|Xs is well defined. Let us assume that there exists a smooth member D of | 2A | such that D does not contain any fiber over S◦. Let σD a holomorphic section of 2A with divisor D. We consider the singular hermitian metric hA := | σD | on A. We set Em := f∗OX(A+mKX/S). Since we have assumed that dimS = 1, Em is a vector bundle for every m ≧ 1. We denote the fiber of the vector bundle over s ∈ S by Em,s. Then we shall define the sequence of 1 A-valued relative volume forms by K̂Am,s := sup{|σ | m ;σ ∈ Em,s, | A · (σ ∧ σ̄) m |= 1} for every s ∈ S◦. This fiberwise construction is different from that in Section 1.2 in the following two points : 1. We use the singular metric hA |Xs instead of a C∞ hermitian metric on A |Xs. 2. We use Em,s instead of Γ(Xs,OXs(A|Xs +mKXs)). We note that the 2nd difference occurs only over at most countable union of proper analytic subsets in S◦. Since hA is singular, at some point s ∈ S◦ and for some positive integer m0, K̂ might be identically 0 on Xs. But for any s ∈ S◦ we find a positive integer m0 such that for every m ≧ m0, we have A |Xs) = OXs holds for every m ≧ m0. Hence even in this case we see that K̂Am,s is not identically 0 for every sufficiently large m. We define the relative |A | 2m valued volume form K̂Am by K̂Am|Xs := K̂Am,s(s ∈ S) and a relative volume form K̂A∞ by K̂A∞|Xs := lim sup A · K̂ m,s(s ∈ S). Of course the above construction of K̂Am,s(s ∈ S◦) works also for C∞ hermitian metric instead of the singular hA as above. The reason why we use the singular hA is that we shall use the variation of Hodge structure to prove the plurisub- harmonic variation of log K̂Am,s.We may use a C ∞ metric with strictly positive curvature on A, instead of the singular hA as above, if we use the plurisubhar- monicity properties of Bergman kernels ([Ber, Theorem 1.2]) instead of Theorem 3.1. See Theorem 4.1 below. We define singular hermitian metrics on A+mKX/S by ĥm,A := the lower envelope of (K̂ Let us fix a C∞ hermitian metric hA,0 on A and we set ĥcan,A := the lower envelope of lim inf A,0 · ĥm,A. Cleary ĥcan,A does not depend on the choice of hA,0 (in this sense, the presence of hA,0 is rather auxilary). Then we define ĥcan := the lower envelope of inf ĥcan,A, where A runs all the ample line bundle on X . At this moment, ĥcan is defined only on KX/S |X◦. The extension of ĥcan to the singular hermitian metric on the whole KX/S will be discussed later. 3.2 Semipositivity of the curvature current of ĥm,A To prove the semipositivity of the curvature of ĥm,A, the following theorem is essential. Theorem 3.1 ([Ka3, p.174,Theorem 1.1] see also [F, Ka1]) φ : M −→ C be a projective morphism with connected fibers from a smooth projective variety M onto a smooth curve C. Let KM/C be the relative canonical bundle. We set F := φ∗OM (KM/C)) and let C◦ denote the nonempty maximal Zariski open subset of C such that φ is smooth over C◦. Let hM/C be the hermitian metric on F | C◦ by hM/C(σ, σ ′) := ( σ ∧ σ′, where n = dimM − 1. Let π : P(F ∗) −→ C be the projective bundle associated with F ∗ and Let L −→ P(F ∗) be the tautological line bundle. Let hL denote the hermitian metric on L | π−1(S◦) induced by hM/C . Then hL has semipositive curvature on π −1(S◦) and hL extends to the sin- gular hermitian metric on L with semipositive curvature current. � We define the pseudonorm ‖σ‖ 1 of σ ∈ Em,s by ‖σ‖ 1 A · (σ ∧ σ̄) m |m2 . We set Em = f∗OX(A+mKX/S) and let Lm be the tautological line bundle on P(E∗m), where E m denotes the dual of Em. By Theorem 3.1 and the branched covering trick, we obtain the following essential lemma. Lemma 3.2 ([Ka1, p.63, Lemma 7 and p.64, Lemma 8]) Let σ ∈ Γ(X,OX(A + mKX/S)). Then ‖ ‖ defines a singular hermitian metric with semipositive curvature on Lm. � Proof of Lemma 3.2. If there were no A, the lemma is completely the same as [Ka1, p.63, Lemma 7 and p.64, Lemma 8]. In our case, we use the Kawamata’s trick to reduce the logarithmic case to the non logarithmic case. Since this trick has been used repeatedly by Kawamata himself (see [Ka2, Ka3] for example), the following argument has no originality. We consider the multivalued relative log canonical form Then there exists a finite cyclic covering µ : Y −→ X such that µ∗( σ√ m is a (single valued) relative canonical form on Y 5. Here the branch locus of µ may be much larger than the union of D∪(σ). But it does not matter. The branch covering is used to reduce the log canonical case to the canonical case. Let π : Ỹ −→ Y be an equivariant resolution of singularities and let f̃ : Ỹ −→ S be the resulting family. We shall denote the composition µ ◦ π : Ỹ −→ X by µ̃. Let U be a Zariski open subset of Sσ such that f̃ is smooth. We note that the Galois group action is isometric on f̃∗OỸ (KỸ /S) with respect to the natural L2-inner product on f̃∗OỸ (KỸ /S). Therefore by Theorem 3.1, we see that ‖ ‖ defines a singular hermitian metric on Lm with semipositive curvature on a nonempty Zariski open of P(E∗m). Again by Theorem 3.1 the singular hermitian metric extends to the whole P(E∗m) preserving semipositive curvature property. We also present an alternative proof indicated by Bo Berndtsson at the workshop at MSRI in April, 2007. Alternative proof of Lemma 3.2(cf. [B-P, Section 6]). We use the eqality | σ | 2m= | σ | | σ |2m−1m and view | σ |2m−1m as a singular hermitian metric on (m− 1)KX/S +A. Then by [T4, Therem 5.4] or [B-P], we see that A · (σ ∧ σ̄) m | 2m defines a singular hermitian metric with semipositive curvature current on Lm. The rest of the proof is identical as the previous one. � 5If we use a C∞ hermitian metric instead of the above hA, we also construct a cyclic covering µ : Y −→ X such that 1 µ∗L is a genuine line bundle on Y and µ∗σ m is a 1 valued canonical form on Y . Remark 3.3 The metric hA can be replaced by a C ∞-hermitian metric with semipositive curvature in the second proof. � Corollary 3.4 (see also [B-P, Section 6]) The curvature Θ ĥm,A −1∂∂̄ log K̂m,A is semipositive everywhere on X◦. � Proof. Let x ∈ Xs(s ∈ S◦) and let Ω be a holomorphic local generatorof KX/S and let eA be a holomorphic local generator of A on a neighbourhood U of x in X◦. Viewing ξ(y) := (e−1A · Ω−m)(y) as an element of the dual of Em,f(y) by σ ∈ Em,f(y) 7→ σ(y) · (e−1A · Ω−m)(y)(y ∈ U), log(K̂m,A(y)· | eA |− m · | Ω |−2 (y)) (y ∈ U) is plurisubharmonic function on U , since | ξ(y) | m ·K̂m,A(y) = sup{ | ξ(y) · σ(y) | 2m ‖ [σ][ξ(y)] ‖ ; σ ∈ Em,f(y), [σ][ξ(y)] 6= 0} holds, where [σ][ξ(y)] denotes the class of σ ∈ Em,f(y) in the fiber Lm,[ξ(y)] at [ξ(y)] ∈ P(E∗m). � Now let us consider the behavior of ĥm,A along X\X◦. Since the problem is local, we may and do assume S is a unit open disk ∆ in C for the time being. For every local holomorphic section σ of Em the function A · (σ ∧ σ̄) is of algebraic growth along S \S◦. More precisely for s0 ∈ S \S◦ as in [Ka1, p.59 and p. 66] there exist positive numbers C,α, β such that A · (σ ∧ σ̄) m |≦ C· |s− s0|−α · | log(s− s0) |β (3) holds. Moreover as [Ka1, p.66] for a nonvanishing holomorphic section σ of Em around p ∈ S \S◦, the pseudonorm ‖σ‖ 1 A (σ ∧ σ̄) has a positive lower bound around every p ∈ S. This implies that ĥm,A is bounded from below by a smooth metric along the boundary X \X◦. By the above estimate, ĥm,A is of algebraic growth along the fiber on X \X◦ by its definition and ĥm,A extends to a singular hermitian metric on A+KX/S with semipositive curvature on the whole X . Now we set ĥcan,A := the lower envelope of lim inf A,0 · ĥm,A, where hA,0 be a C ∞ metric on A (with strictly positive curvature) as in the last subsetion 6. To extend ĥcan,A across S \S◦, we use the following useful lemma. 6One may use hA instead of hA,0 here. But the corresponding limits may be different along D, although the difference is negligible by taking the lower envelopes. Lemma 3.5 ([B-T, Corollary 7.3]) Let {uj} be a sequence of plurisubharmonic functions locally bounded above on the bounded open set Ω in Cm. Suppose further lim sup is not identically −∞ on any component of Ω. Then there exists a plurisubhar- monic function u on Ω such that the set of points {x ∈ Ω | u(x) 6= (lim sup uj)(x)} is pluripolar. � Since ĥm,A extends to a singular hermitian metric on A + KX/S with semipositive curvature current on the whole X and ĥcan,A := the lower envelope of lim inf A,0 · ĥm,A exists as a singular hermitian metric on KX/S on X ◦ = f−1(S◦), we see that ĥcan,A extends as a singular hermitian metric with semipositive curvature cur- rent on the whole X by Lemma 3.5. Repeating the same argument we see that ĥcan is a well defined singular her- mitian metric with semipositive curvature current on KX/S |X◦ and it extends to a singular hermitian metric on KX/S with semipositive curvature current on the whole X . 3.3 Uniqueness of ĥcan,A for singular hA’s In the above construction, we use a singular hermitian metric hA on A instead of a C∞ hermitian metric. We note that hA is singular along the divisor D. Hence the resulting metric may be a little bit different from the original construction apriori. But actually Theorem 2.7 still holds. Our metric hA is defined as as above. Let h′A be a C ∞ hermitian metric on A. Let us fix an arbitrary point s ∈ S◦. Let us fix a Kähler metric on X and let Uε be the ε neighbourhood of D with respect to the metric. By the upper estimate Lemma 2.1, we see that although hA is singular along D, there exists a positive integer m0 and a positive constnat C depending only on s such that for every m ≧ m0 and any σ ∈ Em,s with ‖ σ ‖ 1 A · (σ ∧ σ̄) 2 = 1, Uε∩Xs A · (σ ∧ σ̄) m |≦ C · ε holds. This means that there is no mass concentration around the neighbour- hood of D ∩ Xs. We note that on Xs \Uε the ratio (hA/h′A) m converges uni- formly to 1 as m tends to infinity. Hence by the definitions of K̂Am,s and (K̂ we see that for every s ∈ S◦ and δ > 0, there exists a positive integer m1 such that for every m ≧ m1 (1− δ)(K̂Am,s)′ ≦ K̂Am,s ≦ (1 + δ)(K̂Am,s)′ holds on Xs. Hence we have the following lemma. Lemma 3.6 K̂A∞,s is same as the one defined by a C ∞ hermitian metric on A for every s ∈ S◦. � 3.4 Case dimS > 1 In Sections 3.1,3.2, we have assumed that dimS = 1. In the case of dimS > 1 the same proof works similarly. But there are several minor differences. First there may not exist D ∈| 2A | which does not contain any fibers, hence the restriction of hA may not be well defined on some fibers in this case. But this can be taken care by Lemma 3.6. Namely ĥcan is independent of the choice of D. Hence replacing hA by a C ∞ hermitian metric, we see that K̂A∞ is defined on all fibers over S◦. Second in this case Em = f∗OX(A+mKX/S) may not be locally free on S◦. If Em.s is not locally free at s0 ∈ S◦, then K̂A∞ may be discontinuous at s0. But J := {s ∈ S◦ | Em is not locally free at s for some m ≧ 1} is at most a countable union of proper subvarieties of S◦ and ĥcan,A := the lower envelope of is a well defined singular hermitian metric with semipositive curvature current on X◦, i.e., the construction is indifferent to the thin set J . Hence we may construct ĥcan on X ◦ in this case. The extension of ĥcan as a singular hermitian metric onKX/S with semipositive curvature current can be accomplished just by slicing S by curves. Hence we complete the proof of the assertion 1 in Theorem 1.10. 3.5 Completion of the proof of Theorem 1.10 To complete the proof of Theorem 1.10, we need to show that ĥcan defines an AZD for KXs for every s ∈ S. To show this fact, we modify the construction of K̂Am. Here we do not assume dimS = 1. Let us fix s ∈ S◦ and let h0,s be an AZD constructed as in Section 2.2. Let U be a neighbourhood of s ∈ S◦ in S◦ which is biholomorphic to an open ball in k(k := dimS). By the L2-extension theorem ([O-T, O]), we have the following lemma. Lemma 3.7 Every element of Γ(Xs,OXs(A | Xs+mKXs)⊗I(hm−10,s )) extends to an element of Γ(f−1(U),OX(A+mKX)) for every positive integer m. � Proof of Lemma 3.7. We prove the lemma by induction on m. If m = 1, then the L2-extension theorem ([O-T, O]) implies that every element of Γ(Xs,OXs(A +KXs)) extends to an element of Γ(f−1(U),OX(A +KX)). Let {σ(m−1)1,s , · · · , σ (m−1) N(m−1)} be a basis of Γ(Xs,OXs(A | Xs+(m−1)KXs)⊗I(h̃ 0,s )) for some m ≧ 2. Suppose that we have already constructed holomorphic exten- sions {σ̃(m−1)1,s , · · · , σ̃ (m−1) N(m−1),s} ⊂ Γ(f −1(U),OX(A+ (m− 1)KX)) of {σ(m−1)1,s , · · · , σ (m−1) N(m−1),s} to f −1(U). We define the singular hermitian metric Hm−1 on (A+ (m− 1)KX) | f−1(U) by Hm−1 := 1∑N(m−1) j=1 | σ̃ (m−1) j,s |2 We note that by the choice of A, OXs(A |Xs + mKXs) ⊗ I(hm−10,s ) is globally generated. Hence we see that I(hm0,s) ⊆ I(hm−10,s ) ⊆ I(Hm−1|Xs) hold on Xs. Apparently Hm−1 has a semipositive curvature current. Hence by the L2-extension theorem ([O-T, p.200, Theorem]), we may extend every element of Γ(Xs,OXs(A+mKXs)⊗ I(hm−10,s )) extends to an element of Γ(f−1(U),OX(A+mKX)⊗ I(Hm−1)). This completes the proof of Lemma 3.7 by induction. � Let hA,0 be a C ∞ hermitian metric on A with strictly positive curvature as in the end of the last subsection. We define the sequence of {K̃Am,s} by K̃Am,s := sup{| σ | m ; σ ∈ Γ(Xs,OXs(A | Xs+mKXs)⊗I(hm−10,s )), | A,0·(σ∧σ̄) m |= 1}. By Lemma 3.7, we obtain the following lemma immediately. Lemma 3.8 lim sup A,0 · K̃ m,s ≦ K̂ holds. � Proof. We set K̂A,0m,s = sup{| σ | m ; σ ∈ Em,s, | A,0 · (σ ∧ σ̄) m |= 1}. Then by the definition of K̃Am,s and Lemma 3.7 we have that K̃Am,s ≦ K̂ m,s (4) holds on Xs. On the other hand by Lemma 3.6, we see that lim sup A,0 · K̂ m,s = lim sup A,0 · K̂ m,s = K̂∞,s (5) hold. Hence combining (4) and (5), we complete the proof of Lemma 3.8. � We set h̃m,A,s := (K̃ We have the following lemma. Lemma 3.9 If we define K̃A∞,s := lim sup A,0 · K̃ h̃∞,A,s := the lower envelope of K̃ ∞.A,s, h̃∞,A,s is an AZD of KXs . � Proof. Let h0,s be an AZD of KXs as above. We note that OXs(A |Xs + mKXs) ⊗ I(hm−10,s ) is globally generated by the definition of A. Then by the definition of K̃Am,s, I(hm0,s) ⊆ I(h̃mm,A,s) holds for every m ≧ 1. Hence by repeating the arugument in Section 2.2, similar to Lemma 2.6, we have that h̃∞,A,s ≦ ( h−10,s) · h0,s holds. Hence h̃∞,A,s is an AZD of KXs . � Since by the construction and Lemma 3.6 ĥcan,s ≦ h̃∞,A,s holds on s, we see that ĥcan |Xs is an AZD of KXs . Since s ∈ S◦ is arbitrary, we see that ĥcan |Xs is an AZD of KXs for every s ∈ S◦. This completes the proof of the assertion 2 in Theorem 1.10. We have already seen that the singular hermitian metric ĥcan has semipositive curvature in the sense of current (cf. Section 3.2 expecially Corollary 3.4). We note that there exists the union F of at most countable union of proper subvarieties of S such that for every s ∈ S◦ \F E(ℓ)m,s = Γ(Xs,OX(ℓA+mKXs)) holds for every ℓ,m ≧ 1. Then by the construction and Theorem 2.11(see Remark 2.12)7 for every s ∈ S◦ \F , ĥcan|Xs ≦ ĥcan,s holds, where ĥcan,s is the supercanonical AZD of KXs . This completes the proof of the assertion 3 in Theorem 1.10. 7Theorem 2.11 is used because some ample line bundle on the fiber may not extends to an ample line bundle on X in general. We shall define the singular hermitian metric Ĥcan on KX/S|X◦ by Ĥcan|Xs := ĥcan,s (s ∈ S◦). Then by the construction of ĥcan there exists a subset Z of measure 0 in X such that Ĥcan|X◦ \Z = ĥcan|X◦ \Z holds. Let us set G := {s ∈ S◦ | Xs ∩ Z is not of measure 0 in Xs}. Then since Z is of measure 0, G is of measure 0 in S◦. For s ∈ S \G, by the definition of the supercanonical AZD ĥcan,s of KXs , we see that ĥcan|Xs = ĥcan,s holds. This completes the proof of Theorem 1.10. � Remark 3.10 As above we have used the singular hermitian metric hA to prove Theorem 1.10 and then go back to the case of a C∞ metric by the uniqueness result (Lemma 3.6). This kind of interaction between singular and smooth met- rics have been seen in the convergence of the currents associated with random sections of a positive line bundle to the 1-st Chern form of the positive line bundle (see [S-Z]). My first plan of the proof of Theorem 1.10 was to use the random sections to go to the smooth case from the singular case. Although I cannot justify it, it seems to be interesting to pursue this direction. � 4 Appendix The following theorem is a generalization of Theorem 3.1. Theorem 4.1 φ : M −→ C be a projective morphism with connected fibers from a smooth projective variety M onto a smooth curve C. Let KM/C be the relative canonical bundle. Let (L, hL) be a pseudoeffective singular hermitian line bundle on M Let m be a positive integer. We set F := φ∗OM (mKM/C +L) and let C◦ denote the nonempty maximal Zariski open subset of C such that φ is smooth over C◦. Let π : P(F ∗) −→ C be the projective bundle associated with F ∗ and Let H −→ P(F ∗) be the tautological line bundle. Let hH denote the singular hermitian metric on H | π−1(S◦) defined by hH(σ, σ) := {( L · (σ ∧ σ) m }m2 , where n = dimM − 1. Then hH has semipositive curvature on π−1(S◦) and hH extends to the singular hermitian metric on H with semipositive curvature current. � Proof. The proof is a minor modification of the proof of Lemma 3.2. Let σ be a local holomorphic section of H on π−1(S◦). We consider the multivalued L-valued canonical form m σ and uniformize it by taking a suitable cyclic Galois covering µ : Y −→ X as in Lemma 3.2. Then applying [Ber, Theorem 1.2] or [T4, Theorem 5.4] (see also [B-P]) on Y , as in Lemma 3.2, we see that hH defines a singular hermitian metric on the tautological line bundle on P(E∗m). Hence we see that hH has semipositive curvature on π−1(S◦). The extension of hH to the whole H is also follows from [T4, Theorem 5.4]. This completes the proof of Theorem 4.1. � References [B-T] E. Bedford, B.A. Taylor : A new capacity of plurisubharmonic functions, Acta Math. 149 (1982), 1-40. [Ber] B. Berndtsson: Curvature of vector bundles associated to holomorphic fibra- tions, math.CV/0511225 (2005). [B-P] B. Berndtsson, B. and M. Paun : Bergman kernels and the pseudoeffectivity of relative canonical bundles, math.AG/0703344 (2007). [D] J.P. Demailly : Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1 (1992), no. 3, 361–409. [D-P-S] J.P. Demailly-T. Peternell-M. Schneider : Pseudo-effective line bundles on compact Kähler manifolds, International Jour. of Math. 12 (2001), 689-742. [F] T. Fujita : On Kähler fiber spaces over curves, J. Math. Soc. Japan 30, 779-794 (1978). [Ka1] Y. Kawamata: Kodaira dimension of Algebraic fiber spaces over curves, Invent. Math. 66 (1982), pp. 57-71. [Ka2] Y. Kawamata, Subadjunction of log canonical divisors II, alg-geom math.AG/9712014, Amer. J. of Math. 120 (1998),893-899. [Ka3] Y. Kawamata, On effective nonvanishing and base point freeness, Kodaira’s issue, Asian J. Math. 4, (2000), 173-181. [Kr] S. Krantz : Function theory of several complex variables, John Wiley and Sons (1982). [L] P. Lelong : Fonctions Plurisousharmoniques et Formes Differentielles Positives, Gordon and Breach (1968). [N] A.M. Nadel: Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. 132 (1990),549-596. [O-T] T. Ohsawa, K. Takegoshi: L2-extension of holomorphic functions, Math. Z. 195 (1987),197-204. [O] T. Ohsawa: On the extension of L2 holomorphic functions V, effects of gener- alization, Nagoya Math. J. 161 (2001) 1-21, Erratum : Nagoya Math. J. 163 (2001). [S-Z] B. Shiffman, S. Zelditch :Distribution of zeros of random and quantum chaotic sections of positive line bundles. Comm. Math. Phys. 200 (1999), no. 3, 661–683. [S1] Y.-T. Siu : Invariance of plurigenera, Invent. Math. 134 (1998), 661-673. [S2] Y.-T. Siu : Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not nec- essarily of general type, Collected papers Dedicated to Professor Hans Grauert (2002), pp. 223-277. [T1] H. Tsuji: Analytic Zariski decomposition, Proc. of Japan Acad. 61(1992) 161- [T2] H. Tsuji: Existence and Applications of Analytic Zariski Decompositions, Trends in Math. Analysis and Geometry in Several Complex Variables, (1999) 253-272. http://arxiv.org/abs/math/0511225 http://arxiv.org/abs/math/0703344 http://arxiv.org/abs/math/9712014 [T3] H. Tsuji: Deformation invariance of plurigenera, Nagoya Math. J. 166 (2002), 117-134. [T4] H. Tsuji: Dynamical construction of Kähler-Einstein metrics, math.AG/0606023 (2006). [T5] H. Tsuji: Curvature semipositivity of relative pluricanonical systems, math.AG/0703729 (2007). [T6] H. Tsuji: Kodaira dimension of algebraic fiber spaces, in preparation. Author’s address Hajime Tsuji Department of Mathematics Sophia University 7-1 Kioicho, Chiyoda-ku 102-8554 Japan http://arxiv.org/abs/math/0606023 http://arxiv.org/abs/math/0703729 Introduction Canonical AZD hcan Supercanonical AZD can Variation of the supercanonical AZD can Proof of Theorem 1.7 Upper estimate of mA Lower estimate of mA Independence of can,A from hA Completion of the proof of Theorem 1.7 Comparison of hcan and can Variation of can under projective deformations Construction of can on a family Semipositivity of the curvature current of m,A Uniqueness of can,A for singular hA's Case dimS > 1 Completion of the proof of Theorem 1.10 Appendix

0704.0567

Yield Curve Shapes and the Asymptotic Short Rate Distribution in Affine One-Factor Models

YIELD CURVE SHAPES AND THE ASYMPTOTIC SHORT RATE DISTRIBUTION IN AFFINE ONE-FACTOR MODELS MARTIN KELLER-RESSEL AND THOMAS STEINER Abstract. We consider a model for interest rates, where the short rate is given under the risk-neutral measure by a time-homogenous, one-dimensional affine process in the sense of Duffie, Filipović, and Schachermayer. We show that in such a model yield curves can only be normal, inverse or humped (i.e. endowed with a single local maximum). Each case can be characterized by simple conditions on the present short rate rt. We give conditions under which the short rate process will converge to a limit distribution and describe the risk-neutral limit distribution in terms of its cumulant generating function. We apply our results to the Vasiček model, the CIR model, a CIR model with added jumps and a model of Ornstein-Uhlenbeck type. 1. Introduction We consider a model for the term structure of interest rates, where the short rate (rt)t≥0 is given under the martingale measure by a one-dimensional conserva- tive affine process in the sense of Duffie, Filipović, and Schachermayer [2003]. An affine short rate process of this type will lead to an exponentially-affine structure of zero-coupon bond prices and thus also to an affine term structure of yields and forward rates. We emphasize here that the definition of Duffie et al. [2003] is not limited to diffu- sions, but also includes processes with jumps and even with jumps whose intensity depends in an affine way on the state of the process itself. The class of models we consider naturally includes the Vasiček model, the CIR model and variants of them that are obtained by adding jumps, such as the JCIR-model of Brigo and Mercurio [2006, Section 22.8]. Since they are the best-known, the two ‘classical’ models of Vasiček and Cox-Ingersoll-Ross will serve as the starting point for our discussion of yield curve shapes: A common criticism of the (time-homogenous) CIR and the Vasiček model is that they are not flexible enough to accommodate more complex shapes of yield curves, such as curves with a dip (a local minimum), curves with a dip and a hump, or Date: November 4, 2018. 2000 Mathematics Subject Classification. 60J25, 91B28. Key words and phrases. affine process, term structure of interest rates, Ornstein-Uhlenbeck process, yield curve. Supported by the Austrian Science Fund (FWF) through project P18022 and the START programm Y328. Supported by the module M5 “Modelling of Fixed Income Markets” of the PRisMa Lab, financed by Bank Austria and the Republic of Austria through the Christian Doppler Research Association. Both authors would like to thank Josef Teichmann for most valuable discussions and encour- agement. We also thank various proof-readers at FAM for their comments. http://arxiv.org/abs/0704.0567v2 2 MARTIN KELLER-RESSEL AND THOMAS STEINER other shapes that are frequently observed in the markets. Often these shortcomings are explained by ‘too few parameters’ in the model (cf. Carmona and Tehranchi [2006, Section 2.3.5] or Brigo and Mercurio [2006, Section 3.2]). However if jumps are added to the mentioned models, additional parameters (potentially infinitely many) are introduced through the jump part, while the model still remains in the scope of affine models. It is not clear per se what consequences the introduction of jumps will have for the range of attainable yield curves, and this is one question we intend to answer in this article. Moreover, there seems to be some confusion about what shapes of yield curves are actually attainable even in well-studied models like the CIR-model. While most sources (including the original paper of Cox et al. [1985]) mention inverse, normal and humped shapes, Carmona and Tehranchi [2006, Section 2.3.5] write that ‘tweaking the parameters [of the CIR model] can produce yield curves with one hump or one dip’, and Brigo and Mercurio [2006, Section 3.2] state that ‘some typ- ical shapes, like that of an inverted yield curve, may not be reproduced by the [CIR or Vasiček] model.’ In our main result, Theorem 3.9, we settle this question and prove that in any time-homogenous, affine one-factor model the attainable yield curves are either inverse, normal or humped. The proof will rely only on tools of elementary analysis and on the characterization of affine processes through the generalized Riccati equations of Duffie et al. [2003]. Another related problem is how the shape of the yield curve is determined by the parameters of the model, and also how – when the parameters are fixed – the yield curve is determined by the level of the current short rate. We show in Section 4.2 that also in this respect the CIR model has not been completely understood and discuss a misconception that originates in [Cox et al., 1985] and is repeated for ex- ample in [Rebonato, 1998]. In Section 3.3 we provide conditions under which an affine process converges to a limit distribution. We also characterize the limit distribution in terms of its cumulant generating function, extending results of Jurek and Vervaat [1983] and Sato and Yamazato [1984] for OU-type processes to the class of affine processes. These results can again be interpreted in the context of interest rates, where they can be used to derive the risk-neutral asymptotic distribution of the short rate (rt)t≥0 as t goes to infinity. We conclude our article in Section 4 by applying the theoretical results to several interest rate models, such as the Vasiček model, the CIR model, the JCIR model and an Ornstein-Uhlenbeck-type model. 2. Preliminaries In this section we collect some key results on affine processes from Duffie et al. [2003]. In their article affine processes are defined on the (m+n)-dimensional state space Rm >0 × Rn, and we will try to simplify notation where this is possible in the one-dimensional case. Results on affine processes with state space R>0 can also be found in Filipović [2001]. Definition 2.1 (One-dimensional affine process). A time-homogenous Markov pro- cess (rt)t≥0 with state space D = R>0 or R and its semi-group (Pt)t≥0 are called YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 3 affine, if the characteristic function of its transition kernel pt(x, .), given by p̂t(x, u) = euξ pt(x, dξ) and defined (at least) on {u ∈ C : Reu ≤ 0} if D = R>0 , {u ∈ C : Reu = 0} if D = R , is exponentially affine in x. That is, there exist C-valued functions φ(t, u) and ψ(t, u), defined on R>0 × U , such that (2.1) p̂t(x, u) = exp (φ(t, u) + xψ(t, u)) for all x ∈ D, (t, u) ∈ R>0 × U . For subsequent results the following regularity condition for (rt)t≥0 will be needed: Definition 2.2. An affine process is called regular if it is stochastically continuous and the right hand derivatives ∂+t φ(t, u)|t=0 and ∂+t ψ(t, u)|t=0 exist for all u ∈ U and are continuous at u = 0. Definition 2.3. The parameters (a, α, b, β, c, γ,m, µ) are called admissible for a process with state space R>0 if a = 0, α, b, c, γ ∈ R>0 , β ∈ R , m, µ are Lévy measures on (0,∞), where m satisfies (0,∞) (ξ ∧ 1)m(dξ) <∞ , and admissible for a process with state space R if a, c ∈ R>0 , b, β ∈ R , m is a Lévy measure on R \ {0} , α = 0, γ = 0, µ ≡ 0 . Moreover define the truncation functions hF (ξ) = 0 if D = R>0 if D = R and hR(ξ) = if D = R>0 0 if D = R and finally the functions F (u), R(u) for u ∈ C as F (u) = au2 + bu− c+ D\{0} euξ − 1− uhF (ξ) m(dξ) ,(2.2) R(u) = αu2 + βu− γ + D\{0} euξ − 1− uhR(ξ) µ(dξ) .(2.3) The next result is a one-dimensional version of the key result of Duffie et al. [2003]: 4 MARTIN KELLER-RESSEL AND THOMAS STEINER Theorem 2.4 (Duffie, Filipović, and Schachermayer, Theorem 2.7). Suppose (rt)t≥0 is a one-dimensional regular affine process. Then it is a Feller process. Let A be its infinitesimal generator. Then C∞c (D) is a core of A, C2c (D) ⊆ D(A) and there exist some admissible parameters (a, α, b, β, c, γ,m, µ) such that, for f ∈ C2c (D), Af(x) = (a+ αx)f ′′(x) + (b + βx)f ′(x)− (c+ γx)f(x)+ D\{0} (f(x+ ξ)− f(x)− f ′(x)hF (ξ)) m(dξ)+ D\{0} (f(x+ ξ)− f(x) − f ′(x)hR(ξ)) µ(dξ) .(2.4) Moreover φ(t, u) and ψ(t, u), defined by (2.1), solve the generalized Riccati equa- tions ∂t φ(t, u) = F (ψ(t, u)) , φ(0, u) = 0 ,(2.5a) ∂t ψ(t, u) = R (ψ(t, u)) , ψ(0, u) = u .(2.5b) Conversely let (a, α, b, β, c, γ, µ,m) be some admissible parameters. Then there ex- ists a unique regular affine semigroup (Pt)t≥0 with infinitesimal generator (2.4), and (2.1) holds with φ(t, u) and ψ(t, u) given by (2.5). Closely related to affine processes is the notion of an Ornstein-Uhlenbeck (OU- )type process. These processes are of some importance, since they usually offer good analytic tractability and have been studied for longer than affine processes. Following Sato [1999, Chapter 17] an OU-type process (Xt)t≥0 can be defined as the solution of the Langevin SDE dXt = −λXt dt+ dLt, λ ∈ R, X0 ∈ R, where (Lt)t≥0 is a Lévy process, often called background driving Lévy process (BDLP). In an equivalent definition, an OU-type process is a time-homogenous Markov process, whose transition kernel pt(x, .) has the characteristic function p̂t(x, u) = exp F (e−λsu) ds+ xe−λtu where F (u) is the characteristic exponent of (Lt)t≥0. From the last equation it is immediately seen that every OU-type process is an affine process in the sense of Definition 2.1. It is also seen that in the generalized Riccati equations (2.5) for an OU-type process necessarily R(u) = −λu. Comparing this with (2.3) and Defini- tion 2.3, it is seen that any regular affine process with state space R is a process of OU-type. The reverse, however is not true, as there also exist OU-type processes with state space R>0. We will give an example of such a process in Section 4.4. Naturally we will not only be interested in the process (rt)t≥0 itself, but also in its integral rs ds and in quantities of the type (2.6) Qt f(x) := E rs ds f(rt) ∣∣∣∣ r0 = x where f is a bounded function on D. The next result is an application of the Feynman-Kac formula for Feller semigroups (cf. Rogers and Williams [1994, Sec- tion III.19]) and can be found in Duffie et al. [2003]. It relies on the positivity of (rt)t≥0 and is therefore only applicable if D = R>0. YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 5 Proposition 2.5 (Duffie, Filipović, and Schachermayer, Proposition 11.1). Let (rt)t≥0 be a one-dimensional, regular affine process with state space R>0. Then the fam- ily (Qt)t≥0 defined by (2.6) forms a regular, affine semigroup with infinitesimal generator Bf(x) = Af(x)− xf(x) for all f ∈ C2c (D) . We will make extensive use of the convexity and continuous differentiability of the functions F and R from Definition 2.3. These properties are established in this Lemma: Lemma 2.6. If c = γ = 0 then F, R as defined in Definition 2.3 have the following properties: (i) R(0) = 0 and F (0) = 0. (ii) R(u) <∞ for all u ∈ (−∞, 0]. (iii) If F (u) < ∞ on (c1, c2) ⊆ R, then F is either strictly convex on (c1, c2) or F (u) = bu for all u ∈ R. The same holds for R with b replaced by β. (iv) If F (u) <∞ on (c1, c2) ⊆ R, then F is continuously differentiable on (c1, c2). Also the one-sided derivatives at c1 and c2 are defined but may take the values −∞ (at c1) and +∞ (at c2). The same holds for R. Proof. Property (i) is obvious. If D = R then by Definition 2.3 R(u) = βu such that (ii) follows immediately. If D = R>0 we use the estimate (2.7) |euξ − 1− uhR(ξ)| ≤ |u| O(ξ2) ∧ 1 for all u ∈ (−∞, 0] and ξ ∈ R>0, and (ii) follows from (2.3). For Property (iii) note that by the Lévy-Khintchine formula there exists an infinitely divisible random variableX , such that F is its cumulant generating function, i.e. F (u) = logE for u ∈ (c1, c2). Choosing two distinct numbers u, v ∈ (c1, c2), we apply the Cauchy- Schwarz inequality to = logE 2 · e vX2 ≤ log E[euX ] · E[evX ] = F (u) + F (v) which shows convexity of F . The inequality is strict unless there exists some c 6= 0 such that euX = cevX almost surely. This can only be the case if X is con- stant a.s., in which case F is linear. The same argument applies to R. Property (iv) follows from the convexity and from the fact that F and R are analytic on {u ∈ C : Reu ∈ (c1, c2)} (cf. Lukacs [1960, Chapter 7]). � 3. Theoretical Results We will now use the theory from the last section to calculate bond prices, yields and other quantities in an interest rate model where the short rate follows a one- dimensional regular affine process (rt)t≥0 under the martingale measure. Naturally we will also make the assumption that (rt)t≥0 is conservative, i.e. that pt(x,D) = 1 for all (t, x) ⊆ R>0 ×D. This implies by Duffie et al. [2003, Proposition 9.1] that c = γ = 0 in Definition 2.3. We will need some additional assumptions which are summarized in the following condition: Condition 3.1. The one-dimensional affine process (rt)t≥0 is assumed to be reg- ular and conservative. In addition, if the process has state space R, such that by 6 MARTIN KELLER-RESSEL AND THOMAS STEINER Definition 2.3 R(u) = βu, we require that (3.1) F (u) <∞ for all u ∈ (1/β, 0] if β < 0 , (−∞, 0] else . It will be seen that the condition on F is necessary to guarantee existence of bond prices for all maturities in the term structure model. By Sato [1999, Theorem 25.17] we get an equivalent formulation of Condition 3.1, if we replace F (u) < ∞ by∫ |ξ|>1 euξm(dξ) <∞. Next we define a quantity that will generalize the coefficient of mean reversion from OU-type processes: Definition 3.2 (quasi-mean-reversion). Given a one-dimensional conservative affine process (rt)t≥0, define the quasi-mean-reversion λ as the positive solution of (3.2) R(−1/λ) = 1 . If there is no positive solution we set λ = 0. Since R is by Lemma 2.6 a convex function satisfying R(0) = 0, it is easy to see that (3.2) can have at most one solution and thus λ is well-defined. The name quasi- mean-reversion is derived from the fact that if (rt)t≥0 is a process of OU-type with positive mean-reversion, then R(u) = βu and the quasi-mean-reversion λ = −β is exactly the coefficient of mean reversion of (rt)t≥0. When the process (rt)t≥0 satisfies Condition 3.1, it is seen that F must be defined at least on (−1/λ, 0]. We will encounter several times the condition that λ > 0. The next result gives an equivalent formulation in terms of (α, β, µ): Proposition 3.3. The quasi-mean reversion λ is strictly positive if and only if α > 0, D\{0} hR(ξ)µ(dξ) = ∞, or β − D\{0} hR(ξ)µ(dξ) < 0. Proof. First note that by Lemma 2.6 R(u) < ∞ for all u ∈ (−∞, 0]. Using the estimate (2.7) and a dominated convergence argument it is seen from (2.3) that = α(3.3) R(u)− αu2 = β0 := β − D\{0} hR(ξ)µ(dξ) ,(3.4) where β0 can also take the value −∞. Suppose now that α > 0. Then by (3.3) we get limu→−∞R(u) = ∞. Since R(0) = 0 and R is continuous it follows that there exists a λ > 0 such that R(−1/λ) = 1. Similarly if α = 0, but β0 < 0, it follows from (3.4) that limu→−∞R(u) = ∞ and thus again that λ > 0. Conversely, suppose that α = 0 and β0 ≥ 0. Then R′(u) = lim = β0 ≥ 0 . By the convexity of R it follows that R′(u) ≥ 0 for all u ∈ (−∞, 0). Since R(0) = 0 this implies that R(u) ≤ 0 for all u ∈ (−∞, 0), and consequently that λ = 0. � 3.1. Bond Prices. We consider now the price P (t, t + x) of a zero-coupon bond with time to maturity x, at time t, given by P (t, t+ x) = E ∫ t+x rs ds )∣∣∣∣Ft YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 7 The affine structure of (rt)t≥0 carries over to the bond prices, and we get the following result: Proposition 3.4. Let the short rate be given by a one-dimensional affine process (rt)t≥0 satisfying Condition 3.1. Then the bond price P (t, t+ x) exists for all t, x ≥ 0 and is given by (3.5) P (t, t+ x) = exp (A(x) + rtB(x)) where A and B solve the generalized Riccati equations ∂xA(x) = F (B(x)) A(0) = 0 ,(3.6a) ∂xB(x) = R (B(x)) − 1 B(0) = 0 .(3.6b) Proof. If D = R>0 the assertion follows directly from Proposition 2.5 by noting that P (t, t+ x) = Qx 1. If D = R then, as discussed after Theorem 2.4, (rt)t≥0 is a process of OU-type and R(u) has the simple structure R(u) = βu. By Sato [1999, (17.2) - (17.3)] we obtain in this case directly that (3.7) E ∫ t+x rs ds = exp F (B(s)) ds + rtB(x) with B(x) = (1 − eβx)/β if β 6= 0 and B(x) = −x when β = 0. As a function of x ∈ R>0, B is continuously decreasing from 0 to 1/β if β < 0, and from 0 to −∞ if β ≥ 0. It is therefore seen that the integral on the right side of (3.7) is finite for all x ∈ R>0 if and only if F satisfies (3.1), as required by Condition 3.1. � Corollary 3.5. Let (rt)t≥0 satisfy Condition 3.1 and have quasi-mean-reversion λ. Then the function B(x) from Proposition 3.4 is strictly decreasing and satisfies B(x) = −1/λ . Proof. The result follows from a qualitative analysis of the autonomous ODE (3.6b). Let λ > 0. Since R(−1/λ)−1 = 0 the point x∗ := −1/λ is an critical point of (3.6b). By the convexity of R and the fact that R(0) = 0 it follows that R′(x∗) < 0 such that x∗ is asymptotically stable, i.e. solutions entering a small enough neighborhood of x∗ must converge to x∗. Since R(x)− 1 < 0 for x ∈ (x∗, 0] and there is no other critical point in (x∗, 0], we conclude that B(x) – the solution of (3.6b) starting at 0 – is strictly decreasing and converges to x∗. If λ = 0 then there is no critical point in (−∞, 0] and R(x)−1 < 0 for x ∈ (−∞, 0]. It follows that B(x) is strictly decreasing and diverges to −∞. � 3.2. The Yield Curve and the Forward Rate Curve. The next results are the central theoretical results of this article and describe the global shapes of attainable yield curves in any affine one-factor term structure model. Definition 3.6. The (zero-coupon) yield Y (rt, x) is given by Y (rt, 0) := rt and (3.8) Y (rt, x) := − logP (t, t+ x) = −A(x) for all x > 0 . For rt fixed, we call the function Y (rt, .) the yield curve. The (instantaneous) forward rate f(rt, x) is given by f(rt, 0) := rt and (3.9) f(rt, x) := −∂x logP (t, t+ x) = −A′(x)− rtB′(x) for all x > 0 . For rt fixed, we call the function f(rt, .) the forward rate curve. 8 MARTIN KELLER-RESSEL AND THOMAS STEINER By l’Hospital’s rule and the generalized Riccati equations (3.6) it is seen that both the yield and the forward rate curve are continuous at 0. The first quantity associated to the yield curve that we consider, is the asymptotic level basymp of the yield curve as x → ∞, also known as long-term yield, consol yield or simply ‘long end’. Theorem 3.7. Let the short rate process be given by a one-dimensional affine pro- cess (rt)t≥0 satisfying Condition 3.1 with quasi-mean-reversion λ. If λ > 0 then basymp := lim Y (rt, x) = lim f(rt, x) = −F (−1/λ) . If λ = 0 then basymp = lim −F (u) + rt (1−R(u)) . Proof. From (3.6a) we obtain that (3.10) lim = lim A′(x) = lim F (B(x)) . If λ > 0 then by Corollary 3.5 (3.11) lim B(x) = −1/λ, lim = 0 and lim B′(x) = 0 and the assertion follows by combining (3.8) – (3.11). If λ = 0 then limx→∞B(x) = −∞ and = lim B′(x) = lim R(B(x)) − 1 . By setting u := B(x) we obtain the desired result. � From Theorem 3.7 it is clear that for practical purposes only models with λ > 0 will be useful. So far we know that in this case the short end of the yield curve is given by Y (rt, 0) = rt and the long end by Y (rt,∞) = basymp. We will now examine what happens between these two endpoints. Definition 3.8. The yield curve Y (rt, x) is called • normal if it is a strictly increasing function of x, • inverse if it is a strictly decreasing function of x, • humped if it has exactly one local maximum and no minimum on (0,∞). In addition we call the yield curve flat if it is constant over all x ∈ R>0. This is our main result on the shapes of yield curves in affine one-factor models: Theorem 3.9. Let the risk-neutral short rate process be given by a one-dimensional affine process (rt)t≥0 satisfying Condition 3.1 and with quasi-mean-reversion λ > 0. In addition suppose that F 6= 0 and that either F or R is non-linear. Then the following holds: • The yield curve Y (rt, .) can only be normal, inverse or humped. • Define bnorm := − F ′(−1/λ) R′(−1/λ) and binv := R′(0) if R′(0) < 0 +∞ if R′(0) ≥ 0 . YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 9 The yield curve is normal if rt ≤ bnorm , humped if bnorm < rt < binv and inverse if rt ≥ binv . The above theorem is visualized in Figure 1. For its proof we will use a simple Lemma. We state the Lemma without proof, since it follows in an elementary way from the usual definition of a convex function on R. Lemma 3.10. A strictly convex or a strictly concave function on R intersects an affine function in at most two points. In the case of two intersection points p1 < p2, the convex function lies strictly below the affine function on the interval (p1, p2); if the function is concave it lies strictly above the affine function on (p1, p2). Proof of Theorem 3.9. Define the function H(x) : R>0 → R by (3.12) H(x) := Y (rt, x)x = −A(x)− rtB(x) . We will see that the convexity behavior of H will be crucial for the shape of the yield curve Y (rt, .). From the generalized Riccati equations (3.6) the first derivative of H is calculated as (3.13) ∂xH(x) = −F (B(x)) − rt (R(B(x))− 1) and the second as (3.14) ∂xxH(x) = −B′(x) (F ′(B(x)) + rtR′(B(x))) . Note that F and R are continuously differentiable by Lemma 2.6, and also B by (3.6b), such that the second derivative of H is well-defined and continuous. Since B is strictly decreasing by Corollary 3.5, the factor −B′(x) is positive for all x ∈ R>0. The sign of ∂xxH(x) therefore equals the sign of (3.15) k(x) := F ′(B(x)) + rtR ′(B(x)) . From the fact that B is decreasing and F and R are convex it is obvious that k must be decreasing. We will now show that k has at most a single zero in [0,∞): (a) D = R>0: We have assumed that either F or R is non-linear. By Lemma 2.6 this implies that either F or R is strictly convex, and thus that either F ′ or R′ is strictly increasing. If rt > 0, then it follows that k is strictly decreasing and thus has at most a single zero. If rt = 0, an additional argument is needed: It could happen that F is of the form F = bu such that k(x) = b and k is no longer strictly decreasing. However, by assumption, F 6= 0 such that in this case k has no zero in [0,∞). (b) D = R: In this case, by the admissibility conditions in Definition 2.3, we have necessarily R(u) = βu. Also, since either F or R is non-linear, F must be non-linear and thus by Lemma 2.6 strictly convex. It follows that k(x) = F ′(B(x))+rtβ is strictly decreasing and thus has at most a single zero in [0,∞). We have shown that k is decreasing and has at most a single zero; to determine whether it has a zero for some value of rt, we consider the two ‘endpoints’ k(0) and limx→∞ k(x). First we show that (3.16) k(0) ≥ 0 if and only if rt ≤ binv := R′(0) if R′(0) < 0 +∞ if R′(0) ≥ 0 . Since B(0) = 0 by Proposition 3.4 it follows that k(0) = F ′(0) + rtR ′(0) . 10 MARTIN KELLER-RESSEL AND THOMAS STEINER We distinguish two cases: (a) If R′(0) < 0 then the assertion (3.16) follows immediately. (b) Consider the case that R′(0) ≥ 0: Assume that D = R. Then we have R(u) = βu and R′(0) = β ≥ 0. This, however, stands in contradiction to our assumption λ > 0, which implies that β = −λ < 0 (cf. Definition 3.2). Thus we must have D = R>0 and rt ≥ 0; in this case it follows that k(0) ≥ 0, for all rt ∈ D, and we set binv = +∞. Next we consider the right end of k(x) and show that (3.17) lim k(x) ≤ 0 if and only if rt ≥ bnorm := − F ′(−1/λ) R′(−1/λ) . Since limx→∞B(x) = −1/λ by Corollary 3.5 we have that (3.18) lim k(x) = F ′(−1/λ) + rtR′(−1/λ) . By assumption λ > 0, and by Definition 3.2 it holds that R(−1/λ) = 1. Also R(0) = 0, and by the mean value theorem 1 = R(−1/λ)−R(0) = − 1 R′(ξ) for some ξ ∈ (−1/λ, 0). Since R′ is increasing, it follows that R′(−1/λ) ≤ −λ < 0, and we can deduce (3.17) directly from (3.18). We summarize our results on the function k so far: k stays negative on (0,∞) if rt ≥ binv and positive if rt ≤ bnorm. It has a single zero on (0,∞) if and only if bnorm < rt < binv. If k has a zero on (0,∞), since k is decreasing, the sign of k will be positive to the left of the zero and negative to the right of the zero. Since ∂xxH has the same sign as k, the statements above translate in the obvious way to the convexity behavior of H . We will now use the convexity behavior of H to derive our results about the yield curve. Consider the equation (3.19) H(x) = cx, x ∈ [0,∞) for some fixed c ∈ R. Since H(0) = 0 this equation has at least one solution, x0 = 0. If rt ≥ binv then H(x) is strictly concave on [0,∞), and by Lemma 3.10 the equation (3.19) has at most one additional solution x1. Also, when the solution exists, H(x) crosses cx from above at x1. Similarly if rt ≤ bnorm then H(x) is strictly convex, and there exists at most one additional solution x2 to (3.19) on [0,∞). If the solution exists, then cx is crossed from below at x2. In the last case bnorm < rt < binv, there exists a x∗ – the zero of k(x) – such that H(x) is strictly convex on (0, x∗) and strictly concave on (x∗,∞). Now there can exist at most two additional solutions x1, x2 to (3.19) with x1 < x ∗ < x2, such that cx is crossed from below at x1 and from above at x2. Because of definition (3.12), every solution to (3.19), excluding x0 = 0, is also a solution to (3.20) Y (rt, x) = c, x ∈ (0,∞) with rt fixed. Also the properties of crossing from above/below are preserved since x is positive. This means that in the case rt ≥ binv, equation (3.20) has at most a single solution, or in other words, that every horizontal line is crossed by the yield curve at most in a single point. If it is crossed, it is crossed from above. This YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 11 implies that Y (x) is a strictly decreasing function of x, or following Definition 3.8, that the yield curve is inverse. In the case rt ≤ bnorm we have again that (3.20) has at most a single solution and that every horizontal line is crossed from below by the yield curve, if it is crossed. In other words, the yield curve is normal. In the last case of bnorm < rt < binv, the yield curve crosses every horizontal line at most twice, in which case it crosses first from below, then from above. Thus in this case the yield curve is humped. � Corollary 3.11. Under the conditions of Theorem 3.9 the instantaneous forward rate curve has the same global behavior as the yield curve, i.e. Y (rt, .) is inverse ⇐⇒ f(rt, .) is strictly decreasing Y (rt, .) is humped ⇐⇒ f(rt, .) has exactly one local maximum and no local minimum Y (rt, .) is normal ⇐⇒ f(rt, .) is strictly increasing . In the second case the maximum of the forward rate curve is f(rt, x∗), where x∗ solves (3.21) rt = − F ′(B(x)) R′(B(x)) , x ∈ (0,∞) . Proof. This follows from the fact that ∂xH(x) as given in (3.13) is exactly the forward rate f(rt, x). The derivative of the forward rate is therefore ∂xxH(x), which is given in (3.14) as ∂xf(rt, x) = ∂xxH(x) = −B′(x) · k(x) . The factor −B′(x) 6= 0 is always positive, and the possible sign changes and zeroes of k(x) are discussed in the proof of Theorem 3.9, leading to the stated equivalences. Equation (3.21) is simply the condition k(x∗) = 0. � Corollary 3.12. Under the conditions of Theorem 3.9 it holds that (3.22) bnorm < basymp < binv whenever the quantities are finite. In addition it holds that (3.23) D ∩ (bnorm, binv) 6= ∅ . Remark 3.13. Note that equation (3.23) implies that there is always some rt ∈ D such that the yield curve Y (rt, .) is humped. Proof. By the mean value theorem there exists a ξ ∈ (−1/λ, 0) such that basymp = −F (−1/λ) = F (0)− F (−1/λ) = F ′(ξ) . Since F is convex and thus F ′ increasing, it holds that (3.24) F ′(−1/λ) ≤ basymp ≤ F ′(0) Applying the mean value theorem to R, there exists another ξ ∈ (−1/λ, 0) such 1 = R(−1/λ)−R(0) = − 1 R′(ξ) . 12 MARTIN KELLER-RESSEL AND THOMAS STEINER Since R′ is increasing we deduce that R′(−1/λ) ≤ −λ < 0. Assuming that also R′(0) < 0 we get (3.25) − 1 R′(−1/λ) ≤ ≤ − 1 R′(0) Since either F orR is non-linear, one of the functions is strictly convex by Lemma 2.6. Consequently either both inequalities in (3.24) or in (3.25) are strict. Putting them together we get ′(−1/λ) R′(−1/λ) < basymp < − F ′(0) R′(0) proving (3.22) under the assumption that R′(0) < 0. If R′(0) ≥ 0 then by definition binv = ∞. Equation (3.24) still holds, but in (3.25) only the left inequality sign remains valid. Together this still proves that bnorm < basymp and we have shown (3.22). To prove (3.23) we distinguish two cases: (a) D = R. In this case it is sufficient to prove−∞ < binv and bnorm <∞. Consider first binv. If R ′(0) ≥ 0 then by definition binv = ∞ and nothing is to prove. If R′(0) < 0 then binv = −F ′(0)/R′(0). By convexity F ′(0) > −∞ and the assertion follows. Consider now bnorm = −F ′(−1/λ)/R′(−1/λ). From (3.25) we know that R′(−1/λ) ≤ −λ < 0. By convexity F ′(−1/λ) <∞ and it follows that bnorm <∞. (b) D = R>0. In this case it is sufficient to prove 0 ≤ bnorm and to apply (3.22). As above we have that bnorm = −F ′(−1/λ)/R′(−1/λ) and that R′(−1/λ) ≤ −λ < 0. By Definition 2.3 F ′(−1/λ) = b+ (0,∞) ξe−ξ/λm(dξ) with b ≥ 0. It follows that F ′(−1/λ) ≥ 0, proving the assertion. � The last Corollary of this section shows the interesting fact that the occurrence of a humped yield curve is a necessary and sufficient sign of randomness in the short rate model: Corollary 3.14. Let the risk-neutral short rate process be given by a one-dimensional affine process (rt)t≥0 satisfying Condition 3.1 with F 6= 0 and quasi-mean-reversion λ > 0. Then the following statements are equivalent: (i) There exists a rt ∈ D such that Y (rt, .) is flat. (ii) There exists no rt ∈ D such that Y (rt, .) is humped. (iii) The short rate process (rt)t≥0 is deterministic. (iv) F (u) = bu and R(u) = βu. Proof. Theorem 3.9, together with Corollary 3.12, shows already that ¬(iv) implies ¬(i) and ¬(ii). Also, from the form of the generator in (2.4), it is seen that (iii) and (iv) are equivalent. It remains to show that (iv) implies (i) and (ii). Proceeding as in the proof of Theorem 3.9 we obtain instead of (3.15) simply k(x) = b+ rtβ . The yield curve will be humped if and only if k has a single (isolated) zero in [0,∞). Since k is a constant function, this cannot be the case for any rt ∈ D and we have shown (ii). By the same arguments as in the proof of Theorem 3.9 the yield curve YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 13 binv = − F’(0) R’(0) basymp = − F(− 1 λ) bnorm = − F’(− 1 λ) R’(− 1 λ) Time to Maturity Figure 1. This Figure shows a graphical summary of Theorems 3.7 and 3.9, as well as the definitions of the key quantities bnorm, basymp and binv. In any affine model satisfying the conditions of Theorem 3.9, the shapes of yield curves will follow the picture given here. They will be normal if r0 is below bnorm, humped if r0 is between bnorm and binv and inverse if r0 is above binv. Also all yield curves will tend asymptotically to the same level basymp. is flat if and only if k is constant and equal to 0. This is the case if rt = − bβ . It remains to show that rt ∈ D. Note that β = −λ < 0. In particular β 6= 0, such that for D = R we are already done. If D = R>0 we have by the admissibility conditions in Definition 2.3 that b ≥ 0. Thus rt = − bβ ≥ 0 and we have shown (i). � 3.3. The Limit Distribution of an Affine Process. It is well-known that the Gaussian Ornstein-Uhlenbeck process, for example, converges in law to a limit distribution and that this distribution is Gaussian. The goal of this section is to establish a corresponding result for affine processes. While calculating the marginal distributions of an affine process involves solving the generalized Riccati equations (2.5), it will be seen that the limit distribution is much easier obtained and can be determined directly from the functions F and R. In the interest rate model considered in the preceding section, the short rate follows an affine process under the martingale measure, such that the results will allow us to characterize the risk-neutral asymptotic short rate distribution. Often also the 14 MARTIN KELLER-RESSEL AND THOMAS STEINER limit distribution under the objective measure is of interest, but the affine prop- erty is in general not preserved by an equivalent change of measure, such that the results are not directly applicable. Nevertheless, for the sake of tractability, condi- tions on the measure change can be imposed, such that the model is affine under both the objective and the risk-neutral measure. (See Nicolato and Venardos [2003] for an example from option pricing and Cheridito et al. [2005] for more general re- sults). In such a setting the results can also be applied under the objective measure. Before we state the result, we want to recall that a real-valued random variable L is called self-decomposable if for every c ∈ (0, 1) there exists a random variable Lc, independent of L, such that L = cL+ Lc for all c ∈ (0, 1) . Since self-decomposability is a distributional property, we will identify L and its law, and refer to both as self-decomposable. For OU-type processes, limit distributions have been studied for some time; the first results can be found in Jurek and Vervaat [1983] and Sato and Yamazato [1984]. The next theorem summarizes these results, and can be found in similar form in Sato [1999, Theorem 17.5]: Theorem 3.15. Let (rt)t≥0 be a OU-type process on R. If β < 0 and |ξ|>1 log |ξ|m(dξ) <∞ then (rt)t≥0 converges in law to a limit distribution L which is independent of r0 and has the following properties: (i) L is self-decomposable. (ii) The cumulant generating function κ(u) = log eux dL(x) satisfies (3.26) κ(iu) = F (is) ds for all u ∈ R . Conversely, if L is a self-decomposable distribution on R and β < 0, there exists a unique triplet (a, b,m) satisfying the admissibility conditions of Definition 2.3, such that L is the limit distribution of the affine process (of OU-type) given by the parameters (a, b,m, β). As discussed in Section 2, every regular affine process with state space R is of OU-type, such that the above theorem applies. We now state our corresponding result for affine processes on R>0: Theorem 3.16. Let (rt)t≥0 be a one-dimensional, regular, conservative affine pro- cess with state space R>0. If R′(0) < 0 and log ξ m(dξ) <∞ then (rt)t≥0 converges in law to a limit distribution L which is independent of r0, and whose cumulant generating function κ is given by (3.27) κ(u) = F (s) ds for all u ∈ (−∞, 0] . YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 15 Proof. By Theorem 2.4 the transition kernel pt(x, .) of the process (rt)t≥0 has the characteristic function p̂t(x, u) = exp (φ(t, u) + xψ(t, u)) where φ and ψ satisfy the generalized Riccati equations (2.5) for all u ∈ U , and thus in particular for all u ∈ (−∞, 0]. Since R(0) = 0, 0 is a critical point of the autonomous ODE (2.5b), and by the assumption R′(0) < 0 it is asymptotically stable. By the convexity of R, R′(0) < 0 also implies that R(u) > 0 for all u ∈ (−∞, 0), such that ψ(t, u) is a strictly increasing function in t for all u ∈ (−∞, 0). Since 0 is the only critical point of (2.5b) on (−∞, 0] it also follows that ψ(t, u) = 0 for all u ∈ (−∞, 0] . Consequently, (3.28) lim log p̂t(x, u) = lim φ(t, u) = F (ψ(r, u)) dr = F (s) where the last two equalities follow from (2.5) and the transformation s = ψ(r, u). We will now show that the last integral in (3.28) converges absolutely for all u ∈ (−∞, 0]: Since R(u) ≥ 0 and F (u) ≤ 0 for all u ∈ (−∞, 0] we obtain F (s) ∣∣∣∣ ds = − F (s) ds ≤ − 1 R′(0) F (s) ds, u ∈ (−∞, 0] , where the inequality follows from the fact that the convex function R is supported by its tangent at 0. From the definition of F (u) in (2.2) it is clear that the convergence of the last integral will depend only on the jump part of F , i.e. the integral converges if and only if (3.29) (0,∞) esξ − 1 m(dξ) ds <∞, for all u ∈ (−∞, 0]. Define M(u, ξ) = esξ−1 ds. For a fixed u ∈ (−∞, 0], it is easily verified that M(u, ξ) = O(ξ) as ξ → 0, and that M(u, ξ) = O(log ξ) as ξ → ∞. Since the Lévy measurem(dξ) integrates (ξ∧1) by Definition 2.3, and log ξ ·1{ξ>1} by assumption, it must also integrateM(u, ξ). Applying Fubini’s theorem, (3.29) follows, such that κ(u) := F (s) ds converges for all u ∈ (−∞, 0]. In particular limu↑0 κ(u) = 0, such that the limit in (3.28) is a function that is left-continuous at 0. By standard results on Laplace transforms of probability measures (cf. Steutel and van Harn [2004, Theorem A.3.1]), the pointwise convergence of cumulant generating functions to a function that is left-continuous at 0 implies convergence in distribution of (rt)t≥0 to a limit distribution L with cumulant generating function given by (3.28). � Since the marginal distributions of an affine process are infinitely divisible, also the limit distribution L must be infinitely divisible, if it exists. In Theorem 3.15 a stronger result is given for an affine process on R: In this case L is also self- decomposable. An obvious question is, if this result can be extended to the state space R>0. We will see that the answer is negative. In Section 4.3 an example of an affine process with state space R>0 is given, which converges to an infinitely divisible limit distribution that is not self-decomposable. This result is interesting, since it leaves open the possibility of some unexpected properties of the limit distribution of 16 MARTIN KELLER-RESSEL AND THOMAS STEINER an affine process. For example a self-decomposable distribution is always unimodal, whereas an infinitely divisible distribution might be not. 4. Applications 4.1. The Vasiček model. We apply the results of the last section to the classical Vasiček model (4.1) drt = −λ(rt − θ) dt+ σ dWt, r0 ∈ R where (Wt)t≥0 is a standard Brownian motion under the risk-neutral measure and λ, θ, σ > 0. The Vasiček model is arguably the simplest affine model, and no surprises are to be expected here. In fact all results that we state here can already be found in the original paper of Vasiček [1977]. We advise the reader to view this paragraph as a warm-up for the following examples. Clearly (rt)t≥0 is a conservative affine process with F (u) = λθu + u2 ,(4.2) R(u) = −λu .(4.3) From the quadratic term in F and Definition 2.3, it is seen that (rt)t≥0 has state space R. This property is often criticized, since it allows the short rate to become negative. From Theorem 3.9 we calculate binv = θ and bnorm = θ − such that the yield curve in the Vasiček model is normal if rt ≤ θ − σ2/λ2, inverse if rt ≥ θ and humped in the remaining cases. The long term yield is calculated from (3.7) as basymp = −F (−1/λ) = θ − in this case exactly the arithmetic mean of binv and bnorm. Theorem 3.15 applies and the cumulant generating function κ of the risk-neutral limit distribution L satisfies κ(iu) = − 1 F (is) iθ − σ ds = uiθ − u for u ∈ R. Hence, L is Gaussian with mean θ and variance σ2 4.2. The Cox-Ingersoll-Ross model. The Cox-Ingersoll-Ross (CIR)-model was introduced by Cox et al. [1985]. In this model the short rate process (rt)t≥0 is given by the SDE (4.4) drt = −a(rt − θ)dt+ σ rt dWt, r0 ∈ R>0 where (Wt)t≥0 is a standard Brownian Motion under the risk-neutral measure and a, θ, σ > 0. The process (rt)t≥0 is a conservative affine process with F (u) = aθu ,(4.5) R(u) = u2 − au .(4.6) YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 17 From Definition 2.3 it is seen that (rt)t≥0 has state space R>0. The fact that interest rates stay non-negative in the CIR-model is often cited as an advantage of the model over the Vasiček model. Calculating the quasi-mean-reversion (see Definition 3.2), we find that a2 + 2σ2 + a From Theorem 3.7 we find that the long-term yield is given by basymp = −F (−1/λ) = a2 + 2σ2 + a The boundary between humped and inverse behavior binv is calculated from Theo- rem 3.9 as binv = − F ′(0) R′(0) Both quantities basymp and binv can also be found in [Cox et al., 1985, Eq. (26) and following paragraph]. Before we consider bnorm, we quote (with notation adapted to (4.4)) from page 394 of [Cox et al., 1985] where the shape of the yield curve is discussed: ‘When the spot rate is below the long-term yield [= basymp], the term structure is uniformly rising. With an interest rate in excess of θ [= binv], the term structure is falling. For intermediate values of the interest rate, the yield curve is humped.’ In our terminology, they claim that the yield curve is normal for rt ≤ basymp, humped for basymp < rt < binv and inverse for rt ≥ binv. This stands in clear contradiction to Theorem 3.9 and Corollary 3.12 where we have obtained that yield curves are normal if and only if rt ≤ bnorm and that bnorm < basymp, or – in plain words – that there are yield curves starting strictly below the long-term yield that are still humped. The claims of Cox et al. [1985] are repeated in [Rebonato, 1998, p. 244f], where even several plots of ‘yield surfaces’ (the yield as a function of rt and x) are presented as evidence. However Rebonato fails to indicate the level of basymp in the plots, such that the conclusion remains ambiguous. To clarify the scope of humped yield curves in the CIR-model we calculate bnorm from Theorem 3.9: bnorm = − F ′(−1/λ) R′(−1/λ) = a2 + 2σ2 The relation bnorm < basymp < binv is immediately confirmed by noting that basymp is the harmonic mean of bnorm and binv. For a graphical illustration we refer to the second yield curve from below in Figure 1. The plot actually shows CIR yield curves with parameters a = 0.5, σ = 0.5, θ = 6% plotted over a time scale of 25 years. The second curve from below starts at r0 = 4.2%, i.e. below the long-term yield, but is visibly humped. 18 MARTIN KELLER-RESSEL AND THOMAS STEINER To calculate the limit distribution of (rt)t≥0, we apply Theorem 3.16: The cu- mulant generating function κ(u) of the limit distribution is given by κ(u) = F (s) 1− sσ2/2a ds = − This is the cumulant generating function of a gamma distribution with shape pa- rameter 2aθ/σ2 and scale parameter σ2/2a. Again this result can already be found in Cox et al. [1985, p. 392]. 4.3. An extension of the CIR model. To illustrate the power of the affine setting, we consider now an extension of the CIR model that is obtained by adding jumps to (4.4). We define the risk-neutral short rate process by (4.7) drt = −a(rt − θ)dt+ σ rt dWt + dJt, r0 ≥ 0 where (Jt)t≥0 is a compound Poisson process with intensity c > 0 and expo- nentially distributed jumps of mean ν > 0. This model has been introduced by Duffie and Gârleanu [2001] as a model for default intensity and is used by Filipović [2001] as a short rate model. It can also be found in Brigo and Mercurio [2006] under the name JCIR model. It is easily calculated that F (u) = aθu+ ν − u, u ∈ (−∞, ν) ,(4.8) R(u) = u2 − au .(4.9) Solving the generalized Riccati equations (3.6) for A(x) and B(x) becomes quite tedious, but the quantities binv, basymp, bnorm can be calculated from Theorem 3.7 and Theorem 3.9 in a few lines: The quasi-mean reversion λ stays the same as in the CIR model, since R does not change. From F ′(u) = aθ + (ν − u)2 we derive immediately binv = θ + basymp = ν(a+ ν) + 2 bnorm = γ(σ2ν + γ − a)2 , where γ = a2 + 2σ2. Note that by setting the jump intensity c to zero, the ex- pressions of the (original) CIR model are recovered. Next we calculate the limit distribution of the model. Using the abbreviations ρ := σ2/2 and ∆ := a− νρ we obtain κ(u) = F (s) 1− sρ/a ds+ c (s− ν)(ρs− a) = if ∆ 6= 0 −θν log if ∆ = 0 YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 19 as the cumulant generating function of the limit distribution L under the martin- gale measure. We now take a closer look at the distribution L, since this will answer the question raised at the end of Section 3.3: For certain parameters, L is an example for a limit distribution of an affine process that is infinitely divisible, but not self- decomposable. We consider the case ∆ = 0 and define (4.10) l(x) := νe−νx, x ∈ R>0 . By Frullani’s integral formula (4.11) κ(u) = (eux − 1) l(x) for all u ∈ (−∞, ν). Since l is non-negative on R>0, l(x)/x is the density of a Lévy measure and (4.11) is seen to be the Lévy-Khintchine representation for the cumulant generating function of the infinitely divisible distribution L. In addition, L is self-decomposable if and only if l is non-negative and non-increasing on R>0 (cf. Sato [1999, Corollary 15.11]). In the case of l(x) given by (4.10), it is easily calculated that l(x) has a single maximum at x∗ = 1 . Thus, if c ≤ aθν, then x∗ ≤ 0, such that l is non- increasing on R>0 and L is self-decomposable. If c > aθν then l is increasing in the interval [0, x∗) and the limit distribution L is infinitely divisible, but not self-decomposable. 4.4. The gamma model. Instead of analyzing the properties of a known model, we will now follow a different route and construct a model that satisfies some given properties. We want to construct an affine process on R>0 that has the same limit distribution as the CIR model (i.e. a gamma distribution), but is a process of OU- type. The second property is equivalent to R(u) = βu. Considering Theorem 3.16, we know that if we want to obtain a limit distribution, we need β < 0. To keep with the notation of the Vasiček model, we will write R(u) = −λu where λ > 0. Now by (3.27) the cumulant generating function of the limit distribution is given (4.12) κ(u) = F (s) ds for all u ∈ (−∞, 0] . Let the limit distribution be a gamma distribution with shape parameter k > 0 and scale parameter θ > 0. Then κ(u) = −k log(1 − θu) and by (4.12) F (u) = 1− θu . Setting c = λk and ν = 1/θ it is seen that F (u) is equal to the last term in (4.8). This means that the driving Lévy process of (rt)t≥0 is of the same kind as the process (Jt)t≥0 in (4.7), i.e. (rt)t≥0 is a pure jump OU-type process with exponentially distributed jump heights of mean 1/θ and with jump intensity λk. We interpret the affine process we have constructed as a risk-neutral short rate process. It is clear that the bond prices are of the exponentially-affine form (3.5). From the generalized Riccati equation (3.6b) we obtain B(x) = e−λx − 1 20 MARTIN KELLER-RESSEL AND THOMAS STEINER From equation (3.6a) we calculate A(x) = F (B(s)) ds = θ + λ (log(1− θB(x)) − θx) , such that the bond prices are given by P (t, t+ x) = exp −x λθk θ + λ + rtB(x) (1− θB(x)) θ+λ . The global shape of the yield curve is described by the quantities binv = kθ, basymp = 1/θ+ 1/λ , bnorm = (1/θ + 1/λ)2 and it is seen that for the gamma-OU-process basymp is the geometric average of binv and bnorm. 5. Conclusions In this article we have given, under very general conditions, a characterization of the yield curve shapes that are attainable in term structure models where the risk- neutral short rate is given by a time-homogenous, one-dimensional affine process. Even though the parameter space for this class of models is infinite-dimensional, the scope of attainable yield curves is very narrow, with only three possible global shapes. In addition we have given conditions under which an affine process con- verges to a limit distribution, and we have characterized the limit distribution in terms of its cumulant generating function, extending some known results on OU- type processes. The most obvious question for future research is the extension of these results to multi-factor models. It is evident from numerical results that in two-factor models yield curves with e.g. a dip, or also with a dip and a hump, can be obtained. It would be interesting to see if more complex shapes can also be produced, or if there are similar limitations as in the single-factor case. Also, in the one-factor case the dependence of the yield curve shape on the current short rate is basically described by the intervals D ∩ (−∞, bnorm], (bnorm, binv) and [binv,∞). In the two-factor case the partitioning of the state-space might be more complex, and we expect to see more interesting transitions between yield curve types. Another aspect is, that since affine processes as a general framework become better understood, extensions of classical models e.g. by adding jumps, like in the JCIR model described in Section 4.3, become more feasible and attractive for applications. References Damiano Brigo and Fabio Mercurio. Interest Rate Models - Theory and Practice. Springer Finance. Springer, 2nd edition, 2006. René Carmona and Michael Tehranchi. Interest Rate Models: An Infinite Dimen- sional Stochastic Analysis Perspective. Springer Finance. Springer, 2006. Patrick Cheridito, Damir Filipović, and Marc Yor. Equivalent and absolutely con- tinuous measure changes for jump-diffusion processes. The Annals of Applied Probability, 15(3), 2005. John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross. A theory on the term structure of interest rates. Econometrica, 53(2):385–407, 1985. YIELD CURVE SHAPES IN AFFINE ONE-FACTOR MODELS 21 Darrell Duffie and Nicolae Gârleanu. Risk and valuation of collateralized debt obligations. Financial Analysts Journal, 57(1):41 – 59, 2001. Darrell Duffie, Damir Filipović, and Walter Schachermayer. Affine processes and applications in finance. The Annals of Applied Probability, 13(3):984–1053, 2003. Damir Filipović. A general characterization of one factor affine term structure models. Finance and Stochastics, 5:389–412, 2001. Zbigniew J. Jurek and Wim Vervaat. An integral representation for self- decomposable Banach space valued random variables. Zeitschrift für Wahrschein- lichkeitstheorie und verwandte Gebiete, 62:247–262, 1983. Eugene Lukacs. Characteristic Functions. Charles Griffin & Co Ltd., 1960. Elisa Nicolato and Emmanouil Venardos. Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Mathematical Finance, 13 (4):445–466, 2003. Riccardo Rebonato. Interest-Rate Option Models. Wiley, 2nd edition, 1998. L.C.G. Rogers and David Williams. Diffusions, Markov Processes and Martingales, Volume 1. Cambridge Mathematical Library, 2nd edition, 1994. Ken-iti Sato. Lévy processes and infinitely divisible distributions. Cambridge Uni- versity Press, 1999. Ken-iti Sato and M. Yamazato. Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Processes and Applications, 17:73–100, 1984. Fred Steutel and Klaas van Harn. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker Inc., 2004. Oldrich Vasiček. An equilibrium characterization of the term structure. Journal of Financial Economics, 5:177–188, 1977. Vienna University of Technology, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria E-mail address: [email protected] Vienna University of Technology, Wiedner Hauptstrasse 8–10, A-1040 Wien, Austria E-mail address: [email protected] 1. Introduction 2. Preliminaries 3. Theoretical Results 3.1. Bond Prices 3.2. The Yield Curve and the Forward Rate Curve 3.3. The Limit Distribution of an Affine Process 4. Applications 4.1. The Vasicek model 4.2. The Cox-Ingersoll-Ross model 4.3. An extension of the CIR model 4.4. The gamma model 5. Conclusions References

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Thermally Stimulated Luminescence and Current in new heterocyclic materials for Organic field transistors and organic light emitting diodes

Microsoft Word - Journal Opto _marius.doc Thermally Stimulated Luminescence and Current in new heterocyclic materials for Organic field transistors and organic light emitting diodes Marius Prelipceanu 1,2, Otilia Sanda Prelipceanu 1,2 , Ovidiu-Gelu Tudose 1,2, Sigurd Schrader1,2 1University of Potsdam, Institute of Physics, Condensed Matter Physics, Am Neuen Palais 10, D-14469, Germany 2University of Applied Sciences Wildau, Department of Engineering Physics, D-15745 Wildau, Germany Corresponding author: [email protected] Published in Abstract Book of ICPAM07, Iasi, Romania, June 2004 In the last years progress has been made in the field of organic electronics and in particular organic light emitting devices and organic field effect transistors. In this case the study of localised levels in technologically relevant materials like oxadiazoles and quinoxalines is of fundamental importance. Several scientific tools enable to study localised levels in solids and among others the thermally stimulated techniques give the most direct evidence of their presence. The present work is focused on theoretical and experimental study of localised levels in organic materials suitable for light-emitting devices and field effect transistors by means of thermal techniques. Keywords: thermally stimulated techniques, localised levels, organic materials, light-emitting devices (LED), field effect transistors (FET) 1. Introduction The discussion about the presence of localised states and their density naturally leads to the question about the kind of traps we are dealing with. In a first approach we should distinguish between intrinsic and extrinsic defects. In the first type we should inscribe polymer end groups, grain boundaries, structural defects, conformational disorder up to molecular groups with large permanent dipole moment that could increase the level of energetic disorder [1]. For the second type we should mention the chemical impurities, somehow unavoidable in the synthesis of organic molecules. We can further distinguish the kind of traps in function of their location, interfacial or bulk, or in term of energy, deep traps or shallow traps. Also polarons could be seen, in a simplistic way, like defects caused by an electron plus an induced lattice polarisation [2] followed by a lattice distortion. Traps are into the samples in a great variety and in different proportion, often despite the same preparation procedure. For that reason sometimes their nature is difficult to investigate and the data must be handled with care. In the studies of trapped states because of the above underlined variety of defects in solids the most successful approach is to start introducing a single type of defect in a well-known system in a controlled way. 2. Experimental Part In our work we focused on low molecular compounds as well as on polymers, especially of two classes of materials: oxadiazoles and quinoxalines. Both organic compounds are well know as electron transport materials in OLEDs. PPQs (see figure 1) in general show very high solubility [3] in a variety of common organic solvents, and according to literature they exhibit a glass transition at quite high temperature (250-350 °C). The materials were deposited by spin coating on gold, aluminium or silicon in different speed or concentration for the film optimisation. The layer thickness was controlled by Dektak techniques and Ellypsometry. The thermally stimulated luminescence (TSL) and thermally stimulated current (TSC) are powerful instruments in order to study de-trapping and relaxation processes in organic materials. TSL is a contact less technique that allows to distinguish between deep and shallow trapping states. The proposed mathematical model for the TSL enables to study trap levels and recombination centres inside the band gap. The analytical solution of the rate equations allows two different de-trapping regimes, including or excluding subsequent re-trapping effects. The first order solution kinetic indicates that no re-trapping phenomena are permitted. The electron released from a localised level recombines with a hole in a recombination centre and its re-trapping probability, before to recombine, is negligible. The second order equation deals with the opposite extreme case. The phonon-assisted release of an electron is followed by multiple re-trapping. In this second order kinetic regime the probability of a released electron to get re-trapped is very high. The main factors governing both solutions are the energy depth of the traps calculated with respect to the conduction band edge and the frequency factor. This second important factor in general indicates the attempt-to- escape frequency of electrons from the localised levels. The mathematical model takes also into account the occurrence of distributions of localised levels. In case of a Gaussian distribution of localised states a meaningful parameter is the width of the distribution. Numerical simulations, calculated with the proposed model, show that while a first order peak is characterised by an asymmetric peak shape with a steep decreasing side, the second order kinetic peaks are characterised by a more symmetric shape. The signal is smeared along the whole peak temperature range due to the re-trapping effect. Figure 1. Poly-[2,2’-(1,4-phenylene)-6,6’-bis(3-phenylquinoxaline)] (PPQ IA) The same theoretical description holds for both techniques, TSL and TSC. However, TSC theory requires the presence of an extended conduction band. During a TSC experiment a driving voltage is applied to the sample and the de-trapped charges are extracted at the device contacts. However, the equations describing a TSC peak are similar to equations describing TSL. Additionally, it is possible to determine the density of the trapping states evaluating the area under a TSC peak. Simultaneous TSL and TSC measurements give useful information about the localised states combining the best possibilities of both thermal techniques. Unambiguous information about trap depth, density of states, kinetics order and frequency factor can be extract making use of the full possibilities of the combined measurements. In typical thermally stimulated process experiments a sample is heated in a controlled way and the current, in case of TSC, or the light emission, in case of TSL, or both simultaneously are monitored. The effect appears only when an optical or electrical excitation takes place prior to the heating. TSC, in contrast to TSL, requires the presence of good ohmic contacts. In the following a TSL experiment is described in more detail and the rate equations derived. The sample, in an equilibrium state at room temperature where all the shallow traps are empty, is cooled down to a low temperature. Then, it is illuminated with electromagnetic radiation of certain energy. The incident radiation excites the electrons from the valence band to the conduction band trough the gap. In the case of prompt fluorescence the generated electrons recombine promptly. Otherwise they can form an electron hole pair followed by geminate recombination or by dissociation with subsequent trapping. Charge carriers can get trapped in localised levels that, considering the random fluctuation of the potential in disordered materials, are distributed in energy. From statistical consideration, the distribution type should be generally Gaussian. The thermal emission from traps at low temperature is negligibly small. Therefore, the perturbed equilibrium created by the incident radiation resists for a long time and the electrons are just stored in the localised levels. Temperature is then raised in a controlled way, electrons acquire energy and finally escape from the traps by means of a phonon assisted jump and recombine with holes trough a recombination centre where recombination occurs with subsequent photon emission. By means of spectrally resolved TSL experiments it is possible to get information about recombination centres studying the wavelength of the emitted light as function of temperature and intensity. The above-described processes are illustrated in figure 2. The illustrated scheme for thermoluminescence is simple, but despite of its simplicity it can describe all fundamental features of a thermoluminescence process. Following Chen [4], the electron exchange between the HOMO and LUMO levels, during the trap emptying, can be described by the following three differential equations: h Ann ⋅⋅−= (1) pnAnNn c ⋅−⋅−= )( (2) c AnnAnNnpn ⋅⋅−⋅−−⋅= )( (3) Here nh is the concentration of holes in the recombination centres, nc is the concentration of electrons in the conduction band, Ar is the recombination coefficient for electrons in the conduction band with holes in the recombination centres, n is the concentration of electrons in traps, N is the function describing the concentration of electron traps at depth E below the edge of the conduction band, A is the transition coefficient for electrons in the conduction band becoming trapped and p is the same probability of thermal release of electrons from traps defined in equation (1), which represents in fact their release rate. Equation (2) describes the change of hole density nh in recombination centres versus time. The recombination rate depends both from the concentration of free electrons (nc) and from the concentration of holes already present in the recombination centres trough a probability coefficient (Ar) that depends on the cross section and the thermal velocity72 of electrons. An increase in these parameters results in an increase of the recombination probability. Equation (3) describes the exchange of electrons between conduction band and traps. The first term in the right hand side includes the probability A for an electron to be trapped. That probability A, like Ar, also depends on the thermal velocity of electrons and on the cross section of traps. The second term on the right hand side is the de-trapping term. It is proportional to the concentration of trapped Electron centre Hole centre Conduction band Valence band Recombination centre Figure 2. Energy diagram describing the elementary process of the simple model for TSL electrons and to the Boltzmann’s function, i.e. equation (1). The proportionality factor s, often called frequency factor or pre-exponential factor, should be, when interpreted in terms of attempt to escape of an electron from the potential well, in the order of magnitude of 1010 ÷ 1014 s-1. A saturation effect for carrier release from traps, caused by a limited number of available states in the conduction band, is neglected in this model. In each moment the number of available states in conduction band is much higher than the released amount of electrons from the localised states [5]. Equation (3) describes the variation of electron density in the conduction band and essentially it takes into account the charge neutrality of the whole system. The variation rate of electrons in the conduction band depends on electrons being released - first term on the right hand side-, electrons being trapped - second term - and electrons that recombine - third term. Electrons and holes in that model are generated at the same time - geminate couples, but they are not necessarily still bound. Saturation effects due to filled deeper traps or recombination centres that have already a hole on them are not considered. Complex models have the disadvantage to introduce an increased number of parameters [7]. Actually, several combinations of too many parameters can generate the same shape of a real glow curve, making impossible to find a most probable fit [8]. For that reason it is preferable to deal with a reasonable simple model that involves few reliable parameters. Actually, the proposed simple model can successfully describe the experimental glow curves, but it is necessary to take also the energetic distribution of localised states into account in order to describe the complex behaviour of disordered systems, like amorphous polymers or organic polycrystalline thin films. In such case the total number of traps is represented by the following equation (4). The traps do not have single activation energy, but they are continuously energetically distributed. dEENN (4) Here N(E) can be in principle any kind of distribution, but considering the statistical disorder in organic materials it should have a Gaussian shape. In principle also the frequency factor should have the same energetic distribution. In order to solve the system of differential equations (1)-(4), equation (5), regarding the time dependence of temperature, should be add. tTT ⋅+= β0 (5) In equation (5) β is the experimental constant heating rate. It should be note that as long as T is a well knows function of the time the only real variable is the time. For that reason it is very important, experimentally, to have a perfect control of the temperature linearity. 3. Results and discussion The main peak in figure 3 has the maximum temperature at Tm = 159.7 K and the second, of the roughly half the intensity, at Tm = 230.7 K. The peak at Tm = 159.7 K has a symmetry factor µ = 0.54, very near to the typical value for a second order kinetic. This fact gives a hint that in PPQ IA an electron, before recombination, has high probability to get re-trapped several times. Because of its hidden position the analysis of the minor peak of PPQ IA appearing at Tm = 230.7 K is very difficult [6]. 60 80 100 120 140 160 180 200 220 240 =230.7 K = 159.7 K Temperature (K) 40 60 80 100 120 140 160 180 200 220 240 260 Temperature (K) Figure 3. TSL of PPQ IA Figure 4. TSL glow curve of a PPQ IA sample (red line) compared with the second order numerical simulation of the first peak (green line) Figure 4 shows the numerical fitting of the main TSL peak of PPQ IA. The fit is performed by means of a second order equation characterised by a Gaussian distribution of traps. While the high temperature side perfectly fits the glow curve, the low temperature side do not follow the curve shape. For that reason the numerical simulation is not completely satisfactory. The distribution has a width σ = 0.12 eV and the distribution maximum is at Em = 0.37 eV. The energy maximum Em lies exactly in the middle of the integration limits E1 = 0.25 eV and E2 = 0.49 eV, having the distribution in such case a perfect Gaussian shape. The natural frequency s of this peak can be estimated to be in the order of s = 1x1010 s-1, considering, as is normal, the recombination coefficient / trapping coefficient ratio equal to 1 and a density of traps of 1014 cm- 3. The value of Em, derived by numerical analysis, is far from the expected energy depth of a trap calculated with the initial rise method. However, this mismatching can be explained considering the particular complexity of the peak that could result from the sum of at least two distributed peaks. In effect the initial rise procedure reveals the activation energy of a hidden peak at low temperature. This is an important point to clarify because of the importance of the presence of shallow traps in materials suitable for plastic electronic applications. Shallow traps are, at room temperature, empty and they play a crucial role in the electron transport property of organic materials. For a different thickness we obtain glow curves from figure 5. 40 60 80 100 120 140 160 180 200 220 240 260 Temperature (K) Figure 5. Glow curve of PPQ IA for 1500 nm We made TSC measurements for Poly-3-hexyle-thiophene (P3HT) on SiO2 – treated and untreated in oxygen plasma. The TSC experiment consisted of: (1) Cooling to -180°C (2) Trap filling by light (Mercury lamp) with + 4 V bias voltage (3) Application of readout voltage, heating with 0.10 K/sec. and measurement of detrapping current All measurements were carried out in vacuum (4,5 x 10 –5 mbar). The traps were filled by creating carriers with band-to-band photoexcitation of the samples. The light source was a Mercury lamp (200 W). The thermally stimulated currents were measured by a Keithley 617 electrometer. The TSC and temperature data were stored in a personal computer as described earlier. In a typical experiment, the samples are cooled down to T = 80 K and kept at this temperature for 15 min. Then they are illuminated through front electrode for a 15 minutes at a bias voltage + 4 V. Measurements were started after exposure to light, and samples are then heated with a constant rate (β = 0.1 K/sec.) from 50 up to 240 K. We measured and compared 2 samples of P3HT on SiO2 – treated and untreated in oxygen plasma. Both experiments were performed under the same conditions. The concentration of the traps was estimated using (Manfredotti et. al.) the relation: NT = (6) Here Q is the quantity of charge released during a TSC experiment and can be calculated from the area under the TSC peaks; A and L are the area and the thickness of the sample, respectively; e is the electronic charge and G is the photoconductivity gain, which equals to the number of electrons passing through the sample for each absorbed photon. NT was calculated by assuming G = 1. For that samples L = 60 nm and samples have 2,5 x 2,5 cm, A = 6,25. 10-4 m2. The trap is characterized by the temperature (Tm) corresponding to the peak maximum at the thermally stimulated current. The energy associated with the trap is the thermal energy at Tm given by: mmo KTTTfE ),,( ´βα= (7) 80 100 120 140 160 180 200 220 240 1.54E-010 1.55E-010 1.55E-010 1.56E-010 1.56E-010 Ea = 22.2 meV Ea=20.1 meV Ea = 25.8 meV Ea=17,5 meV =26.79x1014 =4,452x1014Nt=0.2825x10 =48.31x1014 198.56 171.08 155.05 134.43 Temperature (K) 80 100 120 140 160 180 200 220 240 1.00E-013 1.10E-013 1.20E-013 1.30E-013 1.40E-013 1.50E-013 Ea=26 meV =1.368x1012 cm3 =0.608x1012 cm3 =0.84x1012 cm-3 Ea=21 meV Ea=16 meV 202.914 167.161 127.709 Temperature (K) Figure 6. TSC for P3HT on SiO2 untreated sample Figure 7. TSC for P3HT on SiO2 treated sample In this equation, α is a dimensionless model dependent constant. The variable T’ is the temperature at half of the maximum current value on the low temperature side of the current peak. Assuming the Grossweiner model, the constant α and function f are given by: 51.1=α (8) ´ 1),,( TTf mmβ (9) We observed a difference between the trap concentrations in these two samples (treated and untreated) of two orders of magnitude. 4. Conclusions Thermal techniques are a powerful tool in the to study of localised levels in inorganic and organic materials. Thermally stimulated luminescence, thermally stimulated currents and thermally stimulated depolarisation currents allow, when applied in synergy the details shallow of traps and deeper levels to be investigated. They also permit to study, in synergy with dielectric spectroscopy, as polarisation and depolarisation effects. The analysis of the thermograms, emerging from the thermal techniques, can be performed starting from the differential rate equations of the de-trapping phenomena. Such an approach, allowed by the computing power of the modern computers, is not the most fruitful, while the number of free variables involved in the numerical resolution of the rate differential equations is too high. Sometimes completely different sets of parameters can fit the same thermally stimulated peak and ambiguous results are often achieved. 5. Acknowledgements Many thanks for the European Commission (contract number - HPRN-CT-2002-00327 - RTN- EUROFET) for the financial support as well as all co-workers and many friends of EUROFET network. 6. References [1] Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Holt, Rinehart & Winston, New York, 1976). [2] McKeever, S. W. S. Thermoluminescence of Solids (eds. Cahn, R. W., Davis, E. A. & Ward, I. M.) (Cambridge University Press, Cambridge, 1985). [3] Paolo Imperia, Localised States in Organic Semiconductors and their Detection, University of Potsdam, 2003. [4] Bässler, H. Charge Transport in Disordered Organic Photoconductors, a Monte Carlo Simulation Study. phys. stat. sol. (b) 175, 15 (1993). [5] M. Prelipceanu, O.G. Tudose, S. Schrader, Thermally Stimulated Luminescence Investigations of New Materials For OFET’s and OLED’s, Winterschool on Organic Electronics (OEWS’04) Materials, Thin Films, Charge Transport & Device, Planneralm, Austria, 2004. [6] van Turnhout, J. in Electrets (ed. Sessler, G. M.) 81 (Springer Verlag, Berlin, 1980). [7] Schrader, S., Imperia, P., Koch, N., Leising, G. & Falk, B. in Organic Light-Emitting Materials and Devices, (ed. Kafafi, Z. H.) 209 (SPIE, San Diego, California, 1999).

0704.0569

A microfluidic device based on droplet storage for screening solubility diagrams

A mi ro�uidi devi e based on droplet storage for s reening solubility diagrams Philippe Laval, Ni olas Lisai, Jean-Baptiste Salmon, and Mathieu Joani ot LOF, unité mixte Rhodia�CNRS�Bordeaux 1, 178 avenue du Do teur S hweitzer, F�33608 Pessa edex � FRANCE (Dated: O tober 25, 2018) This work des ribes a new mi ro�uidi devi e developed for rapid s reening of solubility diagrams. In several parallel hannels, hundreds of nanoliter-volume droplets of a given solution are �rst stored with a gradual variation in the solute on entration. Then, the appli ation of a temperature gradient along these hannels enables us to read dire tly and quantitatively phase diagrams, on entration vs. temperature. We show, using a solution of adipi a id, that we an measure ten points of the solubility urve in less than 1 hr and with only 250 µL of solution. I. INTRODUCTION Chemistry, biology, and pharma ology, are fa ing al- ways more omplex systems depending on multiple pa- rameters. Therefore their omplete investigations take time and require signi� ant amounts of produ ts. In this ontext, roboti �uidi workstations have already met a great su ess and proved their e� ien y for instan e in the genome sequen ing and analysis [1℄. However, these instruments remain very expensive, need important la- bor, and the volumes involved (≤ mL) are still too large for some spe i� appli ations (e.g. proteomi s) [2, 3℄. Nowadays, other high throughput te hniques based on mi ro�uidi s [4, 5℄ an o�er suitable alternative solutions for the development of rapid s reening tools. Mi ro�uidi devi es are now largely used in biologi al and hemi al �elds for multiple appli ations [6℄ like mole ular sepa- rations and ells sorting [7℄, polymerase hain rea tion [8, 9℄, rapid mi romixing and analysis of hemi al rea - tions [10, 11, 12, 13℄. . . Moreover, the development of mi rovalves and mi romixers has made possible the pro- du tion of highly integrated systems whi h an be used to address individually hundreds of rea tion hambers [14℄. These devi es are well adapted to arry out high throughput s reening of phase diagrams, parti ularly in the ase of protein rystallization investigation. However, their fabri ation and multiplexing are still ompli ated. Another possible strategy is the use of droplets playing the role of nanoliter-sized rea tion ompartments. These droplets an be produ ed in spe i� mi ro�uidi geome- tries [15℄, and their volume and hemi al omposition an be �xed in a ontrolled way. In addition, they also allow a rapid mixing of the di�erent ompounds, prevent from any hydrodynami dispersion and ross ontamination, and an be stored in mi ro hannels (see Ref. [16℄ and referen es therein). Su h a strategy has already proved to be useful for rystallization studies: e.g. s reening of protein rystallization onditions [17, 18℄, or rystal nu leation kineti s measurements [19℄. Figure 1 summarizes the main insights of our work. Ele troni address: philippe.laval-exterieur�eu.rhodia. om We have engineered a new mi ro�uidi hip that allow a dire t and quantitative reading of two-dimensional di- agrams. Hundreds of nanoliter-sized droplets of di�er- ent hemi al ompositions an be stored in parallel mi- ro hannels, and a temperature gradient applied along these hannels enables us to obtain a two-dimensional array of droplets of di�erent on entrations and tempera- tures. For solubility diagram s reening, droplets ontain- ing a given solute are �rst stored with a gradual variation of on entration. Then, rystallization in the droplets is indu ed by ooling, and �nally, the appli ation of an ad- equat temperature gradient dissolves rystals in droplets whose temperature is higher than their solubility tem- perature. As a result, we dire tly read the limit between droplets with and without rystals as shown on Fig. 1( ), whi h gives the solubility temperatures of the solution at the di�erent on entrations. In the materials and methods se tion, we des ribe the mi ro�uidi devi e, the method used to store the droplets in the hannels, and the temperature ontrol setup. We also hara terize the on entration and temperature gra- dients. In the last se tion, we present an experimental proto ol to measure quantitatively solubility diagrams using this devi e. We demonstrate its e� ien y by mea- suring with only 250 µL of solution, the solubility urve of an organi ompound. II. MATERIALS AND METHODS A. Mi rofabri ation The mi ro�uidi devi e is fabri ated in poly(dimethylsiloxane) (PDMS) by using soft- lithographi te hniques [20℄. PDMS (Sili one Elastomer Base, Sylgard 184; Dow Corning) is molded on master fabri ated on a sili on wafer (3-In h-Si-Wafer; Siegert Consulting e.k.) using a negative photoresist (SU-8 2100; Mi roChem). To make molds of 500 µm height, we spin su essively two 250 µm thi k SU-8 layers on the wafer. After ea h spin oating pro ess, the wafer is soft-baked (10 min/65 C and 60 min/95 C). Photolithography is used to de�ne negative images of the mi ro hannels. Eventually, the wafer is hard-baked (25 min/95 C) and http://arxiv.org/abs/0704.0569v1 mailto:[email protected] (a) 1 FIG. 1: (a) Design of the mi ro�uidi devi e ( hannels width 500 µm). Sili one oil is inje ted in inlet 1 and aqueous so- lutions in inlets 2 and 3. The two dotted areas indi ate the positions of the two Peltier modules used to apply tempera- ture gradients ∇T . The three lines of dots mark the positions of temperature measurements. (b) Pi ture of the mi ro�uidi hip made of PDMS sealed with a glass slide (76×52 mm improve larity. Droplets ontaining a olored dye at di�erent on entrations are stored in the ten parallel hannels. ( ) Ex- ample of dire t reading of a solubility diagram. The droplets ontain an organi solute. The dotted line bounding droplets ontaining rystals give an estimation of the solubility limit (see se tion Results for details). developed (SU-8 Developer; Mi roChem). A mixture 10:1 of PDMS is molded on the SU-8 master des ribed above (65 C/60 min). The rossed linked PDMS layer is then peeled o� the mold and holes for the inlets and outlets (1/32 and 1/16 in. o.d.) are pun hed into the material. Then, the PDMS surfa e and a lean sili on wafer surfa e (3-In h-Si-Wafer; 500 µm; Siegert Consulting e.k.) are a tivated for 2 min in a UV ozone apparatus (UVO Cleaner, Model 144AX; Jelight) and brought together. Finally, the devi e is pla ed at 65 for 2 hr to improve the sealing. B. Droplet storage proto ol The devi e, presented on Fig. 1(a), is omposed of three inlets and ten outlets lo ated at the extremities of hannels 1 to 10. As shown on Fig. 1(b), ea h out- let is onne ted to a ≈ 20 m long rigid tubing (FEP 1/16 in.) ended with a pie e of soft PVC tubing (Nal- gene, ≈ 5 m long) inserted in an automated pin h ele - trovalve (105S�01059P; As o Jou omati ). Thanks to this system, ea h outlet an be independently losed or opened by pin hing or not the orresponding PVC tub- ing. However, the pin hing out of a tube leads in a liquid displa ement. To minimize the subsequent liquid distur- ban e in the mi ro hannels, the ele trovalves are pla ed lose to the rigid ones, and the hydrodynami resistan e after the ele trovalves is kept as weak as possible using large tubing. Sili one oil (500 St; Rhodorsil) is inje ted in inlet 1 at onstant �ow rate Q1 ≈ 3 mL hr , and aqueous phases are inje ted at �ow rates Q2 and Q3 ranging from 0 to about 1 mL hr , in inlets 2 and 3 respe tively. All liquids are inje ted with syringe pumps (PHD 2000 infu- sion; Harvard Apparatus). At the interse tion between the oil and the aqueous streams, monodisperse droplets of the aqueous phase in oil are ontinuously produ ed [21℄. Both the droplet volume (about 100�300 nL) and their produ tion frequen y (typi ally between one to ten droplets per se ond) an be tuned by the ratio of oil to aqueous phase �ow rates. The droplet omposition is monitored by the ratio Q2/Q3. Thanks to the possible opening and losing of ea h out- let, we an store droplets of given aqueous ompositions in the di�erent storage hannels i. Several steps are ne - essary to perform su h a �lling. First, all the hannels are initially �lled with sili one oil. Se ondly, the outlet of hannel 1 is opened and all the others are losed. In this on�guration, all the droplets of a given omposition �ow through 1. Finally, on e the �ow is stable, the outlet of 1 is suddenly losed and simultaneously, the outlet of hannel 2 is opened. All the droplets previously present in 1 stay immobilized whereas the other droplets, whose omposition an be hanged, �ow through 2. Su es- sively, in the same way, we an store droplets of various ompositions in all hannels i. C. Chemi al omposition ontrol For solubility investigations, the ontrol of the on- entrations in the droplets is ru ial. However, be ause of PDMS elasti ity and syringe pumps pre ision, an in- a ura y in droplet on entration remains. To estimate this error, we have performed investigations with a on- fo al Raman mi ros ope (HR800 Horiba; Jobin-Yvon). A 50× mi ros ope obje tive was used for fo using a 532 nm wavelength laser beam in the droplets, and for olle ting Raman s attered light, subsequently dispersed with a grating of 600 lines per millimeter. To minimize the out-of-fo us ba kground signals, we �xed the on- fo al pinhole at 500 µm. Experiments were performed on droplets made of two initial aqueous solutions of K4Fe(CN)6 (0.5 M) and K3Fe(CN)6 (0.5 M) inje ted in inlets 2 and 3 respe tively. These two ompounds display strong and distin t Raman signals [22℄. Figure 2 shows three Raman spe tra measured in droplets ontaining di�erent on entration ratios RC=[K4Fe(CN)6℄/[K3Fe(CN)6℄. The two bands entered 2000 2060 2095 2136 2200 wave number (cm−1) FIG. 2: Raman spe tra of droplets ontaining di�erent on- entration ratios RC of potassium ferro yanide K4Fe(CN)6 and potassium ferri yanide K3Fe(CN)6. (a) RC = 0 (b) RC = 1 ( ) RC = 9. at 2060 and 2095 m orrespond to K3Fe(CN)6 and the one at 2136 m orresponds to K4Fe(CN)6. The on entration of ea h ompound an be probed from the area under their spe i� Raman bands by: Ai = KiCitV , (1) where Ai is the area under the Raman band of the om- pound i, Ci its on entration, Ki a spe i� onstant, t the a quisition time, and V the analysis volume. As a onsequen e, the ratio RA of the Raman bands areas of K4Fe(CN)6 and K3Fe(CN)6 is proportional to the on- entrations ratio RC , and does not depend on the a qui- sition parameters. In order to optimize the �lling proto ol, we �rst use Ra- man mi ros opy to follow the kineti s of the on entra- tion stabilization in the droplets after a sudden hange in the aqueous phases �ow rates. Indeed, due to the PDMS elasti ity and the inje tion system (syringe pumps), the �nite response time of the devi e does not allow instan- taneous hange of the on entrations. To estimate this response time, we have performed the following experi- ment: for t < 0 s, Q2 = 0 and Q3 = 500 µL hr , and for t > 0 s, Q2 = Q3 = 250 µL hr . Droplets �rst �ow through hannel 1 whi h is losed after 30 s. Then, droplets are stored in �ve other hannels after 1, 2, 4, 6, and 10 min. Thus, Raman spe tra obtained from the droplets in the di�erent hannels enable us to follow the evolution of RA as a fun tion of time after the �ow rates hange. Figure 3(a) shows it rea hes almost a onstant value after 60 s meaning the on entrations be ome sta- ble after this time. Su h measurements illustrate that 20 min long proto ols are e� ient to store droplets of desired ompositions in the ten hannels (≈ 2 min per hannel). A se ond series of experiments was performed to es- timate and hara terize the on entration gradient we an apply in the devi e. The storage hannels are �lled with droplets of di�erent on entrations in K3Fe(CN)6 and K4Fe(CN)6 set from the �ow rates. In ea h hannel i, to rea h a stable droplets omposition, we maintain the �ow for 90 s before losing the outlet to store them [see Fig. 3(a)℄. By measuring the Raman spe tra of the droplets omposition in the di�erent hannels, we ob- tain the ratio RA as a fun tion of the theoreti al ratio = Q2/Q3. The error bar orresponds to the stan- dard deviation of the measurements performed on the droplets in a given hannel. As an be seen on Fig. 3(b), a linear relationship between RA and R is observed as expe ted. Deviations of a few per ents around the linear law are probably due to the Raman measurements un- ertainties, to the a ura y of the inje tion system, and also to the PDMS elasti ity. These Raman measurements demonstrate that with the developed proto ol, we are able to store hundreds of droplets in ten hannels in about 20 min, and on- suming less than a few hundreds of µL of solution. We believe that more rigid and smaller mi rodevi es om- bined with even more rea tive inje tion system would de rease signi antly the amount of liquids used when �ll- ing the hannels. Other strategies involving for instan e droplet generation thanks to integrated mi rovalves [23℄, may also proved to be useful to de rease the required volumes of solution. 0 1 2 3 4 0 60 200 400 600 t (s) FIG. 3: (a) Evolution of the ratio RA in the droplets after a sudden hange of the aqueous solutions �ow rates Q2 and Q3. Before t = 0 s, Q2 = 0 and Q3 = 500 µL hr . For t > 0 s, Q2 = Q3 = 250 µL hr . Between 5 and 30 s, RA are obtained from three single droplets in hannel 1. After t = 60 s, ea h point is a mean value al ulated on several droplets in a given hannel. (b) Con entration ratio RA in droplets as a fun tion of the on entration ratio R C determined from the aqueous solutions �ow rates. The dotted line orresponds to the linear �t of the data. D. Temperature ontrol The temperature �eld of the hip is ontrolled with two Peltier modules (30×30×3.3 mm ; CP1.4�71�06L; Mel- or) pla ed underneath the wafer at positions marked by the two dotted areas on Fig. 1(a). Sin e the two Peltier modules are independant, we an heat or ool the de- vi e, and also apply important temperature gradients. We use a sili on wafer as hip support to optimize ther- mal transfers and thus to reate regular temperature gra- dients along the storage hannels. Thin thermo ouples (type K, 76 µm o.d., 5SRTC-TTKI-40-1M; Omega) mea- sure the temperature of the devi e along three series of positions parallel to the storage hannels. The �rst series is pla ed above 1, the se ond one between 5 and 6, and the third one below 10 [see Fig. 1(a)℄. To rea h the maxi- mal pre ision on liquid temperature measurements inside the hannels, the thermo ouples are inserted in holes pre- viously pun hed through the PDMS layer and �lled with sili one oil. Thermo ouples signals are pro essed with a data a quisition instrument (USB�9161; National In- struments) and LabView software. Figure 4(a) shows we are able to apply easily temperature gradients up to C on 5 m. To estimate the temperature at any po- 0 10 20 30 40 X (mm) X (mm)C FIG. 4: Temperature pro�les of the hip for a given tem- perature gradient (a) Temperatures measured along the stor- age hannels with thermo ouples inserted through the PDMS layer at di�erent positions shown on Fig. 1(a). (N) measure- ments series above hannel 1; (�) series between 5 and 6; (H) series below 10. (b) Interpolated temperature pro�le of the hip. sitions along the storage hannels, we perform a longi- tudinal and transverse linear interpolation of the three series of measurements. The �nal pro�le obtained after su h interpolation is depi ted on Fig. 4(b). Note that the temperature is not perfe tly homogeneous transversely to the storage hannels. This is due to the size of the Peltier module as ompared to the size of the droplet storage area: smaller storage area, or larger Peltier mod- ules, would give homogeneous temperature pro�les along the transverse dire tion of the hannels. III. RESULTS In the previous se tion we have shown that our mi- rodevi e allows us to build a two-dimensional array of droplets with both on entration and temperature gra- dients. We now present an appli ation of this hip by measuring the solubility urve of an organi solute. Su h measurements are arried out with an adipi a id solution previously prepared in a beaker. It is made of 10.14 g of adipi a id (99%; Aldri h) in 50.66 g of deion- ized water. The solubility temperature of the solution is 63 C. To avoid any rystallization before the droplets formation, the syringe ontaining the solution and the orresponding tubing are heated at about 65 C with two �exible heaters (Min o) ontrolled with temperature on- trollers (Min o). A stereo mi ros ope (SZX12; Olympus) with an obje tive (DF PLFL 0.5× PF; Olympus) enables us to observe the devi e during the solubility study. We inje t the adipi a id solution in inlet 2 and deion- ized water in inlet 3. By hanging the �ow rates ratio we �ll the storage hannels with droplets whose on en- tration in adipi a id varies from 20 g / 100 g of water in hannel 1 down to 6 g / 100 g of water in 10. The massi on entration C in the droplets is al ulated a - ording to: 1 + (1 + C0)Q3/Q2 where C0 is the massi on entration of the initial adipi a id solution, Q2 and Q3 the respe tive �ow rates of the solution and water (we he ked that density variations in- du ed by the presen e of adipi a id are negligible). The mi ro�uidi hip is kept at about 65 C using the Peltier modules to avoid any rystallization during the droplet storage. Before stopping the droplets in a hannel, we maintain it open for 90 s for �ow stabilization. In these onditions, the total �lling of the ten hannels is rea hed in less than 20 min and only 250 µL of solution are spent. After the droplet storage, rystallization is indu ed by ooling. Note that the mean time of rystal nu leation is inversely proportional to the rea tor volume. Indeed, the nu leation frequen y is given by 1/JV where J is the nu leation rate that does no depend on the volume V of the rea tor (see Refs. [19, 24, 25, 26℄ and referen es therein). Crystal nu leation in a droplet of 100 nL is thus 10 times longer than in a vial of 1 mL. To redu e su h long indu tion time, we apply a strong ooling to in rease signi� antly the supersaturation. In our ase, down to ≈ −5◦C, rystals appear in all the droplets after a few minutes. To obtain the solubility urve dire tly on the hip, we then apply a temperature gradient between 32 and 65 after the rystallization step. Crystals dissolve in all the droplets whose temperature is higher than their solubil- ity temperature. In the other droplets, rystals are partly solubilized but still exist (the equilibrium is rea hed in about 20 min). Typi al images of the storage area are FIG. 5: Images of a part of the storage area obtained under rossed polarizers. Droplets of adipi a id solution are stored in the hannels. The on entration in adipi a id was grad- ually hanged between the upper and the bottom hannels. After rystallisation of all the droplets, a temperature gradi- ent is applied (low temperature on the left and high temper- ature on the right). (a) The dotted line separating droplets ontaining rystals from empty droplets give an estimation of the solubility limit. (b) Same image but with a di�erent ontrast displaying the droplets positions. presented on Fig. 5. Sin e adipi a id rystals have bire- fringent properties, they are easily dete ted under rossed polarizers. The smallest dete table rystals size is about 50×50 µm2 at the magni� ation used. Figure 5(a) en- ables us to dire tly observe the limit of rystal presen e. Using interpolated temperature pro�les su h as the one displayed in Fig. 4, allows us to estimate the solubility temperatures for all the ten on entrations (we hoose them in the middle of the two su essive droplets with and without rystals). Figure 6 presents su h solubility temperatures measured with our mi ro�uidi devi e. The error orresponds to the temperatures di�eren e between the two droplets en losing the solubility limit positions. These results are in good agreements with data obtained from literature [27℄. Naturally, the errors done on su h measurements de- pend on the distan e between two su essive droplets, and on the amplitude of the temperature gradient. In our ase, the temperature gradient of 0.7 a typi al distan e of 3 mm between two droplets give an error of ±1◦C. The appli ation of smaller temperature 40 45 50 55 60 T (°C) FIG. 6: (•) Solubility of adipi a id in water measured in the ase of a temperature gradient of 0.7 . (◦) Solubility data from literature, the dotted line is a guideline for eyes. gradients and the redu tion of the distan e between two su essive droplets would give a better a ura y on the solubility limit. For the moment, the maximal temperature whi h an be investigated is limited by the evaporation of water through the PDMS layer [28℄. Simple measurements show that the volume of an aqueous droplet stored in our devi e at 60 C, de reases by ≈ 10% in 4 hr. Su h an e�e t is negligible for the experiments des ribed above (droplet �lling time 20 min at 65 C), but may explain the small dis repan y observed on Fig. 6 at high temper- ature. We believe that the use of non-permeable materi- als su h as glass, instead of PDMS, ould easily broaden the possibilities o�ered by our system. IV. CONCLUSION In this work we have presented a new mi ro�uidi tool to perform rapid s reening of solubility diagrams. The devi e enables us to store hundreds of droplets (≈ 100 nL) of various hemi al ompositions in parallel mi ro hannels, and to apply large temperature gradients. We have demonstrated using a model system (adipi a id in water), that we ould easily and dire tly a ess to ten simultaneous measurements of the solubility urve on a large temperature range, in less than 1 hr, and with only 250 µL of solution. To on lude, we believe our devi e is a suitable tool for solubility diagrams s reen- ing, more rapid, with a better temperature ontrol, and heaper than lassi al roboti workstations. Su h a mi- ro�uidi tool may also be useful for many other appli a- tions, where two-dimensional s reening, temperature vs. omposition, is required. A knowledgments We gratefully thank G. Cristobal, J. Krishnamurti, J. Leng, and F. Sarrazin for fruitful dis ussions and riti- al reading of this manus ript. We also a knowledge Ré- gion Aquitaine for funding and support, and the Atelier Mé anique of the CRPP for their te hni al help. [1℄ G. H. W. Sanders and A. Manz, Trends Anal. Chem. 19, 364 (2000). [2℄ J. R. Luft, J. Wol�ey, I. Jurisi a, J. Glasgow, S. Fortier, and G. T. DeTitta, J. Cryst. Growth 232, 591 (2001). [3℄ D. L. Chen and R. F. Ismagilov, Curr. Opin. Chem. Biol. 10, 226 (2006). [4℄ H. A. Stone, A. D. Stroo k, and A. Ajdari, Annu. Rev. Fluid. Me h. 36, 381 (2004). [5℄ T. M. Squires and S. R. Quake, Rev. Mod. Phys. 77, 977 (2005). [6℄ T. Vilkner, D. Janasek, and A. Manz, Anal. Chem. 76, 3373 (2004). [7℄ N. Min , C. Futterer, K. D. Dorfman, A. Ban aud, C. Gosse, C. Goubault, and J. L. Viovy, Anal. Chem. 76, 3770 (2004). [8℄ J. Khandurina and A. Guttman, J. Chromatogr. A 943, 159 (2002). [9℄ M. Chabert, K. D. Dorfman, P. de Cremoux, J. Roer- aade, and J.-L. Viovy, Anal. Chem. 78, 7722 (2006). [10℄ A. D. Stroo k, S. K. Dertinger, A. Ajdari, I. Mezi , H. A. Stone, and G. M. 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0704.0570

Composite fermion wave functions as conformal field theory correlators

Composite fermion wave functions as conformal field theory correlators T.H. Hansson,1 C,-C. Chang,2 J.K. Jain,2 and S. Viefers3 1Department of Physics, Stockholm University, AlbaNova University Center, SE - 106 91 Stockholm, Sweden 2Physics Department, 104 Davey Lab, The Pennsylvania State University, University Park, Pennsylvania 16802 3 Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway (Dated: October 25, 2018) It is known that a subset of fractional quantum Hall wave functions has been expressed as confor- mal field theory (CFT) correlators, notably the Laughlin wave function at filling factor ν = 1/m (m odd) and its quasiholes, and the Pfaffian wave function at ν = 1/2 and its quasiholes. We develop a general scheme for constructing composite-fermion (CF) wave functions from conformal field theory. Quasiparticles at ν = 1/m are created by inserting anyonic vertex operators, P 1 (z), that replace a subset of the electron operators in the correlator. The one-quasiparticle wave function is identical to the corresponding CF wave function, and the two-quasiparticle wave function has correct fractional charge and statistics and is numerically almost identical to the corresponding CF wave function. We further show how to exactly represent the CF wavefunctions in the Jain series ν = s/(2sp+ 1) as the CFT correlators of a new type of fermionic vertex operators, Vp,n(z), constructed from n free compactified bosons; these operators provide the CFT representation of composite fermions carry- ing 2p flux quanta in the nth CF Landau level. We also construct the corresponding quasiparticle- and quasihole operators and argue that they have the expected fractional charge and statistics. For filling fractions 2/5 and 3/7 we show that the chiral CFTs that describe the bulk wave functions are identical to those given by Wen’s general classification of quantum Hall states in terms of K- matrices and l- and t-vectors, and we propose that to be generally true. Our results suggest a general procedure for constructing quasiparticle wave functions for other fractional Hall states, as well as for constructing ground states at filling fractions not contained in the principal Jain series. PACS numbers: 73.43.-f, 11.25.Hf I. INTRODUCTION The evidence for an intriguing connection between conformal field theory (CFT) and the fractional quantum Hall effect (FQHE) was accumulating in the 1980s. It was realized that the effective low-energy theory of the FQHE is a topological field theory of the Chern-Simons type, where the exchange phases of the anyonic quasiparticles and quasiholes are coded in the braiding properties of the corresponding Wilson loops1. Witten’s subsequent demonstration that the braiding of the Wilson loops are reflected in the correlation functions of certain CFTs2 suggested a CFT- FQHE relationship, which was further strengthened by Wen, who proposed that the gapless chiral edge modes of a FQH-droplet are described by a chiral 1 + 1 dimensional CFT3. It was also noticed that the holomorphic part of the Laughlin wave function takes the form of a correlator of bosonic exponents, or vertex operators, in a two dimensional CFT4,5. The 1991 paper by Moore and Read was particularly important since it synthesized many of these ideas and made an explicit conjecture about the CFT description of quantum Hall (QH) states containing two parts: 1. “Representative” electronic wave functions for the ground state and its quasiparticle and quasihole excitations are correlation functions, or, more precisely, conformal blocks, in a rational conformal field theory (RCFT) where the various particles correspond to different primary fields. 2. The very same RCFT describes the edge excitations of the corresponding FQH droplet. In their paper Moore and Read gave some striking circumstantial arguments to support their conjecture, and they also showed that many FQH states, namely the Laughlin state, the states in the Halperin-Haldane hierarchy, their quasihole excitations, the Halperin spin singlet state6, and the Haldane-Rezayi spin singlet pairing state7, may be represented in terms of conformal blocks. All this might have been criticized for being just a reformulation of old results, but Moore and Read also used the CFT formalism to propose a new ν = 1/2 state, the so-called Pfaffian wave function, which is tentatively assigned to the observed ν = 5/2 FQHE. The quasiholes in this state have charge q = 1/4 rather than q = 1/2 expected from the filling fraction, and exhibit non-Abelian fractional statistics. To establish the latter it was essential to use CFT technology.44 Despite this advance one and a half decades ago, the program of establishing a one-to-one correspondence between QH states and conformal field theory has remained incomplete. No explicit conformal field theory expressions have so far been established for many important FQHE states; in particular, despite interesting progress10, this is the case for the ground state wave functions of the prominent FQHE series ν = s/(2sp ± 1), and their related quasihole or quasiparticle excitations. (Expressions for the states in the Haldane-Halperin hierarchy were given in Ref. 4, but these are indirect, involving multiple integrals over auxillary quasihole coordinates.) Surprisingly, a proper conformal field theory representation does not exist even for the quasiparticles – as opposed to quasiholes – of the FQHE state at ν = 1/m and the Pfaffian wave function at ν = 1/2.45 It is worth reminding ourselves what we can hope to accomplish using CFT techniques: We cannot “derive” the FQHE wave functions, since the CFT does not contain any information about the actual interelectron interaction. It is true that the short distance behavior of the electronic wave functions is reflected in the operator product expansion of the pertinent CFT vertex operators, but only in the simplest cases can this be directly related to a potential of the Haldane-Kivelson-Trugman type. Thus we can only hope to get ”representative wave functions” in the sense of Moore and Read, and any new candidate wave function suggested by the CFT approach must be tested and confirmed against exact solutions of the Schrödinger equation known for small systems. The crucial question is if the CFT wave functions are sufficiently natural and simple to give new insight into the physics of the problem, facilitate computations of quantities like local charge and braiding statistics, and most importantly, inspire new generalizations. Finally, we should point out that we know of no general microscopic principle that requires that the correlated quantum mechanical wave functions of interacting electrons in the lowest Landau level should be expressible as simple correlation functions of certain vertex operators in a two dimensional Euclidean rational conformal field theory. An insight into the general FQHE states comes from the composite fermion (CF) formalism12,13. Here the exper- imentally prominent Jain states at ν = s/(2sp + 1) are formed from s filled Landau levels of “composite fermions,” which are electrons carrying 2p flux quanta. Other CF states, as e.g. the Pfaffian, which is the preferred candidate for the observed ν = 5/2 state, can be formed by various BCS type pairing mechanisms4,8. In the CF description, a quasihole is obtained simply by removing a composite fermion from an incompressible FQHE state, and a quasipar- ticle is a composite fermion in a higher, otherwise empty CF Landau level (LL). (CF Landau levels are also called Λ levels.) Explicit wave functions are constructed for all ground states and their quasiparticle and quasihole excitations. (The asymmetry between quasiparticles and quasiholes occurs since they reside in different CF Landau levels.) The CF approach is very successful, both in comparison with experiments and with numerical studies of two-dimensional electron gases in strong magnetic fields13. The issue of fractional charge and fractional statistics of the composite fermions is a subtle one. The quasiparticles and quasiholes are composite fermions added to or removed from a CF Landau level. From one perspective, they have unit charge and fermionic statistics. Indeed, the addition of one composite fermion increases the number of electrons, and hence the net charge, by one unit, and the fermionic statistics of composite fermions has been confirmed by numerous experiments (e.g. the observation of their Fermi sea). On the other hand, the CF quasiparticles and quasiholes have a fractional “local charge” (where the local charge is the charge measured relative to the background FQHE state) and a fractional braiding statistics13,14,15,16. These properties capture the physics that adding or removing a composite fermion causes nonlocal changes in the state, because the vortex, a constituent of the composite fermion, is a nonlocal object. This should be contrasted with the analogous process in the integral QHE, which is essentially local (the Landau level projection destroys locality only on the scale of the magnetic length `), and can be described by a local, charge-e operator ψ†α(~x), where the subscript denotes the Landau level index. No such local operator can be constructed for the creation of a composite fermion, since the local charge of the quasiparticle differs from that of the electron. The fractional statistics of the quasiparticles also implies that they cannot be described by local operators, as emphasized by Fröhlich and Marchetti17. Even though fractional charge and fractional statistics cannot be read off directly from the CF wave functions, they nonetheless contain that information, not surprising in view of the fact that the CF construction provides a good description of all the low energy states. We mention here the quasiparticles at ν = 1/m, for which the CF wave function differs from that proposed earlier by Laughlin18. The calculation of the Berry phase associated with two-CF quasiparticle exchange, originally performed by Kjønsberg and Leinaas15 and subsequently by Jeon and collaborators16, shows that the braiding statistics for the CF quasiparticles has a sharply defined fractional value; for the Laughlin quasiparticles, in contrast, numerical calculations do not produce a convergent result for the statistical angle19. In this paper we establish a firm connection between CF wave functions and CFT correlators. Specifically: 1. We construct the quasiparticles of ν = 1/m (m odd) using a new kind of anyonic vertex operators P 1 . For a single quasiparticle, the resulting wave function is identical to that obtained using the CF theory. A generaliza- tion to two or more quasiparticles produces wave functions that are very similar to the CF wave functions but not identical. For two quasiparticles at ν = 1/3, the overlap between the two wave functions is typically 99.99% for as many as 40 electrons. 2. We show that the ground state wave functions in the Jain series ν = n/(2np+ 1) are exactly given by sums of CFT correlators of a set of vertex operators, Vnp, which in the CF language correspond to creating composite fermions in higher CF Landau levels. 3. We generalize the construction of the quasiparticle operator P 1 , as well as of the quasihole operators, to higher levels in the Jain sequence; at level n, there are n independent hole operators and one quasiparticle operator. The vertex operator Vn,p at level n is closely related to the quasiparticle operator at level n− 1. 4. We demonstrate that the very CFT that yields the CF wave functions also directly defines an edge theory for the Jain states that is precisely the one expected from the general arguments given by Wen3. Our CFT construction has many advantages. (i) At the technical level, it produces accurate wave functions directly in the lowest Landau level with no need for projection, and the charge and statistics of the quasiparticles are revealed in the algebraic properties of the corresponding operators, just as in the case of the quasiholes of the ν = 1/m states. (ii) Although the effective edge theory for the Jain states was known from general principles, we provide a direct derivation from a CFT where the conformal blocks yield microscopically accurate bulk wave functions. (iii) It gives a new insight and suggests new extensions; a generalization of this work produces natural ansätze for quasiparticle wave functions for more complicated CF states such as the Moore-Read Pfaffian state, as well as for ground states at fractions (e.g., 4/11), which do not belong to the principal Jain series. The paper is organized as follows. In the next section we explain the basic ideas behind our construction and give explicit wave functions for one- and two-quasiparticles, as well as that for a quasiparticle-quasihole pair. The general structure of the CFT description of the states in the Jain series is discussed in section III, while the detailed technical proof for the equivalence between the CF and the CFT wave functions is left for Appendix B. In section IV we explain the construction of the edge theory, and in section V we construct localized quasiparticle states and show how to extract charge and statistics from the relevant Berry phases; the latter can be ascertained analytically if we make a random phase assumption. Some details of the calculations are found in Appendix C. Section V presents numerical calculations supporting our claims in sections II and V and, finally, a summary is found in section VII. A short report on parts of this work has been published previously20. II. ONE AND TWO QUASIPARTICLES IN THE LAUGHLIN STATE A. The ground state and the quasihole states We first review some of the basic formalism of the CFT construction of QHE wave functions, in particular the construction of the ground state and quasihole wave functions at the Laughlin fractions ν = 1/m, where m is an odd integer. Following Moore and Read4, we introduce the normal-ordered vertex operators, V1(z) = : e mϕ1(z) : (1) (η) = : e ϕ1(η) : , (2) where the normal ordering symbol : :, will be suppressed in the following. The free massless boson field, ϕ1, is normalized so as to have the (holomorphic) two point function 〈ϕ1(z)ϕ1(w)〉 = − ln(z − w) , (3) so that the the vertex operators obey the relations eiαϕ1(z)eiβϕ1(w) = eiπαβeiβϕ1(w)eiαϕ1(z) = (z − w)αβeiαϕ1(z)+iβϕ1(w) ∼ (z − w)αβei(α+β)ϕ1(w) (4) where the last line expresses the operator product expansion (OPE) in the limit z → w. From (4) follows V1(z)V1(w)+ V1(w)V1(z) = 0, and H 1 (z)H 1 (w) − eiπ/mH 1 (w)H 1 (z) = 0. The first of these reflects that the electrons are fermions, while the second is appropriate for fractional statistics as discussed in reference [4]. We normalize the (holomorphic) U(1) charge density operator as J(z) = ∂zϕ1(z) (5) so the corresponding charge is given by dz ∂zϕ1(z), (6) where the contour encircles the whole system. The U(1) charges, Q = 1 of the electron and Q = 1/m of the quasihole, can be read directly from the commutators [Q, V1(z)] = V1(z) and [Q, H 1 (η)] = 1 (η). It is noted that Q does not give the electric charge; rather it has the interpretation of vorticity as seen from (4). Introducing a positive vorticity in a homogenous state corresponds to a local depletion of the electron liquid, while a negative vorticity amounts to a local increase in density. Thus the excess electron number compared with the ground state created by an operator with U(1) charge Q is given by ∆n = δn−Q, (7) where the integer δn is the number of electrons added by the operator. If the argument of the operator is an electron coordinate, zi, one electron is added, while no electron is added if the argument is a quasihole coordinate ηi. (The idea of binding of an electron and m vortices was implicit in Laughlin’s original work, and was made explicit by Halperin6, Girvin and MacDonald21 and Read22.) The total electric charge of a particle is given by Qel = −e∆n = e(Q− δn). Note that the excess charge associated with the addition of an electron is zero, as expected, because this expands the droplet without creating any local charge variation. The (un-normalized) ν = 1/m Laughlin wave function can now be written as (for notational convenience, we write Ψ(zi) instead of Ψ({zi})): ΨL(zi) = 〈0|R{V1(z1)V1(z2) . . . V1(zN−1)V1(zN )e−i d2z ϕ1(z)}|0〉 (8) ≡ 〈V1(z1)V1(z2) . . . V1(zN−1)V1(zN )〉1/m (zi − zj)me− |zi|2/4`2 , where R denotes radial ordering. The second line defines the average 〈. . . 〉1/m, and the third follows for the ordering |z1| ≥ |z2| ≥ . . . |zN |, which will be assumed below unless indicated otherwise. In the following, we shall suppress the subscript 1/m whenever it is clear to what ground state we are referring. The exponential operator in (8) corresponds to a constant background particle density, ρm = −ρ0/m, where ρ0 = 1/2π`2 is the density of a filled Landau level. This is necessary since the U(1) charge neutrality condition, known from the Coulomb gas formulation, in the CFT ensures that the correlator vanishes unless N = ρm d2z = ρmA, which defines the area, A, of the system. As explained in reference [4], the background charge will produce the correct gaussian factor e− |zi|2/4`2 characteristic of the lowest Landau level wave function. For a more detailed discussion of this background charge prescription, see Appendix A. The wave function for a collection of Laughlin quasiholes is also easily written: ΨL(η1 . . . ηn; zi) = 〈H 1 (η1)H 1 (η2) . . . H 1 (ηn)V1(z1)V1(z2) . . . V1(zN−1)V1(zN )〉. (9) In this case the charge neutrality condition reads N + n/m = ρmA′, indicating an expansion of the droplet. From the general relation (4) we get H 1 (z)V1(w) + V1(w)H 1 (z) = 0 which guarantees that (9) is uniquely defined and analytic in the electron coordinates. Very little of the rather sophisticated mathematics of CFT will be used in this paper, but a few formal comments are in order. A CFT is in general not defined by a Lagrangian, but by an operator product algebra, or set of fusion rules, together with a specification of the field content defined by the so-called primary fields. The CFTs of interest here are defined by a Lagrangian describing a collection of free bosons, ϕi, compactified on circles of radius Ri = where mi are odd integers. The primary fields are given by the chiral vertex operators V (z) = e ϕi(z) where the integers qi define the charge lattice describing the possible “electric” charges in the Coulomb gas formulation of the CFT. The vertex operators satisfy an extended chiral algebra that, together with the charge lattice, defines the relevant CFT, which in this case is called a “rational torus” with radii mi; this is an example of a rational CFT. Acting on the primary fields with the generators of the conformal group gives families of “descendant fields”, which can be expressed using derivatives of the parent primary fields. Such descendant fields will be important in the construction of quasiparticle operators presented in the next section. The full CFT contains fields of both chiralities and has correlation functions that can be written as (in general a sum over) products of holomorphic and anti-holomorphic factors, so-called conformal blocks. The holomorphic blocks are precisely the correlation functions of chiral vertex operators that we have identified with the electronic wave functions. In general, these blocks also depend parametrically on quasiparticle and quasihole coordinates, and acquire nontrivial phase factors, called monodromies, when these coordinates are transported along closed loops. It is these monodromies that reproduce the braiding phases that also can be calculated from the expectation values of Wilson loops in a Chern-Simons theory. A detailed discussion of the conditions that a CFT has to fulfill in order to describe a QH state can be found in Ref. 23. B. One quasiparticle The most immediate guess4 for a quasiparticle operator would be to simply change the sign in the exponent in the quasihole operator of (2), i.e. to use e− ϕ1(η). That, however, introduces unacceptable singular terms ∼ i(zi−η) in the electronic wave function. Inspired by the CF wave functions, we instead define a quasiparticle operator, P 1 which has a U(1) charge (1 − 1/m), and that will replace one of the the original electron operators V1(z). We can thus think of P (z) as a modified electron operator, but with a different amount of vorticity. The excess electric charge associated with such a modification is the difference between the charges of the operators V1 and P 1 ∆Qel = e((1− 1/m)− 1) = −e/m, as appropriate for a quasiparticle at ν = 1/m. The modified electron operator is given by (z) = ∂ei( m− 1√ )ϕ1(z), (10) and the wave function for a single quasiparticle with angular momentum l is written as Ψ(l)1qp(zi) = A{z −|z1|2/4m`2〈P 1 (z1)V1(z2) . . . . . . V1(zN )〉} (11) (−1)i+1zli e −|zi|2/4m`2〈P 1 j 6=i V1(zj)〉 (−1)izli (zj − zk)m∂i l 6=i (zl − zi)m−1, where A denotes anti-symmetrization of the coordinates. The second line follows by noting that the anti-symmetrized product has the form of a Slater determinant which is then expanded by the first row. From (4) we get P 1 (z)V1(w)− V1(w)P 1 (z) = 0, so the radial reordering of the quasiparticle operator does not give rise to any sign change. The anti-symmetrization with respect to the remaining coordinates is trivial since V1(z)V1(w) + V1(w)V1(z) = 0. The charge neutrality condition now reads N − 1 + (1− 1 ) = ρmA′′, so the droplet has undergone a small contraction, as expected for a quasiparticle. While the exponent of (10) follows naturally from the above charge requirement (and may be viewed as a combination of an electron operator and an “inverse” quasihole operator), the derivative has been put in “by hand”. Without the derivative, the wave function (11) can be shown to be identically zero. Technically, P 1 (z) is a descendant of the primary field, ei( m− 1√ )ϕ(z), a construction that naturally generalizes to more complicated QH states24. Note that the derivative in (11) acts only on the holomorphic part of the wave function.46 The quasiparticle wave function of (11) has a different character than those written above for the ground and the quasihole states, in that it is a sum over correlators, and that it involves prefactors f1(zi) = zlie −|zi|2/4m`2 . The factor zli sets the angular momentum, while the exponential factor is chosen to give the correct lowest Landau level (LLL) electronic wave function: Due to its modified charge, the quasiparticle operator P 1 (zi) gives rise to an exponential factor exp(−|zi|2(1− 1/m)/4`2), and the compensating prefactor ensures that the overall gaussian factor is exp{− j |zj | 2/(4`2)}. Here and in the following, we suppress exponential factors of the correlators whenever convenient, but fully display all prefactors for clarity. It is suggestive that the prefactors f1 precisely constitute the angular momentum l wave function ψl(z) = zle−|z| 2/4m`2 for a charge e/m particle in the LLL. Although we have no formal derivation of this, we find below a similar interpretation in the case of several quasiparticles, where their anyonic nature is also manifest. As pointed out previously, the quasiparticle wave function above is obtained by modifying one of the electron operators, rather than inserting a new operator. This is very suggestive of the CF picture of a quasiparticle as an excitation of a composite fermion to a higher CF Landau level. In fact, what originally led us to construct the operator P 1 was the observation that the wave function (11) is identical to the corresponding CF wave function (Eq. 5 of ref. 25), which is known to have a good variational energy and the correct fractional charge. In spite of this identity, however, there are two differences between the present derivation and the CF construction that deserve to be noted: First, the present formalism is entirely within the lowest Landau level. The CF construction of wave functions, on the other hand, involves placing composite fermions in higher CF Landau levels and subsequently projecting onto the LLL by replacing all z̄:s by derivatives in the resulting polynomial. Technically, of course, when deriving the one-quasiparticle wave function, the derivatives in (11) enter in the exact same places as those due to projection in the CF construction – but no projection is needed in the present formalism26. We return to this point in section IV, where we construct the ground states of the Jain sequences at ν = n/(2np + 1).47 Second, in spite of the close relation to composite fermions, the operator P 1 (z) is not fermionic, as can be seen from the commutation relation (z)P 1 (w)− eiπ(m−2+1/m)P 1 (w)P 1 (z) = 0 or the OPE P 1 (z)P 1 (w) ∼ (z −w)m−4+ 2(m−1)√ ϕ1(w), that follow from (4). The precise connection to composite fermions will be discussed in the section on the ν = 2/5 state below. Although the fractional exponent 1/m suggests fractional statistics, one cannot directly read the statistical angle from the two-point function. This issue is discussed in more detail in section V. C. Two or more quasiparticles Based on the experience with the single quasiparticle case, we expect the wave function for M quasiparticles to be of the form Ψ(l)Mqp(zi) = A{fM (z1 . . . zM )〈P 1m (z1) . . . P 1m (zM )V1(zM+1) . . . . . . V1(zN )〉}. (12) The form of fM is determined by the condition that the final electronic wave function be analytic and antisymmetric, with limiting behavior ∼ (zp−zq)m−1+lpq , with the relative angular momenta lpq ≥ 1 and odd. Because the correlator gives non-analytic factors of the type ∂p∂q(zp − zq)m−2+1/m from all contractions among quasiparticle operators, we choose fM (z1 . . . zM ) = g(Z) (zp − zq)1+lpq−1/me− i |zi| 2/4m`2 , where Z = 1 i=1 zi is the center of mass coordinate. Again, the exponential factors are included to give the correct gaussian factor exp[− j |zj | 2/(4`2)] in the N -electron wave function. As anticipated in the case of one quasiparticle, fM is just the LLL wave function of M anyons with fractional charge e/m. To cast (12) in a form suitable for computation, we will use the following formula, which generalizes the expansion by a row used in (11) above: (zp − zq)1−1/m+lpqP 1 (z1) . . . P 1 (zM )V1(zM+1) . . . . . . V1(zN )} (13) p=1 ipR{ (zip − ziq ) 1−1/m+lpqP 1 (zi1) . . . P 1 (ziM )V1(zīM+1) . . . V1(zīN )}, where the sum is over all subsets {i1 . . . iM} of M of the N integers, and {̄i1 . . . īM} is the conjugate subset of N −M integers. The proof is found in Appendix B 1. Using this result, the wave functions for two quasiparticles with total angular momentum L and relative angular momentum l can be written as Ψ2qp(zi) = = (−1)i+jZLij(zi − zj) 1+l− 1 (|zi|2+|zj |2)〈P 1 (zi)P 1 k 6=i,j V1(zk)〉, (14) where Zij = (zi + zj)/2. Evaluating the correlator we obtain the following explicit form for the wave function for two quasiparticles with relative angular momentum l and center of mass angular momentum L, Ψl,L2qp(zi) = (−1)i+jZLij(zi − zj) 1+l− 1 m ∂zi∂zj (zi − zj) m−2+ 1 m (15) (ij)(zk − zi)m−1 (ij)(zl − zj)m−1 (ij)∏ (zm − zn)m, where the derivatives act on the whole expression to their right, and (ij) = k 6=i,j ∏(kl) i<j = i,j 6=k,l The corresponding wave function in the CF approach is given by25 Ψ̃l,L2qp(zi) = (−1)i+jZLij(zi − zj) l∂zi∂zj (zi − zj) m−1 (16) (ij)(zk − zi)m−1 (ij)(zl − zj)m−1 (ij)∏ (zm − zn)m . The two wave functions differ by terms wherein the derivatives in (16) act on the factor (zi − zj)1− m . It is known25 that the CF wave function in (16) gives the correct fractional charge and statistics of the two-quasiparticle state. The first non-trivial test of our construction is therefore to check whether the CFT wave function (14) shares these good charge and statistics properties. This is indeed the case, as demonstrated by our numerical simulations, which are summarized in section V below. These results show that the two wave functions are essentially identical (for example, their overlap is 99.96% for 50 particles). This can be understood from the following heuristic arguments: First, since the derivatives in (16) act on a function which is a polynomial of order N in both zi and zj , this will generate O(N2) terms. It is unlikely that the few terms picked up by acting on the first factor will be significant. Secondly, these terms are sub-leading in the coordinate difference (zi − zj) between the quasiparticles, and thus unlikely to affect qualitative properties. D. Quasiparticles and quasiholes Wave functions for pairs of quasiparticles and quasiholes can be constructed by inserting pairs of the corresponding operators into the CFT correlator for the Laughlin ground state. The simplest case is a quasiparticle at the origin together with a quasihole at position η, given by Ψqp−qh(zi, η) = A{e−|z1| 2/4m`2〈P 1 (z1)V1(z2) . . . . . . V1(zN )H 1 (η)〉} (17) (−1)i+1 e−|zi| 2/4m`2〈P 1 j 6=i V1(zj)H 1 (−1)i (zj − zk)m j 6=i (zj − η) ∂i l 6=i (zl − zi)m−1 (zi − η)1− where the antisymmetrization acts on the electron coordinates zi only. More generally, a quasiparticle localized at some position η′ away from the origin may be constructed as a coherent superposition of the angular momentum states given in (11). For states with equally many quasiparticles and quasiholes, the background charge does not have to be changed from its ground state value. In this sense, wave functions of this type are the natural low energy bulk excitations that do not require any compensating edge charge. On a closed surface, no fractionally charged states are allowed. III. COMPOSITE FERMION STATES IN THE JAIN SERIES A. The ν = 2/5 composite fermion ground state In the composite fermion picture, the ground state wave functions at fillings ν = n/(2np+ 1) are constructed as n filled Landau levels of composite fermions with 2p flux quanta attached. In particular, the ν = 2/5 state corresponds to filling the lowest two CF Landau levels. This state may thus be viewed as a “compact” state of N/2 quasiparticles, i.e. the CF:s in the second Landau level are in the lowest possible total angular momentum state. To explore the connection to our CFT construction, we generalize the two-quasiparticle wave function (15) of the 1/m state to the M -quasiparticle case, with M = N/2, and consider a maximum density circular droplet obtained by putting all the quasiparticle pairs in their lowest allowed relative angular momentum (` = 1), and with zero angular momentum for the center of mass (L = 0). For simplicity we shall also take m = 3 (and suppress the subscript m on the operators) since the generalization to arbitrary odd m is obvious. Using (13) and evaluating the correlators, the wave function for M quasiparticles reads ΨMqp(zi) = i1<i2<···<iM (zik − zil) 3 ∂zi1∂zi2 . . . ∂ziM k′<l′ − zi′ 3 (18) (i2,i3...iM )(zk1 − zi1) (i1,i3...iM )(zk2 − zi2) 2 . . . (i1,i2...iM )(zkM − ziM ) (i1,i2...iM )∏ (zm − zn)3 . Since the anyonic wave function on the first line has the form of a Jastrow factor, it is natural to introduce a second free bosonic field ϕ2(z). In fact, by defining Ṽ (z) = ei 3ϕ2(z)∂e ϕ1(z) , (19) we find that (18) may be written in the following compact form ΨMqp(zi) = A{〈 Ṽ (zi) j=M+1 V1(zj)〉} (20) i.e. as a sum of correlators of M Ṽ :s and (N −M) V1:s. Again, this expression differs from the corresponding CF wave function only in the ordering of the derivatives and the Jastrow factors in the first line of (18). Indeed, as demonstrated in Appendix B, the CF wave function is obtained simply by moving all the derivatives all the way to the left. Let us therefore define V2(z) = ∂e ϕ1(z)ei 3ϕ2(z) , (21) where the derivative now acts on both the exponentials, and consider the case N = 2M . We then find that the following sum of correlators of M V2:s and M V1:s: ΨCF2/5(zi) = A{〈 V2(zi) j=M+1 V1(zj)〉} (22) i1<i2...iM ī1<ī2...̄iM k ik〈V2(zi1) . . . V2(ziM )V1(zī1) . . . V1(zīM )〉 exactly reproduces the (N = 2M)-electron CF wavefunction for ν = 2/5. The operators V2(zi), as opposed to the P (zi):s, are real fermionic operators in that they anticommute among themselves, but commute with V1(zi):s, just as the P (zi):s. Note that the form of V2 was determined entirely from the form of the maximum density M -quasiparticle wave function, so its fermionic nature was not an input. If we want to interpret V2 as a composite electron operator, it should have the same charge as V1. This is ensured if we redefine the charge density operator as J(z) = ∂ϕ1(z) + ∂ϕ2(z). (23) This construction may seem ad hoc in the sense that we fix the coefficient of ϕ2 by hand so as to obtain the correct charge. However, we shall see below that this choice is consistent, in that it produces the correct charge for the quasiholes in the ν = 2/5 state. Fulfillment of the charge neutrality condition for the vertex operators V2 requires a background charge, which for the maximum density circular droplet can be assumed to be constant. Furthermore, this density must reproduce the correct exponential factor for electrons in the LLL. The latter is achieved by redefining the expectation value as 〈· · · 〉2/5 ≡ 〈0| . . . e−i 15ρ̃3 d2z ϕ2(z)e−i d2z ϕ1(z)|0〉, (24) where ρ̃3 = (1/15)ρ0, so the total background electron density is (1/3 + 1/15)ρ0 = (2/5)ρ0. We stress that this value is not an input, but follows from demanding that V2 describe unit charge particles in the LLL, which was what led us to the above form (23) of the charge density operator. We now show that this state is indeed homogeneous, i.e. that the droplets formed by the N/2 = M V1:s and the M V2:s have the same area. Charge neutrality gives the following conditions on the areas A and à integrated over in (24), 3ρ3A (25)√ 15ρ̃3à , which implies A = à and thus homogeneity. From the perspective of composite fermions, this correponds to two filled CF Landau levels, since the degeneracy is the same in all Landau levels. It would be interesting to redo the above construction on a closed manifold, where we would expect the concept of “filled CF Landau level” to emerge in a natural way from the condition that the correlators do not vanish. Although it is possible to write general many-quasiparticle wave functions similar to the two particle wave function in (14), it is only the maximum density droplet of (18), and more generally the ”compact” CF states13, that allow for a simple expression in terms of conformal blocks as in (20); for general relative angular momenta one still has to explicitly put in compensating (anyonic) wave functions by hand. In this general case, there is also no reason for introducing a constant background charge different from that of the “parent” ν = 1/3, so there is no natural way to obtain non-zero correlators even if we were to introduce the field ϕ2(z). As we see below, this would also be in conflict with the known properties of the charge 1/3 quasiholes. B. The quasihole operators To create quasiholes in the 2/5 state, the operator H 1 (η) of (2) is no longer appropriate since it does not give holomorphic electron wave functions, as is seen from, e.g. , 〈V2(z)H1/3(η)〉 ∼ (z − η)2/3. Instead, it is necessary to include the second Bose field, ϕ2, and construct quasihole operators of the form Hpq(η) = e ϕ1(η)+i ϕ2(η). The coefficients p and q are determined from the requirements that (i) the wave function of any single quasihole be holomorphic, i.e. the power of the correlator between any quasihole operator and V1(z) or V2(z) be a non-negative integer, and (ii) the resulting hole operator not be expressible as a combination (product) of the other quasihole or vertex operators. These conditions uniquely determine the allowed coefficients p and q, and lead to the following two fundamental quasihole operators for the ν = 2/5 state: H01 = e ϕ2(η) (26) H10 = e ϕ1(η)− 2i√15ϕ2(η) . Using the charge operator corresponding to the charge density (23) one verifies that both these operators create quasiholes with charge 1/5. Note that this charge assignment is a prediction of our scheme, rather than an input, since the form of the charge operator (23) was determined independently from demanding V2 to have unit charge. All other allowed vertex operators can be constructed as products of H01(η) and H10(η); the operators in (26) span the charge lattice. It is an easy exercise to construct the explicit electron wave functions obtained by inserting the operators (26) in the correlator (22). Not surprisingly, a direct correspondence with the composite fermion picture is again revealed: Inserting the operator H10(η) (with η = 0 for simplicity) into the ν = 2/5 ground state (22) exactly gives the wave function of a quasihole in the center of the lowest CF Landau level, while H01 gives a quasihole in the second CF Landau level. Taking the product of the two quasihole operators, one obtains a charge-2/5 operator which, in the CF language, reproduces the wave function of a vortex, i.e. (for η = 0) two quasiholes at the origin, one in each CF-Landau level.13 Section V clarifies the relation between these quasihole operators and Wen’s effective bulk and edge theories for the ν = 2/5 quantum Hall state. If we would attempt to use the operators V1 and V2 to describe a 1/3 state with a small number of quasiparticles (e.g. by putting a compensating charge at the edge or at infinity by hand), we would be forced to use the operators (26) for the quasiholes and thus be led either to a wrong charge assignment for the quasiholes or to redefine the charge operator as to make the V2:s carry fractional charge. This again stresses that the form of the charge operator as well as the various vertex operators is intimately tied to the particular ground state under consideration. C. The quasiparticle operator The quasiparticle operator of the ν = 2/5 state is constructed in the same spirit as P 1 given in (10), i.e. as a combination of an “inverse” quasihole operator and one of the electron operators, combined with an appropriate number of derivatives. Since in the 2/5 state there are two independent hole operators (H01 and H10 in (26)) and two electron operators (V1 and V2), it superficially looks as if as there are four quasiparticle candidates. However, it can be shown24 that three of these are excluded as they do not produce non-zero wave functions, and one is left with P2/5(z) = ∂ ϕ1(z)+ ϕ2(z) (27) which corresponds to combining H01 (a quasihole in the second CF Landau level) with V2 (a composite fermion in the second CF Landau level). Again, the two derivatives are necessary in order to produce a non-zero wave function Ψ1qp(zi) = A〈P2/5(z1) V2(zi) 2M+1∏ j=M+2 V1(zj)〉, (28) and (28) is identical to the corresponding CF wave function. Note that, given the connection to composite fermions, it is very natural to have two different quasihole operators but only one quasiparticle operator: There are two filled CF LLs in which to create quasiholes, but the only way (except for higher excitations) to create a quasiparticle is to put one composite fermion in the third CF Landau level. D. The ν = 3/7 state and the Jain series As a final explicit example, let us construct the ground state and quasiholes of the ν = 3/7 state, i.e. the third level of the ν = s/(2s+ 1) Jain sequence. The generalization to the full Jain series is given in Appendix B 3 . The 3/7 state is obtained from a correlator containing an equal number of V1:s, V2:s and the new operator V3: V3(z) = P2/5(z)e ϕ3(z) = ∂2ei[ ϕ1(z)+ ϕ2(z)+ ϕ3(z)] (29) and again, the result is precisely the ν = 3/7 CF wave function (see appendix B 3). The relevant charge density operator, which ensures unit charge of V3, is given by J(z) = ∂ϕ1(z) + ∂ϕ2(z) + ∂ϕ3(z). (30) It is easy to check that V3(z) is fermionic, but commutes with both V1 and V2, and that the wave function written in analogy with (22) has filling fraction ν = 3/7. In the language of composite fermions, this corresponds to filling up three CF Landau levels. In analogy with the 2/5 state, one finds three independent charge-1/7 quasihole operators, which exactly correspond to quasiholes in the third, second, and first CF Landau levels, respectively: H001(η) = e ϕ3(η) H010(η) = e ϕ2(η)− 2√35ϕ3(η) H100(η) = e ϕ1(η)− 2√15ϕ2(η)− ϕ3(η) Operators for excitations with higher charge are obtained as products of these; for example, the product of all three is a charge-3/7 vortex. Again, it is straightforward to check that the operators (31) span the charge lattice. In direct generalization of the ν = 2/5 case, the ν = 3/7 quasiparticle operator is given by a combination of the inverse hole operator in the highest occupied CF Landau level, i.e. H001, and V3, with one additional derivative, P3/7(z) = ∂ i[ 2√ ϕ1(z)+ ϕ2(z)+ ϕ3(z)] . (32) The pattern for construction of higher level operators in the ν = s/(2s + 1) series should now be obvious, and in Appendix B 3 we give the general expressions for the operators Vpn describing the electrons at the nth level in the n/(2np+ 1) series, as well as the corresponding current density operator. The proof that the CF wave functions for n filled CF Landau levels are reproduced by sums of correlators with an equal number of Vpn:s (for fixed p) is outlined in Appendix B 3. The construction of the pertinent quasihole operators should be straightforward, although we have not derived the explicit formulae beyond the ones given above. From the general expressions of the operators, it is easy to see that two operators Vpn(zi) and Vpn(zj) at the same level give a factor (zi − zj)2p+1 in the correlation function, while two operators Vp,n1(zi) and Vp,n2(zj) at different levels produce a factor (zi − zj)2p (see appendix B 3). This gives an alternative way to calculate the filling fraction, and also demonstrates that the limiting value for n→∞ is ν = 1/2p. IV. CONNECTION TO EFFECTIVE CHERN-SIMONS THEORIES AND EDGE STATES Wen has developed a general effective theory formalism for the QH liquids based on representing the currents by two dimensional gauge fields aIµ with a Chern-Simons action3, L = − KII′aIµ∂νaI′λ ε µνλ − Aµ∂νtIaIλε µνλ , (33) where the matrix K and the “charge vector” tT = (t1 . . . tp) have integer elements. The filling fraction is given by ν = tTK−1t. A generic quasiparticle carries integral charges of the aIµ field, and is thus labeled by p integers constituting the vector l = (l1 . . . lp). The electric charge and the statistics of the quasiparticle are given by q = −etTK−1l and θ = πlTK−1l, respectively. This description is not unique; as explained in reference [3], an equivalent description is given by (K ′, t′, l′) = (WKWT ,W t,W l) where W is an element of SL(p, Z), i.e. an integer valued p× p matrix with unit determinant. As an example of the above, the ν = 2/5 state is described by the K matrix and t vector, K2/5 = tT = (1, 1) . (34) This is an example of what Wen refers to as the symmetric basis, where in general tT = (1, 1, . . . , 1). By an SL(2, Z) transformation, we can represent the same state in the “hierarchy basis” (which naturally occurs when constructing states in the Halperin-Haldane hierarchy) characterized by tT = (1, 0, . . . , 0). K ′2/5 = WKW ; t′T = tTWT = (1, 0) ; W = , (35) Starting from the Chern-Simons theory (33) defined on a finite two dimensional domain, one can derive a dynamical theory for the edge excitations. The details can be found in [3] and references therein, and the resulting theory is Sed = dtdx [KIJ∂tφI∂xφJ − VIJ∂xφI∂xφJ + 2eAµ�µν∂νtIφI ], (36) where K and t, as well as the quasiparticle vector l, are the same as in the effective bulk theory (33). This is a multicomponent chiral boson theory with the current operator given by Jµ = − �µνtI∂νφI . (37) The quasiparticle operators (including the electron operator) take the generic form Ψ ∼ ei q lqφq , (38) familiar from abelian bosonization of one-dimensional fermion systems. The numbers VIJ are the non-univeral edge velocities, which depend on the details of the confining potential. In their original paper on the connection between QH liquids and conformal field theories, Moore and Read made two basic claims. The first, which we already have discussed, is that the electronic wave functions can be expressed as conformal blocks of certain CFT:s. The second is that this very same CFT is the one dimensional theory describing the dynamical edge excitations. This last claim should not be taken literally since it is known that the edge dynamics is non-universal. Not only the edge velocities, but also the character, and even the number of edge modes can depend on details of the edge potential. Examples are the polarization edge modes related to edge spin texture27 and the counter-propagating modes resulting from edge reconstruction as first discussed by Shamon and Wen28. Thus we can only hope that the CFT will provide a “minimal” edge theory consistent with the topological properties of the bulk, i.e. that it supports excitations with the same charges. In spite of these limitations, the Moore-Read conjecture about the edge theory has been very fruitful, especially in the search for effective field theories for the non-abelian Pfaffian state29. We shall now demonstrate the connection between the CFT construction of the Jain states and Wen’s K-matrix formulation by explicitly working out the case of ν = 2/5. Led by the Moore-Read conjecture, we will start from our CFT bulk theory, read off the K-matrix and the charge vector, and show that in the basis where (26) are the fundamental quasihole operators, one exactly recovers Wen’s K-matrix and t-vector in the symmetric basis. This is consistent with Read’s earlier result30 that the symmetric basis naturally describes the Jain states. Alternatively, we may choose a basis consisting of either of the charge 1/5 quasiholes in (26), along with the charge 2/5 vortex (i.e. the product of the two 1/5-hole operators); as we shall see, this instead corresponds to the hierarchical basis. The conformal field theory contains the two uncoupled bosonic fields ϕ1 and ϕ2, compacitfied on radii R2 = 3 and 15, respectively. The corresponding action, S = d2xLcft for the full scalar fields φi(x, t) = ϕi(z) + ϕ̄i(z̄) is obtained from the Lagrangian, Lcft = (φ̄1, φ̄2) µ �µν∂νφI ≡ KIJφI∂µ∂ µφJ −AµJµ . (39) where the information about the compactification radii is contained in the charge vector t̃T = (1/ 3, 1/ 15). The Lagrangian (39) contains both right and left moving fields, but these decouple, and it is known that the dynamics of a single chiral component, such as ϕi(z), is described by the first order Lagrangian (36) with the same K matrix and t vector31. In order to directly compare with Wen’s formalism, we rescale the Bose fields such as to obtain an integer charge vector, tT = (1, 1): (ϕ′1, ϕ 2) ≡ (ϕ1/ 3, ϕ2/ 15). Naively, the corresponding K matrix would then be diag(3, 15). It is however important to remember that a CFT is not defined only by the Lagrangian of the fields ϕi, which gives the operator product expansions, or fusion rules, of the primary fields (i.e. the vertex operators), but also by primary field content, i.e. the allowed vertex operators. In the case of the ν = 2/5 state these allowed fields define a charge lattice with the basis vectors given by the quasihole operators (26). Thus, we will change to a basis (χ1, χ2) where the fundamental quasihole operators spanning the charge lattice are given by Hi = eiχi . As can be seen from (26), this is achieved by the field redefinition, ϕ2 = 3ϕ 2 (40) ϕ2 = ϕ 1 − 2ϕ Inverting this transformation and inserting into (39), it is now easy to verify that the resulting K matrix and t vector are precisely the K and t in (34). Alternatively (and equivalently), if we start from a basis of one of the 1/5 quasihole operators, say H10, together with the charge 2/5 “vortex” H11 ≡ H10H01, corresponding to the change of basis χ1 = ϕ 1 + ϕ 2 (41) χ2 = ϕ 1 − 2ϕ we find that the corresponding K-matrix and t-vector are the ones given in (35), i.e. the hierarchical basis. This equivalence, at the effective Chern-Simons theory level, of the Jain states and the hierarchy scheme, has been previously pointed out by several authors30,32. These authors arrive at this result by a general argument, based on similarity between the Jain states and filled Landau levels, that ignores the projection on the lowest Landau level. It is reassuring that the above demonstration, based on explicitly holomorphic wave functions, leads to the same result. This construction straightforwardly carries over to the other fractions in the Jain sequence; for example, in the case of ν = 3/7, one may pick the three charge-1/7 quasihole operators of (31) as basis of the charge lattice, corresponding to the field redefinition ϕ3 (42) Again, this brings us to the symmetric basis, with tT = (1, 1, 1) and the K-matrix given by Kij = 2+δij . Alternatively, we may construct a basis consisting of quasihole operators with charge 1/7, 2/7 and 3/7, respectively, by appropriate combinations of the charge-1/7 quasiholes in (31). As before, this corresponds to the hierarchical basis, with tT = (1, 0, 0) and the same K-matrix as that given by Wen3, Kh3/7 =  3 −1 0−1 2 −1 0 −1 2  . (43) V. LOCALIZED QUASIPARTICLES AND FRACTIONAL CHARGE AND STATISTICS The present formulation already gives a strong hint for fractional charge and fractional statistics of the CF quasi- particles: We have seen from (7) that the operator P 1 (z) corresponds to a localized charge at z, and the presence of the factor (zi − zj) m is suggestive of fractional statistics with angle π . This is not a proof, however. The usual argument for fractional charge and statistics proceeds via the Berry phases produced by adiabatic braidings of local- ized quasiparticles. In this section we construct the wave functions for localized states of one and two quasiparticles, and use these to calculate the Berry phases relevant for charge and statistics within what we call a “random phase assumption”. A localized quasiparticle state is constructed as a coherent superposition of a the angular momentum states given in (11) and (15). For a single quasiparticle at location η̄ we have (putting ` = 1) Ψ1qp(η, η̄; zi) = Ñ1(η̄η)e− 4m |η̄| (2m)ll! Ψl1qp(zi) (44) = Ñ1(η̄η) (−1)ie− 4m (|zi| 2+|η̄|2−2η̄zi)〈P (zi) (i)V (zj)〉 . Notice that the normalization constant N1(η̄η) = Ñ1(η̄η)e− 4m |η̄| only depends on the combination η̄η. Likewise, we construct the wave function for two quasiparticles at positions η̄± = N̄ ± η̄/2 as Ψ2qp(N, N̄, η, η̄; zi) = Ñ2(N, N̄, η, η̄)e− 8m |η̄| 2− 12m |N̄ | l=1,3,... L=0,1,... η)l−1 Ψl,L2qp(zi) (45) = Ñ2 (−1)i+je− 8m (|η̄| 2+|zij |2) sinh η̄zij 2m (|N̄ | 2+|Zij |2−2N̄Z)z ij 〈P 1m (zi)P 1m (zj) (ij)V1(zk)〉 , where zij = zi − zj . For η̄ = 0 and N̄ = η̄ = 0, respectively, these expressions reduce to Ψ01qp and Ψ 2qp, the wave functions with minimum angular momentum. The explicit wave functions obtained by evaluating the correlators in (44) and (45), are very similar, but not identical, to the corresponding CF wave functions. One source of difference is the slight deviation between the angular momentum eigenstates given by (15) and (16), pointed out in section II C, and shown to be numerically insignificant in section VI. The other source of difference can be seen already for the one quasiparticle state. The CF wave function reads ΨCF = (−1)i e− (zj − zk)3 (zi − zn)2 zi − zn  exp |zk|2 , (46) but the term proportional to (1/m−1)η̄ is missing in the corresponding CFT wave function. This difference, however, is a finite size effect. This term contributes to the wave function only when the exponential factor exp(η̄zi/2m) is expanded to the N th power, and thus amounts to a (nonuniversal) boundary term. Before proceeding to the calculation of the charge and statistics of the quasiparticles, it is helpful to recall, as a background, the corresponding calculation for the quasiholes of ν = 1/m. Consider the normalized wave function for one quasihole, given by (9): Ψ(η, η̄; zi) = N ′e− |η|2ΨL(η; zi) (47) where ΨL(η; zi) = i(zi − η1) k<l(zk − zl) |zi|2 is the Laughlin unnormalized wave function for a single quasihole. The plasma analogy shows that N ′ is independent of η; the normalization integral of (47) is the partition function of a Coloumb plasma with a charged impurity, which is independent of the position of the impurity as long as it is farther than a screening length (`) from the edge. The Berry phase associated with a circular loop η = Reiθ; θ ∈ {0, 2π}, is given by (with ~ = c = 1) dθ 〈Ψ(η, η̄)|i∂θ|Ψ(η, η̄)〉 (48) dθ 〈Ψ(η, η̄)|(η̄∂η̄ − η∂η)|Ψ(η̄, η)〉 where A is the area enclosed by the loop, and the last line employs integration by parts and the observation that the only η̄ dependence in the wave function is from the gaussian factor. The result, as expected, is the Aharanov-Bohm phase for a particle of charge e/m. Turning to two quasiholes and again using (9) we have Ψ(η̄a, ηa; zi) = N ′′(η1 − η2) (|η1|2+|η2|2)ΨL(ηa; zi) (49) where ΨL(ηa; zi) = i(zi − η1) j(zj − η2) k<l(zk − zl) |zi|2 is the unnormalized Laughlin wave function for two quasiholes. Naively we would read the fractional statistics parameter as θ = π/m directly form the factor (η1−η2) m in (49) but that gives the correct result only if there is no extra Berry phase (other than the usual Aharonov- Bohm phase) associated with the exchange path (here integrating θ from 0 to π)33. The absence of additional phases can be confirmed by an explicit calculation using the plasma analogy for a system with two impurities separated farther than the screening distance, `0. Taking η1 = −η2 = η, we get γexB = − BπR2, which is nothing but the AB-phase expected from the exchange of two charge e particles through a circular path with radius R. That allows us to read the exchange statistics phase directly from the factor (η1 − η2) m in (49). Wilczek and Nayak33 suggest that it is no coincidence that (9) yields a wave function with no Berry contribution to the statistics angle; they make a conjecture, supported by arguments, that QH wave functions given directly as correlators, or conformal blocks, of a CFT have vanishing Berry phase (forgetting the Aharanov-Bohm contribution) so the exchange statistics can be obtained from the so-called monodromy, which in this simple case is just the phase ei2π/m produced by the factor (η1 − η2) m when the quasiholes braid around each other along closed paths defined via an analytic continuation of the original wave function. In the more general case of non-Abelian fractional statistics, several conformal blocks correspond to the same configuration of quasihole coordinates, and the monodromies are now matrices that encode how these conformal blocks transform into each other under braidings of the coordinates. Rather than using (48), we follow Kjønsberg and Leinaas who showed that for a (normalized) wave function of the form, Ψ(η, η̄; zi) = N (η̄η)Ψh(η, η̄; zi), where N is a real function only of η̄η, the Berry phase is given by34 dθ 〈Ψ(η, η̄)|i∂θ|Ψ(η)〉 = ΘR2 lnN (η̄η)2 . (50) The Laughlin wave function for a single quasihole at η or two quasiholes at ±η has this form. The upper limit is taken to be Θ = 2π for a single quasihole, and Θ = π for an exchange of two quasiholes. From (47) and (49), the normalization constants are given by (up to an η independent factor) N1(η̄η) = e− |η|2 (51) N2(η̄iηi) = |η1 − η2| (|η1|2+|η2|2), so the formula (50) can be applied. (We have assumed sufficiently far separated quasiholes.) For a single quasihole loop of radius R, (50) gives the Berry phase −2πAB(e/m). For two quasiholes at ±η, the Berry phase is (π/m)−AB(e/m) (with Θ = π). The difference, π/m, gives the contribution from fractional statistics. In this case, there is no monodromy, and the the full statistical phase appears as a Berry phase.48 We now turn to the case of quasiparticles, where we need to calculate the relevant normalization constants from the wave functions (44) and (45). Because the sum in (45) extends only over even powers of η̄, the holomorphic part is single valued under η̄ → −η̄, implying an absence of monodromy contribution to the statistical angle, and thus both the charge and statistics can be extracted directly from N1 and N2, provided that they can be chosen as real functions of η̄η only. For quasiparticles, the normalized wave function has the form N (η̄η)Ψ(η̄), and the Berry phase is given by γB = −ΘR2 lnN (η̄η)2 . (52) Unfortunately, the calculation of the normalization factors is more difficult than in the case of the quasiholes, because the electronic wave functions (44) and (45) involve sums over correlators; e.g. for two quasiparticles the relevant integral is ∼ ijkl(−1) i+j+k+l〈P 1 (zi)P 1 (ij)V1(zp)〉〈P 1 (z̄k)P 1 (z̄l) (lm)V1(zj)〉. If, however, we keep only the diagonal terms in the sums, which amounts to a kind of random phase assumption discussed in the next section, then the normalization factors can be calculated as shown in Appendix C. The calculation of the normalization constants, outlined in Appendix C, gives the result: N 21 = Ñ |η|2 ∼ (R2)−1e− R2 ; η = Reiθ (53) N 22 = Ñ |η|2 ∼ (R2) R2 ; η = eiθ (54) Using (52) we can extract the fractional charge from the Berry phase corresponding to a single quasiparticle, Bπ(R2 + 2m`2) = BA+ 2π , (55) so the leading term is precisely the expected AB phase for a charge e/m object. The O(`2) term is a quantum correction to the “classical” R2. Such corrections have been discovered earlier both in the context of CF quasiparticles16 and noncommutative matrix models35. The statistical angle is obtained by subtracting this result from the Berry phase extracted from the exchange path. Remembering that the effective area enclosed by the exchange path is πR2/4, we θ = γexcB − γB = − . (56) This reproduces the result obtained from a direct numerical evaluation of the Berry phases from the composite fermion wave functions (44) and (45)15,16,19. The statistics of the quasiholes and quasiparticles obtained above differ in sign, contrary to the expectation from general considerations based on effective Chern-Simons or the CF theory, and we want to comment upon this. Both normalization constants in (53) are of the form N ∼ (R2)ae−bR , and from the derivation in the Appendix one sees that the constant a is unambiguously determined, and the same holds for the fractional part of b, while the integer part of b could in principle be shifted by using a different prescription for the ordering of the derivatives. However, this does not necessarily mean that the frational part of θ is well determined since it is sensitive to a cancellation of a large number of terms proportional to R2 in the Berry phases. The above result is based on the assumption that we have correctly identified |η| as the distance between the two quasiparticles. This question is discussed in some detail in in Ref. 16 where it is shown that the distance between two quasiparticles is slightly different from |η|; correcting for the distance produces the same statistics as for quasiholes (modulo an integer). Because our wave function is closely related to the CF wave function, similar considerations should apply here as well. VI. NUMERICAL TESTS This section concerns quantitative tests of the CFT quasiparticle wave functions. The numerical tests consist of two parts. In the first part, we compare, at filling factor ν = 1/3, the two-quasiparticle wave functions (15) obtained from the conformal field theory with the standard wave functions from the CF theory by calculating their Coulomb interaction energies and overlap. We will find that the two are practically identical. In the second part, the random phase approximation used in section V is examined. A. Two-quasiparticle wave function The N -composite fermion wave function for two quasiparticles at filling factor ν = 1/3 is constructed by compactly filling the lowest CF-LL by N − 2 composite fermions, and placing the remaining two composite fermions in the second CF-LL. We consider below the state in which the two “excited” composite fermions are in angular momentum orbitals, and occupy the smallest angular momenta. The wave function for this state is written as ΨCF = PLLL ∣∣∣∣∣∣∣∣∣∣∣∣ η1,−1(z1) η1,−1(z2) . . . η1,−1(zN ) η1,0(z1) η1,0(z2) . . . η1,0(zN ) η0,0(z1) η0,0(z2) . . . η0,0(zN ) η0,1(z1) η0,1(z2) . . . η0,1(zN ) η0,N−3(z1) η0,N−3(z2) . . . η0,N−3(zN ) ∣∣∣∣∣∣∣∣∣∣∣∣ (zi − zj)2 exp |zk|2 , (57) where PLLL denotes projection into the lowest Laudau level (LLL), and ηn,m(z) is the single particle eigenstate in symmetric gauge: ηn,m(z, z̄) = Nn,m e −|z|2/4 (−1)k z̄kzk+m, (58) in which n = 0, 1, 2, . . . labels the Landau level index, m = −n,−n+ 1,−n+ 2, . . . is the angular momentum quantum number, k0 = max(0,−m), and Nn,m = 2π2m(n+m)! is the normalization constant. As usual, the magnetic length has been set to unity. The wave function can be shown to be equal to ΨCF = PLLL ∣∣∣∣∣∣∣∣∣∣∣∣∣ z̄1z1 z̄2z2 . . . z̄NzN z̄1 z̄2 . . . z̄N 1 1 . . . 1 z1 z2 . . . zN zN31 z 2 . . . z ∣∣∣∣∣∣∣∣∣∣∣∣∣ (zi − zj)2 exp |zk|2 . (60) Following the standard procedure, the projection procedure is accomplished by expanding the determinant, moving all z̄’s to the left, and replacing z̄ by 2∂/∂z (with the convention that the derivatives do not act on the Gaussian part). A technique developed in Ref. [36] has made it possible to perform the projection in a more convenient manner, which is the one we use below. (This method gives projected states very close to those obtained by “brute force” projection.) The projected wave function is written as ΨCF = (−1)i+j(zi − zj)3 (ij)∏ (zi − zk)2 (ij)∏ (zj − zl)2 (ij)∏ (zm − zn)3  −2(zi − zj)2 + (ij)∑ (zi − zk)(zj − zl) (ij)∑ (zk − zi)(zk − zj)  exp |zk|2 . (61) The symbol (ij) denotes that indices i and j are excluded in the summation or the product. The total angular momentum of (61) is N + 2. (62) Numerical simulations for the wave function in (61) have shown that it produces better variational energy than the two quasiparticles wave function obtained by generalizing Laughlin’s single quasiparticle state25. The CFT wave function for two quasiparticles located at the origin is given by (14). To make contact with the above CF wave function, we set the center of mass angular momentum to zero and put the two quasiparticles in the smallest relative angular momentum channel l = 1; that produces a wave function that has the total angular momentum given in (62) . For these parameters, the CFT ansatz for the two quasiparticles wave function ΨCFT at ν = 1/3 reduces to ΨCFT = (−1)i+j(zi − zj)3 (ij)∏ (zk − zi)2 (ij)∏ (zl − zj)2 (ij)∏ (zm − zn)3 (63) −49 1(zi − zj)2 + 83 (ij)∑ (zk − zi)(zk − zj) (i,j)∑ (zk − zi)(zl − zj)  exp |zk|2 To compare the two-quasiparticle wave functions in Eqs. (63) and (61), we compare their Coulomb interaction energies and also calculate their overlaps. The Coulomb energy (in units of e2/�`0) is defined as ∣∣∣∣∣∣ 12 i 6=j |ri − rj | ∣∣∣∣∣∣Ψ 〈Ψ|Ψ〉 , (64) We do not include here electron-background and background-background contributions; these are not necessary for the present purpose, as they are identical for the two wave functions. The overlap is defined as |〈ΨCFT|ΨCF〉|√ 〈ΨCFT|ΨCFT〉〈ΨCF|ΨCF〉 . (65) Both quantities are evaluated using Metropolis Monte Carlo integration. A single data point is obtained by averaging 100 independent Monte Carlo runs, with ∼ 1.2 × 105 iterations in each run. For N = 50 electrons, the total N ECFT ECF O 10 7.76619(62) 7.76600(62) 0.9999301(2) 20 24.1403(19) 24.1402(19) 0.9999274(4) 30 46.3258(18) 46.3257(18) 0.9999274(3) 40 73.2339(17) 73.2339(17) 0.9999266(2) 50 102.0588(16) 102.0585(16) 0.999613(5) TABLE I: The Coulomb energy ECFT (ECF), quoted in units of e 2/�`, for the CFT (composite fermion) two-quasiparticle wave function ΨCFT (ΨCF). O is the properly normalized overlap between the two candidate wave functions for a number of system sizes N . In ΨCFT, we set L = 0 and l = 1. The definitions of energy and overlap are given in the text. The Monte Carlo statistical uncertainty is shown in brackets. 0 2 4 6 8 10 12 14 16 18 excess charge 0 5 10 15 20 excess charge FIG. 1: Density profiles of ΨCFT and ΨCF for N = 40 (left panel), and 50 (right panel) particles. “GS” denotes the ν = 1/3 background obtained from Laughlin’s wave function. The horizontal dashed line indicates 2/3 units of charge. In both cases, the excess integrated charge for ΨCFT, denoted by thick dashed line, shows the correct quantized value. The statistical undertainty in Monte Carlo is smaller than the widths of the lines. computational time is approximately 100 hours on a single node of a Beowulf-type PC cluster consisting of dual 3.06 GHz Intel Xeon Processors. The results are summarized in Table I. The excellent agreement demonstrates that the two two-quasiparticle wave functions are essentially identical. Figure 1 depicts the density profiles of the two wave functions for N = 40 and 50 electrons. The excess integrated charge (i.e. the charge measured relative to the ν = 1/3 background) for the CFT wave function is also shown. The excess charge has a well-defined plateau at 2/3 before the edge distortion. (The creation of two quasiparticles near the center of the droplet induces two quasiholes of equal charge at the edge.) B. Random Phase Approximation The composite fermion quasiparticles have been shown to possess well-defined fractional braiding statistics16. The comparisons in the previous subsection imply that this property, in principle, carries over to the CFT quasiparticles. However, one of the strengths of the CFT description of the FQHE states is that it reveals the braiding properties in a transparent manner. A calculation of the braiding statistics requires wave functions for spatially localized quasiparticles, which are constructed in the previous section. A key observation is that the braiding statistics can be obtained from the wave function normalization factor, which depends only on the coordinates of localized quasiparticles (c.f. (45) ). However it is difficult to obtain an explicit analytical expression for the normalization factor for localized quasiparticle states, because the localized quasiparticle wave functions are sums over many correlators; for example, the relevant term in the integral for the normalization factor of a single quasiparticle is∑ 〈Pm(zi) k 6=i Vm(zk)〉〈Pm(z̄j) l 6=j Vm(z̄l)〉. (66) The analytic form of the normalization factor can be obtained by assuming that only the “diagonal” elements in (66) are relevant, which we have referred to as the random-phase assumption; the braiding properties of the quasiparticles can be derived under this approximation and are in agreement with the known results. In this section we test the random phase assumption for a single localized quasiparticle. The wave function for a single quasiparticle at ν = 1/m localized at η̄ is written as Ψ1qp(η̄) = (−1)i e− 4m (|η̄| 2+2η̄zi)〈Pm(zi) j 6=i Vm(zj)〉, (67) with the correlator 〈Pm(zi) j 6=i Vm(zj)〉 given by 〈Pm(zi) j 6=i Vm(zj)〉 = (zj − zk)m j 6=i (zi − zj)m−1 j 6=i zi − zj · exp |zl|2 . (68) For simplicity, we define the notation F (zi) ≡ (−1)i e− 2m (η̄zi)〈Pm(zi) j 6=i Vm(zj)〉. (69) Then the wave function Ψ1qp(η̄) can be expressed as Ψ1qp(η̄) = e − 14m |η̄| F (zi). (70) The square of the normalization factor, N1(η, η̄), is given by the integral N 21 (η, η̄) = dzk Ψ 1qp(η̄)Ψ1qp(η̄) 2m |η̄| |F (zi)|2 + [F ∗(zi)F (zj) + F (zi)F ∗(zj)] ≡ Mdiag +Moff−diag. (71) Mdiag andMoff−diag are the “diagonal” and “off-diagonal” contributions to the full normalization factor, respectively. We calculate the ratio Mdiag/N1(η, η̄)2 for several quasiparticle locations for ν = 1/3. The result is shown in Fig. 2. The principal conclusion is that the contribution of the single diagonal term is of the same order as that of a large number of off-diagonal terms. Although not conclusive, this suggests that the diagonal term used to calculate the quasiparticle charge and statistics in section V will be dominant, thus providing a partial justification for the neglect of the off-diagonal terms. VII. SUMMARY AND OUTLOOK In this paper we have extended the class of QH wave functions that can be expressed as correlators in a conformal field theory to include the quasiparticle states at the Laughlin fractions, as well as all the ground states in the positive Jain sequence and their quasiparticle and quasihole excitations. The connection between the CFT operators and composite fermions was explicitly demonstrated by constructing n fermionic vertex operators Vp,n, built from n compactified scalar fields, corresponding to the n filled CF Landau levels at ν = n/(2np+ 1). For these states we also constructed the fractionally charged excitations: At ν = n/(2np + 1), there are n independent hole operators (corresponding to removing a composite fermion from any one of the n filled CF Landau levels) and one unique quasiparticle operator (corresponding to putting a CF in the empty (n + 1)st Landau level). The fermionic vertex operator Vp,n at level n is closely related to the quasiparticle operator of the Jain state with n − 1 filled CF LLs; in this sense, the ground state at the fraction n/(2pn+ 1) of the Jain sequence may be viewed as a condensate of quasiparticles of the state at (n − 1)/[2p(n − 1) + 1]. It should be noted, however, that these quasiparticles obey well defined fractional statistics only in the dilute limit. We also showed that the conformal field theories used to obtain the CF wave functions give precisely the chiral edge theories that are expected from general considerations within the effective field theory scheme developed by Wen, thus giving microscopic support to that approach. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 !=(0.0,0.0) !=(2.0,0.0) !=(4.0,0.0) !=(6.0,0.0) FIG. 2: The ratio Mdiag/N1(η, η̄)2 for several quasiparticle location η. The coordinates are in units of magnetic length. The results for η = (6.0, 0.0) are not plotted for N < 10 because of significant edge effects. An attractive aspect of the methods developed in this paper is that they can be extended and applied to other quantum Hall states. For example, in a recent paper, a straightforward generalization of our vertex operators was employed38 to describe the states observed recently by Pan et.al.39, which do not belong to the principal series ν = n/(2pn + 1) but have been modeled as the FQHE of composite fermions39,40. Generalizing our methods to include the negative Jain sequence, ν = n/(2np − 1), or more generally to states obtained from condensing holes, is a more challenging problem. Another interesting challenge is to find a CFT operator that directly creates a localized quasiparticle, rather than having to construct it as a coherent superposition of angular momentum states as was done in this paper; work on this question is in progress24. Acknowledgements: We would like to thank Jürgen Fuchs, Maria Hermanns, Anders Karlhede and Jon Magne Leinaas for helpful discussions. We thank Prof. B. Janko for hospitality at the ITS, a joint institute of ANL and the University of Notre Dame, funded through DOE contract W-31-109-ENG-38 and Notre Dame Office of Research. This work was supported in part by the Swedish Research Council, by Nordforsk, and by NSF under grant no. DMR-0240458 APPENDIX A: THE BACKGROUND CHARGE Definition of correlators of vertex operators such as (8) requires introduction of a compensating charge to satisfy the neutrality condition implied by the conserved U(1) charge37. This can be done in different ways. The simplest is to put a compensating charge Vbg(z∞) = e−i Nmϕ(z∞) at the position z∞, taken to infinity, and define the correlator by a limiting procedure:41 ΨL(zi) = lim 〈V1(z1)V1(z2) . . . V1(zN−1)V1(zN )〉 = (zi − zj)m . (A1) This prescription does not produce the exponential factor e− |zi|2/4`2 , characteristic of a lowest Landau level wave function. In this paper we use the prescription given in Ref. 4, which corresponds to a smeared background charge given by the operator d2z ϕ(z) , (A2) where ρm = ρ0/m with ρ0 = eB/2π the density of filled Landau level. The difficulty with this prescription is that a direct evaluation of the correlator gives a contribution e−mρm d2z ln(z−zi) , (A3) where the presence of the logarithm makes the imaginary part of the integral undefined. The aim of this appendix is to give a regularized version of the smeared background charge that: i) is well defined; ii) reproduces the pertinent gaussian factor in (8); and iii) differs from (8) only through a well defined (although singular) gauge transformation. The idea behind our regularization is to replace the continuous background field by a lattice of singular flux tubes of strength , (A4) which defines the integer n. First consider a single electron in the presence of a single (fractional) flux tube at the origin. The wave function close to the flux tube behaves as ψ(z) ∼ z− 2π , (A5) so for a single electron in a flux tube lattice, it is natural to consider a wave function of the type ψ(z; {z~n}) = f(z) (z − z~n)− 2π , (A6) which has the correct singular behaviour at the lattice points z~n and satisfies the Laplace equation, ∇2ψ(z; {z~n}) = 4∂z∂z̄ψ(z; {z~n}) = 0 ; z /∈ {z~n}. (A7) This is not a sufficient condition for an acceptable electron wave function – we must also require that ψ(z; {z~n}) is normalizable. As we now show, this will determine the allowed analytic functions f(z). Without any loss of generality, we specialize to a regular lattice of K points z~n = (nx + iny)a with spacing a that covers a total area A - this is our regularized version of a uniform flux Φ = Kδφ corresponding to a uniform field of strength B = Φ/A inside the area A. We will ignore edge effects. Since we have limz→∞ ψ(z; {z~n}) ∼ f(z)z−K 2π , and normalizability requires49 limz→∞ ψ(z; {z~n}) ∼ 1/z, we de- mand f(z) ∼ zk where, = k − ≤ −1, (A8) where we choose A so that Φ/2π is an integer. This can be rewritten as k + 1 ≤ = Aρ0 = N0, (A9) where N0 is the number of states in the lowest Landau level. This result makes it plausible that, in the limit of large n, the functions (A6) with f(z) = N0−1∑ will give a good description of the lowest Landau level at a magnetic field of strength B. To show this, we rewrite ψ ψ(z; {z~n}) = f(z) z − z~n z̄ − z̄~n )− δφ4π ∏ |z − z~n|− 2π , (A10) and approximate |z − z~n|− 2π = exp ln |z − z~n| ≈ exp d2r′ ln |~r − ~r′| . (A11) The last integral was calculated as∫ d2r′ ln |~r − ~r′| = d2r′ ln |~r − ~r′|∇2r′r d2r′ δ2(~r − ~r′)r′2 = , (A12) where we integrated by parts and neglected boundary terms. (The justification is that there is an understood density function ρ(~r′) that rapidly falls to zero outside the area A but is essentially constant inside. This still leaves an edge correction due to the derivatives acting on the profile that we ignore.) Finally we note that δφ = Kδφ 1 = Φ 1 , where ` is the magnetic length corresponding to the field strength B, so the approximate wave functions are of the form ψ(z; {z~n}) = z − z~n z̄ − z̄~n )− δφ4π f(z)e− |z|2 (A13) and are expected to be valid in the limit of a/`→ 0, and z well inside the lattice. We have thus recovered the standard lowest Landau level wave functions, albeit in an unconventional gauge defined by the (arbitrarily chosen) flux lattice. This conclusion is also strongly suggested by the numerical calculations of Pryor, who shows that already for n = 8, at least four flat bands are identifiable in the electron spectrum, corresponding to the four lowest Landau levels42. A generalization of the analytic argument presented above to include higher Landau levels would be interesting. Returning to our original goal of regularizing the operator insertion (A2), we see that d2z ϕ(z) → Vb(z~n) = m2π`2 φ(z~n) (A14) will do the job in the limit a/`2 → 0, because the total U(1) charge of the K vertex operators Vb equals −Ka2eB/2πm = −Aρm = −N , where N is the total number of electrons. Making the replacement (A14) in (8) (A14) in (8) 〈 V1(z1)V1(z2) . . . V1(zN ) m2π`2 φ(z~n) , (A15) and using the same approximation as in (A11), we regain the correct exponential factor (The contraction of V1(zj) ~n Vb(z~n) gives the factor ~n |zj − z~n| −δφ/2π which, according to (A11), gives the Gaussian factor for zj), an unimportant constant from the contractions between the different Vb’s, and also a singular and rapidly changing phase factor just as in (A13). This is the regularized version of the statement in reference 4 that (8) “is trying to give us the answer in a gauge where the vector potential is zero, which means it differs by an everywhere-singular gauge transformation from the usual symmetric gauge vector potential for the uniform background magnetic field.” Here we should also mention that in correlation functions involving the full scalar field, φ(z, z̄), the singular phases will cancel out, so these are well behaved functions even in the limit of vanishing lattice spacing. Finally, we note that the regularization procedure outlined above suffers from a formal difficulty - it introduces vertex operators with charges that do not belong to the charge lattice of the CFT under consideration. Putting a compensating charge at infinity would not suffer from this problem, but would also not correspond to a homogeneous system. APPENDIX B: EQUIVALENCE BETWEEN CFT AND CF WAVE FUNCTIONS In this appendix we provide derivations for some of the formulae in the main text, and prove that the CFT wave functions for the states in the Jain series indeed reproduce the wave functions from the CF framework using a direct projection on the LLL. 1. An identity We begin by deriving the central relation (13). The basic idea is to express the antisymmetrization as a Slater determinant and then use the Laplace expansion of an (N +M)× (N +M) determinant, detA = �PidetBidetCi, (B1) where the sum is over the ways in which a N ×N matrix Bi can be formed from the first N rows of A, Ci is the complementary M ×M matrix and �Pi is the sign of the permutation needed to bring the N columns of Bi followed by the M columns of Ci into the original order.43 This formula generalizes the expansion by a row used to derive (11) for the one quasiparticle case. We have, (zp − zq)1/m+lpqP 1 (z1) . . . P 1 (zM )V1(zM+1) . . . . . . V1(zN )} (B2) (zrp − zrq ) 1/m+lpqP 1 (zr1) . . . P 1 (zrM )V1(zrM+1) . . . . . . V1(zrN ) �{in} sgnPs(zsp − zsq ) 1/m+lpqP 1 (zs1) . . . P 1 (zsM ) sgnPtV1(ztM+1) . . . V1(ztN ) p=1 ipR{ (zip − ziq ) 1/m+lpqP 1 (zi1) . . . P 1 (ziM )V1(zīM+1) . . . V1(zīN )} , where {r1 . . . rN} is a permutation, Pr, of {1 . . . N}; {s1 . . . sM} is a permutation, Ps, of a subset {i1 . . . iM} of the N integers; {tM+1 . . . tN} a permutation, Pt, of the conjugate set {̄i1 . . . īM} of N −M integers; and �Pr etc. the corresponding sign factors. The symbol lpq denotes the relative angular momentum, and R is the radial ordering operator. The sum {in} is over all the N !/M !(N −M)! ways of doing this partition, and �{in} is the sign of the permutation needed to order the M rows of the first partition to the left in the Slater determinant. In the above expression, the second line is the definition, the third row follows from the Laplace expansion, and the last by noting that the prefactor makes the expression explicitly antisymmetric under exchange of coordinates in the subset {in}. Antisymmetry under exchange in the second subset is already guaranteed by the anti-commutation relations for the V1:s. As in the single quasiparticle case, no extra signs are obtained by the final radial ordering because P and V1 commute. 2. Equivalence between the ν = 2/5 CF and CFT wave functions. We now prove that the CFT wave function (22) for ν = 2/5 is identical to that of composite fermions. Recalling that (22) differs from (18) only in that all the derivatives are on the left, we have the following explicit expression (recall that N = 2M so there are M V1:s and M V2:s in the correlator), ΨCFT2/5 (zi) = i1<i2...iM ik∂zi1∂zi2 . . . ∂ziM (zik − zil) 3 (B3) (i2,i3...iM )(zk1 − zi1) (i1,i3...iM )(zk2 − zi2) 2 . . . (i1,i2...iM )(zkN − zin) (i1,i2...iM )∏ (zm − zn)3 . To write this in the CF form, we factor out a full Jastrow factor: ΨCFT2/5 (zi) = i1<i2...iM ik∂zi1∂zi2 . . . ∂ziM (zik − zil) (i1,i2...iM )∏ (zm − zn)1 N=2M∏ (zp − zq)2 (B4) The first two Jastow factors are nothing but the Vandermonde determinants of the subset I = {zi1 . . . ziM } and the conjugate subset J = {zī1 . . . zīM }. Also useful is the operator identity ∂z1∂z2 . . . ∂zM ∣∣∣∣∣∣∣∣∣∣∣ 1 1 . . 1 z1 z2 . . zM z21 z 2 . . z . . . . . zM−11 . . . z ∣∣∣∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣∣∣∣ ∂z1 ∂z2 . . ∂zM ∂z1z1 ∂z2z2 . . ∂zM zM 1 ∂z2z 2 . . ∂zM z . . . . . 1 . . . ∂zM z ∣∣∣∣∣∣∣∣∣∣∣ which follows because each coordinate, as well as the corresponding derivative, appears once and only once in every term when the determinant is expanded. We can now use the Laplace formula, (B1), in the opposite direction to write ΨCFT2/5 (zi) = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∂z1 ∂z2 . . ∂zN ∂z1z1 ∂z2z2 . . ∂zN zN 1 ∂z2z 2 . . ∂zN z . . . . . 1 . . . ∂zN z 1 1 . . 1 z1 z2 . . zN z21 z 2 . . z . . . . . zM−11 . . . z ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (zp − zq)2 = ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ∂z1 ∂z2 . . ∂zN z1∂z1 z2∂z2 . . zN∂zN z21∂z1 z 2∂z2 . . z . . . . . zM−11 ∂z1 . . . z N ∂zN 1 1 . . 1 z1 z2 . . zN z21 z 2 . . z . . . . . zM−11 . . . z ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (zp − zq)2, (B6) where we omitted an unimportant sign. The last expression is, up to an overall normalization factor, the CF wave function as given in reference36. The last identity in (B6) follows because when the derivatives act on the factors in the determinant they give a row that is already present in the lower part of the determinant. For this to be true it is necessary that all the angular momentum states are present and that there are at least as many rows without derivatives as those with derivatives. These conditions correspond to having a maximum density droplet of electrons in the second CF Landau level which is no larger then the droplet formed by the electrons in the lowest CF Landau level. This completes the proof of the statement in the main text. 3. The general CF operators and the Jain series We now extend the previous analysis to a general state in the Jain series. First we give the explicit expressions for the operators Vp,n discussed in section III D: Vp,1(z) = e 2p+1ϕ1(z) Vp,2(z) = ∂e ϕ1(z)e 2p+1ϕ2(z) Vp,3(z) = ∂ ϕ1(z)e (2p+1)(4p+1) ϕ2(z) 4p+1ϕ3(z) (B7) . . . Vp,n(z) = ∂ ϕ1(z)e (2p+1)(4p+1) ϕ2(z) . . . e [2p(n−2)+1][2p(n−1)+1] ϕn−1(z) 2np+1 2(n−1)p+1ϕn(z). Because all ϕi’s commute, we can write Vp,n(z) = ∂ n−1eiϕ̃n(z), (B8) where ϕ̃1 = ϕ1 and ϕ̃n(z) = [2(k − 1)p+ 1](2kp+ 1) ϕk(z) + 2np+ 1 2(n− 1)p+ 1 ϕn(z) ; n ≥ 0 (B9) Using the sum formula: [2p(k − 1) + 1][2pk + 1] 2pn+ 1 (B10) and the charge density operator J(z) = 2p+ 1 ∂zϕ1(z) + (2p+ 1)(4p+ 1) ∂ϕ2(z) · · ·+ [2p(n− 1) + 1][2pn+ 1] ∂ϕn(z) (B11) it can be shown that the operators (B7) satisfy the properties stated in the text vis á vis charge and statistics, and also give the filling fraction ν = n 2pn+1 . We can now construct the wave function for the general ground state in the Jain series by a recursive procedure. For a total of N = nM electrons, it is natural to write ΨCFp,n (zi) = A{〈 Vp,n(zi) j=M+1 Vp,n−1(zj) · · · j=(n−1)M+1 Vp,1(zj)〉} (B12) The proof that this indeed reproduces the ν = n/(2pn + 1) CF wave function, is a straightforward generalization of that given for 2/5 in the previous section. It involves using the Laplace formula (B1) iteratively n− 1 times, breaking the problem down into the n groups (Landau levels) of particles, in analogy with the procedure in section B 1. The generalization of (B3) then contains n− 1 sign factors, one for each additional group of particles, and one can follow the logic of (B4) - (B6) (with n (M ×M) subdeterminants instead of two) to derive the equivalence of the CF and CFT wave functions. APPENDIX C: THE NORMALIZATION FACTORS N1 AND N2 We begin with a single quasiparticle. Using the explicit form (44), and keeping only the diagonal terms in the double sum in the normalization integral we get, |Ñ1(η, η̄)|−2 ∼ d2zi e − 12m |zi−η| j 6=i d2zj 〈P 1 V1(zi)〉 〈P 1 (z̄i) V1(zi)〉∗ . (C1) Here and below we use the sign ∼ to indicate that we neglect η-independent constants. We write P 1 (zi) = ∂iP̂ 1 and P 1 (z̄i) = ∂̄iP̂ 1 (z̄i) and perform the zi integral after making the approximate substitution zi → η in the correlators. This gives |Ñ1(η, η̄)|−2 ∼ j 6=i ∂η〈P̂ 1 j 6=i V1(zj)〉 ∂η̄〈P̂ 1 j 6=i V1(zj)〉∗  (C2) Note that we first moved the derivatives in the operators P 1 (z) outside the expectation values. That this is allowed follows either from a direct calculation, or from noting that P 1 (z) is a descendant of the primary field P̂ 1 (z) and using standard methods to express the correlator of descendant fields as derivatives of correlators of primary fields37. It is important that all sign factors cancel in the diagonal terms. Next we note that the η̄ dependence of each correlator is given by 〈P̂ 1 j 6=i V1(zj)〉 ∼ exp [−(m− 1)|η| 2/(4m)]. This allows us to move the derivatives outside the full two dimensional correlation function. Reintroducing the magnetic length, `, defining Dη = ∂η + c η̄ with c = (m− 1)/(m`2), and noting [Dη, D̄η] = 0, we get, |Ñ1(η, η̄)|−2 ∼ DηD̄η j 6=i d2zj |〈P̂ 1 j 6=i V1(zj)〉|2 (C3) The right hand side of this equation is now in a form where plasma analogy arguments can be applied: the integral is the free energy of an overall neutral plasma with a charged impurity at the fixed postion η. This free energy is independent of the impurity positions because of screening, so finally, using DηD̄η1 = c2 η̄η + c, and noting that all terms in the sum give identical contributions, we conclude that to leading order in `2/|η|2, |Ñ1(η, η̄)|−2 ∼ η̄η, which gives (53) in the main text. The calculation of the two quasiparticle normalization factor, |Ñ2(N, η; N̄ , η̄)| follows in an analogous manner, with some extra complication due to the more complicated exponential factors in the expression (45). Again keeping only the diagonal terms and completing squares in the exponents, we get, |Ñ2(N, η; N̄ , η̄)|−2 ∼ d2zijd 2Zij e |Zij−N |2 [e− 4m |zij−η| 4m |zij+η| − 2 cosϑ e− 4m (|η| 2+|zij |2)] (zijzij) k 6=i,j d2zk [∂i∂j〈P̂ 1 (zi)P̂ 1 k 6=i,j V1(zk)〉][∂i∂j〈P̂ 1 (z̄i)P̂ 1 (z̄j) k 6=i,j V1(zk)〉], (C4) where eiϑ = η̄zij − zijη, and overall constants are suppressed and the derivatives are moved outside the expectation values. Because of the gaussian factors in |Zij −N |, we can approximate the integral by substituting the maximum value Zij = N = 12 (η+ + η−). The third term in the square brackets (∼ cosϑ) is maximum at zij = 0; the integral vanishes with this substitution because of the factor (zijzij)1− m . This term is therefore neglected. The remaining two terms are equal. For the first term, we have zij = η = η+ − η−. Proceeding as before, defining D+ = ∂η+ + c η̄+ etc, and using that the η̄+ dependence of each correlator is ∼ exp[−(m− 1)|η|2/(4m)], we get, |Ñ2(N, η; N̄ , η̄)|−2 ∼ k 6=i,j d2zk [∂i∂j〈P̂ 1 (zi)P̂ 1 k 6=i,j V1(zk)〉][∂̄i∂̄j〈P̂ 1 (z̄i)P̂ 1 (z̄j) k 6=i,j V1(zk)〉] (C5) = D+D−D̄+D̄− k 6=i,j d2zk |〈P̂ 1 (zi)P̂ 1 k 6=i,j V1(zk)〉|2. The integral is now the partition function for a neutral plasma with two impurities at positions η+ and η− and this free energy is again independent of the impurity positions because of screening. We thus have D+D−D̄+D̄−1 = c2[c2|η+|2|η−|2 + c|η+|2 + c|η−|2 + 1]. Finally, taking N = 0 and substituting z± = ±η/2 in the above expressions, and noting that all terms in the sums give identical contributions, we get, to leading order in `2/|η|2, the formula (54) quoted in the text. 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Ardonne, PhD thesis, www.kitp.ucsb.edu/˜ ardonne/thesis/boekje.pdf, which also gives a general discussion of the application of conformal field theory to the quantum Hall effect. 42 C. Pryor, Phys. Rev. B 44, 12473 (1991). 43 M. L. Metha, Matrix Theory, Hindustan Publishing Corporation, (1989). 44 Later the non-Abelian statistics has also been understood in the context of d-wave paired superconductors8 and has also been studied numerically9. 45 An asymmetry in the description of the quasiparticles and quasiholes at ν = 1/m has been a striking feature of other descriptions as well. In Laughlin’s theory, a quasihole at position η is represented as a vortex (zi−η), while a quasiparticle is created by a complicated operator involving many derivatives. The fractional statistics of the quasiholes is easy to derive, while the statistics of the quasiparticles eludes a precise analytical treatment. In the Ginzburg-Landau-Chern-Simons effective theories, the quasiholes and quasiparticles are described by vortices and anti-vortices respectively, and again there is an asymmetry in the description.11 46 A more careful evaluation of the correlators using a regularized background charge (cf. Appendix A) does give a contribution also from the exponential. In order to have holomorphic wave function this must be cancelled by replacing ∂i with a suitable covariant derivative. Since this in the end amounts to a mere change of notation, we simply use the rule that the derivatives do not act on the exponential part in the correlation function. 47 As straightforward projection tends to get computationally heavy in numerical calculations with many particles and a large number of derivatives, slightly different methods of obtaining LLL wave functions have been employed in most of the CF literature13. These, too, are often referred to as projection. For a single quasiparticle the different prescriptions agree, while in the general case they produce very similar but not identical wave functions. It is the brute force projection which exactly matches with the CFT construction for the Jain sequence ground states. 48 The quantity N (η̄η) can be interpreted as the (diagonal element) of the quasihole density matrix, N (η, η̄)2 ≡ ρ(η, η̄) = ψ(η)ψ(η̄)?, where the factor ψ(η1, η2) is the quasihole wave function. 49 This is the correct condition on a sphere (or a compactified plane). On a real plane the last mode will be marginal in the sense that the normalization integral will have a logarithmic divergence. http://arxiv.org/abs/cond-mat/0702516 Introduction One and two quasiparticles in the Laughlin state The ground state and the quasihole states One quasiparticle Two or more quasiparticles Quasiparticles and quasiholes Composite Fermion states in the Jain series The = 2/5 composite fermion ground state The quasihole operators The quasiparticle operator The = 3/7 state and the Jain series Connection to effective Chern-Simons theories and edge states Localized quasiparticles and fractional charge and statistics Numerical tests Two-quasiparticle wave function Random Phase Approximation Summary and Outlook The background charge Equivalence between CFT and CF wave functions An identity Equivalence between the = 2/5 CF and CFT wave functions. The general CF operators and the Jain series The normalization factors N1 and N2 References

0704.0571

$\Bz\to\pip\pim\piz$ Time Dependent Dalitz analysis at BaBar

Untitled 0 → π+π−π0 Time Dependent Dalitz analysis at BaBar. Gianluca Cavoto∗ INFN Sezione di Roma, Piazzale Aldo Moro 2, 00185 Rome, Italy I present here results of a time-dependent analysis of the Dalitz structure of neutral B meson decays to π+π−π0 from a dataset of 346 million BB̄ pairs collected at the Υ (4S) center of mass energy by the BaBar detector at the SLAC PEP-II e+e− accelerator. No significant CP violation effects are observed and 68% confidence interval is derived on the weak angle α to be [75,152] I. INTRODUCTION The time-dependent analysis of the B0 → π+π−π0 Dalitz plot (DP), dominated by the ρ(770) intermedi- ate resonances, extracts simultaneously the strong tran- sition amplitudes and the weak interaction phase α ≡ arg [−VtdV ∗tb/VudV ∗ub] of the Unitarity Triangle [1]. In the Standard Model, a non-zero value for α is respon- sible for the occurrence of mixing-induced CP violation in this decay. ρ±π∓ is not a CP eigenstate, and four flavor-charge configurations (B0(B0) → ρ±π∓) must be considered. The corresponding isospin analysis [2] is un- fruitful with the present statistics since two pentagonal amplitude relations with 12 unknowns have to be solved (compared to 6 unknowns for the π+π− and ρ+ρ− sys- tems). The differential B0 decay width with respect to the Mandelstam variables s+, s− (i.e., the Dalitz plot [3]) reads dΓ(B0 → π+π−π0) = 1 (2π)3 |A3π | ds+ds−, where A3π (A3π) is the Lorentz-invariant amplitude of the three-body decay B0 → π+π−π0 (B0 → π+π−π0). We assume in the following that the amplitudes are dom- inated by the three resonances ρ+, ρ− and ρ0 and we write A3π = f+A + + f−A − + f0A 0 and A3π = f+A+ + −+f0A 0, where the fκ (with κ = {+,−, 0} denoting the charge of the ρ from the decay of the B0 meson) are functions of s+ and s− that incorporate the kinematic and dynamical properties of the B0 decay into a (vec- tor) ρ resonance and a (pseudoscalar) pion, and where the Aκ are complex amplitudes that include weak and strong transition phases and that are independent of the Dalitz variables. With ∆t ≡ t3π−ttag defined as the proper time interval between the decay of the fully reconstructedB03π and that of the other meson B0tag, the time-dependent decay rate ∗Electronic address: [email protected] when the tagging meson is a B0 (B0) is given by |A±3π(∆t)| e−|∆t|/τB0 |A3π |2 + |A3π|2 ∓ |A3π |2 − |A3π|2 cos(∆md∆t) ± 2Im A3πA∗3π sin(∆md∆t) , (1) where τB0 is the mean B 0 lifetime and ∆md is the B oscillation frequency. Here, we have assumed that CP violation in b mixing is absent (|q/p| = 1), ∆ΓBd = 0 and CPT is conserved. Inserting the amplitudes A3π and A3π one obtains for the terms in Eq. (1) |A3π |2 ± |A3π|2 = κ∈{+,−,0} |fκ|2U±κ + κ<σ∈{+,−,0} Re [fκf κσ − Im [fκf∗σ ]U±,Imκσ A3πA∗3π κ∈{+,−,0} |fκ|2Iκ + κ<σ∈{+,−,0} Re [fκf σ ] I κσ + Im [fκf σ ] I , (2) The 27 real-valued coefficients defined in Tab.IV that multiply the fκf σ bilinears are determined by the fit. Each of the coefficients is related in a unique way to phys- ically more intuitive quantities, such as tree-level and penguin-type amplitudes, the angle α, or the quasi-two- body CP and dilution parameters [4] (cf. Section IVB). We determine the quantities of interest in a subsequent least-squares fit to the measured U and I coefficients. II. DALITZ MODEL The ρ resonances are assumed to be the sum of the ground state ρ(770) and the radial excitations ρ(1450) and ρ(1700), with resonance parameters determined by a combined fit to τ+ → ντπ+π0 and e+e− → π+π− data [5]. Since the hadronic environment is different in B decays, we cannot rely on this result and therefore de- termine the relative ρ(1450) and ρ(1700) amplitudes si- multaneously with the CP parameters from the fit. Vari- ations of the other parameters and possible contributions http://arxiv.org/abs/0704.0571v1 0 0.2 0.4 0.6 0.8 1 Interference FIG. 1: Square Dalitz plots for Monte-Carlo generated B0 → π+π−π0 decays.The decays have been simulated without any detector effect and the amplitudes A+, A− and A0 have all been chosen equal to 1 in order to have destructive inter- ferences at equal ρ masses. The main overlap regions be- tween the charged and neutral ρ bands are indicated by the hatched areas. Dashed lines in both plots correspond to√ s+,−,0 = 1.5 GeV/c 2: the central region of the Dalitz plot contains almost no signal event. to the B0 → π+π−π0 decay other than the ρ’s are studied as part of the systematic uncertainties (Section IVA). Following Ref. [5], the ρ resonances are parameterized in fκ by a modified relativistic Breit-Wigner function in- troduced by Gounaris and Sakurai (GS) [6]. Large variations occurring in small areas of the Dalitz plot are very difficult to describe in detail. These re- gions are particularly important since this is where the interference, and hence our ability to determine the strong phases, occurs. We therefore apply the trans- formation ds+ ds− −→ | detJ | dm′ dθ′, which defines the Square Dalitz plot (SDP). The new coordinates are m′ ≡ 1 arccos −mmin , θ′ ≡ 1 θ0, where m0 is the invariant mass between the charged tracks, mmax0 = mB0 − mπ0 and mmin0 = 2mπ+ are the kine- matic limits of m0 and θ0 is the ρ 0 helicity angle; θ0 is defined by the angle between the π+ in the ρ0 rest frame and the ρ0 flight direction in the B0 rest frame. J is the Jacobian of the transformation that zooms into the kinematic boundaries of the Dalitz plot, shown in Fig.1 . III. ANALYSIS DESCRIPTION The U and I coefficients and the B0 → π+π−π0 event yield are determined by a maximum-likelihood fit of the signal model to the selected candidate events. Kinematic and event shape variables exploiting the characteristic properties of the events are used in the fit to discriminate signal from background. A. Signal and background parametrization We reconstruct B0 → π+π−π0 candidates from pairs of oppositely-charged tracks, which are required to form a good quality vertex, and a π0 candidate. In order to ensure that all events are within the Dalitz plot bound- aries, we constrain the three-pion invariant mass to the B mass. A B-meson candidate is characterized kinemat- ically by the energy-substituted mass mES = s+ p0 · pB)2/E20 − p2B] 2 and energy difference ∆E = E∗B − 12 s, where (EB,pB) and (E0,p0) are the four-vectors of the B-candidate and the initial electron-positron system, respectively. The asterisk denotes the Υ (4S) frame, and s is the square of the invariant mass of the electron-positron system. We require 5.272 < mES < 5.288GeV/c 2. The ∆E res- olution exhibits a dependence on the π0 energy and therefore varies across the Dalitz plot. We account for this effect by introducing the transformed quantity ∆E′ = (2∆E − ∆E+ − ∆E−)/(∆E+ − ∆E−), with ∆E±(m0) = c± − (c± ∓ c̄) (m0/mmax0 )2, where m0 is strongly correlated with the energy of π0. We use the val- ues c̄ = 0.045GeV, c− = −0.140GeV, c+ = 0.080GeV, mmax0 = 5.0GeV, and require −1 < ∆E′ < 1. Backgrounds arise primarily from random combina- tions in continuum qq̄ events. To enhance discrimination between signal and continuum, we use a neural network (NN) [7] to combine discriminating topological variables. The time difference ∆t is obtained from the measured distance between the z positions (along the beam direc- tion) of the B03π and B tag decay vertices, and the boost βγ = 0.56 of the e+e− system: ∆t = ∆z/βγc. To deter- mine the flavor of the B0tag we use the B flavor tagging algorithm of Ref. [8]. This produces six mutually exclu- sive tagging categories. Events with multiple B candidates passing the full se- lection occur in 16% (ρ±π∓) and 9% (ρ0π0) of the time, according to signal MC. If the multiple candidates have different π0 candidates, we choose the B candidate with the reconstructed π0 mass closest to the nominal π0 mass; in the case that both candidates have the same π0, we pick the first one. The signal efficiency determined from MC simulation is 24% for B0 → ρ±π∓ and B0 → ρ0π0 events, and 11% for non-resonant B0 → π+π−π0 events. Of the selected signal events, 22% of B0 → ρ±π∓, 13% of B0 → ρ0π0, and 6% of non-resonant events are misreconstructed. Misreconstructed events occur when a track or neutral cluster from the tagging B is assigned to the reconstructed signal candidate. This occurs most often for low-momentum tracks and photons and hence the misreconstructed events are concentrated in the cor- ners of the Dalitz plot. Since these are also the areas where the ρ resonances overlap strongly, it is important to model the misreconstruced events correctly. We use MC simulated events to study the background from other B decays. More than a hundred channels were considered in preliminary studies, of which twenty- nine are included in the final likelihood model. For each mode, the expected number of selected events is com- puted by multiplying the selection efficiency (estimated using MC simulated decays) by the world average branch- ing fraction (or upper limit), scaled to the dataset lumi- nosity (310 fb−1). The selected on-resonance data sample is assumed to consist of signal, continuum-background and B-background components, separated by the flavor and tagging category of the tag side B decay. The sig- nal likelihood consists of the sum of a correctly recon- structed (“truth-matched”, TM) component and a mis- reconstructed (“self-cross-feed”, SCF) component. B. Dalitz and ∆t distribution The Dalitz plot PDFs require as input the Dalitz plot- dependent relative selection efficiency, ǫ = ǫ(m′, θ′), and SCF fraction, fSCF = fSCF(m ′, θ′). Both quantities are taken from MC simulation. Away from the Dalitz plot corners the efficiency is uni- form, while it decreases when approaching the corners, where one of the three particles in the final state is close to rest so that the acceptance requirements on the par- ticle reconstruction become restrictive. Combinatorial backgrounds and hence SCF fractions are large in the corners of the Dalitz plot due to the presence of soft neu- tral clusters and tracks. The width of the dominant ρ(770) resonance is large compared to the mass resolution for TM events (about 8MeV/c2 core Gaussian resolution). We therefore neglect resolution effects in the TM model. Misreconstructed events have a poor mass resolution that strongly varies across the Dalitz plot. It is described in the fit by a 2 × 2-dimensional resolution function, convoluted with signal Dalitz PDF. The ∆t resolution function for signal and B- background events is a sum of three Gaussian distribu- tions, with parameters determined by a fit to fully recon- structed B0 decays [8]. The Dalitz plot- and ∆t-dependent PDFs factorize for the charged-B-backgroundmodes, but not necessarily for the neutral-B background due to B0B0 mixing. The charged B-background contribution to the likeli- hood parametrizes tag-“charge” correlation (represented by an effective flavor-tag-versus-Dalitz-coordinate cor- relation), and therefore possible direct CP violation in these events. The Dalitz plot PDFs are obtained from MC simula- tion and are described with the use of non-parametric functions. The ∆t resolution parameters are determined by a fit to fully reconstructed B+ decays. The neutral-B background is parameterized with PDFs that depend on the flavor tag of the event and, depending on the final states they can show correla- tions between the flavor tag and the Dalitz coordinate. The Dalitz plot PDFs are obtained from MC simulation and are described with the use of non-parametric func- tions. For neutral-B background, the signal ∆t resolution model is assumed. The Dalitz plot of the continuum events is parametrized with an empirical shape. extracted from on-resonance events selected in the mES sidebands and corrected for feed-through from B decays. The contin- uum ∆t distribution is parameterized as the sum of three Gaussian distributions with common mean and three dis- tinct widths that scale the ∆t per-event error, all deter- mined by the fit. IV. RESULTS The maximum-likelihood fit results in a B0 → π+π−π0 event yield of 1847 ± 69, where the error is statistical only. For the U and I coefficients, the results are given together with their statistical and systematic errors in Table IV. The signal is dominated by B0 → ρ±π∓ de- cays. We observe an excess of ρ0π0 events, which is in agreement with our previous upper limit [9], and the lat- est measurement from the Belle collaboration [10]. The result for the ρ(1450) amplitude is in agreement with the findings in τ and e+e− decays [5]. For the relative strong phase between the ρ(770) and the ρ(1450) amplitudes we find (171± 23)◦ (statistical error only), which is compat- ible with the result from τ and e+e− data. A. Systematics studies The most important contribution to the systematic un- certainty stems from the modeling of the Dalitz plot dy- namics for signal. We evaluated this by observing the difference between the true values and Monte Carlo fit re- sults, in which events are generated based on an alterna- tive model. The alternative fit model has, in addition, a uniform Dalitz distribution for the non-resonance events and possible resonances including f0(980), f2(1270), and a low mass S-wave σ. The fit does not find significant number of any of those decays. However, the inclusion of a low mass π+π− S-wave component significantly de- grades our ability to identify ρ0π0 events. . We vary the mass and width of the ρ(770), ρ(1450), and ρ(1700) within ranges that exceed twice the errors found for these parameters in the fits to τ and e+e− data [5], and assign the observed differences in the mea- sured U and I coefficients as systematic uncertainties. To validate the fitting tool, we perform fits on large MC samples with the measured proportions of signal, contin- uum and B-background events. No significant biases are observed in these fits, and the statistical uncertainties on the fit parameters are taken as systematic uncertainties ”Quasi twobody” U±κ = |Aκ|2 ± |Aκ|2 U+0 ρ 0π0 fit fraction 0.237 ± 0.053 ± 0.043 U+− ρ −π+ fit fraction 1.33± 0.11 ± 0.04 U−0 Direct CPV (ρ 0π0) −0.055± 0.098 ± 0.13 U−− Direct CPV (ρ −π+) −0.30± 0.15 ± 0.03 U−+ Direct CPV (ρ +π−) 0.53± 0.15 ± 0.04 ”Quasi twobody” Iκ = Im AκAκ∗ I0 Int. Mixing CPV ρ 0π0 −0.028± 0.058 ± 0.02 I− Int. Mixing CPV ρ −π+ −0.03± 0.10 ± 0.03 I+ Int. Mixing CPV ρ +π− −0.039± 0.097 ± 0.02 ”Interference” U ±,Re(Im) κσ = Re(Im) AκAσ∗ ± AκAσ∗ +− 0.62± 0.54 ± 0.72 +− 0.13± 0.94 ± 0.17 +− 0.38± 0.55 ± 0.28 +− 2.14± 0.91 ± 0.33 +0 0.03± 0.42 ± 0.12 +0 −0.75± 0.40 ± 0.15 +0 −0.93± 0.68 ± 0.08 +0 −0.47± 0.80 ± 0.3 −0 −0.03± 0.40 ± 0.23 −0 −0.52± 0.32 ± 0.08 −0 0.24± 0.61 ± 0.2 −0 −0.42± 0.73 ± 0.28 ”Interference” IReκσ = Re AκAσ∗ −AσAκ∗ IRe+− −0.1 ± 1.9 ± 0.3 IRe+0 0.2 ± 1.1 ± 0.4 IRe−0 0.92± 0.91 ± 0.4 ”Interference” IImκσ = Im AκAσ∗ + AσAκ∗ IIm+− −1.9 ± 1.1 ± 0.1 IIm+0 −0.1 ± 1.1 ± 0.3 IIm−0 0.7 ± 1.0 ± 0.3 TABLE I: Definitions and results for the 26 U and I observ- ables extracted from the fit. We determine the relative values of U and I coefficients to U++ . Another major source of systematic uncertainty is the B-background model. The expected event yields from the background modes are varied according to the uncer- tainties in the measured or estimated branching fractions Since B-backgroundmodes may exhibit CP violation, the corresponding parameters are varied within appropriate uncertainty ranges. Continuum Dalitz plot PDF is extrapolated form mES sideband, and large samples of off-resonance data with loosened requirements on ∆E and the NN are used to compare the distributions of m′ and θ′ between the mES sideband and the signal region. No significant differences are found. We assign as systematic error the effect seen when weighting the continuum Dalitz plot PDF by the ratio of both data sets. This effect is mostly statistical in origin. Other systematic effects due to the signal PDFs com- prise uncertainties in the PDF parameterization, the treatment of misreconstructed events, the tagging per- 0 50 100 150 α (deg) B A B A R P R E L I M I N A R Y FIG. 2: Confidence level functions for α. Indicated by the dashed horizontal lines are the confidence level (C.L.) values corresponding to 1σ and 2σ, respectively. formance, and the modeling of the signal contributions and are estimated using arious data control samples. B. Intepretation of the results The U and I coefficients are related to the quasi-two- body parameters (Tab.IVB) defined in Ref. [4], explic- itly accounting for the presence of interference effects, and are thus exact even for a ρ with finite width. The systematic errors are dominated by the uncertainty on the CP content of the B-related backgrounds. One can transform the experimentally convenient, namely uncor- related, direct CP -violation parameters C and Aρπ into the physically more intuitive quantities A+−ρπ and A−+ρπ . The significance, including systematic uncertainties and calculated by using a mininum χ2 method, for the ob- servation of non-zero direct CP violation is at the 3.0σ level. C = (C+ + C−)/2 0.154 ± 0.090 ± 0.037 S = (S+ + S−)/2 0.01± 0.12 ± 0.028 ∆C = (C+ − C−)/2 0.377 ± 0.091 ± 0.021 ∆S = (S+ − S−)/2 0.06 ± 0.13 ± 0.029 Aρπ = −0.142± 0.041 ± 0.015 A+−ρπ = |κ +−|2−1 |κ+−|2+1 0.03 ± 0.07± 0.03 A−+ρπ = |κ −+|2−1 |κ−+|2+1 −0.38+0.15−0.16 ± 0.07 TABLE II: Quasi twobody parameters definition and results, where C± = and S± = ; κ+− = (q/p)(A−/A+) and κ−+ = (q/p)(A+/A−), so that A+−ρπ (A−+ρπ ) involves only diagrams where the ρ (π) meson is emitted by the W bo- son. A+−ρπ and A−+ρπ are evaluated as − Aρπ+C+Aρπ∆C 1+∆C+AρπC Aρπ−C−Aρπ∆C 1−∆C−AρπC . Their correlation coefficient is 0.62. The measurement of the resonance interference terms allows us to constrain the relative phase δ+− = arg (A+∗A−) between the amplitudes of the decays B0 → ρ−π+ and B0 → ρ+π−. This constraint can be improved with the use of strong isospin symmetry. The amplitudes Aκ represent the sum of tree-level (T κ) and penguin- type (P κ) amplitudes, which have different CKM fac- tors. Here we denote by κ the charge conjugate of κ, where 0 = 0. We define [11] Aκ = T κe−iα + P κ and Aκ = T κe+iα + P κ, where the magnitudes of the CKM factors have been absorbed in the T κ, P κ, T κ and P κ. Using strong isospin symmetry and neglecting isospin- breaking effects, one can identify P 0 = −(P+ + P−)/2 and 9 unknowns have to be determined by the fit. We find for the solution that is favored by the fit δ+− = (34 ± 29)◦, where the errors include both sta- tistical and systematic effects, but only a marginal con- straint on δ+− is obtained for C.L. < 0.05. Finally, following the same procedure, we can also de- rive a constraint on α. The resulting C.L. function versus α is given in Fig. 2 and includes systematic uncertain- ties. Ignoring the mirror solution at α + 180◦, we find α ∈ (75◦, 152◦) at 68% C.L. No constraint on α is achieved at two sigma and beyond. V. CONCLUSIONS We have presented the preliminary measurement of CP -violating asymmetries in B0 → π+π−π0 decays dom- inated by the ρ resonance. The results are obtained from a data sample of 346 million Υ (4S) → BB decays. We perform a time-dependent Dalitz plot analysis. From the measurement of the coefficients of 26 form factor bilin- ears we determine the three CP -violating and two CP - conserving quasi-two-body parameters, where we find a 3.0σ evidence of direct CP violation. Taking advantage of the interference between the ρ resonances in the Dalitz plot, we derive constraints on the relative strong phase between B0 decays to ρ+π− and ρ−π+, and on the an- gle α of the Unitarity Triangle. These measurements are consistent with the expectation from the CKM fit [12]. Acknowledgments The author wishes to thank the conference organizers for an enjoyable and well-organized workshop. This work is supported by the Istituto Nazionale di Fisica Nucle- are (INFN) and the United State Department of Energy (DOE) under contract DE-AC02-76SF00515. [1] H.R. Quinn and A.E. Snyder, Phys. Rev. D48, 2139 (1993). [2] H.J. Lipkin, Y. Nir, H.R. Quinn and A. Snyder, Phys. Rev. D44, 1454 (1991). [3] W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1. [4] BABAR Collaboration (B. Aubert et al.), Phys. Rev. Lett. 91, 201802 (2003); updated preliminary results at BABAR-PLOT-0055 (2003). [5] ALEPH Collaboration, (R. Barate et al.), Z. Phys. C76, 15 (1997); we use updated lineshape fits including new data from e+e− annihilation [13] and τ spectral func- tions [14] (masses and widths in MeV/c2): mρ±(770) = 775.5±0.6, mρ0(770) = 773.1±0.5, Γρ±(770) = 148.2±0.8, Γρ±(770) = 148.0 ± 0.9, mρ(1450) = 1409 ± 12, Γρ(1450) = 500± 37, mρ(1700) = 1749 ± 20, and Γρ(1700) ≡ 235. [6] G.J. Gounaris and J.J. Sakurai, Phys. Rev. Lett. 21, 244 (1968). [7] P. Gay, B. Michel, J. Proriol, and O. Deschamps, “Tag- ging Higgs Bosons in Hadronic LEP-2 Events with Neural Networks.”, In Pisa 1995, New computing techniques in physics research, 725 (1995). [8] BABAR Collaboration, B. Aubert et al., Phys. Rev. D66, 032003 (2002). [9] BABAR Collaboration (B. Aubert et al.), Phys. Rev. Lett. 93, 051802 (2004). [10] Belle Collaboration (J. Dragic et al.), Phys. Rev. D73, 111105 (2006). [11] The BABAR Physics Book, Editors P.F. Harrison and H.R. Quinn, SLAC-R-504 (1998). [12] M. Bona et al., JHEP, 0507 (2005) 028, J. Charles et al., Eur. Phys. J. C41, 1 (2005). [13] R.R. Akhmetshin et al. (CMD-2 Collaboration), Phys. Lett. B527, 161 (2002). [14] ALEPH Collaboration, ALEPH 2002-030 CONF 2002- 019, (July 2002).

0704.0572

New Organic thermally stable materials for optoelectronics devices - A linear spectroscopy study

Microsoft Word - Journal opto-otilia.doc New Organic thermally stable materials for optoelectronics devices - a linear spectroscopy study Otilia Sanda Prelipceanu1,3, Marius Prelipceanu1,3, Ovidiu-Gelu Tudose1,3, Bernd Grimm2,3, Sigurd Schrader1,3 1University of Potsdam, Institute of Physics, Condensed Matter Physics, Am Neuen Palais 10, D-14469, Germany 2IDM, Institute of Thin Film Technology and Micro Sensorics, Kantstr.55, D-14513 Teltow, Germany 3 University of Applied Sciences Wildau, Department of Engineering Physics, D- 15745 Wildau, Germany Contact address: [email protected] Published in Abstract Book of ICPAM07 Conference Iasi, Romania 1. Introduction Thermally stable polymers have attracted a lot of interest due to their potential use as the active component in electronic, optical and optoelectronic applications, such as light-emitting diodes, light emitting electrochemical cells, photodiodes, photovoltaic cells, field effect transistors, optocouplers and optically pumped lasers in solution and solid state. Polymer-based structures are the focus of intensive investigations as mechanically and physically flexible, processible materials for large-area photoemitting and photosensitive devices. Their wide practical application is inhibited by present-day limitations in control over luminescent spectra, sensitivity and efficiency. We report results of our investigations into the use of thermal treatment of poly(p-phenylene vinylene) (PPV) films grown on a variety of substrates (quartz and glass). The samples studied had a thickness in the range 50 - 200 nm. Film thickness, morphology and structural properties were investigated by a range of techniques in particular: atomic force microscope - AFM, DEKTAK method, Ellipsometry and UV-VIS spectroscopy. 2. Experiment Part Thin polymeric films are often used in the microelectronic industry, the development of optoelectronic applications. Homogeneous films with thickness varying from 50 – 200 nm are commonly prepared by spin coating. I this technique, polymers solution is dropped on the substrate surface (our case glass and quartz), which rotates at a given angular velocity during a give period of time. The film thickness is controlled by the concentration of the polymer is solution – 5% PPV in our experiment -, polymer molecular weight, spinning velocity and solvent evaporation rate. The polymers films are annealed at higher temperatures in vacuum and normal atmosphere and after this are investigated and results are compared. This work is concerned with the morphology of the thin films obtained from spin coating when different annealing method. The interactions between substrate, polymer and solvent were qualitatively correlated with the resulting surface morphology of spin coated films and treatment applied. We choose quartz and glass as substrates because this is transparent and easy for the spectroscopic investigations in transmission mode, and P-PPV dissolved in common solvents like toluene and chloroform. Moreover, the determination of the optical absorption and transmission, morphology and stability of the films are important for the development of electronic applications and waveguides[1]. Analytical grade toluene and chloroform were used to prepare the solutions at the polymer concentration 5 mg mL-1. The P-PPV was dissolved in solvents, where no phase separation takes place. The chemical structure for PPV is schematically represented in Figure 1. Cl Cl H2O2 / TeO2 NaOtBu 200oC III IV Figure 1. Synthesis of PPV ( after C.J. Brabec, et al.) 3. Methods and Results Spin coating – The PPV films were prepared by spin coating on commercial quartz and glass substrates. The substrates dimensions of 1 cm x 2 cm were previously cleaned in standard manner and dried under a stream of N2 [2]. All coatings were performed with the spinning velocity of 2000 rpm and the spinning time of 60 seconds. Ellipsometry – The mean thickness and index of refraction (n) of the films were determined by means of ellipsometry in a Plasmos SD2000Automatic ellipsometer, Munich, Germany. The samples characteristics are shown in Table 1. Table 1. Characteristic of PPV films obtained from spin coating. All measurements were performed at 24 ± 2 0 C Dektak measurements – We measured and compared the morphology, thickness and aspect of films before and after treatment. For PPV films before annealing we obtained the thickness of 87 nm witch is shown in figure 3, and in figure 4 after annealing in vacuum we obtained 45 nm and figure 5 and 6 shown aspects layers before and after annealing. [3]. Sample Solvent Thickness (nm) Reflection index P-PPV on quartz (before annealing) Toluene and Chloroform 87 ± 5 1,3 ± 0.05 PPV on Quartz (annealed in vacuum) Toluene and Chloroform 45 ± 5 2,590 ± 0.05 PPV on quartz (annealed in normal atmosphere) Toluene and Chloroform 58 ± 5 2,567 ± 0.05 PPV on quartz (after second annealing in vacuum) Toluene and Chloroform 44 ± 5 2,578 ± 0,05 Figure 2. Spin coating deposition -100 0 100 200 300 400 500 600 700 87 nm distance µm -100 0 100 200 300 400 500 600 700 Distance (µm) Figure 3.PPV film thickness before annealing Figure 4. PPV film thickness after annealing in vacuum 2 h at 200 oC Figure 5. PPV film aspects before annealing Figure 6. PPV film aspects before annealing Atomic Force Microscopy – Measurements were carried out with an instrument from Park Instrument Scientific (Sunnyvale, CA, USA) in non-contact mode in air at room temperature. All AFM images represent unfiltered original data and are displayed in color scale in figure 7, 8 and 9 [4]. Figure 7. AFM image from PPV films after anneling in vacuum at 200 0 C from 2 h Figure 8. AFM image from PPV films after second anneling in vacuum at 200 0 C from 2 Figure 9. AFM image from PPV films after anneling in mormal atmosphere at 200 0 C from 2 h In figure 7 and 8 are shown the image of PPV films after first and second annealing in vacuum at 200 0C for 2 hours. We can see not many changes between films, thichness were almost the same ( 45 nm respectively 44 nm) [5]. Figure 9 shows the surface structure of PPV films after annealing in normal atmophere at 200 0C for 2 hours and the film structure are diferent, compare with films structure which were annealed in vacuum. For all images, films are continuous and smooth with a a root mean square (r.m.s) roughness of 2 – 3 nm, from the annealed in vacuum and 5-7 nm from the normal athmosphere annealed [6]. We get amorphous films in the both case. The main informations observed in Dektak measurements, AFM investigations and ellipsomentry are: the surface roughness of the films depend on the speed of heating, slow heat up raises the roughness, quick heat up leads to more smoth films. The same situation is meet in case of vacuum annealing and normal atmosphere annealing. More over, the thickness of the layers is reduced to about the haltf after annealing in the both case. The PPV layers are not orienteded in the both annealing method [6]. UV/VIS measurements - were made using the Perkin Elmer – UV/VIS Spectrometer Lambda 16. Spectra were acquired from 300 to 900 nm for optical excitation. Figure 9, 10 and 11 shows a set of absorption spectra of PPV films obtained from spin coating converted by heating under vacuum and normal atmosphere at 200oC for 2 hours [7]. Spectra were normalized by dividing absorption spectrum of each individual sample by its absorption at the maximum. In this way relative changes within the spectrum and between the spectra are easily observed. One can notice differences in the position of the absorption maxima of PPV films prepared in different annealing method. The changes in the optical spectra of PPV films obtained from the precursors prepared in vacuum conditions and normal atmosphere condition are consistent with earlier observations in figure 10, 11 and 12 [8]. 4. Conclusions We summarize our findings as follows: (i) Annealing of PPV films causes ordering of polymer chains and, as a result, change in the luminescence intensity and spectra. (ii) spectral characteristics of the converted PPV-precursor strongly dependent on the preparation condition of the precursor (iii) the thickness of layers is reduced to about the half after annealing. (iv) The surface roughness of the films depends on the speed of heating: slow heat up raises the roughness; quick heat up leads to more smooth films. (v) PPV is thermally stable up to more than 500 0C measured by TGA. (vi) We get amorphous films in spin coating deposition. 5. Acknowledgements Financial support of the European Commission under contract number: FP6 – 505478-1 ODEON - Project and RTN EUROFET – Project is gratefully acknowledged. 300 400 500 600 700 800 900 1000 after annealing in normal atmosphere before annealing after annealing in vacuum wavelength (nm) 100 200 300 400 500 600 700 800 900 1000 before annealing after 1st annealing (in vacuum) after 2nd annealing (in vacuum) X Axis Title Figure 10. UV-VIS spectra of PPV films obtained in vacuum conversion and normal atmosphere conversion Figure 11. UV-VIS spectra of PPV films obtained in vacuum conversion made several times 300 400 500 600 700 800 900 before annealing after annealing wavelength (nm) Figure 12. UV-VIS spectra comparation of PPV films after and before vacuum conversion 6. References [1] L. Bakueva, E.H. Sargent, R. Resendes, A. Bartole, I. Manners, J. Mater. Sci.: Mater. Electron. 12 (2001) 21. [2] M. Pope, C.E. Swenberg, Electronic Processes in Organic Crystals and Polymers, Oxford Science Publications, Oxford, 1999. [3] L. Bakueva, S. Musikhin, E.H. Sargent, A. Shik, 2001. MRS Fall Meeting, Boston, November 26–30, 2001 Book of Abstracts. [4] D. Moses, A. Dogariu, A.J. Heeger, Synth. Met. 116 (2001) 19. [5] B. Hu, F.E. Karaz, Chem. Phys. 227 (1998) 263. [6] X.-R. Zeng, T.-M. Ko, J. Polym. Sci. B 35 (1997) 1993. [7] C.E. Lee, C.-H. Jin, Synthet. Met. 117 (2001) 27. [8] D.F.S. Petri, J.Braz.Chem.Soc. vol. 13, no 5, 695-699,2002.

0704.0573

Elativistic treatment in}$D$ - Dimensions to a spin-zero particle with noncentral equal scalar and vector ring-shaped Kratzer potential

arXiv:0704.0573v1 [quant-ph] 4 Apr 2007 Relativistic treatment in D-dimensions to a spin-zero particle with noncentral equal scalar and vector ring-shaped Kratzer potential Sameer M. Ikhdair∗ and Ramazan Sever† ∗Department of Physics, Near East University, Nicosia, North Cyprus, Mersin 10, Turkey †Department of Physics, Middle East Technical University, 06531 Ankara, Turkey. (November 4, 2018) Abstract The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer potential plus ring-shaped potential is solved analytically by means of the conventional Nikiforov-Uvarov method. The exact energy bound-states and the corresponding wave functions of the Klein-Gordon are obtained in the presence of the noncentral equal scalar and vector potentials. The results obtained in this work are more general and can be reduced to the standard forms in three-dimensions given by other works. Keywords: Energy eigenvalues and eigenfunctions, Klein-Gordon equa- tion, Kratzer potential, ring-shaped potential, non-central potentials, Niki- forov and Uvarov method. PACS numbers: 03.65.-w; 03.65.Fd; 03.65.Ge. ∗[email protected] †[email protected] http://arxiv.org/abs/0704.0573v1 I. INTRODUCTION In various physical applications including those in nuclear physics and high energy physics [1,2], one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein- Gordon and Dirac wave equations are frequently used to describe the particle dynamics in relativistic quantum mechanics. The Klein-Gordon equation has also been used to under- stand the motion of a spin-0 particle in large class of potentials. In recent years, much efforts have been paid to solve these relativistic wave equations for various potentials by using different methods. These relativistic equations contain two objects: the four-vector linear momentum operator and the scalar rest mass. They allow us to introduce two types of potential coupling, which are the four-vector potential (V) and the space-time scalar potential (S). Recently, many authors have worked on solving these equations with physical potentials including Morse potential [3], Hulthen potential [4], Woods-Saxon potential [5], Pösch-Teller potential [6], reflectionless-type potential [7], pseudoharmonic oscillator [8], ring-shaped har- monic oscillator [9], V0 tanh 2(r/r0) potential [10], five-parameter exponential potential [11], Rosen-Morse potential [12], and generalized symmetrical double-well potential [13], etc. It is remarkable that in most works in this area, the scalar and vector potentials are almost taken to be equal (i.e., S = V ) [2,14]. However, in some few other cases, it is considered the case where the scalar potential is greater than the vector potential (in order to guar- antee the existence of Klein-Gordon bound states) (i.e., S > V ) [15-19]. Nonetheless, such physical potentials are very few. The bound-state solutions for the last case is obtained for the exponential potential for the s-wave Klein-Gordon equation when the scalar potential is greater than the vector potential [15]. The study of exact solutions of the nonrelativistic equation for a class of non-central po- tentials with a vector potential and a non-central scalar potential is of considerable interest in quantum chemistry [20-22]. In recent years, numerous studies [23] have been made in analyzing the bound states of an electron in a Coulomb field with simultaneous presence of Aharanov-Bohm (AB) [24] field, and/or a magnetic Dirac monopole [25], and Aharanov- Bohm plus oscillator (ABO) systems. In most of these works, the eigenvalues and eigen- functions are obtained by means of seperation of variables in spherical or other orthogonal curvilinear coordinate systems. The path integral for particles moving in non-central poten- tials is evaluated to derive the energy spectrum of this system analytically [26]. In addition, the idea of SUSY and shape invariance is also used to obtain exact solutions of such non- central but seperable potentials [27,28]. Very recently, the conventional Nikiforov-Uvarov (NU) method [29] has been used to give a clear recipe of how to obtain an explicit exact bound-states solutions for the energy eigenvalues and their corresponding wave functions in terms of orthogonal polynomials for a class of non-central potentials [30]. Another type of noncentral potentials is the ring-shaped Kratzer potential, which is a combination of a Coulomb potential plus an inverse square potential plus a noncentral angu- lar part [31,32]. The Kratzer potential has been used to describe the vibrational-rotational motion of isolated diatomic molecules [33] and has a mixed-energy spectrum containing both bound and scattering states with bound-states have been widely used in molecular spec- troscopy [34]. The ring-shaped Kratzer potential consists of radial and angular-dependent potentials and is useful in studying ring-shaped molecules [22]. In taking the relativistic effects into account for spin-0 particle in the presence of a class of noncentral potentials, Ya- suk et al [35] applied the NU method to solve the Klein-Gordon equation for the noncentral Coulombic ring-shaped potential [21] for the case V = S. Further, Berkdemir [36] also used the same method to solve the Klein-Gordon equation for the Kratzer-type potential. Recently, Chen and Dong [37] proposed a new ring-shaped potential and obtained the exact solution of the Schrödinger equation for the Coulomb potential plus this new ring- shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. This type of potential used by Chen and Dong [37] appears to be very similar to the potential used by Yasuk et al [35]. Moreover, Cheng and Dai [38], proposed a new potential consisting from the modified Kratzer’s potential [33] plus the new proposed ring-shaped potential in [37]. They have presented the energy eigenvalues for this proposed exactly-solvable non-central potential in three dimensional (i.e., D = 3)- Schrödinger equation by means of the NU method. The two quantum systems solved by Refs [37,38] are closely relevant to each other as they deal with a Coulombic field interaction except for a slight change in the angular momentum barrier acts as a repulsive core which is for any arbitrary angular momentum ℓ prevents collapse of the system in any dimensional space due to the slight perturbation to the original angular momentum barrier. Very recently, we have also applied the NU method to solve the Schrödinger equation in any arbitrary D- dimension to this new modified Kratzer-type potential [39,40]. The aim of the present paper is to consider the relativistic effects for the spin-0 parti- cle in our recent works [39,40]. So we want to present a systematic recipe to solving the D-dimensional Klein-Gordon equation for the Kratzer plus the new ring-shaped potential proposed in [38] using the simple NU method. This method is based on solving the Klein- Gordon equation by reducing it to a generalized hypergeometric equation. This work is organized as follows: in section II, we shall present the Klein-Gordon equation in spherical coordinates for spin-0 particle in the presence of equal scalar and vector noncentral Kratzer plus the new ring-shaped potential and we also separate it into radial and angular parts. Section III is devoted to a brief description of the NU method. In section IV, we present the exact solutions to the radial and angular parts of the Klein- Gordon equation in D-dimensions. Finally, the relevant conclusions are given in section II. THE KLEIN-GORDON EQUATION WITH EQUAL SCALAR AND VECTOR POTENTIALS In relativistic quantum mechanics, we usually use the Klein-Gordon equation for de- scribing a scalar particle, i.e., the spin-0 particle dynamics. The discussion of the relativistic behavior of spin-zero particles requires understanding the single particle spectrum and the exact solutions to the Klein Gordon equation which are constructed by using the four-vector potential Aλ (λ = 0, 1, 2, 3) and the scalar potential (S). In order to simplify the solution of the Klein-Gordon equation, the four-vector potential can be written as Aλ = (A0, 0, 0, 0). The first component of the four-vector potential is represented by a vector potential (V ), i.e., A0 = V. In this case, the motion of a relativistic spin-0 particle in a potential is described by the Klein-Gordon equation with the potentials V and S [1]. For the case S ≥ V, there exist bound-state (real) solutions for a relativistic spin-zero particle [15-19]. On the other hand, for S = V, the Klein-Gordon equation reduces to a Schrödinger-like equation and thereby the bound-state solutions are easily obtained by using the well-known methods developed in nonrelativistic quantum mechanics [2]. The Klein-Gordon equation describing a scalar particle (spin-0 particle) with scalar S(r, θ, ϕ) and vector V (r, θ, ϕ) potentials is given by [2,14] 2 − [ER − V (r, θ, ϕ)/2] + [µ+ S(r, θ, ϕ)/2] ψ(r, θ, ϕ) = 0, (1) where ER,P and µ are the relativistic energy, momentum operator and rest mass of the particle, respectively. The potential terms are scaled in (1) by Alhaidari et al [14] so that in the nonrelativistic limit the interaction potential becomes V. In this work, we consider the equal scalar and vector potentials case, that is, S(r, θ, ϕ) = V (r, θ, ϕ) with the recently proposed general non-central potential taken in the form of the Kratzer plus ring-shaped potential [38-40]: V (r, θ, ϕ) = V1(r) + V2(θ) V3(ϕ) r2 sin2 θ , (2) V1(r) = − , V2(θ) = Cctg 2θ, V3(ϕ) = 0, (3) where A = 2a0r0, B = a0r 0 and C is positive real constant with a0 is the dissociation energy and r0 is the equilibrium internuclear distance [33]. The potentials in Eq. (3) introduced by Cheng-Dai [38] reduce to the Kratzer potential in the limiting case of C = 0 [33]. In fact the energy spectrum for this potential can be obtained directly by considering it as special case of the general non-central seperable potentials [30]. In the relativistic atomic units (h̄ = c = 1), the D-dimensional Klein-Gordon equation in (1) becomes [41] sin θ sin θ sin2 θ − (ER + µ) V1(r) + V2(θ) V3(ϕ) r2 sin2 θ E2R − µ ψ(r, θ, ϕ) = 0. (4) with ψ(r, θ, ϕ) being the spherical total wave function separated as follows ψnjm(r, θ, ϕ) = R(r)Y j (θ, ϕ), R(r) = r −(D−1)/2g(r), Y mj (θ, ϕ) = H(θ)Φ(ϕ). (5) Inserting Eqs (3) and (5) into Eq. (4) and using the method of separation of variables, the following differential equations are obtained: dR(r) j(j +D − 2) + α22 α21 − R(r) = 0, (6) sin θ sin θ m2 + Cα22 cos sin2 θ + j(j +D − 2) H(θ) = 0, (7) d2Φ(ϕ) +m2Φ(ϕ) = 0, (8) where α21 = µ−ER, α 2 = µ+ER, m and j are constants and with m 2 and λj = j(j+D−2) are the separation constants. For a nonrelativistic treatment with the same potential, the Schrödinger equation in spherical coordinates is sin θ sin θ sin2 θ ENR − V1(r)− V2(θ) V3(ϕ) r2 sin2 θ ψ(r, θ, ϕ) = 0. (9) where µ and ENR are the reduced mass and the nonrelativistic energy, respectively. Besides, the spherical total wave function appearing in Eq. (9) has the same representation as in Eq. (5) but with the transformation j → ℓ. Inserting Eq. (5) into Eq. (9) leads to the following differential equations [39,40]: dR(r) ENR + R(r) = 0, (10) sin θ sin θ m2 + 2µC cos2 θ sin2 θ + ℓ(ℓ+D − 2) H(θ) = 0, (11) d2Φ(ϕ) +m2Φ(ϕ) = 0, (12) where m2 and λℓ = ℓ(ℓ + D − 2) are the separation constants. Equations (6)-(8) have the same functional form as Eqs (10)-(12). Therefore, the solution of the Klein-Gordon equation can be reduced to the solution of the Schrödinger equation with the appropriate choice of parameters: j → ℓ, α21 → −ENR and α 2 → 2µ. The solution of Eq. (8) is well-known periodic and must satisfy the period boundary condition Φ(ϕ + 2π) = Φ(ϕ) which is the azimuthal angle solution: Φm(ϕ) = exp(±imϕ), m = 0, 1, 2, ..... (13) Additionally, Eqs (6) and (7) are radial and polar angle equations and they will be solved by using the Nikiforov-Uvarov (NU) method [29] which is given briefly in the following section. III. NIKIFOROV-UVAROV METHOD The NU method is based on reducing the second-order differential equation to a gener- alized equation of hypergeometric type [29]. In this sense, the Schrödinger equation, after employing an appropriate coordinate transformation s = s(r), transforms to the following form: ψ′′n(s) + τ̃(s) ψ′n(s) + σ̃(s) σ2(s) ψn(s) = 0, (14) where σ(s) and σ̃(s) are polynomials, at most of second-degree, and τ̃ (s) is a first-degree polynomial. Using a wave function, ψn(s), of the simple ansatz: ψn(s) = φn(s)yn(s), (15) reduces (14) into an equation of a hypergeometric type σ(s)y′′n(s) + τ(s)y n(s) + λyn(s) = 0, (16) where σ(s) = π(s) φ′(s) , (17) τ(s) = τ̃(s) + 2π(s), τ ′(s) < 0, (18) and λ is a parameter defined as λ = λn = −nτ ′(s)− n (n− 1) σ′′(s), n = 0, 1, 2, .... (19) The polynomial τ(s) with the parameter s and prime factors show the differentials at first degree be negative. It is worthwhile to note that λ or λn are obtained from a particular solution of the form y(s) = yn(s) which is a polynomial of degree n. Further, the other part yn(s) of the wave function (14) is the hypergeometric-type function whose polynomial solutions are given by Rodrigues relation yn(s) = [σn(s)ρ(s)] , (20) where Bn is the normalization constant and the weight function ρ(s) must satisfy the con- dition [29] w(s) = w(s), w(s) = σ(s)ρ(s). (21) The function π and the parameter λ are defined as π(s) = σ′(s)− τ̃ (s) σ′(s)− τ̃ (s) − σ̃(s) + kσ(s), (22) λ = k + π′(s). (23) In principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in (22) can be arranged to be the square of a polynomial of first degree [29]. This is possible only if its discriminant is zero. In this case, an equation for k is obtained. After solving this equation, the obtained values of k are substituted in (22). In addition, by comparing equations (19) and (23), we obtain the energy eigenvalues. IV. EXACT SOLUTIONS OF THE RADIAL AND ANGLE-DEPENDENT EQUATIONS A. Separating variables of the Klein-Gordon equation We seek to solving the radial and angular parts of the Klein-Gordon equation given by Eqs (6) and (7). Equation (6) involving the radial part can be written simply in the following form [39-41]: d2g(r) (M − 1)(M − 3) − α22 + α21α g(r) = 0, (24) where M = D + 2j. (25) On the other hand, Eq. (7) involving the angular part of Klein-Gordon equation retakes the simple form d2H(θ) + ctg(θ) dH(θ) m2 + Cα22 cos sin2 θ − j(j +D − 2) H(θ) = 0. (26) Thus, Eqs (24) and (26) have to be solved latter through the NU method in the following subsections. B. Eigenvalues and eigenfunctions of the angle-dependent equation In order to apply NU method [29,30,33,35,36,38-40,42-44], we use a suitable transforma- tion variable s = cos θ. The polar angle part of the Klein Gordon equation in (26) can be written in the following universal associated-Legendre differential equation form [38-40] d2H(s) 1− s2 dH(s) (1− s2)2 j(j +D − 2)(1− s2)−m2 − Cα22s H(θ) = 0. (27) Equation (27) has already been solved for the three-dimensional Schrödinger equation through the NU method in [38]. However, the aim in this subsection is to solve with different parameters resulting from the D-space-dimensions of Klein-Gordon equation. Further, Eq. (27) is compared with (14) and the following identifications are obtained τ̃(s) = −2s, σ(s) = 1− s2, σ̃(s) = −m′2 + (1− s2)ν ′, (28) where ν ′ = j′(j′ +D − 2) = j(j +D − 2) + Cα22, m ′2 = m2 + Cα22. (29) Inserting the above expressions into equation (22), one obtains the following function: π(s) = ± (ν ′ − k)s2 + k − ν ′ +m′2, (30) Following the method, the polynomial π(s) is found in the following possible values π(s) =   m′s for k1 = ν ′ −m′2, −m′s for k1 = ν ′ −m′2, m′ for k2 = ν −m′ for k2 = ν ′. Imposing the condition τ ′(s) < 0, for equation (18), one selects k1 = ν ′ −m′2 and π(s) = −m′s, (32) which yields τ(s) = −2(1 +m′)s. (33) Using equations (19) and (23), the following expressions for λ are obtained, respectively, λ = λn = 2ñ(1 +m ′) + ñ(ñ− 1), (34) λ = ν ′ −m′(1 +m′). (35) We compare equations (34) and (35), the new angular momentum j values are obtained as j = − (D − 2) (D − 2)2 + (2ñ+ 2m′ + 1)2 − 4Cα22 − 1, (36) j′ = − (D − 2) (D − 2)2 + (2ñ+ 2m′ + 1)2 − 1. (37) Using Eqs (15)-(17) and (20)-(21), the polynomial solution of yn is expressed in terms of Jacobi polynomials [39,40] which are one of the orthogonal polynomials: Hñ(θ) = Nñ sin m′(θ)P (m′,m′) (cos θ), (38) where Nñ = (ñ+m′)! (2ñ+2m′+1)(ñ+2m′)!ñ! is the normalization constant determined by [Hñ(s)] ds = 1 and using the orthogonality relation of Jacobi polynomials [35,45,46]. Besides ñ = − (1 + 2m′) (2j + 1)2 + 4j(D − 3) + 4Cα22, (39) where m′ is defined by equation (29). C. Eigensolutions of the radial equation The solution of the radial part of Klein-Gordon equation, Eq. (24), for the Kratzer’s potential has already been solved by means of NU-method in [39]. Very recently, using the same method, the problem for the non-central potential in (2) has been solved in three dimensions (3D) by Cheng and Dai [36]. However, the aim of this subsection is to solve the problem with a different radial separation function g(r) in any arbitrary dimensions. In what follows, we present the exact bound-states (real) solution of Eq. (24). Letting ε2 = α21α 2, 4γ 2 = (M − 1)(M − 3) + 4Bα22, β 2 = Aα22, (40) and substituting these expressions in equation (24), one gets d2g(r) −ε2r2 + β2r − γ2 g(r) = 0. (41) To apply the conventional NU-method, equation (41) is compared with (14), resulting in the following expressions: τ̃ (r) = 0, σ(r) = r, σ̃(r) = −ε2r2 + β2r − γ2. (42) Substituting the above expressions into equation (22) gives π(r) = 4ε2r2 + 4(k − β2)r + 4γ2 + 1. (43) Therefore, we can determine the constant k by using the condition that the discriminant of the square root is zero, that is k = β2 ± ε 4γ2 + 1, 4γ2 + 1 = (D + 2j − 2)2 + 4Bα22. (44) In view of that, we arrive at the following four possible functions of π(r) : π(r) =   εr + 1 4γ2 + 1 for k1 = β 2 + ε 4γ2 + 1, εr + 1 4γ2 + 1 for k1 = β 2 + ε 4γ2 + 1, εr − 1 4γ2 + 1 for k2 = β 2 − ε 4γ2 + 1, εr − 1 4γ2 + 1 for k2 = β 2 − ε 4γ2 + 1. The correct value of π(r) is chosen such that the function τ(r) given by Eq. (18) will have negative derivative [29]. So we can select the physical values to be k = β2 − ε 4γ2 + 1 and π(r) = 4γ2 + 1 , (46) which yield τ(r) = −2εr + (1 + 4γ2 + 1), τ ′(r) = −2ε < 0. (47) Using Eqs (19) and (23), the following expressions for λ are obtained, respectively, λ = λn = 2nε, n = 0, 1, 2, ..., (48) λ = δ2 − ε(1 + 4γ2 + 1). (49) So we can obtain the Klein Gordon energy eigenvalues from the following relation: 1 + 2n+ (D + 2j − 2)2 + 4(µ+ ER)B µ−ER = A µ+ ER, (50) and hence for the Kratzer plus the new ring-shaped potential, it becomes 1 + 2n+ (D + 2j − 2)2 + 4a0r20(µ+ ER) µ−ER = 2a0r0 µ+ ER, (51) with j defined in (36). Although Eq. (51) is exactly solvable for ER but it looks to be little complicated. Further, it is interesting to investigate the solution for the Coulomb potential. Therefore, applying the following transformations: A = Ze2, B = 0, and j = ℓ, the central part of the potential in (3) turns into the Coulomb potential with Klein Gordon solution for the energy spectra given by ER = µ 2q2e2 q2e2 + (2n+ 2ℓ+D − 1)2 , n, ℓ = 0, 1, 2, ..., (52) where q = Ze is the charge of the nucleus. Further, Eq. (52) can be expanded as a series in the nucleus charge as ER = µ− 2µq2e2 (2n + 2ℓ+D − 1)2 2µq4e4 (2n+ 2ℓ+D − 1)4 −O(qe)6, (53) The physical meaning of each term in the last equation was given in detail by Ref. [36]. Besides, the difference from the conventional nonrelativistic form is because of the choice of the vector V (r, θ, ϕ) and scalar S(r, θ, ϕ) parts of the potential in Eq. (1). Overmore, if the value of j obtained by Eq.(36) is inserted into the eigenvalues of the radial part of the Klein Gordon equation with the noncentral potential given by Eq. (51), we finally find the energy eigenvalues for a bound electron in the presence of a noncentral potential by Eq. (2) as 1 + 2n+ (2j′ +D − 2)2 + 4(a0r20 − C)(µ+ ER) µ−ER = 2a0r0 µ+ ER, (54) where m′ = m2 + C(µ+ ER) and ñ is given by Eq. (39). On the other hand, the solution of the Schrödinger equation, Eq. (9), for this potential has already been obtained by using the same method in Ref. [39] and it is in the Coulombic-like form: ENR = − 8µa20r 2n+ 1 + (2ℓ′ +D − 2)2 + 8µ(a0r20 − C) ]2 , n = 0, 1, 2, ... (55) 2ℓ′ +D − 2 = (D − 2)2 + (2ñ + 2m′ + 1)2 − 1, (56) wherem′ = m2 + 2µC. Also, applying the following appropriate transformation: µ+ER → 2µ, µ−ER → − ENR, j → ℓ to Eq. (54) provides exactly the nonrelativistic limit given by Eq. (55). In what follows, let us now turn attention to find the radial wavefunctions for this potential. Substituting the values of σ(r), π(r) and τ(r) in Eqs (42), (45) and (47) into Eqs. (17) and (21), we find φ(r) = r(ζ+1)/2e−εr, (57) ρ(r) = rζe−2εr, (58) where ζ = 4γ2 + 1. Then from equation (20), we obtain ynj(r) = Bnjr −ζe2εr rn+ζe−2εr , (59) and the wave function g(r) can be written in the form of the generalized Laguerre polyno- mials as g(ρ) = Cnj )(1+ζ)/2 e−ρ/2Lζn(ρ), (60) where for Kratzer’s potential we have (D + 2j − 2)2 + 4a0r20(µ+ ER), ρ = 2εr. (61) Finally, the radial wave functions of the Klein-Gordon equation are obtained R(ρ) = Cnj )(ζ+2−D)/2 e−ρ/2Lζn(ρ), (62) where Cnj is the normalization constant to be determined below. Using the normalization condition, R2(r)rD−1dr = 1, and the orthogonality relation of the generalized Laguerre polynomials, zη+1e−z [Lηn(z)] (2n+η+1)(n+η)! , we have Cnj = µ2 − E2R )1+ ζ (2n+ ζ + 1) (n+ ζ)! . (63) Finally, we may express the normalized total wave functions as ψ(r, θ, ϕ) = µ2 − E2R )1+ ζ (ñ+m′)! √√√√(2ñ+ 2m ′ + 1)(ñ+ 2m′)!ñ!n! 2π (2n+ ζ + 1) (n + ζ)! (ζ+2−D) 2 exp(− µ2 − E2Rr)L µ2 − E2Rr) sin m′(θ)P (m ′,m′) n (cos θ) exp(±imϕ). (64) where ζ is defined in Eq. (61) and m′ is given after the Eq. (54). V. CONCLUSIONS The relativistic spin-0 particle D-dimensional Klein-Gordon equation has been solved easily for its exact bound-states with equal scalar and vector ring-shaped Kratzer potential through the conventional NU method. The analytical expressions for the total energy levels and eigenfunctions of this system can be reduced to their conventional three-dimensional space form upon setting D = 3. Further, the noncentral potentials treated in [30] can be introduced as perturbation to the Kratzer’s potential by adjusting the strength of the coupling constant C in terms of a0, which is the coupling constant of the Kratzer’s potential. Additionally, the radial and polar angle wave functions of Klein-Gordon equation are found in terms of Laguerre and Jacobi polynomials, respectively. The method presented in this paper is general and worth extending to the solution of other interaction problems. This method is very simple and useful in solving other complicated systems analytically without given a restiction conditions on the solution of some quantum systems as the case in the other models. We have seen that for the nonrelativistic model, the exact energy spectra can be obtained either by solving the Schrödinger equation in (9) (cf. Ref. [39] or Eq. (55)) or by applying appropriate transformation to the relativistic solution given by Eq. (54). Finally, we point out that these exact results obtained for this new proposed form of the potential (2) may have some interesting applications in the study of different quantum mechanical systems, atomic and molecular physics. ACKNOWLEDGMENTS This research was partially supported by the Scientific and Technological Research Coun- cil of Turkey. S.M. 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0704.0574

On the potential of transit surveys in star clusters: Impact of correlated noise and radial velocity follow-up

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 4 November 2018 (MN LATEX style file v2.2) On the potential of transit surveys in star clusters: Impact of correlated noise and radial velocity follow-up Suzanne Aigrain1,2 and Frédéric Pont3 1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom 2School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom 2Observatoire Astronomique de l’Université de Genève, 51, chemin des Maillettes, CH-1290 Sauverny, Switzerland Accepted . . . Received . . . ; in original form. . . ABSTRACT We present an extension of the formalism recently proposed by Pepper & Gaudi to evaluate the yield of transit surveys in homogeneous stellar systems, incorporating the impact of correlated noise on transit time-scales on the detectability of transits, and simultaneously incorporating the magnitude limits imposed by the need for radial velocity follow-up of transit candidates. New expressions are derived for the different contributions to the noise budget on transit time-scales and the least-squares detection statistic for box-shaped transits, and their behaviour as a function of stellar mass is re-examined. Correlated noise that is constant with apparent stellar magnitude implies a steep decrease in detection probability at the high mass end which, when considered jointly with the radial velocity requirements, can severely limit the potential of oth- erwise promising surveys in star clusters. However, we find that small-aperture, wide field surveys may detect hot Neptunes whose radial velocity signal can be measured with present-day instrumentation in very nearby (< 100 pc) clusters. Key words: planetary systems – surveys – techniques: photometric – open clusters and associations: general. 1 INTRODUCTION Open Clusters have long been used as laboratories to test our understanding of star formation and stellar evolution, as each contains a (sometimes large) sample of stars with relatively well-known and common properties (age, composition, environment) but spanning a wide range of masses. For the same reasons, since the discovery of the first extra-solar planet around a Sun-like star in the field just over a decade ago (Mayor & Queloz 1995), the possibility of discovery extra-solar planets in open clusters has been a tantalising goal. Open Clusters tend to be relatively distant, and their members relatively faint compared to the field stars usually targeted by radial velocity surveys. Most of the projects searching for extra-solar planets in Open Clusters therefore employ the transit technique, which is particularly well suited to dense stellar environments and has the additional advantage of providing a direct measurement of the planet to star radius ratio. Recent or ongoing transit surveys in Open Clusters include the UStAPS (the University of St Andrews Planet Search, Street et al. 2003; Bramich et al. 2005; Hood et al. 2005), EXPLORE- OC (von Braun et al. 2005), PISCES (Planets in Stellar Clusters Extensive Search, Mochejska et al. 2005, 2006), STEPSS (Survey for Transiting Extra-solar Planets in Stellar Systems, Burke et al. 2006) and the Monitor project (Aigrain et al. 2007). Understanding the factors that affect the yield of such a survey is vital not only to maximise its detection rate, but also to enable the interpretation of the results of the survey, including in the case of non-detections, in terms of constraints on the incidence and parameter distributions of planetary companions. Recently, Pepper & Gaudi (2005a) (hereafter PG05a) introduced an analytical formalism to estimate the rate of detection of exo-planets via the transit method in stellar systems. One particularly interesting result is the fact that the probability that transits of a given system in a given cluster are detectable, if they occur, is a very slowly varying function of stellar mass in the regime where the photometric performance is dominated by the source photon noise, but drops sharply with stellar mass in the background-dominated regime. This implies c© 0000 RAS http://arxiv.org/abs/0704.0574v1 2 S. Aigrain et al. that the number of detections to be expected from a given survey is roughly proportional to the number of stars with source photon counts above the sky background level in that survey. In a follow-up paper, Pepper & Gaudi (2005b) (hereafter PG05b) applied the aforementioned formalism to young open clusters, showing that transit surveys focusing on these systems have the potential to detect transiting Neptune and even Earth-sized planets, by making use of the fact that low-mass stars are relatively bright at early ages, and that their smaller radius gives rise to deeper transits for a given planet radius. This opens up the tantalising possibility of detecting transits of terrestrial planets from the ground, and what is more of doing so in young systems, where one might obtain particularly interesting constraints on the formation and evolution of extra-solar planets (Aigrain et al. 2007). The formalism of PG05a assumes that the photometric errors on each star and in each observation are independent of each other (i.e. that the noise is white). However, an a posteriori analysis of the detection threshold of the OGLE transit survey in the light of their RV follow-up observations of OGLE candidate transits (Pont et al. 2005) demonstrated that the effective detection threshold is significantly higher than that expected for white noise only, suggesting that correlated noise on transit time-scales might be present in OGLE light curves. Pont, Zucker & Queloz (2006) (hereafter PZQ06) since developed a set of methods for evaluating the amount of correlated noise on transit time-scales in the light curves of transit surveys, and applied them to the OGLE light curves to show the latter do indeed contain correlated noise at the level of a few mmag. Similar analysis of light curves from other transit surveys (see e.g. Smith et al. 2006 and Irwin et al. 2007, and Pont 2007 for an overview) has since shown that they are also affected by correlated noise at a similar level. Because correlated noise does not average out as more observations of a given transit event are obtained, as white noise does, it is generally correlated noise which dominates over white noise in determining the detectability of transits around all but the faintest stars in a given field survey. A number of effects can give rise to correlated noise, including seeing-dependent contamination of the flux measured for a given star by flux from neighbouring stars, pointing drifts combined with flat-fielding errors, and imperfect sky subtraction – some or all of which may be important in a given survey depending on the telescope/instrument combination used and the observing strategy. Photometry alone does not allow the mass of the companion causing the transits to be ascertained, and radial velocity measurements are thus generally needed to confirm the planetary nature of a transit event. As pointed out by PG05a, this effectively imposes an apparent magnitude limit for transit detections to be confirmable, as accurate radial velocity measurements of faint stars are extremely expensive in terms of large telescope time. As noise that is correlated on transit timescales reduces the detectability of transits around the brightest stars in a given survey, but the need to perform radial velocity follow-up implies that only around the brightest stars can transits be confirmed, both effects must be incorporated in the scaling laws used to estimate the number of detections expected from a given survey. The present paper attempts to do this by extending the formalism of PG05a to include correlated noise and by translating the magnitude limit imposed by radial velocity follow-up into a cluster-specific mass limit. Section 2 briefly sketches out the basics of the formalism of PG05a and describes how one or more additional noise terms representing correlated noise terms can be incorporated in it. The impact of these modifications on the noise budget on transit time-scales and on the transit detection probability as a function of mass are investigated in Section 3. Considerations external to the transit search itself, including radial velocity follow-up, are introduced in Section 4. Finally, the practical implications of the resulting formalism for Open Cluster transit searches are briefly explored in Section 5. 2 INTRODUCING RED NOISE TERMS 2.1 Overall formalism This formalism is described in detail in PG05a, and only its major characteristics are sketched out here, so as to allow the modifications implied by the presence of correlated noise to be made clear. PG05a compute the number of transiting planets with periods between P and P + dP and radii between r and r + dr that can be detected around stars with masses between M and M + dM in a given stellar system as d3Ndet dM dr dP = N⋆fp dr dP Ptot(M, P, r) . (1) where Ndet is the number of detected transiting planets, N⋆ is the total number of stars in the system, d 2p/dr dP is the probability that a planet around a star in the system has a period between P and P + dP and a radius between r and r+ dr, fp is the fraction of stars in the system with planets, Ptot(M, P, r) is the probability that a planet of radius r and orbital period P will be detected around a star of mass M , and dn/dM is the mass function of the stars in the system, normalised over the mass range corresponding to N⋆. Following Gaudi (2000), PG05a separate Ptot(M, P, r) into three factors: Ptot(M, P, r) = Ptr(M, P )PS/N(M, P, r)PW(P ) (2) c© 0000 RAS, MNRAS 000, 000–000 Transit Surveys in star clusters 3 Ptr is the probability that a planet transits its parent star, PS/N is the probability that, should a transit occur during a night of observing, it will yield a signal-to-noise ratio (S/N) that is higher than some threshold value, and PW is the window function that describes the probability that more than one transit will occur during the observations. PG05a’s expression for the transit probability is used without modification Ptr = where R is the star radius and a the orbital distance. The S/N of a set of transits is S/N = ∆χ2tr , where ∆χ2tr is the difference in χ 2 between a constant flux and a boxcar transit fit to the data. PZQ06 give the general expression: d2 n2 where d is the transit depth, σd is the uncertainty on the transit depth, n is the number of in-transit data points and C is the covariance matrix the in-transit flux measurements1. If the noise is uncorrelated, the non-diagonal elements of C are zero, Cij = σ2i where σi is the uncertainty on the i th flux measurement. Additionally, if this uncertainty is constant, i.e. σi = σw, Equation (4) further reduces to: tr = n , (5) which, for single transits, is equivalent to Equation (4) in PG05a. Note that the notation adopted here matches that of PZQ06, and thus differs that of PG05a,b. In particular, the symbols Ntr and n, used in PG05a,b to represent the number of in-transit points and the number of transits respectivtely, are inverted here. We also use σ0 where PG05a,b used σ, and d where they used δ. 2.2 Modifying the detection statistic to account for red noise In an attempt to account for the saturation of the rms. noise level that is seen at the bright end of all transit surveys, PG05a introduced in their Section 4.3 thqe concept of a minimum observational error σsys, which is added in quadrature to the error contribution σphot from the sky and source photon noise to give the error estimate σind for each data point: σw = (σ phot + σ 2 . (6) This expression for σw is then simply inserted into Equation 5). However, detailed analysis of the light curves of various transit surveys (PZQ06, Pont 2007) shows that they systematically contain noise that is correlated on transit timescales (2–3 h for a Hot Jupiter transit), i.e. the non-diagonal elements of the covariance matrix are non-zero. As a result, σd no longer decays as n −1/2 as expected for uncorrelated (white) noise. PZQ06 propose a single parameter description of the covariance, assuming the noise can be separated into purely uncorrelated (white) and purely correlated (red) components, the former decaying as n−1/2 but the latter independent of n: r (7) where σw and σr reresent the white and red noise components respectively. This single parameter description of the correlated noise assumes that the degree of correlation remains unchanged on all timescales up to the maximum transit duration. It is equivalent to approximating the covariance matrix with Cii = σ 0 ≡ σ2w + σ2r in the diagonal, Cij = σ2r for two data points in the same transit, and Cij = 0 otherwise (see Section for the treatment of multiple transits). There is evidence that the correlation timescale in transit survey light curves is finite (Gould et al. 2006). If this correlation timescale is shorter than the maxium transit duration, the above expression would underestimate the significance of long- duration transit events. This does not appear to be the case for light curves analysed by PZQ06 and Pont (2007), where the noise remains correlated up to 3 h timescales. Nevertheless, it is interesting to investigate the impact of finite correlation timescales through a simple example. We consider a transit with a depth of 1% lasting 2 h and observed with 15min time sampling, i.e. n = 8. In the white noise only case, if σw = 2mmag, σd = 0.71mmag and ∆χ tr = 200. If correlated noise is present, with σr = 1mmag, the single parameter approximation gives σd = 1.22mmag and ∆χ tr = 67. If on the other hand the noise is correlated only over timescales up to 1 h or 4 data points, i.e. Cij = 0 for |i − j| ≥ 4, σd = 1.10mmag and ∆χ2tr = 82. In general, even if the characteristic correlation timescale of the noise is shorter than a transit duration, we expect 1 One can show that the estimate of d which minimises the χ2 of the fit is the inverse-variance weighted average of the in-transit flux-measurements, and σd is thus the uncertainty on this average. c© 0000 RAS, MNRAS 000, 000–000 4 S. Aigrain et al. the single parameter correlated noise approximation adopted here to give an estimate of the transit significance that is much nearer to the true value than that obtained with the white noise approximation. We therefore adopt Equation (7) for what follows. The white noise is assumed to be equal to the photon noise and modelled as of contributions from the source and the sky background: w = σ source + σ back = Ns +Nb , (8) where Ns and Nb are the number of photons from the source and the sky detected in the photometric aperture. PZQ06 found that the distribution of the rms. of OGLE light curves over a typical transit time-scale of 2.5 h is consistent with a constant red noise level of σsys ∼ 3mmag, independent of apparent magnitude. Processing the light curves with a systematics removal algorithm such as Sys-Rem reduces σsys to ∼ 1.5mmag for the best objects. The work of the International Space Science Institute (ISSI) working group on transiting planets (Pont 2007) has shown that similar values are also typical of other surveys, with a correlated noise value of ∼ 1.5mmag for the best objects. We therefore adopt σsys ∼ 1.5mmag throughout the following calculations, which would correspond to very good ground-based photometry. While it is theoretically possible to reduce the level of correlated systematics further, this value is used because it is considered representative of the leading surveys currently in operation. In addition to this systematics term, a red noise component proportional to the white noise level (as a function of magnitude) is present in some surveys. This dominates over the systematics term in the domain where background photon noise dominates the white noise and is thus likely to be somehow associated with background subtraction. We therefore label it σsub. For the purposes of the present calculations, it is modelled as a term proportional to the background noise: σsub = kσb (9) For the purposes of the present work we assume k = 0.2. This is the kind of values the ISSI team found for the correlated noise in the HAT and SuperWASP surveys. Most cluster surveys, such as the University of St Andrews Planet Search (Street et al. 2003; Bramich et al. 2005; Hood et al. 2005), EXPLORE–OC (von Braun et al. 2005), STEPSS (Burke et al. 2006), PISCES (Mochejska et al. 2005; Hartman et al. 2005; Mochejska et al. 2006) or Monitor (Aigrain et al. 2007; Irwin et al. 2007), have ‘better spatial sampling, and lower values of k might therefore be expected to apply, though preliminary analysis of test light curves indicates that k ∼ 0.2 is also appropriate, if not an underestimate, for at least some of these surveys. In any case, this value is used here to illustrate the effects of noise of this type when it dominates the overall noise budget. The overall red noise budget is thus r = σ sys + σ sub = σ sys + , (10) 2.3 Multiple transits As correlated noise does not average out over transit time-scales, but does average out over repeated transit events, it is particularly important to consider the repeatability of transits in the detection process when one suspects correlated noise might dominate. In an appendix, PG05a derived an expression for PS/N for multiple transits, which is based on the equation tr(multiple transits) = Ntr ∆ tr(single transits) = Ntr where Ntr is the number of observed transits. This equation remains valid in the presence of correlated noise provided there is no correlation over long timescales (similar to the planet’s orbital period). In PG05a, PW has to be calculated separately for each value of the number of transits. This assumes that the number of data points in each observed transit is the same, and in practice one must therefore choose a minimum value for the number of data points in a partially observed transit above which that transit contributes to PW, and below which it does not. PZQ06 provide a general formula which accounts for the number of data points in each observed transit: tr(multiple transits) = n n2kV(nk) where ntot = = 1Ntr is the total number of in-transit points and V(nk) ≡ nk×nk block Cij (13) is the noise integrated over the kth observed transit. PW then becomes a multi-dimensional quantity dependent on not only Ntr but for each Ntr, on the set of nk. As with the PG05a formalism, it must be evaluated numerically. In the present work, we make the assumption of homogeneous phase coverage, which allows us to ignore differences between nk for the different transits, and enables us to (roughly) estimate the number of transits observed as a function of c© 0000 RAS, MNRAS 000, 000–000 Transit Surveys in star clusters 5 period given the time sampling and survey duration. This can then be incorporated into Equation (11) directly, therefore alleviating the need to compute PW separately. One can approximate Ntr as Ntr = Nn tnight where ttot is the total time spent on target, which is the product of the number of nights Nn and the average duration of a night tnight, and P is orbital period. Reality diverges strongly from the homoegeneous phase coverage assumption close to harmonics of the daily interruptions in the observations, but it follows the global 1/P trend (see Fig. 1 of PG05a). 3 IMPACT ON THE NOISE BUDGET AND DETECTION STATISTIC 3.1 Noise budget on transit time-scales Useful insights regarding the dominant noise sources, and how to mitigate those that have the largest impact on the tran- sit detection performance, can be gained by exploring the dependency of the the various noise components on the stellar parameters. We start from the following expressions, given by PG05a, for Ns, Nb and n (recalling that n is called Ntr in PG05a): LX,⊙10 −0.4AX texpπ = Ns,⊙ , (15) Nb = Ssky,XΩtexpπ , (16) 1− b2R⊙ )1/3 ( )α− 1 1− b2 neq,⊙ )α− 1 where M is the stellar mass; α is the index of the (power-law) mass-radius relation; βX is the index of the (power- law) mass-luminosity relation in the filter X under consideration; D is the telescope aperture; texp is the exposure time; d is the distance to the cluster; AX is the extinction to the cluster. PG05a adopt the distance-dependent extinction law AI = 0.5(d/kpc); LX,⊙ is the Sun’s photon luminosity in the filter of interest, which we compute, following PG05a, as LX,⊙ = 8π2cR2⊙λ X,c∆λX exp (hc/λX,ckT⊙)− 1 , (18) where λX,c and ∆λX are the filter central wavelength and FWHM respectively; Ssky,X is the sky photon flux per unit solid angle; Ω is the effective area of the seeing disk, which we compute, following PG05a, as see (19) where θsee is the FWHM of the PSF; b is the impact parameter of the transit; δt is the interval between consecutive measure- ments. In PG05a, δt = texp + tread (20) where tread is the readout time, which can be generalised to include any time spent off-target; Ns,⊙ is the number of source photons in the aperture for a solar-mass star; neq,⊙ is the number of points in an equatorial transit for a solar-mass star. Substituting for Ns and Nb from Equations (15) and (16) into Equation (8) gives 1 + C2 . (21) where we have introduced, following PG05a, 4πd2Ssky,XΩ LX,⊙10−0.4AX which is the ratio of sky to source flux in the aperture for a solar mass star. Taking σr from Equation (10), σw from Equation (21) and n from Equation (17) and substituting into Equation (7), 1− b2 )−1/2 Ns,⊙neq,⊙ −α−βX 1 +C2 sys + )−2βX . (23) To simplify this expression we introduce two new constants c© 0000 RAS, MNRAS 000, 000–000 6 S. Aigrain et al. Figure 1. Error budget on individual data points (left) and over a the duration of an equatorial transit (right) for the fixed and fiducial parameters of PG05a, assuming σsys = 1.5mmag and k = 0.2. The black dashed, dot-dash, triple dot-dash and dotted lines show the source photon noise, background photon noise, background subtraction noise and systematics terms respectively, and the solid black line shows the total noise budget. The grey line on the right panel shows what the behaviour the total noise would have it all the components behaved as white noise. The grey vertical dotted lines show mark transitions between the different regimes, as defined in Equations (27) to (30). C4 = Ns,⊙ neq,⊙ (24) which is the square of the background subtraction component for a Sun-like star, and which is the total number of source photons collected during an equatorial transit for a Sun-like star. (Note that C3 is defined in PG05a but not used here.) Equation (23) then becomes 1− b2 )−1/2 −α−βX 1 +C2 sys +C5 )−2βX . (26) The form of σeq, the depth uncertainty for equatorial transits (b = 0) and of the different terms that compose it is illustrated in Figure 1 (right panel). Also shown for comparison is the noise level per data point (left panel), or V(1). The relative importance of the red noise components is clearly enhanced over the transit time-scale. While the systematics term is the same for all stellar masses, the source photon noise is a steeply decreasing function of stellar mass, the background subtraction noise is even steeper, and the background photon noise is the steepest. There may thus be up to four noise regimes, starting with the systematics-limited regime at the highest masses, followed by the source noise-limited regime, the subtraction-limited regime, and finally the background noise-limited regime at the lowest masses. Equating each pair of components and solving for M yields the mass regimes in which each component dominates. This exercise was done by PG05a to obtain Msky, the transition mass between the source and background noise-limited regimes, which for clarity we rename Ms,b. Ms,b = C 2 M⊙ (27) Given the two additional noise terms that have been introduced, the relevant transitions are now: Msys,s = 1−3α−3βX M⊙ (28) Ms,sub = (C4 C5) 1−3α+3βX M⊙ (29) Msub,b = C4 C5 M⊙ (30) c© 0000 RAS, MNRAS 000, 000–000 Transit Surveys in star clusters 7 However, it is very easy for the source noise-limited regime to disappear altogether, because the source photon noise averages out over the duration of the transit whereas the systematics and background subtraction noise do not. Even if one ignores the background subtraction term, the source-limited regime disappears if Ms,b ≥ Msys,s. Given the set of fixed and fiducial parameters adopted by PG05a, the source limited regime exists only if σsys < 0.5mmag. Adopting a more realistic value of 1.5mmag, there is a direct transition between the systematics- and subtraction-limited regime, which occurs at Msys,sub = σ2sys M⊙. (31) For the subtraction-dominated regime to exist requires k to be relatively large (k ≥ 0.2). If this is not the case, there is a direct transition between the systematics and background limited regimes: Msys,b = 1−3α−2βX M⊙. (32) 3.2 Detection probability PS/N The detection probability is derived following the same method as PG05a, although we consider multiple, rather than single transits. A transit observed Ntr times is assumed to be detectable if it gives rise to a detection statistic ∆χ tr ≥ ∆χ2min. If equatorial transits of a given system are detectable, one can derive a maximum impact parameter bmax up to which transits of such a system are also discoverable (PG05a). This arises because n = teq 1− b2/δt, where b is the impact parameter of the transit and teq the duration of an equatorial transit: teq = R where R is the radius and M the mass of the star. We have assumed that the planet radius r ≪ R and ignored limb-darkening, which allows us to ignore grazing transits as both extremely rare and hard to detect, and to write δ = (r/R)2 where r is the planet radius. Therefore, the probability that transits of such a system are detectable, i.e. that ∆χ2tr ≥ ∆χ2min, reduces to PS/N = bmax when integrated over b, assuming the impact parameters are uniformly distributed between 0 and 1. However, the statement: tr = ∆χeq 1− b2, (34) which is valid in PG05a, no longer holds here, because ∆χ2tr is no longer simply proportional to n. Instead, Equations (4) and (7) imply: tr = Ntr δ = Ntr δ 1− b2 . (35) The expression for bmax is found by setting the left hand side of Equation (35) to ∆χ min and solving for b. This yields: PS/N = bmax = Ntr δ2 ∆χ2min − σ2r Inserting the expressions for the different noise terms derived above gives: PS/N = −α−βX 1 +C2 − σ2sys − C5 )−2βX . (37) This expression reduces, in the case of white noise only – i.e. when σsys and k vanish – to PG05a’s Equation (15). A similar expression for ∆χ2eq, the detection statistic for equatorial transits, also ensues: −α−βX 1 + C2 + σ2sys + C5 . (38) If ∆χ2eq < ∆χ min for a particular star-planet system, transits of that systems are not detectable, whatever the inclination. c© 0000 RAS, MNRAS 000, 000–000 8 S. Aigrain et al. Figure 2. Detection statistic ∆χ2eq for an equatorial transit, for individual transits and assuming the systematics are white (as in PG05a, left) or for multiple transits and incorporating both red noise terms with σsys = 1.5mmag and k = 0.2 (this work, right). The line styles have the same meaning as in Figure 1. The grey horizontal dotted line marks the detection threshold ∆χ2 = 30 adopted by PG05a. The grey vertical dotted lines mark the lower and, if applicable, upper mass limits between which the transits are detectable. The grey vertical dashed lines mark the mass range where RV follow-up is feasible with FLAMES+UVES. The overall behaviour of ∆χ2eq as a function of mass is illustrated in Figure 2 (right panel). Also shown for comparison is the single-transit ∆χ2eq obtained following PG05a, i.e. assuming the systematics are white (left panel). PG05a point out that, using α = 1 and βI = 3.5, ∆χ eq ∝ M1/6 and M11/3 in the source and background noise-limited regimes respectively, which has the remarkable implication that the detectability of planetary transits is virtually independent of mass for all stars above sky, while it decreases rapidly for stars below sky. In white, source limited noise only, the number of detections from a given survey is thus roughly proportional to the number of unsaturated stars above sky. In the red noise- limited regimes, ∆χ2tr no longer depends on n, i.e. on b (provided the transit is not grazing, a given transit event contributes the same amount to the detectability no matter what the number of observations in that transit). Transits of a given system are thus detectable, whatever the inclination (i.e. the transit duration), if they are deep enough (δ) and enough of them are observed (Ntr). Using the same values of α and βI as in PG05a, ∆χ eq ∝ M−4 and M3 in the systematics and background- limited regimes respectively, with the remarkable implication that transits are detectable only around stars below a certain mass, determined by the systematics term. The reason is that the degree of correlation of the noise lowers the advantage of having longer transits and lower photon noise (a larger and brighter primary) compared to that of having deeper transits (a smaller primary). The combined effect of the different noise terms across the entire stellar mass range is to give rise to a peak in ∆χ2eq versus M , as illustrated by Figure 2. This immediately points to a potentially very simple way of evaluating wether a given type of planet is detectable at all in a given cluster with a given observational setup: find the ‘peak mass’, or stellar mass at which ∆χ2eq is maximised, by differentiating Equation (38) with respect to M and setting the derivative to zero: −α−βX 5α+ βX − 5α+ 2βX − + C5 (1 + 2βX ) )−2βX sys = 0. (39) The solution of Equation (39) could then be plugged back into Equation (38), to yield ∆χ2peak. If ∆χ peak > ∆χ min, detections are possible in the cluster under consideration. In that case, one can also compute the limits Mlow and Mup of the stellar mass range over which detections are possible by setting the left hand side of Equation (38) to ∆χ2min and solving for M . In practice, both equations cannot be solved analytically in the general case. If one is interested in calculating precise values of ∆χ2peak, Mlow and Mp, the simplest way is to to compute ∆χ eq for a range of M and find the quantities of interest numerically. However, as discussed in Section 3.1, it is not uncommon for a single component to dominate over a significant portion of the mass regime. By considering dependence of ∆χ2eq on each of the components one at a time, one can obtain useful insights c© 0000 RAS, MNRAS 000, 000–000 Transit Surveys in star clusters 9 into what limits the transit survey’s performance, and what mass range will be accessible. Each of the source, background and background subtraction terms imply a minimum mass around which a given type of transit is detectable: Mlow,s = −9α+3βX−1 M⊙ (40) Mlow,b = −9α+6βX−1 M⊙ (41) Mlow,sub = 1 C4 C5 4α−2βX M⊙ (42) where C′1 is a multiple-transit equivalent of PG05a’s C1: ∆χ2min Ntr Ns,⊙ neq,⊙ and Mlow,s and Mlow,b are equivalent to PG05a’s Mth,s and Mth,b. On the other hand, the systematics term implies a maximum mass: Mup = C′1 C4 σ M⊙ (44) Note that PG05a’s white systematic noise term also induces an upper mass limit detection, but it is typically larger than 3M⊙. As long as Mup is above the peak of the mass function, correlated systematic noise will not significantly affect the total number of detections in a given survey. However, it does have the effect of preventing detections around the brightest stars, which are arguably the most interesting, because of their enhanced potential for follow-up. In general, more than one component contributes near the limits of the range of masses of stars around which transits of a given type of planet are detectable, and the expressions for Mlow and Mup are rather complex. Again, an alternative is to compute each term in ∆χ2eq for an array of stellar masses and to find the values of M between which ∆χ eq > ∆χ 4 ADDITIONAL CONSIDERATIONS 4.1 Turnoff mass Following PG05a, we compute the turnoff mass Mto = Lbol,⊙A )1/(β−1) where ǫ is the efficiency of Hydrogen burning and β is the bolometric mass-luminosity index. 4.2 Saturation mass Also following PG05a’s expression for the number of photons at the peak of the PSF of a given star, the saturation mass is 0.4AX )−2 NFW texpπ − Ssky,Xθ2pix 1− exp − ln 2 where NFW is the full-well capacity of the detector and θpix is the angular size of the pixels. 4.3 Radial velocity follow up Radial velocity (RV) follow-up is necessary to confirm the planetary nature of any detected transits and to measure companion masses. In this section,we examine the range of stellar masses over which this is feassible for a planet of a given mass and period. PG05a used a fixed magnitude limit (V = 17 or V = 18) beyond which planets were considered undetectable by the radial velocity method. This is approximately suitable for planetary companions to Sun-like stars: it is extremely difficult to measure radial velocities with precisions of a few tens of m/s level beyond V ∼ 18 even with the largest telescopes available at present (Pont et al. 2005). However, in cluster transit searches, many of the detections occur around lower-mass stars, where planetary companions may induce significantly larger radial velocity modulations, and a more detailed treatment is needed. For a star of a given magnitude, the minimum detectable radial velocity amplitude Kmin is highly instrument depen- dent, and we examine two representative telescope / instrument combination: the UV-Visual Echelle Spectrograph (UVES) coupled to the Fibre Large Area Multi-Element Spectrograph (FLAMES) on the Very Large Telescope (VLT) – hereafter FLAMES+UVES – and the High Accuracy Radial velocity Planet Searcher on the 3.6m telescope at La Silla – hereafter c© 0000 RAS, MNRAS 000, 000–000 10 S. Aigrain et al. HARPS. High precision measurements tend to be limited by instrument stability rather than by photon noise, in the sense that, if deemed interesting enough, a given (short-period) object can be observed as long as necessary, binning the phase-folded measurements to reduce the photon noise contribution. However, for each telescope / instrument combination there is also a magnitude limit YRV, beyond which the signal-to-noise ratio achievable in a single exposure drops below a critical level, and high precision measurements are no-longer feasible in reasonable exposure times. As the spectral region used typically covers the V and R-bands, Y should be either V or R, depending on which filter the object under consideration is brightest in. The radial velocity semi-amplitude induced by a given planet scales as K ∝ mP−1/3M−2/3 (47) where m is the planet mass and we have assumed that m ≪ M and that the inclination of the system is edge-on. All planets giving rise to K ≥ Kmin are then assumed to be detectable around stars with apparent magnitude down to YRV, beyond which it is assumed that high precision radial velocity measurements are not feasible at all with the telescopes/instruments under consideration. For HARPS, we use Y = V and YRV = 14, for FLAMES+UVES we use Y = R and YRV = 18. Both are relatively optimistic limits. This means that in each cluster, there is a lower mass limit MRV,min = 10 MY,⊙−YRV+5 log d−5+AY /2.5βY (48) where Y is R or V , below which no radial velocity measurements are feasible with a given instrument, and above which the minimum detectable planet mass is mmin = mref 3 days )1/3( where mref = MNeptune for HARPS and MJupiter for FLAMES+UVES. If considering a particular planet mass m across a range of stellar masses, one can derive a maximum stellar mass MRV,max around which such a planet produces a detectable RV signal by setting Mmin in Equation (49) to m: MRV,max = 3 days )−1/2 M⊙ (50) which is independent of the cluster properties and depends on the planet mass and period only. Planets with mass m and period P can thus be confirmed by radial velocity with present observational means only if they orbit stars with MRV,min < M < MRV,max. Note that, for the sake of simplicity, we have ignored a number of important factors, including morphological differences in the spectra of stars of different types and the impact of rotation, which broadens the lines and degrades the radial velocity precision. 5 APPLICATIONS One can roughly evaluate the mass range [Mmin;Mmax] within which planets of a given radius and period in a given cluster produce detectable transits and RV modulations: Mmin = max (Mlow,MRV,min) (51) Mmax = min (Mup,Mto,Msat,MRV,max) (52) 5.1 PG05a’s fiducial cluster Going back to the fiducial cluster of PG05a, under the relatively optimistic assumption that σsys = 1.5mmag, the detection of transits alone for a 1MJupiter planet in a 2.5-d orbit is possible around stars with masses between 0.28 and 1.49M⊙. However, using FLAMES+UVES on the VLT, for which we assume that Kmin corresponds to a Jupiter mass planet in the same orbit around the same star and that RRV = 18, MRV,min = 1.13M⊙ and MRV,max = 1.22M⊙, so that the mass range where such a planet can be detect via transits and radial velocity is only 0.11M⊙. The combination of correlated systematics and follow-up requirements imposes very stringent limits on the potential of transit surveys in open clusters. In practice, it implies an even stronger dependence on cluster distance that illustrated in the bottom right panel of PG05a’s Figure 8. 5.2 Example galactic open clusters In a subsequent paper, PG05b applied the formalism of PG05a to a number of well-studied Galactic open clusters and, on this basis, made the prediction that close-in Neptune- or even Earth-sized planets should be detectable in some of these clusters c© 0000 RAS, MNRAS 000, 000–000 Transit Surveys in star clusters 11 Name Distance Age Aperture texp Mlow (Cause) Mup (Cause) ∆M (pc) (Myr) (m) (s) (M⊙) (M⊙) (M⊙) Hyades 46 625 1.8 15 0.15 (lim) 0.55 (sat) 0.47 Praesepe 175 800 1.8 45 0.15 (lim) 0.87 (sat) 0.79 NGC 2682 (M67) 783 4000 3.6 45 0.15 (lim) 1.36 (TO) 1.28 NGC 2168 (M35) 912 180 3.6 45 0.15 (lim) 1.49 (sys) 1.41 NGC 2323 (M50) 1000 130 3.6 45 0.15 (lim) 1.49 (sys) 1.41 NGC 2099 (M37) 1513 580 3.6 45 0.15 (lim) 1.49 (sys) 1.41 NGC 6819 2500 2900 6.5 45 0.15 (lim) 1.49 (sys) 1.41 NGC 1245 2850 960 6.5 45 0.15 (lim) 1.49 (sys) 1.39 NGC 6791 4800 8000 6.5 45 0.15 (lim) 1.08 (TO) 0.92 Table 1. Masses ranges over which transits of Jupiter-sized planets in 2 d orbits are detectable in selected Galactic open clusters, using the observational parameters of PG05b. Columns 7 and 9 give the primary cause of the upper and lower limits (sat: saturation; TO: turn-off; sys: systematics; lim: lower limit of mass range considered). Name Aperture texp Mlow (Cause) Mup (Cause) ∆M (m) (s) (M⊙) (M⊙) (M⊙) Hyades 0.1 30 0.15 (lim) 0.79 (sat) 0.70 Praesepe 0.1 30 0.18 (RV) 1.22 (RV) 1.04 NGC 2682 (M67) 0.8 15 0.57 (RV) 1.22 (RV) 0.66 NGC 2168 (M35) 1.3 15 0.64 (RV) 1.22 (RV) 0.59 NGC 2323 (M50) 1.5 15 0.66 (RV) 1.22 (RV) 0.57 NGC 2099 (M37) 2.7 15 0.75 (RV) 1.22 (RV) 0.48 NGC 6819 5.0 15 0.84 (RV) 1.22 (RV) 0.38 NGC 1245 6.5 18 0.93 (RV) 1.22 (RV) 0.30 NGC 6791 6.5 65 1.07 (RV) 1.08 (TO) 0.01 Table 2. Masses ranges over which transits and radial velocity modulations of Jupiter-sized planets in 2 d orbits are detectable in selected Galactic open clusters, using SuperWASP for the Hyades and Praesepe and the observational parameters of PG05b for the other clusters, and using FLAMES+UVES for radial velocity follow-up. Columns 7 and 9 give the primary cause of the upper and lower limits (RV: radial velocity). via transit surveys from ground-based 2- to 6-m class telescopes. If so, transit surveys in open clusters might not only enable the detection of planets around well characterised stars of known age and metallicity, but may also lead to the first radius measurements for terrestrial planets. It is therefore interesting to investigate the detectability of Jupiter-sized and smaller planets in these clusters in the presence of red noise. We use the same test sample of 9 clusters (the Hyades, Praesepe, M67, M35, M50, M37, NGC6819, NGC1245 and NGC6791) as PG05b, from which we take the cluster parameters (distance, age, extinction) and the observational parameters, which are similar to those of PG05a except that the night duration is tnight = 8h, the telescope apertures are 1.8m (Pan- STARRS), 3.6m (CFHT) and 6.5m (MMT) depending on the cluster (as selected by PG05b), and the exposure time is texp = 45 s for all clusters except the Hyades for which texp = 15 s. Table 1 shows the range of stellar masses between which transits Jupiter-sized planets in 2 d orbits produce detectable transits. For all but the most distant cluster, transits are detectable right down to the minimum stellar mass considered (0.15Modot, well below the limit of 0.3M⊙ adopted by PG5b), i.e. the addition of a correlated background subtraction term component does not affect the results. For all but the nearest clusters, the upper limit comes from the systematics term (Mup = 1.49Msun, independent of the cluster and observational parameters). For the Hyades and Praesepe, the upper limit is saturation with the observational setup considered here, but this can be raised by using shorter exposure times and / or smaller telescopes, which have the added advantage of providing wider fields of view. For example, the SuperWASP project Pollacco et al. (2006) uses multiple 11 cm apertures and has 13.5 as pixels and an effective bandpass similar to R (Ssky ∼). As it cycles between fields, δ = 8min. With the standard exposure times of 30 s, we obtain Msat = 0.79M⊙ for the Hyades and 1.71M⊙ for Praesepe. These represent a significant gain, and hereafter we adopt these observational parameters for these two clusters. Note that one could decrease the exposure time for the Hyades to increase Msat further, but this would conflict with the primary goal of SuperWASP, namely to search for transits around field stars. Overall, correlated noise does not strongly affect the detectability of transits of hot Jupiters in these clusters. We now incorporate the limits induced by radial velocity follow-up with FLAMES+UVES in the calculations. The results are shown in Table 2. Radial velocity or turnoff are now the limiting factors in almost all cases, and imply a stronger distance c© 0000 RAS, MNRAS 000, 000–000 12 S. Aigrain et al. Name Aperture texp M (Cause) M (Cause) Mup (Cause) ∆M (m) (s) (M⊙) (M⊙) (M⊙) (M⊙) Hyades 0.1 30 0.08 (lim) 0.28 (RV) 0.50 (sys) 0.42 Praesepe 1.8 15 0.08 (lim) 0.63 (RV) 0.52 (sys) 0.44 NGC 2682 (M67) 1.8 15 0.18 (back) 1.06 (RV) 0.52 (sys) 0.34 NGC 2168 (M35) 3.6 15 0.16 (back) 1.19 (RV) 0.52 (sys) 0.36 NGC 2323 (M50) 3.6 15 0.18 (back) 1.25 (RV) 0.52 (sys) 0.34 NGC 2099 (M37) 3.6 15 0.32 (back) 1.49 (RV) 0.49 (sys) 0.17 NGC 6819 6.5 15 0.39 (back) 1.70 (RV) 0.46 (sys) 0.07 Table 3. Masses ranges over which transits of Neptune-sized planets in 2 d orbits are detectable (1) and confirmable with HARPS (2) in selected Galactic open clusters, using SuperWASP for the Hyades and the observational parameters of PG05b for the other clusters. dependence of the planet yield than transits. It is interesting to note, however, that confirmed detections of transiting hot Jupiters are possible down to very low stellar masses in the nearest clusters. We also investigate the detectability of Neptune-sized planets, using HARPS for radial velocity follow-up, still with a period of 2 d. For such planets, the systematics term implies an upper mass limit of Mup = 0.53M⊙ independent of the observational set-up. We use the same observational setup as before except for Praesepe, where a greater photon-collecting capacity than SuperWASP’s is needed to offset the smaller planet radius, so we revert to Pan-STARRS. The results are shown in Table 3. We have omitted NGC1245 and NGC6791 because transits of Neptune-sized planets are not detectable at all in these clusters. The systematics term severely limits the maximum stellar mass around which transits of hot Neptunes can be detected, while the need for radial velocity measurements limits the minimum mass around which they can be confirmed, and it is only in the Hyades that confirmed transiting Neptunes are expected to be detectable. We stress that these limits are relatively independent of theobservational setup. For hot Earths, the systematics term implies a very stringent upper limit of Mup = 0.13M⊙, and the formalism adopted here also precludes radial velocity confirmation around any stars in the clusters condidered (although it may be feasible to detect the radial velocity signal from a hot Earth around a bright star using HARPS by observing many repetitions of the orbit). 6 CONCLUSIONS Simple modifications have been made to the formalism of PG05a to account for correlated noise and the need for RV follow-up. These should lead to more realistic estimates of the efficiency of Open Cluster transit surveys, while retaining the analytic nature of the original formalism, which affords useful insights into the behaviour of the detection probability as a function of mass. Two types of correlated noise were considered: systematics, which are constant with apparent stellar magnitude, and background subtraction noise, which scales with the background photon noise level. The latter behaves in a similar fashion to background photon noise itself, though its contribution to the total noise budget on transit time-scales has a slightly less steep dependence on the stellar mass, and therefore it does not significantly modify the yield of a survey unless extreme assumptions are adopted. However, the former implies a detection probability that steeply decreases with increasing mass and therefore curtails detections at the bright end. This effect is much stronger than the loss of sensitivity implied by a white minimum observational error of similar magnitude. In the course of evaluating the impact of correlated noise on the detectability of transits, we made a number of simplifying assumptions, and these should be borne in mind when comparing the predictions of the present formalism to the yield of real cluster transit surveys. First, we have assumed that the noise budget is the same for all stars of a given magnitude, and that every data point in a given light curve is affected by the same noise level. In fact, both white and correlated noise typically affect some objects and/or nights more than others, as they depend on factors which vary from object to object (e.g. crowding, position on the detector, colour) and time (e.g. weather, instrumental problems). Additionally, the way we compute the number of observed transits does not take into account the very strong features close to integer multiples of a day that are present in the window function of most ground-based surveys. As a result, while the scaling laws derived here apply for the majority of the objects in a given survey, the most significant detections in a real survey may well occur in special cases where the time sampling and the noise characteristics were particularly favourable. On the other hand, the radial velocity modulation induced by the companion in the primary, in order to measure the companion’s mass, is only detectable given present day instrumentation over a certain stellar mass range which can be close to, if not above, the maximum mass implied by the systematics term for typical targets and observational set-ups. Thus, even though correlated systematics may not affect the yield of Open Cluster transit surveys significantly in terms of transit c© 0000 RAS, MNRAS 000, 000–000 Transit Surveys in star clusters 13 detection alone (because transits usually remain detectable around stars close to the peak of the mass function), it has a very serious impact on the yield in terms of transits whose planetary nature can be confirmed and where the companion mass can be measured. While the specific colour-magnitude relation followed by the members of a given cluster may enable one to exclude many of the astrophysical false positives which affect all transit surveys without actually detecting the radial velocity modulation of the primary, the scientific impact of any detection of a transiting planet will be significantly lowered if its mass cannot be measured. To illustrate a possible application of this modified formalism, it was applied to a selection of well-studied Galactic Open Clusters, which were used by PG05b to show that transits of Hot Neptunes, and even Hot Earths, should be detectable from the ground in nearby young Open Clusters. While correlated noise alone has little effect on the detectability of hot Jupiters in these clusters, we find that radial velocity follow-up severely limits the minimum mass around which their masses can be measured, which makes the confirmation of even Jupiter-mass planets in the more distant clusters difficult. Additionally, correlated systematics at the level of 1.5mmag affecting all stars in a 20 night survey imply that transits of hot Neptunes are only detectable around stars with masses below 0.5M⊙. For such low stellar masses, the planetary radial velocity signal will only be measurable in very nearby clusters (< 100 pc) with present-day facilities. If hot Neptunes are abundant around M-stars, some could be detected by the combination of small aperture, wide field surveys such as SuperWASP and state of the art radial velocity instruments such as HARPS. The same level of systematics limits the detection of transits of Hot Earths to stars with masses below 0.13M⊙, irrespective of the cluster properties or observational setup. It is thus vital to achieve lower systematics (e.g. by going to space with CoRoT and Kepler) to detect transits of terrestrial planets, and particularly to detect them around stars bright enough that it may be possible to measure their radial velocity signal with future instrumentation. A general trend that emerges from this work is that the combination of correlated noise and RV follow-up requirements severely limits the choice of suitable target clusters, and effectively imposes a rather stringent distance limit. Additionally, we note that, for a given cluster, the optimal observational setup differs depending on the type of planet considered. ACKNOWLEDGEMENTS This work was initiated in the course of the meetings of the International Team on Transiting Planets set up in 2005 under the auspices of the International Space Science Institute (ISSI), and we are indebted to all the members of this team for invigorating discussions and useful feedback. SA gratefully acknowledges support from a PPARC Postdoctoral Research Fellowship. This work has been done in the context of the preparation to the CoRoT data analysis, for which FP acknowledges support from a PRODEX grant. The authors are also grateful to the referee, Scott Gaudi, for his careful reading of the manuscript and his useful comments and suggestions. REFERENCES Aigrain S., Hodgkin S., Irwin J., Hebb L., Irwin M., Favata F., Moraux E., Pont F., 2007, MNRAS, in press Bramich D. M., Horne K., Bond I. A., Street R. A., Cameron A. C., Hood B., Cooke J., James D., Lister T. A., Mitchell D., Pearson K., Penny A., Quirrenbach A., Safizadeh N., Tsapras Y., 2005, MNRAS, 359, 1096 Burke C. J., Gaudi B. S., DePoy D. L., Pogge R. W., 2006, AJ, 132, 210 Gaudi B. S., 2000, ApJL, 539, L59 Gould A., Dorsher S., Gaudi B. S., Udalski A., 2006, AcA, 56, 1 Hartman J. D., Stanek K. Z., Gaudi B. S., Holman M. J., McLeod B. A., 2005, AJ, 130, 2241 Hood B., Cameron A. C., Kane S. R., Bramich D. M., Horne K., Street R. A., Bond I. A., Penny A. J., Tsapras Y., Quirrenbach A., Safizadeh N., Mitchell D., Cooke J., 2005, MNRAS, 360, 791 Irwin J., Irwin M., Aigrain S., Hodgkin S., Hebb L., Moraux E., 2007, A&A, in press Mayor M., Queloz D., 1995, Nature, 378, 355 Mochejska B. J., Stanek K. Z., Sasselov D. D., Szentgyorgyi A. H., Adams E., Cooper R. L., Foster J. B., Hartman J. D., Hickox R. C., Lai K., Westover M., Winn J. N., 2006, AJ, 131, 1090 Mochejska B. J., Stanek K. Z., Sasselov D. D., Szentgyorgyi A. H., Bakos G. Á., Hradecky J., Devor V., Marrone D. P., Winn J. N., Zaldarriaga M., 2005, AJ, 129, 2856 Pepper J., Gaudi B. S., 2005a, ApJ, 631, 581 Pepper J., Gaudi B. S., 2005b, Acta Astronomica, 56, 183 Pollacco D. L., Skillen I., Cameron A. C., Christian D. J., Hellier C., Irwin J., Lister T. A., Street R. A., West R. G., Anderson D., Clarkson W. I., Deeg H., Enoch B., Evans A., Fitzsimmons A., Haswell C. A., Hodgkin S., Horne K., Kane S. R. e. a., 2006, PASP, 118, 1407 c© 0000 RAS, MNRAS 000, 000–000 14 S. Aigrain et al. Pont F., 2007, in Afonso C., Weldrake D., Henning T., eds, Transiting Extrasolar Planets ASP Conf. Ser., Detection capacities of ground-based transit surveys (the ISSI team on exoplanet transits) Pont F., Bouchy F., Melo C., Santos N. C., Mayor M., Queloz D., Udry S., 2005, A&A, 438, 1123 Pont F., Zucker S., Queloz D., 2006, MNRAS, 373, 231 Smith A. M. S., Collier Cameron A., Christian D., Clarkson W. I., Enoch B., Evans A., Haswel C., Hellier C., Horne K., J. I., Kane S. R., Lister T. A., Norton J., et al. 2006, MNRAS, 373, 1151 Street R. A., Horne K., Lister T. A., Penny A. J., Tsapras Y., Quirrenbach A., Safizadeh N., Mitchell D., Cooke J., Cameron A. C., 2003, MNRAS, 340, 1287 von Braun K., Lee B. L., Seager S., Yee H. K. C., Mallén-Ornelas G., Gladders M. D., 2005, PASP, 117, 141 c© 0000 RAS, MNRAS 000, 000–000 Introduction Introducing red noise terms Overall formalism Modifying the detection statistic to account for red noise Multiple transits Impact on the noise budget and detection statistic Noise budget on transit time-scales Detection probability PS/N Additional considerations Turnoff mass Saturation mass Radial velocity follow up Applications PG05a's fiducial cluster Example galactic open clusters Conclusions

0704.0575

A non-perturbative proof of Bertrand's theorem

7 A non-perturbative proof of Bertrand’s theorem F C Santos ∗ V Soares† A C Tort ‡ Instituto de F́ısica Universidade Federal do Rio de Janeiro Caixa Postal 68.528; CEP 21941-972 Rio de Janeiro, Brazil September 17, 2018 Abstract We discuss an alternative non-perturbative proof of Bertrand’s theorem that leads in a concise way directly to the two allowed fields: the newtonian and the isotropic harmonic oscillator central fields. PACS: 45.50.Dd; 45.00.Pk 1 Introduction In 1873, J. Bertrand[1] published a short but important paper in which he proved that there are of only two central fields for which all orbits radially bounded are closed, namely: The newtonian field and the isotropic harmonic oscillator field. Because of this additional degenerescency it is no wonder that the properties of those two fields have been under close scrutiny since Newton’s times. Newton addresses to the isotropic harmonic oscillator in proposition X Book I, and to the inverse-square law in proposition XI [2]. Newton shows that both fields give rise to an elliptical orbit with the difference that in the first case the force is directed towards the geometrical centre of the ellipse and in the second ∗e-mail: [email protected] †e-mail: [email protected] ‡e-mail: [email protected]. http://arxiv.org/abs/0704.0575v1 case the force is directed to one of the foci. Bertrand’s result, also known as Bertrand’s theorem, continues to fascinate old and new generations of physicists interested in classical mechanics and unsurprisingly papers devoted to it continue to be produced and published. Bertrand’s proof concise and elegant and contrary to what one may be led to think by a number of perturbative demonstrations that can be found in modern literature, textbooks and papers on the subject, it is fully non-perturbative. As examples of perturbative demonstrations the reader can consult references [3, 4, 5]. We can also find in the literature demonstrations that resemble the spirit of Bertrand’s original work as for example [6]. As far as the present authors are aware of all those demonstrations have a restrictive feature, i.e., they set a limit on the number of possibilities of the existence of central fields with the property mentioned above to a finite number and finally show explicitly that among the surviving possibilities only two, the newtonian and the isotropic harmonic oscillator, are really possible. In his paper, Bertrand proves initially by taking into consideration the equal radii limit that a central force f(r) acting on a point-like body able of generating radially bounded orbits must necessarily be of the form f (r) = κ r(1/m where r is the radial distance to center of force, κ is a constant and m a rational number. Next, making use of this particular form of the law of force and considering also an additional limiting condition, Bertrand finally shows that only for m = 1 and m = 1/2, which correspond to Newton’s gravitational law of force f (r) = − and to the isotropic harmonic oscillator law of force f (r) = −κ r, respectively, we can have orbits with the properties stated in the theorem. However, we can also prove that for these laws of force all bounded orbits are closed. Here we offer an alternative non-perturbative proof of Bertrand’s theorem that leads in a more concise way directly to the two allowed fields. 2 Bertrand’s theorem In a central field one can introduce a potential function V (r), through the property f = −∇V (r) . (1) in such a way that the mechanical energy of a point-like body of mass µ v2 + V (r) , (2) is conserved. For radially bounded orbits there are two extreme radii rmax e rmin, the so called apsidal points ra, that are determined by the condition ṙa = 0, and between which the particle oscillates indefinitely. Moreover, the conservation of the angular momentum of the particle under the action of a central field obliges the motion to take place on a fixed plane and allows the introduction of the effective potential U(r) = V (r) + , (3) with the help of which it is possible to reduce this problem to an equivalent unidimensional one. This procedure can be found in several textbooks at the undergraduate and gradu- ated level, see for example [7]. In terms of the effective potential orbits radially bounded are characterised by apsidal distances rmax e rmin that satisfy the condition E = U (ra). Evidently there is an intermediate point r0 where the effective potential has a minimum that satisfies U ′ (r0) = V ′ (r0)− = 0. (4) The angular displacement of the particle between two successive apsidal points, the apsidal angle ∆θa, is determined by ∆θa = ∫ rmax [E − U (r)] . (5) By considering the effective potential U as the independent variable and by making use of the inverse function r (U), Tikochinsky[3] produced a very ingenious proof of Bertrand’s theorem. The inversion of the equation (3), however, is not possible in all the domain on which the radial coordinate r is defined because the function is not one-to-one in the field of the real numbers. To circumvent this difficulty we define two one-to-one branches of the function U (r), namely, one to the left and the other to the right of the point r0. Then we introduce the inverse functions r1 = r1 (U) and r2 = r2 (U), defined to the left and to the right of the point r0, respectively, see Figure 1 . We express initially the angular displacement, equation when the particle moves from the point of minimum radial distance rmin to the point r0 in terms of the variable U ∆θ1 = [E − U ] [E − U ] . (6) radial distance r r0rmin Figure 1: General form of the effective potential energy. By the same token we will also have ∆θ2 = [E − U ] [E − U ] , (7) for the angular displacement from r0 to the point of maximum radial distance r2. Upon adding up equations (6) and (7) we obtain the angular displacement between two succes- sive apsidal points ∆θa = F (U) E − U , (8) where F (U) = . (9) Equation (8) is Abel’s integral equation the solution of which can be found, for ex- ample, in Landau’s well known book on classical mechanics [8]. A beautiful and straight- forward solution of this equation is the one by Oldham and Spanier [9]. Abel’s solution reads ∆θa (E)√ U − E dE, (10) where the explicit dependency of the apsidal angle on the energy was stressed. If all bounded orbits are closed then the apsidal angle ∆θa (E), for these orbits, cannot change when the energy changes in a continual manner otherwise the continual changes would inevitably lead to open orbits. Taking this fact into account let us determine the central potentials that produce the same apsidal angle for all radially bounded orbits. After integrating equation (10) we obtain 2m∆θa U − U0. (11) Equation (11) was derived in Ref. [3] where a perturbative technique applied on a circular orbit leads to Bertrand’s result. The functions r1 (U) and r2 (U) being the inverse function of the function U (r) are not independent of each other, and combined as they are in equation (11), do not allow an efficient manipulation and hide the unique inverse we are looking for. At this point we perform an analytical continuation of the function U (r) such that we can consider its inverse function r = r (U). Therefore we write 2µ∆θa U − U0 + Φ(U, U0) , (12) where Φ (U, U0) is an analytical function of the complex variable U in an open neighbor- hood of U0 satisfying the condition Φ (U0, U0) = 1/r0, and whose analytical continuation cannot have poles but can have other ramification points. Notice that it s not necessary to make use of the symbol ± before the second term of equation 12) because the square root has two branches. The positive sign corresponds to r < r0 and the negative one to r > r0. Taking equation (3) into equation (12) we obtain 2m∆θa r2V (r) + − U0r2 + Φ(U, U0) . (13) The left-hand side of the identity (13) represents a meromorphic function with a single pole at r = 0 and the right hand side of this same identity contains several terms but only one can spoil the analyticity of the complete function at some point not equal to r = 0, namely the term that depends on the square root that generates a branch point at r = r0. To avoid this it is mandatory to undo the branching effect inherent to the square root. This is possible only if the radicand is the square of an analytical function with a zero at r = r0. In this way we identify two possibilities for the potential V (r), to wit V (r) = − , newtonian potential, (14) V (r) = κ r2, isotropic harmonic oscillator potential; (15) for which the apsidal angle is independent of the energy. We can calculate the corre- sponding constant apsidal angles for those two potentials as follows. For the newtonian potential the effective potential, equation (3), is given by U = − . (16) Solving equation (16) with respect to 1/r we obtain . (17) Making use of equation (4) with the effective potential given by equation (16) we find r0 = ℓ 2/(µκ) and the corresponding minimum energy U0 = −µκ2/(2ℓ2). Now we can recast equation (17) into the form U − U0. (18) Comparing equation (12) with equation (18) we can finally determine the apsidal angle for the newtonian potential which reads ∆θa = π. (19) The procedure employed with the newtonian potential can be also applied with a little bit more of effort to the case of the isotropic harmonic oscillator. The effective potential is now given by kr2 + . (20) This equation is a quartic equation in 1/r, biquadratic more precisely, and its solution is given by − 1. (21) Factoring out the right hand side of the equation (21) we have U − U0 + U + U0, (22) where now we have made use of the relations r2 µκ and U0 = ℓ κ/µ. Comparing equations (12) and (22) we obtain 2µ∆θa , ∴ ∆θa = . (23) We can see that both potentials for which the apsidal angle is constant the orbits are closed. For the newtonian case the radius oscillates only once in a complete cycle and for the oscillator case the radius oscillates twice. 3 Final Remarks In this brief paper we derived Bertrand’s theorem in a non-perturbative way. We have shown that simple analytical function techniques applied to the problem of finding the only central fields that allow an entire class of bounded, closed orbits with a minimum number of restrictions leads in a concise, straightforward way directly to the two allowed fields. We believe that the derivation discussed here is a valid alternative to a non-perturbative proof of Bertrand’s theorem and can be presented at the undergraduate and graduate level or assigned as a problem for classroom discussion. References [1] Bertrand J 1873 C.R. Acad. Sci. Paris 77 849 [2] Newton I 1687 Philosophiae Naturalis Principia Mathematica (London: Royal Soci- ety). English translation by A Motte revised by F Cajori 1962 (University of Cali- fornia Press, Berkeley CA) [3] Tikochinsky Y 1988 Am. J. Phys. 56 1073 [4] Brown L S 1978 Am. J. Phys. 46 930 [5] Zarmi Y 2002 Am. J. Phys. 70 446 [6] Arnol’d V I 1976 Les Méthodes Mathématiques de la Mécanique Classique (Mir: Moscou) [7] Goldstein H, Poole C and Safko J 2002 Classical Mechanics 3rd edn (Reading: Addison-Wesley) [8] Landau L and Lifchitz E 1969 Mècanique 3e èdition revue (Mir: Moscou) [9] Oldham K B and Spanier J 1974 The Fractional Calculus (London: Academic Press) Introduction Bertrand's theorem Final Remarks

0704.0576

Neutron-Capture Elements in the Double-Enhanced Star HE 1305-0007: a New s- and r-Process Paradigm

[1] [2] (Received 12 January 2007) Neutron-Capture Elements in the Double-Enhanced Star HE 1305-0007: a New s- and r-Process Paradigm∗ CUI Wen-Yuan()1,2,3, CUI Dong-Nuan()1, DU Yun-Shuang()1, ZHANG Bo()1,2 Department of Physics, Hebei Normal University, Shijiazhuang 050016 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012 Graduate School of the Chinese Academy of Sciences, Beijing 100049 The star HE 1305-0007 is a metal-poor double-enhanced star with metallicity [Fe/H] = −2.0, which is just at the upper limit of the metallicity for the observed double-enhanced stars. Using a parametric model, we find that almost all s-elements were made in a single neutron exposure. This star should be a member of a post-common-envelope binary. After the s-process material has experienced only one neutron exposure in the nucleosynthesis region and is dredged-up to its envelope, the AGB evolution is terminated by the onset of common-envelope evolution. Based on the high radial-velocity of HE 1305-0007, we speculate that the star could be a runaway star from a binary system, in which the AIC event has occurred and produced the r-process elements. PACS numbers: 97.10.Cv,26.45.+h,97.10.Tk The discovery that several stars show enhancements of both r-process and s-process elements (s+r stars hereafter)[1,2] is puzzling, as they require pollution from both an AGB star and a supernova. In 2003, Qian and Wasserburg[3] proposed a theory, i.e. accretion-induced collapse(AIC), for the possible creation of s+r-process stars. Another possible s+r scenario is that the AGB star transfers s-rich matter to the observed star but not suffer from a large mass loss and at the end of the AGB phase, the degenerate core of low-metallicity, high-mass AGB star may reach the Chandresekhar mass, leading to type-1.5 supernova.[4] Because the initial-final-mass re- lation flats at higher metallicity,[4] the degenerate cores of high-metallicity AGB stars are smaller than those of the low-metallicity stars, the formation of AIC or SN1.5 is more difficult in the high-metallicity binary system, which can explain the upper limit of the metallicity ([Fe/H] < −2.0) for the observed r+s stars.[5] Recently, Barbuy et al.[6] and Wanajo et al.[7] suggested massive AGB stars (M = 8 ∼ 12M⊙) to be the origin of these double enhancements. Such a large mass AGB star could possibly provide the observed enhancement of s-process elements in the first phase, and explode or collapse pro- viding the r-process elements. However, the modeling of the evolution of such a large mass metal-poor star is a difficult task, an amount of the s-process material is pro- duced and its abundance distribution is still uncertain.[7] The generally favoured s-process model till now is as- sociated with the partial mixing of protons (PMP here- after) into the radiative C-rich layers during thermal pulses.[8−11] PMP activates the chain of reactions 12C(p, γ)13N(β)13C(α, n)16O, which likely occurs in a narrow mass region of the He intershell (i.e. 13C-pocket) during the interpulse phases of an AGB star. The nucleosynthe- sis of neutron-capture elements in the carbon-enhanced metal-poor stars (CEMP stars hereafter)[12] can be in- vestigated by abundance studies of s-rich or r-rich stars. In 2006, Goswami et al.[13] analysed the spectra of the s- and r-rich metal-poor star HE 1305-0007, and concluded that the observed abundances could not be well fit by a scaled solar system r-process pattern nor by the s-process pattern of an AGB model. This star shows that the en- hancements of the neutron-capture elements Sr and Y are much lower than the enhancement of Ba and the abundances ratio [Pb/Ba] is only about 0.05. Because of the Na overabundance, which is believed to be formed through deep CNO-burning, Goswami et al.[13] have also speculated that this star should be polluted by a massive AGB star. Clearly, the restudy of elemental abundances in this object is still very important for well understand- ing the nucleosynthesis of neutron-capture elements in metal-poor stars. The chemical abundance distributions of the very metal-poor double-enhanced stars are excellent informa- tion to set new constraints on models of neutron-capture processes at low metallicity. The metallicity of HE 1305- 0007 is [Fe/H] = −2.0, which is just at the upper limit of the metallicity for the observed double-enhanced stars. There have been many theoretical studies of s-process nucleosynthesis in low-mass AGB stars. Unfortunately, the precise mechanism for chemical mixing of protons from the hydrogen-rich envelope into the 12C-rich layer to form a 13C-pocket is still unknown.[14] It is interest- ing to adopt the parametric model for metal-poor stars presented by Aoki et al.[15] and developed by Zhang et al.[5] to study the physical conditions which could repro- duce the observed abundance pattern found in this star. In this Letter, we investigate the characteristics of the nucleosynthesis pathway that produces the special abun- dance ratios of s- and r-rich object HE 1305-0007 using the s-process parametric model.[5] The calculated results are presented. We also discuss the characteristics of the s-process nucleosynthesis at low metallicity. We explored the origin of the neutron-capture elements in HE 1305-0007 by comparing the observed abundances with predicted s- and r-process contribution. For this http://arxiv.org/abs/0704.0576v1 purpose, we adopt the parametric model for metal-poor stars presented by Zhang et al.[5] The ith element abun- dance in the envelope of the star can be calculated by Ni(Z) = CsNi,s + CrNi,r10 [Fe/H], (1) where Z is the metallicity of the star, Ni,s is the abun- dance of the i-th element produced by the s-process in the AGB star and Ni,r is the abundance of the ith element produced by the r-process (per Si = 106 at Z = Z⊙), Cs and Cr are the component coefficients that correspond to contributions from the s-process and r-process respec- tively. There are four parameters in the parametric model of s- and r-rich stars. They are the neutron exposure per thermal pulse ∆τ , the overlap factor r, the component co- efficient of the s-process Cs and the component coefficient of the r-process Cr. The adopted initial abundances of seed nuclei lighter than the iron peak elements were taken to be the solar-system abundances, scaled to the value of [Fe/H] of the star. Because the neutron-capture-element component of the interstellar gas to form very mental- deficient stars is expected to consist of mostly pure r- process elements, for the other heavier nuclei we use the r-process abundances of the solar system,[16] normalized to the value of [Fe/H]. The abundances of r-process nu- clei in Eq. (1) are taken to be the solar-system r-process abundances[16] for the elements heavier than Ba, for the other lighter nuclei we use solar-system r-process abun- dances multiplied by a factor of 0.4266.[5,17] Using the observed data in the sample star HE 1305-0007, the pa- rameters in the model can be obtained from the para- metric approach. Figure 1 shows our calculated best-fit result. For this star, the curves produced by the model are consistent with the observed abundances within the error limits. The agreement of the model results with the observations provides strong support to the validity of the paramet- ric model. In the AGB model, the overlap factor r is a fundamental parameter. In 1998, Gallino et al.[8] (G98 hereafter) have found an overlap factor of r ≃ 0.4−0.7 in their standard evolution model of low-mass (1.5−3.0M⊙) AGB stars at solar metallicity. The overlap factor calcu- lated for other s-enhanced metal-poor stars lies between 0.1 and 0.81.[5] The overlap factor deduced for HE 1305- 0007 is about r = 0+0.17 −0.00, which is much smaller than the range presented by G98. This just implies that iron seeds could experience only one neutron exposure in the nucleosynthesis region.[18] For the third dredge-up and the AGB model, sev- eral important properties depend primarily on the core mass.[19−21] In the core-mass range 0.6 ≤ Mc ≤ 1.36, an analytical formula for the AGB stars was given by Iben[19] showing that the overlap factor increases with decreasing core mass. Combing the formula and the initial-final mass relations,[4] Cui and Zhang [22] obtained the overlap factor as a function of the initial mass and metallicity. In an evolution model of AGB stars, a small r may be realized if the third dredge-up is deep 40 50 60 70 80 = 0.71(mbarn 1) r = 0.00 Cr = 67.43 Cs= 0.0047 = 1.290 FIG. 1: Best fit to observational result of HE 1305-0007. The black circles with appropriate error bars denote the ob- served element abundances, the solid line represents predic- tions from s-process calculations considering r-process contri- bution (taken from Ref. [13]). enough for the s-processed material to be diluted by ex- tensive admixture of unprocessed material. Karakas[21] and Herwig[23,24] have found that the third dredge-up is more efficient for the AGB stars with larger core masses, confirming the low values of r obtained by Iben[19] in these cases. In AGB stars with initial mass in the range M = 1.0−4.0M⊙, the core mass Mc lies between 0.6 and 1.2M⊙ at [Fe/H] = −2.0. According to the formula pre- sented by Iben,[19] the corresponding values of r would range between 0.76 and 0.26. Obviously, the overlap fac- tor of HE 1305-0007 is smaller than this range. We have extensively explored the convergence of the abundance distribution of s-process elements through re- current neutron exposures. All elements, including Pb, were found to be made in the first neutron exposure. This is consistent with the small overlap factor r ≃ 0 deduced in our best-fit model. Thus the possibility that the s-process material has experienced only one neutron exposure in the nucleosynthesis region is existent. In 2000, Fujimoto, Ikeda and Iben[25] have proposed a scenario for the extra-metal-poor AGB stars with [Fe/H]< −2.5 in which the convective shell triggered by the thermal runaway develops inside the helium layer. Once this occurs,12C captures proton to synthesize13C and other neutron-source nuclei. The thermal runaway continues to heat material in the thermal pulse so that neutrons produced by the 22Ne(α, n)25Mg reaction as well as the 13C(α, n)16O reaction may contribute. In this case, only one episode of proton mixing into He intershell layer occurs in metal-poor stars.[25,15,45] After the first two pulses no more proton mixing occurs although the third dredge-up events continue to repeat, so the abun- dances of the s-rich metal-poor stars can be characterized by only one neutron exposure. Obviously, the metallicity of HE 1305-0007 is higher than the range of metallicity for this scenario. 0.1 1 FIG. 2: Best fit to observational result of metal-deficient star HE 1305-0007 shows the calculated abundances logε(Pb), logε(Ba) and logε(Sr) and reduced χ2 (bottom)as a function of the neutron exposure ∆τ in a model with Cr = 67.4, Cs = 0.0047 and r = 0. These are compared with the ob- served abundances of HE 1305-0007. 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 3: The same as those in Fig. 2 but as a function of the overlap factor r in a model with ∆τ = 0.71. One major goal of this work is to explore the charac- teristics of the binary system that HE 1305-0007 origin belongs to. The enhancement of the neutron-capture ele- ments Ba and Pb suggests that in a binary system a mass- transfer episode from a former AGB star took place. The radial-velocity measurement indicates that HE 1305-0007 is a high-velocity star, with a radial-velocity of 217.8 km s−1. From the high velocity of HE 1305-0007, we could speculate that the star could be a runaway star from a bi- nary system, which has experienced the AIC event. The strong overabundance of r-process elements for HE 1305- 0007 (Cr = 67.4) should be a significant evidence for the AIC scenario. In this case, the orbital separation must be small enough to allow for capture of a sufficient amount of material to create the formation of this event. As- suming that HE 1305-0007 is formed in a binary system, the AGB connection strongly suggests that this star is a member of a post-common-envelope binary. This must be the case if the overabundances of s-process elements are attributed to mass-transfer from an AGB star. We can only speculate about the effects of common-envelope phase on the nuclear signatures in a metal-poor star that was formed from this mechanism. One case could in- volve several thermal pulses with dredge-up causing the observed abundance distribution corresponding to larger overlap factor. However, after the s-process material has experienced only one neutron exposure in the nucleosyn- thesis region and is dredged-up to its envelope, the AGB evolution is terminated by the onset of common-envelope evolution. This could explain the characteristic of single neutron exposure in this star. In addition, based on the Na overabundance, Goswami et al.[13] have speculated that HE 1305-0007 should be polluted by a massive AGB star, which has a large core-mass and favours the forma- tion of AIC. Clearly, a detailed theoretical investigation of this scenario is highly desirable. The neutron exposure per pulse, ∆τ , is another funda- mental parameter in the AGB model. In 2006, Zhang et al.[5] have deduced the neutron exposure per pulse for other s-enhanced metal-poor stars which lies between 0.45 and 0.88 mbarn−1. The neutron exposure deduced for HE 1305-0007 is about ∆τ = 0.71+0.06 −0.04 mbarn −1. Fig- ures 2 and 3 show the calculated abundances logε(Pb), logε(Ba) and logε(Sr) as versus the neutron exposure ∆τ in a model with Cr = 67.4, Cs = 0.0047 and r = 0 and versus overlap r with ∆τ = 0.71 mbarn−1, respec- tively. These are compared with the observed abun- dances of HE 1305-0007. There is only one region in Fig. 2, ∆τ = 0.71+0.06 −0.04 mbarn −1, in which all the ob- served ratios of three representative elements can be ac- counted for within the error limits. The bottom panel in Fig. 2 displays the reduced χ2 value calculated in our model with all detected elemental abundances be- ing taken into account and there is a minimum, with χ2 = 1.290, at ∆τ = 0.71 mbarn−1. From Fig. 3, we find that the abundances logε(Pb), logε(Ba) and logε(Sr)are insensitive to the overlap factor r in a wider range, 0 ≤ r ≤ 0.17. The uncertainties of the parameters for the star HE 1305-0007 are similar to those for metal- poor stars LP 625-44 and LP 706-7 obtained by Aoki et al.[15] In addition, it is worth further commenting on the be- haviour of logε(Sr), logε(Ba) and logε(Pb) as a function of the neutron exposure ∆τ seen in Fig. 2. The non- linear trends displayed in the plot reveal the complex dependence on the neutron exposure. The trends can be illustrated as follows. Starting from low neutron expo- sure and moving toward higher neutron exposure values, they show how the Sr peak elements are preferentially produced at nearly ∆τ∼ 0.4mbarn−1. At larger neutron exposure (e.g., ∆τ∼ 0.7mbarn−1), the Ba-peak elements become dominant. In fact, the higher neutron exposure favors large amounts of production of the heavier ele- ments such as Ba, La, etc. and less Sr, Y, etc.,[22] which is the reason of the abundance pattern of the s-process elements in HE 1305-0007, i.e. the enhancements of the neutron-capture elements Sr and Y are much lower than the enhancement of Ba and the abundances ratio [Pb/Ba] is only about 0.05. Then a higher value of logε(Pb)∼ 4 follows at ∆τ = 1.5 mbarn−1. In this case, the s-process flow extends beyond the Sr-peak and Ba-peak nuclei to cause an accumulation at 208Pb. Clearly, logε(Pb) is very sensitive to the neutron exposure. The r- and s-process component coefficients of HE 1305-0007 are about 67.4 and 0.0047, which implies that this star belongs to s+r stars. Recently, Zhang et al.[5] have calculated 12 s+r stars with 0.0005 ≤ Cs ≤ 0.0060. The s-process component coefficient of HE 1305-0007 lies in this range. The Ba and Eu abundances are most use- ful for unraveling the sites and nuclear parameters asso- ciated with the s- and r-process corresponding to those in extremely metal-poor stars, polluted by material with a few times of nucleosynthesis processing. In the Sun, the elemental abundances of Ba and Eu consist of signif- icantly different combinations of s- and r-process isotope contributions, with s:r ratios for Ba and Eu of 81:19 and 6:94, respectively.[16] From Eq. (1), we can obtain the s:r ratios for Ba and Eu are 95.7:4.3 and 30.1:69.9, which are obviously larger than the ratios in the solar system. From Fig. 1 we find that our model cannot explain the larger errors of some neutron-capture elements, such as Y and Zr in HE 1305-0007. This implies that our un- derstanding of the true nature of s-process or r-process is incomplete for at least some of these elements.[27] In conclusion, the star HE 1305-0007 is an s+r star with metallicity [Fe/H] = −2.0, which is just at the upper limit of the metallicity for the observed double-enhanced stars. Theoretical predictions for abundances starting with Sr fit well the observed data for the sample star, providing an estimation for neutron exposure occurred in AGB star. The calculated results indicated that al- most all s-elements were made in the first neutron expo- sure. Once this happens, after only one time dredge-up, the observed abundance profile of the s-rich stars may be reproduced in a single neutron exposure. From the high radial-velocity of HE 1305-0007, we speculate that the star could be a runaway star from a binary system, which has experienced the AIC event. The r-process el- ements in HE 1305-0007 (Cr = 67.4) should come from the AIC event. Because the orbital separation must be small enough to allow for capture of a sufficient amount of material to create the formation of AIC, this star should be a member of a post-common-envelope binary. After the s-process material has experienced only one neutron exposure in the nucleosynthesis region and is dredged-up to its envelope, the AGB evolution is terminated by the onset of common-envelope evolution. Clearly, such an idea requires a more detailed high-resolution study and long-term radial-velocity monitoring in order to reach a definitive conclusion. More in-depth theoretical and ob- servational studies of this scenario is highly desirable. References [1] Hill V et al 2000 Astron. Astrophys. 353 557 [2] Cohen J G et al 2003 Astrophys. J. 588 1082 [3] Qian Y Z and Wasserburg G J 2003 Astrophys. J. 588 1099 [4] Zijlstra A A 2004 Mon. Not. R. Astron. Soc. 348, [5] Zhang B, Ma K and Zhou G D 2006 Astrophys. J. 642 1075 [6] Barbuy B et al 2005 Astron. Astrophys. 429 1031 [7] Wanajo S et al 2005 Astrophys. J. 636 842 [8] Gallino R et al 1998 Astrophys. J. 497 388 [9] Gallino R et al 2003 Nucl. Phys. A 718 181 [10] Straniero O et al 1995 Astrophys. J. 440 L85 [11] Straniero O, Gallino R and Cristallo S 2006 Nucl. Phys. A 777 311 [12] Cohen J G et al 2005 Astrophys. J. 633 L109 [13] Aruna Goswami et al 2006 Mon. Not. R. Astron. Soc. 372 343 [14] Busso M et al 2001 Astrophys. J. 557 802 [15] Aoki W et al 2001 Astrophys. J. 561 346 [16] Arlandini C et al 1999 Astrophys. J. 525 886 [17] Cui W Y et al 2007 Astrophys. J. 657 1037 [18] Ma K, Cui W Y and Zhang B 2007 Mon. Not. R. Astron. Soc. 375 1418 [19] Iben I Jr 1977 Astrophys. J. 217 788 [20] Groenewegen M A T and de Jong T 1993 Astron. Astrophys. 267 410 [21] Karakas A I, Lattanzio J C and Pols O R 2002 PASA 19 515 [22] Cui W Y and Zhang B 2006Mon. Not. R. Astron. Soc. 368 305 [23] Herwig F 2000 Astron. Astrophys. 360 952 [24] Herwig F 2004 Astrophys. J. 605 425 [25] Fujimoto M Y, Ikeda Y and Iben I Jr 2000 Astro- phys. J. 529 L25 [26] Iwamoto N et al 2003 Nucl. Phys. A 718 193 [27] Travaglio C et al 2004 Astrophys. J. 601 864 [1] ∗Supported by the National Natural Science Foundation of China under Grant Nos 10373005, 10673002 and 10778616. [2] ∗∗To whom correspondence should be addressed. Email: [email protected] References

0704.0577

Membrane in M5-branes Background

Membrane in M5-brane Background Wei-shui Xua and Ding-fang Zengb Institute of Theoretical Physics P.O. Box 2735, Beijing 100080, P. R. China College of Applied Science, Beijing University Of Technology e-mail: [email protected], [email protected] Abstract In this paper, we investigate the properties of a membrane in the M5-brane back- ground. Through solving the classical equations of motion of the membrane, we can understand the classical dynamics of the membrane in this background. April 2007 http://arxiv.org/abs/0704.0577v3 1 Introduction In eleven-dimensional M theory, there exists two extended brane solutions, i.e membrane and M5-brane. The membrane was recovered in [1] as an elementary solution of D = 11 supergravity which preserves half of the spacetime supersymmetry, which is a electric source of four-form field. While, the M5-brane was found in [2] as a soliton solution of D = 11 supergravity also preserving half of the spacetime supersymmetry, but is magnetic source of the same four-form field. These extended brane solutions can be related to the corresponding brane solutions in ten-dimensional string theory. After performing the compactification and some dualities, these branes can be reduced to D-branes or other brane solutions in string theory [3]. In this paper, we will investigate the properties of M2-brane in the M5-brane back- ground. Here, we will not investigate the cases of the brane intersection. Instead, we are mainly concerned with the classical dynamics of membrane in the given background. As will be illustrated, due to the gravity force of M5-brane, the membrane evolves nontriv- ially. In the 11-dimensional supergravity, the classical solution of N coincident M5-brane reads ds2 = H− 3 ηµνdx µdxν +H 3 δijdx idxj, H = 1 + πNl3p (xi)2 = r2 + x11 2, µ, ν = 0, 1, · · · , 5, i, j = 6, 7, 8, 9, 11 (1.1) and the 4-form field strength takes the form F4 = dA3 = 3πNl pdvS4 (1.2) where the dvS4 denotes the volume form of a unit S 4 and lp is the Planck length in the 11-dimensional theory. The N coincident M5-brane are parallel to the xµ directions and located at R = 0 in the transverse space. In the near horizon limit R → 0, the harmonic form H will become H = πNl3p , and the other parts will choose the same forms as in the equations (1.1) and (1.2). As in [4], if we suppose that there are a periodic configuration of N coincident M5- brane along the x11 direction at intervals of 2πR11, and take the limit of 1 ≪ r/R11, then our background metric and the 4-form field strength will become ds2 = f− 3ηµνdx µdxν + f 3 δijdx idxj + f 3 (dx11)2, f = 1 + R11r2 2Nℓ3p dvS3 ∧ dx11, (xi)2, x11 = R11φ, (1.3) where µ, ν = 0, 1, · · · , 5, i, j = 6, 7, 8, 9 and 0 ≤ φ ≤ 2π . We can see this metric has an so(4) symmetry group of rotations in the directions transverse to the M5-brane. In the near horizon limit, the harmonic function f becomes f = R11r2 . While, the other parts of background (1.3) remain unchanged. Actually, if letting the radius of x11 coordinate approach zero, then the metric (1.3) can reduce to the N coincident NS5-brane solution in ten-dimensional string theory [5]. Here we will mainly focus on the classical dynamics of a M2-brane in the above back- grounds (1.1) and (1.3). The dynamics of this single membrane can be described by the Nambu-Goto and Wess-Zumino type effective action. However, for the coincident mem- branes, unlike the coincident D-brane in string theory which can be described by the effective action [6], their worldvolume action is still not very clear [7]. We choose the worldvolume coordinates of membrane as x0, x1, x2, and those of M5-brane as x0, · · · , x5. Hence M2-brane is “parallel” to the M5-brane, i.e it is extended in some of the M5-brane worldvolume directions xµ, and point-like in the directions transverse to the M5-brane (x6, x7, x8, x9, x11). Indeed, this configuration breaks supersymmetry completely. We can label the worldvolume coordinates of the M2-brane by ξµ, µ = 0, 1, 2, and use reparame- terization invariance on the worldvolume of the M2-brane to set ξµ = xµ. The position of the M2-brane in the transverse directions, (x6, · · · , x9, x11), give rise to scalar fields on the worldvolume of the M2-brane, (X6(ξµ), · · · , X9(ξµ), X11(ξµ)). A single M2-brane world- volume action [8] is given by the sum of the Nambu-Goto action and the Wess-Zumino type term in the following form SM2 = −T2 − detP [G]µν + T2 P [A] (1.4) where the tension of the M2-brane is expressed as T2 = 1/4π 2l3p, and P [· · ·] means the pullback operation P [G]µν = GMN(X), P [A] = AMNL(X) . (1.5) The indices M,N,L run over the whole eleven dimensional spacetime. And the fields GMN , AMNL denote the metric and form field in eleven dimensions. In the following sections, we will discuss the M2-brane classical dynamics in the above backgrounds, and suppose that the transverse coordinates of M5-brane only depend on the time coordinate. In this case the Wess-Zumino term in the membrane action will vanish. 2 Classical dynamics of membrane Now let us consider the membrane dynamics in the background (1.1). Since we have supposed that the directions transverse to the M5-brane X i are only the function of time t, where i = 6, 7, 8, 9, 11, the pullback quantities are as following P [G]tt = −H− 3 Ẋ iẊ i, P [G]x1x1 = H P [G]x2x2 = H 3 , P [A] = 0. (2.1) after substituting the above equations (2.1) into the M2-brane action (1.4), we get SM2 = −V T2 H−1 − Ẋ iẊ i (2.2) where V is the space volume of the M2-brane, also i = 6, · · · , 9, 11. We can find it is very similar to the corresponding one in [9] which is the DBI action of D-brane in the N NS5 brane background. Then through using the Legendre transformation, the Hamiltonian is H−1 − Ẋ iẊ i ≡ V E (2.3) where the E denotes the energy density. And the equation of motion will be H−1 − ẊjẊj H−1 − ẊjẊj . (2.4) Using this equation of motion (2.4), one can check that the Hamiltonian is conserved. To solve the (2.4), we need the initial conditions that it is ~X(t = 0) and ~̇X(t = 0). These two vectors define a plane in R5. By an SO(5) rotation, we can rotate this plane into the (x6, x7) plane. Then the motion will remain in the (x6, x7) space for all time. Thus, without loss of generality, we can study trajectories in this space. We choose the polar coordinates X6 = R cos θ, X7 = R sin θ. (2.5) Then the energy density (2.3) will become H−1 − Ṙ2 − R2θ̇2 , (2.6) and the angular momentum density will be H−1 − Ṙ2 − R2θ̇2 . (2.7) We can find this angular momentum of the M2-brane is conserved as well. From the membrane action (2.2), we can obtain energy momentum tensor. The components of Tµν are listed in the following T00 = − H−1 − Ẋ iẊ i Tij = −T2δij H−1 − Ẋ iẊ i, (2.8) and the other components of stress tensor are zero. From the angular momentum Lθ equation (2.7) and energy density E (2.6), we can get the equations of the coordinates R and θ Ṙ2 = T 22 + , (2.9) EH(R)R2 . (2.10) For simplicity, we can first consider Lθ = 0 case, then the radial equation is Ṙ2 = . (2.11) The right hand of the above equation can’t be smaller than zero, so we get a constraint on the coordinate R is πNl3p − 1. (2.12) From the above equation, we can see if the energy density E is larger than the tension of a M2-brane, T2, the constraint (2.12) is empty and the M2-brane can escape to infinity. However, for E < T2, the M2-brane does not have enough energy to overcome the grav- itational pull of the M5-brane, and then will fall down to the M5-brane from an initial position. Choosing the near horizon limit, hence the harmonic function becomes H = πNl3p/R Then the equation (2.11) will be Ṙ2 = πNl3p π2N2E2l6p R6. (2.13) Since the left hand of the equation (2.13) is nonnegative, the coordinate R has a maximal value πNE2l3p/T . Also from this equation, the minimal value of R is zero. Except for these two, there are no other extremum. But there is one inflexion between points R = 0 and πNE2l3p/T . We can regard the M2-brane is at the maximal value πNE2l3p/T at the initial time. Due to the gravitational force of M5-brane, the M2- brane then will roll down to the M5-brane. As the time t → ∞, the radial coordinate R approaches to zero. We can calculate the energy momentum tensor which is Tij = EH(R) as the R → 0, the Tij will approach to zero. It may regard as the pressure decreasing to zero. But we need to mention that the coordinate R can’t reach zero, since at this point the supergravity background will be not reliable. Then the classical dynamics of the membrane near R = 0 from the above analysis will become incorrect. Thus, in order to use the supergravity approximation, we must constrain the coordinate R to be larger than the planck length lp. Now we begin to consider the nonzero case of angular momentum. From the radial equation of motion (2.9), and after substituting the harmonic function H = 1+πNl3p/R we can get the constraint on the radial coordinate R is πNE2l3p πNl3p . (2.14) If choosing the equal case of the above equation, the constraint will become πNE2l3p πNl3p = 0. (2.15) The above equation only has one real root which is the maximal distance that M2-brane is separated from M5-brane. For simplicity, we choose the near horizon limit, then the equation of motion for the radial coordinate will become Ṙ2 = πNl3p π2N2E2l6p π2N2E2l6p R6. (2.16) We find that the equation (2.16) is still very difficult to solve. Instead, here, we take some analysis for this equation. If letting Lθ = 0, then this equation will reduce to the equation (2.13). We let the left hand of the equation (2.16) to zero, then we can get the extremal value for R. Actually, there are two only two real extremal values of the radial coordinate R. One is R = 0, the other is 108πNlp 3E2T2 + 12 6 + 27π2N2lp 6E4T2 − 12Lθ2 108π Nlp 3E2T2 + 12 6 + 27π2N2lp 6E4T2 . (2.17) When Lθ = 0, the above R value will reach the πNE2l3p/T . As the same in the Lθ = 0 case, between the R = 0 and (2.17) there exists a inflexion. We can suppose that the M2-brane is at the maximal value (2.17) at the initial time, then under the gravitational pull of M5-brane, it will monotonic approach to M5-brane. Of course for the Lθ nonzero case, the equation (2.10) for the θ coordinate in the near horizon background is θ̇ = Lθ = RLθ πNEl3p . Thus, if the radial coordinate R reaches the value (2.17), the angular velocity will choose the maximum, and as the R → 0, the angular velocity does also approach to zero. The energy momentum tensor satisfies Tij = −δij EH(R) = −δij πNEl3p Thus, it again goes to zero as in the Lθ = 0 case. As mentioned in the above, near the region R = 0, the classical background will be instability due to the strong interaction. Hence the above supergravity analysis will become unreliable in this region. From the first section, we already know that, after compactifying a periodic circle of coordinate x11, the metric (1.1) will become background (1.3). In the following, we study the membrane dynamics in this background (1.3). Here, we still suppose the directions transverse to the M5-brane X i and X11 are only the function of time t, where i = 6, 7, 8, 9, then the pullback quantities take the form as follows P [G]tt = −f− 3 + f 3 Ẋ iẊ i + f 3 Ẋ11Ẋ11, P [G]x1x1 = f P [G]x2x2 = f 3 , P [A] = 0. (2.18) After inserting (2.18) into the M2-brane action (1.4), we can get SM2 = −V T2 f−1 − Ẋ iẊ i − Ẋ11Ẋ11 (2.19) where V is the space volume of the M2-brane. This action is also very similar to action in [9] except for the harmonic function and dimension. From the Lagrangian (2.19), we can derive the equations of motion for the membrane in this background as followes f−1 − ẊjẊj − R211φ̇2 f−1 − ẊjẊj −R211φ̇2 , (2.20) R11φ̇ f−1 − ẊjẊj − R211φ̇2  = 0. (2.21) Due to some symmetry of this system, there are also some conserved charges. Time translation invariance implies that the energy H = PiẊ i + Pφφ̇− L (2.22) is conserved. The momentum is obtained by varying the Lagrangian L, δẊ i T2V Ẋi f−1 − ẊjẊj − R211φ̇2 , (2.23) T2V R f−1 − ẊjẊj − R211φ̇2 . (2.24) Substituting (2.23) into (2.22), we find that the energy is given by f−1 − Ẋ iẊ i − R211φ̇2 ≡ V E. (2.25) And since the harmonic function f = 1 + R11r2 , then ∂if(r) = X if ′(r)/r, and one of the equations of motion (2.20) can be rewritten as f−1 − ẊjẊj − R211φ̇2 X if ′ 2rf 2 f−1 − ẊjẊj − R211φ̇2 , (2.26) the other one is unchanged. To solve these equations, we need to specify some initial conditions for the coordinates. One condition is ~X(t = 0) and ~̇X(t = 0). These two vectors define a plane in R4. By an SO(4) rotation symmetry, we can rotate this plane into the (x6, x7) plane. The other one is φ(t = 0) and φ̇(t = 0). Then the motion of the membrane will remain in the (x6, x7, φ) space for all time. Thus, without loss of generality, we can study trajectories in this space. In addition to the energy, the angular momentum of the M2-brane is conserved as well. It is given by (X6P 7 −X7P 6). (2.27) Using the expression for the momentum, (2.23), we find that Lθ = T2 X6Ẋ7 −X7Ẋ6 f−1 − ẊjẊj −R211φ̇2 . (2.28) Another interest quantity is the stress tensor Tµν associated with the moving M2- brane. The component T00 denotes the energy density, so it is given by expression (2.25) for E, with the factor of the volume stripped off. We list the components of Tµν in the following equations T00 = − f−1 − Ẋ iẊ i −R211φ̇2 Tij = −T2δij f−1 − Ẋ iẊ i − R211φ̇2, Tφφ = −T2R211 f−1 − Ẋ iẊ i −R211φ̇2, (2.29) and the other components of stress tensor are zero. Due to the so(4) rotation symmetry in the transverse directions of M5-brane, it is convenient to change to the polar coordinates X6 = r cos θ, X7 = r sin θ. (2.30) In these coordinates, the expressions of the energy density and angular momentum density becomes f−1 − ṙ2 − r2θ̇2 − R211φ̇2 , (2.31) f−1 − ṙ2 − r2θ̇2 − R211φ̇2 , (2.32) f−1 − ṙ2 − r2θ̇2 − R211φ̇2 . (2.33) One can check directly that Lθ and Lφ are conserved by using the equations of motion (2.26) and (2.21). In order to solve the equations of motion for the given energy and angular momentum densities E, Lθ and Lφ, we would like to solve the equation (2.32) for θ̇, and then substitute this solution into the (2.31). Then the equation for the θ̇ is . (2.34) Inserting it into (2.31), (2.32) and solving for ṙ, we find ṙ2 = E2f 2 T 22 + . (2.35) Also we have the equation of φ̇ EfR211 . (2.36) In the next, we would like to study the solutions of the equations of motion (2.34), (2.35) and (2.36). Firstly, we consider the angular momentum Lθ = 0 case. Then Equation (2.34) implies that θ is constant, while the radial equation (2.35) takes the form ṙ2 = E2f 2 T 22 + . (2.37) Since the right hand side of the equation(2.37) is non-negative, then we can get the condition 1 T 22 + ≥ 0. After substituting the harmonic function f , (1.3), into it, we find the constraint on r (for fixed energy density E) R11r2 − 1 (2.38) where we can define the effective M2-brane tension is T 2e = T (2.39) From the equation of constraint (2.38), obviously, if the energy density E is larger than the effective tension of a M2-brane, Te, the constraint (2.38) is empty and the M2-brane can escape to infinity. For E < Te, the M2-brane does not have enough energy to escape the gravitational pull of the M5-brane, which means that it cannot exceed some maximal distance from the M5-brane. Under the near horizon limit, the harmonic function f will become f = R11r2 . Then the equation (2.38) will be R11r2 . Thus, if r << , the effective tension of membrane Te satisfies the constraint Te/E >> 1. However, r >> , the case will be otherwise. Indeed, in this near horizon case, we can solve for the trajectory r(t), φ(t) exactly. Substituting the harmonic function f = R11r2 into (2.37), we find the equation of motion ṙ2 = E2N2l6p T 22 + r4. (2.40) Then the solution can be obtained L2φ +R NR11E2l3p t (2.41) where we choose t = 0 to be the time at which the M2-brane reaches its maximal distance from the M5-brane. For an observer living on M5-brane, the M2-brane reaching r = 0 will take an infinite time. Also, the M2 radial motion is similar to D-brane’s motion in And the equation of motion (2.24) becomes φ̇2 = r2 − ṙ2 11 + T . (2.42) Substituting the solution r into equation (2.42), we can get the equation L2φ + T . (2.43) Then after solving this equation, the solution can be obtained L2φ + T t. (2.44) It is interesting to calculate the energy momentum tensor of the M2-brane in this case. The energy density T00 is constant and equal to E throughout the time evolution. However, for the parts Tij and Tφφ, we can find Tij = −δij Tφφ = − R211T . (2.45) We see that the pressure goes smoothly to zero as r → 0, since f(r) ∼ 1/r2. But again as the analysis in the background (1.1), this may be unreliable near the r = 0 region. So far we have discussed the trajectories with vanishing angular momentum density (2.32). A natural question is whether anything qualitatively new occurs for non-zero Lθ. Just as [9], we can think as follows, the radial equation of motion (2.35) can be thought of as describing a particle with mass m = 2, moving in one dimension r in the effective potential Veff(r) = E2f 2 T 22 + (2.46) with zero energy. Now we discuss the properties of this effective potential Veff . In the small r region, it will behave as Veff(r) ≃ E2Nl3p r2. (2.47) For large r, the leading terms of this potential will be Veff(r) ≃ − 1. (2.48) If the energy density of the M2-brane is smaller than the effective tension of a M2-brane, E < Te, then the effective potential Veff approaches to a positive constant (2.48) as r → ∞, which means the membrane cannot escape to infinity. From the equation (2.47), we can find that in order to have trajectories at non-zero r, the angular momentum must satisfy the constraint NEl3p . (2.49) If the constraint (2.49) is not satisfied, the only solution is r = 0. But, if the condition (2.49) is satisfied, the trajectory of the M2-brane is qualitatively similar to that in the Lθ = 0 case. It will approach the M5-brane and does not have stable orbits at finite r. For the case Te >> E, the whole trajectory lies again in the region r << Nl3p/R11, and one can approximate the harmonic function (1.3) by f = R11r2 . Then the equation (2.35) for ṙ will be ṙ2 = Nl3pE E2N2l4p T 22 + r4 , (2.50) with the solution L2φ +R R11NE2l3p − R11L2θ NR11E2l3p − R211L2θ NEl3p t. (2.51) We can find that the non-zero angular momentum can slow down the exponential decrease of r as t → ∞. In the near horizon limit f(r) = Nl , the solution of the equation (2.34) for θ is R11Lθ ENl3p t. (2.52) The solution (2.51) and (2.52) mean that the M2-brane in the background (1.3) will be spiralling towards the origin, circling around it an infinite number of times in the process. The equation about φ is Lφ(NE 2l3p −R11L2θ) ENl3p(L cosh R11E2Nl3p − R211L2θ ENl3p . (2.53) and the solution of the above equation reads L2φ +R NE2l3p − L2θ R11NE2l3p − R211L2θ ENl3p t. (2.54) At t = 0, the φ = 0, however, the time t → ∞, then, φ → Lφ NE2l3p−L Thus, the non-zero angular momentum Lθ slows down the variation of φ. From these three solutions, we know that the M2-brane is circling along the θ direction, varying along the φ and falling down towards the M5-brane in the process. Also, the energy momentum tensor Tij and Tφφ will approach to zero as r → 0, since f(r) ∼ 1/r2. But we must mention that, near the r = 0 region, the discussion may be incorrect due to the strong coupling. In the background (1.3), the results about the dynamics of a M2-brane have some similar properties as studying in [9]. This can be understood that the D2-brane and NS5-brane in IIA can be got by compactified one transverse dimension of M2-brane and M5-brane in M theory. The solutions of equation of motion describe the M2-brane falling towards the M5-brane. In the non-zero angular momentum Lθ, the M2-brane is spiralling towards the M5-brane. But both in this two case, M2-brane has a angular momentum Lφ. We need to mention that the background (1.3) is only correct in the limit of 1 ≪ r/R11. Therefore, as the M2-brane approaches the M5-brane, the energy momentum tensor Tij and Tφφ approaching zero may be unreliable. Since here the radial coordinate r is smaller than the radius R11. So we are not sure whether the membrane will have the same behavior just like the late time behavior of unstable D-brane [10, 11, 12, 13, 14]. In the above sections, we investigated the membrane classical dynamics in various M5-brane backgrounds. There may be some generalizations, since under the Penrose limit, the N coincident M5-brane solution (1.1) will reduce to the AdS7 × S4 geometry. Hence one can investigate the membrane dynamics in this geometry. For the (1.1), (1.3) and their near horizon background geometry, after calculating the classical equations of motion of membrane from the membrane action (1.4), we can analyze the moving trajec- tories of membrane. In some particular cases, we can get the exact solution of trajectories of membrane. However, generally, the equations of motion is very difficult to solve. But through analyzing these equations, we still can obtain some qualitative information about the motion of membrane. Consequently, in the M5-brane background, the membrane will be falling and spiralling towards to the M5-brane by the gravitational force of M5-brane. In the near M5-brane region, i.e R (or r) being of the order of the planck length lp, the above analysis of the classical dynamics of membrane may not be trusted, since the method of the supergravity approximation is unreliable. Acknowledgements We would like to thank Yi-hong Gao for the useful suggestions and discussions. References [1] M. J. Duff and K. Stelle, “Multimembrane solutions of d = 11 supergravity,” Phys. Lett. B 253 (1991) 113. [2] R. Gueven, “Black p-brane solutions of D = 11 supergravity theory,” Phys. Lett. B 276 (1992) 49. [3] J. Polchinski, “String Theory (Vol. I, Vol. II),” Cambridge Press, 1998. [4] Y. Hyakutake, “ Expanded Strings in the Background of NS5-branes via a M2-brane, a D2-brane and D0-branes,” hep-th/0112073. [5] C. G. Callan, J. A. Harvey and A. Strominger, “Worldbrane actions for string soli- tons,” Nucl. Phys. B367: 60-82, 1991; “World sheet approach to heterotic instantons and solitons,” Nucl. Phys. B359: 611-634, 1991. [6] R. C. Myers, “Dielectric branes,” JHEP 9912: 022, 1999 [hep-th/9910053]. [7] A. Basu and J. A. Harvey, “The M2-M5 brane system and a generalized Nahm’s equation,” Nucl. Phys. B713: 136-150, 2005 [hep-th/0412310]. [8] E. Bergshoeff, E. Sezgin and P. K. Townsend, “Properties Of The Eleven-Dimensional Super Membrane Theory,” Annals Phys 185 (1988) 330. [9] D. Kutasov, “D-Brane Dynamics Near NS5-Branes,” hep-th/0405058; “A Geometric interpretation of the open string tachyon,” hep-th/0408073; K. L. Panigrahi, “D- Brane Dynamics in Dp-Brane Background,” hep-th/0407134. [10] A. Sen, “Tachyon dynamics in open string theory,” Int. J. Mod. Phys. A20: 5513- 5656, 2005 [hep-th/0410103]. [11] A. Sen, “Rolling tachyon,” JHEP 0204, 048 (2002) [hep-th/0203211]. [12] F. Larsen, A. Naqvi and S. Terashima, “Rolling tachyons and decaying branes,” hep-th/0212248. [13] T. Okuda and S. Sugimoto, “Coupling of rolling tachyon to closed strings,” Nucl. Phys. B647, 101 (2002) [hep-th/0208196]. [14] N. Lambert, H. Liu and J. Maldacena, “Closed strings from decaying D-branes,” hep-th/0303139. Introduction Classical dynamics of membrane

0704.0578

3D photospheric velocity field of a Supergranular cell

Astronomy & Astrophysics manuscript no. delmoro07˙LR c© ESO 2021 November 10, 2021 3D photospheric velocity field of a Supergranular cell Del Moro, D., Giordano, S. and Berrilli, F. Dipartimento di Fisica, Università di Roma “Tor Vergata”, I-00133 Roma, Italy Received date will be inserted by the editor; accepted date will be inserted by the editor ABSTRACT Aims. We investigate the plasma flow properties inside a Supergranular (SG) cell, in particular its interaction with small scale magnetic field structures. Methods. The SG cell has been identified using the magnetic network (CaII wing brightness) as proxy, applying the Two-Level Structure Tracking (TST) to high spatial, spectral and temporal resolution observations obtained by IBIS. The full 3D velocity vector field for the SG has been reconstructed at two different photospheric heights. In order to strengthen our findings, we also computed the mean radial flow of the SG by means of cork tracing. We also studied the behaviour of the horizontal and Line of Sight plasma flow cospatial with cluster of bright CaII structures of magnetic origin to better understand the interaction between photospheric convection and small scale magnetic features. Results. The SG cell we investigated seems to be organized with an almost radial flow from its centre to the border. The large scale divergence structure is probably created by a compact region of constant up-flow close to the cell centre. On the edge of the SG, isolated regions of strong convergent flow are nearby or cospatial with extended clusters of bright CaII wing features forming the knots of the magnetic network. Key words. Sun:photosphere – Sun:magnetic fields – Methods:data analysis 1. Introduction Solar research is currently working on understanding how turbulent convection on the Sun transports mass and en- ergy through the convective zone, how it couples with the magnetic field and how it manages to deposit in the higher parts of the solar atmosphere the energy released from the corona. Among the different approaches to these questions, observations of the solar photosphere are essential, as they provide the only direct look at what is happening just below the solar surface. The hierarchy of surface features found on the photosphere are the visible representation of the plasma flows beneath the photosphere and are customar- ily classified by size and lifetime as patterns of granula- tion (1 Mm, 0.2 hr), mesogranulation (5-10 Mm, 5 hr) and supergranulation (15-35 Mm, 24 hr). These features have been initially regarded as direct manifestation of var- ious sized convection cells existing in the convection zone (Schrijver et al., 1997; Raju et al., 1999); lately, the idea is consolidating that meso and supergranulation are sig- natures of a collective interaction of granular cells (Rast, 2003; Roudier et al., 2003; Berrilli et al., 2005). Despite years of intensive studies, the character of their motions remains not completely understood (Beck & Duvall, 2000; Krishan et al., 2002; Berrilli et al., 2004; Del Moro et al., 2004; DeRosa & Toomre, 2004). The aim of the study we present is to investigate the origin of the supergranular (SG) flow field: directly convective or a collective interaction of smaller convective features. The study performed by Simon & Leighton (1964) initi- ated the campaign to characterize supergranular flows. Outflows on SG scales have been observed to sweep embedded granules and magnetic flux elements toward Send offprint requests to: [email protected] convergence lanes between cells (Leighton, 1964; Zwaan, 1978; Rimmele, 1989; Shine et al., 2000). Such behaviour causes the chromospheric transition CaII k line to be a good proxy for the network of intercellular lanes due to the higher magnetic elements density in the SG perimeters. The advent of full-disk Doppler imaging, pro- vided by the MDI onboard SOHO spacecraft, has con- siderably improved our capability to study such fea- tures (Hathaway et al., 2002; DeRosa & Toomre, 2004; Paniveni et al., 2004; Meunier et al., 2007); but direct ob- servations of supergranular flows are still hindered by the fact that there is no contrast on supergranular scales in visi- ble light, observations in CaII only provide the cell network boundaries and Doppler images show SG only away from disk centre. At present, the only methods to reconstruct the full 3D vector velocity field are direct Doppler measurement in combination with a tracking type measure for the velocity horizontal component (above the τ = 1 surface) or Local Helioseismology (below the τ = 1 surface). To gain com- plete insight of the dynamics of the plasma flows inside a SG structure, we need a spatial and temporal resolution still not reached by local helioseismology, while to obtain the 3D velocity field through the other method, observations with very high spatial, spectral and temporal resolution are necessary. With the assumption that granule motions are mainly driven by plasma flows (Rieutord et al., 2001), it is possible to employ the TST to infer the horizontal velocity field. In this work we reconstruct the 3D velocity field of a sin- gle SG structure and investigate in detail its plasma flow using data acquired with the IBIS spectrometer, trying to discern whether the SG pattern has a convective nature or is originated by small scale structure interaction. http://arxiv.org/abs/0704.0578v2 2 Del Moro et al.: 3D photospheric velocity field of a SG cell Fig. 1. A representative synoptic panel from the 16th October 2003 dataset. Upper left panel: Ca II wing intensity image. Upper middle panel: Doppler velocity field computed from FeI 709.0 nm line scan. Upper right panel: FeI 709.0 nm line core intensity Lower left panel: Doppler velocity field computed from FeII 722.4 nm line scan. Lower middle panel: FeII 722.4 nm line core intensity. Lower right panel: Continuum (near 709.0 nm) intensity image. Line Wavelength zTcore FWHMRFI zVline FWHMRFI [nm] [km] [km] [km] [km] FeI 709.0 ≃100 ∼300 ≃140 ∼300 FeII 722.4 ≃50 ∼200 ≃70 ∼200 Table 1. Line RF peak depths. Depths are in km above the level τ500nm = 1. 2. Observations The data utilized in this analysis have been acquired with the IBIS (Interferometric BIdimensional Spectrometer) 2D spectrometer (Cavallini et al., 2001; Cavallini, 2006) on October 16, 2003 (from 14:24 UT to 17:32 UT). We imaged a roundish network cell near the solar disk cen- tre (SLAT=7.8N, SLONG=3.6E). When observed in MDI high-resolution magnetograms, all the features outlining the cell exhibit negative polarity and seem to survive for at least 10 hours, with little or no evolution. The full dataset consists of 600 sequences, containing a 16 image scan of the FeI 709.0 nm line, a 14 image scan of the FeII 722.4 nm line and 5 spectral images in the wing (line centre + 12 nm) of the CaII 854.2 nm line, imag- ing a round Field of View (FoV) of about 80” diameter. Each monochromatic image was acquired with a 25 ms ex- posure time by a 12bit CCD detector, whose pixel scale was 0.17′′·pixel−1. The time required for the acquisition of a single sequence was 19 s, thus setting the temporal res- olution. Each image was reduced with the standard IBIS pipeline (Janssen & Cauzzi, 2006; Giordano et al., 2007), correcting for CCD non linearity effects, dark current, gain table and blue shift. The Line of Sight (LoS) velocity fields were computed for the Fe I and Fe II lines by means of Doppler shifts, evaluated, pixel by pixel, fitting a Gaussian on the line profile. In order to remove the orbital contribu- tion, we set to zero the average value of each LoS velocity image. The 5-minutes oscillations were removed applying a 3D Fourier filter in the kh −ω domain with a cut-off veloc- ity of 7 km·s−1 both on intensity and velocity image series. After the whole reduction process and selecting only the period of good seeing we are left with a 30 minutes dataset imaging a square FoV of ∼ 50”. An example of the im- ages of this reduced dataset is shown in Fig. 1. The mean resolution due to the seeing of the CaII images is 0.35”; the mean resolution of the LoS velocity images, the inten- sity continuum images and the line core images is instead 0.45”, somewhat degraded by both the reduction pipeline and the kh − ω filtering. In order to obtain information about the depth dependence of a photospheric quantity by associating a suitable ‘for- mation zone’ with a line, it is possible to consider its ef- Del Moro et al.: 3D photospheric velocity field of a SG cell 3 fect on the line characteristic as linear perturbations and to study the Response Function RF of the emergent line characteristic at the observed wavelengths within the line. In particular, the RF Ip is, at each depth, the function we must use to weigh the perturbation p in order to get the variation of the emergent intensity I (Caccin et al., 1977). This approach has been employed to derive the RF IT and Fig. 2. Core intensity fields of FeII 722.4 nm (z≃50 km) and FeI 709.0 nm (z≃100 km ) in comparison with con- tinuum image (z≃0 km) and CaII 854.2 nm wing intensity field (z≃150 km) (Qu & Xu , 2002). The z axis is greatly exaggerated with respect to the x-y axes in order to allow a better visualization. V for the spectral lines FeI 709.0 nm and FeII 722.4 nm (Del Moro, 2005). In Table 1 we report the photospheric depths of the line core RF IT maximum (zTcore), and of the mean RF V maxi- mum (zVline) for the two spectral lines. We also report the RF full width at half maximum for the two spectral lines: these rather large values imply broad formation zones for both the FeI 709.0 nm and FeII 722.4 nm Doppler velocity and core intensity signals. 3. 3D Velocity Field The TST procedure (Del Moro, 2004) has been applied on the continuum image series and on both the FeII 722.4 nm and FeI 709.0 nm Doppler field series, in order to retrieve the horizontal velocity field at different depths of the solar atmosphere. To minimize the effect of the proper motion of the granules, which are used as trackers of the mean plasma flow, we computed the horizontal velocity field using all the structures that were tracked, so that statistically at least one tracker is present in each interpolated horizontal velocity field pixel, as suggested by Behan (2000). This means we used a grid step of ∼ 1.5 Mm and a temporal window of ∼ 30 min. Possibly, this would not completely remove the noise associated with granule proper motions or residuals from the 5-min oscillation filtering, but should minimize it. Combining the horizontal velocity fields retrieved from Fig. 3. 3D representation of velocity vectors extracted from continnum (z≃0 km), FeII 722.4 nm (z≃70 km) and FeI 709.0 nm (z≃140 km). Cone size is proportional to the ve- locity vector module: the yellowish cone corresponds to 1 km·s−1. The z axis is greatly exaggerated with respect to the x-y axes in order to allow a better visualization. the Dopplergrams by the TST and the Doppler LoS velocity, the 3D vector field has been reconstructed for the FeII 722.4 nm and FeI 709.0 nm lines. We are aware that associating the vector fields to precise heights in the photosphere is an oversemplification, as can be readily understood from the large FWHM reported in Table 1, nevertheless, we did it for the sake of a good visualization: in Fig. 3 we show the mean 3D velocity field associated to the the dataset. In both the 3D fields we retrieved, the vector velocity appears to be structured quite coherently with the SG feature visible in the CaII wing images. In order to further investigate the structuring of the velocity field, we computed the average continuum (upper panel of Fig. 4) and CaII wing (bottom panel of Fig. 4) intensity images, the average Dopplergram from FeII 722.4 nm (middle panel of Fig. 4) and the average Dopplergram from FeI 709.0 nm (upper panel of Fig. 4) and correlated them. While the average continuum image does not show any evident signal, there is a significant correlation between strong downflows and bright CaII features, in particular for the complex cluster of features in the lower part of the FoV. This issue can be at least partially explained by the coherence of the 3D velocity field with the SG structure. We expanded this study by comparing the averaged images with the horizontal velocity field extracted by the TST from granules as seen in the continuum and up-flows from the FeI 709.0 nm Doppler images. We excluded from this analysis the FeII 722.4 nm Doppler images because we found its horizontal velocity field to be not as reliable as the others. This is due to the TST finding less than optimal number of features to track because of the shallowness of the FeII 722.4 nm line. A shallow line Doppler shift is much harder to measure by the LoS velocity reconstruction procedure, resulting in a more noisy dopplergram. This noise is interpreted by the TST as a fast variation of the structures, therefore causing a lot of them to be rejected for the tracking. As a consequence, the TST finds too few trackers in the 722.4 Doppler for the divergence field to be reliably reconstructed. 4 Del Moro et al.: 3D photospheric velocity field of a SG cell The horizontal velocity fields extracted by the TST are Fig. 4. Upper panel: average continuum image with the horizontal velocity field (obtained by tracking granules) represented as red arrows. The granules were tracked by ap- plying the TST to the continuum image time series. Lower panel: divergence field computed from the interpolated hor- izontal velocity field. shown superimposed on the average images in the left panels of Fig. 4 and Fig. 5, above the associated 2D divergence images. The values of the divergence fields range from +0.25 km s−1 Mm−1 in the brightest part of the image to −0.25 km s−1 Mm−1 in the darkest parts. The continuum granules and FeI 709.0 nm up-flow fields show a divergent flow from the centre of the SG structure and convergent flows in the border of the SG structure. In detail, these two fields agree very well, showing a single, large divergent feature in the centre of the SG, whose mean value is about +0.1 km s−1 Mm−1, almost completely surrounded by convergent flows of the same magnitude. The peak divergence signals we retrieved both in the centre and in the periphery of the SG cell are an order of magnitude larger than the averaged values reported by Meunier et al. (2007). This discrepancy probably stems mainly from the different temporal and spatial averaging processes in the divergence reconstruction and marginally from the different resolution of the two datasets. The structuring of the divergence field is very compatible with a net flow from the centre of the SG to its border. Fig. 5. Upper panel: average FeI 709.0 nm Doppler velocity image with the horizontal velocity field (obtained by track- ing up-flows) represented as red arrows. The up-flows were tracked by applying the TST to the FeI 709.0 nm Doppler velocity field time series. Lower panel: divergence field com- puted from the interpolated horizontal velocity field. Moreover, examining the LoS velocity fields, we found a strong and stable up-flow region nearby the divergence maximum, with a mean FeII 722.4 nm Doppler veloc- ity value of Vc ∼200 m·s −1 for almost the whole time span. This last region is liable to be the origin of the divergence signal we measured, possibly as suggested by Rieutord et al. (2000); Roudier et al. (2003). Observing the divergence images (bottom panels of Fig. 4 and Fig. 5) extracted from the horizontal velocity fields, the supergranule is outlined by convergences on ∼ 66% of its circumference, while the bright cluster area clearly visible in the CaII wing image does not seem to be a region of strong convergence, despite the fact that it is mostly formed of down-flows. This region has a mean FeII 722.4 nm Doppler velocity value Vc ∼-100 m·s −1 for the whole dataset time span (T ≃ 0.5 hour), and it seems to mantain similar values also for the part of the observations discarded for the loss of spatial resolution due to worsening seeing condition. A similar cluster of bright CaII structures is present in the upper-left part of the FoV, but its associated downflow shows a much smaller coherence: it has a mean FeII 722.4 nm Doppler velocity value Vc ∼-50 m·s −1 for more than half Del Moro et al.: 3D photospheric velocity field of a SG cell 5 of the time span, then it drops to ∼-20 m·s−1. Whether or not downflow regions like these may be organizing the SG pattern, as predicted by Rast (2003), is a question we cannot address due to the short duration of our dataset. 4. Horizontal Flow Analysis via Cork Tracking To further extract information about the plasma motion inside the SG structure, we tracked the evolution of tracers (corks) passively advected by instantaneous velocity and intensity fields. The corks, initially randomly spread over the FoV, are moved following the local gradient towards sites of minimum intensity or of minimum velocity in the case of intensity or velocity fields, respectively. We will com- pare the final and initial positions of the corks, which will give us information about the motion of downflows in the field of view. Corks are tracked for ∼ 16 minutes (a time sufficiently longer than the characteristic time scale of pho- tospheric fields (Müller et al., 2001; Berrilli et al., 2002) to let the cork settle in a downflow feature and track it for a while) and their initial and final position are stored. As corks tend to accumulate in long lasting downflow struc- tures, new corks are added each ∼ 5 minutes in order to also track structures forming during the observations. In Fig. 6 we report the result of the cork tracing for a contin- uum image series and for both the FeII 722.4 nm and the FeI 709.0 nm Doppler fields. In particular, we plotted the difference between the final and initial distances from the image centre of the corks versus their initial distances from the image centre. The alignment effect of the scatter plot is due to an inverse linear relationship between the ρstart and the δρ of corks with different initial postions which end in the same ‘attractor’ and therefore share their final position. As the image is centred on the SG structure, this will give us information about a possible difference of mean flows inside and outside the SG. In order to investigate the properties of the distribution, we fit on the scatter plots a sigmoidal function: A1 −A2 1 + e(x−x0)/dx +A2 (1) so that x0 will tell where the transition between the two values A1 and A2 of the distribution takes place and dx will tell how fast this transition is. The parameters of the fit are retrieved by a recursive Levenberg-Marquardt min- imization algorithm. The fits agree in retrieving positive values of A1 and near zero values for A2 (Table 2). This means that the corks inside a circle of radius x0 from the image centre tend to increase their radial distance, while the corks outside have no preferred direction in their mo- tion. Several simulations on randomly generated velocity fields showed that we can neglect the contribute from corks whose initial position is so near to the image centre that they are biased towards positive radial displacement. Finally, we tested the robustness of these results against the initial guesses and against the SG centre position in the FoV. The retrieved parameters do not depend on these factors, as long as the initial guesses are of the same order of magnitude of the convergence values or the FoV shift is less than 2.5 Mm. In Fig. 4 we show the mean intensity fields associated to the plots in Fig. 6, with superimposed the location of the change of the A value represented as an annulus of mean Fig. 6. Cork displacements versus initial positions. Top Panel: results of the cork tracking for the intensity field from continuum images. Central Panel: results of the cork tracking for the vertical velocity field extracted from FeII 722.4 nm. Bottom Panel: results of the cork tracking for the vertical velocity field extracted from FeI 709.0 nm. Each scatter plot has been fitted with a sigmoidal function (equa- tion 1). The retrieved fits are overplotted on the relative scatter plots. radius x0 and thickness 2dx. The three annular shapes essentially agree in retrieving the same SG diameter of ∼ 25 Mm. The width of the annuli, instead, seems to depend on the atmospheric altitude. In the upper panel of Fig. 8 we report the value of dx computed by the sigmoidal fits versus the photospheric height. Error bars represent the standard deviation from the fit. Recently, Berrilli et al. (2002) found a similar height dependence of the statistical properties of granular flows. In particular, they reported an intense braking in the first ∼ 120 km of the photosphere, confirmed by Puschmann et al. (2005) and a damping effect that filtered 6 Del Moro et al.: 3D photospheric velocity field of a SG cell A1 A2 x0 dx [Mm] [Mm] [Mm] [Mm] Continuum Intensity 0.49 ± 0.02 0.08 ± 0.01 11.3± 0.2 0.3± 0.2 FeII 722.4 LoS Velocity 0.61 ± 0.03 −0.09± 0.02 12.0± 0.2 1.1± 0.2 FeI 709.0 LoS Velocity 0.67 ± 0.08 −0.09± 0.03 10.9± 0.7 2.9± 0.6 Table 2. Parameters of the sigmoidal fits to the scatter plots reported in Fig. 6. out small features in higher atmospheric layers, letting only large flow features penetrate into the upper photosphere. The same process can explain the broadening of the SG border we found: in higher layers the corks are collected in larger and fewer downflow structures. As more corks are collected by the same structures, the number of indepen- dent tracers is decreased and similarly the precision of the retrieval of the boundary is decreased. Instead, we can exclude that such a smoothing effect is due to data reduction or seeing, because in that case the SG border retrieved from the FeI 709.0 nm LoS field would have been thinner than the one retrieved from the FeII 722.4 nm LoS field, as the latter shows lower contrast features, more prone to be degraded by the loss of spatial resolution. Due to the form of equation 1, the difference A1 − A2, divided for the time allotted to the corks to move, will give the mean radial velocity experienced by the corks. We plot in the bottom panel of Fig. 8 the radial velocity retrieved from the three scatter plots as a function of the photospheric altitude. Error bars represent the standard deviation from the fit. To account for these results, we assume that the large and more coherent features present in the LoS dopplergrams are reliable to retrieve the radial velocity measure, while the measure from the continuum images is somewhat reduced by the presence of tiny structures which are more turbulent in their motion. Such structures are not present in the higher layers dopplergrams because of the damping effect already discussed. We therefore discard the value obtained from the WL dataset because it is probably smeared by the turbulent motions of very small scale features and take into account only the two values retrieved from the higher layers, retrieving a mean velocity of 0.75± 0.05 km s−1. Such a value for the flows from the SG structure centre is consistent with the literature (Simon & Leighton, 1964; Hathaway et al., 2002; Paniveni et al., 2004; Meunier et al., 2007). 5. Conclusions The study of the full 3D velocity field of a SG shows that strong downflows are located on the border of the super- granular structure, but also that the mean granular flow regresses from the centre to the periphery of the SG. The divergence images show that the SG structure is out- lined by convergence sites on ∼ 66% of its border. The retrieved divergence values show a nearly radial flow of ∼ 0.1 km s−1 Mm−1 from the centre of the SG and con- vergent flows of the same magnitude in its border. The analysis of the evolution of passive tracers on inten- sity and velocity fields shows that inside the SG structure there is a preferential radial flow towards the SG border of 0.75± 0.05 km s−1. The height behaviour of the thickness of the SG border, again retrieved via cork tracing, shows an increase of the border width with height. This is probably due to a filter- ing effect with height, which preferentially allows large flow features to penetrate into the upper photospheric layers. The large and CaII bright cluster of structures in the lower part of the FoV, is not a site of strong convergence, but is a site of long-lasting downflows. We also found a strong and stable upflow nearby the centre of the cell, liable to organize the CaII bright structures by sweeping them out of the SG cell. The result presented in this paper are extracted from a sub- set of a longer timeseries of excellent spectral and temporal resolution, but varying spatial quality due to seeing. The used 30 min subset is characterized by a constant and good spatial resolution. This allowed us to detect precisely the flow associated with the SG. Anyhow, our analysis would have greatly benefited from a longer time sequence and other SG structures to analyze. In the future, we plan to apply this analysis to a collection of SG structures, so as to derive some statistical describer and possibly generalize the results. Acknowledgements. We thank the referee, T. Roudier, for suggestions and comments that have signicantly improved this paper. Part of this work was supported by Rome “Tor Vergata” University Physics Department grants. The data were acquired by instruments operated by the National Solar Observatory. The National Solar Observatory is a Division of the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. DDM thanks the High Altitude Observatory for support and C. Sormani for helpful comments. The authors aknowl- edge k. Janssen for the development of the IBIS data reduction pipeline, V. Penza for the calculation of the line RFs and M. Rast for very useful discussions and comments. 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Bottom Panel: mean CaII 854.2 nm wing image with SG extracted from the intensity continuum images (∼ 0 km). 8 Del Moro et al.: 3D photospheric velocity field of a SG cell -20 0 20 40 60 80 100 120 140 160 180 200 Photospheric Altitude (km) -20 0 20 40 60 80 100 120 140 160 180 200 Photospheric Altitude [km] Fig. 8. Top panel: dx (annulus width) versus photospheric height. Bottom panel: radial velocity versus photospheric height. Introduction Observations 3D Velocity Field Horizontal Flow Analysis via Cork Tracking Conclusions

0704.0579

Substructures in WINGS clusters

Astronomy & Astrophysics manuscript no. babbage c© ESO 2021 November 8, 2021 Substructures in WINGS clusters ⋆ M. Ramella1, A. Biviano1, A. Pisani2, J. Varela3, D. Bettoni3, W.J. Couch4, M. D’Onofrio5, A. Dressler6, G. Fasano3, P. Kjærgaard7, M. Moles8, E. Pignatelli3, and B.M. Poggianti3 1 INAF/Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34131, Trieste, Italy 2 Istituto di Istruzione Statale Classico Dante Alighieri, Scientifico Duca degli Abruzzi, Magistrale S. Slataper, viale XX settembre 11, I-34170 Gorizia, Italy 3 INAF/Osservatorio Astronomico di Padova, vicolo Osservatorio 5, I-35122, Padova, Italy 4 School of Physics, University of New South Wales, Sydney 2052, Australia 5 Dipartimento di Astronomia, Università di Padova, vicolo Osservatorio 2, I-35122 Padova, Italy 6 Observatories of the Carnegie Institution of Washington, Pasadena, CA 91101, USA 7 Copenhagen University Observatory. The Niels Bohr Institute for Astronomy Physics and Geophysics, Juliane Maries Vej 30, 2100 Copenhagen, Denmark 8 Instituto de Astrofı́sica de Andalucı́a (C.S.I.C.) Apartado 3004, 18080 Granada, Spain Received / Accepted ABSTRACT Aims. We search for and characterize substructures in the projected distribution of galaxies observed in the wide field CCD images of the 77 nearby clusters of the WIde-field Nearby Galaxy-cluster Survey (WINGS). This sample is complete in X-ray flux in the redshift range 0.04 < z < 0.07. Methods. We search for substructures in WINGS clusters with DEDICA, an adaptive-kernel procedure. We test the procedure on Monte-Carlo simulations of the observed frames and determine the reliability for the detected structures. Results. DEDICA identifies at least one reliable structure in the field of 55 clusters. 40 of these clusters have a total of 69 substructures at the same redshift of the cluster (redshift estimates of substructures are from color-magnitude diagrams). The fraction of clusters with subclusters (73%) is higher than in most studies. The presence of subclusters affects the relative luminosities of the brightest cluster galaxies (BCGs). Down to L ∼ 1011.2 L⊙, our observed differential distribution of subcluster luminosities is consistent with the theoretical prediction of the differential mass function of substructures in cosmological simulations. Key words. Galaxies: clusters: general – Galaxies: kinematics and dynamics 1. Introduction According to the current cosmological paradigm, large struc- tures in the Universe form hierarchically. Clusters of galaxies are the largest structures that have grown through mergers of smaller units and have achieved near dynamical equilibrium. In the hierarchical scenario, clusters are a rather young population, and we should be able to observe their formation process even at rather low redshifts. A signature of such process is the presence of cluster substructures. A cluster is said to contain substruc- tures (or subclusters) when its surface density is characterized by multiple, statistically significant peaks on scales larger than the typical galaxy size, with “surface density” being referred to the cluster galaxies, the intra-cluster (IC) gas or the dark matter (DM hereafter; Buote 2002). Studying cluster substructure therefore allows us to investi- gate the process by which clusters form, constrain the cosmo- logical model of structure formation, and ultimately test the hi- erarchical paradigm itself (e.g. Richstone et al. 1992; Mohr et al. 1995; Thomas et al. 1998). In addition, it also allows us to better understand the mechanisms affecting galaxy evolution in clus- ters, which can be accelerated by the perturbative effects of a cluster-subcluster collision and of the tidal field experienced by Send offprint requests to: Massimo Ramella, [email protected] ⋆ Figure 6 is only available in electronic form via http://www.edpsciences.org a group accreting onto a cluster (Bekki 1999; Dubinski 1999; Gnedin 1999). If clusters are to be used as cosmological tools, it is important to calibrate the effects substructures have on the estimate of their internal properties (e.g. Schindler & Müller 1993; Pinkney et al. 1996; Roettiger et al. 1998; Biviano et al. 2006; Lopes et al. 2006). Finally, detailed analyses of clus- ter substructures can be used to constrain the nature of DM (Markevitch et al. 2004; Clowe et al. 2006). The analysis of cluster substructures can be performed us- ing the projected phase-space distribution of cluster galaxies (e.g. Geller & Beers 1982), the surface-brightness distribution and temperature of the X-ray emitting IC gas (e.g. Briel et al. 1992), or the shear pattern in the background galaxy distribu- tion induced by gravitational lensing, that directly samples sub- structure in the DM component (e.g. Abdelsalam et al. 1998). None of these tracers of cluster substructure (cluster galaxies, IC gas, background galaxies) can be considered optimal. The iden- tification of substructures is in fact subject to different biases depending on the tracer used. In X-rays projection effects are less important than in the optical, but the identification of sub- structures is more subject to a z-dependent bias, arising from the point spread function of the X-ray telescope and detector (e.g. Böhringer & Schuecker 2002). Moreover, the different cluster components respond in a different way to a cluster-subcluster collision. The subcluster IC gas can be ram-pressure braked and stripped from the colliding subcluster and lags behind the sub- cluster galaxies and DM along the direction of collision (e.g. http://arxiv.org/abs/0704.0579v1 http://www.edpsciences.org 2 M. Ramella et al.: Substructures in the WINGS clusters Roettiger et al. 1997; Barrena et al. 2002; Clowe et al. 2006). Hence, it is equally useful to address cluster substructure analy- sis in the X-ray and in the optical. Traditionally, the first detections of cluster substructures were obtained from the projected spatial distributions of galax- ies (e.g. Shane & Wirtanen 1954; Abell et al. 1964), in com- bination, when possible, with the distribution of galaxy ve- locities (e.g. van den Bergh 1960, 1961; de Vaucouleurs 1961). Increasingly sophisticated techniques for the detection and characterization of cluster substructures have been developed over the years (see Moles et al. 1986; Perea et al. 1986a,b; Buote 2002; Girardi & Biviano 2002, and references therein). In many of these techniques substructures are identified as de- viations from symmetry in the spatial and/or velocity distri- bution of galaxies and in the X-ray surface-brightness (e.g. West et al. 1988; Fitchett & Merritt 1988; Mohr et al. 1993; Schuecker et al. 2001). In other techniques substructures are identified as significant peaks in the surface density distribu- tion of galaxies or in the X-ray surface brightness, either as residuals left after the subtraction of a smooth, regular model representation of the cluster (e.g. Neumann & Böhringer 1997; Ettori et al. 1998), or in a non-parametric way, e.g. by the tech- nique of wavelets (e.g. Escalera et al. 1994; Slezak et al. 1994; Biviano et al. 1996) and by adaptive-kernel techniques (e.g. Kriessler & Beers 1997; Bardelli et al. 1998a, 2001). The performances of several different methods have been evaluated both using numerical simulations (e.g. Mohr et al. 1995; Crone et al. 1996; Pinkney et al. 1996; Buote & Xu 1997; Cen 1997; Valdarnini et al. 1999; Knebe & Müller 2000; Biviano et al. 2006) and also by applying different methods to the same cluster data-sets and examine the result differ- ences (e.g. Escalera et al. 1992, 1994; Mohr et al. 1995, 1996; Kriessler & Beers 1997; Fadda et al. 1998; Kolokotronis et al. 2001; Schuecker et al. 2001; Lopes et al. 2006). Generally speaking, the sensitivity of substructure detection increases with both increasing statistics (e.g. more galaxies or more X-ray pho- tons) and increasing dimensionality of the test (e.g. using galaxy velocities in addition to their positions, or using X-ray tempera- ture in addition to X-ray surface brightness). Previous investigations have found very different fractions of clusters with substructure in nearby clusters, depending on the method and tracer used for substructure detection, on the cluster sample, and on the size of sampled cluster re- gions (e.g. Geller & Beers 1982; Dressler & Shectman 1988; Mohr et al. 1995; Girardi et al. 1997; Kriessler & Beers 1997; Jones & Forman 1999; Solanes et al. 1999; Kolokotronis et al. 2001; Schuecker et al. 2001; Flin & Krywult 2006; Lopes et al. 2006). Although the distribution of subcluster masses has not been determined observationally, it is known that subclusters of ∼ 10% the cluster mass are typical, while more massive subclusters are less frequent (Escalera et al. 1994; Girardi et al. 1997; Jones & Forman 1999). The situation is probably dif- ferent for distant clusters which tend to show massive sub- structures more often than nearby clusters clearly suggesting hierarchical growth of clusters was more intense in the past (e.g. Gioia et al. 1999; van Dokkum et al. 2000; Haines et al. 2001; Maughan et al. 2003; Huo et al. 2004; Rosati et al. 2004; Demarco et al. 2005; Jeltema et al. 2005). Additional evidence for the hierarchical formation of clus- ters is provided by the analysis of brightest cluster galaxies (BCGs hereafter) in substructured clusters. BCGs usually sit at the bottom of the potential well of their host cluster (e.g. Adami et al. 1998b). When a BCG is found to be significantly displaced from its cluster dynamical center, the cluster displays evidence of substructure (e.g. Beers et al. 1991; Ferrari et al. 2006). From the correlation between cluster and BCG luminosi- ties, Lin & Mohr (2004) conclude that BCGs grow by merg- ing as their host clusters grow hierarchically. The related evo- lution of BCGs and their host clusters is also suggested by the alignement of the main cluster and BCG axes (e.g. Binggeli 1982; Durret et al. 1998). Both the BCG and the cluster axes are aligned with the surrounding large scale structure dis- tribution, where infalling groups come from. These infalling groups are finally identified as substructures once they enter the cluster environment (Durret et al. 1998; Arnaud et al. 2000; West & Blakeslee 2000; Ferrari et al. 2003; Plionis et al. 2003; Adami et al. 2005). Hence, substructure studies really provide direct evidence for the hierarchical formation of clusters. Concerning the impact of subclustering on global cluster properties, it has been found that subclustering leads to over- estimating cluster velocity dispersions and virial masses (e.g. Perea et al. 1990; Bird 1995; Maurogordato et al. 2000), but not in the general case of small substructures (Escalera et al. 1994; Girardi et al. 1997; Xu et al. 2000). During the collision of a subcluster with the main cluster, both the X-ray emitting gas distribution and its temperature have been found to be signifi- cantly affected (e.g. Markevitch & Vikhlinin 2001; Clowe et al. 2006). As a consequence, it has been argued that substruc- ture can explain at least part of the scatter in the scaling rela- tions of optical-to-X-ray cluster properties (e.g. Fitchett 1988; Girardi et al. 1996; Barrena et al. 2002; Lopes et al. 2006). As far as the internal properties of cluster galaxies are concerned, there is observational evidence that a higher frac- tion of cluster galaxies with spectral features characteristic of recent or ongoing starburst episodes is located in sub- structures or in the regions of cluster-subcluster interactions (Caldwell et al. 1993; Abraham et al. 1996; Biviano et al. 1997; Caldwell & Rose 1997; Bardelli et al. 1998b; Moss & Whittle 2000; Miller et al. 2004; Poggianti et al. 2004; Miller 2005; Giacintucci et al. 2006). In this paper we search for and characterize substructures in the sample of 77 nearby clusters of the WIde-field Nearby Galaxy-cluster Survey (WINGS hereafter, Fasano et al. 2006). This sample is an almost complete sample in X-ray flux in the redshift range 0.04 < z < 0.07. We detect substructures from the spatial, projected distribution of galaxies in the cluster fields, us- ing the adaptive-kernel based DEDICA algorithm (Pisani 1993, 1996). In Sect. 2 we describe our data-set; in Sect. 3 we de- scribe the procedure of substructure identification; in Sect. 4 we use Monte Carlo simulations in order to tweak our procedure; in Sect. 5 we describe the identification of substructures in our data-set; in Sect. 6 the catalog of identified substructures is pro- vided. In Sect. 7 we investigate the properties of the identified substructures, and in Sect. 8 we consider the relation between the BCGs and the substructures. We provide a summary of our work in Sect. 9. 2. The Data WINGS is an all-sky, photometric (multi-band) and spectro- scopic survey, whose global goal is the systematic study of the local cosmic variance of the cluster population and of the prop- erties of cluster galaxies as a function of cluster properties and local environment. The WINGS sample consists of 77 clusters selected from three X-ray flux limited samples compiled from ROSAT All- Sky Survey data, with constraints just on the redshift (0.04 < z < 0.07) and distance from the galactic plane (|b| ≥20 deg). The core M. Ramella et al.: Substructures in the WINGS clusters 3 of the project consists of wide-field optical imaging of the se- lected clusters in the B and V bands. The imaging data were col- lected using the WFC@INT (La Palma) and the WFI@MPG (La Silla) in the northern and southern hemispheres, respectively. The observation strategy of the survey favors the uniformity of photometric depth inside the different CCDs, rather than com- plete coverage of the fields that would require dithering. Thus, the gaps in the WINGS optical imaging correspond to the phys- ical gaps between the different CCDs of the mosaics. During the data reduction process, we give particular care to sky subtraction (also in presence of crowded fields including big halo galaxies and/or very bright stars), image cleaning (spikes and bad pixels) and star/galaxy classification (obtained with both automatic and interactive tools). According to Fasano et al. (2006) and Varela et al. (2007), the overall quality of the data reported in the WINGS photomet- ric catalogs can be summarized as follows: (i) the astrometric errors for extended objects have r.m.s. ∼0.2 arcsec; (ii) the av- erage limiting magnitude is ∼24.0, ranging from 23.0 to 25.0; (iii) the completeness of the catalogs is achieved (on average) up to V ∼22.0; (iv) the total (systematic plus random) photometric r.m.s. errors, derived from both internal and external compar- isons, vary from ∼0.02 mag, for bright objects, up to ∼0.2 mag, for objects close to the detection limit. 3. The DEDICA Procedure We base our search for substructures in WINGS clusters on the DEDICA procedure (Pisani 1993, 1996). This procedure has the following advantages: 1. DEDICA gives a total description of the clustering pattern, in particular the membership probability and significance of structures besides geometrical properties; 2. DEDICA is scale invariant; 3. DEDICA does not assume any property of the clusters, i.e. it is completely non-parametric. In particular it does not re- quire particularly rich samples to run effectively. The basic nature and properties of DEDICA are described in Pisani (1993, 1996, and references therein). Here we summarize the main structure of the algorithm and how we apply it to our data sample. The core structure of DEDICA is based on the as- sumption that a structure (or a “cluster” in the algorithm jargon) corresponds to a local maximum in the density of galaxies. We proceed as follows. First we need to estimate the proba- bility density function Ψ(ri) (with i = 1, . . .N) associated with the set of N galaxies with coordinates ri. Second, we need to find the local maxima in our estimate of Ψ(ri) in order to iden- tify clusters and also to evaluate their significance relatively to the noise. Third and finally, we need to estimate the probability that a galaxy is a member of the identified clusters. 3.1. The probability density DEDICA is a non-parametric method in the sense that it does not require any assumption on the probability density function that it is aimed to estimate. The only assumptions are that Ψ(ri) must be continuous and at least twice differentiable. The function f (ri) is an estimate of Ψ(ri) and it is built by using an adaptive kernel method given by: fka(r) = K(ri, σi; r) (1) where we use the two dimensional Gaussian kernel K(ri, σi; r) centered in ri with size σi. The most valuable feature of DEDICA is the procedure to se- lect the values of kernel widths σi. It is possible to show that the optimal choice for σi, i.e. with asymptotically minimum vari- ance and null bias, is obtained by minimizing the distance be- tween our estimate f (ri) and Ψ(ri). This distance can be eval- uated by a particular function called the integrated square error IS E( f ) given by: IS E( f ) = [Ψ(r) − f (r)]2dr (2) Once the minimum IS E( f ) is reached we have obtained the DEDICA estimate of the density as in Eq.1. 3.2. Cluster Identification The second step of DEDICA consists in the identification of the local maxima in fka(r). The positions of the peaks in the density function fka(r) are found as the solutions of the iterative equa- tion: rm+1 = rm + a · ∇ fka(rm) fka(rm) where a is a scale factor set according to optimal convergence requirements. The limit R of the sequence rm defined in Eq.3 depends on the starting position rm=1. rm = R(rm=1) (4) We run the sequence in Eq.3 at each data position ri. We label each data point with the limit Ri = R(rm=1 = ri). These limits Ri are the position of the peak to which the i− th galaxy belong. In the case that all the galaxies are members of a unique cluster, all the labels Ri are the same. At the other extreme each galaxy is a one-member cluster and all Ri have different values. All the members of a given cluster belong to the same peak in fka(r) and have the same Ri. We identify cluster members by listing galaxies having the same values of R. We end up with ν different clusters each with nµ (µ = 1, . . . , ν) members. In order to maintain a coherent notation, we identify with the label µ = 0 the n0 isolated galaxies considered a system of background galaxies. We have: n0 = N − µ=1 nµ. 3.3. Cluster Significance and of Membership Probability The statistical significance S µ (µ = 1, . . . , ν) of each cluster is based on the assumption that the presence of the µ − th cluster causes an increase in the local probability density as well as in the sample likelihood LN = Πi[ fka(ri)] relatively to the value Lµ that one would have if the members of the µ− th cluster were all isolated, i.e. belonging to the background. A large value in the ratio LN/Lµ characterizes the most im- portant clusters. According to Materne (1979) it is possible to estimate the significance of each cluster by using the likelihood ratio test. In other words 2 ln(LN/Lµ) is distributed as a χ 2 vari- able with ν− 1 degrees of freedom. Therefore, once we compute the value of χ2 for each cluster (χ2S ), we can also compute the significance S µ of the cluster. Here we assume that the contribution to the global density field fka(ri) of the µ − th cluster is Fµ(ri). The ratio between the value of Fµ(ri) and the total local density fka(ri) can be used to estimate the membership probability of each galaxy relatively to 4 M. Ramella et al.: Substructures in the WINGS clusters the identified clusters. This criterion also allows us to estimate the probability that a galaxy is isolated. At the end of the DEDICA procedure we are left with a) a catalog of galaxies each with information on position, mem- bership, local density and size of the Gaussian kernel, b) a cat- alog of structures with information on position, richness, the χ2S parameter, and peak density. For each cluster we also com- pute from the coordinate variance matrix the cluster major axis, ellipticity and position angle. 4. Tweaking the Algorithm with Simulations In this section we describe our analysis of the performance of DEDICA and the guidelines we obtain for the interpretation of the clustering analysis of our observations. We build simulated fields containing a cluster with and with- out subclusters. The simulated fields have the same geometry of the WFC field and are populated with the typical number of objects we will analyze. For simplicity we consider only WFC fields. Because DEDICA is scale-free, a different sampling of the same field of view has no consequence on our analysis. In the next section we limit our analysis to MV,lim ≤ −16. At the median redshift of the WINGS cluster, z ≃ 0.05, this absolute magnitude limit corresponds to an apparent magnitude Vlim ≃ 21. Within this magnitude limit the representative number of galaxies in our frames is Ntot= 900. We then consider Ntot= Nmem+ Nbkg, with Nmemthe number of cluster members and Nbkgthe number of field – or background – galaxies. We set Nbkg= 670, close to the average number of background galaxies we expect in our frame based on typical observed fields counts, e.g. Berta et al. (2006) or Arnouts et al. (1997). With this choice, we have Nmem= 230. We distribute uniformly at random Nbkgobjects. We dis- tribute at random the remaining Nmem= 230 objects in one or more overdensities depending on the test we perform. We popu- late overdensities according to a King profile (King 1962) with a core radius Rcore = 90 kpc, representative of our clusters. We then scale Rcore with the number of members of the substructure, NS . We use Rcore = 250 NC + NS where NC is the number of objects in the cluster with Nmem= NS + NC . This scaling of Rcore with cluster richness is from Adami et al. (1998a) assuming direct proportionality between cluster richness and luminosity (e.g. Popesso et al. 2006). As far as the relative richnesses of the cluster and subcluster are concerned, we consider the following richness ratios rcs = NC/NS = 1, 2, 4, 8. With these richness ratios, the number of objects in the cluster are NC = 115, 153, 184, 204, and those in subclusters are NS = 115, 77, 46, 26 respectively. In a first set of simulated fields we place the substructure at 2731 pixels (15 arcmin) from the main cluster so that they do not overlap. In a second set of simulations, we place main cluster and substructure at shorter distances, 683 and 1366 pixels, in order to investigate the ability of DEDICA to resolve structures. At each of these shorter distances we build simulations with both rcs= 1 and 2. For each richness ratio and/or distance between cluster and subcluster we produce 16 simulations with different realizations of the random positions of the data points representing galaxies. In order to minimize the effect of the borders on the detection of structures we add to the simulation a “frame” of 1000 pixel. 5 10 15 simulation Fig. 1. Fraction of recovered members of each substructure for different rcs. The solid line connects substructures with rcs = 2 and 4 We fill this frame with a grid of data points at the same density as the average density of the field. The first result we obtain from the runs of DEDICA on the simulations with varying richness ratio is the positive rate at which we detect real structures. We find that we always recover both cluster and substructure even when the substructure only contains NS = 1/8 NC objects, i.e. 26 objects (on top of the uni- form background). In other words, if there is a real structure DEDICA finds it. We also check how many original members the procedure assigns to structures it recovers. The results are summarized in Fig. 1. In the diagram, the fraction of recovered members of each substructure is represented by the values of its rcs. The solid line connects substructures with rcs = 2 and 4. From Fig. 1 it is clear that our procedure recovers a large fraction of members, almost irrespective of the richness of the original structure. It is also interesting to note that the fluctu- ations identified as substructures are located very close to the center of the corresponding simulated substructures. In almost all cases the distance between original and detected substructure is significantly shorter than the mean inter-particle distance. The second important result we obtain from the simulations is the false positive rate, i.e. the fraction of noise fluctuations that are as significant as the fluctuations corresponding to real structures. First of all we need to define an operative measure of the reliability of the detected structures. In fact DEDICA provides a default value S µ (µ = 1, . . . , ν) of the significance (see Sect. 3.3). However, S µ has a relatively small dynamical range, in particular for highly significant clusters. Density or richness both allow a reasonable ”ranking” of structures. However, both large low-density noise fluctuations (often built up from more than one noise fluctuation) and very high density fluctuations produced by few very close data points could be mistakenly ranked as highly significant structures ac- cording to, respectively, richness and density criteria. M. Ramella et al.: Substructures in the WINGS clusters 5 0 20 40 60 80 Fig. 2. χ2S of simulated noise fluctuations (solid line). Labels are the rcsof simulated structures at the abscissa corresponding to their χ2S and at arbitrary ordinates. We therefore prefer to use the parameter χ2S which stands at the base of the estimate of S µ and which is naturally provided by DEDICA. The main characteristic of χ2S is that it depends both on the density of a cluster relative to the background and on its rich- ness. Using χ2S we classify correctly significantly more structures than with either density or richness alone. In Fig. 2 we plot the distribution of χ2S of noise fluctuations (solid line). In the same plot we also mark the rcsof real struc- tures as detected by our procedure. We use labels indicating rcsand place them at the abscissa corresponding to their χ S and at arbitrary ordinates. Fig. 2 shows that the structures detected with rcs= 1, 2 are al- ways distinguishable from noise fluctuations. Substructures with rcs= 4 or higher, although correctly detected, have χ S values that are close to or lower than the level of noise. With the second set of simulations, we test the minimum dis- tance at which cluster and subcluster can still be identified as separate entities. We place cluster and substructure (rcs= 1, 2) at distances dcs = 683 and 1366 pixel. These distances are 1/4 and 1/2 respectively of the distance between cluster and substructure in the first set of simulations. Again we produce 16 simulations for each of the 4 cases. We find that at dcs = 1366 pixel cluster and substructure are always correctly identified. At the shorter distance dcs = 683 pixel, DEDICA merges cluster and substructure in 1 out of 16 cases for rcs= 1 and in 8 out of 16 cases for rcs= 2. With our density profile, dcs = 683 pixel corresponds to dcs ≃ Rc + Rs with Rc, Rs the radii of the main cluster and of the subcluster respectively. In order to verify the results we obtain for 900 data points we produce more simulations with Ntot= 450, 600 and 1200. In all these simulations RC and RS are the same as in the set with Ntot= 900. We vary Nbkgand Nmemso that Nmem/ Nbkgis the same as in the case Ntot= 900. 0.5 1 1.5 2 2.5 Fig. 3. Small symbols correspond to χ2S as a function of the num- ber of members of noise fluctuations. Crosses, circles, dots and triangles are χ2S for the noise fluctuations of the simulations with Ntot= 450, 600, 900, and 1200 respectively. Large symbols are χ2S of simulated clusters and subclusters with rcs= 1. Horizontal lines mark the levels of χ2S ,threshold. These simulations confirm the results we obtain in the case Ntot= 900, and allow us to set a detection threshold, χ2S ,threshold(Ntot), for significant fluctuations in the analysis of real clusters. We summarize the behavior of the noise fluctuations in our simulations in Fig. 3. In this figure, the small symbols corre- spond to χ2S as a function of the number of members of noise fluctuations. In particular, crosses, circles, dots and triangles are χ2S for the noise fluctuations of the simulations with Ntot= 450, 600, 900, and 1200 respectively. The larger symbols are the χ2S of the fluctuations correspond- ing to simulated clusters and subclusters of equal richness (rcs= The 4 horizontal lines mark the level of χ2S ,threshold, i.e. the average χ2S of the 3 most significant noise fluctuations in each of the 4 groups of simulations with Ntot= 450, 600, 900, and 1200. The expected increase of χ2S ,thresholdwith Ntotis evident. We note that the only significant difference with these find- ings we obtain from the simulations with rcs= 2 is that χ simulated clusters and subclusters is closer to χ2S ,threshold(but still higher). We fit χ2S ,thresholdwith Ntotand obtain log(χ2S ,threshold) = 1/2.55 log(Ntot) + 0.394 (5) in good agreement with the expected behavior of the poissonian fluctuations. As a final test we verify that infra-chip gaps do not have a dramatic impact on the detection of structures in the cases rcs= 1 and 2. We place a 50 pixel wide gap where it has the maximum impact, i.e. where the kernel size is shortest. Even if the infra- 6 M. Ramella et al.: Substructures in the WINGS clusters chip gap cuts through the center of the structures, DEDICA is able to identify these structures correctly. We summarize here the main results of our tests on simulated clusters with substructures: – DEDICA successfully detects even the poorest structures above a uniform poissonian noise background. – DEDICA recovers a large fraction (typically > 3/4) of the real members of a substructure, almost irrespective of the richness of the structure. – DEDICA is able to distinguish between noise fluctuations and true structures only if these structures are rich enough. In the case of our simulations, structures have to be richer than 1/4th of the main structure. – DEDICA is able to separate neighboring structures provided they do not overlap. – infra-chip gaps do not threaten the detection of structures that are rich enough to be reliably detected. – the χ2S threshold we use to identify significant structures is a function of the total number of points and can be scaled within the whole range of numbers of galaxies observed within our fields. In the next section we apply these results to the real WINGS clusters. 5. Substructure detection in WINGS clusters We apply our clustering procedure to the 77 clusters of the WINGS sample. The photometric catalog of each cluster is deep, reaching a completeness magnitude Vcomplete ≤ 22. The num- ber of galaxies is correspondingly large, from Ngal ≃ 3, 000 to Ngal ≃ 10, 000. The large number of bright background galaxies (faint ap- parent magnitudes) dilutes the clustering signal of local WINGS clusters. We perform test runs of the procedure on several clus- ters with magnitude cuts brighter than Vcomplete. Based on these tests, we decide to cut galaxy catalogs to the absolute magnitude threshold MV = −16.0. With this choice a) we maximize the signal-to-noise ratio of the detected subclusters and b) we still have enough galaxies for a stable identification of the system. At the median redshift of WINGS clusters, z ≃ 0.0535, our absolute magnitude cut corresponds to an apparent magnitude V ≃ 21.2. This apparent magnitude also approximately corresponds to the magnitude where the contrast of our typical cluster relative to the field is maximum (this estimate is based on the average cluster luminosity function of (Yagi et al. 2002; De Propris et al. 2003) and on the galaxy counts of (Berta et al. 2006)). The number of galaxies that are brighter than the threshold MV = −16.0 is in the range 600 < Ntot < 1200 for a large fraction of clusters observed with either WFC@INT or with [email protected]. In order to proceed with the identification of significant structures within WINGS clusters, we need to verify that our simulations are sufficiently representative of the real cases. In practice we need to compare the observed distributions of χ2S values of noise fluctuations with the corresponding simulated distributions. In the observations it is impossible to identify indi- vidual fluctuations as noise. In order to have an idea of the distri- butions of χ2S of noise fluctuations we consider that our fields are centered on real clusters. As a consequence, on average, fluctua- tions in the center of the frames are more likely to correspond to real systems than those at the borders. 0 20 40 60 Fig. 4. χ2S distributions for border (thick solid histogram) and central (thick dashed histogram) observed fluctuations. The thin solid line is the normalized distribution of χ2S of the noise fluctu- ations in our simulations We therefore consider separately the fluctuations within the central regions of the frames and all other fluctuations (borders). We define the central regions as the central 10% of WFC and WFI areas. We plot in Fig. 4 the two distributions. The thick solid histogram is for the border and the thick dashed histogram for the center of the frames. The difference between ”noise” and ”signal” is clear. In the same figure we also plot the normal- ized distribution of χ2S of the noise fluctuations in our simulations (thin solid line). The distributions of χ2S of the observed and sim- ulated fluctuations are in reasonable agreement considering a) the simple model used for the simulations and that b) in the ob- servations we can not exclude real low-χ2S structures among noise fluctuations. We conclude that for our clusters we can adopt the same reliability threshold χ2S ,thresholdwe determine from our sim- ulations (Eq. 5). 6. The Catalog of Substructures We detect at least one significant structure in 55 (71%) clus- ters. We find that 12 clusters (16%) have no structure above the threshold (undetected). In the case of another 10 (13%) clusters we find significant structures only at the border of the field of view. In absence of a detection in the center of the frame, we con- sider these border structures unrelated to the target cluster. We also verify that in the Color-Magnitude Diagram (CMD) these border structures are redder than expected given the redshift of the target cluster. We consider also these 10 clusters undetected. Here we list the 22 undetected clusters: A0133, A0548b, A0780, A1644, A1668, A1983, A2271, A2382, A2589, A2626, A2717, A3164, A3395, A3490, A3497, A3528a, A3556, A3560, A3809, A4059, RX1022, Z1261. We note that undetected clusters are real physical systems ac- cording to their x-ray selection. From an operative point of view, the fact that these clusters are not detected by DEDICA is the M. Ramella et al.: Substructures in the WINGS clusters 7 Fig. 5. Isodensity contours (logarithmically spaced) of the Abell 85 field. The title lists the coordinates of the center. The orien- tation is East to the left, North to the top. Galaxies belonging to the systems detected by DEDICA are shown as dots of different colors. Black, light green, blue, red, magenta, dark green are for the main system and the subsequent substructures ordered as in Table A.1. Large symbols are for galaxies with MV ≤ −17.0 that lie where local densities are higher than the median local density of the structure the galaxy belongs to. Open symbols mark the positions of the first- and second-ranked cluster galaxies, BCG1 and BCG2 respectively. Similar plots for the 55 analysed clusters are available in the electronic version of this Journal. result of the division into too many structures of the total avail- able clustering signal in the field (or of a too large fraction of the clustering signal going into border structures). Several phys- ical situations could be at the origin of missed detections. One possibility is an excess of physical substructures of comparable richness. Another possibility is that these clusters are embedded in regions of the large scale structure that are highly clustered. We do not try to recover these structures because they can not be prominent enough. Since our analysis is bidimensional, we can only detect and use confidently the most prominent struc- tures. Redshifts are needed for a more detailed analysis of cluster substructures. We list the 55 clusters with significant structures in Table A.1. We give, for each substructure: (1) the name of the parent cluster; (2) the classification of the structure as main (M), sub- cluster (S), or background (B) together with their order number; (3) right ascension (J2000), and (4) declination (J2000) in deci- mal degrees of the DEDICA peak; the parameters of the ellipse we obtain from the variance matrix of the coordinates of galaxies in the substructure, i.e. (5) major axis in arcminutes, (6) elliptic- ity, and (7) position angle in degrees; (8) luminosity (see the next section); (9) χ2S . We make available contour plots of the number density fields of all clusters in Fig. 6 of the electronic version of this Journal. In Fig. 5 we show an example of these plots. Isodensity contours are drawn at ten logarithmic intervals. Galaxies belonging to the systems detected by DEDICA are shown as dots of different col- ors. We use large symbols for brighter galaxies (MV ≤ −17.0) that lie where local densities are higher than the median local density of the structure the galaxy belongs to. We also mark with open symbols the positions of the first- and second-ranked cluster galaxies, BCG1 and BCG2 respectively. Color coding is black, light green, blue, red, magenta, dark green for the main system and the subsequent substructures ordered as in Table A.1. We describe and analyze in detail our catalog in the next sec- tion. 7. Properties of substructures The first problem we face in order to study the statistical and physical properties of substructures is to determine their asso- ciation with the main structure. In fact, the main structure itself has to be identified among the structures detected by DEDICA in each frame. In most cases it is easy to identify the main structure of a cluster since it is located at the center of the frame and it has a high χ2S . In two cases (A0168 and A1736) the choice of the main structure is complicated because there are several similar structures near the center of the frame. In these cases we select the main structure for its highest χ2S . At this point we limit our analysis to members of the struc- ture that a) have an absolute magnitude MV ≤ −17 (corrected for Galactic absorption) and that b) are in the upper half of the distri- bution of DEDICA-defined local galaxy densities of the system they belong to. The galaxy density threshold we apply allows us to separate adjacent structures whose definition becomes more uncertain at lower galaxy density levels. The magnitude cut in- creases the relative weight of the galaxies we use to evaluate the nature of structures in the CMD. After having identified the main structure, we need to deter- mine which structures in the field of view of a given cluster have to be considered background structures. We consider a structure a physical substructure (or subcluster) if its color-magnitude re- lation (CMR hereafter) is identical, within the errors, to the CMR of the main structure. As a first step we define the color-magnitude relation (CMR) of the “whole cluster”, i.e. of galaxies in the main structure to- gether with all other galaxies not assigned to any structure by DEDICA. We compute the (B − V) CMR of the Coma cluster from published data (Adami et al. 2006). Then we keep fixed the slope of the linear CMR of Coma and shift it to the mean redshift of the cluster. In order to determine that the main structure and a substruc- ture are at the same redshift, we evaluate the fraction of back- ground (red) galaxies, fbg, that each structure has in the CMD. If these fractions are identical within the errors (Gehrels 1986), we consider the two structures to be at the same redshift. In practice we determine fbg by assigning to the background those galaxies of a structure that are redder than a line parallel to the CMR and vertically shifted (i.e. redwards) by 2.33 times the root-mean square of the colors of galaxies in the CMR. We note that the probability that a random variable is greater than 2.33 in a Gaussian distribution is only 1%. The result of the selection of main structures and substruc- tures is the following: 40 clusters have a total of 69 substruc- tures at the same redshift as the main structure, only 15 clus- ters are left without substructures. A total of 35 systems are found in the background. Considering a) the number density of poor-to-rich clusters (Mazure et al. 1996; Zabludoff et al. 1993), b) the average luminosity function of clusters (Yagi et al. 2002; 8 M. Ramella et al.: Substructures in the WINGS clusters Fig. 7. Cumulative distributions of the two different indicators of subclustering: left panel Nsub, right panel fLsub. De Propris et al. 2003), c) the total area covered by the 55 clus- ter fields, and d) the limiting apparent magnitude corresponding to our absolute magnitude threshold MV = −16.0, we expect to find ∼ 0.5± 0.2 background systems per cluster field, 28± 11 in total. This estimate is consistent with the 35 background systems we find. The fraction of clusters with subclusters (73%) is higher than generally found in previous investigations (typically∼ 30%, see, e.g., Girardi & Biviano 2002; Flin & Krywult 2006; Lopes et al. 2006, and references therein). Even if we count all undetected clusters as clusters without substructures, this fraction only de- creases to 52% (40/77). It is however acknowledged that the fraction of substructured clusters depends, among other factors, on the algorithm used to detect substructures, on the quality and depth of the galaxy catalog. For example Kolokotronis et al. (2001) using optical and X-ray data find that the fraction of clus- ters with substructures is ≥ 45%, Burgett et al. (2004) using a battery of tests detect substructures in 84% of the 25 clusters of their sample. Having established the “global” fraction of substructured clusters, we now investigate the degree of subclustering of in- dividual clusters, i.e. the distribution of the number of substruc- tures Nsub we find in our sample. We find 15 (27%) clusters without substructures; 22 (40%) clusters with Nsub = 1; 10 (18%) clusters with Nsub = 2; 6 (11%) clusters with Nsub = 3; and 2 (3%) clusters with Nsub = 4. We plot in the left panel of Fig. 7 the integral distribution of Nsub. The distribution of the level of subclustering does not change when we measure it as the fractional luminosity of subclusters, fLsub, relative to the luminosity of the whole cluster (see Fig. 7, right panel). The luminosities we estimate are background cor- rected using the counts of Berta et al. (2006). We use the ellipses output from DEDICA (see previous section) as a measure of the area of subclusters. We find that Nsub and fLsub are clearly correlated according to the Spearman rank-correlation test. We now consider the distribution of subcluster luminosities and plot the corresponding histogram in Fig. 8. In the same fig- ure we also plot with arbitrary scaling the power-law∝ L−1. This relation is the prediction for the differential mass function of substructures in the cosmological simulations of De Lucia et al. (2004). Fig. 8. Observed differential distribution of subcluster lumi- nosities (histogram) and theoretical model (arbitrary scaling; De Lucia et al. 2004). Our observations are consistent to within the uncertainties with the theoretical prediction of De Lucia et al. (2004) down to L ∼ 1011.2 L⊙. The disagreement at lower luminosity is ex- pected since: a) below this limit galaxy-sized halos become im- portant among the simulated substructures, and b) only above this limit we expect our catalog to be complete. In fact only subclusters with luminosities brighter than L = 1011.2 L⊙ have always richnesses that are ≥ 1/3 of the main structure. This richness limit approximately corresponds to the completeness limit of DEDICA detections according to our simulations (see Sect. 4). 8. Brightest Cluster Galaxies Here we investigate the relation between BCGs and cluster struc- tures. We find that, on average, BCG1s are located close to the den- sity peak of the main structures. In projection on the sky, the bi- weight average (see Beers et al. 1990) distance of BCG1s from the peak of the main system is 72 ± 11 kpc. If we only consider the 44 BCG1s that are on the CMR and are assigned to main systems by DEDICA, the average distance decreases to 56 ± 8 kpc. The fact that BCG1s are close to the center of the system is consistent with current theoretical view on the formation of BCGs (e.g. Dubinski 1998; Nipoti et al. 2004). BCG2s are more distant than BCG1s from the peak of the main system: the biweight average distance is 345 ± 47 kpc. If we only consider the 26 BCG2s that are on the CMR and are assigned to main systems by DEDICA, the average distance de- creases to 161 ± 34 kpc. Projected distances of BCG2s from density peaks remain larger than those of BCG1s even when we consider the density peak of the structure or substructure they belong to. In Fig. 9 we plot the cumulative distributions of the distances of BCG1s (solid line) and BCG2s (dashed line) from the density peak of their systems. The distributions are different at the > 99.99% level according to a Kolmogorov-Smirnov test (KS-test). Now we turn to luminosities and find that the magnitude dif- ference between BCG1s and BCG2s, ∆M12, is larger in clus- ters without substructures than in clusters with substructures. In Fig. 10 we plot the cumulative distributions of ∆M12 for clusters with (dashed line) and without (solid line) subclusters. M. Ramella et al.: Substructures in the WINGS clusters 9 Fig. 9. Cumulative distributions of distances of BCG1 (solid line) and BCG2 (dashed line) from the density peak of their sys- Fig. 10. Cumulative distributions of the magnitude difference be- tween BCG1 and BCG2 in clusters with (dashed line) and with- out subclusters (solid line). The two distributions are different according to a KS-test at the 99.1% confidence level. We note that Lin & Mohr (2004) find that ∆M12 is independent of cluster properties. These authors however do not consider subclustering. In order to determine whether the higher values of ∆M12 in clusters without subclusters are due to an increased luminosity of the BCG1 (L1) or to a decreased luminosity of the BCG2 (L2), we consider the luminosity of the 10 th brightest galaxy (L10) as a reference. The biweight average luminosity ratios are < L1/L10 >= 8.6 ± 1.0 and < L2/L10 >= 3.3 ± 0.3 in clus- ters without substructures, and < L1/L10 >= 7.1 ± 0.4 and < L2/L10 >= 3.4 ± 0.2 in clusters with substructures. We then conclude that the ∆M12-effect is caused by a brightening of the BCG1 relative to the BCG2 in clusters without substructures. The fact that ∆M12 is higher in clusters without substruc- tures can be interpreted, at least qualitatively, in the framework of the hierarchical scenario of structure evolution. Clusters with- out substructures are likely to be evolved after several merger phases. Their BCG1s have already had time to accrete many galaxies, in particular the more massive ones, which slow down and sink to the cluster center as the result of dynamical friction. Some of these galaxies may even have been BCGs of the merg- ing structures. The simulations by De Lucia & Blaizot (2006) show that the BCG1s continue to increase their mass via can- nibalism even at recent times, and that there is a large vari- ance in the mass accretion history of BCG1s from cluster to cluster. The result of such a cannibalism process is an increase of the BCG1 luminosity with respect to other cluster galaxies, and in extreme cases may lead to the formation of fossil groups (Khosroshahi et al. 2006). However, according to these simulations, only 15% of all BCG1s have accreted > 30% of their mass over the last 2 Gyr, while another 15% have accreted <3% of their mass over the same period. Our results indicate that about 60% of the BCG1s are more than 1 magnitude brighter than the corresponding BCG2s. Given the size and generality of the luminosity dif- ferences it would seem that cannibalism alone, even if present along the merging history of a given cluster, cannot account for it. Most of the BCG1s should have then been assembled in early times, as pointed out in the downsizing scenario for galaxy for- mation (Cowie et al. 1996) and entered that merging history al- ready with luminosity not far form the present one. 9. Summary In this paper we search for and characterize cluster substructures, or subclusters, in the sample of 77 nearby clusters of the WINGS (Fasano et al. 2006). This sample is an almost complete sample in X-ray flux in the redshift range 0.04 < z < 0.07. We detect substructures in the spatial projected distribution of galaxies in the cluster fields using DEDICA (Pisani 1993, 1996) an adaptive-kernel technique. DEDICA has the following advantages for our study of WINGS clusters: a) DEDICA gives a total description of the clustering pat- tern, in particular membership probability and significance of structures besides geometrical properties. b) DEDICA is scale invariant c) DEDICA does not assume any property of the clusters, i.e. it is completely non-parametric. In particular it does not require particularly rich samples to run effectively. In order to test DEDICA and to set guidelines for the in- terpretation of the results of the application of DEDICA to our observations we run DEDICA on several sets of simulated fields containing a cluster with and without subclusters. We find that: a) DEDICA always identifies both cluster and subcluster even when the substructure richness ratio cluster- to-subcluster is rcs= 8, b) DEDICA recovers a large fraction of members, almost irrespective of the richness of the original structure (>∼ 70% in most cases), c) structures with richness ra- tios rcs<∼ 3 are always distinguishable from noise fluctuations of the poissonian simulated field. These simulations also allow us to define a threshold that we use to identify significant structures in the observed fields. We apply our clustering procedure to the 77 clusters of the WINGS sample. We cut galaxy catalogs to the absolute magni- tude threshold MV = −16.0 in order to maximize the signal-to- noise ratio of the detected subclusters. We detect at least one significant structure in 55 (71%) clus- ter fields. We find that 12 clusters (16%) have no structure above the threshold (undetected). In the remaining 10 (13%) clusters we find significant structures only at the border of the field of view. In absence of a detection in the center of the frame, we consider these border structures unrelated to the target cluster. We also verify that in the CMD these border structures are redder 10 M. Ramella et al.: Substructures in the WINGS clusters than expected given the redshift of the target cluster. We consider also these clusters undetected. We provide the coordinates of all substructures in the 55 clusters together with their main properties. Using the CMR of the early-type cluster galaxies we sep- arate ”true” subclusters from unrelated background structures. We find that 40 clusters out of 55 (73%) have a total of 69 sub- structures with 15 clusters left without substructures. The fraction of clusters with subclusters (73%) we identify is higher than most previously published values (typically ∼ 30%, see, e.g., Girardi & Biviano 2002, and references therein). It is however acknowledged that the fraction of substructured clus- ters depends, among other factors, on the algorithm used to de- tect substructures, on the quality and depth of the galaxy catalog (Kolokotronis et al. 2001; Burgett et al. 2004). Another important result of our analysis is the distribution of subcluster luminosities. In the luminosity range where our sub- structure detection is complete (L ≥ 1011.2 L⊙), we find that the distribution of subcluster luminosities is in agreement with the power-law ∝ L−1 predicted for the differential mass function of substructures in the cosmological simulations of De Lucia et al. (2004). Finally, we investigate the relation between BCGs and clus- ter structures. We find that, on average, BCG1s are located close to the den- sity peak of the main structures. In projection on the sky, the bi- weight average distance of BCG1s from the peak of the main system is 72±11 kpc. BCG2s are significantly more distant than BCG1s from the peak of the main system (345 ± 47 kpc). The fact that BCG1s are close to the center of the system is consistent with current theoretical view on the formation of BCGs (Dubinski 1998). A more surprising result is that the magnitude difference be- tween BCG1s and BCG2s, ∆M12, is significantly larger in clus- ters without substructures than in clusters with substructures. This fact may be interpreted in the framework of the hierarchical scenario of structure evolution (e.g. De Lucia & Blaizot 2006). 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Ramella et al.: Substructures in the WINGS clusters Appendix A: The catalog of substructures We provide here the catalog of substructures. In Table A.1 we give, for each substructure: (1) the name of the parent cluster; (2) the classification of the structure as main (M), subcluster (S), or background (B) together with their order number; (3) right ascension (J2000), and (4) declination (J2000) in decimal degrees of the DEDICA peak; the parameters of the ellipse we obtain from the variance matrix of the coordinates of galaxies in the substructure, i.e. (5) major axis in arcminutes, (6) ellipticity, and (7) position angle in degrees; (8) luminosity; (9) χ2S . ID class αJ2000 δJ2000 a e PA L χ (deg) (deg) (arcmin) (deg) (1012 L⊙) A0085 M 10.4752 -9.3025 2.0 0.23 -17. 0.41536 48.4 A0085 S1 10.4410 -9.4430 1.8 0.35 -39. 0.17649 42.9 A0085 S2 10.3947 -9.3501 2.3 0.40 -72. 0.12337 32.8 A0119 M 14.0625 -1.2630 4.1 0.44 -65. 0.83955 63.4 A0119 S1 14.1183 -1.2106 4.6 0.60 -23. 0.26847 50.8 A0119 S2 14.0267 -1.0441 3.4 0.34 80. 0.03592 32.0 A0119 B1 13.9402 -1.4979 3.1 0.39 46. – 23.5 A0147 M 17.0648 2.2033 3.9 0.45 79. 0.31392 45.2 A0147 S1 16.8673 2.1393 4.1 0.25 -50. 0.05638 24.8 A0147 S2 17.1925 1.9284 4.4 0.38 55. 0.05052 21.4 A0147 B1 17.0753 2.3174 4.4 0.37 75. – 58.0 A0151 M 17.2186 -15.4219 1.7 0.26 -16. 0.47344 39.9 A0151 S1 17.3516 -15.3652 2.1 0.37 -58. 0.13761 42.9 A0151 S2 17.2632 -15.5564 1.6 0.26 -53. 0.19762 40.9 A0151 B1 17.1375 -15.6116 1.5 0.08 -4. – 59.0 A0160 M 18.2344 15.5126 3.6 0.37 82. 0.55525 66.7 A0160 S1 18.2483 15.3138 5.0 0.41 85. 0.03120 38.1 A0160 S2 18.1141 15.7501 3.0 0.16 86. 0.15196 28.3 A0160 S3 17.9981 15.4150 3.9 0.41 0. 0.06315 27.8 A0168 M 18.7755 0.3999 3.1 0.32 -11. 0.24492 30.5 A0168 S1 18.8799 0.2993 2.0 0.33 4. 0.06871 28.6 A0193 M 21.2894 8.6994 2.1 0.08 36. 0.61982 105.7 A0193 B1 20.9945 8.6119 4.9 0.45 -1. – 39.1 A0311 M 32.3793 19.7722 2.3 0.19 43. 0.43320 44.0 A0376 M 41.4276 36.9517 1.7 0.07 -67. 0.13477 40.8 A0376 S1 41.5569 36.9214 4.4 0.49 -22. 0.24350 33.6 A0500 M 69.6476 -22.1308 2.0 0.31 16. 0.41203 45.5 A0500 S1 69.5915 -22.2377 2.3 0.19 36. 0.20520 47.3 A0602 M 118.3638 29.3528 1.8 0.55 -46. 0.20112 55.4 A0602 S1 118.1848 29.4145 2.3 0.52 31. 0.08470 34.8 A0671 M 127.1237 30.4269 1.6 0.24 -51. 0.68582 69.8 A0671 S1 127.2241 30.4342 2.0 0.40 -5. 0.19736 44.3 A0671 S2 127.1617 30.2967 1.9 0.24 -90. 0.13778 43.0 A0754 M 137.1073 -9.6370 2.0 0.25 53. 0.56063 46.8 A0754 S1 137.3707 -9.6760 3.2 0.53 -8. 0.30590 54.9 A0754 S2 137.2619 -9.6367 1.7 0.14 76. 0.23734 51.2 A0957x M 153.4095 -0.9259 2.0 0.09 -83. 0.42106 38.6 A0957x B1 153.5517 -0.7023 2.2 0.44 -63. – 37.9 A0970 M 154.3595 -10.6921 1.5 0.27 -30. 0.46130 62.5 A0970 S1 154.2369 -10.6422 1.7 0.15 15. 0.13660 42.3 A0970 B1 154.1833 -10.6771 1.8 0.23 -76. – 32.2 A1069 M 159.9418 -8.6883 2.8 0.31 52. 0.37270 50.2 A1069 S1 159.9286 -8.5506 2.4 0.23 88. 0.18532 32.7 A1069 B1 159.7678 -8.9262 3.5 0.55 77. – 54.7 A1291 M 173.0467 56.0255 2.5 0.51 -11. 0.25272 32.1 A1291 S1 172.9090 56.1872 1.4 0.48 -82. 0.03530 37.6 A1631a M 193.2410 -15.3413 1.4 0.35 40. 0.20077 33.9 A1736 M 202.0097 -27.3131 3.1 0.35 58. 0.41824 52.1 A1736 S1 201.7305 -27.0170 2.8 0.32 9. 0.24023 42.6 A1736 S2 201.7662 -27.4067 3.4 0.28 7. 0.14528 42.3 A1736 S3 201.5672 -27.4291 2.7 0.44 73. 0.16926 40.4 A1736 S4 201.9057 -27.1600 3.5 0.21 -1. 0.24192 32.9 A1736 S5 201.7036 -27.1236 3.0 0.39 -12. 0.40395 31.7 A1795 M 207.1911 26.5586 0.6 0.17 55. 0.12341 52.4 M. Ramella et al.: Substructures in the WINGS clusters 13 ID class αJ2000 δJ2000 a e PA L χ (deg) (deg) (arcmin) (deg) (1012 L⊙) A1795 S1 207.2329 26.7362 1.3 0.38 82. 0.05123 46.7 A1831 M 209.8120 27.9714 1.9 0.43 9. 1.08418 56.0 A1831 S1 209.7356 28.0636 2.1 0.34 41. 0.36295 59.7 A1831 B1 209.5725 28.0206 1.7 0.25 -10. – 47.7 A1991 M 223.6405 18.6390 2.3 0.54 -78. 0.28195 40.3 A1991 S1 223.7575 18.7812 2.5 0.32 41. 0.11412 49.9 A1991 B1 223.7683 18.7022 1.7 0.31 71. – 36.6 A2107 M 234.9497 21.8075 2.7 0.19 48. 0.50994 61.1 A2107 B1 235.0699 22.0127 4.3 0.48 83. – 32.9 A2107 B2 235.1409 21.8276 2.4 0.10 55. – 20.4 A2124 M 236.2400 36.0990 1.3 0.24 32. 0.41727 43.3 A2124 B1 236.0207 36.1779 1.6 0.22 34. – 59.7 A2149 M 240.3723 53.9406 1.5 0.46 -10. 0.37347 48.7 A2169 M 243.4867 49.1875 0.6 0.22 72. 0.15358 34.2 A2256 M 255.9260 78.6412 1.9 0.29 -86. 1.46563 95.6 A2256 B1 256.3094 78.4886 2.2 0.48 75. – 48.2 A2256 B2 256.6024 78.4283 2.0 0.12 -88. – 46.8 A2399 M 329.3693 -7.7772 3.5 0.64 -26. 0.40505 38.8 A2415 M 331.3829 -5.5444 2.3 0.23 -60. 0.36780 44.8 A2415 S1 331.5610 -5.3960 2.3 0.33 52. 0.05032 33.9 A2415 B1 331.3800 -5.4017 1.9 0.34 32. – 41.4 A2415 B2 331.3295 -5.3890 1.6 0.41 4. – 37.3 A2457 M 338.9462 1.4765 4.3 0.50 -84. 0.88720 107.3 A2457 S1 339.0392 1.6459 4.1 0.65 -50. 0.05960 23.1 A2457 B1 339.0667 1.3266 5.6 0.53 77. – 73.8 A2572a M 349.3192 18.7197 2.9 0.39 23. 0.44749 47.8 A2572a S1 349.1122 18.5320 4.1 0.25 8. 0.07320 34.5 A2572a S2 349.3851 18.5395 2.6 0.31 67. 0.05345 25.1 A2572a S3 349.0037 18.7220 3.2 0.34 86. 0.00884 20.5 A2593 M 351.0766 14.6539 1.1 0.25 58. 0.28333 33.8 A2593 S1 351.0677 14.4048 2.2 0.42 80. 0.09810 27.0 A2622 M 353.7384 27.3856 3.1 0.09 76. 0.48920 68.1 A2622 S1 353.4880 27.2877 4.2 0.35 46. 0.03070 35.1 A2622 B1 353.7837 27.3182 3.0 0.49 -5. – 53.7 A2622 B2 353.8009 27.6217 2.6 0.38 68. – 29.9 A2657 M 356.1725 9.1818 4.5 0.47 22. 0.27061 49.6 A2657 S1 356.2755 9.1799 3.2 0.35 10. 0.20771 34.0 A2657 B1 355.9569 8.9422 2.8 0.06 -10. – 38.6 A2665 M 357.7050 6.1582 3.6 0.26 71. 0.67950 121.8 A2665 S1 357.4003 5.8659 4.9 0.64 84. 0.01780 17.5 A2665 B1 357.8218 6.3522 3.5 0.44 -50. – 32.7 A2734 M 2.8363 -28.8652 3.4 0.33 3. 0.48700 56.3 A2734 S1 2.6950 -28.7728 4.0 0.27 -7. 0.18970 55.0 A2734 S2 2.6987 -29.0394 3.2 0.24 55. 0.03030 43.3 A2734 S3 2.5727 -29.0562 3.4 0.51 84. 0.03100 33.1 A2734 B1 2.7701 -28.6488 3.7 0.39 14. – 28.0 A3128 M 52.4825 -52.5764 2.2 0.17 -36. 0.40452 37.2 A3128 S1 52.7366 -52.7089 4.1 0.44 -9. 0.25240 51.8 A3128 S2 52.6655 -52.4413 2.3 0.32 -65. 0.19646 51.0 A3128 S3 52.3697 -52.7570 3.2 0.47 81. 0.17169 39.0 A3158 M 55.7477 -53.6334 2.6 0.62 2. 0.70205 52.8 A3158 S1 55.8382 -53.6780 3.4 0.58 10. 0.45553 53.4 A3266 M 67.7893 -61.4637 1.1 0.27 -72. 0.42993 63.5 A3376 M 90.1628 -39.9950 2.5 0.14 -43. 0.33708 43.1 A3376 S1 90.4344 -39.9776 2.7 0.20 -4. 0.21279 59.8 A3376 S2 90.4712 -39.7946 2.1 0.39 -88. 0.00904 31.2 A3528b M 193.5928 -29.0136 1.3 0.04 -24. 0.65638 66.4 A3528b S1 193.6030 -29.0721 1.3 0.26 10. 0.16706 59.0 A3530 M 193.9098 -30.3606 1.9 0.26 33. 0.53043 34.9 A3532 M 194.3035 -30.3732 3.6 0.52 -44. 0.76920 51.1 A3532 B1 194.0413 -30.2130 3.8 0.38 66. – 56.3 14 M. Ramella et al.: Substructures in the WINGS clusters ID class αJ2000 δJ2000 a e PA L χ (deg) (deg) (arcmin) (deg) (1012 L⊙) A3558 M 201.9587 -31.4892 4.9 0.54 49. 1.14860 64.1 A3558 B1 202.2501 -31.6887 2.6 0.49 44. – 37.6 A3562 M 203.4603 -31.6812 2.5 0.18 82. 0.39087 42.4 A3562 S1 203.1622 -31.7742 4.0 0.40 -86. 0.15010 51.1 A3562 S2 203.3137 -31.6953 3.6 0.40 76. 0.11706 41.0 A3562 S3 203.6982 -31.7171 2.5 0.21 -50. 0.07820 36.7 A3562 S4 203.6541 -31.5969 4.0 0.55 -81. 0.06542 30.3 A3667 M 303.1637 -56.8598 2.1 0.14 23. 0.56803 42.0 A3667 S1 303.5297 -56.9660 3.0 0.39 -86. 0.27086 39.2 A3667 S2 302.7241 -56.6674 1.5 0.17 75. 0.28948 34.0 A3667 S3 302.7081 -56.7557 2.3 0.39 12. 0.09961 33.7 A3716 M 312.9910 -52.7677 4.7 0.38 36. 0.76450 77.9 A3716 S1 312.9769 -52.6434 3.5 0.22 -6. 0.49159 56.9 A3716 B1 312.7735 -52.8976 4.1 0.40 28. – 48.4 A3716 B2 313.1888 -52.4785 3.2 0.23 21. – 22.6 A3880 M 336.9796 -30.5474 3.8 0.33 -50. 0.25840 44.1 A3880 B1 336.8684 -30.8171 2.5 0.30 64. – 30.9 A3880 B2 336.7356 -30.7839 2.7 0.35 46. – 28.4 IIZW108 M 318.4443 2.5706 2.6 0.38 42. 0.49940 33.4 IIZW108 S1 318.6247 2.5533 3.2 0.36 -14. 0.08565 42.3 IIZW108 B1 318.3335 2.7751 1.6 0.24 43. – 44.5 IIZW108 B2 318.5190 2.8039 2.5 0.44 22. – 33.6 MKW3s M 230.3916 7.7281 2.3 0.30 -8. 0.37614 48.7 MKW3s S1 230.4576 7.8769 3.3 0.46 -22. 0.04585 39.3 MKW3s B1 230.7349 7.8882 2.0 0.40 16. – 25.5 RX0058 M 14.5875 26.8816 2.4 0.22 -31. 0.31967 44.2 RX0058 S1 14.7652 27.0424 3.8 0.60 64. 0.31661 50.6 RX0058 B1 14.4012 26.7041 3.3 0.08 -12. – 28.9 RX1740 M 265.1398 35.6416 2.8 0.21 -26. 0.17896 42.1 RX1740 S1 265.2600 35.4366 3.3 0.22 38. 0.01946 31.7 RX1740 S2 264.8688 35.6053 3.7 0.52 28. 0.01340 27.9 RX1740 S3 265.0744 35.8116 3.5 0.44 -7. 0.01166 21.8 Z2844 M 150.7281 32.6483 2.9 0.20 -85. 0.10143 48.3 Z2844 S1 150.6524 32.7621 5.2 0.58 63. 0.04930 50.5 Z2844 S2 150.5821 32.8890 2.6 0.39 0. 0.00395 23.1 Z8338 M 272.7447 49.9078 3.0 0.37 -67. 0.45876 43.1 Z8338 S1 272.8606 49.7916 3.2 0.11 62. 0.05549 31.9 Z8338 S2 272.6903 49.9737 3.1 0.67 62. 0.07089 25.7 Z8338 B1 272.4479 49.6815 1.7 0.18 9. – 25.8 Z8852 M 347.6024 7.5824 2.7 0.41 -45. 0.76110 67.9 Z8852 S1 347.5926 7.3999 5.8 0.56 62. 0.12022 32.4 Z8852 S2 347.6986 7.8018 2.3 0.13 81. 0.02493 25.5 Z8852 B1 347.7381 7.6808 2.1 0.25 -73. – 35.9 Z8852 B2 347.4951 7.8165 2.4 0.21 72. – 27.0 M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 1 Online Material M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 2 Fig. 6. Isodensity contours (logarithmically spaced) of the 55 clusters with significant structures. The title lists the coordinates of the center. The orientation is East to the left, North to the top. Galaxies belonging to the systems detected by DEDICA are shown as dots of different colors. Black, light green, blue, red, magenta, dark green are for the main system and the subsequent substructures ordered as in Table A.1. Large symbols are for galaxies with MV ≤ −17.0 that lie where local densities are higher than the median local density of the structure the galaxy belongs to. Open symbols mark the positions of the first- and second-ranked cluster galaxies, BCG1 and BCG2 respectively. M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 3 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 4 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 5 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 6 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 7 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 8 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 9 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 10 Fig. 6. (continued) M. Ramella et al.: Substructures in the WINGS clusters, Online Material p 11 Fig. 6. (continued) Introduction The Data The DEDICA Procedure The probability density Cluster Identification Cluster Significance and of Membership Probability Tweaking the Algorithm with Simulations Substructure detection in WINGS clusters The Catalog of Substructures Properties of substructures Brightest Cluster Galaxies Summary The catalog of substructures

0704.0580

Ising-like dynamics and frozen states in systems of ultrafine magnetic particles

Ising-like dynamics and frozen states in systems of ultrafine magnetic particles Stefanie Russ1 and Armin Bunde1 Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany (Dated: February 28, 2022) We use Monte-Carlo simulations to study aging phenomena and the occurence of spinglass phases in systems of single-domain ferromagnetic nanoparticles under the combined influence of dipolar interaction and anisotropy energy, for different combinations of positional and orientational disorder. We find that the magnetic moments oriente themselves preferably parallel to their anisotropy axes and changes of the total magnetization are solely achieved by 180 degree flips of the magnetic moments, as in Ising systems. Since the dipolar interaction favorizes the formation of antiparallel chain-like structures, antiparallel chain-like patterns are frozen in at low temperatures, leading to aging phenomena characteristic for spin-glasses. Contrary to the intuition, these aging effects are more pronounced in ordered than in disordered structures. PACS numbers: 75.75.+a, 75.40.Mg, 75.50.Lk, 75.50.Tt INTRODUCTION In the last decade, systems of ultrafine magnetic nanoparticles have received considerable interest, due both to their important technological applications (mainly in magnetic storage and recordings) and their rich and often unusual experimental behavior, which is related to their role as a complex mesoscopic system [1, 2]. It has been discussed controversially in the past, under which circumstances these systems are able to show spin-glass phases. While experiments on disor- dered magnetic materials present indications of a spin- glass phase [2, 3, 4] or of a glassy-like random anisotropy system [5], the situation is less clear on the theoretical side. Simulations of the zero-field cooling (ZFC) and field-cooling susceptibility showed no indication of a spin- glass phase [6, 7]. In contrast, simulations on aging [8] (on a simplified system, where the dipolar interaction was only considered up to a cut-off radius) and magnetic relaxation [9, 10] favorize the spin-glass hypothesis, but the structure of the frozen history-dependent states as well as the actual mechanism leading to them has not yet been clarified. In this letter, in order to clarify these questions, we use Monte Carlo simulations [11] to study aging phe- nomena on a large variety of systems of ultrafine mag- netic nanoparticles (see Fig. 1). Our simulations do not only point to the existence of frozen history-dependent states at low temperatures that are characteristic for spin glasses, but also yield an insight into the structure of the frozen states and the underlying dynamics. We find that under the combined influence of dipolar and anisotropy energy, the magnetic moments have a tendency to align in an Ising-like manner either parallel or antiparallel to their anisotropy axes and change their directions by 180 degree flips as in Ising systems. This way, chain-like structures are formed where all magnetic moments point into the same direction and neighboring chains have the tendency to oriente themselves in an antiparallel way. (a) (b) (c) (d) FIG. 1: Two-dimensional sketches of the geometries consid- ered in this paper: (a) cubic arrangement of the particles and all anisotropy axes aligned into the z-direction, (b) liquid-like arrangement and all axes arranged, (c) cubic arrangement and all axes randomly oriented and (d) liquid-like arrange- ment and all axes randomly oriented. In the simulations, the systems were three-dimensional (64 particles per cube). These topological chains that freeze in at low tempera- tures, form simple straight lines, when the particles are arranged on the sites of a cubic lattice [10] and form com- plex winded curves, when the arrangement of the parti- cles is liquid-like. As a consequence, if a small external magnetic field is applied, the magnetic moments can fol- low the field more easily in a disordered system than in the ordered configuration. This leads, contrary to the in- tuition, to more pronounced aging effects (characteristic for spin glasses) in ordered than in disordered structures. http://arxiv.org/abs/0704.0580v1 MODEL SYSTEM AND NUMERICAL SIMULATIONS For the numerical calculations, we focus on the same model as in earlier papers [6, 9], which (i) assumes a coherent magnetization rotation within the anisotropic particles, and (ii) takes into account the magnetic dipo- lar interaction between them. Every particle i of volume Vi is considered to be a single magnetic domain ~µi with all its atomic magnetic moments rotating coherently and the Vi are taken from a Gaussian distribution of width σV = 0.4 and 〈V 〉 = 1 (see also [6, 9]). This results in a constant absolute value |µi| = MsVi of the total mag- netic moment of each particle, whereMs is the saturation magnetization. The energy of each particle consists of three contributions: anisotropy energy, dipolar interac- tion and magnetic energy of an external field. We assume a temperature independent uniaxial anisotropy energy A = −KVi((~µi~ni)/|~µi|) 2, where K is the anisotropy constant and the unit vector ~ni denotes the easy direc- tions. Eventually, the magnetic moments are coupled to an external field H leading to the additional field energy H = −~µi ~H . Finally, the energy of the magnetic dipo- lar interaction between two particles i and j separated by ~rij is given by E (i,j) D = (~µi~µj)/r ij − 3(~µi~rij)(~µj~rij)/r Adding up the three energy contributions and summing over all N particles we obtain the total energy j 6=i (i,j) . (1) In the Monte Carlo simulations we concentrate on sam- ples of N = L3 particles placed inside a cube of side length L = 4 and average over 1000 configurations. Dur- ing the simulations, both, the positions of the particles and their easy axes are kept fixed. The unitless concen- tration c is defined as the ratio between the total vol- Vi occupied by the particles and the volume Vs of the sample. Here, we focus on the concentra- tion c/c0 ≈ 0.3, where c0 = 2K/M s is a dimensionless material-dependent constant, c0 ∼ 1.4 for iron nitride and c0 ∼ 2.1 for maghemite nanoparticles [9]. We also tested systems with higher concentrations c/c0 ≈ 0.4 and (the extremely high concentration) c/c0 ≈ 0.6 and found that the results remain qualitatively unchanged. The temperature is measured in units of the reduced temper- ature T̃ ≡ 1/(2βKV ), where 2KV is the height of the anisotropy barrier and β = 1/(kBT ). Similarly, the mag- netic field is measured in units of the anisotropy field Ha = 2K/Ms. The relaxation of the individual magnetic moments is simulated by the standard Metropolis algo- rithm [12]. In contrast to [8], where dipole interactions between the particles were only considered up to a cut-off radius, we calculate the interaction energies by the Ewald sum method with periodic boundary conditions in x, y and z-direction [6, 13] and thus are able to account fully (a) (b) (c) (d) FIG. 2: (Colors online) The magnetization m(τ ) after wait- ing times tw = 0 (filled symbols) and tw = 10000 Monte Carlo steps (open symbols) is plotted versus τ (number of Monte Carlo steps with applied external field) for (a) cubic lattice and aligned axes, (b) liquid-like system and aligned axes, (c) cubic system and random axes and (d) liquid- like systems and random axes for the reduced temperatures T̃ = kBT/(2KV ) = 5 (black symbols, circles), T̃ = 1/10 (red symbols, squares) and T̃ = 1/40 (blue symbols, diamonds). for the long-range character of the dipole forces. The magnetic moment ~µi is characterized by the spherical angles θi and ϕi relative to a coordinate frame, where the z-axis is parallel to the external field [9, 14, 15]. To study the magnetic relaxation we determine as a function of time t (number of Monte Carlo steps) for each particle i the angle θi between the magnetic moment ~µi and the z-axis, from which we obtain the relevant quantities, as e.g. the normalized magnetization, m(t) = cos θi(t). (2) To obtain the orientation of ~µi relative to ~ni, we intro- duce the ”orientational order parameter” Oµ ≡ 〈|~µi~ni|〉, i.e. the average of the absolute values of the scalar prod- uct ~µi~ni over all N particles and all configurations. Oµ does not distinguish between the parallel and the an- tiparallel alignment. It is equal to zero when all ~µi are perpendicular to their axes ~ni and equal to 1 if they are all parallel or antiparallel to them. To study aging phenomena, we determine the magne- tization in a ZFC simulation. First, starting in a ran- dom configuration of the magnetic moments, the system is cooled down in the absence of an external field, from T = ∞ to a reduced temperature T̃ with a constant cool- ing rate of ∆β/∆t = 0.1, corresponding to 400 Monte Carlo steps for T̃ = 1/40 and 10 steps for T̃ = 1. Sec- ond, the cooling process is stopped at T̃ and the system is allowed to relax for a certain waiting time tw. Finally, in the third step, a small external field h = 0.1Ha is applied in z-direction. The magnetization m(τ) is determined as a function of τ ≡ t− tw (number of Monte Carlo steps af- ter switching on the field). Aging effects are represented by differences between the m(τ)-curves for different tw and occur, when many different relaxation rates exist in the system, so that after a given waiting time tw, the sys- tem has only partly relaxed towards equilibrium. Experi- mentally, aging effects have already been found in several spin-glasses, as e.g. in Permalloy/alumina granular films [16], rare-earth manganates [17], CuMn spin-glasses [18], multilayer systems [19] and in Fe3N nanoparticle sys- tems [20]. NUMERICAL RESULTS Figure 2 shows m(τ) for the systems of Fig. 1 without waiting time, tw = 0 (filled symbols), and for tw = 10 (open symbols). The different colors (and symbols) stand for three different temperatures T̃ = 1/5, 1/10 and 1/40. Clearly, all curves show aging effects, similar to the ex- perimental results of Refs. [16, 17, 18]. Systems with no or only small tw follow the external field faster than the systems with longer waiting times, indicating that the longer relaxation leads to more stable chains. The ag- ing effects are most pronounced for those systems where all anisotropy axes are oriented into the direction of the external field (Fig. 2(a,b)) and less pronounced but still visible for the systems with disordered anisotropy axes (Fig. 2(c,d)). In these systems with orientational disor- der, the m(τ) curves coincide for small τ and show aging effects only after a certain crossover time (close to 102 Monte Carlo steps). This indicates that in these systems a certain fraction of dipoles does not belong to quasi- stable chain-like structures and can follow the external field nearly instantaneously, independentely of the wait- ing time and thus dominate the short-time behavior. The aging effects decrease with increasing T̃ , when the order is destroyed by the thermal fluctuations. In order to understand the dynamical behavior in a more microscopic way, we compare m(τ) with the time- dependence of the corresponding orientational order pa- rameters Oµ(τ). Figure 3 shows Oµ in the 3rd step of the aging process for tw = 0 and tw = 10000 (filled and open symbols, respectively) and for the same geometries as before (see Fig. 1). The figure shows that quite con- trary to the expectation, apart from a slight minimum at intermediate τ , Oµ is constant in time for the systems of tw = 10000. Without waiting time, the curves start at much smaller values of Oµ, but increase rapidly un- til they reach at a crossover time τc of about 10 3 Monte Carlo steps the common plateau value. In the plateau regime, the dipolar moments ~µi are either oriented par- (a) (b) (c) (d) FIG. 3: (Colors online) The order parameter Oµ after a wait- ing time tw = 0 (filled symbols) and tw = 10000 (open sym- bols) is plotted versus τ (number of Monte Carlo steps) for the same geometries, temperatures, system parameters and symbols and colors as in Fig. 2. (a) (b) (c) (d) FIG. 4: (Colors online) The percentage Nup of particles per system pointing upwards after waiting times tw = 0 (filled symbols) and tw = 10000 (open symbols) is plotted versus τ (number of Monte Carlo steps) for the same geometries, temperatures, system parameters and symbols and colors as in Fig. 2. allel or antiparallel to their easy axes ~ni and do therefore flip only between these two directions. Accordingly, the value of Oµ does neither depend on the external field nor on the functional form of m(τ). Since Oµ(τ) stays con- stant for large tw or τ > τc, while m(τ) increases with time (see Fig. 2), the ~µi have already reached their par- FIG. 5: (Colors online) The magnetization m(τ ) after a wait- ing time tw = 0 (filled symbols) and tw = 10000 (open sym- bols) for T̃ = 1/10 (red symbols) and T̃ = 1/40 (blue symbols) of systems with aligned and randomly oriented anisotropy axes (red circles and diamonds, respectively for T̃ = 1/10 and blue squares and triangles respectively for T̃ = 1/40) are plotted versus τ (number of Monte Carlo steps) for systems without dipole-interaction. allel or antiparallel position and can only perform spin flips by 180 degrees, thereby increasing m(τ) and leav- ing Oµ unchanged. To make this point still clearer, we plot in Fig. 4 the percentage Nup of particles pointing upwards, i.e. with ϑi < π/2, again for the geometries of Fig. 1. The similarity between Fig. 4 and Fig. 2 is obvi- ous, showing that the number of the magnetic moments oriented upwards determine the shape of m(τ). We therefore arrive at a remarkably simple Ising-like dynamics of these ultrafine magnetic particles. The amount of aging is directly related to the degree of or- der a system can achieve during tw. In the fully ordered system of Figs. 1-3(a), after a long waiting time tw, the ~µi prefer to be aligned in stable chains [10] along the z- direction and thus cannot follow an external field easily. Single magnetic moments inside a chain will hardly flip to the other side and flips of whole chains possess extremely large relaxation times. Without waiting time, on the other hand, the ~µi are in unstable positions which allows them to follow the external field quite rapidly, leading to large aging effects in ordered systems. As Figs. 2(b-d) show, the situation is different in systems with positional and/or orientational disorder. The relaxation times for spin flips decrease with the amount of disorder, in partic- ular with the amount of orientational disorder. When the chains are winded and aligned into different directions, they are less stable and possess a large variety of inter- mediate positions to flip to the other side. Accordingly, aging effects become weaker with increasing disorder. For illustration, we visualize the aging process in Fig. 6 for the system with the highest order and the strongest aging effects, i.e. for the cubic system with aligned anisotropy axes. For this visualization, we follow the definition of the transversal order parameter of Ref. [10]: each of the L2 sites in the xy plane can be either a + site or a − site, if a chain has already been formed and all magnetic moments in the chain point into the positive or negative z direction, respectively (white sites). If this is not the case, the site is a 0 site (grey sites). The figure shows that chains are quite obviously formed in the sec- ond step of the aging process during the waiting time tw, as can most easily be seen by comparing Fig. 6(a), where tw = 10000 with 6(d) where tw = 0. In (a), many chains are formed during tw that appear to be quite stable in the following 3rd step of the aging process (Fig. 6(b,c)), when an external field is applied in the + direction. We can see that most of the − chains persist in spite of the external field. The situation is different in Fig. 6(d-f), where only few chains exist at the end of the 2nd step of the aging process (Fig. 6(d)). Here, after switching on the external magnetic field, new chains can be built from the 0 sites and the system therefore follows the field much easier than in Fig. 6(a-c). Recently, it has been argued that also a broad distri- bution of anisotropy energy barriers might lead to aging effects in superparamagnetic systems [20]. To show that these kinds of aging effects are in fact negligible com- pared with systems where both energy contributions are present, we have studied systems without dipole interac- tion (solely anisotropy energy) at temperatures T̃ = 1/10 and 1/40. In this case, the particle positions play no role, so that the geometry of Fig. 1(a) and (b) as well as (c) and (d) are physically identical. The results of m(τ) for these two geometries are shown in Fig. 5 for the same aging procedure as before. The figure shows that the dif- ferences between the curves for tw = 0 and tw = 10000 are orders of magnitude smaller than in the systems with dipolar interaction. It is interesting to note that also for systems with only dipolar interaction, some kind of ag- ing can be seen, but orders of magnitude smaller than for systems with both energy contributions. In summary, analyzing the microscopic dynamics of ul- trafine magnetic particles, we found that irrespective of the strength of the dipolar interaction, the dipoles ori- ente themselves either parallel or antiparallel to their anisotropy axes. We therefore arrive at a remarakably simple picture of the dipole dynamics, where similar to the Ising model, the ~µi perform ”spin flips” between these two orientations. Aging effects occur when after a certain waiting time, the magnetic dipoles have arranged them- selves in stable configurations and flips of single magnetic moments are suppressed. These aging effects increase in a counter-intuitive way with the order of the system and are thus most pronounced in completely orderded sys- tems with cubic arrangement of the particles and axes aligned into the direction of the magnetic field. (a) (b) (c) (d) (e) (f) FIG. 6: Visualization of the chains perpendicular to the xy- plane in the cubic system with aligned anisotropy axes at T̃ = 1/5 for one typical system. The complete chains are indicated by white sites and by + or − signs, depending on the direction of the chain. Sites, where chains have not (yet) been built are indicated by the grey shade. (a-c) System with waiting time tw = 10000, i.e. (a) after the cooling process and tw = 10000 (b,c) after an external magnetic field in the + direction has been applied for (b) 1000 and (c) 10000 Monte Carlo steps. (d-f) System without waiting time (tw = 0), i.e. (d) after the cooling process (and tw = 0) (e,f) after an external magnetic field in the + direction has been applied for (e) 1000 and (f) 10000 Monte Carlo steps. ACKNOWLEDGEMENTS We gratefully acknowledge very valuable discussions with W. Kleemann and financial support from the Deutsche Forschungsgemeinschaft. [1] X. Batlle and A. Labarta, J. Phys. D 35, R15 (2002). [2] Xi Chen, S. Sahoo, W. Kleemann, S. Cardoso and P. P. Freitas, Phys. Rev. B 70, 172411 (2004). [3] T. Jonsson, J. Mattsson, C. Djurberg, F. A. Khan, P. Nordblad, and P. Svedlindh, Phys. Rev. Lett. 75, 4138 (1995). [4] R. W. Chantrell, M. El-Hilo, and K. O Grady, IEEE Trans. Magn. 27, 3570 (1991). [5] W. Luo, S. R. Nagel, T. F. Rosenbaum, and R. E. Rosensweig, Phys. Rev. Lett. 67, 2721 (1991). [6] J. Garcia-Otero, M. Porto, J. Rivas, and A. Bunde, Phys. Rev. Lett. 84, 167 (2000). [7] M. Porto, Eur. Phys. J. B 45, 369 (2005). [8] J.-O. Andersson et al., Phys. Rev. B 56, 13983 (1997). [9] M. Ulrich, J. Garcia-Otero, J. Rivas, and A. Bunde; Phys. Rev. B 67, 024416 (2003). [10] S. Russ, A. Bunde, Phys. Rev. B 74, 064426 (2006). [11] U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. 84, 163 (2000). [12] In every step, we select a particle i at random and gen- erate an attempted orientation of its magnetization, cho- sen in a spherical segment around the present orientation with an aperture angle dθ (see also Ref. [6]). By varying dθ, i.e. the maximum jump angle, it is possible to modify the rate of acceptance and to optimize the simulation. As a compromise between simulations at low and high temperatures, we chose dθ = 0.1 for all simulations, in- dependent of temperature, which refers to an accecptance rate between 0.5 and 0.8 for T̃ between 1/40 and 1/5. We also tested larger values of dθ with considerably lower ac- ceptation rates and found that they did not change the final states significantly. [13] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon, Oxford, 1987). [14] R. V. Chamberlin, G. Mozurkewich, and R. Orbach, Phys. Rev. Lett. 52, 867 (1984). [15] K. L. Ngai and U. Strom, Phys. Rev. B 38, 10350 (1988). [16] E. Vincent, Y. Yuan, J. Hamman, H. Hurdequint and F. Guevara; J. of Mag. and Mag. Mat. 161209 (1996). [17] A. K. Kundu, P. Nordblad and C. N. R. Rao; Phys. Rev. B 72, 144423 (2005). [18] L. Lundgren, P. Svendlindh, P. Nordblad and O. Beck- mann; Phys. Rev. Lett. 51, 811 (1983). [19] S. Bedanta, O. Petracic, E. Kentzinger, W. Kleemann, U. Rücker, A. Paul, Th. Brückel, S. Cardoso and P. P. Freitas, Phys. Rev. B 72, 024419 (2005). [20] M. Sasaki, P.E. Jönsson, H. Takayama and H. Mamiya; Phys. Rev. B 71, 104405 (2005).

0704.0581

Counting characters in linear group actions

arXiv:0704.0581v1 [math.RT] 4 Apr 2007 Counting characters in linear group actions Thomas Michael Keller Department of Mathematics Texas State University 601 University Drive San Marcos, TX 78666 e–mail: [email protected] 2000 Mathematics Subject Classification: 20C15. http://arxiv.org/abs/0704.0581v1 Abstract. Let G be a finite group and V be a finite G–module. We present upper bounds for the cardinalities of certain subsets of Irr(GV ), such as the set of those χ ∈ Irr(GV ) such that, for a fixed v ∈ V , the restriction of χ to 〈v〉 is not a multiple of the regular character of 〈v〉. These results might be useful in attacking the non–coprime k(GV )–problem. 1 Introduction Let G be a finite group and V be a finite G–module of characteristic p. If (|G|, |V |) = 1, then in [3, Theorem 2.2] R. Knörr presented a beautiful argument showing how to obtain strong upper bounds for k(GV ) (the number of conjugacy classes of GV ) by using only information on CG(v) for a fixed v ∈ V . Note that his result immediately implies the important special case that if G has a regular orbit on V (i.e., there is a v ∈ V with CG(v) = 1), then k(GV ) ≤ |V |, which was a crucial result in the solution of the k(GV )–problem. In this note we give a much shorter proof of this result (see Proposition 3.1 below). The main objective of the paper, however, is to modify and generalize Knörr’s argument in various directions to include non–coprime situations. This way we obtain a number of bounds on certain subsets of Irr(GV ), such as the following: Theorem A. Let G be a finite group and let V be a finite G–module of characteristic p. Let v ∈ V and C = CG(v) and suppose that (|C|, |V |) = 1. Then the number of irreducible characters whose restriction to 〈v〉 is not a multiple of the regular character of 〈v〉 is bounded above by |CV (ci)|, where the ci are representatives of the conjugacy classes of C. Theorem B. Let G be a finite group and V be a finite G–module. Let g ∈ G be of prime order not dividing |V |. Then the number of irreducible characters of GV whose restriction to A = 〈g〉 is not a multiple of the regular character of 〈g〉 is bounded above by |CG(g)| n(CG(g), CV (g)), where n(CG(g), CV (g)) denotes the number of orbits of CG(g) on CV (g). Stronger versions and refinements of these results are proved in the paper. It is hoped that these results prove useful in solving the non–coprime k(GV )–problem, as discussed, for instance, in [2] and [1]. Theorem A and B will be proved in Sections 3 and 4 below respectively. In Section 2, we will generalize a recent result of P. Schmid [5, Theorem 2(a)] stating that in the situation of the k(GV )–problem, if G has a regular orbit on V , then k(GV ) = |V | can only hold if G is abelian. We prove Theorem C. Let G be a finite group and V a finite faithful G–module with (|G|, |V |) = 1. Suppose that G has a regular orbit on V . Then k(GV ) ≤ |V | − |G|+ k(G). Our proof is different from the approach taken in [5], and we actually will prove a slightly stronger result including some non–coprime actions. Notation: If the group A acts on the set B, we write n(A,B) for the number of orbits of A on B. All other notation is standard or explained along the way. 2 k(GV) = |V| and regular orbits In this paper we often work under the hypothesis of the k(GV )–problem which is the following. 2.1 Hypothesis. Let G be a finite group and let V be a finite faithful G–module such that (|G|, |V |) = 1. Write p for the characteristic of V . In [5, Theorem 2(a)] P. Schmid proved that under Hypothesis 2.1, if G has a regular orbit on V , V is irreducible, and k(GV ) = |V |, then G is abelian, and from this it follows easily that either |G| = 1 and |V | = p, or G is cyclic of order |V | − 1. The proof in [5] is somewhat technical. The goal of this section is to give a short proof of a generalization of Schmid’s result based on a beautiful argument of Knörr [3]. We word it in such a way that we even do not need the coprime hypothesis, so that the result may even be useful to study the non–coprime k(GV )–problem. To do this, for any group X and x ∈ X we introduce the set Irr(X,x) = {χ ∈ Irr(G)| χ|〈x〉 is not an integer multiple of the regular character of 〈x〉} and write k(X,x) = |Irr(X,x)|. 2.2 Theorem. Let G be a finite group and let V be a finite G–module such that G possesses a regular orbit on V . Let v ∈ V be a representative of such an orbit. Then k(GV, v) ≤ |V | − |G|+ k(G) Proof. Let p be the characteristic if V . We proceed exactly as in Case (ii) of the proof of [3, Theorem 2.2]. Write C = CG(v). As C = 1, we see that for A = 〈v〉 we trivially have that |C| and |A| are coprime, and so that proof yields (1) (p − 1)|V | = τ∈Irr(GV ) (τη, τ)A where η is the character of A defined by η = p1A − ρA with ρA being the regular character of A. Now for any τ ∈ Irr(GV ) we have (2) (τη, τ)A = τ(a)(p − ρA(a))τ(a) 16=a∈A |τ(a)|2 = 0 if τ |A is an integer multiple of ρA ≥ p− 1 otherwise where the last step follows from [4, Corollary 4]. Next observe that if τ ∈ Irr(GV ) with V ≤ ker τ , then τ ∈ Irr(G) and clearly τ |A is not a multiple of ρA, and then clearly (3) (τη, τ)A = 16=a∈A |τ(a)|2 = 16=a∈A τ(1)2 = (p − 1)τ(1)2. Thus with (1), (2), and (3) we get (p − 1)|V | = τ∈Irr(G) (τη, τ)A + τ ∈ Irr(GV ), V 6≤ ker τ (τη, τ)A τ∈Irr(G) (p− 1)τ(1)2 + (k(GV, v) − k(G))(p − 1) which yields |V | ≥ τ∈Irr(G) τ(1)2 + k(GV, v) − k(G) = |G|+ k(GV, v) − k(G). This implies the assertion of the theorem, and we are done. ✸ The following consequence implies Schmid’s result [5, Theorem 2(a)]. 2.3 Corollary. Assume Hypothesis 2.1 and that G has a regular orbit on V . Then k(GV ) ≤ |V | − |G|+ k(G). In particular, if k(GV ) = |V |, then G is abelian. Proof. By Ito’s theorem and as (|G|, |V |) = 1, we know that χ(1) divides |G| for every χ ∈ Irr(GV ), so in particular p does not divide χ(1). Thus for any v ∈ V # we see that χ|〈v〉 cannot be an integer multiple of ρ〈v〉. Therefore k(GV, v) = k(GV ). Now the assertion follows from Theorem 2.2. ✸ 3 Bounds for k(GV) In this section we study more variations of Knörr’s argument in [3, Theorem 2.2] and generalize it to some non-coprime situations. We begin, however, by looking at a classical application of it. An important and immediate consequence of Knörr’s result is that if under Hypothesis 2.1 G has a regular orbit on V , then k(GV ) ≤ |V |. This important result can be obtained in the following shorter way. 3.1 Proposition. Let G be a finite group and let V be a finite faithful G–module. Let v ∈ V . k(GV, v) ≤ |CG(v)||V |, in particular, if (|G|, |V |) = 1 and G has a regular orbit on V , then k(GV ) ≤ |V |. Proof. PutA = 〈v〉. If τ ∈ Irr(GV, v), then by [4, Corollary 4] we know that 16=a∈A |τ(a)|2 ≥ p−1. With this and well–known character theory we get (p− 1)k(GV, v) ≤ k(GV, v) min τ∈Irr(GV,v) 16=a∈A |τ(a)|2 τ∈Irr(GV ) 16=a∈A |τ(a)|2 16=a∈A τ∈Irr(GV ) τ(a)τ(a) 16=a∈A |CGV (a)| 16=a∈A |CG(v)||V | = (p− 1)|CG(v)||V | This implies the first result. If (|G|, |V |) = 1, then by Ito’s result τ(1)||G| for all τ ∈ Irr(GV ), so p cannot divide τ(1), and thus k(GV, v) = k(GV ), and the second result now follows by choosing v to be in a regular orbit of G on V . ✸ Now we turn to generalizing Knörr’s argument. We discuss various ways to do so. 3.2 Remark. Let G be a finite group and let V be a finite faithful G–module of characteristic p. Let v ∈ V and put C = CG(v) and A = 〈v〉. Let Irr(GV,C, v) := Irr0(GV ) := Irr(GV )−{χ ∈ Irr(GV ) | χ|C×〈v〉 = τ ×ρA for a character τ of C} Irrp′(GV ) = {χ ∈ Irr(GV ) | p does not divide χ(1)}, so that clearly Irrp′(GV ) ⊆ Irr0(GV ). Note that if (|G|, |V |) = 1, then by Ito Irr(GV ) = Irrp′(GV ). To work towards our next result, we again proceed somewhat similarly as in [3, Theorem 2.2]. In the following we work under the hypothesis that (|C|, |V |) = 1. Let N = NG(A). Then |N : C| divides p − 1. Moreover, from Knörr’s proof we know that if ci (i = 1, . . . , k(C)) with c1 = 1 are representatives of the conjugacy classes of C and aj (j = 1, . . . , |N :C| ) are representatives of the N–conjugacy classes of A − 1 then, the ciaj are representatives of those conjugacy classes of GV which intersect C × (A− 1) nontrivially. Moreover recall from Knörr’s proof that for c ∈ C, 1 6= a ∈ A, g ∈ G, u ∈ V we know that (ca)gu ∈ C ×A if and only if g ∈ N and u ∈ CV (c Now define a character η on C ×A by η = 1C × (p1A − ρA). Then for c ∈ C, a ∈ A we have η(ca) = p, if a 6= 1 0, if a = 1 Therefore ηGV vanishes on all conjugacy classes of GV which intersect C × (A − 1) trivially, whereas for c ∈ C, 1 6= a ∈ A we have that ηGV (ca) = |C ×A| g ∈ G u ∈ V η̇((ca)gu) u∈CV (c η(cgag) |CV (c |CV (c)| = |N : C| |CV (c)|. Thus if xi (i = 1, . . . , k(GV )) are representatives of the conjugacy classes of GV , then we get k(GV ) ηGV (xi) = |N:C| ηGV (ciaj) |N:C| |N : C| |CV (ci)| |N : C| |N : C| |CV (ci)| = (p− 1) |CV (ci)|, and thus (p − 1) |CV (ci)| = k(GV ) ηGV (xi) = τ∈Irr(GV ) (τηGV , τ)GV τ∈Irr(GV ) (τη, τ)C×A (1). Now if τ ∈ Irr(GV ), as in [3] write τ |C×A = λ∈Irr(A) τλ × λ (2) where τλ is a character of C or τλ = 0. Then as in [3] we see that (τη, τ)C×A = |C ×A| c ∈ C a ∈ A τ(ca)η(ca)τ(ca) c ∈ C 1 6= a ∈ A τ(ca)τ(ca) ((τλ − τµ), (τλ − τµ))C (3) where ”≤” is some arbitrary ordering on Irr(A). Now if τλ − τµ is a nonzero multiple of ρC , then (τλ − τµ, τλ − τµ)C ≥ |C| (4) and thus (τη, τ)C×A ≥ |C|. Moreover, note that if τ ∈ Irr0(GV ), then not all τλ− τµ can be equal to 0 as otherwise from (2) we see that τ |C×A would be equal to τλ × ρA for any λ. So we can partition the set Irr(A) into two disjoint nonempty subsets Λ1 = {λ ∈ Irr(A) | τλ = τ1} and Λ2 = {λ ∈ Irr(A) | τλ 6= τ1}, and thus as in [3] we see that |Λ1| |Λ2| ≥ p − 1, so there are at least p − 1 pairs λ, µ ∈ Irr(A) such that τλ − τµ 6= 0. Thus (τη, τ)C×A ≥ p− 1 for all τ ∈ Irr0(GV ). (5) Therefore by (1) and (5) we get that (p− 1) |CV (ci)| = τ∈Irr(GV ) (τη, τ)C×A ≥ τ∈Irr0(GV ) (τη, τ)C×A ≥ (p − 1)|Irr0(GV )| and thus |Irr0(GV )| ≤ |CV (ci)|. (6) From now on we assume that C > 1. Now we repeat the arguments of this proof, but replace η by η1 = (|C|1C − ρC)× (p1A − ρA), so for c ∈ C and a ∈ A we have η1(ca) = |C|p if c 6= 1 and a 6= 1 0 if c = 1 or a = 1 Now from the above we know that the ciaj (i = 2, . . . , k(C), j = 1, . . . , |N :C| ) are representatives of those conjugacy classes which intersect (C − 1)× (A− 1) nontrivially. Clearly ηGV1 vanishes on all conjugacy classes of GV which intersect (C − 1)× (A− 1) trivially, whereas for 1 6= c ∈ C, 1 6= a ∈ A, if (|C|, |V |) = 1, we have that ηGV1 (ca) = |C ×A| g ∈ G u ∈ V η̇1((ca) u∈CV (c = |N | |CV (c)|. Next we conclude that k(GV ) ηGV1 (xi) = |N:C| ηGV1 (ciaj) = (p− 1)|C| |CV (ci)|, and so as in (1) we see that (p− 1)|C| |CV (ci)| = τ∈Irr(GV ) (τη1, τ)C×A (7). Now with (2) similarly as in [3] we see that (τη1, τ)C×A = |C ×A| c ∈ C a ∈ A τ(ca)η1(ca)τ(ca) 1 6= c ∈ C 1 6= a ∈ A τ(ca)τ(ca) 1 6= c ∈ C 1 6= a ∈ A λ∈Irr(A) τλ(c)λ(a) µ∈Irr(A) τµ(c)µ(a) λ,µ∈Irr(A) 16=c∈C τλ(c)τµ(c) 16=a∈A λ(a)µ(a) = (p − 1) λ∈Irr(A) 16=c∈C τλ(c)τλ(c) − λ, µ ∈ Irr(A) λ 6= µ 16=c∈C τλ(c)τµ(c) λ∈Irr(A) 16=c∈C τλ(c)τλ(c)− λ,µ∈Irr(A) 16=c∈C τλ(c)τµ(c) 16=c∈C (τλ(c) − τµ(c))(τλ(c)− τµ(c)) 16=c∈C |τλ(c) − τµ(c)| 2 (8) for some arbitrary ordering ≤ on Irr(A). Now recall that if τ ∈ Irr0(GV ), then not all of the τλ − τµ can be 0. So choose λ, µ ∈ Irr(C) such that τλ − τµ 6= 0. If all the τµ (µ ∈ Irr(A)) are integer multiples of ρC then put Λ1 = {φ ∈ Irr(A) | τφ = τλ} and Λ2 = {φ ∈ Irr(A) | τφ 6= τλ}, so Λ1 6= ∅ and Λ2 6= ∅ and from 0 ≤ (|Λ1| − 1)(|Λ2| − 1) we clearly deduce that |Λ1||Λ2| ≥ p− 1, so there are at least p− 1 pairs (φ1, φ2) ∈ Irr(A) × Irr(A) such that τφ1 − τφ2 is a nonzero multiple of ρC . So next we assume that τλ is not a multiple of ρC . Then put Γ1 = {φ ∈ Irr(A) | τλ − τφ is a multiple of ρC} Γ2 = {φ ∈ Irr(A) | τλ − τφ is not a multiple of ρC}. Clearly λ ∈ Γ1, so Γ1 6= ∅. If Γ2 = ∅, then Irr(A) = Γ1, and if we define Λ1, Λ2 as in the previous argument, we see that there are at least (p − 1) pairs (φ1, φ2) ∈ Irr(A) × Irr(A) such that τφ1 − τφ2 is a nonzero multiple of ρC . So now suppose Γ2 6= ∅. Then |Γ1| + |Γ2| = p, and if φ1 ∈ Γ1 and φ2 ∈ Γ2, then τφ1 − τφ2 = (τφ1 − τλ) + (τλ − τφ2) clearly is not a multiple of ρC , and by the same argument as used before we see that |Γ1||Γ2| ≥ p − 1, so there are at least (p − 1) pairs (φ1, φ2) ∈ Irr(A) × Irr(A) such that τφ1 − τφ2 is not a multiple of ρC . Altogether we thus have shown that for any τ ∈ Irr0(GV ) one of the following holds: (A) There are at least (p− 1) pairs (φ1, φ2) ∈ Irr(A)× Irr(A) such that τφ1 − τφ2 is a nonzero multiple of ρC , or (B) there are at least (p − 1) pairs (φ1, φ2) ∈ Irr(A)× Irr(A) such that τφ1 − τφ2 is not a multiple of ρC . Now it remains to consider two cases: Case 1: At least half of the τ ∈ Irr0(GV ) satisfy (A). Then for any of these τ by (3) and (4) we have (τη, τ)C×A = ((τλ − τµ), (τλ − τµ))C ≥ (p − 1)|C| and so by (1) we see that (p − 1) |CV (ci)| ≥ τ∈Irr0(GV ) (τη, τ)C×A ≥ |Irr0(GV )|(p− 1)|C| which implies |Irr0(GV )| ≤ |CV (ci)| (9). Case 2: At least half of the τ ∈ Irr0(GV ) satisfy (B). Then for any of these τ by (8) and [4, Corollary 4] we have (τη1, τ)C×A ≥ (p− 1)(k(C) − 1). Thus by (7) we have that (p− 1)|C| |CV (ci)| ≥ τ∈Irr0(GV ) (τη1, τ)C×A ≥ |Irr0(GV )|(p − 1) · (k(C)− 1) whence |Irr0(GV )| ≤ k(C)− 1 |CV (ci)| (10). Now we drop the assumption (|C|, |V |) = 1 and work towards a general bound for |Irr0(GV )|. For this, fix g0 ∈C such that g0 is of prime order q and put C0 = 〈g0〉 andN0 = NG(C0). Trivially there are at most |C0|(p − 1) = q(p − 1) conjugacy classes of GV that intersect C0 × (A − 1) nontrivially, and given 1 6= c ∈ C0, 1 6= a ∈ A, we see that for g ∈ G, u ∈ V (ca)gu = cg[cg, u]ag ∈ C0 ×A first implies c g ∈ C0, i.e., g ∈ N0, and for each fixed g ∈ N0, the equation [c g, u]ag ∈ A implies [cg, u] ∈ Aa−g which has at most |CV (c g)| |Ag−1| = p|CV (g0)| solutions u. Moreover, if c = 1, then (ca)gu = agu = ag implies g ∈ NG(A) = N and u ∈ V. Now we define the character η2 on C0 ×A by η2 = 1C0 × (p1A − ρA). Thus η 2 vanishes on all conjugacy classes of GV which intersect C0×(A−1) trivially, whereas for 1 6= c ∈ C0, 1 6= a ∈ A we get ηGV2 (ca) = |C0 ×A| g ∈ G u ∈ V η̇ ((ca)gu) p|CV (g0)|p |N0||CV (g0)|, and for c = 1, 1 6= a ∈ A we get ηGV2 (ca) = η 2 (a) = |V |p = |N ||V |. Thus if xi (i = 1, . . . , k(GV )) are representatives of the conjugacy classes of GV , then k(GV ) ηGV2 (xi) ≤ (p − 1) |N ||V |+ (q − 1)(p − 1) |N0||CV (g0)| and as in (1) we see that k(GV ) ηGV2 (xi) = τ∈Irr(GV ) (τη2, τ)C0×A. Now arguing as in (2), (3), (5) and (6) above will yield |Irrp′(GV )| ≤ k(GV, v) ≤ |Irr(GV,C0, v)| ≤ (|N ||V |+ (q − 1)p|N0||CV (g0)|), where Irr(GV,C0, v) is as defined at the beginning of Remark 3.2. Putting the main results together, altogether we have proved the following: 3.3 Theorem. Let G be a finite group and let V be a finite faithful G–module of characteristic p. Let v ∈ V and put C = CG(v). If ci (i = 1, . . . , k(C)) are representatives of the conjugacy classes of C, then the following hold: (a) If (|C|, |V |) = 1, then |Irr0(GV )| ≤ |CV (ci)| and if C > 1, then |Irr0(GV )| ≤ max |CV (ci)|, k(C)− 1 |CV (ci)| (b) If (|G|, |V |) = 1, then Irr0(GV ) = Irr(G), so k(GV ) = |Irr0(GV )| and the bounds in (a) hold true for k(GV ) instead of |Irr0(GV )|. (c) In general, if g ∈ C such that o(g) = q is a prime, then |Irrp′(GV )| ≤ k(GV, v) ≤ |NG(〈v〉)||V |+ (q − 1)p|NG(〈g〉)||CV (g)| 4 The dual approach In the previous section, we always fixed v ∈ V and obtained bounds on the size of suitable subsets of Irr(GV ) in terms of properties of the action of CG(v) on V . In this section we consider a ”dual” approach: We fix g ∈ G and find bounds in terms of the action of CG(g) on CV (g). For this, put Irrg(GV ) = {χ ∈ Irr(G) | χ|〈g〉×CV (g) cannot be written as ρ〈g〉×ψ for a character ψ of CV (g)}. In particular, Irr(GV, g) ⊆ Irrg(GV ). 4.1 Theorem. Let G be a finite group and V be a finite G–module. Let g ∈ G such that (o(g), |V |) = 1. Write A = 〈g〉, N = NG(A) and C = CV (g). Then (a) |Irrg(GV )| ≤ (n(N,A)−1)n(CG(A),C) (|A|−1)|C| max16=a∈A(|NG(〈a〉)||CV (a)|) (b) if g is of prime order, then |Irrg(GV )| ≤ |CG(A)|n(CG(A), C) (c) there are X,Y ⊆ Irrg(GV ) such that Irrg(GV ) is a disjoint union of X and Y and |X| ≤ (n(N,A)− 1)n(CG(A), C) (|A| − 1)|C|2 16=a∈A (|NG(〈a〉)||CV (a)|) and |Y | ≤ (n(N,A)− 1)(n(CG(A), C)− 1) (|A| − 1)|C| 16=a∈A (|NG(〈a〉)||CV (a)|) (d) if g is of prime order and X,Y are as in (c), then |X| ≤ |CG(A)|n(CG(A), C) and |Y | ≤ |CG(A)|(n(CG(A), C)− 1) Proof. If a1, a2 ∈ A and c1, c2 ∈ C − {1}, then it is straightforward to see that (a1, c1) (a2, c2) GV implies that aG1 = a 2 . Hence if T is a set of representatives of the orbits of N on A−{1}, then every conjugacy class of GV that intersects nontrivially with (A−{1})×C has a representative ac for some a ∈ T and some c ∈ C. Moreover, for each a ∈ T we have that if c3, c4 ∈ C are CG(A)–conjugate, then ac3 and ac4 are CG(A)–conjugate and thus (ac3) G = (ac4) This shows that for each a ∈ T there are at most n(CG(A), C) conjugacy classes of GV inter- secting nontrivially with {a} × C. Hence altogether we see that there are at most |T |n(CG(A), C) = (n(N,A) − 1)n(CG(A), C) (1) conjugacy classes of GV which intersect (A− {1}) × C nontrivially. Moreover observe that for 1 6= a ∈ A, c ∈ C, h ∈ G and u ∈ V we have (ac)hu ∈ A× C if and only if h ∈ NG(〈a〉), c h ∈ C and u ∈ CV (a) because the condition (ac)hu = ah[ah, u]ch ∈ A × C first forces ah ∈ A which implies (as A is cyclic) ah ∈ 〈a〉, so h ∈ NG(〈a〉), and then as c ∈ C ≤ CV (〈a〉), it follows that c h ∈ CV (〈a〉) and [ah, u] ∈ [〈a〉, V ]. Now as by our hypothesis we have V = CV (〈a〉) × [〈a〉, V ], we see that (ac)hu ∈ A× C now forces [ah, u] = 1 and ch ∈ C. Hence u ∈ CV (a h) = CV (a). Note that the direct product A×C is a subgroup of GV . We now define a generalized character η on A× C by η = (|A| · 1A − ρA)× 1C where ρA is the regular character of A. So for a ∈ A, c ∈ C we have η(ac) = 0, a = 1 |A|, a 6= 1 Therefore ηGV vanishes on all conjugacy classes of GV which intersect (A− {1}) × C trivially, whereas for c ∈ C and 1 6= a ∈ A we have ηGV (ac) = |A× C| h ∈ G u ∈ V η̇((ac)hu) |A||C| h ∈ NG(〈a〉) with ch ∈ C u∈CV (a) η((ac)hu) |A||C| h ∈ NG(〈a〉) with ch ∈ C u∈CV (a) η(ahch) |CV (a)| |A||C| h ∈ NG(〈a〉) with ch ∈ C |NG(〈a〉)||CV (a)| Thus if {xi | i = 1, . . . , k(GV )} is a set of representatives for the conjugacy classes of GV , then by (1) and (2) we see that (n(N,A) − 1)n(CG(A), C) · 16=a∈A (|NG(〈a〉)||CV (a)|) ≥ k(GV ) ηGV (xi) τ∈Irr(GV ) (τηGV , τ)GV τ∈Irr(GV ) (τη, τ)A×C (3). Observe that in case that A is of prime order, then n(N,A)− 1 = |A| − 1 |N : CG(A)| (|A| − 1)|CG(A)| and max 16=a∈A (|NG(〈a〉)||CV (a)|) = |N ||C|, so that (3) becomes |CG(A)|(|A| − 1)n(CG(A), C) ≥ τ∈Irr(GV ) (τη, τ)A×C (3a) Since A× C is a direct product, we can write τA×C = λ∈Irr(C) (τλ × λ), where τλ is a character of A or τλ = 0. Then (τη, τ)A×C = |A× C| a ∈ A c ∈ C τ(ac)η(ac)τ(ac) |A||C| 1 6= a ∈ A c ∈ C τ(ac)|A|τ(ac) 1 6= a ∈ A c ∈ C λ∈Irr(C) τλ(a)λ(c) µ∈Irr(C) τµ(a)µ(c) 16=a∈A λ,µ∈Irr(C) τλ(a)τµ(a) λ(c)µ(c) 16=a∈A λ,µ∈Irr(C) τλ(a)τµ(a)(λ, µ)C As (λ, µ)C = 1, λ = µ 0, λ 6= µ , we further obtain (τη, τ)A×C = 16=a∈A λ∈Irr(C) τλ(a)τλ(a) λ∈Irr(C) 16=a∈A |τλ(a)| 2 (4) Now observe that τ(1) = λ∈Irr(C) τλ(1). If all the τλ are multiples of ρA, then clearly τ1 6∈ Irrg(GV ), and so if τ ∈ Irrg(GV ), then by [4, Corollary 4] with (4) we see that (τη, τ)A×C ≥ |A| − 1 (5) So (3) and (5) yield |Irrg(GV )| ≤ (n(N,A)− 1)n(CG(A), C) (|A| − 1)|C| 16=a∈A (|NG(〈a〉)||CV (a)|), (6) and if g is of prime order, then (3a) and (5) yield |Irrg(GV )| ≤ |CG(A)|n(CG(A), C). (6a) Now as in Section 3, we now repeat the same arguments, but use η1 = (|A|1A − ρA)× (|C|1C − ρC) instead of η. One can then easily check that (n(N,A)− 1)(n(CG(A), C) − 1) · 16=a∈A (|NG(〈a〉)||CV (a)|) ≥ τ∈Irr(GV ) (τη1, τ)A×C (3b) and if g is of prime order, then |CG(A)|(|A| − 1)(n(CG(A), C) − 1) ≥ τ∈Irr(GV ) (τη1, τ)A×C (3c) Moreover it is easily seen that (τη1, τ)A×C = 1 6= a ∈ A 1 6= c ∈ C τ(ac)τ(ac) 16=a∈A λ,µ∈Irr(C) τλ(a)τµ(a) 16=c∈C λ(c)µ(c), and as 16=c∈C λ(c)µ(c) = −1, if λ 6= µ |C| − 1, if λ = µ , it follows that (τη1, τ)A×C = 16=a∈A |τλ(a)− τµ(a)| 2 (7) where ”≤” is an arbitrary ordering on Irr(C). Next suppose that there are exactly a characters τ ∈ Irrg(GV ) such that there is a character ψ of A (depending on τ) and there are aλ ∈ ZZ (λ ∈ Irr(C)) such that τλ = ψ + aλρA for all λ ∈ Irr(C) and ψ is not a multiple of ρA. Then by (4) and [4, Corollary 4] we know that (τη, τ)A×C = λ∈Irr(C) 16=a∈A |ψ(a)|2 ≥ |C|(|A| − 1) and hence by (3) we get (n(N,A)− 1)n(CG(A), C) (|A| − 1)|C|2 16=a∈A (|NG(〈a〉)||CV (a)|), (8) and if g is of prime order, then by (3a) even |CG(A)|n(CG(A), C) Now let b be the number of τ ∈ Irrg(GV ) such that there is no such ψ. Then there exist λ, µ ∈ Irr(C) with 16=a∈A |τλ(a)− τµ(a)| 2 6= 0, and thus by [4, Corollary 4] we have (τη1, τ) ≥ |A| − 1 (9) So (3b) and (9) yield (n(N,A)− 1)(n(CG(A), C) − 1) |C|(|A| − 1) 16=a∈A (|NG(〈a〉)||CV (a)|) (10) and, if g is of prime order, then by (3c) b ≤ |CG(A)|(n(CG(A), C) − 1), (10b) and clearly a+ b = |Irrg(GV )|, and hence all the assertions follow and we are done. ✸ References [1] R. Guralnick, P. H. Tiep, The non–coprime k(GV )–problem, J. Algebra 279 (2004), 694– [2] T. M. Keller, Fixed conjugacy classes of normal subgroups and the k(GV )–problem, J. Algebra 305 (2006), 457–486. [3] R. Knörr, On the number of characters in a p–block of a p–solvable group, Illinois J. Math 28 (1984), 181–209. [4] G. R. Robinson, A bound on norms of generalized characters with applications, J. Algebra 212 (1999), 660–668. [5] P. Schmid, Some remarks on the k(GV )–theorem, J. Group Theory 8 (2005), 589–604.

0704.0582

Continuous interfaces with disorder: Even strong pinning is too weak in 2 dimensions

Continuous interfaces with disorder: Even strong pinning is too weak in 2 dimensions Christof Külske ∗and Enza Orlandi† November 4, 2018 Abstract We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta-pinning of strength ε at height zero. There is a detailed mathematical understanding of the depinning transition in 2 dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for ε > 0 and diverges like | log ε| for ε ↓ 0 How does the presence of a quenched disorder term in the Hamiltonian modify this transition? We show that an arbitarily weak random field term is enough to beat an ar- bitrarily strong delta-pinning in 2 dimensions and will cause delocalization. The proof is based on a rigorous lower bound for the overlap between local magnetiza- tions and random fields in finite volume. In 2 dimensions it implies growth faster than the volume which is a contradiction to localization. We also derive a simple complementary inequality which shows that in higher dimensions the fraction of pinned sites converges to one with ε ↑ ∞. AMS 2000 subject classification: 60K57, 82B24,82B44. 1 Introduction 1.1 The setup The study of lattice effective interface models, continous and discrete, has a long tradi- tion in statistical mechanics [14, 5, 9, 10, 13, 2, 3, 4]. The model we study is given in terms of variables ϕi ∈ R which, physically speaking, are thought to represent height variables of a random surface at the sites i ∈ Zd. Mathematically speaking they are just continuous unbounded (spin) variables. The model is defined in terms of: a pair potential V , a quenched random term, and a pinning term at interface height zero. More precisely, we are interested in the behavior of the quenched finite-volume Gibbs measures in a finite volume Λ⊂Zd with fixed boundary condition at height zero, given University of Groningen, Department of Mathematics and Computing Sciences, Blauwborgje 3, 9747 AC Groningen, The Netherlands [email protected], http://www.math.rug.nl/∼kuelske/ Dipartimento di Matematica, Universit degli Studi ”Roma Tre”, Largo San Leonardo Murialdo, 1, 00146 Roma, ITALY, [email protected] , http://www.mat.uniroma3.it/users/orlandi/ http://arxiv.org/abs/0704.0582v1 http://www.math.rug.nl/~kuelske/ µε,Λ[η](dϕΛ) 〈i,j〉∈Λ V (ϕi−ϕj)− i∈Λ,j∈Λc,|i−j|=1 V (ϕi)+ i∈Λ ηiϕi i∈Λ(dϕi + εδ0(dϕi)) Zε,Λ[η] where the partition function Zε,Λ[η] denotes the normalization constant that turns the last expression into a probability measure. The Dirac-measures at the interface height zero are multiplied with the parameter ε, having the meaning of a coupling strength. The disorder configuration η = (ηi)i∈Rd denotes an arbitrary fixed configuration of external fields, modelling a ”quenched” (or frozen) random environment. What do we expect for such a model? Recall that the variance of a free massless interface in a finite box diverges like the logarithm of the sidelength when there are no random fields. Adding an arbitrarily small pinning ε (without disorder) always localizes the interface uniformly in the volume, with the variance of the field behaving on the scale | log ε| when ε tends to zero. Indeed, there is a beautiful and complete mathematical understanding of the model without disorder, in the case of both Gaussian and uniformly elliptic potentials (see [1, 7]) with precise asymptotics as the pinning force tends to zero. These results follow from the analysis of the distribution of pinned sites and the random walk (arising from the random walk representation of the covariance of the ϕi’s) with killing at these sites. In this sense there is already a random system that needs to be analyzed even without disorder in the original model. What do we expect if we turn on randomness in the model and add the ηi’s ? Let us review first what we know about the same model without a pinning force. In d = 2 we recently proved the deterministic lower bound µΛN [η](|ϕ0| ≥ t logL) ≥ c exp(−ct2) uniformly for any fixed disorder configuration η, for general potentials V (assuming not too slow growth at infinity) [12]. So, it is not possible to stabilize an interface by cleverly choosing a random field configuration (one could think e.g. that this might be possible with a staggered field). As this result holds at any arbitrary fixed configuration here we don’t need any assumptions on the distribution of random fields. This result clearly excludes the existence of an infinite-volume Gibbs measure describing a two dimensional interface in infinite volume in the presence of random fields. In another paper [8] the question of existence of gradient Gibbs measures (Gibbs distributions of the increments of the interface) in infinite volume was raised. Note that while interface states may not exist in the infinite volume such gradient states may very well exist, as the example of the two-dimensional Gaussian free field shows, by computation. (For existence beyond the Gaussian case which is far less trivial, see [10, 11].) It was proved in [8] that there are no such gradient Gibbs measures in the random model in dimension d = 2. Now, turn to the full model in d = 2. In view of the localization taking place at any positive pinning force ε without disorder, a natural guess might be that with disorder at least at very large ε there would be pinning. However, we show as a result of the present paper that this is not the case, somewhat to our own surprise, and an arbitrarily strong pinning does not suffice to keep the interface bounded. 1.2 Main results Delocalization in d = 2 - superextensivity of the overlap Denote by ΛL the square of sidelength 2L+ 1 centered at the origin. In this subsection we consider the disorder average of the overlap in ΛL showing that it grows faster than the volume. This in particular implies that in two dimensions there is never pinning, for arbitrarily weak random field and arbitrarily large pinning forces ε. Here is the result. Theorem 1.1 Assume that supt V ′′(t) ≤ 1, lim inf |t|↑∞ log V (t) log |t| > 1, and let ηi be sym- metrically distributed, i.i.d. with finite second moment. Let d = 2. Then there is a constant a > 0, independent of the distribution of the random fields and the pinning strength ε ≥ 0, such that lim inf L2 logL ηiµε,ΛL [η](ϕi) ≥ aE(η20). (2) Note that the growth condition on V includes the quadratic case and ensures the finiteness of the integrals appearing in (1) for all arbitrarily fixed choices of η, even at ε = 0. Generalizations to interactions that are non-nearest neighbor are obvious; all results go through e.g. for finite range and we skip them in this presentation for the sake of simplicity. We like to exhibit the case of Gaussian random fields (and not necessarily Gaussian potential V ) since the bound acquires a form that looks even more striking because it becomes independent of the size of the variance of the ηi’s (as long as this is strictly positive). Corollary 1.2 Let us assume that the random fields ηi have an i.i.d. Gaussian distri- bution with mean zero and strictly positive variance of arbitrary size. Then, with the same constant a as above, we have the bound lim inf L2 logL µε,ΛL [η](ϕ i )− µε,ΛL [η](ϕi) ≥ a > 0 (3) for any 0 ≤ ε < ∞. (3) follows from (2) by partial integration w.r.t. the Gaussian disorder average (transforming the overlap into the variance of the ϕi’s). Note that, even in the unpinned case of ε = 0, Theorem 1.1 is not entirely trivial in the case of general potentials V . Here it provides an alternative simple way to see the delocalization in the presence of random fields (while the explicit lower bound on the tails of [12] provides more information.) Lower bound on overlap in d ≥ 3 The analogue of Theorem 1.1 for higher dimensions is the following. Theorem 1.3 Let d ≥ 3 and let ε ≥ 0 be arbitrary and assume the same conditions on V and ηi as in Theorem 1.1. Then there are positive constants B1, B2 < ∞, independent of the distribution of the random fields and the pinning strength ε ≥ 0, such that lim inf ηiµε,ΛL[η](ϕi) E(η20) (−∆−1)0,0 − log(B1 +B2ε) (4) where the positive constant (−∆−1)0,0 is the diagonal element of the inverse of the infinite-volume lattice Laplace operator whose existence is guaranteed in d ≥ 3. Lower bound on the pinned volume in d ≥ 3 We complement the previous lower bounds on the overlaps which are depinning-type of results by a pinning-type result. It is a lower bound on the disorder average of the quenched Gibbs-expectation of the fraction of pinned sites. While we needed an upper bound on the interaction potential V before we are assuming now a lower bound on V . Theorem 1.4 Let d ≥ 3. Assume that inft V ′′(t) = c− > 0 and let ηi be symmetrically distributed, i.i.d. with finite second moment. Then there exist dimension-dependent constants C1, C2 > 0, independent of the distribution of the disorder, such that, for all ε and for all volumes Λ, the disorder average of the fraction of pinned sites obeys the estimate µε,Λ[η](ϕi = 0) ≥ 1− C1 + C2E(η log ε . (5) This shows pinning for the large ε regime in the ”thermodynamic sense” that the fraction of pinned sites can be made arbitrarily close to one, uniformly in the volume. As usual this result does not allow to make statement about the Gibbs measure itself. The proofs follows from ”thermodynamic reasoning”. The first ”depinning-type” result follows from taking the log of the partition function and differentiating and in- tegrating back w.r.t. the coupling strength of the random fields. Exploiting the linear form of the random fields, convexity, comparison of non-Gaussian with the Gaussian partition functions, and asymptotics of Green’s functions the results follow, see Chapter 2 Proof of Depinning-type results The estimates in formulas (2), (3), and (4) are immediate consequences of the following fixed-disorder estimate. Proposition 2.1 For any dimension d, there are constants CnG,d < ∞ and cG,d > 0 such that, for all fixed configurations of local fields η, we have i,j∈Λ (−∆Λ)−1i,j ηiηj − |Λ| log CnG,d + ε ηiµε,Λ[η](ϕi). (6) Proof of the Proposition: Let us see what comes out when we differentiate and integrate back the free energy in finite volume w.r.t. strength of the random fields. logZε,Λ[hη] = ηiµε,Λ[hη](ϕi). (7) At every fixed η, this quantity is a monotone function of h, which is seen by another differentiation w.r.t. h which produces the variance. We have Zε,Λ[η] Zε,Λ[0] dhηiµε,Λ[hη](ϕi) ≤ ηi µε,Λ[η](ϕi). (8) We note the lower bound on the numerator which we get by dropping the pinning term, giving us Zε,Λ[η] ≥ Zε=0,Λ[η] ≥ ZGaussε=0,Λ[η] = exp i,j∈Λ (−∆Λ)−1i,j ηiηj ZGaussε=0,Λ[0] ≥ exp i,j∈Λ (−∆Λ)−1i,j ηiηj Here we have denoted by ZGaussε=0,Λ[η] the Gaussian partition function with potential V (t) = Further we used that the lower bound on V (t) taken from the hypothesis implies that, for any partition function in any volume D, we have Zε=0,D[0] ≤ C |D|nG,d. This gives Zε,Λ[0] = ε|A|Zε,Λ\A[0] ε|A|C |Λ\A| nG,d = (CnG,d + ε) So the desired estimate on the overlap follows from (8),(9),(10). This concludes the proof of the Proposition. � It is easy to obtain the Theorems 1.1 and 1.3 from the proposition. Indeed, taking a disorder average we have E(η20) (−∆Λ)−1i,i − |Λ| log CnG,d + ε ηiµε,Λ[η](ϕi) . (11) Now use the asymptotics of the Green’s-function in a square (−∆ΛL) i,i ∼ logL at fixed i to get the first theorem. The proof of the case d ≥ 3 follows from the existence of the infinite-volume Green’s-function in d ≥ 3. Finally let us note in passing that a constant magnetic field is always winning against an arbitrarily strong pinning, and even more strongly than a random field. Indeeed, let d ≥ 2, let ηi = h ≥ 0 for all sites i and let ε ≥ 0 be arbitrary. Then, there is a constant cd > 0, independent of h and ε, such that lim inf µε,ΛL [h](ϕi) ≥ cdh. (12) This again follows from the Proposition, using i,j∈Λ(−∆ΛL) i,j ∼ Ld+2. 3 Proof of Pinning-type results To prove the lower bound on the fraction of pinning sites in dimension d ≥ 3 given in Theorem 1.4 we will in fact prove the following fixed-disorder lower bound: For all finite volumes Λ and for all realizations η we have, for any ε0 > 0 µε,Λ[η](ϕi = 0) log ε 2c−|Λ| i,j∈Λ (−∆Λ)−1i,j ηiηj with a constant CG,d defined in (21). Taking a disorder-expectation (5) follows by the finiteness of Green’s function in the infinite volume (−∆ )−10,0 with ε0 = 1. � Proof of (13): The proof is based on the trick to differentiate and integrate back the log of the partition function, now w.r.t. ε: Differentiation gives logZε,Λ[η] = µε,Λ[η](ϕi = 0). (14) We integrate this relation back, and it will be important for us to do it starting from a positive ε0 > 0. So we get Zε,Λ[η] Zε0,Λ[η] µε̃,Λ[η](ϕi = 0) ≤ log µε,Λ[η](ϕi = 0) (15) where we have used that i∈Λ µε̃,Λ[η](ϕi = 0) is a monotone function of ε̃. Note that the integrand itself is not a monotone function. (Compare [6] for a related non-random pinning scenario, with back-integration from zero.) Now we have the trivial lower bound obtained by keeping only the contribution in the expansion where all sites are pinned, i.e. Zε,Λ[η] ≥ ε|Λ|. (16) For the upper bound on the partition function of the full model (at ε0) we first use the lower bound on the potential V (t) ≥ c−t giving us a comparison with a Gaussian partition function with curvature c−: Zε0,Λ[η] ≤ Z Gauss,c− [η]. (17) It is a simple matter to rescale the Gaussian curvature away Gauss,c− [η] = c− 2 ZGauss 2 η] (18) where the partition function on the r.h.s. is taken with unity curvature potential. For the Gaussian partition function we claim the upper bound (writing again in the original parameters) of the form ZGaussε,Λ [η] ≤ ZGaussε=0,Λ[η]. (19) Here is an elementary proof: We will replace successively the single-site integrations involving the Dirac measure by integrations only over the Lebesgue measure with the appropriately adjusted prefactor. Indeed, consider one site i and compute the contri- bution to the partition function while fixing the values of ϕj for j not equal to i. Then use that dϕi + εδ0(dϕi) ϕj + ηi)ϕi = (2π) 2 exp j∼i ϕj + ηi) 2 exp j∼i ϕj + ηi) dϕi exp ϕj + ηi)ϕi and iterate over the sites. For the Gaussian unpinned partition function use ZGaussε=0,Λ[η] = exp i,j∈Λ (−∆Λ)−1i,j ηiηj ZGaussε=0,Λ[0] ≤ exp i,j∈Λ (−∆Λ)−1i,j ηiηj with a suitable constant. From here (5) follows from (15,16,17,18,19,21) � Acknowledgements: The authors thank Pietro Caputo for an interesting discus- sion and Aernout van Enter for comments on a previous draft of the manuscript. C.K. thanks the university Roma Tre for hospitality. References [1] E. Bolthausen, Y. Velenik, Critical behavior of the massless free field at the depinning transition. Comm. Math. Phys. 223, 161-203, 2001. [2] M. Biskup and R. Kotecký, Phase coexistence of gradient Gibbs states. Published Online in Probab. Theory Rel. Fields DOI 10.1007/s00440-006-0013-6, 2007. [3] A. Bovier and C. Külske, A rigorous renormalization group method for interfaces in random media. Rev. Math. Phys. 6, 413–496, 1994. [4] A. Bovier and C. Külske, There are no nice interfaces in (2 + 1)-dimensional SOS models in random media, J. Statist. Phys., 83: 751–759, 1996. [5] J. Bricmont, A. El Mellouki, and J. Fröhlich, Random surfaces in statistical mechanics: roughening, rounding, wetting, . . . J. Statist. Phys. 42, 743–798, 1986. [6] P. Caputo, Y. Velenik, A note on wetting transition for gradient fields. Stochastic Process. Appl. 87, 107–113, 2000. [7] J.-D. Deuschel, Y. Velenik, Non-Gaussian surface pinned by a weak potential. Probab. Theory Related Fields 116, 359-377, 2000. [8] A. C. D. van Enter, C. Külske, Non-existence of random gradient Gibbs measures in contin- uous interface models in d = 2., math.PR/0611140, to be published in Annals of Applied Probability [9] G. Forgacs, R. Lipowski and Th.M. Nieuwenhuizen, The Behaviour of Interfaces in Ordered and Disordered Systems, in Phase Transitions and Critical Phenomena, vol. 14, edited by C. Domb and J.L. Lebowitz, Academic Press, 1986. [10] T. Funaki, Stochastic Interface models. 2003 Saint Flour lectures, Springer Lecture Notes in Mathematics, 1869, 103–294, 2005. [11] T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau ∇ϕ inter- face model. Comm. Math. Phys. 185, 1–36, 1997. [12] C. Külske, E. Orlandi, A simple fluctuation lower bound for a disordered massless random continuous spin model in d = 2. Electronic Comm. Probab. 11 200-205 (2006) [13] S. Sheffield, Random surfaces, large deviations principles and gradient Gibbs measure clas- sifications. arXiv math.PR/0304049, Asterisque 304, 2005. [14] Y. Velenik, Localization and delocalization of random interfaces. Probability Surveys 3, 112-169, 2006. http://arxiv.org/abs/math/0611140 Introduction The setup Main results Proof of Depinning-type results Proof of Pinning-type results

0704.0583

On the KK-theory of strongly self-absorbing C*-algebras

ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS MARIUS DADARLAT AND WILHELM WINTER Abstract. Let D and A be unital and separable C∗-algebras; let D be strongly self- absorbing. It is known that any two unital ∗-homomorphisms from D to A ⊗ D are approximately unitarily equivalent. We show that, if D is also K1-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital en- domorphism of D is asymptotically inner. Moreover, the space of automorphisms of D is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space X, the set of homotopy classes [X,Aut (D)] reduces to a point. The re- spective statement holds for the space of unital endomorphisms of D. As an application, we give a description of the Kasparov group KK(D, A⊗D) in terms of ∗-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group KK(D, A⊗D) is isomorphic to K0(A⊗D). 0. Introduction A unital and separable C∗-algebra D 6= C is strongly self-absorbing if there is an isomorphism D → D ⊗ D which is approximately unitarily equivalent to the inclusion map D → D ⊗ D, d 7→ d ⊗ 1D ([14]). Strongly self-absorbing C ∗-algebras are known to be simple and nuclear; moreover, they are either purely infinite or stably finite. The only known examples of strongly self-absorbing C∗-algebras are the UHF algebras of infinite type (i.e., every prime number that occurs in the respective supernatural number occurs with infinite multiplicity), the Cuntz algebras O2 and O∞, the Jiang–Su algebra Z and tensor products of O∞ with UHF algebras of infinite type, see [14]. All these examples are K1-injective, i.e., the canonical map U(D)/U0(D) → K1(D) is injective. It was observed in [14] that any two unital ∗-homomorphisms σ, γ : D → A ⊗ D are approximately unitarily equivalent, were A is another unital and separable C∗-algebra. If D is K1-injective, the unitaries implementing the equivalence may even be chosen to Date: August 3, 2021. 2000 Mathematics Subject Classification. 46L05, 47L40. Key words and phrases. Strongly self-absorbing C∗-algebras, KK-theory, asymptotic unitary equivalence, continuous fields of C∗-algebras. Supported by: The first named author was partially supported by NSF grant #DMS-0500693. The second named author was supported by the DFG (SFB 478). http://arxiv.org/abs/0704.0583v1 2 MARIUS DADARLAT AND WILHELM WINTER be homotopic to the unit. When D is O2, O∞, it was known that σ and γ are even asymptotically unitarily equivalent – i.e., they can be intertwined by a continuous path of unitaries, parametrized by a half-open interval. Up to this point, it was not clear whether the respective statement holds for the Jiang–Su algebra Z. Theorem 2.2 below provides an affirmative answer to this problem. Even more, we show that the path intertwining σ and γ may be chosen in the component of the unit. We believe this result, albeit technical, is interesting in its own right, and that it will be a useful ingredient for the systematic further use of strongly self-absorbing C∗-algebras in Elliott’s program to classify nuclear C∗-algebras by K-theory data. In fact, this point of view is our main motivation for the study of strongly self-absorbing C∗-algebras; see [8], [10], [16], [17], [18] and [15] for already existing results in this direction. For the time being, we use Theorem 2.2 to derive some consequences for the Kasparov groups of the form KK(D, A ⊗ D). More precisely, we show that all the elements of the Kasparov group KK(D, A ⊗ D) are of the form [ϕ] − n[ι] where ϕ : D → K ⊗ A ⊗ D is a ∗-homomorphism and ι : D → A ⊗ D is the inclusion ι(d) = 1A ⊗ d and n ∈ N. Moreover, two non-zero ∗-homomorphisms ϕ,ψ : D → K⊗A⊗D with ϕ(1D) = ψ(1D) = e have the same KK-theory class if and only if there is a unitary-valued continuous map u : [0, 1) → e(K ⊗ A ⊗ D)e, t 7→ ut such that u0 = e and limt→1 ‖ut ϕ(d)u t − ψ(d)‖ = 0 for all d ∈ D. In addition, we show that KKi(D,D ⊗A) ∼= Ki(D ⊗A), i = 0, 1. One may note the similarity to the descriptions of KK(O∞,O∞ ⊗ A) ([8],[10]) and KK(C,C ⊗ A). However, we do not require that D satisfies the universal coefficient theorem (UCT) in KK-theory. In the same spirit, we characterize O2 and the universal UHF algebra Q using K-theoretic conditions, but without involving the UCT. As another application of Theorem 2.2 (and the results of [7]), we prove in [4] an automatic trivialization result for continuous fields with strongly self-absorbing fibres over finite dimensional spaces. The second named author would like to thank Eberhard Kirchberg for an inspiring conversation on the problem of proving Theorem 2.2. 1. Strongly self-absorbing C∗-algebras In this section we recall the notion of strongly self-absorbing C∗-algebras and some facts from [14]. 1.1 Definition: Let A, B be C∗-algebras and σ, γ : A → B be ∗-homomorphisms. Suppose that B is unital. ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 3 (i) We say that σ and γ are approximately unitarily equivalent, σ ≈u γ, if there is a sequence (un)n∈N of unitaries in B such that ‖unσ(a)u n − γ(a)‖ for every a ∈ A. If all un can be chosen to be in U0(B), the connected component of 1B of the unitary group U(B), then we say that σ and γ are strongly approximately unitarily equivalent, written σ ≈su γ. (ii) We say that σ and γ are asymptotically unitarily equivalent, σ ≈uh γ, if there is a norm-continuous path (ut)t∈[0,∞) of unitaries in B such that ‖utσ(a)u t − γ(a)‖ for every a ∈ A. If one can arrange that u0 = 1B and hence (ut ∈ U0(B) for all t), then we say that σ and γ are strongly asymptotically unitarily equivalent, written σ ≈suh γ. 1.2 The concept of strongly self-absorbing C∗-algebras was formally introduced in [14, Definition 1.3]: Definition: A separable unital C∗-algebra D is strongly self-absorbing, if D 6= C and there is an isomorphism ϕ : D → D ⊗D such that ϕ ≈u idD ⊗ 1D. 1.3 Recall [14, Corollary 1.12]: Proposition: Let A and D be unital C∗-algebras, with D strongly self-absorbing. Then, any two unital ∗-homomorphisms σ, γ : D → A⊗D are approximately unitarily equivalent. In particular, any two unital endomorphisms of D are approximately unitarily equivalent. We note that the assumption that A is separable which appears in the original statement of [14, Corollary 1.12] is not necessary and was not used in the proof. 1.4 Lemma: Let D be a strongly self-absorbing C∗-algebra. Then there is a sequence of unitaries (wn)n∈N in the commutator subgroup of U(D ⊗ D) such that for all d ∈ D ‖wn(d⊗ 1D)w n − 1D ⊗ d‖ → 0 as n→ ∞. Proof: Let F ⊂ D be a finite normalized set and let ε > 0. By [14, Prop. 1.5] there is a unitary u ∈ U(D⊗D) such that ‖u(d⊗1D)u ∗−1D⊗d‖ < ε for all d ∈ F . Let θ : D⊗D → D be a ∗-isomorphism. Then ‖(θ(u∗) ⊗ 1D)u(d ⊗ 1D)u ∗(θ(u) ⊗ 1D) − 1D ⊗ d‖ < ε for all d ∈ F . By Proposition 1.3 θ ⊗ 1D ≈u idD⊗D and so there is a unitary v ∈ U(D ⊗ D) such that ‖θ(u∗) ⊗ 1D − vu ∗v∗‖ < ε and hence ‖(θ(u∗) ⊗ 1D)u − vu ∗v∗u‖ < ε. Setting w = vu∗v∗u we deduce that ‖w(d ⊗ 1D)w ∗ − 1D ⊗ d‖ < 3ε for all d ∈ F . 1.5 Remark: In the situation of Proposition 1.3, suppose that the commutator subgroup of U(D) is contained in U0(D). This will happen for instance if D is assumed to be K1- injective. Then one may choose the unitaries (un)n∈N which implement the approximate 4 MARIUS DADARLAT AND WILHELM WINTER unitary equivalence between σ and γ to lie in U0(A⊗D). This follows from [14, (the proof of) Corollary 1.12], since the unitaries (un)n∈N are essentially images of the unitaries (wn)n∈N of Lemma 1.4 under suitable unital ∗-homomorphisms. 2. Asymptotic vs. approximate unitary equivalence It is the aim of this section to establish a continuous version of Proposition 1.3. 2.1 Lemma: Let D be separable unital strongly self-absorbing C∗-algebra. For any finite subset F ⊂ D and ε > 0, there are a finite subset G ⊂ D and δ > 0 such that the following holds: If A is another unital C∗-algebra and σ : D → A⊗D is a unital ∗-homomorphism, and if w ∈ U0(A⊗D) is a unitary satisfying ‖[w, σ(d)]‖ < δ for all d ∈ G, then there is a continuous path (wt)t∈[0,1] of unitaries in U0(A ⊗ D) such that w0 = w, w1 = 1A⊗D and ‖[wt, σ(d)]‖ < ε for all d ∈ F , t ∈ [0, 1]. Proof: We may clearly assume that the elements of F are normalized and that ε < 1. Let u ∈ D ⊗D be a unitary satisfying (1) ‖u(d ⊗ 1D)u ∗ − 1D ⊗ d‖ < for all d ∈ F . There exist k ∈ N and elements s1, . . . , sk, t1, . . . , tk ∈ D of norm at most one such that (2) ‖u− si ⊗ ti‖ < (3) δ := k · 10 (4) G := {s1, . . . , sk} ⊂ D. Now let w ∈ U0(A⊗D) be a unitary as in the assertion of the lemma, i.e., w satisfies (5) ‖[w, σ(si)]‖ < δ for all i = 1, . . . , k. We proceed to construct the path (wt)t∈[0,1]. By [14, Remark 2.7] there is a unital ∗-homomorphism ϕ : A⊗D ⊗D → A⊗D ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 5 such that (6) ‖ϕ(a⊗ 1D)− a‖ < for all a ∈ σ(F) ∪ {w}. Since w ∈ U0(A⊗D), there is a path (w̄t)t∈[ 1 ,1] of unitaries in A⊗D such that (7) w̄ 1 = w and w̄1 = 1A⊗D. For t ∈ [1 , 1] define (8) wt := ϕ((σ ⊗ idD)(u) ∗(w̄t ⊗ 1D)(σ ⊗ idD)(u)) ∈ U(A⊗D); then (wt)t∈[ 1 ,1] is a continuous path of unitaries in A ⊗ D. For t ∈ [ , 1] and d ∈ F we ‖[wt, σ(d)]‖ = ‖wtσ(d)w t − σ(d)‖ < ‖wtϕ(σ(d) ⊗ 1D)w t − ϕ(σ(d) ⊗ 1D)‖+ 2 · ≤ ‖((σ ⊗ idD)(u)) ∗(w̄t ⊗ 1D)((σ ⊗ idD)(u(d ⊗ 1D)u ∗))(w̄∗t ⊗ 1D) ·((σ ⊗ idD)(u)) − ((σ ⊗ idD)(d⊗ 1D))‖ + < ‖((σ ⊗ idD)(u)) ∗(w̄t ⊗ 1D)((σ ⊗ idD)(1D ⊗ d))(w̄ t ⊗ 1D) ·((σ ⊗ idD)(u)) − ((σ ⊗ idD)(d⊗ 1D))‖ + = ‖(σ ⊗ idD)(u ∗(1D ⊗ d)u− d⊗ 1D)‖+ 6 MARIUS DADARLAT AND WILHELM WINTER where for the last equality we have used that the w̄t are unitaries and that σ is a unital ∗-homomorphism. Furthermore, we have (7),(8) = ‖ϕ(((σ ⊗ idD)(u)) ∗(w ⊗ 1D)((σ ⊗ idD)(u))) − w‖ < ‖ϕ(((σ ⊗ idD)(u)) ∗(w ⊗ 1D)( σ(si)⊗ ti))− w‖+ ≤ ‖ϕ(((σ ⊗ idD)(u)) σ(si)⊗ ti)(w ⊗ 1D))− w‖ ‖[w, σ(si)]‖ · ‖ti‖+ (5),(4),(2) < ‖ϕ(w ⊗ 1D)− w‖+ k · δ + 2 · (6),(3) + 2 · The above estimate allows us to extend the path (wt)t∈[ 1 ,1] to the whole interval [0, 1] in the desired way: We have ‖w 1 w∗ − 1D‖ < < 2, whence −1 is not in the spectrum of w 1 w∗. By functional calculus, there is a = a∗ ∈ A ⊗ D with ‖a‖ < 1 such that w∗ = exp(πia). For t ∈ [0, 1 ) we may therefore define a continuous path of unitaries wt := (exp(2πita))w ∈ U(A⊗D). It is clear that w0 = w and wt → w 1 as t→ (1 )−, whence (wt)t∈[0,1] is a continuous path of unitaries in A satisfying w0 = w and w1 = 1A ⊗D. Moreover, it is easy to see that ‖wt − w‖ ≤ ‖w 1 − w‖ < for all t ∈ [0, 1 ), whence ‖[wt, σ(d)]‖ < ‖[w 1 , σ(d)]‖ + for t ∈ [0, 1 ), d ∈ F . We have now constructed a path (wt)t∈[0,1] ⊂ U(A) with the desired properties. 2.2 Theorem: Let A and D be unital C∗-algebras, with D separable, strongly self- absorbing and K1-injective. Then, any two unital ∗-homomorphisms σ, γ : D → A⊗D are strongly asymptotically unitarily equivalent. In particular, any two unital endomorphisms of D are strongly asymptotically unitarily equivalent. ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 7 Proof: Note that the second statement follows from the first one with A = D, since D ∼= D ⊗D by assumption. Let A be a unital C∗-algebra such that A ∼= A ⊗ D and let σ, γ : D → A be unital ∗-homomorphisms. We shall prove that σ and γ are strongly asymptotically unitarily equivalent. Choose an increasing sequence F0 ⊂ F1 ⊂ . . . of finite subsets of D such that Fn is a dense subset of D. Let 1 > ε0 > ε1 > . . . be a decreasing sequence of strictly positive numbers converging to 0. For each n ∈ N, employ Lemma 2.1 (with Fn and εn in place of F and ε) to obtain a finite subset Gn ⊂ D and δn > 0. We may clearly assume that (10) Fn ⊂ Gn ⊂ Gn+1 and that δn+1 < δn < εn for all n ∈ N. Since σ and γ are strongly approximately unitarily equivalent by Proposition 1.3 and Remark 1.5, there is a sequence of unitaries (un)n∈N ⊂ U0(A) such that (11) ‖unσ(d)u n − γ(d)‖ < for all d ∈ Gn and n ∈ N. Let us set wn := u n+1un, n ∈ N. Then wn ∈ U0(A) and ‖[wn, σ(d)]‖ = ‖wnσ(d)w n − σ(d)‖ ≤ ‖u∗n+1unσ(d)u nun+1 − u n+1γ(d)un+1‖ +‖u∗n+1γ(d)un+1 − σ(d)‖ for d ∈ Gn, n ∈ N. Now by Lemma 2.1 (and the choice of the Gn and δn), for each n there is a continuous path (wn,t)t∈[0,1] of unitaries in U0(A) such that wn,0 = wn, wn,1 = 1A and (12) ‖[wn,t, σ(d)]‖ < εn for all d ∈ Fn, t ∈ [0, 1]. Next, define a path (ūt)t∈[0,∞) of unitaries in U0(A) by ūt := un+1wn,t−n if t ∈ [n, n+ 1). 8 MARIUS DADARLAT AND WILHELM WINTER We have that (13) ūn = un+1wn = un and that ūt → un+1 as t → n + 1 from below, which implies that the path (ūt)t∈[0,∞) is continuous in U0(A). Furthermore, for t ∈ [n, n+ 1) and d ∈ Fn we obtain ‖ūtσ(d)ū t − γ(d)‖ = ‖un+1wn,t−nσ(d)w n,t−nu n+1 − γ(d)‖ < ‖un+1σ(d)u n+1 − γ(d)‖ + εn (11),(10) < 2εn. Since the Fn are nested and the εn converge to 0, we have (14) ‖ūtσ(d)ū t − γ(d)‖ for all d ∈ n=0Fn; by continuity and since n=0Fn is dense in D, we have (14) for all d ∈ D. Since ū0 ∈ U0(A) we may arrange that ū0 = 1A. 3. The group KK(D, A⊗D) and some applications 3.1 For a separable C∗-algebra D we endow the group of automorphisms Aut (D) with the point-norm topology. Corollary: Let D be a separable, unital, strongly self-absorbing and K1-injective C algebra. Then [X,Aut(D)] reduces to a point for any compact Hausdorff space X. Proof: Let ϕ,ψ : X → Aut (D) be continuous maps. We identify ϕ and ψ with unital ∗-homomorphisms ϕ,ψ : D → C(X) ⊗ D. By Theorem 2.2, ϕ is strongly asymptotically unitarily equivalent to ψ. This gives a homotopy between the two maps ϕ,ψ : X → Aut (D). 3.2 Remark: The conclusion of Corollary 3.1 was known before for D a UHF algebra of infinite type and X a CW complex by [13], for D = O2 by [8] and [10], and for D = O∞ by [2]. It is new for the Jiang–Su algebra. 3.3 For unital C∗-algebras D and B we denote by [D, B] the set of homotopy classes of unital ∗-homomorphisms from D to B. By a similar argument as above we also have the following corollary. ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 9 Corollary: Let D and A be unital C∗-algebras. If D is separable, strongly self-absorbing and K1-injective, then [D, A⊗D] reduces to a singleton. 3.4 For separable unital C∗-algebras D and B, let χi : KKi(D, B) → KKi(C, B) ∼= Ki(B), i = 0, 1 be the morphism of groups induced by the unital inclusion ν : C → D. Theorem: Let D be a unital, separable and strongly self-absorbing C∗-algebra. Then for any separable C∗-algebra A, the map χi : KKi(D, A ⊗ D) → Ki(A ⊗ D) is bijective, for i = 0, 1. In particular both groups KKi(D, A⊗D) are countable and discrete with respect to their natural topology. Proof: Since D is KK-equivalent to D ⊗ O∞, we may assume that D is purely infinite and in particular K1-injective by [11, Prop. 4.1.4]. Let CνD denote the mapping cone C algebra of ν. By [3, Cor. 3.10], there is a bijection [D, A⊗ D] → KK(CνD, SA⊗ D) and hence KK(CνD, SA⊗D) = 0 for all separable and unital C ∗-algebras A as a consequence of Corollary 3.3. Since KK(CνD, A ⊗ D) is isomorphic to KK(CνD, S 2A ⊗ D) by Bott periodicity and the latter group injects in KK(CνD, SC(T) ⊗ A ⊗ D) = 0, we have that KKi(CνD,D ⊗ A) = 0 for all unital and separable C ∗-algebras A and i = 0, 1. Since KKi(CνD,D ⊗A) is a subgroup of KKi(CνD,D ⊗ Ã) = 0 (where à is the unitization of A) we see that KKi(CνD,D ⊗ A) = 0 for all separable C ∗-algebras A. Using the Puppe exact sequence, where χi = ν KKi+1(CνD, A⊗D) // KKi(D, A⊗D) // KKi(C, A⊗D) // KKi(CνD, A⊗D) we conclude that χi is an isomorphism, i = 0, 1. The map χi = ν ∗ is continuous since it is given by the Kasparov product with a fixed element (we refer the reader to [12], [9] or [1] for a background on the topology of the Kasparov groups). Since the topology of Ki is discrete and χi is injective, it follows that the topology of KKi(D, A⊗D) is also discrete. The countability of KKi(D, A⊗D) follows from that of Ki(A⊗D), as A⊗D is separable. 3.5 Remark: In contrast to Theorem 3.4, if D is the universal UHF algebra, then KK(D,C) ∼= Ext(Q,Z) ∼= QN has the power of the continuum [6, p. 221]. 3.6 Let D and A be as in Theorem 3.4 and assume in addition that D is K1-injective and A is unital. Let ι : D → A⊗D be defined by ι(d) = 1A ⊗ d. Corollary: If e ∈ K ⊗ A⊗ D is a projection, and ϕ,ψ : D → e(K ⊗ A ⊗ D)e are two unital ∗-homomorphisms, then ϕ ≈suh ψ and hence [ϕ] = [ψ] ∈ KK(D, A⊗D). Moreover: KK(D, A⊗D) = {[ϕ]− n[ι] |ϕ : D → K⊗A⊗D is a ∗-homomorphism, n ∈ N}. 10 MARIUS DADARLAT AND WILHELM WINTER Proof: Let ϕ, ψ and e be as in the first part of the statement. By [14, Cor. 3.1], the unital C∗-algebra e(K⊗A⊗D)e is D-stable, being a hereditary subalgebra of a D-stable C∗-algebra. Therefore ϕ ≈suh ψ by Theorem 2.2. Now for the second part of the statement, let x ∈ KK(D, A ⊗ D) be an arbitrary element. Then χ0(x) = [e]−n[1A⊗D] for some projection e ∈ K⊗A⊗D and n ∈ N. Since e(K ⊗ A ⊗ D)e is D-stable, there is a unital ∗-homomorphism ϕ : D → e(K ⊗ A ⊗ D)e. χ0([ϕ] − n[ι]) = [ϕ(1D)]− n[ι(1D)] = [e]− n[1A⊗D] = χ0(x), and hence [ϕ]− n[ι] = x since χ0 is injective by Theorem 3.4. In the remainder of the paper we give characterizations for the Cuntz algebra O2 and for the universal UHF-algebra which do not require the UCT. The latter result is a variation of a theorem of Effros and Rosenberg [5]. 3.7 Proposition: Let D be a separable unital strongly self-absorbing C∗-algebra. If [1D] = 0 in K0(D), then D ∼= O2. Proof: Since D must be nuclear (see [14]), D embeds unitally in O2 by Kirchberg’s theorem. D is not stably finite since [1D] = 0. By the dichotomy of [14, Thm. 1.7] D must be purely infinite. Since [1D] = 0 in K0(D), there is a unital embedding O2 → D, see [11, Prop. 4.2.3]. We conclude that D is isomorphic to O2 by [14, Prop. 5.12]. 3.8 Proposition: Let D, A be separable, unital, strongly self-absorbing C∗-algebras. Suppose that for any finite subset F of D and any ε > 0 there is a u.c.p. map ϕ : D → A such that ‖ϕ(cd) − ϕ(c)ϕ(d)‖ < ε for all c, d ∈ F . Then A ∼= A⊗D. Proof: By [14, Thm. 2.2] it suffices to show that for any given finite subsets F of D, G of A and any ε > 0 there is u.c.p. map Φ : D → A such that (i) ‖Φ(cd)− Φ(c)Φ(d)‖ < ε for all c, d ∈ F and (ii) ‖[Φ(d), a]‖ < ε for all d ∈ F and a ∈ G. We may assume that ‖d‖ ≤ 1 for all d ∈ F . Since A is strongly self-absorbing, by [14, Prop. 1.10] there is a unital ∗- homomorphism γ : A⊗A→ A such that ‖γ(a⊗1A)−a‖ < ε/2 for all a ∈ G. On the other hand, by assumption there is a u.c.p. map ϕ : D → A such that ‖ϕ(cd) − ϕ(c)ϕ(d)‖ < ε for all c, d ∈ F . Let us define a u.c.p. map Φ : D → A by Φ(d) = γ(1A ⊗ ϕ(d)). It is clear that Φ satisfies (i) since γ is a ∗-homomorphism. To conclude the proof we check now that Φ also satisfies (ii). Let d ∈ F and a ∈ G. Then ‖[Φ(d), a]‖ ≤ ‖[Φ(d), a − γ(a⊗ 1A)]‖+ ‖[Φ(d), γ(a ⊗ 1A)]‖ ≤ 2‖Φ(d)‖‖a − γ(a⊗ 1A)‖+ ‖[γ(1A ⊗ ϕ(d)), γ(a ⊗ 1A)]‖ < 2ε/2 + 0 = ε. ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 11 3.9 Proposition: Let D be a separable, unital, strongly self-absorbing C∗-algebra. Sup- pose that D is quasidiagonal, it has cancellation of projections and that [1D] ∈ nK0(D) for all n ≥ 1. Then D is isomorphic to the universal UHF algebra Q with K0(Q) ∼= Q. Proof: Since D is separable unital and quasidiagonal, there is a unital ∗-representation π : D → B(H) on a separable Hilbert space H and a sequence of nonzero projections pn ∈ B(H) of finite rank k(n) such that limn→∞ ‖[pn, π(d)]‖ = 0 for all d ∈ D. Then the sequence of u.c.p. maps ϕn : D → pnB(H)pn ∼= Mk(n)(C) ⊂ Q is asymptotically multiplicative, i.e limn→∞ ‖ϕn(cd) − ϕn(c)ϕn(d))‖ = 0 for all c, d ∈ D. Therefore Q ∼= Q⊗D by Proposition 3.8. In the second part of the proof we show that D ∼= D ⊗Q. Let En : Q → Mn!(C) ⊂ Q be a conditional expectation onto Mn!(C). Then limn→∞ ‖En(a)− a‖ = 0 for all a ∈ Q. By assumption, for each n there is a projection e in D ⊗Mm(C) (for some m) such that n![e] = [1D] in K0(D). Let ϕ : Mn!(C) → Mn!(C) ⊗ e(D ⊗ Mm(C))e be defined by ϕ(b) = b ⊗ e. Since D has cancellation of projections and since n![e] = [1D], there is a partial isometry v ∈ Mn!(C) ⊗ D ⊗Mm(C) such that v ∗v = 1Mn!(C) ⊗ e and vv e11⊗1D⊗e11. Therefore b 7→ v ϕ(b) v ∗ gives a unital embedding ofMn!(C) into D. Finally, ψn(a) = v (ϕ ◦ En(a)) v ∗ defines a sequence of asymptotically multiplicative u.c.p. maps Q → D. Therefore D ∼= D ⊗Q by Proposition 3.8. 3.10 Remark: Let D be a separable, unital, strongly self-absorbing and quasidiagonal C∗- algebra. Then D ⊗Q ∼= Q by the first part of the proof of Proposition 3.9. In particular K1(D) ⊗ Q = 0 and K0(D) ⊗ Q ∼= Q by the Künneth formula (or by writing Q as an inductive limit of matrices). References [1] M. Dadarlat. On the topology of the Kasparov groups and its applications., J. Funct. Anal. 228 (2005), 394–418. [2] M. Dadarlat. Continuous fields of C∗-algebras over finite dimensional spaces , arXiv preprint math.OA/0611405 (2006). [3] M. Dadarlat. The homotopy groups of the automorphism group of Kirchberg algebras, J. Noncomm. Geom. 1 (2007), 113–139. [4] M. Dadarlat and W. Winter. Trivialization of C(X)-algebras with strongly self-absorbing fibres, preprint (2007). [5] E. G. Effros and J. Rosenberg. C∗-algebras with approximately inner flip, Pacific J. Math. 77 (1978), 417–443. [6] L. Fuchs. Infinite abelian groups, vol. 1, Academic Press, New York and London, 1970. [7] I. Hirshberg, M. Rørdam and W. Winter. C0(X)-algebras, stability and strongly self-absorbing C algebras, arXiv preprint math.OA/0610344 (2006). To appear in Math. Ann. [8] E. Kirchberg. The classification of purely infinite C∗-algebras using Kasparov’s theory, preprint (1994). http://arxiv.org/abs/math/0611405 http://arxiv.org/abs/math/0610344 12 MARIUS DADARLAT AND WILHELM WINTER [9] M. V. Pimsner. A topology on the Kasparov groups, draft. [10] N. C. Phillips. A classification theorem for nuclear purely infinite simple C∗-algebras, Documenta Math. 5 (2000), 49–114. [11] M. Rørdam. Classification of Nuclear C∗-Algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002. [12] C. Schochet. The fine structure of the Kasparov groups I. Continuity of the KK-pairing, J. Funct. Anal. 186 (2001), 25–61. [13] K. Thomsen. The homotopy type of the group of automorphisms of a UHF-algebra, J. Funct. Anal. 72 (1987), 182–207. [14] A. Toms andW.Winter. Strongly self-absorbing C∗-algebras, arXiv preprint math.OA/0502211 (2005). To appear in Trans. Amer. Math. Soc. [15] A. Toms and W. Winter. Z-stable ASH algebras, arXiv preprint math.OA/0508218 (2005). To appear in Can. J. Math. [16] W. Winter. On the classification of simple Z-stable C∗-algebras with real rank zero and finite decom- position rank, J. London Math. Soc. 74 (2006), 167–183. [17] W. Winter. Simple C∗-algebras with locally finite decomposition rank, J. Funct. Anal. 243 (2007), 394–425. [18] W. Winter. Localizing the Elliott conjecture, in preparation. Department of Mathematics, Purdue University, West Lafayette,, IN 47907, USA E-mail address: [email protected] Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany E-mail address: [email protected] http://arxiv.org/abs/math/0502211 http://arxiv.org/abs/math/0508218 0. Introduction 1. Strongly self-absorbing C*-algebras 2. Asymptotic vs. approximate unitary equivalence 3. The group KK(D,AD) and some applications References

0704.0584

Effective interactions from q-deformed inspired transformations

Effective interactions from q-deformed inspired transformations V. S. Timóteo a C. L. Lima b aCentro Superior de Educação Tecnológica, Universidade Estadual de Campinas, 13484-370, Limeira, SP, Brasil bInstituto de F́ısica, Universidade de São Paulo, CP 66318, 05315-970, São Paulo, SP, Brazil Abstract From the mass term for the transformed quark fields, we obtain effective contact interactions of the NJL type. The parameters of the model that maps a system of non-interacting transformed fields into quarks interacting via NJL contact terms are discussed. It is very common in physics to use transformations that make one particular system mathematically simpler, yet describing the same phenomena. A clear example is the use of canonical transformations in classical mechanics. q-Deformed algebras provide a nice framework to incorporate, in an effective way, interactions not originally contained in the Lagrangian of a particular system. In hadron physics, the NJL model is a very simple effective model for strong interactions that describes important features like the dynamical mass genera- tion, spontaneous chiral symmetry breaking, and chiral symmetry restoration at finite temperature. In recent works, we have been investigating possible applications of quantum algebras in hadronic physics. In general, we observed that when we deform the underlying algebra, the system is affected with correlations between its constituents. We have studied in detail the NJL model under the influence of a quantum su(2) algebra. The question we approach in this letter is: is it possible to obtain a transfor- mation connecting the NJL model to a simpler non-interacting system? We verified that we can indeed obtain the same dynamics of the NJL interaction Preprint submitted to Elsevier 4 November 2018 http://arxiv.org/abs/0704.0584v1 with a simple transformation of the quark fields, inspired in the q-deformed quark fields of previous works [2], [3], [6]. Mass term We start by writing a mass term for the transformed quark fields Lmassq =−M ΨΨ Ψ1Ψ1 +Ψ2Ψ2 UU +DD where  . (2) The transformed quark fields can be written in terms of the standard fields as Ψ1=ψ1 + (q −1 − 1) ψ1ψ2γ0ψ2 , (3) Ψ2=ψ2 + (q −1 − 1) ψ2ψ1γ0ψ1 , (4) U = u+ (q−1 − 1) udγ0d , (5) D= d+ (q−1 − 1) duγ0u , (6)  . (7) Here both components are modified in the same way, so that the above ex- pressions are different from the ones used in [2,3], where only one component is affected. Extending the transformation to both components is required to obtain a set of terms that will form an interaction of the NJL type. This im- plies that the anti-commutation relations for the deformed fields Ψ will also be different from the ones in [2,3]. Since obtaining the new anti-commuation relations is not in the scope of this work, we focus on the effective interactions contained in the non-interacting Lagrangian. Using Eqs. (5) and (6), we can re-write the Lagragian Eq.(1) in terms of the standard quark fields UU = uu+Q uud†d+Q d†duu+Q2 dduudd , (8) DD= dd+Q ddu†u+Q u†udd+Q2 uudduu , (9) where Q = (q−1 − 1). We can re-write the above equations as follows 1 + 2Q d†d dduudd+ dduudd , (10) 1 + 2Q u†u uudduu+ uudduu , (11) so that we identify the contact interactions between the quarks contained in the non-interacting deformed fields Lagrangian. Figure 1 shows the six-point contact interactions contained in the mass term for the q-deformed quark fields. We can reduce the six-point interactions to four-point contact terms in a mean field approach [5], so that we have UU +DD 1 + 2Q uu+ dd dduu+ dddd+ uudd+ uuuu , (12) where 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 = ρv, = 〈uu〉 = = ρs, and A = A(T ; q) has the same dimension of the condensate and will be determined later in this letter. The reduction of the six-point to four-point contact terms by closing one fermion line is also shown in Figure 1. Now we can write the mass term for the transformed quark fields Lmassq = −MΨΨ = −M 1 + 2 ψψ − M Γ2 ψψψψ , (13) with Γ = Q/A Kinetic energy term Accordingly, the kinetic energy term for the transformed fields, Ψγµ∂µΨ, can be written in terms of the standard ones as Ψγµ∂µΨ=Uγ µ∂µU +Dγ = uγµ∂µu+Q dγ0duγ µ∂µu+ uγ µ∂µudγ0d + dγµ∂µd+Q uγ0udγ µ∂µd+ dγ µ∂µduγ0u dγ0duγ µ∂µudγ0d uγ0udγ µ∂µduγ0u By using an extreme mean field approximation, namely, substituting every- where in the kinetic energy contribution 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 → ρv, and = 〈uu〉 = → ρs, we obtain Ψγµ∂µΨ= uγ µ∂µu (1 + 2Γρv) + dγµ∂µd (1 + 2Γρv) + (uγµ∂µu) Γ 2ρv + dγµ∂µd uγµ∂µu+ dγ (1 + Γρv) This corresponds to a usual kinetic energy with a shifted momentum p → p (1 + Γρv) The full Lagrangian The treatment of the density dependence of the kinetic energy term is rather cumbersome and will be postponed to a further contribution. We will consider the influence of this momentum dependent kinetic energy term in an effective way. Therefore, we will study a class of Lagrangians of the type L′q = (1 + Γρv) Lq = ψγµ∂µψ −M 1 + 2 (1 + Γρv) (1 + Γρv) ψψψψ (16) This representative of the full Lagrangian Lq = Ψγµ∂µΨ + Lmassq , when writ- ten in terms of the standard quark fields, can be identified with the NJL Lagrangian LNJL = ψγµ∂µψ −m0 ψψ +G ψψψψ . (17) The conditions for both Lagrangians, LNJL and L′q, to be equivalent for any values of T and q are (1 + Γρv) (1 + 2Γρv) m0 , (18) G = −M (1 + Γρv) . (19) Inserting Eq. (18) in Eq. (19), we obtain an equation for Γ Γ2 − 2αρv Γ− α = 0 , (20) where α = − 2G > 0. (21) This equation has two solutions Γ± = αρv . (22) The mass of the transformed fermion fields, M , has to be positive, so we associate the two solutions Γ− and Γ+ with the two regimes q < 1 and q > 1, respectively. The quantity A will be negative in both cases. The scalar (ρs) and vector (ρv) densities were calculated from the NJL model at finite temperature: [1− n− n] , (23) dpp2 [n− n] , (24) where n(p, T, µ) = 1 + exp [β (E − µ)] , (25) n(p, T, µ) = 1 + exp [β (E + µ)] , (26) are the fermions and anti-fermions distribution functions respectively with p2 +m2. First we solve the set of coupled gap equations for m, µ, and ρv (Eqs. 27 and 24, respectively) in the NJL model at finite temperature and chemical potential m = m0 − 2Gρs , µ = µ0 − GNcρv . The next step is to calculate the scalar and vector densities entering in the equation for Γ for a given value of the transformation parameter q. In this way we obtain A(T ; q), which in turn is used to obtain M . The numerical results are displayed in Figures 2 and 3, where we show the quantity A and the mass M as a function of both temperature and transformation parameter in the q > 1 and q < 1 regimes. It is worth to note that the mass of the transformed fermion fields does not depend on the transformation parameter. The well known results of the NJL model are mapped through A(T ; q) from the non-interacting transformed fermion fields Lagrangian. It is worth to note that the mass of the q-deformed fermion fields does not depend on the deformation of the algebra. The quantity A (T ; q) maps the simple non-interacting model into the NJL model. It represents, in an effective way, the correlations introduced by the transformations, when we write the non-interacting Lagrangian in terms of the standard quark fields. These correlations, in a mean field approximation, are effectively represented by contact interactions of the NJL type. It is also important to mention that it inherits the phase transition. When the con- densate and the dynamical mass vanishes with increasing T , the quantity A also experiences the phase transition. This is an expected behavior, since it depends on the dynamical mass. For a given temperature, T , and transfor- mation parameter, q, there is a value of the mapping function, A(T ; q), that makes the Lagrangians Eq.(16) and Eq.(17) equivalent. Summarizing, we have shown that it is possible to describe the dynamics of an interacting system of the NJL type with a simple non-interacting system by using a set of quantum algebra inspired transformations and a mapping function. Acknowledgments C. L. L. thanks Profs. D. Galetti and B. M. Pimentel for most helpful discus- sions. This work was partially supported by FAPESP Grant No. 2002/10896-7. V.S.T. would like to thank FAEPEX/UNICAMP for financial support. References [1] D. Galetti and B. M. Pimentel, An. Acad. Bras. Ci. 67 (1995) 7; S. S. Avancini, A. Eiras, D. Galetti, B. M. Pimentel, and C. L. Lima, J. Phys. A: Math. Gen. 28 (1995) 4915; D. Galetti, J. T. Lunardi, B. M. Pimentel, and C. L. Lima, Physica A242 (1997) 501. [2] M. Ubriaco, Phys. Lett. A 219 (1996) 205. [3] L. Tripodi and C. L. Lima, Phys. Lett. B 412 (1997) 7. [4] Y. Nambu and G. Jona-Lasinio, Physical Review 122 (1961) 345. [5] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195. [6] V. S. Timóteo and C. L. Lima, Phys. Lett. B 448 (1999) 1. [7] V. S. Timóteo and C. L. Lima, Mod. Phys. Lett. A 15 (2000) 219. [8] V. S. Timóteo and C. L. Lima, nucl-th/0509089. http://arxiv.org/abs/nucl-th/0509089 d dd d Gq Gq Gq Gq u u d d d d u u 〈uu〉〈dd〉 〈dd〉〈uu〉 Fig. 1. Contact interactions generated by the mass term for the q-deformed fermion fields and their reduction from six-point to four-point by closing one fernion line. 0.2 1 T (GeV) 0.2 0.40.5 qT (GeV) Fig. 2. The quantity A, in units of the chiral condensate at zero temperature ρs(T = 0) = −1.42 × 10−2 GeV3, as a function of temperature and q-deformation for the q > 1 and q < 1 regimes. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 T (GeV) Fig. 3. The mass of the q-deformed quark fields, in units of the current quark mass m0 = 5 MeV, as a function of T for both q > 1 and q < 1 regimes. For small temperatures, M = m0. References

0704.0585

Magnetospectroscopy of epitaxial few-layer graphene

Magnetospectroscopy of epitaxial few-layer graphene M.L. Sadowski a G. Martinez a, M. Potemski a, C. Berger b,c, W.A. de Heer b aGrenoble High Magnetic Field Laboratory, Grenoble,France bGeorgia Institute of Technology, Atlanta, Georgia, USA cInstitut Néel, CNRS, Grenoble, France Abstract The inter-Landau level transitions observed in far-infrared transmission experiments on few-layer graphene samples show a behaviour characteristic of the linear disper- sion expected in graphene. This behaviour persists in relatively thick samples, and is qualitatively different from that of thin samples of bulk graphite. Key words: Graphene, Cyclotron resonance, PACS: 71.70.Di, 76.40.+b, 78.30.-j, 78.67.-n The interest in two-dimensional graphite is fuelled by its particular band struc- ture and ensuing dispersion relation for electrons, leading to numerous differ- ences with respect to “conventional” two-dimensional electron gases (2DEG). Single graphite layers (graphene) have long been used as a starting point in band structure calculations of bulk graphite [1,2,3] and, more recently, carbon nanotubes [4]. The band structure of a single graphene sheet is considered to be composed of cones located at two inequivalent Brillouin zone corners at which the conduction and valence bands merge. In the vicinity of these points the electron energy depends linearly on its momentum, which implies that free charge carriers in graphene are governed not by Schrödinger’s equation, but rather by Dirac’s relativistic equation for zero rest mass particles, with an effective velocity c̃, which replaces the speed of light [5,6]. The recent appearance of ultrathin graphite layers (few-layer graphene, FLG), obtained by epitaxial [7,8,9] and exfoliation techniques [10], followed by sin- gle graphene and its unusual sequence of quantum Hall states [11,12] has re-ignited this interest. The prospects of studying quantum electrodynamics Preprint submitted to Solid State Communications 4 November 2018 http://arxiv.org/abs/0704.0585v1 100 200 300 400 500 600 700 Energy (cm-1) 0.4 T 1.9 K Fig. 1. Transmission spectrum of epitaxial graphene at 0.4 T. The inset shows a schematic of the assignations of the observed transitions. in solid state experiments on the one hand and the possibility of future ap- plications in carbon-based electronics on the other are currently driving a considerable research effort. The majority of the published literature remains theoretical; the extremely small lateral dimensions (≈ 10µm) of the graphene flakes used in the above-mentioned transport experiments makes them difficult objects for experimental studies. Moreover, due to the somewhat hit-and-miss character of the exfoliation method, as well as the inherent difficulty of ob- taining large numbers of samples, it appears to be an unlikely candidate for possible applications. Epitaxial methods on the other hand offer the opportu- nity of obtaining relatively large, high quality two-dimensional graphite [13]. In the following, we present optical measurements of the characteristic disper- sion relation of FLG, confirming directly its linear (“relativistic”) character. A number of epitaxial graphene samples have been studied by means of far- infrared magnetotransmission measurements. The samples were about 4 × 4 mm2 in area, grown by sublimating SiC substrates at high temperatures [9,13]. The experimental details and part of the results have been described elsewhere [14]. A representative transmission spectrum of a three-graphene-layer sample is shown in Fig. 1 for a weak magnetic field of 0.4 T. When the magnetic field is increased, all the features visible in this figure are displaced towards higher energies. Furthermore, their strength increases [14] and more features become visible at higher energies. The positions of the features observed for the sample containing 3 graphene layers are plotted versus the square root of the magnetic field in Fig. 2. It may be seen that the resonant energies observed evolve proportionally to the square root of the magnetic field. The oscillator strength Line Slope in units of c̃ 2e~ Transition 1 L1 → L2 B 1 L0 → L1(L−1 → L0) −1 → L2(L−2 → L1) −2 → L3(L−3 → L2) −3 → L4(L−4 → L3) −4 → L5(L−5 → L4) −5 → L6(L−6 → L5) −6 → L7(L−7 → L6) −7 → L8(L−8 → L7) Table 1 Observed lines and their assignments of the transition labelled B in Fig 1 has also been shown [14] to increase linearly with the square root of the magnetic field. These results are, in a first approximation, in excellent agreement with predic- tions arising from a simple single-particle model of non-interacting massless Dirac fermions. Using appropriate graphene wavefunctions [4] and the Hamiltonian commonly used to describe electrons in a single graphene layer, it is fairly straightforward to work out the optical selection rules [15]. It may then be shown that the allowed transitions are Ln → Lm such that |m| = |n| − 1 for the “+” circular polarisation and |m| = |n|+1 in the “-” circular polarisation. For unpolarised radiation, used in the current experiment, the allowed transitions are simply those between states n,m such that |m| = |n| ± 1. The Landau level energies are obtained as En = c̃ 2~eB|n| (1) where c̃ is the effective velocity of the Dirac fermions, B is the magnetic field and n = 0,±1,±2 ... is the Landau level index (the electron and hole levels being identical). The energies of the allowed optical transitions may then be concisely written as 2~eB( |n+ 1| ± |n|) (2) The positions of the transitions shown in Fig. 2 are summarised in Table 1. It should be stressed that all the positions of all the observed lines are described 0 1 2 3 4 4000 I H G F E )( 2/1TB Fig. 2. Evolution with magnetic field of transitions observed in transmission. The letters correspond to those used in Fig. 1; the shaded region corresponds to the range where the substrate is opaque. by a single fitting parameter - the effective light velocity c̃. We should add, for the sake of completeness, that the present experiment, using unpolarised light, does not distinguish between electrons and holes, which are expected to be identical in terms of the effective mass and dispersion relation. Thus, transition A, attributed to the L1 → L2 process, could also be due to the corresponding −2 → L−1 one. While a p-type character appears to be unlikely, it cannot be ruled out on the basis of the experiment in question. The striking agreement of the experimental data obtained using several lay- ers of graphene with expectations for a single layer is surprising, given that calculations suggest a completely different behaviour already for a graphene bilayer [17]. On the other hand, it has long been known that particles with a linear dispersion exist in bulk graphite as well - a minority pocket of carriers in the vicinity of the H point of the Brillouin zone were shown to give rise to electronic transitions following a square root dependence on the magnetic field [18]. The question therefore is posed: at what point, if at all, does epitaxial FLG become bulk graphite? Early work on epitaxial graphene [7] suggested that the process of baking SiC substrates led to a single graphene layer floating above a graphite layer. More recent calculations [22] suggest that the first carbon layer on top of an SiC substrate has an electronic structure different from that of graphene, and acts as a buffer, allowing subsequent layers to behave like graphene. A strong dependence of the electronic structure of FLG on the type of stacking has also been suggested [23]. The common Bernal, or AB, stacking found for example in HOPG graphite is usually assumed for all FLG structures as well; this is 100 200 300 400 500 600 700 60 layers 9 layers Energy (cm-1) 3 layers Fig. 3. Transmission spectra at 4T for epitaxial FLG samples (top three) of varying thickness and, for comparison, of HOPG graphite at the same magnetic field. not necessarily the case. Also, let us note that the HOPG interlayer distance of 3.354 Å may not be the correct value for epitaxial graphene. In order to elucidate the effect of multiplying layers on the transmission spec- trum, samples of varying thickness were studied and compared with a layer of HOPG obtained by exfoliation. The details of this study shall be presented elsewhere [19]; for the time being let us note the qualitative differences in the spectra, shown in Fig. 3. Four spectra are shown, at a magnetic field of 4T: for sample consisting of 3, 9 and 60 layers of graphene on SiC, and for the HOPG sample. The dominant feature in the epitaxial samples is always the L0 → L1 −1 → L0) transition; we can see that it grows stronger as the number of layers is increased, and is several times stronger for the sample containing 60 layers. In this sample one can also see the appearance of other features at lower energies, which were absent in the thinner samples, and which appear to correspond to bulk-like features visible in the lowest (HOPG) trace in the figure. On the other hand, the L0 → L1 (L−1 → L0) transition, which has a square root dispersion even in the 60 layer sample, is absent from the HOPG spectrum. The observed persistence of the Dirac fermion-like behaviour of the carriers in epitaxial FLG up to relatively thick ( 19 nm) structures appears to suggest that the structure of this material is in fact different from that of bulk HOPG. The simplest explanation would be a far weaker interaction between adjacent graphene layers, leading to a sequence of graphene layers instead of bulk, or even multilayer, graphene. More studies are necessary to elucidate this question. The GHMFL is a “Laboratoire conventionné avec l’UJF et l’INPG de Greno- ble”. The present work was supported in part by the European Commission through grant RITA-CT-2003-505474 and by grants from the Intel Research Corporation and the NSF: NIRT “Electronic Devices from Nano-Patterned Epitaxial Graphite”. References [1] P.R. Wallace, Phys. Rev. 71, 622 (1947) [2] J.W. McClure, Phys. Rev. 104, 666 (1956) [3] J.C. Slonczewski and P.R. Weiss, Phys. Rev.109, 272 (1958) [4] T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005) [5] F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988) [6] Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 (2002) [7] I. Forbeaux, J.-M. Themlin, and J.-M. Debever, Phys. Rev. B 58, 16396 (1998) [8] A.Charrier, A. Coati, T. Argunova, F. Thibaudau, Y. Garreau, R. Pinchaux, I. Forbeaux, J.-M. Debever, M. Sauvage-Simkin, J.-M. Themlin, J. Appl. Phys. 92, 2479 (2002) [9] C. Berger, Z. Song, T. Li, X. Li, A.Y. Ogbazghi, R. Feng, Z. Dai, A.N. Marchenkov, E.H. Conrad, P.N. First, and W.A. de Heer, J. Phys. Chem. 108, 19912 (2004). [10] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, and A.A. Firsov, Science 306, 666 (2004) [11] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov, Nature 438, 197 (2005). [12] Y. Zhang, Y.-W. Tan, H.L. Stormer and P. Kim, Nature 438, 201 (2005). [13] C. Berger, Z. Song, T. Li, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, A.N. Marchenkov, E.H. Conrad, P.N. First, and W.A. de Heer, Science 312, 1191 (2006) [14] M.L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W.A. de Heer, Phys. Rev. Lett 97, 266405 (2006). [15] M.L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W.A. de Heer, Int. J. Mod. Phys. B, in press. [16] V.P. Gusynin, S.G.Sharapov, and J.P. Carbotte, J. Phys.: Condens. Matter 19, 026222 (2007) [17] D.S.L. Abergel and V.I. Fal’ko, cond-mat/0610673 [18] W.W. Toy, M.S. Dresselhaus, and G. Dresselhaus, Phys. Rev. 15, 4077 (1977) [19] M.L. Sadowski et al., to be published [20] T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, Science 313, 951 (2006) [21] B. Partoens and F.M. Peeters, Phys. Rev. B 74,075404 (2006) [22] F. Varchon, R. Feng, J. Hass, X. Li, B.N. Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, E.H. Conrad, and L. Magaud, cond-mat/0702311 [23] F. Guinea, A.H. Castro Neto, N.M.R. Peres, Phys. Rev. B 73, 245426 (2006) http://arxiv.org/abs/cond-mat/0610673 http://arxiv.org/abs/cond-mat/0702311 References

0704.0586

Dust Formation and Survival in Supernova Ejecta

Mon. Not. R. Astron. Soc. 000, 1–11 (2007) Printed 26 October 2018 (MN LATEX style file v2.2) Dust Formation and Survival in Supernova Ejecta Simone Bianchi and Raffaella Schneider 1 INAF - Istituto di Radioastronomia, Sezione di Firenze, Largo Enrico Fermi 5, 50125 Firenze, Italy 2 INAF - Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy 3 April 2007 ABSTRACT The presence of dust at high redshift requires efficient condensation of grains in SN ejecta, in accordance with current theoretical models. Yet, observations of the few well studied SNe and SN remnants imply condensation efficiencies which are about two orders of magnitude smaller. Motivated by this tension, we have (i) revisited the model of Todini & Ferrara (2001) for dust formation in the ejecta of core collapse SNe and (ii) followed, for the first time, the evolution of newly condensed grains from the time of formation to their survival - through the passage of the reverse shock - in the SN remnant. We find that 0.1 - 0.6 M⊙ of dust form in the ejecta of 12 - 40 M⊙ stellar progenitors. Depending on the density of the surrounding ISM, between 2-20% of the initial dust mass survives the passage of the reverse shock, on time-scales of about 4−8×104 yr from the stellar explosion. Sputtering by the hot gas induces a shift of the dust size distribution towards smaller grains. The resulting dust extinction curve shows a good agreement with that derived by observations of a reddened QSO at z = 6.2. Stochastic heating of small grains leads to a wide distribution of dust temperatures. This supports the idea that large amounts (∼ 0.1M⊙) of cold dust (T ∼ 40K) can be present in SN remnants, without being in conflict with the observed IR emission. Key words: dust, extinction - shock waves - supernova remnants - supernovae: individual (Cassiopeia A) 1 INTRODUCTION In the last few years, mm and submm observations of sam- ples of 5 < z < 6.4 quasars have provided a powerful way of probing the very existence and properties of dust within 1 Gyr of the Big Bang. The inferred far-IR luminosities are consistent with thermal emission from warm dust (T < 100 K), with dust masses > 108M⊙ (Bertoldi et al. 2003; Robson et al. 2004; Beelen et al. 2006; Hines et al. 2006). Despite the uncertainties due to the poorly constrained dust temperatures and absorption coefficients, the estimated dust masses are huge, implying a high abundance of heavy ele- ments at z ≈ 6, consistent with the super-solar metallicities inferred from the optical emission-line ratios for many of these systems (Pentericci et al. 2002; Freudling et al. 2003; Maiolino et al. 2003). Although high redshift quasars are extreme and rare objects, hardly representative of the dominant star forming galaxies, the above observations show that early star forma- tion leads to rapid enrichment of the Interstellar Medium (ISM) with metals and dust. It is difficult for the dust to have originated from low- mass evolved stars at z > 5 as their evolutionary timescales (108 to 109 yr) are comparable to the age of the Universe at that time (Morgan & Edmunds 2003; Marchenko 2006). Thus, if the observed dust at z > 5 is the product of stellar processes, grain condensation in supernova (SN) ejecta pro- vides the only viable explanation for its existence. This sce- nario has recently been tested through the observation of the reddened quasar SDSSJ1048+46 at z = 6.2 (Maiolino et al. 2004). The inferred extinction curve of the dust responsible for the reddening is different with respect to that observed at z < 4 (Small Magellanic Cloud-like, Hopkins et al. 2004), and it shows a very good agreement with the extinction curve predicted for dust formed in SN ejecta. Theoretical models, based on classical nucleation the- ory, predict that a few hundred days after the explo- sions silicate and carbon grains can form in expanding SN ejecta, with condensation efficiencies in the range 0.1-0.3 (Kozasa et al. 1991; Todini & Ferrara 2001; Clayton et al. 2001). Direct observational evidences for dust production have been collected only for a limited number of SNe, such as 1987A (Wooden et al. 1993) 1999em (Elmhamdi et al. 2003), and 2003gd (Sugerman et al. 2006). With the excep- tion of 2003gd, the dust masses derived from the IR emission are ≈ 10−3M⊙, corresponding to condensation efficiencies which are two orders of magnitude smaller than what the- ory predicts. A fraction of dust could escape detection if it is cold and concentrated in clumps. This has been con- c© 2007 RAS http://arxiv.org/abs/0704.0586v1 2 S. Bianchi & R. Schneider firmed to be the case for SN 2003gd where a radiative trans- fer code has been used to simultaneosly fit the optical ex- tinction and IR emission, leading to an estimated dust mass of 2× 10−2M⊙ (Sugerman et al. 2006). However, when ap- plied to SN 1987A, the same numerical model gives dust mass estimates which do not differ significantly from previ- ous analytic results (Ercolano et al. 2007). Similar low dust masses have been inferred from in- frared observations of galactic SN remnants with Spitzer and ISO satellites (Hines et al. 2004; Krause et al. 2004; Green et al. 2004). The consistent picture that emerges is that the mid- and far-IR excess observed is due to emission from small amounts of warm dust, with indicative tempera- tures T ∼ 80−270 K and masses 3×10−3−10−5M⊙ for Cas A, and temperatures T ∼ 50 K and masses 3×10−3−0.02M⊙ for the Crab nebula. Cold dust has also been detected through far-IR and submm observations of these remnants (Dunne et al. 2003; Krause et al. 2004). However, the inter- pretation of these data is complicated by the strong contam- ination from cold dust along the line of sight, providing so far only upper limits of 0.2M⊙ on the amount of cold dust associated to the SN remnants. The aim of the present paper is to critically assess the model developed by Todini & Ferrara (2001) exploring a wider range of initial conditions and model assumptions. We then follow the evolution of dust condensed in SN ejecta on longer timescales with respect to previous theoretical mod- els. In particular, we are interested in understanding how the passage of the reverse shock affects the newly formed grain size distributions and masses, so as to make predictions for the expected dust properties from the time of formation in the ejecta (a few hundred days after the explosion) to its survival in the SN remnant, hundreds of years later. So far this process has received little attention, most of the stud- ies being dedicated to the destruction of ISM dust grains caused by the SN forward shock (Draine & Salpeter 1979; Jones et al. 1994; Nozawa et al. 2006), with the notable ex- ception of Dwek (2005), who, on the basis of timescale con- siderations, finds that the reverse shock is able to destroy much of the initially formed dust. The paper is organised as follows: Sect. 2 revisits the dust formation models based on the nucleation theory; Sect. 3 describes the model adopted for the propagation of the reverse shock into the ejecta and shows the effect of sputtering on the grain size distribution and total mass; in Sect. 4 we compare the extinction and emission properties of the surviving SN dust with observations. Finally, the results are summarised in Sect. 5. 2 SN DUST FORMATION REVISITED Models of dust formation in the ejecta of core collapse SNe typically predict that large masses of dust (0.1−1.0M⊙) are formed within 1000 days from the onset of the explosion, when the ejecta are still compact (radius of order 1016cm; Kozasa et al. 1991; Todini & Ferrara 2001; Nozawa et al. 2003). If these freshly formed dust grains were distributed homogeneously within the ejecta, their opacity would be very high, with center-to-edge optical depths of order 102 − 104 in optical wavelengths, depending on the grain material and size distribution. The ejecta would thus be opaque to radiation produced within it (Kozasa et al. 1991). Observations of recent SNe, instead, reveal extinc- tions smaller than a couple of magnitudes, which imply dust masses of only 10−4 − 10−2M⊙ (Sugerman et al. 2006; Ercolano et al. 2007). The dust mass derived from extinc- tion measures could be underestimated if grains are dis- tributed in clumps with a small volume filling factor: for a given amount of grains, a clumpy distribution would pro- duce a lower effective extinction. However, the comparison between observations of dust extinction/emission and radia- tive transfer models shows that the neglect of clumping can only produce a moderate underestimation of the dust mass in the ejecta (Ercolano et al. 2007). To check whether the dust production in SNe is overesti- mated, we have reconsidered the model of Todini & Ferrara (2001). In the model, dust formation is investigated in the framework of standard nucleation theory: when a gas be- comes supersaturated, particles (monomers) aggregate in a seed cluster which subsequently grows by accretion of other monomers (Feder et al. 1966). For grain materials whose molecules are not present in the gas phase, the key species approach is adopted (Kozasa & Hasegawa 1987). Six materials where considered in the original work: amor- phous carbon (AC), iron, corundum (Al2O3), magnetite (Fe3O4), enstatite (MgSiO3) and forsterite (Mg2SiO4). Fol- lowing Schneider et al. (2004), we have added the formation of SiO2 grains. SiC grains, found in meteorites and consid- ered to be of SN origin from their anomalous isotopic ratios (Clayton & Nittler 2004), are not considered since their for- mation is impeded by the formation of AC and Si-bearing grains (Nozawa et al. 2003). The model of Todini & Ferrara also considers the formation and destruction of SiO and CO molecules: while the first is necessary to study the formation of Si-bearing grains, the second may be a sink for carbon atoms that otherwise would accrete on grains. The ejecta are taken to have a uniform composition and density, with initial temperature and density chosen to match the observations of SN1987A. The initial composition depends on the metallicity and mass of the progenitor star, Mstar, while the dynamic is given by the mass of the ejecta Meje and the kinetic energy of the explosion Ekin: the models of Woosley & Weaver (1995) were used. In the models of Todini & Ferrara, the gas becomes su- persaturated after a few hundred days from the explosion. The nucleation process starts at temperature between 1800K (for AC) and 1200K (for Si-bearing materials). At the be- ginning the gas is moderately supersaturated and large seed clusters, made of N monomers, tend to form. However, their formation rate per unit volume (the nucleation current) is small. As the volume of the ejecta increases, the supersat- uration rate grows and smaller clusters aggregate with a larger formation rate. This occurs until the gas becomes sufficiently rarified (because of expansion and/or exhaus- tion of monomers in the gas phase) and the formation rate drops. The nucleation process, together with accretion, re- sults in a typical log-normal grain size distributions (see, e.g., Todini & Ferrara 2001; Nozawa et al. 2003). For materials apart from AC, the supersaturation rate increases quickly during the ejecta expansion, and the seed clusters can become very small. In Todini & Ferrara (2001) seed clusters were allowed to be of any size. In this paper we c© 2007 RAS, MNRAS 000, 1–11 Dust Formation and Survival in SN ejecta 3 Figure 1. Size distribution for grains formed in the ejecta of a SN with a progenitor star of solar metallicity and mass Mstar = 20M⊙ (N = 2). The distributions of Al2O3 and Mg2SiO4 are shown with dashed lines for ease of identification. consider only clusters with N > 2, and introduce discrete accretion of monomers. While these two (more physical) re- quirements have a limited effect on AC grains, they alter the size distributions and masses of grains composed by the other materials. In Fig. 1 we show the size distribution of grains formed in the ejecta of a SN with a progenitor star of solar metallicity and Mstar = 20M⊙. Only AC grains retain the usual log-normal distribution. Instead, the size distribu- tion of grains of other materials lacks the low-radius tail. Compared to the results of Todini & Ferrara (2001), their total number is reduced (since larger seed clusters have a smaller formation rate) and their mean size is larger (since the monomers not allowed to form the smaller clusters are now available to accrete on the larger). It is to be noted, however, that the use of the standard nucleation theory is questionable when clusters are made by N . 10 monomers (Draine 1979; Gail et al. 1984). To check what influence this limit has on the results, we have run models in which the formation of clusters with N < 10 is suppressed. The resulting size distributions confirm the same trend: less non-AC grains form, and of larger mean size. Again AC is unaffected. In Fig. 2 we show Mdust, the mass of dust formed in the ejecta of SNe of solar metallicity, as a function of Mstar. The solid line refer to the models with N > 2. Though reduced with respect to Todini & Ferrara, still considerable masses of dust are formed, predominantly of AC and Fe3O4 grains. If Mstar 6 25M⊙, all the available carbon condenses in dust grains. In the more massive models, roughly equal amounts of carbon goes in grains and in CO, since the molecule destruction mechanism provided by 56Co decay is reduced because of its low yield in the ejecta. Results are similar (within a factor of two) if the metallicity of the progenitor stars is below solar. The only distinction is the model with zero metallicity, where stars with Mstar > 35M⊙ produce no dust (Schneider et al. 2004). No substantial differences are found if a different thermal history of the ejecta is assumed: Mdust is still of the same order of magnitude if densities and temperatures follow the evolution adopted by Nozawa et al. (2003). As already seen in Fig. 1, imposing N > 10 results Figure 2. Mass of dust formed in the ejecta of a SN as a func- tion of the mass of the progenitor star, for models with different minimum cluster size and sticking coefficient. The metallicity of the progenitors is solar. in a great reduction of the number of non-AC grains: the dust mass in these models is entirely due to AC, which is unaffected by the limit (Fig. 2, long-dashed line). Dust formation models depend strongly on the sticking coefficient α. In most of the published models, and in the results presented so far, it is assumed that all gas particles colliding on a grain will stick to it (α = 1). However, theory predicts that α depends on the impact energy, on the grain internal energy, and on the material involved: for the gas temperature at which most grains form α is significantly re- duced (Leitch-Devlin & Williams 1985). Indeed, laboratory experiments on the formation of cosmic dust analogs shows that α ≈ 0.1 for Si-bearing grains (Gail 2003). Thus, we have also run models assuming α = 0.1 for all the species considered. By reducing α, monomers stay in the gas phase longer and dust formation is delayed to times when super- saturation is larger: typically, smaller seed clusters form. For N > 2, the number of non-AC grains is further reduced and their mass becomes negligible compared to that of AC. Again all available carbon is locked in AC grains, but their size distribution is shifted towards lower radii and seed clus- ters form with N < 10. For N > 10 (Fig. 2, dashed line) their mass reduces and the size distribution becomes similar to those of non-AC grains in Fig. 1. At least for low mass progenitors, the predicted Mdust are closer to the values in- ferred by observations. Clearly, the thermodynamic properties of the ejecta are at the limits of applicability of classical nucleation theory. A different approach may be needed, expecially if realistic α values are taken into account. Unless otherwise stated, in the following we will study the evolution of dust grains resulting from models with solar metallicity for the SN pro- genitors and N > 2, α = 1 (so as to conform to most works in literature). However, we will also discuss the results for models with different assuptions on N and α. c© 2007 RAS, MNRAS 000, 1–11 4 S. Bianchi & R. Schneider 3 SURVIVAL IN THE REVERSE SHOCK As the ejecta expands, a forward shock is driven into the ISM, which compresses and heats the ambient gas. The ISM becomes an hostile environment for the survival of dust grains preexisting the SN event, mainly because of sputter- ing by collisions with gas particles (Draine & Salpeter 1979; Jones et al. 1994; Nozawa et al. 2006). In turn, the shocked ambient gas drives a reverse shock in the ejecta, which, by about 1000 years, has swept over a considerable fraction of its volume. The dust within the SNe, then, has to face hos- tile conditions inside what had previously been its cradle. We study the process in this Section. 3.1 Dynamics of the reverse shock Truelove & McKee (1999) have studied the dynamics of a SN remnant through its nonradiative stages, the ejecta dom- inated and the Sedov-Taylor. They provide analytic approxi- mations for velocity and position of the reverse and forward shocks, as a function of the kinetic energy Ekin and mass Meje of the ejecta, and of the ISM density ρISM. We use here their solution for a uniform density distribution inside the ejecta. The values for Ekin and Meje are the same that were used in the dust formation models: Ekin = 1.2 × 10 erg and 10M⊙ . Meje . 30M⊙ for stellar progenitor masses in the range 12-40 M⊙ and metallicities between zero and solar (Woosley & Weaver 1995). We study the effect of three different ISM environments, with ρISM = 10 −25, 10−24 and 10−23 g cm−3. For each model, we have divided the ejecta into Ns spherical shells. We have assumed that all shells have the same initial width ∆R = Reje/Ns, with Reje the initial radius of the ejecta. The mass of each shell is conserved throughout the evolution. For the j shell (counting shells outwards), the initial velocity of the gas at its inner boundary is given by homologous expansion, vj = veje , veje = , (1) where Rj is the initial radius of the inner shell boundary and veje is the velocity of the external boundary for ejecta of uni- form density. For practical purposes, we start our simulation at a time t0 (ideally, t0 → 0), and we set Reje = vejet0. The results do not depend on the exact value of t0, provided it is taken small enough (we use a value of order a few tens of years). After setting the initial conditions, we study the evo- lution of the ejecta with time. At each time step, the re- verse shock goes inward through a single shell. Thus, at time ti, the reverse shock has travelled inward through i shells, and lies at the inner boundary of shell jrs = Ns − i − 1. Shells that have not been visited by the reverse shock (for 0 6 j 6 jrs − 1) continue to follow homologous expansion, i.e. the inner and outer radii grow linearly with time, with velocity given by Eq. 1. Following the shell expansion (in- crease in the shell volume Vj), the shell gas density decreases as ρj ∝ V j . For an adiabatic expansion, the shell tempera- ture scales as Tj ∝ V j , with γ = 5/3 (Truelove & McKee 1999). Since the shock is strong, the results are independent on the initial choice for the gas temperature in the ejecta. For the shell j = jrs that has been swept over by the Figure 3. Velocity, density and temperature of the ejecta at t = tch as a function of the radius, for the model with progen- itor star of solar metallicity and mass Mstar = 20M⊙ (Meje = 18M⊙) expanding in a ISM with density ρISM = 10 −24 g cm−3. All quantities are normalised to their characteristic val- ues (Truelove & McKee 1999). For the model shown here tch = 5800 yr, Rch = 10.7 pc, Tch = 5.7 × 10 7 K, vch = 1800 km s It is also ρch = ρISM. The contact discontinuity marks the border between the shocked ejecta and the ISM swept by the forward shock. shock at time ti, we apply the standard Rankine-Hugoniot jump conditions for a strong adiabatic shock. The density, velocity and temperature change as γ + 1 γ − 1 vj = v γ + 1 ṽrs, Tj = 2 γ − 1 (γ + 1)2 where ρ′j and v j are the density and velocity before the shock (i.e. following the same evolution as for shells with j < jrs), ṽrs is the velocity of the reverse shock in the reference frame of the unshocked ejecta (provided by Truelove & McKee 1999), m is the mean particle mass and k the Boltzmann’s constant. To ensure mass conservation, the volume of shell j = jrs is reduced by a factor (γ − 1)/(γ + 1). For the shells jrs < j < Nrs shocked at earlier times t < ti, we compute the velocity vj by interpolating between the velocity of the j = jrs shell and the velocity of the for- ward shock (in the ambient rest frame), as a function of the logarithm of the shell inner radius. Velocity and position of the forward shock are also given by Truelove & McKee (1999). As for shells with j < jrs, the evolution of density and temperature is derived from the condition of adiabatic expansion and conservation of the shell mass. The typical trends for velocity, density and temperature around the re- verse shock are shown in Fig. 3. We checked the results (in particular the assumption for the evolution of j > jrs shells) with the 1-D hydrody- namical models of SN blast waves of van der Swaluw et al. (2001) and with simulations kindly provided by L. Del Zanna c© 2007 RAS, MNRAS 000, 1–11 Dust Formation and Survival in SN ejecta 5 (based on the code described in Del Zanna et al. 2003). However crude, our approximation provide a simple and fast solution for the density and temperature evolution of the ejecta during the passage of the reverse shock. Chosing an adequate number of shells (we use Ns = 400), it agrees with the complete hydrodynamical solution within a factor of 2. 3.2 Dust grain survival We assume that dust grains are distributed uniformly within the ejecta, and that the size distribution is the same every- where. In the shells that have been visited by the reverse shock, dust grains are bathed in a gas heated to high tem- perature (of order 107 − 108 K for the cases studied here). Also, the gas is slowed down and dust grains decouple from it, attaining a velocity relative to the gas vdj = γ + 1 ṽrs. Gas particles thus impact on dust grains transferring ther- mal and kinetic energy, which are of the same order of mag- nitude (both depending on the reverse shock velocity ṽrs, which is of order 103 km s−1). Thermal and non-thermal sputtering result, which erode the dust grain, reducing its size. Eventually, the gas drag due to direct and Coulomb collisions slows the grain and non-thermal sputtering weak- ens. In this work we consider both thermal and non-thermal sputtering, but we neglect the gas drag and the grain charge: once passed through the reverse shock, the grain retains its velocity relative to the gas. We can thus provide upper limits on the influence of non-thermal sputtering. The number of atoms that are sputtered off a dust grain per unit time is given by the sputtering rate dN/dt, a complex function of the gas density, temperature and of the nature of the dust/gas (target/projectiles) interaction (full expressions for dN/dt can be found elsewere, see e.g. Bianchi & Ferrara 2005). The sputtering rate depends on the sputtering yield, Y , the fraction of atoms that leave the target per projectile collision, which is a function of the en- ergy of the impact. We use here the Y functions described in Nozawa et al. (2006), and we consider collisions of dust grains with H, He and O atoms in the ejecta. The grain radius decreases with sputtering as , (2) where, q is the number of atoms in a molecule of the grain material, and am is the molecule radius, computed from the material density and the molecule mass. The values for am can be derived easily from the a0j values of Table 2 in Nozawa et al. (2003). At each time step, we reduce the grain size according to Eq. 2 in all shells that have been swept by the reverse shock. We follow the evolution until the reverse shock arrives near the center of the ejecta: this is the limit of validity of the approximations in Truelove & McKee (1999). After that, we simply assume that the ejecta expands adi- abatically, and we end the simulations when the sputtering rate becomes negligible. Since we do not include gas drag and grain charge, grains do not attain differential veloci- ties for different sizes. Thus, we have neglected destruction due to grain-grain collisions. However, sputtering dominates Figure 4. Changes in the size distribution of AC and Fe3O4 grains. For each material, the thick line is the initial size distribu- tion (the same as in Fig. 1). The thin line is the size distribution after the passage of the reverse shock through the ejecta. over this process for the high shock velocities considered here (Jones et al. 1994). Dust grains in the ionized shocked gas are heated mainly by collisions with electrons. If the grains are small, heating is stochastic and an equilibrium temperature does not exist. Instead, a broad temperature distribution P (Td) establishes, peaking at low temperature but extending also to high val- ues (Dwek 1986). The temperature may be so high that dust grains sublimate (Guhathakurta & Draine 1989). For the cases studied here, however, sublimation is negligible. Details on the calculation are presented in Appendix A. In Fig. 4 we show the initial (thick lines) and final (thin lines) size distributions for AC and Fe3O4 grains in the ejecta of a star with Mstar = 20M⊙ expanding in a medium with ρISM = 10 −24 g cm−3. As it is evident for the (initially) more peaked size distributions of magnetite, sputtering pro- duces a leaking towards smaller sizes. The evolution of the size distribution is analogous to that of ISM grains destroyed by the forward shock (Nozawa et al. 2006). In Fig. 5 we show the mass of dust that survives the pas- sage of the reverse shock. For the reference model (dashed line), the erosion caused by sputtering reduces the dust mass to about 7% of its initial value, almost independently of the stellar progenitor model. Most of the dust (about 70% in mass) is consumed within one characteristic time tch from the explosion, when 95% of the original volume of the ejecta has been reached by the reverse shock (tch = 4−8×10 4 yr for the ejecta discussed here; Truelove & McKee 1999). Dust in the inner shells is less affected by erosion, because the sput- tering rate is lower. A minor fraction of the dust mass, less than 10%, is consumed after the reverse shock bounces at the center of the ejecta (for t & 2.6tch; Truelove & McKee 1999). If the SN explodes in a denser ISM, the reverse shock would travel faster inside the ejecta and would encounter a gas at higher density. This increases the effect of sputtering. In Fig. 5 (long-dashed line) we see the fraction of dust mass that survives when ρISM = 10 −23 g cm−3: only about 2% of the dust mass survives. Conversely, for a lower density ISM, a larger fraction is left: for ρISM = 10 −25 g cm−3, it is 20% c© 2007 RAS, MNRAS 000, 1–11 6 S. Bianchi & R. Schneider Figure 5. Mass of dust that survives the passage of the reverse shock in the ejecta, as a function of the mass of the progenitor star and of the density of the surrounding ISM. The solid line shows the initial dust mass (same as the solid line in Fig. 2). (dotted line). While the number of surviving grains changes with the ISM density, the shape of the size distributions remain similar in all cases, with the typical patterns shown in Fig. 4. No substancial change is observed in models where the dust was produced by progenitors of metallicity different from solar. Dust destruction is instead enhanced in models where a smaller sticking coefficient is adopted. If α = 0.1 (Sect. 2), only 10, 3 and 1% of the original dust mass sur- vives, respectively, for ρISM = 10 −25, 10−24 and 10−23 g cm−3 (compared to 20, 7 and 2% for α = 1.0). This is be- cause for smaller values of α, the dust distribution is made by grains of smaller radii, which are more easily destroyed. 4 EXTINCTION AND EMISSION FROM SN Maiolino et al. (2004) measured the reddening in the rest- frame UV spectrum of a z = 6.2 QSO and found it to be different from that of the SMC, typically used to dered- den the spectra of lower redshift QSOs. The measured red- dening is instead compatible with the extinction law from the Todini & Ferrara (2001) SN dust model. We repeat here the same analysis using the updated dust formation mod- els of Sect. 2 and the final distributions after the reverse shock passage of Sect. 3. As in Maiolino et al. (2004), we derive the extinction properties from the grain sizes using the Mie’s (1908) theory for spherical dust grains and refrac- tive indexes for dust materials from the literature (references are provided in Table A1). The procedure is analogous to that adopted by Hirashita et al. (2005) when modelling the dust extinction from the SNe dust models of Nozawa et al. (2003). In Fig. 6 we show the results for dust formed in SNe from progenitors of solar metallicity. The grain size distri- butions from progenitors of different masses have been aver- Figure 6. Extinction law for SN dust. The thick solid line is the extinction law for dust freshly formed in the ejecta. The thin solid line is the extinction law from dust processed by the re- verse shock. The curves are computed from the IMF-averaged size distributions of grains formed in SNe from solar metallicity progenitors (see text for details). The gray line and shaded area are the extinction law measured on a z = 6.2 QSO and its un- certainty (Maiolino et al. 2004). The dashed line is the extinction law of the SMC (Pei 1992). aged over a stellar Initial Mass Function (IMF): we adopted the Salpeter IMF, but the results do not depend heavily on this choice (Maiolino et al. 2004). The thick solid line repre- sents the extinction law of dust as formed within the ejecta, without taking into account the grain processing caused by the reverse shock. The SN dust extinction law is still flatter than the SMC extinction law, but the agreement with the observations (shaded area) is worse than in Maiolino et al. (2004). This is mainly due to a change in the grain materials that contribute to extinction: apart from AC, present in both the old and new model, the rise at λ < 2000 Å was due to Mg2SiO4 grains, with a minor contribution from Fe3O4. In the new model, Mg2SiO4 contribution is insignificant, while Fe3O4 grains (larger than in the original model) cause the far UV rise. The bump at λ ≈ 2500 Å is due to AC grains and it is typical of the optical properties derived from amor- phous carbon formed in an inert athmosphere (the ACAR sample of Zubko et al. (1996)). During the passage of the reverse shock, Fe3O4 grains are consumed more effectively than AC grains. The resulting extinction law (thin solid line in Fig. 6) becomes flatter, lead- ing to an excellent agreement with observations at λ 6 1600 Å. These results apply for ejecta expanding in a medium with ρISM = 10 −24 g cm−3. There is no significant change in the extinction law if different ISM densities are considered, since the size distributions are similar in all cases (though the extinction at any given wavelength is smaller for higher ρISM, because less grains survive). It is worth noting that grains with a . 20 Å though as abundant as larger grains, c© 2007 RAS, MNRAS 000, 1–11 Dust Formation and Survival in SN ejecta 7 do not contribute to the extinction law because of their re- duced extinction cross section. As in Maiolino et al. (2004), we find that if progenitors of metallicity lower than solar are considered, the difference in the resulting extinction laws are small and lie within 0.1 y-axis units from the lines plotted in Fig. 6. Extending calculations to the infrared, we have derived the dust emissivity. For all the IMF averaged size distribu- tions, the emissivity in the wavelength range 10 6 λ/µm 6 1000 is rather featureless, and can be well described by a power law in wavelength of index -1.4 with κ(100µm) = 40 cm2 g−1 for models where all dust has been processed by the reverse shock. Emissivities for dust formed from progen- itors of a given mass are within 10% of the IMF averaged value, while the emissivity at the end of dust condensation, before any significant sputtering has occurred, is found to be about 20% lower. No significant dependence is found on the metallicity of the progenitor and on ρISM. In all cases, the emissivity is almost entirely due to the large AC grains1. The amount of shock-heated dust in the ejecta can be derived from infrared observations of SN remnants. A no- table (and debated) case is that of Cas A, the remnant from an historical SN which shows infrared emission from the re- gion between the forward and reverse shocks. The identity of Cas A’s progenitor is still highly debated. A star of 15-25M⊙ that loses its hydrogen envelope through winds (Chevalier 2006) or binary interactions (Young et al. 2006) and then undergoes an energetic explosion can match all the avail- able observational constraints. In particular, the age and dynamics suggest a mass for the ejecta of 3 M⊙, with about the same amount of gas reached by the reverse shock in the ejecta and swept by the forward shock in the surrounding ISM (Truelove & McKee 1999). Given these uncertainties, and the dependence of the predicted dust masses on the stellar progenitor (see Fig. 5), we can only give a tentative estimate of the amount of dust predicted for Cas A by our model. An ejecta evolution compatible with observations can be obtained for a 12 M⊙ progenitor, provided we neglect the hydrogen mass. In such a model, ≈0.1 M⊙ of dust forms. By the age of the remnant (∼325 yr), ≈0.05 M⊙ survives in the region reached by the reverse shock, where it is heated by the hot gas. We also need to consider the contribution to emission from dust in the ISM reached by the forward shock. Typically, dust in the shocked ISM is exposed to a gas of similar density and temperature to those in the reverse shock (van der Swaluw et al. 2001). For a standard value of the ISM gas-to-dust mass ratio, one would roughly expect a similar mass of emitting dust in the ISM. Thus, a model for Cas A remnant would have about 0.1 M⊙ of emitting dust. This mass appears to be more than an order of magni- tude larger than what could be derived fitting the observed Spectral Energy distribution (SED) of Cas A (Hines et al. 2004). Using the emissivity predicted for SN dust, the flux in the wavelength range 10 6 λ/µm 6 100 can be reasonably well reproduced with a single modified blackbody with tem- 1 For the same reason, increasing the minimum cluster size N and/or decreasing the sticking coefficient α does not affect the predicted extinction laws and emissivities, which are similar to those found for our reference model after the passage of the re- verse shock. Figure 7. Synchrotron-subtracted SED of dust emission in CasA. Data points are from Hines et al. (2004). The solid line is a one- component modified blackbody fit to the data for λ 6 100µm (T ≈ 100K, Md ≈ 4.0 × 10 −3M⊙). The dashed line is a two- component fit with T ≈ 110K, Md ≈ 3.0×10 −3M⊙ and T ≈ 35K, Md ≈ 0.1M⊙. The dotted line is the spectrum from stochastically heated dust in our model. See text for details. perature T = 100K, and a dust mass of 4×10−3 M⊙ (Fig. 7, solid line). Hines et al. (2004) obtain for the cold, more mas- sive component a similar dust mass with T = 80K. However, the large uncertainties and the limited FIR coverage allow to fit, equally well, a two-component model with temperatures 110 and 35K and masses, respectively, of 3 × 10−3 and 0.1 M⊙ (Fig. 7, dashed line). Unfortunately Cas A lies on the line of sight of dense molecular clouds which do not allow a reliable estimate of the cold dust mass from observations at longer wavelengths in the FIR and sub-mm. Still, upper limits on the dust mass in the remnant are compatible with our model predictions (Krause et al. 2004). A broad span of temperatures is clearly needed for a reliable estimate of the dust mass in the remnant. In Fig. 7 we also show the SED of the shock heated dust in the CasA model (dotted line). Because of stochastic heat- ing (Appendix A), grains have temperatures mainly ranging from 10 to 100K. The SED cannot be easily modelled using a 2-component modified blackbody: the longer wavelength side could be described with a cold component of T ≈ 60K, which would underestimate the dust mass by about a factor 5; instead, a hot component at T ≈ 150K would leave a sub- stantial residual in the fit at λ . 10µm. When comparing to the data for CasA, it appears that dust temperature in our models is overestimated. This could be due to an over- estimate of the dust stochastic heating, to a reduction of smaller grains with respect to the dust formation model, or to differences between the emission properties of true and modelled materials. However, the uncertainties in the ther- mal/dynamical history of the ejecta of CasA and the im- possibility of discriminating between ISM and ejecta dust emission in the spectrum prevent a more detailed analysis. c© 2007 RAS, MNRAS 000, 1–11 8 S. Bianchi & R. Schneider 5 SUMMARY In the present work we have revisited the model of Todini & Ferrara (2001) for dust formation in the ejecta of core collapse SNe and followed the evolution of newly condensed grains from the time of formation to their survival through the passage of the reverse shock. The main results can be summarized as follows: (i) The new features introduced in the dust formation model have only a minor impact on AC grains but signif- icantly affect other species (Si-bearing grains, Al2O3, and Fe3O4). For 12 - 40 M⊙ stellar progenitors with Z = Z⊙, the predicted Mdust ranges between 0.1 - 0.6 M⊙; comparable values (within a factor 2) are found if the progenitors have Z < Z⊙. The dominant grain species are AC and Fe2O3. (ii) We identify the most critical parameters to be the minimum number of monomers, N , which define a critical seed cluster, and the value of the sticking coefficient, α. As- suming N > 10 (below which the application of standard nucleation theory is questionable) results in a great reduc- tion of non-AC grains because these species nucleate when the gas in the ejecta is highly super-saturated and smaller seed clusters form. This effect is further enhanced if α < 1: for α = 0.1 and stellar progenitor masses Mstar < 20M⊙, the total mass of dust is reduced to values in the range 0.001- 0.1 M⊙, comparable to those inferred from the IR emission at 400-700 days after the explosion for 1987A and 2003gd, the only two core-collapse SNe for which these data were available. (iii) Using a semi-analytical model to describe the dy- namics of the reverse shock, we have found that thermal and non-thermal sputtering produce a shift of the size dis- tribution function towards smaller grains; the resulting dust mass reduction depends on the density of the surrounding ISM: for ρISM = 10 −25, 10−24, 10−23 g cm−3, about 20%, 7%, and 2% (respectively) of the initial dust mass survives. Most of dust consumption occurs within one characteristic time from the explosion, about 4−8×104 yr for core-collapse SNe. Thus, the impact of the reverse shock needs to be taken into account when comparing model predictions with obser- vations of young SN remnants. (iv) Averaging over a Salpeter IMF, we have derived dust extinction and emissivity. We find that the extinction curve is dominated by AC and Fe3O4 grains with radii larger than 20 Å. As a result, it is relatively flat in the range 1500-2500Å and then rises in the far UV. Thus, the peculiar behaviour of the extinction produced by SN dust, which has been suc- cessfully used to interpret observations of a reddened QSO at z = 6.2 (Maiolino et al. 2004), is preserved in the present model, and it is further amplified by the modifications in- duced by the passage of the reverse shock. (v) Using dust emissivity predicted by the model, we can reproduce the observed IR flux from the young SN rem- nant CasA adopting a single modified black-body of tem- perature T = 100 K, which implies a mass of warm dust of 4 × 10−3M⊙, consistent with Hines et al. (2004). How- ever, the limited observational coverage in the FIR allows to equally well reproduce the data adding a cold component with temperature T = 35 K and dust mass of 0.1M⊙. Ac- cording to our model, such a mass of dust is what would be produced by a single 12 M⊙ star that has exploded after losing its hydrogen envelope, a plausible candidate for the highly debated CasA’s progenitor. Because of the stochastic heating of small grains by collisions with hot gas electrons, dust in the shocked gas is predicted to have temperatures ranging from 10 to 100K. We conclude that our study supports the idea that core-collapse SNe can be major dust factories. At the same time, it shows that our knowledge of dust condensation and its survival in SN ejecta still lacks to control some critical parameters, which prevent reliable estimates of condensa- tion efficiencies, especially for the less massive progenitors. Within these uncertainties, the model can accomodate the still sparse observational probes of the presence of dust in SN and SN remnants. ACKNOWLEDGMENTS We are grateful to A. Ferrara for profitable discussions and suggestions, and to L. Del Zanna for kindly providing us the results of 1-D hydrodynamical simulations. We also acknowl- edge DAVID members2 for fruitful comments and Cristiano Porciani for precious help. REFERENCES Beelen A., Cox P., Benford D. J., Dowell C. 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A., Fryer C. L., Hungerford A., Arnett D., Rock- efeller G., Timmes F. X., Voit B., Meakin C., Eriksen K. A., 2006, ApJ, 640, 891 Zubko V. G., Mennella V., Colangeli L., Bussoletti E., 1996, MNRAS, 282, 1321 APPENDIX A: STOCHASTIC HEATING FROM ELECTRON COLLISIONS We have derived the temperature distribution P (Td) follow- ing the method of Guhathakurta & Draine (1989), to which we refer for a more detailed description. We have divided the range of possible dust tempera- tures into Nb bins. For the case studied here, we found suf- ficient to define Nb = 500 bins, logarithmically spaced in the range 2 < Td/K< 2000. The i-bin has temperature Td,i, energy Ei and energy width ∆Ei. The energy corresponding to each value of Td is defined as E(Td) = C(T )dT, where C is the specific heat, which is derived by fitting ex- perimental data. We have adopted a piecewise power-law of the form C(T ) = A T The fitted values for A and B are given in Table A1. c© 2007 RAS, MNRAS 000, 1–11 10 S. Bianchi & R. Schneider Table A1. Thermal and optical properties of SNe dust materials Materials Specific heat Refractive index A (erg cm3 K−1) B range in T (K) Refs. Refs. Al2O3 4.44 3 (0, 110] Ditmars et al. (1982) Koike et al. (1995, ISAS) 1.22 102 2.29 (110, 200] Chase et al. (1985) Begemann et al. (1997, compact) 7.00 105 0.66 (200, 500] 1.45 107 0.17 (500, 2000] 5.44 107 0 (2000,∞) Fe3O4 22.5 3 (0, 10] Chase et al. (1985) Mukai (1989) 7.65 3.47 (10, 30] Shepherd et al. (1985) 3.74 102 2.32 (30, 80] Koenitzer et al. (1989) 1.04 105 1.04 (80, 300] 3.95 107 0 (300,∞) MgSiO3 9.22 3 (0, 20] Kelley (1943) Jäger et al. (2003) 2.01 3.51 (20, 50] Chase et al. (1985) 7.94 102 1.98 (50, 130] Krupka et al. (1985) 1.32 105 0.93 (130, 400] 3.47 107 0 (400,∞) Mg2SiO4 22.7 3 (0, 60] Kelley (1943) Jäger et al. (2003) 6.45 102 2.18 (60, 120] Chase et al. (1985) 1.40 105 1.06 (120, 300] 1.42 107 0.25 (300, 2000] 9.44 107 0 (2000,∞) AC 3.82 102 2 (0, 70] Chase et al. (1985) Zubko et al. (1996, ACAR) 3.27 103 1.50 (70, 300] Draine & Li (2001) 5.58 104 1.00 (300, 700] 1.09 107 0.19 (700, 3000] 5.09 107 0 (3000,∞) SiO2 9.95 10 2 2 (0, 60] Chase et al. (1985) Philipp (1985) 5.50 104 1.02 (60, 500] Leger et al. (1985) Henning & Mutschke (1997) 3.11 107 0 (500,∞) The function P (Td) is computed from the transition matrix between the initial and final internal energy states, Af,i. The probability per unit time that a dust grain in the energy state Ei is heated to the energy state Ef (with f > i) by collisions with electrons is given by Af,i = n πa2∆Ef× 2(Ef − Ei) f(Ef − Ei) + if Ef − Ei < E⋆ 2Eeff f(Eeff) 1− (Ef − Ei)/Eeff 1− (Ef − Ei)/Eeff 0 otherwise. In the above equation, n andm are the electron number den- sity and mass, respectively, and f(E) is the Maxwell distri- bution function defined such that nf(E)∆E is the number of electrons of energy E per unit volume. E⋆ is the maximum energy that an electron can transfer to a dust grain (Dwek 1986). If Ef −Ei > E⋆ it is not possible to jump from stage i to stage f through electron collisions; if Ef − Ei < E⋆ the jump is allowed for all electrons with energy Ef − Ei (which will transfer their entire energy to the grain) and for the electrons with energy Eeff (which will transfer an energy Eeffζ(Eeff) = Ef − Ei; the function ζ(E) is the fraction of the electron energy which is deposited on a dust grain when E > E⋆, as defined in Dwek 1986). As in Guhathakurta & Draine (1989) only cooling terms from level f+1 to level f are considered: Af,f+1 = Ef+1 − Ef Qabs(a, λ) Bλ(Td,f+1)dλ, where Bλ is the Planck function and Qabs(a, λ) the absorp- tion efficiency of a grain of radius a. Values for Qabs(a, λ) were derived using the Mie’s (1908) theory for spherical dust grains, adopting the refractive index from the references in Table A1. In Fig. A1 we show the temperature distributions for AC and Fe3O4 dust grains of radii a=10, 50 and 200Å. Grains are exposed to a gas with T = 108K and n = 10 electrons cm−3, typical conditions encountered in the ejecta swept by the reverse shock. Smaller grains have broader tem- perature distributions, with a high temperature tail. Using the sublimation rate of Draine & Hao (2002), we have derived a sublimation temperature Ts, defined as the temperature necessary to completely consume a grain in about 20 yr, the typical time step of our simulation. For the grain sizes encountered in our work, we have Td & 1200K. At c© 2007 RAS, MNRAS 000, 1–11 Dust Formation and Survival in SN ejecta 11 Figure A1. Temperature distributions P (Td) for AC and Fe3O4 dust grains in a hot gas with T = 108K and n = 10 electrons cm−3. P (Td) is shown for radii a = 10 Å (broader distributions), 50 Å and 200 Å (distributions spanning a smaller range of Td). any time in the ejecta evolution, only a negligible fraction of all grains would exceed that temperature. As a result, dust sublimation is insignificant in our models. c© 2007 RAS, MNRAS 000, 1–11 Introduction SN dust formation revisited Survival in the reverse shock Dynamics of the reverse shock Dust grain survival Extinction and emission from SN dust Summary Stochastic heating from electron collisions

0704.0587

Preferential interaction coefficient for nucleic acids and other cylindrical poly-ions

Preferential interaction coefficient for nucleic acids and other cylindrical poly-ions Emmanuel Trizac∗ CNRS; Univ. Paris Sud, UMR8626, LPTMS, ORSAY CEDEX, F-91405 and Center for Theoretical Biological Physics, UC San Diego, 9500 Gilman Drive MC 0374 - La Jolla, CA 92093-0374, USA Gabriel Téllez† Departamento de F́ısica, Universidad de Los Andes, Apartado Aéreo 4976, Bogotá, Colombia The thermodynamics of nucleic acid processes is heavily affected by the electric double-layer of micro-ions around the polyions. We focus here on the Coulombic contribution to the salt- polyelectrolyte preferential interaction (Donnan) coefficient and we report extremely accurate ana- lytical expressions valid in the range of low salt concentration (when polyion radius is smaller than the Debye length). The analysis is performed at Poisson-Boltzmann level, in cylindrical geometry, with emphasis on highly charged poly-ions (beyond “counter-ion condensation”). The results hold for any electrolyte of the form z−:z+. We also obtain a remarkably accurate expression for the electric potential in the vicinity of the poly-ion. Coulombic interactions between salt and poly-anions play a key role in the equilibrium and kinetics of nucleic acid processes [1]. A convenient quantity quantifying such interactions and allowing for the analysis and inter- pretation of their thermodynamics consequences, is the so called preferential interaction coefficient. Several def- initions have been proposed and their interrelation stud- ied, see e.g. [2, 3, 4]. In the present work, they are defined as the integrated deficit (with respect to bulk conditions) of co-ions concentration around a rod-like poly-ion. Our goal is to provide analytical expressions describing the effect of salt concentration and poly-ion structural pa- rameters on the preferential interaction coefficient, for a broad class of asymmetric electrolytes. For symmetric electrolytes, it will be shown that our formulas improve upon existing analytical results. For other asymmetries, they seem to have no counterpart in the literature. Our analysis holds for highly (i.e. beyond counter-ion conden- sation [5, 6]) and uniformly charged cylindrical poly-ions, and is explicitly limited to the low salt regime (i.e. when the poly-ion radius a is smaller than the Debye length 1/κ). These conditions are most relevant for RNA or DNA in their single, double, or triple strand forms. As in several previous approaches [7, 8, 9, 10], we adopt the mean-field framework of Poisson-Boltzmann equation, in a homogeneous dielectric background of per- mittivity ε. The same starting point has proven relevant for related structural physical chemistry studies of nu- cleic acids [11]. In a z−:z+ electrolyte, the dimensionless electrostatic potential φ = eϕ/kT (with e > 0 the ele- mentary charge and kT thermal energy) then obeys the equation [12] z+ + z− ez−φ − e−z+φ , (1) ∗Electronic address: [email protected] †Electronic address: [email protected] where r is the radial distance to the rod axis. The va- lencies z+ and z− of salt ions are both taken positive. Denoting derivative with a prime, the boundary condi- tions read rφ′(r) = 2ξ > 0 at the polyion radius (r = a) and φ → 0 for r → ∞. The latter condition expresses the infinite dilution of poly-ion limit and ensures that the whole system is electrically neutral, since it (indi- rectly) implies that rφ′ → 0 for r → ∞. We consider a negatively charged poly-anion for which φ < 0 and the line charge density reads λ = −eξ/ℓB < 0, where ℓB = e 2/(εkT ) denotes the Bjerrum length (0.71 nm in water at room temperature). Finally, the Debye length is defined from the bulk ionic densities n∞+ and n − through κ2 = 4πℓB(z + + z The Coulombic contribution to the anionic preferential interaction coefficient is defined as [7, 8, 9, 10, 13] Γ = κ2 (ez−φ − 1) rdr, (2) while its cationic counterpart follows from electro- neutrality. This quantity –which provides a measure of the Donnan effect [14]– can be expressed in closed form as a function of the electrostatic potential, see Appendix A. As can be seen in (A3) and (A4), Γ depends expo- nentially on the surface potential φ0, so that deriving a precise analytical expression is a challenging task. Fur- thermore, we are interested here in the limit κa < 1 (including the regime κa≪ 1) which is analytically more difficult than the opposite high salt situation where to leading order, the charged rod behaves as an infinite plane, and curvature corrections can be perturbatively included [15, 16, 17]. We will proceed in two steps. Focusing first on the surface potential φ0 = φ(a), we make use of recent re- sults [18] that have been obtained from a mapping of Eq. (1) onto a Painlevé type III problem [19, 20, 21]. The exact expressions thereby derived only hold for 1:1, 1:2 and 2:1 electrolytes, but may be written in a way that is electrolyte independent. This remarkable feature is spe- http://arxiv.org/abs/0704.0587v1 mailto:[email protected] mailto:[email protected] z+/z− 1/10 1/3 1/2 1 2 3 10 C -2.51 -1.94 -1.763 -1.502 -1.301 -1.21 -1.06 TABLE I: Values of C appearing in Eq. (4) as a function of electrolyte asymmetries. For z+/z− = 1, 1/2 and 2, C is known analytically from the results of [18]. The corresponding values are recalled in Appendix B. For other values of z+/z−, C has been determined numerically, see in particular Fig. 6 of Appendix B. cific to the short distance behaviour of φ and has been overlooked so far, since not only short distance but also large distance properties have been studied [18]. We are then led to conjecture that the corresponding expression holds for any binary electrolyte z−:z+, and we explicitly check the relevance of our assumption on several specific examples. Technical details are deferred to the appendices. It is in particular concluded in Appendix B that the surface potential may be written e−z+φ0 ≃ 2(z+ + z−) z+(κa)2 (z+ξ − 1) 2 + µ̃2 where log(κa) + C − (z+ξ − 1)−1 . (4) Expression (4) is valid for κa < 1 and z+ξ > 1 [in fact z+ξ > 1 + O(1/| log κa|)]. These conditions are easily fulfilled for nucleic acids. The “constant” C appearing in (3) depends smoothly on the ratio z+/z− but is oth- erwise salt and charge independent. We report in Table I its values for several electrolyte asymmetries. The de- crease (in absolute value) of C when z+/z− increases is a signature of more efficient (non-linear) screening with counter-ions of higher valencies. From Eq. (3) and the results of Appendix B, our ap- proximation for Γ takes a simple form Γ ≃ − (1 + µ̃2). (5) This expression is tested in Figures 1 and 2 against the “true” numerical results that serve as a benchmark. In Fig. 1 which corresponds to a monovalent salt (or more generally a z:z electrolyte), we also show the prediction of Ref. [9], which is, to our knowledge, the most accurate existing formula for a 1:1 salt. For the technical reasons discussed in Appendix B, and that are evidenced in Fig- ure 6, our expression improves that of Shkel, Tsodikov and Record [9], particularly at lower salt content. For 1:2 and 2:1 salts, we expect Eq. (5) to be also accurate, since it is based on exact expansions. The situation of other salt asymmetries is more conjectural (see Appendix B), but Eq. (5) is nevertheless in remarkable agreement with the full solution of Eq. (1), see Fig. 2. To be spe- cific, in both Figures 1 and 2, the relative accuracy of our approximation is better than 0.2% for κa = 10−2 (for both ss and ds RNA parameters). At κa = 0.1, the accuracy is on the order of 1%. As illustrated in Fig. 3, approximation (4) assumes that z+ξ > 1. The corresponding expression for Γ there- fore breaks down when ξ is too low. More general expres- sions, still for κa < 1, may be found in appendix C. The inset of Fig. 3 offers an illustration and shows that the limitations of approximation (4) may be circumvented at little cost, providing a quasi-exact value for Γ. Moreover, it is shown in this appendix that for z+ξ = 1, µ̃ reads log(κa) + C . (6) On the other hand, Eq. (3) still holds. The corresponding Γ is shown in Fig. 4. We provide in Appendix C a general expression of the short scale (i.e valid up to κr ∼ 1) radial dependence of the electric potential, see Eq. (C1). The bare charge should not be too low [more precisely, one must have ξ > ξc with ξc given by Eq. (C5)], and µ̃ –which encodes the dependence on ξ– follows from solving Eq. (C2). In general, the corresponding solution should be found nu- merically. However, one can show a) that µ̃ vanishes for ξ = ξc, b) that µ̃ takes the value (6) when z+ξ = 1 and c) that µ̃ is given by (4) when z+ξ exceeds unity by a small and salt dependent amount. In practice, for DNA and RNA, we have ξ > 2 and Eq. (4) provides 0.001 0.01 0.1 1 0.001 0.01 0.1 FIG. 1: Preferential interaction coefficient for a 1:1 salt. The main graph corresponds to ss-RNA with reduced line charge ξ = 2.2 while the inset is for ds-RNA (ξ = 5). The circles correspond to the value of (2) following from the numerical solution of Eq. (1). The prediction of Eq. (5) with eµ given by (4) and C ≃ −1.502, shown with the continuous curve, is compared to that of Ref. [9], shown with the dashed line. As in all other figures, the opposite of Γ is displayed, to consider a positive quantity. 0.001 0.01 0.1 0.001 0.01 0.1 2.5 dsRNA ssRNA FIG. 2: Same as Figure 1 for a 1:3 and a 3:1 electrolyte. From Table I, we have C ≃ −1.21 in the 1:3 case and conversely C ≃ −1.94 in the 3:1 case. The symbols correspond to the numerical solution of Eq. (1) and the continuous curves show the results of Eq. (5) with again eµ given by (4). 1 1.5 2 2.5 3 1 1.5 2 2.5 FIG. 3: Preferential interaction coefficient for a 1:1 salt (hence C ≃ −1.502) and κa = 10−2. The circles show the numerical solution of PB theory (1), the continuous curve is for (5) with (4) and the dashed line is the prediction of Ref. [9]. Although approximation (4) breaks down at low ξ, the inset shows that eµ following from the solution of Eq. (C2) gives through (5) a Γ (continuous curve), that is in excellent agreement with the “exact one”, shown with circles as in the main graph. excellent results whenever κa < 0.1. To illustrate this, we compare in Figure 5 the potential following from the analytical expression (C1) to its numerical counterpart. We do not display 1:1, 1:2 and 2:1 results since in these cases, Eq. (C1) is obtained from an exact expansion and fully captures the r-dependence of the potential. For the asymmetry 1:3, Fig. 5 shows that the relatively simple form (C1) is very reliable. A similar agreement has been found for all couples z−:z+ sampled, with the trend that 0.001 0.01 0.1 0.001 0.01 0.1 FIG. 4: Same as Fig. 1 for ξ = 1 and z+/z− = 1. The same quantities are shown: our prediction for Γ [Eqs. (5) and (6) with C ≃ −1.502] is compared to that of Ref. [9]. The inset shows −z+Γ/z− for a 1:2 salt such as MgCl2 where C takes the value -1.301. Circles : numerical data; curve : our prediction. the validity of (C1) extends to larger distances as z+/z− is decreased. In this respect, the agreement shown in Fig. 5 for which z+/z− is quite high (3), is one of the “worst” observed. 0.01 0.1 1 0 0.5 1 0.01 0.1 1 FIG. 5: Opposite of the electric potential versus radial dis- tance in a 1:3 electrolyte with κa = 10−2. The continuous curve shows the prediction of Eq. (C1) with eµ given by (4) ; the circles show the numerical solution of Eq. (1). The po- tential for ξ = 2.2 is shown in the main graph on a log-linear scale, and on a linear scale in the lower inset. The upper inset is for ξ = 5. Conclusion. The poly-ion ion preferential interaction coefficient Γ describes the exclusion of co-ions in the vicinity of a polyelectrolyte in an aqueous solution. We have obtained an accurate expression for Γ in the regime of low salt (κa < 1). The present results are particu- larly relevant for highly charged poly-ions (z+ξ > 1, that is beyond the classical Manning threshold [22]), but are somewhat more general and hold in the range ξc < ξ < 1, where ξ stands for the line charge per Bjerrum length and ξc is a salt dependent threshold, given by Eq. (C5). Our formulae have been shown to hold for arbitrary mixed salts of the form z−:z+ (magnesium chloride, cobalt hex- amine etc). They have been derived from exact expan- sions valid in 1:1,1:2 and 2:1 cases, from which a more general conjecture has been inferred. The validity of this conjecture, backed up by analytical arguments, has been extensively tested for various values of z+/z−, poly-ion charge and salt content. These tests have provided the numerical value of the constant C reported in Table I, which only depends of the ratio z+/z−. As a byprod- uct of our analysis, we have obtained a very accurate expression for the electric potential in the vicinity of the charged rod (r < κ−1). It should be emphasized that the validity of our mean-field description relying on the non-linear Poisson- Boltzmann equation depends on the valency of counter- ions (z+), and to a lesser extent to the value of z− [12, 23]. For the 1:1 case in a solvent like water at room temper- ature, micro-ionic correlations can be neglected up to a salt concentration of 0.1M [8]. For z+ ≥ 2 or in sol- vents of lower dielectric permittivity, they play a more important role. Our results however provide mean-field benchmarks from analytical expressions, from which the effects of correlations may be assessed in cases where they cannot be ignored (see e.g. [8] for a detailed discussion). Acknowledgments This work was supported by a ECOS Nord/COLCIENCIAS action of French and Colom- bian cooperation. G. T. acknowledge partial financial support from Comité de Investigaciones, Facultad de Ciencias, Universidad de los Andes. This work has been supported in part by the NSF PFC-sponsored Center for Theoretical Biological Physics (Grants No. PHY-0216576 and PHY-0225630). APPENDIX A In order to explicitly relate the preferential coefficient Γ in (2) to the electric potential, we follow a procedure similar to that which leads to an analytical solution in the cell model, without added salt [24]. Implicit use will be made of the boundary conditions associated to (1). First, integrating Eq. (1), one gets [r′φ′(r′)]ra = z+ + z− e−z+φ − ez−φ r′dr′, (A1) where the notation [F (r′)]ra = F (r) − F (a) has been in- troduced. Then, multiplying Eq. (1) by r2φ′ and inte- grating, we obtain z+ + z− (r′φ′)2 e−z+φ + r′2 e−z+φ dr′.(A2) Combining both relations with adequate weights, in order to suppress the integral over counter-ion (+) density, we r′(ez−φ − 1)dr′ = 2(z+ + z−) ez−φ0 − 1 e−z+φ0 − 1 where φ0 = φ(a) is the surface potential. Equation (A3) will turn useful in the formulation of a general conjec- ture concerning the surface potential φ0, see Appendix B. We also note that for the systems under investigation here, the surface potential is quite high, and a very good approximation to (A3) is r′(ez−φ − 1)dr′ ≃ a2z−e −z+φ0 2(z+ + z−) APPENDIX B We start by analyzing a 1:1 electrolyte, for which it has been shown [19, 20] that the short distance behaviour reads eφ/2 = 2µ log − 2Ψ(µ) + O (κr) where Ψ denotes the argument of the Euler Gamma func- tion Ψ(x) = arg[Γ(ix)] [19, 20]. In (B1), µ denotes the smallest positive root of tan [2µ log(κa/8)− 2Ψ(µ)] = ξ − 1 . (B2) Expressions (B1) and (B2) require that ξ exceeds a salt dependent threshold [denoted ξc below and given by Eq. (C5)] that is always smaller than 1 [18]. They thus always hold for ξ ≥ 1 and in particular encompass the interesting limiting case ξ = 1, which is sufficient for our purposes. For large ξ, we have proposed in [18] an approximation which amounts to linearizing the argument of the tangent in (B1) in the vicinity of −π, and similarly linearizing Ψ to first order: Ψ(x) ≃ −π/2 − γx + O(x3) where γ is the Euler constant, close to 0.577. It turns out however that finding accurate expressions for exp(−z+φ0), which is useful for the computation of the preferential interac- tion coefficient, requires to include the first non-linear correction in the expansion of the tangent. After some algebra, we find : log(κa) + C − (ξ − 1)−1 6(log(κa) + C − (ξ − 1)−1)4 (ξ − 1)3 ψ(2)(1) where the constant C = C1:1 reads C1:1 = γ − log 8 ≃ −1.502 and ψ(2)(1) = d3 ln Γ(x)/dx3|x=1. From (B3) and (B1) where the sinus is expanded to third order, we ob- (κa)2e−φ0 ≃ 4[(ξ − 1)2 + µ̃2] (B4) where µ̃ is given by log(κa) + C − (z+ξ − 1)−1 . (B5) In writing (B5), we have introduced the change of vari- able µ̃ = 2µ [25]. The reason is that similar changes for other electrolyte asymmetries allows to put the final re- sult in a “universal” (electrolyte independent) form, see below. A similar reason holds for introducing z+, here equal to 1, in the denominator of (B5). The functional proximity between our expressions and those reported in [9] in the very same context is striking. We note however that our µ̃ (denoted β in [9]) involves a different constant C. More importantly, the functional form of (B1) differs from that given in [9]. The compari- son of the performances of our results with those of [9] is addressed below, and is also discussed in the main text. Performing a similar analysis as above in the 1:2 case where z+ = 2 and z− = 1, we obtain from the expressions derived in [18]: (κa)2e−z+φ0 ≃ 3[(ξ − 1)2 + µ̃2] (B6) and similarly, in the 2:1 case (z+ = 1, z− = 2): (κa)2e−z+φ0 ≃ 6[(ξ − 1)2 + µ̃2]. (B7) In both cases, provided again that ξ is not too low (see below) µ̃ is given by (B5) [26], with however a different numerical value for C [C1:2 = γ− (3 log 3)/2− (log 2)/3 ≃ −1.301 and C2:1 = γ − (3 log 3)/2− log 2 ≃ −1.763]. The similarity of expressions (B4), (B6) and (B7) leads to conjecture that this form holds for any z−:z+ elec- trolyte : (κa)2e−z+φ0 ≃ A[(z+ξ − 1) 2 + µ̃2]. (B8) We then have to determine the prefactor A as a function of z+ and z−. To this end, we make use of the exact relation (A3) [or equivalently (A4)], where in the limit of large ξ, the lhs is finite while the two terms on the rhs diverge. This yields the leading order behaviour : (κa)2 exp(−z+φ0) z+ + z− (z+ξ − 1) 2. (B9) It then follows that A = 2(z+ + z−)/z+ so that our gen- eral expression (B8) takes the form: (κa)2e−z+φ0 ≃ 2 z+ + z− (z+ξ − 1) 2 + µ̃2 . (B10) This expression holds regardless of the approximation used for µ̃. If Eq. (B5) is used, then z+ξ should not be too close to unity (see appendix C for more general results including the case z+ξ = 1). In order to test the accuracy of (B10) in conjunction with (B5), we have solved numerically Eq. (1) for several values of κa < 1 and electrolyte asymmetry and checked that for several different values of z+ξ > 1, the quantity Q = −π (κa)2e−z+φ0 2(z+ + z−) − (z+ξ − 1) ]−1/2 − log(κa) + (z+ξ − 1) −1 (B11) is a constant C, which only depends on z+/z− but not on salt and ξ [it should be borne in mind that Eq. (B5) is a small κa and large ξ expansion, which becomes increas- ingly incorrect as κa is increased and/or ξ lowered]. This is quite a stringent test (since the two terms on the rhs of (B11) are large and close] which requires high numerical accuracy. This is achieved following the procedure out- lined in [27]. In doing so, we confirm the validity of (B10) and collect the values of C given in Table I. In the 1:1 case, we predict that C = γ− log 8 ≃ −1.507, in excellent agreement with the numerical data of Figure 6. On the other hand, the prediction of Ref. [9] that Q reaches a constant close to -1.90 (shown by the horizontal dashed line in Fig 6) is incorrect. Figure 6 shows that the quality of expression (B10) deteriorates when κa increases, as ex- pected. It is noteworthy however that for κa = 10−1, its accuracy is excellent whenever ξ > 2. The inset of Fig. 6 shows the validity of (B10) for a 3:1 electrolyte. When z+ξ is close to 1, Eq. (B5) becomes an irrelevant approx- imation to the solution of (B2), and can therefore not be inserted into the general formula (B10). This explains the large deviations between Q and the asymptotic value C observed in Fig. 6 for the lower values of ξ reported. We come back to this point in Appendix C. The present results hold for z+ξ > 1 +O(1/| log κa|). In this regime, our analysis shows that Eq. (B10) [with µ̃ given by (B5)] is correct up to order 1/ log4(κa) for any (z−, z+). On the other hand the results of [9] ,valid in the 1:1 case, appear to be correct to order 1/ log2(κa). In addition, our expression for the surface potential may be generalized to a broader range of ξ values and an ex- pression for the short distance dependence of the electric potential may also be provided. This is the purpose of appendix C. 1 2 3 4 5 1 2 3 4 Q = -1.90 FIG. 6: Plot of the quantity Q defined in (B11) versus line charge ξ for a 1:1 electrolyte at κa = 10−3 (continuous curve) and κa = 10−1 (dashed curve). The value reached at large ξ is compared to the prediction of [9] Q → eγ + log 2− γ ≃ −1.90 (horizontal dashed-dotted line) whereas Eqs. (B10) and (B5) imply Q → γ− log 8 ≃ −1.50, shown by the horizontal dotted line. The inset shows the same quantity for a 3:1 electrolyte at κa = 10−5 [such a very low value is required to determine precisely the value of the asymptotic constant C, that can subsequently be used at experimentally relevant (higher) salt concentrations]. Here, we obtain Q → −1.94 (dotted line) which is the value reported for C in Table I. APPENDIX C In Appendix B, the “universal” results valid for all (z+, z−) have been unveiled partly by a change of variable µ → µ̃ from existing expressions [18]. In light of these results, and of their accuracy (assessed in particular by the precision reached for the preferential interaction co- efficient), it is tempting to go further without invoking approximations of (B2), or related expressions for other asymmetries than 1:1. Inspection of the results given in [18] for the 1:1, 1:2 and 2:1 cases lead, with again the help of (A4), to the conjecture that ez+φ/2 ≃ 2(z+ + z−) sin [µ̃ log(κr) + µ̃ C] (C1) tan [ µ̃ log(κa) + µ̃ C ] = z+ξ − 1 . (C2) We emphasize that (C1), much as (B1), is a short dis- tance expansion and typically holds for κr < 1 (hence the requirement that κa < 1). In appendix D we give further analytical support for conjecture (C1). A typical plot showing the accuracy of (C1) is provided in the main text (Fig. 5). For κr < 0.1, the agreement with the ex- act result is better than 0.1%, and becomes progressively worse at higher distances (20% disagreement at κr = 1). From (C1), it follows that the integrated charge q(r) in a cylinder of radius r [that is q(r) = −rφ′(r)/2] reads z+q(r) = −1 + µ̃ tan µ̃ log where the so-called Manning radius [18, 28, 29] is given κRM = exp . (C4) The Manning radius is a convenient measure of the coun- terion condensate thickness. It is the point r where not only z+q(r) = 1 but also where q(r) versus log r exhibits an inflection point [30]. For high enough ξ, the logarith- mic dependence of 1/µ̃ with salt [see (B5)] is such that RM ∝ κ −1/2. The two relations (C1) and (C2) encompass those given in Appendix B and allow to investigate the regime z+ξc < z+ξ, and in particular the case z+ξ = 1, the so-called Manning threshold [5]. However, (C1) and (C2) are not valid for ξ < ξc, with z+ξc = 1 + log κa+ C . (C5) Note that ξc < 1, since the constant C is negative and that salt should fulfill κa < 1. For κa = 10−2 and z+/z− = 1, we obtain ξc ≃ 0.836. This is precisely the point where −Γ = 1 in the inset of Fig. 3. This inset also shows that the value of Γ resulting from the use of the solution of (C2) is remarkably accurate. At this point, it seems useful to investigate the Man- ning threshold case z+ξ = 1 (which corresponds to the onset of counterion condensation when κa → 0 [5, 18, 30]). It is readily seen that the solution of (C2) reads z+ξ=1 log(κa) + C , (C6) which should be inserted in (C1) to obtain the potential profile, or in (5) to get the interaction coefficient. APPENDIX D In this appendix we give further support for the con- jecture (C1) which gives the short-distance expansion of the electric potential. Let us suppose initially that the charge is below the Manning threshold ξ < ξc. It is straightforward to verify that Poisson–Boltzmann equation (1) admits solutions which behave as φ(r) = −2A ln(κr) + lnB + o(1) for κr ≪ 1. Injecting this ex- pansion into equation (1) allows us to compute higher order terms. To study the regime beyond the Man- ning threshold, we compute all higher order terms of the form r2n(1+z+A) (for a negatively charged macroion) and r2n(1−z−A) (for a positively charged macroion), with n a positive integer. These terms turn out to present them- selves as the series expansion of the logarithm, thus re- summing them we obtain φ(r) = −2A ln(κr) + lnB (D1) −z+ (κr)2(1+z+A) 8(z+ + z−)(1 + z+A)2 − (κr)2(1−z−A) 8(z+ + z−)(1 − z−A)2 + · · · The dots represent terms of order r2n(1+z+A)+2m(1−z−A) with n and m two nonzero positive integers. When the Manning threshold is approached, z+A + 1 = 0 for neg- atively charged macroion, the terms r2n(1+z+A) (second line of Eq. (D1)) become of order one, but the rest of the terms (third line of Eq. (D1) and dots) remain higher or- der: a change in the small distance behavior of φ occurs. A similar situation is reached for 1 − z−A = 0 which is the Manning threshold for a positively charged macroion. A and B in the previous equations are constants of in- tegration, which should be determined with the bound- ary conditions rφ′(r) = 2ξ at the polyion radius (r = a) and φ → 0 for r → ∞. Thus to proceed further, we have to connect the long and the short distance behavior of φ. This connection problem has been only solved in the cases 1:1, 1:2 and 2:1 in Refs. [19, 31]. In particular, once A has been chosen (notice that for a = 0, A = −ξ), B should be one and only one function of A in order to satisfy φ→ 0 for r → ∞. The results from [19, 31] show B = 26Aγ ((1 +A)/2) (1 : 1) (D2) B = 33A22Aγ (2(1 +A)/3)γ ((1 +A)/3) (1 : 2) B = 33A22Aγ ((1 + 2A)/3)γ ((2 +A)/3) (2 : 1) where γ(x) = Γ(x)/Γ(1 − x). B turns out to have some interesting properties in the cases 1:1, 1:2 and 2:1, where its exact expression (D2) is known. Namely, at the Man- ning threshold 1 + z+A = 0, A→−1/z+ 8(z+ + z−)(1 + z+A)2 = 1 (D3) Furthermore if we put 1 + z+A = iµ̃, and define e2iΨ(eµ) = 8(z+ + z−)(1 + z+A)2 then for µ̃ ∈ R, Ψ(µ̃) ∈ R is a real function of µ̃, with Ψ(0) = 0. Let us now study the regime beyond the Manning threshold for a negatively charged macroion. From Eq. (D1) we can write ez+φ(r)/2 ∼ (κr)−z+ABz+/2 −z+ (κr)2(1+z+A) 8(z+ + z−)(1 + z+A)2 neglecting terms of higher order when z+A is close to −1. Let us conjecture that the properties of B as a function of A presented above hold in the general case z− : z+. Then using the parameter µ̃ defined above we find after some simple algebra ez+φ(r)/2 = 2(z+ + z−) sin [µ̃ log(κr) + Ψ(µ̃)] +O(r3+2z−/z+) (D6) Recalling that |µ̃| ≪ 1 we can approximate Ψ(µ̃) ≃ µ̃C, where C = Ψ′(0). Replacing this approximation into (D6) and imposing the boundary condition aφ′(a) = 2ξ leads to (C1) and (C2). Numerical values obtained for the constants C are reported in Table I, for different charge asymmetries z− : z+. The previous analysis shows that analytical predictions for C could be made if the con- nection problem is solved and the equivalent of expres- sions (D2) are found for the general case z− : z+. [1] C.F. Anderson and M.T. Record Jr, Annu. Rev. Phys. Chem. 33, 191 (1984). [2] H. Eisenberg, Biological Macromolecules and Polyelec- trolytes in Solution, Clarendon, Oxford (1976). [3] J.A. Schellman, Biophys. Chem. 37, 121 (1990). [4] S.M. Timasheff, Biochemistry 31, 9857 (1992). [5] G.S. Manning, J. Chem. Phys. 51, 924 (1969). [6] F. Oosawa, Polyelectrolytes, Dekker, New York (1971). [7] K.A. Sharp, Biopolymers 36, 227 (1995). [8] H. Ni, C.F. Anderson and M.T. Record Jr, J. Phys. Chem. B 103, 3489 (1999). [9] I.A. Shkel, O.V. Tsodikov and M.T. Record Jr, Proc. Natl. Acad. Sci. USA 99, 2597 (2002). [10] C.H. Taubes, U. Mohanty and S. Chu, J. Phys. Chem. B 109, 21267 (2005). [11] M. Gueron, J.-Ph. Demaret and M. Filoche, Biophys. Journal 78, 1070 (2000). [12] Y. Levin, Rep. Prog. Phys. 65, 1577 (2002). [13] In the 1:1 case, our definition differs from the more stan- dard one as found e.g. in [9] by a factor 4ξ. The reason for doing so is that this allows easier comparison of the salt dependence of Γ for different values of the poly-ion charge. [14] F.G. Donnan, Chem. Rev. 1, 73 (1924). [15] I.A. Shkel, O.V. Tsodikov and M.T. Record Jr, J. Phys. Chem. B 104, 5161 (2000). [16] M. Aubouy, E. Trizac, L. Bocquet, J. Phys. A: Math. Gen. 36, 5835 (2003). [17] G. Tellez and E.Trizac, Phys. Rev. E 70, 011404 (2004). [18] E. Trizac and G. Téllez, Phys. Rev. Lett 96, 038302 (2006) ; G. Téllez and E. Trizac, J. Stat. Mech. P06018 (2006). [19] B.M. McCoy, C.A. Tracy and T.T. Wu, J. Math. Phys. 18, 1058 (1977). [20] J.S. McCaskill and E.D. Fackerell, J. Chem. Soc., Fara- day Trans. 2 84, 161 (1988). [21] C.A. Tracy and H. Widom, Physica A 244, 402 (1997). [22] We emphasize that accurate results for Γ, φ etc may be obtained for ξ < ξc from the results given in [18]. We did not investigate this regime here, since it is of little relevance for nucleic acids. [23] A.Y. Grosberg, T.T. Nguyen and B.I. Shklovskii, Rev. Mod. Phys. 74, 329 (2002). [24] R.M. Fuoss, A. Katchalsky and S.F. Lifson, P. Natl. Acad. Sci. USA 37, 579 (1951). [25] It then appears that the expression given for eµ = 2µ in (B5) corresponds to the dominant term only in (B3) (the first one on the rhs). [26] Compared to the expressions given in [18] where a pa- rameter µ plays a key role, the corresponding change of variables should be performed: eµ = 3µ (1:2 case) and eµ = 3µ/2 for 2:1 electrolytes. [27] E. Trizac, L. Bocquet, M. Aubouy and H.H. von Grünberg, Langmuir 19, 4027 (2003). [28] M. Gueron and G. Weisbuch, Biopolymers 19, 353 (1980). [29] B. O’Shaughnessy and Q. Yang, Phys. Rev. Lett. 94, 048302 (2005). [30] M. Deserno, C. Holm and S. May, Macromolecules 33, 199 (2000). [31] C.A. Tracy and H. Widom, Commun. Math. Phys. 190, 697 (1998).

0704.0588

A new approach to mutual information

A NEW APPROACH TO MUTUAL INFORMATION FUMIO HIAI 1 AND DÉNES PETZ 2 Abstract. A new expression as a certain asymptotic limit via “discrete micro- states” of permutations is provided to the mutual information of both continuous and discrete random variables. Introduction One of the important quantities in information theory is the mutual information of two random variables X and Y which is expressed in terms of the Boltzmann-Gibbs entropy H(·) as follows: I(X ∧ Y ) = −H(X, Y ) +H(X) +H(Y ) when X, Y are continuous variables. For the expression of I(X∧Y ) of discrete variables X, Y , the aboveH(·) is replaced by the Shannon entropy. A more practical and rigorous definition via the relative entropy is I(X ∧ Y ) := S(µ(X,Y ), µX ⊗ µY ), where µ(X,Y ) denotes the joint distribution measure of (X, Y ) and µX⊗µY the product of the respective distribution measures of X, Y . The aim of this paper is to show that the mutual information I(X∧Y ) is gained as a certain asymptotic limit of the volume of “discrete micro-states” consisting of permu- tations approximating joint moments of (X, Y ) in some way. In Section 1, more gener- ally we consider an n-tuple of real bounded random variables (X1, . . . , Xn). Denote by ∆(X1, . . . , Xn;N,m, δ) the set of (x1, . . . ,xn) of xi ∈ R N whose joint moments (on the uniform distributed N -point set) of order up tom approximate those of (X1, . . . , Xn) up to an error δ. Furthermore, denote by ∆sym(X1, . . . , Xn;N,m, δ) the set of (σ1, . . . , σn) of permutations σi ∈ SN such that (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N,m, δ) for some x1, . . . ,xn ∈ R ≤ , where R ≤ is the R N -vectors arranged in increasing order. Then, the asymptotic volume log γ⊗nSN ∆sym(X1, . . . , Xn;N,m, δ) under the uniform probability measure γSN on SN is shown to converge as lim supN→∞ (also lim infN→∞) and then limm→∞,δց0 to −H(X1, . . . , Xn) + H(Xi) 1 Supported in part by Grant-in-Aid for Scientific Research (B)17340043. 2 Supported in part by the Hungarian Research Grant OTKA T068258. AMS subject classification: Primary: 62B10, 94A17. http://arxiv.org/abs/0704.0588v1 2 F. HIAI AND D. PETZ as long as H(Xi) > −∞ for 1 ≤ i ≤ n. Thus, we obtain a kind of discretization of the mutual information via symmetric group (or permutations). The approach can be applied to an n-tuple of discrete random variables (X1, . . . , Xn) as well. But the definition of the ∆sym-set of micro-states for discrete variables is somewhat different from the continuous variable case mentioned above, and we discuss the discrete variable case in Section 2 separately. The idea comes from the paper [3]. Motivated by theory of mutual free information in [6], a similar approach to Voiculescu’s free entropy is provided there. The free entropy is the free probability counterpart of the Boltzmann-Gibbs entropy, and RN -vectors and the symmetric group SN here are replaced by Hermitian N ×N matrices and the unitary group U(N), respectively. In this way, the “discretization approach” here is in some sense a classical analog of the “orbital approach” in [3]. 1. The continuous case For N ∈ N let RN≤ be the convex cone of the N -dimensional Euclidean space R consisting of x = (x1, . . . , xN ) such that x1 ≤ x2 ≤ · · · ≤ xN . The space R N is naturally regarded as the real function algebra on the N -point set. Let SN be the symmetric group of order N (i.e., the permutations on {1, 2, . . . , n}). Throughout this section let (X1, . . . , Xn) be an n-tuple of real random variables on a probability space (Ω,P), and assume that the Xi’s are bounded (i.e., Xi ∈ L ∞(Ω;P)). The Boltzmann-Gibbs entropy of (X1, . . . , Xn) is defined to be H(X1, . . . , Xn) := − · · · p(x1, . . . , xn) log p(x1, . . . , xn) dx1 · · · dxn if the joint density p(x1, . . . , xn) of (X1, . . . , Xn) exists; otherwise H(X1, . . . , Xn) = −∞. Note that the above integral is well defined in [−∞,∞) since the density p is compactly supported. Definition 1.1. The mean value of x = (x1, . . . , xN) in R N is given by κN(x) := For each N,m ∈ N and δ > 0 we define ∆(X1, . . . , Xn;N,m, δ) to be the set of all n-tuples (x1, . . . ,xn) of xi = (xi1, . . . , xiN ) ∈ R N , 1 ≤ i ≤ n, such that |κN(xi1 · · ·xik)− E(Xi1 · · ·Xik)| < δ for all 1 ≤ i1, . . . , ik ≤ n with 1 ≤ k ≤ m, where xi1 · · ·xik means the pointwise product, i.e., xi1 · · ·xik := (xi11 · · ·xik1, xi12 · · ·xik2, . . . , xi1N · · ·xikN ) ∈ R and E(·) denotes the expectation on (Ω,P). For each R > 0, define ∆R(X1, . . . , Xn; N,m, δ) to be the set of all (x1, . . . ,xn) ∈ ∆(X1, . . . , Xn;N,m, δ) such that xi ∈ [−R,R]N for all 1 ≤ i ≤ n. Heuristically, ∆(X1, . . . , Xn;N,m, δ) is the set of “micro-states” consisting of n- tuples of discrete random variables on the N -point set with the uniform probability A NEW APPROACH TO MUTUAL INFORMATION 3 such that all joint moments of order up to m give the corresponding joint moments of X1, . . . , Xn up to an error δ. For x ∈ RN write ‖x‖p := (N j=1 |xj | p)1/p for 1 ≤ p < ∞ and ‖x‖∞ := max1≤j≤N |xj| while ‖X‖p denotes the L p-norm of a real random variable X on (Ω,P). The next lemma is seen from [4, 5.1.1] based on the Sanov large deviation theorem, which says that the Boltzmann-Gibbs entropy is gained as an asymptotic limit of the volume of the approximating micro-states. Lemma 1.2. For every m ∈ N and δ > 0 and for any choice of R ≥ max1≤i≤n ‖Xi‖∞, the limit log λ⊗nN ∆R(X1, . . . , Xn;N,m, δ) exists, where λN is the Lebesgue measure on R N . Furthermore, one has H(X1, . . . , Xn) = lim m→∞,δց0 log λ⊗nN ∆R(X1, . . . , Xn;N,m, δ) independently of the choice of R ≥ max1≤i≤n ‖Xi‖∞. In the following let us introduce some kinds of mutual information in the discretiza- tion approach using micro-states of permutations. Definition 1.3. The action of SN on R N is given by σ(x) := (xσ−1(1), xσ−1(2), . . . , xσ−1(N)) for σ ∈ SN and x = (x1, . . . , xN) ∈ R N . For each N,m ∈ N, δ > 0 and R > 0 we denote by ∆sym,R(X1, . . . , Xn;N,m, δ) the set of all (σ1, . . . , σn) ∈ S N such that (σ1(x1), . . . , σn(xn)) ∈ ∆R(X1, . . . , Xn;N,m, δ) for some (x1, . . . ,xn) ∈ (R n. For each R > 0 define Isym,R(X1, . . . , Xn) := − lim m→∞,δց0 lim sup log γ⊗nSN ∆sym,R(X1, . . . , Xn;N,m, δ) where γSN is the uniform probability measure on SN . Define also Isym,R(X1, . . . , Xn) by replacing lim sup by lim inf. Obviously, 0 ≤ Isym,R(X1, . . . , Xn) ≤ Isym,R(X1, . . . , Xn). Moreover, ∆sym,∞(X1, . . . , Xn;N,m, δ) is defined by replacing ∆R(X1, . . . , Xn;N,m, δ) in the above by ∆(X1, . . . , Xn;N,m, δ) without cut-off by the parameter R. Then Isym,∞(X1, . . . , Xn) and Isym,∞(X1, . . . , Xn) are also defined as above. Definition 1.4. For each 1 ≤ i ≤ n we choose and fix a sequence ξi = {ξi(N)} of ξi(N) ∈ R ≤ , N ∈ N, such that κN (ξi(N) k) → E(Xki ) as N → ∞ for all k ∈ N, i.e., ξi(N) → Xi in moments. For each N,m ∈ N and δ > 0 we define ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ) to be the set of all (σ1, . . . , σn) ∈ S N such that (σ1(ξ1(N)), . . . , σn(ξn(N))) ∈ ∆(X1, . . . , Xn;N,m, δ). 4 F. HIAI AND D. PETZ Define Isym(X1, . . . , Xn : ξ1, . . . , ξn) := − lim m→∞,δց0 lim sup log γ⊗nSN ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ) and Isym(X1, . . . , Xn : ξ1, . . . ξn) by replacing lim sup by lim inf. The next proposition asserts that the quantities in Definitions 1.3 and 1.4 are all equivalent. Lemma 1.5. For any choice of R ≥ max1≤i≤n ‖Xi‖∞ and for any choices of approxi- mating sequences ξ1, . . . , ξn one has Isym,∞(X1, . . . , Xn) = Isym,R(X1, . . . , Xn) = Isym(X1, . . . , Xn : ξ1, . . . , ξn), (1.1) Isym,∞(X1, . . . , Xn) = Isym,R(X1, . . . , Xn) = Isym(X1, . . . , Xn : ξ1, . . . , ξn). (1.2) Proof. It is obvious that ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ) is included in ∆sym,∞(X1, . . . , Xn;N,m, δ) for any approximating sequences ξi. Moreover, for each 1 ≤ i ≤ n an approximating sequence ξi can be chosen so that ‖ξi(N)‖∞ ≤ ‖Xi‖∞ for all N ; then ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ) ⊂ ∆sym,R(X1, . . . , Xn; N,m, δ) for any R ≥ R0 := max1≤i≤n ‖Xi‖∞. Hence it suffices to prove that for any approximating sequences ξi and for every m ∈ N and δ > 0, there are an m ′ ∈ N, a δ′ > 0 and an N0 ∈ N so that ∆sym,∞(X1, . . . , Xn;N,m ′, δ′) ⊂ ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ) for all N ≥ N0. Choose a ρ ∈ (0, 1) with m(R0 + 1) m−1ρ < δ/2. By [5, Lemma 4.3] (also [4, 4.3.4]) there exist an m′ ∈ N with m′ ≥ 2m, a δ′ > 0 with δ′ ≤ min{1, δ/2} and an N0 ∈ N such that for every 1 ≤ i ≤ n and every x ∈ R ≤ with N ≥ N0, if |κN (x k) − E(Xki )| < δ ′ for all 1 ≤ k ≤ m′, then ‖x − ξi(N)‖m < ρ. Suppose N ≥ N0 and (σ1, . . . , σn) ∈ ∆sym,∞(X1, . . . , Xn;N,m ′, δ′); then (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N,m ′, δ′) for some (x1, . . . ,xn) ∈ (R n. Since |κN(x i ) − E(X i )| < δ for all 1 ≤ k ≤ m′, we get ‖xi − ξi(N)‖m ≤ ρ and ‖xi‖m ≤ ‖xi‖2m = κN(x < (E(X2mi ) + 1) ≤ (R2m0 + 1) 1/2m ≤ R0 + 1. Therefore, |κN(σi1(ξi1(N)) · · ·σik(ξik(N)))− E(Xi1 · · ·Xik)| ≤ |κN(σi1(ξi1(N)) · · ·σik(ξik(N)))− κN(σi1(xi1) · · ·σik(xik))| + |κN(σi1(xi1) · · ·σik(xik))− E(Xi1 · · ·Xik)| ≤ m(R0 + 1) m−1ρ+ δ′ < δ for all 1 ≤ i1, . . . , ik ≤ n with 1 ≤ k ≤ m. The above latter inequality follows from the Hölder inequality. Hence (σ1, . . . , σn) ∈ ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ), and the result follows. � A NEW APPROACH TO MUTUAL INFORMATION 5 Consequently, we denote all the quantities in (1.1) by the same Isym(X1, . . . , Xn) and those in (1.2) by Isym(X1, . . . , Xn). We call Isym(X1, . . . , Xn) and Isym(X1, . . . , Xn) the mutual information and upper mutual information of (X1, . . . , Xn), respectively. The terminology “mutual information” will be justified after the next theorem. In the continuous variable case, our main result is the following exact relation of Isym and Isym with the Boltzmann-Gibbs entropy H(·), which says that Isym(X1, . . . , Xn) is formally the sum of the separate entropiesH(Xi)’s minus the compoundH(X1, . . . , Xn). Thus, a naive meaning of Isym(X1, . . . , Xn) is the entropy (or information) overlapping among the Xi’s. Theorem 1.6. H(X1, . . . , Xn) = −Isym(X1, . . . , Xn) + H(Xi) = −Isym(X1, . . . , Xn) + H(Xi). Proof. If the coordinates si of s ∈ R N are all distinct, then s is uniquely written as s = σ(x) with x ∈ RN≤ and σ ∈ SN . Note that the set of s ∈ R N with si = sj for some i 6= j is a closed subset of λN -measure zero. Under the correspondence s ∈ RN ←→ (x, σ) ∈ RN≤ × SN , s = σ(x) (well defined on a co-negligible subset of RN), the measure λN is transformed into the product of λN |RN and the counting measure on SN . In the following proof we adopt, due to Lemma 1.5, the description of Isym and Isym as Isym,R(X1, . . . , Xn) and Isym,R(X1, . . . , Xn) with R := max1≤i≤n ‖Xi‖∞. For each N,m ∈ N and δ > 0, suppose (s1, . . . , sn) ∈ ∆R(X1, . . . , Xn;N,m, δ) and write si = σi(xi) with xi ∈ R ≤ and σi ∈ SN . Then it is obvious that (x1, . . . ,xn; σ1, . . . , σn) ∆R(Xi;N,m, δ) ∩ R ×∆sym,R(X1, . . . , Xn;N,m, δ). By Lemma 1.2 and the fact stated at the beginning of the proof, we obtain H(X1, . . . , Xn) ≤ lim log λ⊗nN ∆R(X1, . . . , Xn;N,m, δ) ≤ lim inf log λN ∆R(Xi;N,m, δ) ∩ R + log#∆sym,R(X1, . . . , Xn;N,m, δ) = lim inf log λN ∆R(Xi;N,m, δ) − n logN ! 6 F. HIAI AND D. PETZ + log#∆sym,R(X1, . . . , Xn;N,m, δ) log λN ∆R(Xi;N,m, δ) + lim inf log γ⊗nSN ∆sym,R(X1, . . . , Xn;N,m, δ) This implies that H(X1, . . . , Xn) ≤ H(Xi)− Isym(X1, . . . , Xn). (1.3) Conversely, for each m ∈ N and δ > 0, by [5, Lemma 4.3] (also [4, 4.3.4]) there are an m′ ∈ N with m′ ≥ m, a δ′ > 0 with δ′ ≤ δ/2 and an N0 ∈ N such that for every N ∈ N and for every x,y ∈ RN≤ , if ‖x‖∞ ≤ R and |κN(x k) − κN(y k)| < 2δ′ for all 1 ≤ k ≤ m′, then ‖x− y‖1 < δ/2m(R + 1) m−1. Suppose N ≥ N0 and (x1, . . . ,xn; σ1, . . . , σn) ∆R(Xi;N,m ′, δ′) ∩ RN≤ ×∆sym,R(X1, . . . , Xn;N,m ′, δ′) so that (σ1(y1), . . . , σn(yn)) ∈ ∆R(X1, . . . , Xn;N,m ′, δ′) for some (y1, . . . ,yn) ∈ (R Since |κN(x i )− κN(y i )| ≤ |κN(x i )− E(X i )|+ |κN(y i )− E(X i )| < 2δ for all 1 ≤ k ≤ m′, we get ‖xi − yi‖1 < δ/2m(R + 1) m−1 for 1 ≤ i ≤ n. Therefore, |κN (σi1(xi1) · · ·σik(xik))− E(Xi1 · · ·Xik)| ≤ |κN(σi1(xi1) · · ·σik(xik))− κN(σi1(yi1) · · ·σik(yik))| + |κN(σi1(yi1) · · ·σik(yik))− E(Xi1 · · ·Xik)| ≤ m(R + 1)m−1 max 1≤i≤n ‖xi − yi‖1 + δ + δ′ ≤ δ for all 1 ≤ i1, . . . , ik ≤ n with 1 ≤ k ≤ m. This implies that (σ1(x1), . . . , σn(xn)) ∈ ∆R(X1, . . . , Xn;N,m, δ). By Lemma 1.2 we obtain H(Xi)− Isym(X1, . . . , Xn) log λN ∆R(Xi;N,m ′, δ′) + lim sup log γ⊗nSN ∆sym,R(X1, . . . , Xn;N,m ′, δ′) = lim sup log λN ∆R(Xi;N,m ′, δ′) ∩ RN≤ A NEW APPROACH TO MUTUAL INFORMATION 7 + log#∆sym,R(X1, . . . , Xn;N,m ′, δ′) ≤ lim sup log λ⊗nN ∆R(X1, . . . , Xn;N,m, δ) This implies by Lemma 1.2 once again that H(Xi)− Isym(X1, . . . , Xn) ≤ H(X1, . . . , Xn). (1.4) The result follows from (1.3) and (1.4). � Let µ(X1,...,Xn) be the joint distribution measure on R n of (X1, . . . , Xn) while µXi is that of Xi for 1 ≤ i ≤ n. Let S(µ(X1,...,Xn), µX1 ⊗· · ·⊗µXn) denote the relative entropy (or the Kullback-Leibler divergence) of µ(X1,...,Xn) with respect to the product measure µX1 ⊗ · · · ⊗ µXn , i.e., S(µ(X1,...,Xn), µX1 ⊗ · · · ⊗ µXn) := dµ(X1,...,Xn) d(µX1 ⊗ · · · ⊗ µXn) dµ(X1,...,Xn) if µ(X1,...,Xn) is absolutely continuous with respect to µX1 ⊗ · · · ⊗ µXn ; otherwise S(µ(X1,...,Xn), µX1 ⊗ · · · ⊗ µXn) := +∞. When H(Xi) > −∞ for all 1 ≤ i ≤ n, one can easily verify that S(µ(X1,...,Xn), µX1 ⊗ · · · ⊗ µXn) = −H(X1, . . . , Xn) + H(Xi). Thus, the above theorem yields the following: Corollary 1.7. If H(Xi) > −∞ for all 1 ≤ i ≤ n, then Isym(X1, . . . , Xn) = Isym(X1, . . . , Xn) = S(µ(X1,...,Xn), µX1 ⊗ · · · ⊗ µXn). Corollary 1.8. Under the same assumption as the above corollary, Isym(X1, . . . , Xn) = 0 if and only if X1, . . . , Xn are independent. In particular, the originalmutual information I(X1∧X2) of two real random variables X1, X2 is normally defined as I(X1 ∧X2) := S(µ(X1,X2), µX1 ⊗ µX2). Hence we have I(X1 ∧X2) = Isym(X1, X2) = Isym(X1, X2) as long as H(X1) > −∞ and H(X2) > −∞ (and X1, X2 are bounded). For this reason, we gave the term “mutual information” to Isym. Finally, some open problems are in order: (1) Without the assumption H(Xi) > −∞ for 1 ≤ i ≤ n, does Isym(X1, . . . , Xn) = Isym(X1, . . . , Xn) hold true? 8 F. HIAI AND D. PETZ (2) More strongly, does the limit such as log γ⊗nSN (∆sym,R(X1, . . . , Xn;N,m, δ)) log γ⊗nSN (∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N,m, δ)) exist as in Lemma 1.2? (3) Without the assumption H(Xi) > −∞ for 1 ≤ i ≤ n, does Isym(X1, . . . , Xn) = S(µ(X1,...,Xn), µX1⊗· · ·⊗µXn) hold true? Also, is Isym(X1, . . . , Xn) = 0 equivalent to the independence of X1, . . . , Xn? (4) Although the boundedness assumption for X1, . . . , Xn is rather essential in the above discussions, it is desirable to extend the results in this section to X1, . . . , Xn not necessarily bounded but having all moments. 2. The discrete case Let Y be a finite set with a probability measure p. The Shannon entropy of p is S(p) := − p(y) log p(y). For each sequence y = (y1, . . . , yN) ∈ Y N , the type of y is a probability measure on Y given by νy(t) := Ny(t) where Ny(t) := #{j : yj = t}, t ∈ Y . The number of possible types is smaller than (N + 1)#Y . If ν is a type and TN(ν) denotes the set of all sequences of type ν from YN , then the cardinality of TN(ν) is estimated as follows: (N + 1)#Y eNS(ν) ≤ #TN (ν) ≤ e NS(ν) (2.1) (see [1, 12.1.3] and [2, Lemma 2.2]). Let p be a probability meausre on Y . For each N ∈ N and δ > 0 we define ∆(p;N, δ) to be the set of all sequences y ∈ YN such that |νy(t)−p(t)| < δ for all t ∈ Y . In other words, ∆(p;N, δ) is the set of all δ-typical sequeces (with respect to the measure p). Then the next lemma is well known. Lemma 2.1. S(p) = lim log#∆(p;N, δ). In fact, this easily follows from (2.1). Let PN,δ be the maximizer of the Shannon entropy on the set of all types νy, y ∈ Y N , such that |νy(t) − p(t)| < δ for all t ∈ Y . We can use the Shannon entropy of the type class corresponding to PN,δ to estimate the cardinality of ∆(p;N, δ): (N + 1)−#YeNS(PN,δ) ≤ #∆(p;N, δ) ≤ eNS(PN,δ)(N + 1)#Y . A NEW APPROACH TO MUTUAL INFORMATION 9 It follows that log#∆(p;N, δ) = sup{S(q) : q is a probability meausre on Y such that |q(t)− p(t)| < δ, t ∈ Y}, and the lemma follows. We consider the case where p is the joint distribution of an n-tuple (X1, . . . , Xn) of discrete random variables on (Ω,P). Throughout this section we assume that the random variables X1, . . . , Xn have their values in a finite set X = {t1, . . . , td}. Definition 2.2. Let p(X1,...,Xn) denote the joint distribution of (X1, . . . , Xn), which is a measure on X n while the distribution pXi of Xi is a measure on X , 1 ≤ i ≤ n. We write ∆(Xi;N, δ) for ∆(pXi;N, δ) and ∆(X1, . . . , Xn;N, δ) for ∆(p(X1,...,Xn);N, δ). Next, we introduce the counterparts of Definitions 1.3 and 1.4 in the discrete variable case. Definition 2.3. The action of SN on X N is similar to that on RN given in Defintion 1.3. For N ∈ N let XN≤ denote the set of all sequences of length N of the form x = (t1, . . . , t1, t2, . . . , t2, . . . , td, . . . , td). Oviously, such a sequence x is uniquely determined by (Nx(t1), . . . , Nx(td)) or the type of x. That is, XN≤ is regarded as the set of all types from X N . For each N ∈ N and δ > 0 we denote by ∆sym(X1, . . . , Xn;N, δ) the set of all (σ1, . . . , σn) ∈ S N such that (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N, δ) for some (x1, . . . ,xn) ∈ (X n. Define Isym(X1, . . . , Xn) := − lim lim sup log γ⊗nSN (∆sym(X1, . . . , Xn;N, δ)), and Isym(X1, . . . , Xn) by replacing lim sup by lim inf. Moreover, for each 1 ≤ i ≤ n, choose a sequence ξi = {ξi(N)} of ξi(N) = (ξi(N)1, . . . , ξi(N)N) ∈ X ≤ such that νξi(N) → pXi as N → ∞. We then define ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N, δ), Isym(X1, . . . , Xn : ξ1, . . . , ξn) and Isym(X1, . . . , Xn : ξ1, . . . , ξn) as in Definition 1.4. Lemma 2.4. For any choices of approximating sequences ξ1, . . . , ξn one has Isym(X1, . . . , Xn) = Isym(X1, . . . , Xn : ξ1, . . . , ξn), Isym(X1, . . . , Xn) = Isym(X1, . . . , Xn : ξ1, . . . , ξn). Proof. It suffices to show that for each δ > 0 there are a δ′ > 0 and an N0 ∈ N such ∆sym(X1, . . . , Xn;N, δ ′) ⊂ ∆sym(X1, . . . , Xn : ξ1(N), . . . , ξn(N);N, δ) (2.2) for all N ≥ N0. Choose δ ′ > 0 so that 3ndn+1δ′ ≤ δ, where d = #X . Suppose (σ1, . . . , σn) is in the left-hand side of (2.2) so that (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn; N, δ′) for some (x1, . . . ,xn), xi = (xi1, . . . , xiN) ∈ X ≤ . Since |ν(σ1(x1),...,σn(xn))(z1, . . . , zn)− p(X1,...,Xn)(z1, . . . , zn)| < δ ′, (z1, . . . , zn) ∈ X n, (2.3) 10 F. HIAI AND D. PETZ νxi(t) = z1,...,zi−1,zi+1,...,zn∈X ν(σ1(x1),...,σn(xn))(z1, . . . , zi−1, t, zi+1, . . . , zn), t ∈ X , pXi(t) = z1,...,zi−1,zi+1,...,zn∈X p(X1,...,Xn)(z1, . . . , zi−1, t, zi+1, . . . , zn), t ∈ X , it follows that |νxi(t)− pXi(t)| < d n−1δ′ (2.4) for any 1 ≤ i ≤ n and t ∈ X . Now, choose an N0 ∈ N so that |νξi(N)(t)− pXi(t)| < δ and hence |νξi(N)(t)− νxi(t)| < 2d n−1δ′ (2.5) for any 1 ≤ i ≤ n and t ∈ X and for all N ≥ N0. Since |(Nξi(N)(t1) + · · ·+Nξi(N)(tl))− (Nxi(t1) + · · ·+Nxi(tl))| ≤ |Nξi(N)(t1)−Nxi(t1)|+ · · ·+ |Nξi(N)(tl)−Nxi(tl)| < 2Ndnδ′ for every 1 ≤ l ≤ d thanks to (2.5), it is easily seen that j ∈ {1, . . . , N} : ξi(N)j 6= xij < 2Ndn+1δ′ for any 1 ≤ i ≤ n. Hence we get |ν(σ1(ξ1(N)),...,σn(ξn(N)))(z1, . . . , zn)− ν(σ1(x1),...,σn(xn))(z1, . . . , zn)| ∣#{j : ξ1(N)σ−1 (j) = z1, . . . , ξn(N)σ−1n (j) = zn} −#{j : x1σ−1 (j) = z1, . . . , xnσ−1n (j) = zn} #{j : ξi(N)j 6= xij} < 2nd n+1δ′ so that thanks to (2.3) |ν(σ1(ξ1(N)),...,σn(ξn(N)))(z1, . . . , zn)− p(X1,...,Xn)(z1, . . . , zn)| < 3nd n+1δ′ ≤ δ for every (z1, . . . , zn) ∈ X n. Therefore, (σ1, . . . , σn) is in the right-hand side of (2.2), as required. � The next theorem is the discrete variable version of Theorem 1.6. Theorem 2.5. Isym(X1, . . . , Xn) = Isym(X1, . . . , Xn) = −S(X1, . . . , Xn) + S(Xi). Proof. For each sequence (N1, . . . , Nd) of integers Nl ≥ 0 with l=1Nl = N , let S(N1, . . . , Nd) denote the subgroup of SN consisting of products of permutations of {1, . . . , N1}, {N1 + 1, . . . , N1 +N2}, . . . , {N1 + · · ·+Nd−1 + 1, . . . , N}, and let SN/S(N1, . . . , Nd) be the set of left cosets of S(N1, . . . , Nd). For each x ∈ X ≤ and σ ∈ SN we write [σ]x for the left coset of S(Nx(t1), . . . , Nx(td)) containing σ. Then it is clear that A NEW APPROACH TO MUTUAL INFORMATION 11 every s ∈ XN is represented as s = σ(x) with a unique pair (x, [σ]x) of x ∈ X ≤ and [σ]x ∈ SN/S(Nx(t1), . . . , Nx(td)). For any ε > 0 one can choose a δ > 0 such that for every 1 ≤ i ≤ n and every probability measure p on X , if |p(t)−pXi(t)| < δ for all t ∈ X , then |S(p)−S(pXi)| < ε. This implies that for each N ∈ N and 1 ≤ i ≤ n, one has |S(νx) − S(pXi)| < ε whenever x ∈ ∆(Xi;N, δ). Notice that ∆sym(X1, . . . , Xn;N, δ/d n−1) is the union of [σ1]x1 × · · · × [σn]xn for all (x1, . . . ,xn; [σ1]x1 , . . . , [σn]xn) of xi ∈ X ≤ and [σi]xi ∈ SN/S(Nxi(t1), . . . , Nxi(td)) such that (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N, δ/d n−1). Now, suppose (x1, . . . ,xn) ∈ (X n, (σ1, . . . , σn) ∈ S N and (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N, δ/d n−1). Then, for each 1 ≤ i ≤ n we get xi ∈ ∆(Xi;N, δ), i.e., |νxi(t)− pXi(t)| < δ for all t ∈ X as (2.4). Hence we have [σ1]x1 × · · · × [σn]xn x∈∆(Xi;N,δ) Nx(t)! (2.6) so that #∆sym(X1, . . . , Xn;N, δ/d ≤ #∆(X1, . . . , Xn;N, δ/d n−1) · x∈∆(Xi;N,δ) Nx(t)! Therefore, log γ⊗nSN ∆sym(X1, . . . , Xn;N, δ/d log#∆(X1, . . . , Xn;N, δ/d x∈∆(Xi;N,δ) logNx(t)! logN !. (2.7) For each 1 ≤ i ≤ n and for any x ∈ ∆(Xi;N, , δ), the Stirling formula yields logNx(t)!− logN ! Nx(t) logNx(t)− Nx(t) − logN + 1 + o(1) = −S(νx) + o(1) ≤ −S(pXi) + ε+ o(1) as N →∞ (2.8) thanks to the above choice of δ > 0. Here, note that the o(1) in the above estimate is uniform for x ∈ ∆(Xi;N, δ). Hence, by (2.7), (2.8) and by Lemma 2.1 applied to p(X1,...,Xn) on X n, we obtain −Isym(X1, . . . , Xn) ≤ S(p(X1,...,Xn))− S(pXi) + nε and hence Isym(X1, . . . , Xn) ≥ −S(X1, . . . , Xn) + S(Xi). (2.9) 12 F. HIAI AND D. PETZ Next, we prove the converse direction. For any ε > 0 choose a δ > 0 as above. For N ∈ N let Ξ(N, δ/dn−1) be the set of all (x1, . . . ,xn) ∈ (X n such that (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N, δ/d for some (σ1, . . . , σn) ∈ S N . Furthermore, for each (x1, . . . ,xn) ∈ Ξ(N, δ/d n−1), let Σ(x1, . . . ,xn;N, δ/d n−1) be the set of all ([σ1]x1 , . . . , [σn]xn) ∈ SN/S(Nxi(t1), . . . , Nxi(td)) such that (σ1(x1), . . . , σn(xn)) ∈ ∆(X1, . . . , Xn;N, δ/d n−1). Then it is obvious that #∆(X1, . . . , Xn;N, δ/d n−1) ≤ (x1,...,xn)∈Ξ(N,δ/dn−1) #Σ(x1, . . . ,xn;N, δ/d n−1). (2.10) When (x1, . . . ,xn) ∈ Ξ(N, δ/d n−1), we get xi ∈ ∆(Xi;N, δ) as (2.4) for 1 ≤ i ≤ n. Hence it is seen that #Ξ(N, δ/dn−1) ≤ #∆(Xi;N, δ) (N1, . . . , Nd) : Nl ≥ 0 is an integer in N(pXi(tl)− δ), N(pXi(tl) + δ) for 1 ≤ l ≤ d < (2Nδ + 1)nd. (2.11) For any fixed (x1, . . . ,xn) ∈ Ξ(N, δ/d n−1), suppose ([σ1]x1 , . . . , [σn]xn) ∈ Σ(x1, . . . ,xn; N, δ/dn−1); then we get [σ1]x1 × · · · × [σn]xn x∈∆(Xi;N,δ) Nx(t)! similarly to (2.6). Therefore, #∆sym(X1, . . . , Xn;N, δ/d ([σ1]x1 ,...,[σn]xn )∈Σ(x1,...,xn;N,δ/d [σ1]x1 × · · · × [σn]xn ≥ #Σ(x1, . . . ,xn;N, δ/d n−1) · x∈∆(Xi;N,δ) Nx(t)! . (2.12) By (2.10)–(2.12) we obtain #∆(X1, . . . , Xn;N, δ/d n−1) ≤ #∆sym(X1, . . . , Xn;N, δ/d n−1) · (2Nδ + 1)nd minx∈∆(Xi;N,δ) t∈X Nx(t)! A NEW APPROACH TO MUTUAL INFORMATION 13 so that log#∆(X1, . . . , Xn;N, δ/d log γ⊗nSN ∆sym(X1, . . . , Xn;N, δ/d x∈∆(Xi;N,δ) logNx(t)! logN ! + log(2Nδ + 1). Since it follows similarly to (2.8) that logNx(t)! + logN ! ≤ S(pXi) + ε+ o(1) as N →∞ with uniform o(1) for all x ∈ ∆(Xi;N, δ), we obtain S(p(X1,...,Xn)) ≤ −Isym(X1, . . . , Xn) + S(pXi) + nε by Lemma 2.1 again, and hence Isym(X1, . . . , Xn) ≤ −S(X1, . . . , Xn) + S(Xi). (2.13) The conclusion follows from (2.9) and (2.13). � In particular, the mutual information I(X1 ∧ X2) of X1 and X2 is equivalently ex- pressed as I(X1 ∧X2) = S(p(X1,X2), pX1 ⊗ pX2) = −S(p(X1,X2)) + S(pX1) + S(pX2) = Isym(X1, X2) = Isym(X1, X2). Similarly to the problem (2) mentioned in the last of Section 1, it is unknown whether the limit log γ⊗nSN ∆sym(X1, . . . , Xn;N, δ) exists or not. References [1] T. M. Cover and J. A. Thomas, Elements of Information Theory, Second edition, Wiley- Interscience, Hoboken, NJ, 2006. [2] I. Csiszár and P. C. Shields, Information Theory and Statistics: A Tutorial, in “Foundations and Trends in Communications and Information Theory,” Vol. 1, No. 4 (2004), 417-528, Now Publishers. [3] F. Hiai, T. Miyamoto and Y. Ueda, Orbital approach to microstate free entropy, preprint, 2007, math.OA/0702745. [4] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs, Vol. 77, Amer. Math. Soc., Providence, 2000. [5] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, II, Invent. Math. 118 (1994), 411–440. [6] D. Voiculescu, The analogue of entropy and of Fisher’s information measure in free probability theory VI: Liberation and mutual free information, Adv. Math. 146 (1999), 101–166. http://arxiv.org/abs/math/0702745 14 F. HIAI AND D. PETZ Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980- 8579, Japan Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13-15, Hungary Introduction 1. The continuous case 2. The discrete case References

0704.0589

Analysis of the real estate market in Las Vegas: Bubble, seasonal patterns, and prediction of the CSW indexes

Analysis of the real estate market in Las Vegas: Bubble, seasonal patterns, and prediction of the CSW indexes 1 Wei-Xing Zhou a, Didier Sornette b,∗ aSchool of Business, School of Science, and Research Center of Systems Engineering, East China University of Science and Technology, Shanghai 200237, China bD-MTEC, ETH Zurich, CH-8032 Zurich, Switzerland Abstract We analyze 27 house price indexes of Las Vegas from Jun. 1983 to Mar. 2005, corresponding to 27 different zip codes. These analyses confirm the existence of a real-estate bubble, defined as a price acceleration faster than exponential, which is found however to be confined to a rather limited time interval in the recent past from approximately 2003 to mid-2004 and has progressively transformed into a more normal growth rate comparable to pre-bubble levels in 2005. There has been no bubble till 2002 except for a medium-sized surge in 1990. In addition, we have identified a strong yearly periodicity which provides a good potential for fine-tuned prediction from month to month. A monthly monitoring using a model that we have developed could confirm, by testing the intra-year structure, if indeed the market has returned to “normal” or if more turbulence is expected ahead. We predict the evolution of the indexes one year ahead, which is validated with new data up to Sep. 2006. The present analysis demonstrates the existence of very significant variations at the local scale, in the sense that the bubble in Las Vegas seems to have preceded the more global USA bubble and has ended approximately two years earlier (mid 2004 for Las Vegas compared with mid-2006 for the whole of the USA). Key words: Econophysics, Real estate market, Periodicity, Power law, Prediction ∗ Corresponding author. Address: KPL F 38.2, Kreuzplatz 5, ETH Zurich, CH-8032 Zurich, Switzerland. Phone: +41 44 632 89 17, Fax: +41 44 632 19 14. Email addresses: [email protected] (Wei-Xing Zhou), [email protected] (Didier Sornette). URL: http://www.er.ethz.ch/ (Didier Sornette). 1 We are grateful to Signature Homes (Las Vegas) for providing us the data. This work was partially supported by the NSFC (Grant 70501011), the Fok Ying Tong Preprint submitted to International Journal of Forecasting 23 October 2018 http://arxiv.org/abs/0704.0589v1 1 Introduction Zhou and Sornette (2003) analyzed the deflated quarterly average sales prices p(t) from December 1992 to December 2002 of new houses sold in all the states in the USA and by regions (northeast, midwest, south and west) and found that, while there was undoubtedly a strong growth rate, there was no evidence of a bubble in the latest six years (as qualified by a super-exponential growth). Then, Zhou and Sornette (2006) analyzed the quarterly average sale prices of new houses sold in the USA as a whole, in the northeast, midwest, south, and west of the USA, in each of the 50 states and the District of Columbia of the USA up to the first quarter of 2005, to determine whether they have grown faster-than-exponential (which is taken as the diagnostic of a bubble). Zhou and Sornette (2006) found that 22 states (mostly Northeast and West) exhibit clear-cut signatures of a fast growing bubble. From the analysis of the S&P 500 Home Index, they concluded that the turning point of the bubble would probably occur around mid-2006. The specific statement found at the bottom of page 306 of Ref.[Zhou and Sornette (2006)] is: “We observe a good stability of the predicted tc ≈ mid-2006 for the two LPPL models (2) and (3). The spread of tc is larger for the second-order LPPL fits but brackets mid- 2006. As mentioned before, the power-law fits are not reliable. We conclude that the turning point of the bubble will probably occur around mid-2006.” It should be stressed that these studies departed from most other reports by analysts and consulting firms on real estate prices in that Zhou and Sornette (2003, 2006) did not characterize the housing market as overpriced in 2003. It is only in 2004-2005 that they confirmed that the signatures of an unsustainable bubble path has been revealed. Let us briefly analyze how this prediction has fared. The upper panel of Figure 1 shows the quarterly house price indexes (HPIs) in the 21 states and in the District of Columbia (DC) from 1994 to the fourth quarter of 2006 released by the OFHEO. It is evident that the growth in most of these 22 HPIs has slowed down or even stopped during the year of 2006. When we look at the S&P Case-Shiller Home Indexes of the 20 major US cities, as illustrated in the lower panel of Figure 1, we observe that the majority of the S&P/CSIs had a maximum denoted by a solid dot in the middle of 2006, validating the pre- diction of Zhou and Sornette (2006). Specifically, the times of the maxima are respectively 2006/06/01, 2006/09/01, 2005/11/01, 2006/05/01, 2006/08/01, 2006/05/01, 2006/12/01, 2006/07/01, 2006/08/01, 2006/09/01, 2005/09/01, 2005/12/01, 2006/09/01, 2006/09/01, 2006/08/01, 2006/06/01, 2006/07/01, 2006/09/01, 2006/08/01, 2006/12/01, 2006/06/01, and 2006/07/01 for the 20 cities shown in the legend of the lower panel. The only two cities with a max- Education Foundation (Grant 101086), and the Alfred Kastler Foundation which supported W.-X. Zhou for a visiting position in France. imum occurring later towards the end of 2006 (2006/12/01) are Miami and Seattle. However their growth rates decreased remarkably in 2006 as shown in the figure. Furthermore, the S&P/CS Home Price Composite-10 reached its historical high 226.29 on 2006/06/01 and the Composite-20 culminated to 206.53 on 2006/07/01, again confirming remarkably well the validity of the forecast of Zhou and Sornette (2006). 1994 1996 1998 2000 2002 2004 2006 2008 2010 2000 2001 2002 2003 2004 2005 2006 2007 Phoenix − AZ Los Angeles San Diego San Francisco Denver Washington Miami Tampa − FL Atlanta − GA Chicago Boston Detroit − MI Minneapolis − MN Charlotte − NC Las Vegas New York Cleveland − OH Portland − OR Dallas − TX Seattle − WA Fig. 1. Evaluation of the prediction of Zhou and Sornette (2006) that “the turning point of the bubble will probably occur around mid-2006” using the OFHEO HPI data (upper panel) and the S&P CSI data (lower panel). In this note, we provide a more regional study of the diagnostic of bubbles and the prediction of their demise. Specifically, we analyze the Case-Shiller- Weiss (CSW) Zip Code Indexes of 27 different Las Vegas regions calculated with a monthly rate from June-1983 to March-2005. The CSW Indexes are based on the so-called repeat sales methods which directly measure house price appreciations. The key to these data is that they are observations of multiple transactions on the same property, repeated over many properties and then pooled in an index. Prices from different time periods are combined to create “matched pairs,” providing a direct measure of price changes for a given property over a known period of time. Bailey et al. (1963) proposed the basic repeat sales method over four decades ago, but only after the work by Case and Shiller (1987, 1989, 1990) did the idea receive significant attention in the housing research community. Studying the Las Vegas database is particular suitable since Las Vegas belongs to a state which was identified by Zhou and Sornette (2006) as one of the 22 states with a fast growing bubble in 2005. With access to 27 different CSW Zip Code Indexes of Las Vegas, we are able to obtain more reliable and fine-grained measures, which both confirm and extend the previous analyses of Zhou and Sornette (2003, 2006). The next section recalls the conceptual background underlying our empirical approach. Then, section 3 analyzes the regional CSW indexes for Las Vegas, showing that there is a regime shift separated by a bubble around year 2004. Section 4 identifies and then analyzes the yearly periodicity and intra-year pattern detected in the growth rate of the regional CSW indexes. Section 5 offers a preliminary forecast based on the periodicity analyses in Sec. 4. Section 6 concludes. 2 Conceptual background of our empirical analysis 2.1 Humans as social animals and herding Humans are perhaps the most social mammals and they shape their envi- ronment to their personal and social needs. This statement is based on a growing body of research at the frontier between new disciplines called neuro- economics, evolutionary psychology, cognitive science, and behavioral finance. This body of evidence emphasizes the very human nature of humans with its biases and limitations, opposed to the previously prevailing view of ratio- nal economic agents optimizing their decisions based on unlimited access to information and to computation resources. Here, we focus on an empirical question (the existence and detection of real- estate bubbles) which, we hypothesize, is a footprint of perhaps the most robust trait of humans and the most visible imprint in our social affairs: im- itation and herding. Imitation has been documented in psychology and in neuro-sciences as one of the most evolved cognitive process, requiring a de- veloped cortex and sophisticated processing abilities. In short, we learn our basics and how to adapt mostly by imitation all along our life. It seems that imitation has evolved as an evolutionary advantageous trait, and may even have promoted the development of our anomalously large brain (compared with other mammals). It is actually “rational” to imitate when lacking suffi- cient time, energy and information to take a decision based only on private information and processing, that is..., most of the time. Imitation, in obvious or subtle forms, is a pervasive activity of humans. In the modern business, economic and financial worlds, the tendency for humans to imitate leads in its strongest form to herding and to crowd effects. Based on a theory of cooperative herding and imitation, we have shown that imitation leads to positive feedbacks, that is, an action leads to consequences which themselves reinforce the action and so on, leading to virtuous or vicious circles. We have formalized these ideas in a general mathematical theory which has led to observable signature of herding, in the form of so-called log-periodic power law acceleration of prices. A power law acceleration of prices reflects the positive feedback mechanism. When present, log-periodicity takes into account the competition between positive feedback (self-fulfilling sentiment), negative feedbacks (contrariant behavior and fundamental/value analysis) and inertia (everything takes time to adjust). Sornette (2003) presented a general introduction, a synthesis and examples of applications. 2.2 Definition and mechanism for bubbles The term “bubble” is widely used but rarely clearly defined. Following Case and Shiller (2003), the term “bubble” refers to a situation in which excessive public ex- pectations of future price increases cause prices to be temporarily elevated. During a housing price bubble, homebuyers think that a home that they would normally consider too expensive for them is now an acceptable purchase be- cause they will be compensated by significant further price increases. They will not need to save as much as they otherwise might, because they expect the increased value of their home to do the saving for them. First-time home- buyers may also worry during a housing bubble that if they do not buy now, they will not be able to afford a home later. Furthermore, the expectation of large price increases may have a strong impact on demand if people think that home prices are very unlikely to fall, and certainly not likely to fall for long, so that there is little perceived risk associated with an investment in a home. What is the origin of bubbles? In a nutshell, speculative bubbles are caused by “precipitating factors” that change public opinion about markets or that have an immediate impact on demand, and by “amplification mechanisms” that take the form of price-to-price feedback, as stressed by Shiller (2000). A number of fundamental factors can influence price movements in housing mar- kets. On the demand side, demographics, income growth, employment growth, changes in financing mechanisms or interest rates, as well as changes in loca- tion characteristics such as accessibility, schools, or crime, to name a few, have been shown to have effects. On the supply side, attention has been paid to construction costs, the age of the housing stock, and the industrial organiza- tion of the housing market. The elasticity of supply has been shown to be a key factor in the cyclical behavior of home prices. The cyclical process that we observed in the 1980s in those cities experiencing boom-and-bust cycles was caused by the general economic expansion, best proxied by employment gains, which drove demand up. In the short run, those increases in demand encountered an inelastic supply of housing and developable land, inventories of for-sale properties shrank, and vacancy declined. As a consequence, prices accelerated. This provided an amplification mechanism as it led buyers to anticipate further gains, and the bubble was born. Once prices overshoot or supply catches up, inventories begin to rise, time on the market increases, vacancy rises, and price increases slow down, eventually encountering down- ward stickiness. The predominant story about home prices is always the prices themselves (see Shiller, 2000; Sornette, 2003); the feedback from initial price increases to further price increases is a mechanism that amplifies the effects of the precipitating factors. If prices are going up rapidly, there is much word-of- mouth communication, a hallmark of a bubble. The word of mouth can spread optimistic stories and thus help cause an overreaction to other stories, such as stories about employment. The amplification can also work on the downside as well. Price decreases will generate publicity for negative stories about the city, but downward stickiness is encountered initially. 2.3 Was there a bubble? Status of the argument based on the ratio of cost of owning versus cost of renting In recent years, there has been increasing debates on whether there was a real estate bubble or not in the United States of America. Case and Shiller (2003), Shiller (2006) and Smith and Smith (2006) argued that the house prices over the period 2000-2005 were not abnormal as they reflected only the convergence of the prices to their fundamentals from below. In contrast, Zhou and Sornette (2006) and Roehner (2006) have suggested that there was a bubble, which be- came identifiable only after 2003, that is, after the work of Zhou and Sornette (2003). In this context, it is instructive to comment on the study by Himmelberg et al. (2005), from the Federal Reserve Bank of New York , as it reflects the never ending debate between tenants of the fundamental valuation explanation and those invoking speculative bubbles. We are resolutely part of the second group. Himmelberg et al. (2005) constructed measures of the annual cost of single- family housing for 46 metropolitan areas in the United States over the last 25 years and compared them with local rents and incomes as a way of judging the level of housing prices. In a nutshell, they claimed in 2005 that conventional metrics like the growth rate of house prices, the price-to-rent ratio, and the price-to-income ratio can be misleading and lead to incorrect conclusions on the existence of the real-estate bubble. Their measure showed that, during the 1980s, houses looked most overvalued in many of the same cities that subsequently experienced the largest house price declines. But they found that from the trough of 1995 to 2004, the cost of owning rose somewhat relative to the cost of renting, but not, in most cities, to levels that made houses look overvalued. The rosy conclusion of Himmelberg et al. (2005), that 2004-2005 prices were justifiable and that there was no risk of deflation as no bubble was present, is based on a particularly curious comparison between cost of owning and cost of renting, as noticed by Jorion (2005), in a letter to the Wall Street Journal. In- deed, they candidly revealed however that their “cost of owning” calculations imply an “expected appreciation on the property” coefficient. The value for this factor is no doubt derived from figures for appreciation as currently ob- served on the housing market, meaning they regarded the current appreciation level as a reasonable assumption for what would indeed happen next – which is precisely what our analyses and that of others question. In other words, the authors had unwittingly hard-wired into their model the assertion that there was no housing bubble; little wonder then that this is also what they felt au- thorized to conclude. The circularity of their reasoning is particularly obvious in an illustration they gave for San Francisco where for more than 60 years the price-to-rent ratio has exceeded the national average, which, so they claimed, “does not necessarily make owning there more expensive than renting.” The reason why is that “high financing costs are offset by above-average expected capital gains.” Translated, this means that as long as there is a bubble, prices will go up and investing in a house remain a profitable operation. This trivial statement is hollow; the real question is whether the trend that is observed now remains sustainable. In addition to this criticism put forward by Jorion (2005), there are other reasons to doubt the validity of the conclusion of Himmelberg et al. (2005). In the own words of Himmelberg et al. (2005), “the ratio of the cost of owning to the cost of renting is especially sensitive to the real long-term interest rates.” They are right in their rosy conclusion... as long as the long-term interest rates remain exceptionally low. It is particularly surprising that their estimation of the ratio of the cost of owning to the cost of renting was based on the most recent rates over the preceding year of their analysis (2004), while the price of a house is a long-term investment: what will be the long-term rates in 10, 20, 30, or 50 years? Another problem is that their analysis was “mono-dimensional”: they proposed that everything depends only on the ratio of the cost of owning to the cost of renting. But they missed the interest rates as an independent variable. As a consequence, it is not reasonable to compare the 1980s and the present time, as the long-term interest rates had nothing in common. Another problem with their analysis is that they assumed “equilibrium,” while people are sensitive to the history-dependent path followed by the prices. In other words, people are sensitive to the way prices reach a certain level, if there is an acceleration that can self-fuel itself for a while, while Himmelberg et al. (2005) discussed only the mono-dimensional level of the price, and not how it got there. We think that this general error made by “equilibrium” economists constitutes a fundamental flaw which fails to capture the real nature of the organization of human societies and their decision process. In the sequel, we actually focus our attention on signatures of price trajectories that highlight the importance of history dependence for prediction. This discussion is reminiscent of the proposition by Mauboussin and Hiler (1999), offered close to the peak of the Internet and new technology bubble that culminated in 2000, that better business models, the network effect, first- to-scale advantages, and real options effect could account rationally for the high prices of dot.com and other New Economy companies. These interest- ing views expounded in early 1999 were in synchrony with the bull market of 1999 and preceding years. They participated in the general optimistic view and added to the strength of the herd. Later, after the collapse of the bubble, these explanations seem less attractive. This did not escape the then U.S. Fed- eral Reserve chairman Alan Greenspan (1997), who said : “Is it possible that there is something fundamentally new about this current period that would warrant such complacency? Yes, it is possible. Markets may have become more efficient, competition is more global, and information technology has doubt- less enhanced the stability of business operations. But, regrettably, history is strewn with visions of such new eras that, in the end, have proven to be a mirage. In short, history counsels caution.” 3 Regime shift in the CSW Zip Code Indexes of Las Vegas 3.1 Description of the data We now turn to the analysis of the CSW indexes of 27 different Las Vegas zip regions obtained with a monthly rate. The 27 monthly CSW data sets start from June-1983 and end in March-2005. Figure 2 shows the price trajectories of all the 27 CSW indexes. Visual inspection shows (i) a very similar behavior of all the different zip codes and (ii) a sudden increase of the indexes since Mid-2003. Let us now analyze this data quantitatively. 3.2 Power law fits The simplest mathematical equation capturing the positive feedback effect and herding is the power law formula (see Broekstra et al., 2005, for a simple introduction in a similar context) I(t) = A+B|tc − t| m , (1) 1980 1985 1990 1995 2000 2005 2010 Fig. 2. Time evolution of the Case-Shiller-Weiss (CSW) Zip Code Indexes of 27 Las Vegas zip regions from June-1983 to March-2005. where B < 0 and 0 < m < 1 or B > 0 and m < 1. Others cases do not qualify as a power law acceleration. For B < 0 and 0 < m < 1 or B > 0 and m < 0, the trajectory of I(t) described by (1) expresses the existence of an accelerating bubble, which is faster than exponential. This is taken as one hallmark of the existence of a bubble. Notice also that this formula expresses the existence of a singularity at time tc, which should be interpreted as a change of regime (the mathematical singu- larity does not exist in reality and is rounded off by so-called finite-size effects and the appearance of a large susceptibility to other mechanisms). This criti- cal time tc must be interpreted as the end of the bubble and the time where the regime is transiting to another state through a crash or simply a plateau or a slowly moving correction. We have fitted each of the 27 individual CSW indexes using the pure power law model (1). The data used for fitting is from Dec-1995 to Jun-2005. We do not show the results as the signature of a power law growth is not evident, essentially because the acceleration is only over a rather short period of time from approximately 2002 to 2004. As a consequence, power law fits give unre- liable critical time tc too much in the future (like 2008 and beyond). We have thus redone the fits of the 27 CSW indexes over a shorter time interval from Aug-2001 to Jun-2005. A typical example is shown in Fig 3. All other 26 CSW are very similar, with some variations of the parameters, but the message is the same: while there is a clear faster-than-exponential acceleration over most of the time interval, the price trajectory has clearly transitioned into another regime in the latter part of the time interval considered here. The transition occurred smoothly from mid-2004 to mid-2005 (the end of the time period analyzed here). It is important to recognize that the power law regime is expected only rela- 2001 2002 2003 2004 2005 2006 = 2012.94; m = −12; χ = 0.007679 Fig. 3. Typical evolution of a CSW index from Aug-2001 to Jun-2005 and its fit by a power law, showing both the faster-than-exponential growth up to mid-2004 and the smooth transition to a much slower growth at later times. The root-mean-square χ of the residuals of the fit as well at tc and m are given inside the figure. tively close to the critical time tc, while other behaviors are expected far from tc. The simplest model is to consider that, far from tc, the price follows an exponential growth with an approximately constant growth rate µ: I(t) = a+ beµt . (2) A fuller description is thus to consider that formula (2) holds from the begin- ning of the time series up to a cross-over time t∗, beyond which expression (1) takes over. Any given price trajectory should thus be fitted by (2) from some initial time tstart to time t ∗ and then by (1) from t∗ to the end of the time series. Technically, t∗ is known from the parameters a, b, µ, A,B, tc, m by the condition of continuity of I(t) at t = t∗, that is, both formulas give the same value at t = t∗. We can further determine one of the parameters a, b or µ by imposing a condition of differentiability at t∗, that is, the first time-derivative of I(t) is continuous at t∗. This approach is known in numerical analysis as “asymptotic matching” (see Bender and Orszag, 1978). A simplified description of such a cross-over between a standard exponential growth and the power law super-exponential acceleration is obtained by using a more compact formulation I(t) = A+B tanh[(tc − t)/τ ] m , (3) where tanh denotes the hyperbolic tangent function. This expression derives from a study of the transition from the non-critical to critical regime in rup- ture processes (of which bubbles and their terminal singularity belong to) conducted by Sornette and Andersen (1998). This expression has the virtue of providing automatically a smooth transition between the exponential be- havior (2) and the pure power law (1), since tanh[(tc − t)/τ ] ≈ (tc − t)/τ for tc− t < τ and tanh[(tc− t)/τ ] ≈ 1− 2e 2(t−tc)/τ for tc− t > τ . In this later case tc − t > τ , expression (3) becomes of the form (2) with m = 1 and a=A+B , (4) b=−2Be−2tc/τ , (5) µ=1/τ . (6) In contrast, for tc − t < τ , expression (3) becomes of the form (1) with the correspondence B/τm → B. Expression (3) has only five free parameters, in contrast with the model involving the cross-over from (2) to (1) which has 7 free parameters (a, b, µ, A,B, tc, m) while t ∗ is determined by the asymptotic matching). The pure power law formula (1) has 4 parameters while the ex- ponential law (2) has just 3 parameters. The problem with expression (3) is that it does not recover a pure exponential growth even for tc − t > τ , when m 6= 1. Thus, expression (3) is limited in fully describing a possible cross-over from a standard mild exponential growth and an super-exponential power law acceleration. Our tests (not shown) find that a fit with model (3) retrieve the pure power law model (1) with the same critical time tc and exponent m and the same root-mean-square residual r.m.s. (the fit adjusts the parameter τ to a very large value, ensuring that the fit is always in the regime tc−t ≪ τ so that the hyperbolic tangential model reduces to the pure power law model). Thus, contrary to our initial hopes, this approach does not provide any additional insight. Inspired by these tests, we could propose the following modified model I(t) = a+ beµt(tc − t) m . (7) It has 5 adjustable parameters, like model (3), but it seems more flexible to describe the looked-for cross-over: for large tc− t, the power law term (tc− t) changes slowly, especially for 0 < m < 1 as is expected here; for small tc−t, the power law term changes a lot while the exponential term is basically constant. But, this model is correct for a critical point only if m < 0 so that b > 0; otherwise, if 0 < m < 1, b < 0 and for tc− t large, the exponential term which dominate does not describe a growth but an exponentially accelerating decay. For 0 < m < 1, we thus need a different formulation. We propose I(t) = a+ beµt + c(tc − t) m . (8) We have fitted this formula to the data over the four periods 1983 - Oct. 2004, 1991 - Oct. 2004, 1983 - Mar. 2005, 1991 - Mar. 2005 and, while the fits are reasonable, the critical time tc is found to overshoot to 2007-2008, which is a typical signature that the model is not predictive. In conclusion of this first preliminary study, the presence of a bubble (faster- than-exponential growth) is confirmed but the determination of the end of this phase is for the moment unreliable. 3.3 Dependence of the growth rate on the index value The monthly growth rate g(t) of a given CSW index at time t is defined by g(t) = ln[p(t)/p(t− 1)] , (9) where p(t) is the price of that CSW index at time t. Figure 4 shows the evolution of the growth rates of the 27 CSW indexes from June-1983 to March- 2005. While there are some variations, all 27 CSW indexes follow practically the same pattern. We clearly observe a large peak of growth over the period 2003-2005. Notice that this recent peak is much larger and coherent than the previous one ending in 1991, which was followed by a price stabilization and even a price drop in certain cases. This figure stresses that the acceleration in growth rate is a very localized event which occurred essentially in 2003-2004 and the subsequent growth rate has leveled off to pre-bubble times. We can conclude that there has been no bubble from 1990 to 2002, approximately, then a short-lived bubble until mid-2004 followed by a smoothed transition back to normal. 1980 1985 1990 1995 2000 2005 2010 −0.04 −0.02 Fig. 4. Evolution of the growth rates of the 27 regional CSW indexes from June-1983 to March-2005. Fig. 5 plots the price growth rate g(t) versus the price p(t) itself for the 27 CSW indexes. A linear regression of the data points on Fig. 5, shown as the red straight line, gives a correlation coefficient of 0.494. If we perform lin- ear regression for each index, then we find an average correlation coefficient 0.503 ± 0.036, confirming the robustness of this estimation of the correlation between growth rate and price level. The obtained relation between g and p obtained from this correlation analysis is captured by the following mathe- matical regression g = 0.00922× − 0.00747 . (10) In words, if p is large, then g is large on average, which confirms the concept of a positive feedback of price on its further growth. The continuous time limit of g(t) defined by (9) is g(t) = d ln p . (11) This last equation together with (10), that we write as g(t) = αp − β (with α = 0.00922/100 and β = 0.00747), implies the following ordinary differential equation = αp2 − βp , (12) which indeed gives a power law acceleration p(t) ∼ 1/(tc − t) asymptotically close to the critical time tc. Note that this critical time is determined by the initial conditions, and is called in mathematics a movable singularity. We conclude from this first analysis that the rough linear growth of the growth rate confirms the existence of a bubble growing faster than exponential according to an approximate power law. But of course, the exponent of this power law is poorly constrained, in particular from the fact that the growth rate g(t) exhibits significant variability and furthermore nonlinearity, as can be seen in Fig. 5. 50 100 150 200 250 300 350 400 450 −0.04 −0.02 Jul. 1983 − Sep. 2003 Oct. 2003 − Sep. 2004 Oct. 2004 − Mar. 2005 Fig. 5. Dependence on the data price p for all CSW indexes of its growth rate g. The overall correlation coefficient is 0.494. The red line is the linear fit of the data points. It is useful to refine this analysis by separating the whole time interval into three distinct intervals. The corresponding plot of the growth rate g as a function of price is shown in Fig. 5 with different symbols: period 1 is Jul. 1983 to Sept. 2003, period 2 is from Oct. 2003 to Sept. 2004, and period 3 is from Oct. 2004 to Mar. 2005. An anomaly can be clearly outlined, associated with the red dots which correspond to the anomalous peak in the growth rate in the period from Oct. 2003 to Sept. 2004. Notice also that the most recent time interval from Oct. 2004 to Mar. 2005 shows practically the same behavior as the first period before 2003. In other words, when removing the data in red for the period from Oct. 2003 to Sept. 2004, the growth rate g(t) is practically independent of p, which qualifies the normal regime. We can thus conclude that this so-called “phase-portrait” of the growth rate versus price has identified clearly an anomalous time interval associated with extremely fast accelerating prices followed by a more recent period where the price growth has resumed a more normal regime. 4 Yearly periodicity and intra-year structure 4.1 Yearly periodicity from superposed year analysis and spectral analysis In Fig. 4, the time dependence of the monthly growth rate exhibits a clear sea- sonality (or periodicity), which appear visually to be predominantly a yearly phenomenon. This visual observation is made quantitative by performing a spectral Fourier analysis. The power spectrum of a typical CSW index is shown in Fig. 6 (all CSW indexes show the same power spectrum). Since the unit of time used here is one year, the frequency f is in unit of 1/year. A periodic behavior with period one year should translate into a peak at f = 1 plus all its harmonics f = 2, 3, 4, · · · , which is indeed observed in Fig. 6. Note also that the spectrum has large peaks at f = 4 and f = 8 among the harmonics of f = 1, which indicates a weak periodicity with period of one quarter. This is consistent with Fig. 7, where four oscillations in the averaged monthly growth rates can be observed. 0 1 2 3 4 5 6 7 8 9 10 11 12 Fig. 6. Spectrum analysis to confirm the strong periodicity in g(t). Note that the power spectrum itself is periodic with a period of 12, which is the sampling frequency, equal to the double of the Nyquist frequency. There are also many peaks in the low-frequency region (larger that one-year time scale) close to f = 0, which are associated with the time scales of the global trends produced by the big peaks in g(t) around year 2004 as well as around 1990. To further explore this seasonal variability of the price growth rates, we cal- culate the averages of the growth rates for given months, where the average is performed over all years. Consider for instance the month of January: we look up the growth rate for all the data over all years for the month of January and take the average. We do the same for each successive months. The result is shown in Figure 7 for two time periods, which gives the average growth rate 〈g〉 for different months of the year. The red dash line and circles give the resultant 〈g〉 for all the data and the black dash line and triangles give the standard deviation σg for all data (which is a measure of the variability from year to year and from zip code to zip code around the average). The difference between the two time periods is precisely the time interval from June 2003 to March 2005: this period is responsible for a significant increase of the average growth rate (compare the red dashed line (filled circles) with the red continu- ous line (open circles)) and an even larger increase of the variability (compare the black continuous line (filled triangles) with the dashed black line (open triangles)), again confirming the evidence of an anomalous behavior in that period. In 2005, it appears that the growth rate relaxed back to the normal level (according to the historical record). Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Fig. 7. Monthly average growth rate (circles) and its standard deviation (triangles) as a function of the month within the year. Dash: results obtained over all 27 indexes over the period from Jun. 1983 to Mar. 2005; Solid: results obtained over all 27 indexes over the period from Jun. 1983 to May 2003. 4.2 Yearly periodicity and intra-year structure with a scale and translation modulated model Inspired by these results, we propose the following quantitative model. Con- sider a time t in units of month. We write t = 12T +m, where T is the year and m is the month within that year and thus goes from 1 (January) to 12 (December). For instance, t = 26 corresponds to T = 2 and m = 2 (Febru- ary), while t = 38 corresponds to T = 3 and again the same month m = 2 (February) within the year. We propose to model the intra-year structure of the growth rate g(t) together with possible yearly variations by the following expression g(t = 12T +m) = f(T )h(m) + j(T ) . (13) In words, the growth rate has an intra-year structure h(m) modulated from year to year in amplitude by f(T ) up to a possible overall translation j(T ) which can also vary from year to year. We can expect f(T ) and j(T ) to be approximately constant for most years, except around 1990 and 2004 for which we should see an anomaly in either or both of them, since these two periods had bubbles. Note that this model (13) gives an exact yearly periodicity if f(T ) and j(T ) are constant. A non-constant f(T ) describes an amplitude modulation of the yearly periodicity. In particular, we expect a strong peak around T = 2004. With this model, we can focus on predicting f(T ) and j(T ) only, because we have removed the complex intra-year structure. We have thus fitted the model (13) to three subsets of the whole available time series for the growth rate g(t) and also to the whole set taken globally, in order to test for the robustness of the model. For this, we use the cost function Tmax∑ [g(t = 12T +m)− f(T )h(m)− j(T )]2 (14) which is minimized with respect to the 12 unknown variables h(1), ..., h(12) and the 2 × Tmax variables [f(1), j(1)], ..., f(Tmax), j(Tmax). There are 12Tmax terms in the sum and 12 + 2 × Tmax unknown variables. This shows that the system is well-constrained as soon as Tmax ≥ 2. For instance for Tmax = 20, we have 52 unknown variables to fit and 240 terms in the sum to constrain the Figure 8 illustrates the result of the fit of model (13) to the growth rate over the whole time interval from 1985 to 2005. As expected, we can observe a clear peak in the amplitude f(T ) corresponding to the year 2004, while there is not appreciable peak around 1990. This means that the recent bubble appears significantly stronger than any other episodes in the last 20 years and dwarfs them. The anomalous nature of the recent bubble is reinforced by the existence of a peak in j(T ) for the same year 2004, showing that both the amplitude and translation components of the growth rates has been completely anomalous in 2004. The middle graph of the top panel of figure 8 shows the intra-year pattern captured by the model, which is in remarkable agreement with the pattern shown in figure 7: one can observe a peak in March, May, August and December, the largest peak being in May. The bottom panel of figure 8 shows visually how well (or badly) the model fits the actual data. The quality of the fit is excellent, except in 2004-2005. In other words, we clearly identify a very anomalous or exceptional behavior in 2004-2005, again providing a confirmation that something exceptional or anomalous has occurred during that period. 1985 1990 1995 2000 2005 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec −0.025 −0.02 −0.015 −0.01 1985 1990 1995 2000 2005 1980 1985 1990 1995 2000 2005 2010 −0.03 −0.02 −0.01 Fig. 8. Upper panels: three graphs showing the three functions f(T ), h(m) and j(T ) fitted on the growth rate over the whole time interval from 1985 to 2005. Lower panel: Comparison between the growth rate data (empty blue circles) and the model (13) (red line). Figure 9 is the same as figure 8 for the period from 1985 to 1990. One can clearly here observe a peak in the scaling amplitude f(T ) at T =1988 and in the translation term j(T ) at T =1986, suggesting that the first bubble of the 1985-2005 period occurred over a relatively large time period 1985-1990, with two successive contributions. The intra-year structure h(m) has also its peaks on March, May, August and December, but this intra-year structure is weaker than for other sub-periods. The lower panel of figure 9 shows that the model captures very well the overall trend as well as the intra-year structure. The main discrepancies are in the amplitude of the large peaks and valleys, which are not fully predicted. 1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5 1989 1989.5 1990 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0.185 0.195 0.205 1985 1985.5 1986 1986.5 1987 1987.5 1988 1988.5 1989 1989.5 1990 −0.29 −0.28 −0.27 1984 1985 1986 1987 1988 1989 1990 1991 −0.03 −0.02 −0.01 Fig. 9. Same as figure 8 for the period from 1985 to 1990. Figure 10 is the same as figure 8 for the period from 1991 to 2000. One can clearly here observe a peak in the scaling amplitude f(T ) at T =1995 and in the translation term j(T ) at T =1994. This thus identifies a small bubble in the mid-1990s. The intra-year structure h(m) has also its peaks on March, May, August and December, with very large amplitudes. The lower panel of figure 10 shows a truly excellent fit. 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec −0.12 −0.115 −0.11 −0.105 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 0.155 0.165 1990 1992 1994 1996 1998 2000 2002 −0.01 −0.005 0.005 0.015 0.025 Fig. 10. Same as figure 8 for the period from 1991 to 2000. Figure 11 is the same as figure 8 for the period from 2001 to 2005. One can clearly here observe a peak in the scaling amplitude f(T ) at T =2004 and in the translation term j(T ) also at T =2004. This thus clearly identifies the bubble as peaking in 2004. The intra-year structure h(m) has also its peaks on March, May, August and December, with very large amplitudes and very good agreement with the other three figures. The lower panel of figure 11 shows an excellent fit up to the early 2003 and then a rather large discrepancy starting early 2003 all the way to the last data point approaching mid-2005. In particular, note that the intra-year structure is washed out by the anomalous growth rate culminating in mid-2004. Symmetrically, the intra-year structure is also absent in the fast decay of the growth rate back to normal. We do not have enough data to ascertain if the growth rate has resumed its normal intra-year pattern. We believe that this is a very important diagnostic to characterize abnormal behavior and this could be a very useful variable to monitor on a monthly basis. 2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec −0.025 −0.02 −0.015 −0.01 2001 2001.5 2002 2002.5 2003 2003.5 2004 2004.5 2005 2000 2001 2002 2003 2004 2005 2006 −0.01 Fig. 11. Same as figure 8 for the period from 2001 to 2005. The four figures 8-11 validate model (13): in particular, they show the very robust intra-year structure with peaks in March, May, August and December. One possible contribution to this quarterly periodicity comes from the con- struction of the CSI: the monthly indexes use a three-month moving average algorithm. Home sales pairs are accumulated in rolling three-month periods, on which the repeat sales methodology is applied. The index point for each reporting month is based on sales pairs found for that month and the pre- ceding two months. For example, the December 2005 index point is based on repeat sales data for October, November and December of 2005. This av- eraging methodology is used to offset delays that can occur in the flow of sales price data from county deed recorders and to keep sample sizes large enough to create meaningful price change averages. A three month rolling window construction corresponds in general to a convolution of the bare price with a kernel which possesses a three month periodicity (or size). The Fourier transform of the convolution is the product of Fourier transforms. Thus the spectrum of the signal should contain the peaks of the Fourier spectrum of the kernel, which by construction contains a peak at three months. However, our synthetic tests (not shown) suggest that this effect is by far too small to explain the strong amplitude of the observed quarterly periodicity. It would be important to understanding why such intra-year structure develops: is it the result of a natural intra-day organization of buyers’ behaviors associated with taxes/ income constraints or a problem of reporting or perhaps the effect of other calendar regularities? Or is it the result of patterns coming from the supply part of the equation, namely home-builders, developers, and perhaps in the time modulation of the rates of allocated permits? Answering these questions is important to determine how much emphasis one should give to these results. But if indeed the intra-day structure is a genuine non-artificial phenomenon, we believe that it offers a remarkable opportunity for monitoring in real time the normal versus abnormal evolution of the market and also for developing forecasts on a month time horizon. 4.3 Intra-year pattern from signs of growth rate increments The existence of a strong and robust intra-year structure in the price growth rate can be further demonstrated by studying the sign of g(t + 1) − g(t). A positive (negative) sign mean that the growth rate tends to increase (decrease) from one month to the next. Based on the seasonality of the growth rate, we are able to answer the following question: given the current growth rate g(t), will the growth rate increase or decrease at time t+1? This amounts to asking what is the sign of g(t+1)−g(t)? Technically, we construct the (unconditional) number of times the sign of the increment g(t + 1) − g(t) is positive or negative irrespective of what is g(t). From Fig. 4, we obtain a sequence of signs: −−+−+−−+−−++. For each month, we calculate the percentage of positive and negative signs, respectively. The second and the third rows of Table 1 gives the percentage of positive and negative signs for each month. The third and fourth rows gives the signs and the associated percentages. For instance, the table says that the “probability” of the sign of g(t = Feb)− g(t = Jan) being “-” is about 92.1%. If we know g(t = Jan), we can say that it is very probable that the growth rate of February will be less than this January value. Thus, this table has predictive power in the sense that the probabilities to predict the signs are much higher than the value of 75% obtained under the null hypothesis that g(t) is a white noise process (see Sornette and Andersen, 2000). This table is another way to rephrase and expand on our preceding analysis on the yearly periodicity by identifying a very strong and robust intra-year structure. Table 1 Analysis of the signs of g(t + 1) − g(t). The second and the third rows gives the percentage of positive and negative signs for each month. The third and fourth rows give the sign for each month that dominates and the associated percentages. Mon Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec +% 7.91 17.2 88.0 5.64 97.7 8.47 8.47 91.4 6.57 8.92 84.2 82.2 -% 92.1 82.8 12.0 94.4 2.29 91.5 91.5 8.59 93.4 91.1 15.8 17.8 sign - - + - + - - + - - + + % 92.1 82.8 88.0 94.4 97.7 91.5 91.5 91.4 93.4 91.1 84.2 82.2 Since our initial analysis performed in the summer of 2005 which used data up to March 2005, new data for the 27 CSW indexes has become available which covers the interval from Apr. 2005 to Sept. 2006. It is very interesting to check if the sign of the growth variations obtained in Table 1 using the data until March 2005 still applies to the new data. The realized signs of the newly available months are calculated and the sequence of signs is the following: - (Apr. 2005, 27 CSW indexes out of 27), + (May. 2005, 27 out of 27), - (Jun. 2005, 27 out of 27), - (Jul. 2005, 27 out of 27), + (Aug. 2005, 27 out of 27), - (Sep. 2005, 27 out of 27), - (Oct. 2005, 27 out of 27), + (Nov. 2005, 27 out of 27), + (Dec. 2005, 21 out of 27), - (Jan. 2006, 27 out of 27), - (Feb. 2006, 27 out of 27), + (Mar. 2006, 27 out of 27), - (Apr. 2006, 27 out of 27), + (May. 2006, 27 out of 27), - (Jun. 2006, 27 out of 27), - (Jul. 2006, 27 out of 27), + (Aug. 2006, 27 out of 27), and - (Sep. 2006, 27 out of 27). Thus, table 1 predicts exactly the signs of the growth rate variations of all 27 CSW indexes for all months except for Dec. 2005 for which there are 6 errors: table 1 predicts that the growth rate variation from Dec. 2005 to Jan. 2006 should be +, which is correct for 21 CSW indexes out of 27, corresponding to a success ratio of 77% (close to the white noise case). This score is slightly lower than the previously estimated probability of 82.2% for the month of December, which is the lowest among all months. Overall, the success rate is remarkably high, adding further evidence that the Las Vegas property market has returned to a more normal phase (no bubble from April 2005 to Sept. 2006). 5 Predicting the monthly growth rate Conditional of the evidence that the anomalous faster than exponential growth has ended, let us attempt to predict the future evolution of the CSW indexes based only on the strong seasonality of the growth rate. Figure 12 presents the predictions one year ahead for the 27 regional CSW indexes. Two different prediction schemes are used. The RED lines are based on the average growth rate obtained from all 27 indexes, while the MAGENTA lines are based on the average growth rate obtained from the individual index under investigation. There is not discernable difference. A similar prediction of the Clark County (Las Vegas MSA) indexes (NVC003Q and NVC003C) has also been made using the average growth rates obtained from all 27 regional indexes. Since these two indexes are only available from July-2000 to March-2005, we do not have enough data to calculate the average growth rates using the indexes themselves. The results are shown in Fig. 13. 2002 2003 2004 2005 2006 2007 Fig. 12. Predicting regional CSW indexes one year ahead. Red lines: Prediction using average growth rate obtained from all 27 indexes; Magenta lines: Prediction using average growth rate obtained from the individual index under investigation. The two kinds of prediction are almost undistinguishable. 2000 2001 2002 2003 2004 2005 2006 2007 NVC003C: Raw data NVC003C: Prediction NVC003Q: Raw data NVC003Q: Prediction Fig. 13. Predicting Clark County (Las Vegas MSA) indexes (NVC003Q and NVC003C) one year ahead. 6 Conclusion We have analyzed 27 house price indexes of Las Vegas from Jun. 1983 to Mar. 2005, corresponding to 27 different zip codes. These analyses confirm the existence of a real-estate bubble, defined as a price acceleration faster than exponential. This bubble is found however to be confined to a rather limited time interval in the recent past from approximately 2003 to mid-2004 and has progressively transformed into a more normal growth rate in 2005. The data up to mid-2005 suggests that the current growth rate has now come back to pre-bubble levels. We conclude that there has been no bubble from 1990 to 2002 except for a medium-sized surge in 1995, then a short-lived but very strong bubble until mid-2004 which has been followed by a smoothed transition back to what appears to be normal. It thus seems that, while the strength of the real-estate bubble has been very strong over the period 2003- 2004, the price appreciation rate has returned basically to normal. In addition, we have identified a strong yearly periodicity which provides a good potential for fine-tuned prediction from month to month. As the intra- year structure is likely a genuine non-artificial phenomenon, it offers a re- markable opportunity for monitoring in real time the normal versus abnormal evolution of the market and also for developing forecasts on a monthly time horizon. In particular, a monthly monitoring using a model that we have de- veloped here could confirm, by testing the intra-year structure, if indeed the market has returned to “normal” or if more turbulence is expected ahead. In addition, it would provide a real-time observatory of upsurges and other anomalous behavior at the monthly scale. This requires additional technical developments and tests beyond this report. Compared with previous analysis of Zhou and Sornette (2003, 2006) at the scale of states and whole regions (northeast, midwest, south and west), the present analysis demonstrates the existence of very significant variations at the local scale, in the sense that the bubble in Las Vegas seems to have preceded the more global USA bubble and has ended approximately two years earlier (mid 2004 for Las Vegas compared with mid-2006 for the whole of the USA). References Bailey, M., Muth, R., Nourse, H., 1963. A regression method for real estate price index construction. Journal of the American Statistical Association 58, 933–942. Bender, C., Orszag, S. A., 1978. Advanced Mathematical Methods for Scien- tists and Engineers. McGraw-Hill, New York. Broekstra, G., Sornette, D., Zhou, W.-X., 2005. Bubble, critical zone and the crash of Royal Ahold. Physica A 346, 529–560. Case, K. E., Shiller, R. J., 1987. Prices of single-family homes since 1970: New indexes for four cities. New England Economic Review Sep/Oct, 45–56. Case, K. E., Shiller, R. J., 1989. The efficiency of the market for single-family homes. American Economic Review 79, 125–137. Case, K. E., Shiller, R. J., 1990. Forecasting prices and excess returns in the housing market. AREUEA J. 18, 253–273. Case, K. E., Shiller, R. J., 2003. Is there a bubble in the housing market. Brookings Papers on Economic Activity (2), 299–362. Greenspan, A., 1997. Federal ReserveÕs semiannual monetary policy report, before the Committee on Banking, Housing, and Urban Affairs, U.S. Senate, February 26. Himmelberg, C., Mayer, C., Sinai, T., September 2005. Assessing high house prices: Bubbles, fundamentals, and misperceptions. Tech. Rep. Staff Report no. 218, Federal Reserve Bank of New York. Jorion, P., 2005. Is housing market surge really sustainable? The Wall Street Journal September 22, A17. Mauboussin, M. J., Hiler, R., 1999. Rational Exuberance? Equity research report of Credit Suisse First Boston, January 26. Roehner, B. M., 2006. Real estate price peaks: A comparative overview. Evo- lutionary and Institutional Economics Review in press, physics/0605133. Shiller, R. J., 2000. Irrational Exuberance. Princeton University Press, New York. Shiller, R. J., 2006. Long-term perspectives on the current boom in home prices. Economists’ Voice 3(4), Art. 4. Smith, M. H., Smith, G., 2006. Bubble, bubble, where’s the housing bubble? Brookings Papers on Economic Activity (1), 1–67. Sornette, D., 2003. Why Stock Markets Crash: Critical Events in Complex Financial Systems. Princeton University Press, Princeton. Sornette, D., Andersen, J.-V., 1998. Scaling with respect to disorder in time- to-failure. European Physical Journal B 1, 353–357. Sornette, D., Andersen, J.-V., 2000. Increments of uncorrelated time series can be predicted with a universal 75% probability of success. International Journal of Modern Physics C 11, 713–720. Zhou, W.-X., Sornette, D., 2003. 2000-2003 real estate bubble in the UK but not in the USA. Physica A 329, 249–263. Zhou, W.-X., Sornette, D., 2006. Is there a real-estate bubble in the US? Physica A 361, 297–308. Biographies: Wei-Xing Zhou is a Professor of Finance at the School of Business in the East China University of Science and Technology. He received his PhD in Chemical Engineering from the East China University of Science and Technology in 2001. His current research interest focuses on the modeling and prediction of catastrophic events in complex systems. Didier Sornette holds the Chair of Entrepreneurial Risks at the Depart- ment of Management, Technology and Economics of ETH Zurich. He received his PhD in Statistical Physics from the University of Nice, France. His current research focuses on the modeling and prediction of catastrophic events in com- plex systems, with applications to finance, economics, seismology, geophysics and biology. Introduction Conceptual background of our empirical analysis Humans as social animals and herding Definition and mechanism for bubbles Was there a bubble? Status of the argument based on the ratio of cost of owning versus cost of renting Regime shift in the CSW Zip Code Indexes of Las Vegas Description of the data Power law fits Dependence of the growth rate on the index value Yearly periodicity and intra-year structure Yearly periodicity from superposed year analysis and spectral analysis Yearly periodicity and intra-year structure with a scale and translation modulated model Intra-year pattern from signs of growth rate increments Predicting the monthly growth rate Conclusion

0704.0590

A Low Complexity Algorithm and Architecture for Systematic Encoding of Hermitian Codes

A Low Complexity Algorithm and Architecture for Systematic Encoding of Hermitian Codes Rachit Agarwal∗†, Ralf Koetter‡ and Emanuel M. Popovici∗§ ∗ Department of Microelectronic Engineering, University College Cork, Cork, Ireland † Microelectronics Application Integration Group, Tyndall National Institute, Cork, Ireland ‡ Institute for Communications Engineering, Technische Universitaet, Muenchen, Germany § Claude Shannon Institute for Discrete Mathematics, Coding and Cryptography, Ireland Email: [email protected], [email protected], [email protected] Abstract— We present an algorithm for systematic encoding of Hermitian codes. For a Hermitian code defined over GF (q2), the proposed algorithm achieves a run time complexity of O(q2) and is suitable for VLSI implementation. The encoder architecture uses as main blocks q varying-rate Reed-Solomon encoders and achieves a space complexity of O(q2) in terms of finite field multipliers and memory elements. I. INTRODUCTION Algebraic-Geometric (AG) codes [1] offer desirable prop- erties such as large code lengths over small finite fields, the potential to find a large selection of codes and good error- correction at high code rates [2]. In recent years, an important class of one-point AG codes, called Hermitian codes, has been frequently discussed [3]-[6]. For a Hermitian code defined over GF (q2), a brute-force way to design an encoder is to multiply the information vector by a generator matrix. The space complexity of a serial-in serial-out architecture for this systematic encoder is O(q5) in terms of finite field multipliers and O(q3) in terms of memory elements. The encoder requires 2n clock cycles to generate a codeword of length n, thus, the latency is n. By considering a Hermitian code as a superposition of several generalized Reed Solomon (RS) codes, an encoding scheme is introduced in [4]. In [5], an encoding algorithm by forming a bivariate information polynomial and evaluating this polynomial at every finite rational point on the Hermitian curve is proposed. However, both such schemes are nonsystematic and involve the evaluation of bivariate polynomials at n finite rational points, thus, they may not have efficient hardware architecture for implementations. A computationally efficient approach for systematic encod- ing was proposed in [7]. A serial-in serial-out architecture for this approach was proposed in [8]. This architecture requires n clock cycles to encode a codeword of length n. The space complexity for this architecture is O(q3), both in terms of finite field multipliers and memory elements. In this paper, we present an algorithm for systematic encod- ing and syndrome computation of Hermitian codes. We give an outline for the encoder architecture, which uses q varying-rate RS encoders as main blocks and requires n2/3 clock cycles for encoding a codeword of length n. The space complexity of the architecture is O(q2) in terms of both, memory elements and finite field multipliers. II. HERMITIAN CODES AND SYNDROME COMPUTATION We consider codes from a Hermitian curve χ : xq+1 = yq + y over a finite field Fq2 . The space L(mP∞) consists of all functions on χ that have a pole of multiplicity at most m only at the unique point at infinity. For L(mP∞), we choose a basis L(mP∞) = 〈xayb : aq + b(q + 1) ≤ m, 0 ≤ a, 0 ≤ b < q〉 Let y0 be an element of Fq2 such that y0 + y0q = 1. The affine rational points on χ are of the form Pα,β = (α, α q+1(y0 + β) + δ(α)β), where δ is the Kronecker-delta and α and β represent arbitrary elements in Fq2 and Fq respectively. Let � be a primitive element in Fq2 and let γ be a primitive element in Fq . We label the positions in a codeword by the corresponding elements α = �i, β = γj and we thus naturally consider a codeword as a q × q2 matrix c. Occasionally we will index elements in this array by elements of the fields Fq2 and Fq , otherwise we index starting with 0. A Hermitian code C(m) is defined as {c ∈ Fq cβ,αf(Pα,β) = 0, ∀f ∈ L(mP∞)} For an in-depth treatment of AG codes we refer to [9]. Throughout this paper we consider m and thus the Hermitian code as being fixed. Given a q × q2 matrix r we can check if r is a codematrix in a Hermitian code by computing the syndromes Sa,b(r) = rβ,α(x(Pα,β)) a(y(Pα,β)) r is a code-matrix iff Sa,b(r) is zero for all xayb ∈ L(mP∞). Substituting the explicit form of the points we get Sa,b(r) = αa(α(q+1)(y0 + β) + δ(α)β) brβ,α These equations can further be developed to give specific forms as shown in (1) and (2). From the structure of (2), it comes naturally to define a matrix as in (3) to convert the expression into a matrix multiplication. Similarly we define a matrix A′ as  1 (γ0)0 (γ1)0 . . . (γq−2)0 0 (γ0)1 (γ1)1 . . . (γq−2)1 0 (γ0)2 (γ1)2 . . . (γq−2)2 . . . 0 (γ0)q−1 (γ1)q−1 . . . (γq−2)q−1  For later use we give here the following Lemma. A. Lemma 1 The l×l submatrices of A consisting of the elements indexed by i, j and that of A′ consisting of the elements indexed by j, i, i = q − l, q − l + 1, . . . , q − 1, j = 0, 1, . . . , l − 1 are non-singular. Proof: This lemma follows in both cases from the properties of Vandermonde matrices. It will be convenient to define an array A of matrices of type A and A′. A = (A0, A1, . . . , Aq2−1), Ai = A′ i = q2-1 A otherwise Given a q× q2 array r with columns rj , we define a q× q2 matrix r̃ with columns r̃j as r̃j = Ajrj One of the main ingredients in both the syndrome calcu- lation and a systematic encoding is the use of techniques for cyclic codes which are extended by one extra position. Let â(b) = b(m − (q − 1 − b)(q + 1))/qc = max(a : xayb ∈ L(mP∞)). B. Definition 1 Let an ordered set < = {ξ0, ξ1, . . . } of elements from Fq be given. We define the code EC(<, q) as {c ∈ Fqq : i = 0, ∀ξ ∈ <\{ξ0}, cq−1 + i = 0} For the natural indexing of elements in Fq and Fq2 induced by γ and � we have the following Lemma. C. Lemma 2 Let a q × q2 matrix r be given. The matrix r is a code matrix in the Hermitian code C(m) iff the ith row of r̃ is a codeword in EC((�0+i(q+1), �1+i(q+1), . . . , �â(i)+i(q+1)), q2). Proof: The proof follows immediately from the syn- drome definition. Codes of type EC((�i(q+1), �1+i(q+1), . . . , �â(i)+i(q+1)), q2) will play a central roll in the sequel. We define codes Ei as Ei = EC((� 0+i(q+1), �1+i(q+1), . . . , �â(i)+i(q+1)), q2) From Lemma 2 we can derive an efficient way to compute the syndrome for a Hermitian code. Given a received matrix r we obtain a matrix r̃ with columns r̃j = Ajrj . Given r̃ we can easily solve the task of computing syn- dromes provided we can compute the corresponding syn- dromes for codes Ei, i = 0, 1, . . . , q − 1. III. SYSTEMATIC ENCODING The idea behind the systematic encoding of Hermitian codes is to use the well known techniques for the systematic encoding of cyclic codes. Lemma 2 almost immediately gives a nonsystematic encoding procedure for Hermitian codes. To this end let r̃ be a q×q2 matrix such that the jth row of r̃ is a codeword in Ej . It follows from Lemma 2 that we can obtain a code-matrix for a Hermitian code by multiplying the columns of r̃ with matrices A−1 and A′−1 respectively. We can obtain such a matrix r̃ using eg. systematic encoding procedures for codes of type Ej , j = 0, . . . , q − 1. We will need A−1 and A′−1. A. Lemma 3 The matrices A and A′ have inverses given in (4) and (5). Proof: The inverse of A′ is straight forward to verify. We only show the inverse of A. The rows of A−1 and the columns of A may be thought of as being indexed by elements of Fq . Sa,b(r) = q2\{0} β∈Fq α aαb(q+1)(y0 + β) brβ,α a 6= 0∑ β∈Fq (α b(q+1)(y0 + β)b + δ(α)βb)rβ,α a = 0 Sa,b(r) = { ∑q2−2 i=0 � i(a+b(q+1))(yb0r0,i + j=0 (y0 + γ rj+1,i) a 6= 0∑q2−2 i=0 � ib(q+1)(yb0r0,i + j=0 (y0 + γ rj+1,i) + j=0 γ jbrj+1,q2−1 a = 0  (y0 + 0)0 (y0 + γ0)0 (y0 + γ1)0 . . . (y0 + γq−2)0 (y0 + 0)1 (y0 + γ0)1 (y0 + γ1)1 . . . (y0 + γq−2)1 (y0 + 0)2 (y0 + γ0)2 (y0 + γ1)2 . . . (y0 + γq−2)2 . . . (y0 + 0)q−1 (y0 + γ0)q−1 (y0 + γ1)q−1 . . . (y0 + γq−2)q−1  (3) A−1 =  1− (y0 + 0)q−1 (y0 + 0)q−2 . . . (y0 + 0)0 1− (y0 + 1)q−1 (y0 + 1)q−2 . . . (y0 + 1)0 1− (y0 + γ)q−1 (y0 + γ)q−2 . . . (y0 + γ)0 . . . 1− (y0 + γq−2)q−1 (y0 + γq−2)q−2 . . . (y0 + γq−2)0  (4) A′−1 =  1 0 0 . . . 0 −1 0 −(1)q−2 −(1)q−3 . . . −(1)1 −1 0 −(γ)q−2 −(γ)q−3 . . . −(γ)1 −1 . . . 0 −(γq−2)q−2 −(γq−2)q−3 . . . −(γq−2)1 −1  (5) Let C be the matrix obtained as C = A−1A. The entry Ci,j is thought of as being indexed by µ, ν ∈ Fq . Cµ,ν = (1− (y0 + µ)q−1(y0 + ν)0 − (y0 + µ)q−2(y0 + ν)1 · · · − (y0 + µ)0(y0 + ν)q−1) i=0 (y0 + µ) q−1−i(y0 + ν)i (y0 + µ)q − (y0 + ν)q y0 + µ− (y0 + ν) 1− (µ− ν)q−1 = 1 µ = ν 0 µ 6= ν We note that we are entirely free to choose ”virtual information symbols” in matrix r̃. Let a sequence of information symbols be given that are to be encoded systematically in a codeword of a Hermitian code. The trick in obtaining a systematic encoding procedure is to choose the information symbols in r̃ so that the mapping with A−1 and A′−1 respectively, gives the primary information symbols that we really want to encode. Before we derive a systematic encoding procedure for Hermitian codes, we treat a somewhat simpler case, which will elucidate the idea of systematic encoding. Let Ĉ be a code on a Hermitian curve defined as Ĉ = {c ∈ Fq : Sa,b(c) = 0, a = 0, 1, . . . , â < q2 − 1, b = 0, 1, . . . , q − 1} The code Ĉ has dimension (q2 − â − 1)q. The following algorithm may be used for systematic encoding of code Ĉ. B. Algorithm 1 1) Write the (q2− â−1)q information symbols in an array d of size q × (q2 − â− 1) 2) Compute r̂ = AdT 3) Encode the ith row of r̂ independently in a systematic way into codewords of the code EC((�0+i(q+1), �1+i(q+1), . . . , �â(i)+i(q+1)), q2) Denote the resulting q × q2 matrix with r̂′. 4) Compute columns ci = A Algorithm 1 yields a systematic encoding procedure for the code Ĉ because c is a code matrix by Lemma 2 and the first (q2− â−1)×q symbols are the original information symbols. The first (q2 − â− 1) columns of d determine the first (q2 − â− 1) columns of r̂. It is the first (q2 − â− 1) columns of r̂ that contain the virtual information symbols for the encoding of the cyclic codes. The situation for Hermitian codes is complicated by the fact that the codes Ei have different rates. Thus at some instance of the algorithm we have to process the columns that are in one part determined by information symbols and the other part is determined by redundancy symbols generated by the systematic encoders of the codes Ei. For simplicity, we will restrict our attention to codes C(m) of dimension k that is less than (q3 − g − q). Let φi : F q2−â(i)−1 −→ Fq be a systematic encoder for a code Ei. The input sequence to the encoder φi are symbols from an array r̃ = r̃i,j for j = 0, 1, . . . , q2 − â(i)− 2. We want to construct an algorithm that takes as input an array d of size q × q2 with arbitrarily chosen symbols in po- sitions (a, b) : b = 0, 1, . . . , q − 1; a = 0, 1, . . . , q2 − â(b)− 2 and zero in the remaining positions and that produces as output a code-array c. Let b̂(j) be defined as the number of information symbols in the jth column of d. The columns of d thus have the form dj = (d0,j ,d1,j , . . . ,db̂(j)−1, 0, 0, . . . , 0). We give a systematic encoder procedure in the following algorithm. During the procedure we also construct an array r̃. The ith row of r̃ is a codeword in Ei. Thus the first q2 − â(i) − 1 positions in the ith row of r̃ determine the ith row of r̃ completely. C. Algorithm 2 The algorithm is shown in (6). Theorem 1: Algorithm 2 computes a code array c of the Hermitian code C = (evD(mP∞))⊥. Proof: The matrix r̃ in the algorithm satisfies the conditions r̃j = Aj cj and the i-th row of r̃ is a codeword in Ei. Thus c is a code-array by Lemma 2. Algorithm 2 outlines the mathematical procedure to achieve systematic encoding of a Hermitian code. The real difficulty Algorithm 2 (6) Input: An q × q2 array d. An empty q × q2 array r̃. Iterations: For j = 0, 1, . . . , q2 − 1 1) Compute the known part of r̃j for i = 0, 1, . . . , q − 1− b̂(j) as r̃i,j = (φi((r̃i,0, r̃i,1, . . . , r̃i,q2−â(i)−2)))j 2) Solve the equation Aj(d0,j ,d1,j , . . . ,db̂(j)−1,j , yb̂(j),j , yb̂(j)+1,j , . . . , yq−1,j) T = (r̃0,j , r̃1,j , . . . , r̃q−1−b̂(j),j , uq−b̂(j),j , uq−b̂(j)−1,j , . . . , uq−1) for y b̂(j),j b̂(j)+1,j , . . . , yq−1,j , uq−b̂(j),j , uq−b̂(j)−1,j , . . . , uq−1 3) Set ci = (d0,j ,d1,j , . . . ,db̂(j)−1,j , yb̂(j),j , yb̂(j)+1,j , . . . , yq−1,j) r̃i = (r̃0,j , r̃1,j , . . . , r̃q−1−b̂(j),j , uq−b̂(j),j , uq−b̂(j)−1,j , . . . , uq−1) lies in an efficient implementation of the algorithm. We give such an implementation in Section IV but before proceeding we will need a simple lemma. Let A be any n × n matrix with inverse A−1. We assume that the submatrix of A indexed by elements i, j, i = n − l, . . . , n − 1 and j = 0, 1, . . . , l − 1 is nonsingular. This will always be true for the cases that we are interested in by Lemma 1. Let Il denote the l × l matrix and let D(l) be a n× n matrix of the following form: D(l) = such that A−1D = for a l × l matrix P̃ . We note that D(l)T is just a systematic encoding matrix for a code which has a parity check matrix the first l rows of A−1. D. Lemma 5 Let x0, x1, . . . , xn−l−1 and v0, v1, . . . , vl−1 be given. The solution for yn−l, yn−l+1, . . . , yn−1 and ul, ul+1, . . . , un−1 to the linear system of equations A(x0, x1, . . . , xn−l−1, yn−l, yn−l+1, . . . , yn−1)T = (v0, v1, . . . , vl−1, ul, ul+1, . . . , un−1)T can be found with the following algorithm. E. Algorithm 3 Algorithm 3 is given as shown in (7). Proof: A−1b̃T = A−1Db̂T which proves that y0, y1, . . . , yn−l−1 equal zero. Now it follows that A(x0, x1, . . . , xn−l−1, yn−l, yn−l+1, . . . , yn−1)T = b̃T + bT = A(A−1Db̂T ) + bT = Db̂T + bT = D(v0, v1, . . . , vl−1, 0, 0, . . . , 0)T −D(b0, b1, . . . , bl−1, 0, 0, . . . , 0)T + bT = (v0, v1, . . . , vl−1, ul, ul+1, . . . , un−1)T IV. EFFICIENT IMPLEMENTATION OF A SYSTEMATIC ENCODER Inspecting Algorithm 2 and Lemma 4, we see that we need modules for multiplication of an array with matrix A, A−1, systematic encoding of codes Ej , and a systematic encoding module for codes Dl defined as Dl = {d ∈ F A−1i,j dj = 0, i = 0, 1, . . . , l − 1} Before describing the modules in detail we give a black box description and the overall description of the implementation. A. Module A: Multiplication with Matrix A,A′ The module has as parallel input a vector d of length q and produces as serial output the numbers (AdT )i, i = 0, 1, . . . , q − 1 during the next q clock cycles. B. Module B: Multiplication with Matrix A−1, A′−1 Module B has as serial input a vector d of length q. After q clock cycles the parallel output is a vector A−1dT . C. Module C: Systematic Encoding of Codes Ei The module has a serial input of q2− â(i)−1 symbols and produce one symbol per clock cycle. The clocking frequency is 1/q of the overall clock rate. Fig. 1 OVERALL OUTLINE OF THE ENCODER CIRCUIT. SWITCHES a & b ARE SYNCHRONIZED AND ROTATE EVERY CLOCK CYCLE. THE CIRCUIT IS DESCRIBED IN THE TEXT. D. Module D: Systematic Encoding of Codes Dl Module D takes a serial input of length l and produces as serial output a codeword of length Dl. E. Encoder Figure 1 outlines the overall implementation. When the left hand input becomes valid, the output of Module A is added to the negative output of Module C, effectively implementing steps 1 and 2 of Algorithm 3. The sum is fed to Module D which implements step 3 of Algorithm 3. The output of Module D is combined with the output of Module A to implement step 4 of Algorithm 3. Simultaneously it is fed to module B of the implementation. After q clock cycles the output of Module B is added to the input thus implementing step 5 of Algorithm 3. Module C can be implemented as an obvious modification of a systematic encoding circuits for RS codes [10]. Module D implements systematic encoding of a code with parity check matrix given by first l rows of matrix A−1. From the form of matrices A−1, we see that code Dl may be defined Dl = {d ∈ F j=0 dq−1−j(xi) j = d0, x0 = y0, xs+1 = (y0 + γs), s = 0, 1, . . . , l − 2} and we can use standard encoding techniques for shortened cyclic codes which are modified in the obvious way. V. FINAL REMARKS A low complexity algorithm for systematic encoding and syndrome computation of Hermitian codes has been presented. The algorithm has a run time complexity of O(n2/3) and is suitable for VLSI implementation. We give an outline for the encoder architecture, which uses as main blocks, q varying- rate Reed Solomon encoders. The architecture achieves a much lower space complexity in terms of finite field multipliers and memory elements when compared to earlier reported works. VI. ACKNOWLEDGEMENT The authors would like to thank Science Foundation Ireland, Claude Shannon Institute and Deutsche Forschungsgemein- schaf for supporting parts of this research work. REFERENCES [1] V. D. Goppa, Codes on Algebraic Curves, Soviet math. Dokl., 1981, 24, pp. 75-91 [2] B. E. Wahlen, and J. Jimenez, Performance Comparison of Hermitian and Reed-Solomon Codes, Proc. MILCOM, 1997. [3] J. H. van Lint and T. A. Springer, Generalized Reed-Solomon Codes from Algebraic Geometry, IEEE Trans. Inform. Theory, vol. IT-33, pp. 305-309, May 1987 [4] T. Yaghoobian and I. F. Blake, Hermitian Codes as Generalized Reed- Solomon Codes, Design, Codes, Cryptography, vol. 2, pp. 5-17, 1992 [5] B. Z. Shen, On Encoding and Decoding of the codes from Hermitian Curves, Proc., Cryptography and Coding III, vol. 45, M. Ganley, Ed., Oxford, UK, pp. 337-356, 1993 [6] J. Little, K. Saints and C. Heegard, On the structure of Hermitian Codes, Journal of Pure and Applied Algebra, vol. 121, pp. 293-314, 1997 Algorithm 3 (7) 1) bT = A(x0, x1, . . . , xn−l−1, 0, 0, . . . , 0)T 2) b̂T = (v0, v1, . . . , vl−1, 0, 0, . . . , 0)T − (b0, b1, . . . , bl−1, 0, 0, . . . , 0)T 3) b̃T = Db̂T 4) (x0, x1, . . . , xn−l−1, yn−l, yn−l+1, . . . , yn−1)T = (x0, x1, . . . , xn−l−1, 0, 0, . . . , 0)T +A−1b̃ 5) (v0, v1, . . . , vl−1, ul, ul+1, . . . , un−1)T = b̃T + bT [7] C. Heegard, J. Little and K. Saints, “Systematic Encoding via Gröbner bases for a class of Algebraic-Geometric Codes,” IEEE Trans. Info. Theory, vol. IT-41, pp. 1752-1762, Nov. 1995 [8] J. Chen and C. Lu, ”A Serial-In-Serial-Out Hardware Architecture for Systematic Encoding of Hermitian Codes via Gröbner Bases”, IEEE Trans. Info. Theory, Vol. 52, No. 8, pp. 1322-1332, August 2004 [9] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, Germany 1993. [10] G. Fettewis and M. Hassner, “A Combined Reed-Solomon Encoder and Syndrome Generator with Small Hardware Complexity” Proc. Int. Symp. Circuits and Systems, pp. 1871-1874, 1992. Introduction Hermitian Codes and Syndrome Computation Lemma 1 Definition 1 Lemma 2 Systematic Encoding Lemma 3 Algorithm 1 Algorithm 2 Lemma 5 Algorithm 3 Efficient Implementation of a Systematic Encoder Module A: Multiplication with Matrix A, A' Module B: Multiplication with Matrix A-1, A'-1 Module C: Systematic Encoding of Codes Ei Module D: Systematic Encoding of Codes Dl Encoder Final Remarks Acknowledgement References

0704.0591

Quantum criticality and disorder in the antiferromagnetic critical point of NiS$_{2}$ pyrite

Quantum criticality and disorder in the antiferromagnetic critical point of NiS2 pyrite N. Takeshita1, S. Takashima2, C. Terakura1, H. Nishikubo2, S. Miyasaka3, M. Nohara2, Y. Tokura1,4, and H. Takagi1,2 1Correlated Electron Research Center (CERC), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba 305-8562, Japan 2Department of Advanced Materials, University of Tokyo, Kashiwa 277-8561, Japan 3Department of Physics, Osaka University, Toyonaka 560-0043, Japan 4Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan (Dated: November 11, 2021) A quantum critical point (QCP) between the antiferromagnetic and the paramagnetic phases was realized by applying a hydrostatic pressure of ∼ 7 GPa on single crystals of NiS2 pyrite with a low residual resistivity, ρ0, of 0.5 µΩcm. We found that the critical behavior of the resistivity, ρ, in this clean system contrasts sharply with those observed in its disordered analogue, NiS2−xSex solid-solution, demonstrating the unexpectedly drastic effect of disorder on the quantum criticality. Over a whole paramagnetic region investigated up to P = 9 GPa, a crossover temperature, defined as the onset of T2 dependence of ρ, an indication of Fermi liquid, was suppressed to a substantially low temperature T ∼ 2 K and, instead, a non Fermi liquid behavior of ρ, T 3/2-dependence, robustly showed up. PACS numbers: A hallmark of strongly correlated electron systems is the presence of a rich variety of phases often competing with each other. When two phases meet with each other in the T = 0 limit by tuning a phase controlling parame- ter such as pressure and chemical doping, quantum fluc- tuations often give rise to exotic states of electrons, which has been attracting considerable interest in condensed matter research [1]. One of the most fascinating cases is a breakdown of the Fermi liquid at magnetic quantum crit- ical points (QCP) in itinerant magnets, which has been believed to be captured by self-consistent renormalization theory [2] and scaling analysis [3, 4]. The onset temper- ature of Fermi liquid coherence, probed by a quadratic temperature dependence of resistivity ρ(T ) ∝ T 2, is pre- dicted to be suppressed by the presence of low lying spin fluctuations near QCP and, right at QCP, a non Fermi liquid ground state is realized which manifests itself as a non-trivial power law behavior of resistivity ρ(T ) ∝ T n down to the T = 0 limit, where n = 3/2 for antiferro- magnetic (AF) QCP and n = 5/3 for ferromagnetic (F) QCP [5]. A V-shaped recovery of Fermi liquid behavior, T 2-resistivity, around QCP is anticipated as a function of phase tuning parameter. The critical behavior of ρ(T ) near the AF QCP in NiS2−xSex solid solution is a textbook demonstration of standard theories for QCP. NiS2 crystallizes in the pyrite structure. Ni is divalent and therefore accommodates two electrons in doubly degenerate eg orbitals (t g). Due to a strong onsite Coulomb repulsion among Ni eg electrons, the system is a S = 1 Mott insulator [6, 7]. By substitut- ing S with Se, the effective bandwidth can be increased due to the increase of p-d hybridization. With increasing x in NiS2−xSex, the system experiences a weakly first or- der transition to an AF metal with a collinear spin struc- ture [8] at x ∼ 0.4 and then a second order transition to a paramagnetic metal at x = 1.0. In the AF metal phase of NiS2−xSex, the AF transi- tion temperature is TN ∼ 90 K at x = 0.5 and, with increasing x, continuously goes down to T = 0 at x = 1.0, giving rise to a well defined AF QCP. The T 3/2 de- pendence of ρ(T ), expected for AF QCP, is observed at least down to 1.7 K at x = 1.0. On going away from x = 1.0, T 2-behavior of ρ(T ) quickly recovers and a V-shaped region with T 2 resistivity is identified around x = 1.0. It is known that the application of pressure is equivalent to Se substitution in that it increases the band width. By applying pressure P on an AF metal NiS1.3Se0.7, sup- pression of TN analogous to Se substitiution was indeed observed and, eventually, QCP was approached with P = 2 GPa [9]. The phase diagram and the critical behavior of the resistivity in pressurized NiS1.3Se0.7 were essen- tially identical with Se content x simply replaced with P Recently, however, there has been growing evidence that, the above mentioned textbook picture is violated in a variety of intermetallic systems. In a helical magnet MnSi [10] and a weak ferromagnet ZrZn2 [11], a non triv- ial power law behavior of resistivity, ρ(T ) ∝ T 3/2, dom- inates the resistivity down to a very low temperature, not only right at the QCP but also over a wide range of paramagnetic phase. At the AF QCP in CePd2Si2, with increasing the purity of sample, the exponent of the power law resistivity was found to deviate significantly from the standard value of 3/2 [12]. The common feature among these systems is that they are clean with a low residual resistivity of ρ0 < 1 µΩcm [10, 11]. In contrast, in NiS2−xSex solid solution where textbook example of critical behavior of ρ(T ) is observed, Se substitution in- herently gives rise to disorder. Indeed, ρ0 of NiS2−xSex around QCP is as large as several 10 µΩcm. Questions immediately arise. Does the non-trvial behavior in the intermetallics represent a generic property of magnetic http://arxiv.org/abs/0704.0591v1 QCP in clean systems? Does standard behavior of QCP shows up only when the system is disordered? To tackle these questions experimentally, we attempted to realize a “clean” QCP in NiS2−xSex. The parent compound of NiS2−xSex, NiS2, is pure and presumably clean. If one can approach the QCP of pure NiS2 by pressure without relying on Se substitution, a clean analogue of AF QCP in NiS2−xSex can be explored and the impact of disor- der on QCP can be captured. Recent progress of high pressure technique enabled us to do so. In this Letter, we address the issue of criticality and disorder by examining the critical behavior of resistivity of pure NiS2 under pressures. The AF QCP of NiS2 was reached at ∼ 7 GPa, where the system was found very clean with a low residual resistivity ρ0 of ∼ 0.5 µΩcm. Not only right at the QCP but over an entire range of the paramagnetic phase investigated, the recovery of Fermi liquid T 2 of ρ(T ) is suppressed substantially to a very low temperature below ∼ 2 K and non Fermi liquid behavior with T 3/2 dependence of ρ(T ) dominated. We demon- strate the drastic influence of disorder on this AF QCP by contrasting the result with previous pressure work on NiS1.3Se0.7 with a residual resistivity of 60 µΩcm [9]. NiS2 sample used in this study was prepared by a va- por transport technique. The resistivity measurement was performed by a conventional four probe technique under hydrostatic pressure up to ∼ 10 GPa in a cubic anvil type pressure system down to 3 K and also in a modified Bridgman-type pressure cell down to 180 mK. The results obtained by the two different pressure setups agreed reasonably in the temperature range of overlap, indicating a very good homogeneity of pressure. Pres- sure was calibrated by measuring the superconducting transition temperature of Pb [13]. The inset of Fig. 1 demonstrates ρ(T ) of NiS2 at rel- atively low pressures below 4 GPa. With applying pres- sure, the insulating behavior of ρ(T ) switches into metal- lic behavior, indicating the occurrence of metal-insulator transition. In between 2.6-3.4 GPa, we observe a discon- tinuous jump of resistivity as a function of temperature, which corresponds to a first order metal-insulator transi- tion line on the phase diagram in Fig. 1. The discontinu- ous jump appears to terminate around 200 K, indicating the presence of a critical end point. In the phase dia- gram of NiS2−xSex solid solution, the first order phase line terminates at much lower temperature and is hard to identify [14]. This difference appears to suggest the strong influence of disorder on the Mott criticality. As seen in the inset of Fig. 1, ρ(T ) of pure NiS2 showed metallic behavior above P = 2.6 GPa. The residual re- sistivity at the critical point was as low as ∼ 0.5 µΩcm, demonstrating that the system is indeed very clean. Mag- netic ordering in the AF metal phase manifests itself as a very weak but sharp kink in ρ(T ) at TN as indicated by the arrows. The antiferromagnetic transition tempera- ture TN thus determined systematically goes down upon 1086420 P (GPa) 3002001000 T (K) 0, 2.6, 3.0, 3.2, 3.3, 3.36, 3.4 GPa TNInsulator 100500 T (K) 4.0 GPa 5.0 GPa 6.2 GPa 7.5 GPa FIG. 1: The electronic phase diagram of clean NiS2 pyrite as a function of pressure. PM and AFM denote paramagnetic metal and AF metal, respectively. The inset shows the tem- perature dependent resistivity under pressures, P = 0 - 3.4 GPa (left) and P = 4.0 - 7.5 GPa (right). pressure and approaches T = 0 somewhere around 7-7.5 GPa. No superconductivity was observed between P = 6 and 9.1 GPa down to 180 mK, in spite of the low resid- ual resistivity. This appears to suggest that realizing an AF QCP in clean systems alone is not enough to achieve exotic superconductivity as observed in heavy Fermion compounds [15, 16, 17, 18] and that additional ingredi- ents such as Kondo physics must be invoked. The pressure dependence of TN , determined by the kink in ρ(T ), together with the first order metal insulator transition, is summarized as a phase diagram in Fig. 1. TN appears to decrease almost linearly in contradiction to (Pc − P ) 2/3 dependence expected from self consistent renormalization theory [5]. Unusual linear suppression of the magnetic transition temperature was also observed analogously for a helical magnet MnSi [10] and a weakly ferromagnet ZrZn2 [11] when the sample is very clean. It may be interesting to infer that, in these clean system, the magnetic transition as a function of pressure is re- ported to be a first order rather than a second order. In the clean NiS2, we cannot rule out the possibility of a first order transition at this stage, because ρ(T ) is not very sensitive to TN near the critical point. The signature of AF criticality in this clean system was explored. The inset of Fig. 2 demonstrates ρ(T ) below 30 K, plotted as ρ vs. T 3/2. In the antiferromagnetic phase at P = 6.2 GPa, ρ - T 3/2 curve is linear above TN but shows superlinear behavior below TN . The temperature dependence below TN is found to be approximately T indicative of the formation of coherent quasi particles. In the paramagnetic phase above ∼ 7 GPa, however, the ρ - T 3/2 curve shows a linear behavior down to very low T (K) T 3/2 6.8 GPa 2001000 T 3/2 (K3/2) 6.2 GPa FIG. 2: Temperature dependent part of resistivity ρ−ρ0 as a function of temperature under pressure above ∼ 7 GPa, where the system is paramagnetic, plotted as log(ρ− ρ0) vs. log T . The inset shows ρ vs. T 3/2 plot. temperature which is expected for the antiferromagnetic critical point due to low lying spin fluctuations. It is remarkable to see T 3/2 behavior characteristic of the an- tiferromagnetic critical point over such a wide range of pressure from ∼ 7 GPa to ∼ 9 GPa. ρ(T ) is surprisingly insensitive to the pressure in the paramagnetic region above 7 GPa and it is hard to find a signature of crit- icality. This is analogous to those observed in a helical magnet MnSi [10] and a weakly ferromagnetic magnet ZrZn2 [11] when the sample is clean, implying that un- usual critical behavior in clean systems is ubiquitous. To investigate the details of unusual temperature de- pendence in the paramagnetic phase further, we plotted the temperature dependence of ρ− ρ0 as log(ρ− ρ0) vs. logT in the main panel of Fig. 2. The residual resistivity ρ0 was determined by extrapolating ρ - T 2 curve to T = 0 limit. We note here that the temperature dependent part ρ− ρ0 is comparable to ρ0 at ∼ 3 K and, therefore, the ambiguity originating from the estimate of ρ0 does not influence the temperature dependence of ρ − ρ0 at least above 1 K. It is again clear that the slope is appar- ently smaller than 2 and instead close to 3/2 above ∼ 2 K. At the lowest temperatures below ∼ 2 K, however, the slope becomes steeper and T 2-resitivity appears to recover eventually below 2 K. This crossover tempera- ture to T 2-resistivity is again insensitive to pressure and always 2-3 K up to 9 GPa. Close inspection of data indi- cates that the crossover temperature is the lowest around 7.5 GPa but only slightly lower than the other pressures. 987654 P (GPa) 6543210 P (GPa) NiS1.3Se0.7 FIG. 3: Contour plot of the exponent n of power low depen- dence of resistivity on pressure-temperature plane, demon- strating the criticality observed in the temperature depen- dence of resistivity. The main panel is data for clean NiS2 and the inset shows data for dirty NiS2−xSex. The Néel tem- perature determined by resistivity anomaly was shown by the white line. This strong suppression of the crossover to T 2 behav- ior in ρ(T ), over a remarkably wide range of pressure from ∼ 7 GPa up to ∼ 9 GPa, contrasts sharply with the observation in Se-doped samples, where the recovery of T 2 resistivity was clearly observed not only for the magnetic side but also for the paramagnetic side. As a function of Se-doping, a metal-insulator transition and antiferromagnetic critical point occurs at around Se con- tent x = 0.4 and 1.0, respectively, while ∼ 2.5 GPa and ∼ 7 GPa are required to reach a metal-insulator transition and QCP, respectively. This yields a conversion ratio of phase controlling parameters, ∼ 0.15 Se/1 GPa. In this disordered NiS2−xSex, the T 3/2-dependence of ρ(T ) dominates at the QCP of x = 1.0. With further doping of Se up to x = 1.33 which is equivalent of additional pressure of ≃ 2 GPa, however, the T 2 resistivity is fully recovered and can be observed below ∼ 80 K [9]. Analo- gously, in a Se doped NiS1.3Se0.7 crystal under pressure, on going from the QCP at P ∼ 2 GPa to P = 4 GPa, T 2 resistivity recovers quickly and shows up below 80 K with increase of 2 GPa. These should be compared with the low crossover temperature of 2-3 K, approximately 2 GPa above the critical point. To visually illustrate these points, we plotted the ex- ponent of power law dependence of ρ(T ), n as a contour map on the pressure-temperature plane in Fig. 3. The exponent was determined by taking the derivative of the log(ρ − ρ0) - logT curve in Fig. 2. It is clear that the V-shaped recovery of Fermi liquid behavior around QCP is absent in clean NiS2. The recovery can be seen only on the antiferromagnetic side below 7 GPa, where the region with n = 2 (T 2) develops below TN . Above the critical point of P ∼ 7 GPa, it is clear that the n = 1.5 (T 3/2) region predominantly occupies a majority of the paramagnetic phase. A thin region with a different color is lying at the T = 0 limit. This corresponds to the marginal recovery of Fermi liquid behavior below ∼ 2 K. In the inset of Fig. 3, we have constructed the contour map also for the NiS1.3Se0.7 data under pressure from a previous report [7]. Note again the V-shaped recovery on the temperature scale of 100 K over ∼ 2 GPa pressure. The remarkable contrast in the critical behavior be- tween pure NiS2 and NiS2−xSex, visually demonstrated in Fig. 3, indicates that the influence of disorder on quantum criticality is surprisingly drastic, since the only difference between the two systems is the disorder pro- duced by Se substitution. In NiS1.3Se0.7 solid solution, the residual resistivity ρ0 is approximately 60 µΩcm, which is larger than those of pure NiS2 by two orders of magnitude. When the samples are disordered, we do see a canonical behavior of the QCP as predicted by stan- dard theories [3, 4, 5]. To our surprise, once the system becomes clean, the textbook behavior is gone and the Fermi liquid coherence seen in ρ(T ) is dramatically sup- pressed. We should note that the magnitude of ρ(T )−ρ0 is roughly 50 µΩcm in the temperature range below 100 K at around QCP. In the Se doped crystal, inelastic scat- tering is always weaker than or at most comparable to elastic scattering due to disorder below 100 K. In the pure NiS2, in contrast, the same situation, ρ − ρ0 < ρ0 occurs only below 2-3 K, where we observed crossover to T 2-resistivity. This suggests that disorder might be controlling the appearance of T 2-resistivity. One of the plausible scenarios for the strong influence of disorder and robust non-Fermi liquid behavior might be a dichotomy of the Fermi surface [19]. It is natural that a specific part of Fermi surface, a “hot spot”, is coupled strongly with a critical antiferromagnetic fluctu- ation with a characteristic wave vector Q. There may remains a region with well defined quasiparticles free from critical fluctuations, a cold spot. The transport is then determined by an interplay of these two contri- butions at high temperatures but eventually a cold spot with T 2-dependence should dominate the conduction at very low temperatures. This phase separation in k-space might explain in part the unusual temperature depen- dence observed in pure NiS2 but it is not clear whether the robustness of non Fermi liquid behavior can be prop- erly described. In this scenario, the strong influence of disorder can be naturally understood. The strong elas- tic scatting should mix up the hot spot and cold spot and the inelastic scattering therefore becomes effectively isotropic, which might be close to the situation implic- itly assumed in standard theories [5]. Another scenario might be a phase separation and the resultant domain or bubble formation in real space as discussed in clean MnSi where the helical spin order disappears discontinuously as a first order transition [10]. These bubbles and domains have been proposed to be responsible for the robust non Fermi liquid behavior in the paramagnetic phase. It is worth checking the possible first order transition carefully checking the magnetism at ∼ 7 GPa. In conclusion, we have demonstrated the sharp con- trast in the quantum critical behavior of ρ(T ) between the clean and the disordered systems by examining a sin- gle crystal of NiS2 with a low residual resistivity of ∼ 0.5 µΩcm. Previously, the V-shaped recovery of Fermi liquid behavior (T 2-behavior of resistivity) around the antifer- romagnetic critical point was clearly observed as a func- tion of pressure and Se content in the dirty NiS2−xSex systems with ρ − ρ0 < ρ0. In sharp contrast, we found a robust non Fermi liquid behavior over a wide pressure range in the paramagnetic side of a QCP in the clean sys- tem with ρ−ρ0 ≫ ρ0. These results clearly demonstrate that our understanding of the quantum critical point is still far from complete and some important ingredient must be missing. The authors would like to thank M. Imada and H. Fukuyama for discussion. This work was partly sup- ported by a Grant-in-Aid for Scientific Research (No. 18043009) from the Ministry of Education, Culture, Sports, Science and Technology of Japan and CREST, Japan Science and Technology Agency (JST). [1] P. Coleman and A. J. Schofield, Nature 433, 226 (2005). [2] T. Moriya, Spin Fluctuations in Itinerant Electron Mag- netism (Springer-Verlag, Berlin, 1985). [3] J. A. Hertz, Phys. Rev. B. 14, 1165 (1976). [4] A. J. Millis, Phys. Rev. B. 48, 7183 (1993). [5] T. Moriya and K. Ueda, Adv. Phys., 49, 555 (2000). [6] S. Ogawa, J. Appl. Phys., 50, 2308 (1979). [7] J. A. Wilson, The Metallic and Non-metallic States of Matter, pp.215-260 (Taylor & Francis, London, 1985). [8] T. Miyadai et al., J. Phys. Soc. Jpn., 38, 115 (1975). [9] S. Miyasaka et al., J. Phys. Soc. Jpn., 69, 3166 (2000). [10] C. Pfleiderer, S. R. Julian, and G. G. Lonzarich, Nature 414, 427 (2001). [11] S. Takashima et al., J. Phys. Soc. Jpn. 76, 043704 (2007). [12] F. M. Grosche et al., J. Phys.: Condens. Matter, 12, L533 (2000). [13] N. Môri, H. Takahashi, and N. Takeshita, High Pressure Research, 24, 225 (2004). [14] G. Czjzek et al., J. Magn. Mag. Mat., 3, 58 (1976). [15] D. Jaccard, K. Behnia, and J. Sierro, Phys. Lett. A 163, 475 (1992). [16] S. R. Julian et al., J. Phys.: Condens. Matter, 8, 9675 (1996). [17] C. Petrovic et al., J. Phys.: Condens. Matter, 13, L337 (2001). [18] S. S. Saxena et al., Nature 406, 587 (2000). [19] A.Rosch, Phys. Rev. B, 62, 4945 (2000).

0704.0592

Spin coherence of holes in GaAs/AlGaAs quantum wells

Spin coherence of holes in GaAs/AlGaAs quantum wells M. Syperek1,2, D. R. Yakovlev1,†, A. Greilich1, J. Misiewicz2, M. Bayer1, D. Reuter3, and A. D. Wieck3 Experimentelle Physik II, Universität Dortmund, D-44221 Dortmund, Germany Institute of Physics, Wroc law University of Technology, 50-370 Wroc law, Poland and Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (Dated: August 2, 2021) The carrier spin coherence in a p-doped GaAs/(Al,Ga)As quantum well with a diluted hole gas has been studied by picosecond pump-probe Kerr rotation with an in-plane magnetic field. For resonant optical excitation of the positively charged exciton the spin precession shows two types of oscillations. Fast oscillating electron spin beats decay with the radiative lifetime of the charged exciton of 50 ps. Long lived spin coherence of the holes with dephasing times up to 650 ps. The spin dephasing time as well as the in-plane hole g factor show strong temperature dependence, underlining the importance of hole localization at cryogenic temperatures. PACS numbers: 42.25.Kb, 78.55.Cr, 78.67.De Recently the investigation of the coherent spin dynam- ics in semiconductor quantum wells (QW) and quantum dots has attracted much attention, due to the possible use of the spin degree of freedom in novel fields of solid state research such as spin-based electronics or quantum information processing [1, 2, 3]. Until now the inter- est has been mostly focused on the spin coherence of electrons, while experimental information about the spin coherence of holes is limited [4]. The hole as a Luttinger spinor has properties strongly different from the electron spin, such as a strong spin-orbit coupling, a strong direc- tional anisotropy, etc. It plays an important role also in coherent control of electron spins, since in many optical schemes charged electron-hole complexes are proposed as intermediate manipulation states [5]. Earlier, the hole spin dynamics in GaAs-based QWs has been measured by optical orientation detecting photoluminescence (PL) either time-integrated or time- resolved [4, 6, 7, 8, 9]. Experimental studies addressed the longitudinal spin relaxation time T1 [6, 7, 8] and the dephasing time T ∗ , exploiting the observation of hole spin quantum beats [4]. The reported relaxation times vary from 4 ps [6] up to ∼1 ns [4, 8] demonstrating strong de- pendence on doping level, doping density and excitation energy. A major drawback of PL techniques is, however, that the spin coherence can be traced only as long as both electrons and holes are present and photolumines- cence can occur. Further, they work only for studying the spin dynamics of minority carriers in a sea of ma- jority carriers and are therefore restricted to undoped or n-type doped QWs. However, then the holes can interact with electrons, providing additional relaxation channels via exchange or shake-up processes [8, 10]. These mech- anisms can be excluded for p-doped structures if the hole spin relaxation occurs on time scales longer than the ra- diative annihilation of electrons. A pump-probe Kerr ro- tation (KR) technique using resonant excitation allows to realize such measurements, which up to now have been reported only for bulk p-type GaN [11] and not yet for low-dimensional systems. The theoretical analysis of the hole spin dynamics in QWs has been focused on free holes [10, 12, 13, 14] by considering different relaxation mechanisms: (i) a Dyakonov-Perel like mechanism, (ii) an acoustic phonon assisted spin-flip due to spin mixing of valence band states, (iii) an exchange induced spin-flip due to scatter- ing on electrons, which resembles the Bir-Aronov-Pikus mechanism, but for holes. Recently the attention has been drawn on the role of hole localization and the de- phasing caused by fluctuations of the in-plane g factor has been calculated [15]. In this paper we use time-resolved pump-probe Kerr rotation [16] to investigate the spin coherence of holes in a p-doped GaAs/Al0.34Ga0.66As single QW with a low hole density. We find spin dephasing times reaching al- most the ns-range at a temperature T = 1.6 K with a hole in-plane g factor of about 0.01. Both quantities de- crease strongly with increasing temperature, suggesting the important influence of hole localization. We discuss also a mechanism that provides generation of spin co- herence for the hole gas under resonant excitation of the positively charged exciton. The structure was fabricated by molecular-beam epi- taxy on a (100) oriented GaAs substrate. A 15 nm- wide GaAs QW was grown on top of a 380 nm- thick Al0.34Ga0.66As barrier and overgrown by a 190 nm-thick Al0.34Ga0.66As layer. 21 nm-thick layers with Al0.34Ga0.66As effective composition realized by GaAs/AlAs short-period superlattices were deposited on both sides of the QW in order to improve interface pla- narity. Two δ-doped layers with Carbon acceptors were positioned symmetrically at 110 nm distance from the QW. The hole gas concentration and mobility in the QW are 1.51× 1011 cm−2 and 1.2× 105 cm2/Vs, respectively, as determined by Hall measurements at T = 4.2 K. It was possible to deplete the hole density in the QW by above barrier illumination and even invert the majority carrier type, resulting in a diluted electron gas. The sam- ple temperature was varied from 1.6 to 6 K. A mode-locked Ti:Sapphire laser with a repetition rate of 75.6 MHz and a pulse duration of ∼1.5 ps (∼1 meV full width at half maximum) was used for optical excitation. http://arxiv.org/abs/0704.0592v1 FIG. 1: (a) KR traces for a p-type 15 nm-wide GaAs/Al0.34Ga0.66As QW vs time delay between pump and probe pulses at B = 0 and 7 T with field tilted by ϑ = 4◦ out of QW plane. Laser at 1.5365 eV is resonant with T+ line. Power was set to 5 and 1 W/cm2 for pump and probe, respectively. Bottom trace was recorded with additional laser illumination at 2.33 eV. T = 1.6 K. (b) PL spectra for DHG (excitation at 1.579 eV) and DEG regime (above barrier ex- citation at 2.33 eV). (c) Comparison of KR traces for ϑ = 0 and 4◦. The laser beam was split into a circularly polarized pump and a linearly polarized probe beam. Both beams where focused on the sample surface to a spot diameter of ∼100 µm. Magnetic fields B ≤ 10 T were applied about per- pendicular to the structure growth z-axis (Voigt geome- try). In a pump-probe KR experiment the pump pulse coherently excites carriers with spins polarized along the z axis. The subsequent coherent evolution of the spins in form of a precession about the magnetic field is tested by the probe pulse polarization. To detect the change of the linear probe polarization plane (the KR angle), a homodyne technique based on phase-sensitive balanced detection was used. Photoluminescence spectra excited above and below the band gap of the Al0.34Ga0.66As barriers are shown in Fig. 1(b) at B = 0 and 7 T. A single PL line correspond- ing to the positively charged trion T+ (consisting of two holes and one electron) is seen for the regime of diluted hole gas (DHG) established for below-barrier excitation. After inverting the type of majority carriers to the DEG FIG. 2: Top trace: KR signal measured at B = 7 T for ϑ = 4◦. Bottom traces are obtained by separating electron and hole contributions (see text). Excitation conditions as in Fig. 1. regime by above barrier illumination the PL spectra con- sist of the exciton (X) and negatively charged trion (T−) lines. The type of majority carriers in the QW can be iden- tified by the KR signals measured at B = 7 T, with the laser energy tuned to the trion resonance. The bottom trace in Fig. 1(a) was measured with additional above- barrier illumination (DEG regime) and shows long-lived electron spin beats with a dephasing time of 2.5 ns which is considerably longer than the radiative decay time of resonantly excited trions of about 50 ps. The precession frequency corresponds to a g factor | ge |= 0.285± 0.005, which is typical for electrons in GaAs-based QWs. Without above-barrier illumination in the DHG regime, fast electron precession is observed only during ∼ 200 ps after pump pulse arrival [see middle trace in Fig. 1(a)]. This signal is caused by the coherent preces- sion of the electron in T+ and disappears with the trion recombination. The electron beats are superimposed on the hole beats with a much longer precession period. The hole beats decay with a time constant of about 100 ps and can be followed up to 500 ps delay. At these long times the KR signal is solely given by coherent hole precession. Experimentally, it is difficult to observe the hole spin quantum beats due to the very small in-plane hole g fac- tor. To enhance the visibility we tilted the magnetic field slightly out of the plane by an angle ϑ = 4◦ to increase the hole g factor by mixing the in-plane com- ponent (gh,⊥) with the one parallel to the QW growth axis (gh,‖), which typically is much larger: gh(ϑ) = sin2 ϑ+ g2 h,⊥ cos 2 ϑ [17]. The strong change of the hole beat frequency with the tilt angle is seen in Fig. 1(c). The precession frequency is analyzed by ωh = µB | gh | B/~, where µB is the Bohr magneton, and gives | gh,⊥ |= 0.012±0.005 for ϑ = 0 ◦ and | gh |= 0.048±0.005 for ϑ = 4◦. The electron and hole contributions to the KR ampli- FIG. 3: Hole component of KR signal at different magnetic fields and ϑ = 4◦. Top inset: Magnetic field dependence of the hole dephasing time T ∗2 . Solid line is 1/B fit to data. Bottom inset: Hole spin precession frequency vs magnetic fields. Line is guide to the eye. In inserts closed and open circles show the data measured for pump to probe powers of 1 to 5 W/cm2 and 5 to 1 W/cm2, respectively. T = 1.6 K. tude, ΘK, can be separated by fitting the experimental data with a superposition form of exponentially damped harmonic functions for electrons and holes: i=e,h Ai exp(− ) cos(ωi∆t). (1) Ai are the corresponding signal amplitudes at pump-to- probe delay ∆t = 0, and T ∗ 2,i are the spin dephasing times. An example for a decomposition of the KR signal in the DHG regime is shown in Fig. 2. Let us turn now to the hole coherence. Figure 3 shows the hole contribution to the KR signal for different B at T = 1.6 K. From the fit by Eq.(1) we have obtained the dephasing time T ∗ , which is plotted versus B in the inset. A very long lived hole spin coherence with T ∗2 = 650 ps is found at B = 1 T. With increasing B up to 10 T it short- ens to 70 ps. The field dependence can be well described by a 1/B-form (see the line in the inset), from which we conclude that the dephasing shortening arises from the inhomogeneity of the hole g factor ∆gh = 0.0007 in the QW, which is translated into a spread of the preces- sion frequency: ∆ωh = ∆ghµBB/~. Since T ∝ 1/∆ωh, this explains the 1/B-dependence of the dephasing time. The magnetic field dependence of the hole precession fre- quency in the lower inset of Fig. 3 shows an approximate B-linear dependence up to 7 T. For higher fields a super linear increase is seen which indicates a change of the hole g factor due to mixing between heavy and light hole states, induced by the field. Two sets of experimental data measured for pump to FIG. 4: Temperature dependence of the hole KR signal at B = 7 T and ϑ = 5◦. Pump and probe powers are set to 1 and 5 W/cm2, respectively. Inset: Hole spin dephasing time T ∗2 vs temperature. probe powers 1 to 5 W/cm2 and 5 to 1 W/cm2 are com- pared in the insets of Fig. 3. The very similar results demonstrate performance of the experiment in the linear regime for both pump and probe beams with power not exceeding 5 W/cm2. Insight into the origin of the long hole spin coherence can be taken from KR at varying temperatures. The data in Fig. 4 measured at ϑ = 5◦ show that (i) the dephasing time T ⋆ decreases from 110 down to 60 ps when increasing the temperature from 1.6 to 6 K (see also inset), and (ii) simultaneously the precession frequency decreases notably corresponding to a g factor decrease from 0.057 to 0.030. These results can be naturally explained by hole lo- calization in the QW potential relief due to monolayer well width fluctuations. The localization energy does not exceed 0.5 meV, which is comparable with the thermal energy at T = 6 K. Free holes are expected to have a short spin coherence time T2 limited by the efficient spin relaxation mechanisms due to the spin-orbit interaction [10, 13, 14]. For localized holes these mechanisms are mostly switched off and one can expect long T2 times. However, in a KR experiment we do not measure the T2 time, but rather the ensemble dephasing time T (Ref. 15), as confirmed by the 1/B dependence in the inset of Fig. 3. The T ∗2 time gives a lower boundary for the spin coherence time T2. Therefore we can conclude, that the T2 for localized holes is at least 650 ps. Thermal delocalization of holes on the one hand decreases the role of inhomogeneities and reduces ∆gh, which should lead to longer dephasing times. On the other hand, then the fast decoherence of free holes becomes the limiting factor for the spin beats dynamics. Mixing of heavy and light hole states in a QW is en- hanced by localization effects. This should be detectable by an increase of the in-plane hole g factor, which is close to zero for free holes [4, 15, 18]. The decrease of the hole g factor with increasing temperature shown in Fig. 4 is consistent with a hole delocalization scenario. We turn now to discussing the mechanism for opti- cal generation of hole spin coherence in a QW with a DHG. The generation mechanism is similar to the one suggested for singly charged quantum dots [19, 20] and QWs with a DEG [21]. In our experiment pump and probe are resonant in energy with the positively charged trion T+. Due to the considerable heavy-light hole split- ting, the circularly polarized pump creates holes and elec- trons with well-defined spin projections, Jh,z = ±3/2 and Se,z = ±1/2, respectively, according to the optical selec- tion rules [12]. Therefore, |⇑⇓↓〉 (|⇑⇓↑〉) trions can be generated by a σ+ (σ−) polarized pump. Here the thick and thin arrows give the spin states of holes and elec- trons, respectively. The pump pulse duration is much shorter than the spin coherence and the electron-hole recombination times. If in addition the pump duration is shorter than the charge coherence time of the trion state the pulse creates a co- herent superposition of a resident hole from the DHG and a hole singlet trion T+. The spin state of the resident hole with arbitrary spin orientation before excitation can be described by α |⇑〉+ β |⇓〉, where |α|2 + |β|2 = 1. With- out magnetic field and for fields oriented normal to the z-axis, the net spin polarization of the hole ensemble is zero, so that the ensemble averaged coefficients are equal: α = β. For σ+ polarized excitation, for which injection of an |⇑↓〉 electron-hole pair is possible, the excited superposi- tion is given by α |⇑〉+β cos(Θ/2) |⇓〉+iβ sin(Θ/2) |⇑⇓↓〉. Here Θ = d · E(t)dt/~ is the dimensionless pulse area with the pump laser electric field E(t) and the dipole transition matrix element d. In general, the hole-trion superposition state may be driven coherently by varying the pulse area, giving rise to Rabi-oscillations as reported recently for (In,Ga)As quantum dots [20]. Such oscilla- tions have not been found yet in QWs, most probably due to the fast carrier dephasing, in particular for strong ex- citation. Dephasing of the superposition occurs shortly after the pulse on a time scale of a few ps, converting the coherent polarization into a population consisting of holes with original spins ⇑ and ⇓ and trions with ⇑⇓↓. In a simplified picture, the spin coherence generation can be described as follows: The σ+ polarized pump cre- ates with certain efficiency trions T+ of spin configura- tion |⇑⇓↓〉. By this process |⇓〉 holes are pumped out of the DHG, leaving behind holes with opposite spin |⇑〉. Right after the pump pulse the KR signal is contributed by the |⇑〉 hole from the DHG and |↓〉 electron of the T+. The further evolution of the coherent signal depends on the strength of external magnetic field applied perpen- dicular to the z-axis. AtB=0, the carrier spins experience no Larmor preces- sion. The electron spin relaxation time usually exceeds the lifetime of trions, which is limited by radiative decay, by one-two orders of magnitude. Trion recombination re- turns the hole to the DHG with the same spin orientation as it was pumped out, if no electron spin scattering oc- curred in the meantime. This compensates the induced spin polarization and nullifies the KR signal at delays exceeding the trion lifetime. Indeed, the KR signal in the top trace in Fig. 1(a) shows a fast decay with a time constant of ∼50 ps, which is characteristic for radiative trion recombination in GaAs/(Al,Ga)As QWs [22]. The long-lived tail of the signal has a very small amplitude and is due to hole coherence provided by weak spin re- laxation of electrons in T+ and/or hole relaxation in the DHG during the trion lifetime. In finite magnetic fields, the carrier spins start to pre- cess about B. Due to the electron spin precession in T+, the hole spin returned to the DHG after trion re- combination will not compensate the spin polarization of the resident holes. Therefore, a long-lived hole coherence with considerable amplitude will be induced. This co- herence is observed in the KR signal as spin beats with low frequency (see Figs. 1 and 3). Note that the Larmor precession of the resident holes may also contribute to generation of hole spin coherence, but the effect is pro- portional to the ratio of the hole and electron Larmor frequencies and therefore will be rather small. Let us compare the spin coherence generation for QWs with DHG and DEG resonantly excited in the T+ and T− states, respectively. We are interested in a long-lived spin coherence which goes beyond the trion lifetime, i.e. in spin coherence induced for the resident carriers. In both cases the amplitude of the KR signal is controlled by the ratio of the electron spin beat period to the trion lifetime. Nevertheless, the two cases are quite different as for DHG the precessing electron is bound in the T+ trion, while for DEG the background electron precesses. In the latter case the electron precession in T− is blocked due to the singlet spin character of the trion ground state. To conclude, a long-lived spin coherence has been found for localized holes in a GaAs/(Al,Ga)As QW with a diluted hole gas. The spin coherence time exceeds 650 ps and is still masked by the spin dephasing due to g factor inhomogeneities. Localization of holes suppresses most spin relaxation mechanisms inherent for free carri- ers. It is also worth to note, that due to the p-type Bloch wave functions the holes do not interact with the nuclear spins, which provides the most efficient spin relaxation mechanism for localized electrons [23]. Acknowledgements. This work was supported by the BMBF program ’nanoquit’. [†] Also at Ioffe Physico-Technical institute, Russian Academy of Sciences, St. Petersburg, Russia. [1] Semiconductor Spintronics and Quantum Computation, ed. by D. D. Awschalom, D. Loss, and N. Samarth, (Springer-Verlag, Heidelberg 2002). [2] I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [3] D. P. DiVincenzo, Science 270, 255 (1995); D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [4] X. Marie et al., Phys. Rev. B 60, 5811 (1999). [5] A. Imamoglu et al., Phys. Rev. Lett. 83, 4204 (1999). [6] T. C. Damen, L. Viña, J. E. Cunningham, J. E. Shah, and L. J. Sham, Phys. Rev. Lett. 67, 3432 (1991). [7] Ph. Roussignol et al., Surf. Sci. 305, 263 (1994). [8] B. Baylac et al., Sol. State Comm. 93, 57 (1995). [9] B. Baylac et al., Surf. Sci. 326, 161 (1995). [10] T. Uenoyama and L. J. Sham, Phys. Rev. Lett. 64, 3070 (1990); Phys. Rev. B 42, 7114 (1990). [11] C. Y. Hu et al., Phys. Rev. B 72, 121203(R) (2005). [12] Optical Orientation, ed. by F. Meier and B. P. Za- kharachenya (North-Holland, Amsterdam 1984), Ch. 2. [13] R. Ferreira and G. Bastard, Phys. Rev. B 43, 9687 (1991). [14] C. Lü, J. L. Cheng, and M. W. Wu, Phys. Rev. B 73, 125314 (2006). [15] Y. G. Semenov, K. N. Borysenko, and K. W. Kim, Phys. Rev. B 66, 113302 (2002). [16] J. J. Baumberg, D. D. Awschalom, N. Samarth, H. Luo, and J. K. Furdyna, Phys. Rev. Lett. 72, 717 (1994). [17] The value | gh,‖ |= 0.60 ± 0.01 was determined from the Zeeman splitting of PL lines at B = 7 T applied along the QW growth axis. [18] R. Winkler, S. J. Papadakis, E. P. De Poortere, and M. Shayegan, Phys. Rev. Lett. 85, 4574 (2000). [19] A. Shabaev, Al. L. Efros, D. Gammon, and I. A. Merkulov, Phys. Rev. B 68, 201305(R) (2003). [20] A. Greilich et al., Phys. Rev. Lett. 96, 227401 (2006). [21] T. A. Kennedy et al., Phys. Rev. B 73, 045307 (2006). [22] G. Finkelstein et al., Phys. Rev. B 58, 12637 (1998). [23] I. A. Merkulov, Al. L. Efros and M. Rosen, Phys. Rev. B 65, 205309 (2002).

0704.0593

Local-field effects in radiatively broadened magneto-dielectric media: negative refraction and absorption reduction

Local-field effects in radiatively broadened magneto-dielectric media: negative refraction and absorption reduction Jürgen Kästel and Michael Fleischhauer Fachbereich Physik, Technische Universität Kaiserslautern, D-67663 Kaiserslautern, Germany Gediminas Juzeliūnas Institute of Theoretical Physics and Astronomy, Vilnius University, A Goštauto 12, Vilnius 01108, Lithuania (Dated: August 11, 2021) We give a microscopic derivation of the Clausius-Mossotti relations for a homogeneous and isotropic magneto-dielectric medium consisting of radiatively broadened atomic oscillators. To this end the diagram series of electromagnetic propagators is calculated exactly for an infinite bi-cubic lattice of dielectric and magnetic dipoles for a lattice constant small compared to the resonance wavelength λ. Modifications of transition frequencies and linewidth of the elementary oscillators are taken into account in a selfconsistent way by a proper incorporation of the singular self-interaction terms. We show that in radiatively broadened media sufficiently close to the free-space resonance the real part of the index of refraction approaches the value -2 in the limit of ρλ3 ≫ 1, where ρ is the number density of scatterers. Since at the same time the imaginary part vanishes as 1/ρ local field effects can have important consequences for realizing low-loss negative index materials. PACS numbers: INTRODUCTION It is well known that in dense dielectric materials the induced polarization P alters the field strength Eloc act- ing on the constituents (i.e. the local field) compared to the average macroscopic field Em. Macroscopic consider- ations show that in systems with high symmetry such as a cubic lattice the two fields are related to each other ac- cording to Eloc = Em +P/(3ε0) [1, 2]. This leads to the well-known Clausius-Mossotti relation for the permittiv- ity ε(ω) ε(ω) = 1 + ρα(ω)/ε0 1− ρα(ω)/(3ε0) where ρ is the density and α(ω) the polarizability of the oscillators. Similar arguments hold for a purely magnetic material [3], except that the required densities are usu- ally much higher due to the smallness of magnetic dipole moments and polarizabilities. In linear response α(ω) is well described by a damped-oscillator model [1] α(ω) = α′ + i α′′ = ω20 − ω2 − iγω . (2) The corresponding (real-valued) parameters such as the oscillator strength α0, the resonance frequency and width, ω0 and γ, are determined by the microscopic model. In general the linewidth γ contains radiative as well as non-radiative contributions. For purely ra- diative interaction these parameters are strongly af- fected by the renormalization of energy levels and spon- taneous emission processes caused by the interaction with the vacuum electromagnetic field in the medium [4, 5, 6, 7, 8, 9, 10, 11]. Since the mode structure of the electromagnetic field inside a dense medium can be substantially modified compared to free space, one would expect that the polarizability entering eq.(1) is different from that in free space. In a macroscopic approach α(ω) is however an input function and no conclusion can be drawn about possible changes due to the different struc- ture of the vacuum modes inside the medium. To take into account the modification of transition frequencies and radiative linewidth in a dense medium in a self- consistent way requires a microscopic approach. In the present paper we develop a microscopic ap- proach to local field effects in dense materials with simul- taneous dielectric and magnetic response using Greens- function techniques similar to those used by deVries and Lagendijk for purely dielectric materials [12]. To this end we consider an infinitely extended bi-cubic lattice of elec- tric and magnetic point dipoles with isotropic response with a lattice constant small compared to the transition wavelength. We however do not make use of the as- sumptions made in [12] to renormalize the singular self- interaction contributions to the lattice T -matrix which eliminated radiative contributions to linewidth and tran- sition frequencies altogether. We show that instead the self-interaction contributions can be summed to yield the dressed t-matrix of an isolated oscillator interacting with the vacuum modes of the electromagnetic field in free space. In this way we derive Clausius-Mossotti relations for general, radiatively broadened, isotropic magneto- dielectrica. Apart from non-radiative broadenings, the electric and magnetic polarizabilities entering these equa- tions are shown to be exactly those of free space. We then show that simultaneous local-field corrections to electric and magnetic fields in purely radiatively broad- ened magneto-dielectrica have a surprising and poten- tially important effect: For sufficiently large densities the http://arxiv.org/abs/0704.0593v2 real part of the refractive index saturates at the level of −2. At the same time, the imaginary part of the com- plex index approaches zero inversely proportional to the density. Thus the medium becomes transparent and left- handed i.e. displays a negative index of refraction with low absorption. LOCAL-FIELD EFFECTS AND RENORMALIZATION OF RADIATIVE SELF-INTERACTION IN DIELECTRIC MEDIA We start by developing a microscopic scattering ap- proach to local-field effects in dielectric media taking into account possible material induced modifications of radiative linewidth and transition frequencies in a self- consistent way. To this end we consider a simple cubic lattice of electric point dipoles with isotropic bare polar- izability αb αb(r) = αb δ(r−R), (3) whereR denote lattice vectors. The dipoles interact with the quantized electromagnetic field Ê which obeys the vector Helmholtz equation ~∇× ~∇× Ê(r, ω)− ω Ê(r, ω) = µ0ω P̂ . (4) In the weak-excitation, i.e. linear response limit, the op- erator of the microscopic electric polarization P̂ has the form P̂(r) = αb(r)Ê(r, ω). Solving eq.(4) we can deter- mine the (isotropic) dispersion relation k = k(ω) from which the permittivity ε(ω) can be extracted. In the linear response limit the solution of the quantum me- chanical interaction problem can most easily be obtained by means of Greensfunction techniques. In particular it is sufficient to calculate the scattering T -matrix of the oscillator lattice. The dispersion relation can then be obtained via [13, 14, 15] det T−1 = 0. (5) The scattering T -matrix obeys a linear Dyson equation T = V + V G(0)V + · · · = V + V G(0)T, (6) where G(0)(r, r′, ω) is the free-space retarded propagator of the electric field which is a solution to the classical vector Helmholtz equation ~∇× ~∇× G(0)(r, r′, ω)− ω G(0)(r, r′, ω) = = 1 δ(r− r′), (7) V (r, ω) = −ω 2αb(r) is a linear, isotropic point vertex. Note that integration over spatial variables was suppressed in eq.(6) for nota- tional simplicity. For a cubic lattice of isotropic scatterers, the series can be summed up to yield [16] T (k,k′) = − ei(k−k R 6=0 RG(0)(R) where G(0)(R) stands for G(0)(r, r+R, ω0) which due to the discrete translation invariance is independent on r. The single-particle scattering t-matrix t(ω) is determined by the bare polarizability [12] t(ω)−1 = + G(0)(0). (10) Note that G(0)(0) is diagonal and isotropic. In eq.(9) we have separated the contribution of the lattice ( R 6=0) from the multiple scattering events at the same oscilla- tor (G(0)(0)). This separation is crucial since G(0)(0) is singular. Rather than eliminating this singularity by a regularization procedure as done in [12], we note that ex- pression (10) gives the single-particle scattering t-matrix t(ω) dressed by the interaction with the vacuum field in free space. This quantity is experimentally observable and is related to the single-particle polarizability α(ω) in free space: α(ω) = t(ω) ε0 (11) αb on the other hand is not observable and thus only a theoretical notion. At this point other broadening mech- anisms can be incorporated by adding appropriate non- radiative decay rates γnon-rad to the polarizability α(ω) (11) (cf. equation (2) and discussion thereafter). Obviously, for the radiative part separating the sum∑ RG(0)(R) into G(0)(0)+ R 6=0 e ik′RG(0)(R) does the trick of writing the full lattice T -matrix in terms of the known free space t-matrix. As a drawback we are left with the sum over the lattice vectors R 6= 0. Unfortunately this sum can not be evaluated exactly and has to be treated approximately. According to Poisson’s summation formula f(n) = dxf(x)e−2πikx (12) the sum over R 6= 0 can be expressed in terms of a real space integral and a sum over inverse lattice vectors K of the Fourier transform of the free space Greensfunction G̃(0)(p) R 6=0 eikRG(0)(R) = Ξ(|r|) (2πa)3 ei(p+k−K)rG̃(0)(p) Here Ξ(|r|) is some smooth function with Ξ(0) = 0 and Ξ(|r| > 0) → 1 introduced to prevent the integral from touching the excluded singular point r = 0. In the following we restrict the discussion to lattices with a lattice constant much smaller than the resonant wavelength, i.e. ka ≪ 1. In this limit the lattice of oscillators behaves essentially as a homogeneous medium. Contributions from large K-vectors to the sum, which reflect the discreteness of the lattice, can be neglected as long as the singular contribution from the origin has been excluded. Therefore we only keep the term K = 0 and assume a Gaussian cutting function Ξ(|r|) = 1− e−r2/δ2 , with δ ≪ a. This yields R 6=0 eikRG(0)(R) ≈ 1 G̃(0)(k) π3/2δ3 (2π)3 dp p2e− (k2+p2)e− k·p̂G̃(0)(p), where p̂ = p/|p|. Apart from the Gaussian p-integral which provides a smooth cut-off in reciprocal space, δ can be treated as a small parameter. That allows to carry out the integration analytically which in leading order of δ yields R 6=0 eikRG(0)(R) ≈ 1 G̃(0)(k)− 1 3ω2/c2 1. (15) The free-space Greentensor G̃(0)(k) is given by [12] G̃(0)(k) = 1− |k|2∆k with ∆k = 1 − k̂ ⊗ k̂ being a projector to directions orthogonal to k. With this we are ready to evaluate eq. (5) which reads in the limit ka ≪ 1 ρα(ω)/ε0 1− |k|2∆k Solving eq. (17) for the (isotropic) dispersion k = k(ω) with k(ω) = ε(ω)ω2/c2 finally yields ε(ω) = 1 + ρα(ω)/ε0 1− ρα(ω)/3ε0 . (18) This is the well-known Clausius-Mossotti relation where for purely radiatively broadened systems α(ω) is the dressed polarizability of an isolated oscillator interacting with the free-space electromagnetic vacuum field. LOCAL-FIELD EFFECTS FOR MAGNETO-DIELECTRICS We now extend the above discussion to the case of a bi-cubic lattice of electric and magnetic dipole oscillators. The microscopic, space-dependent bare electric polariz- ability αbe(r) is then given by αbe(r) = αbe δ(r−R) = αbe eiKr (19) and, similarly, the bare magnetic polarizability by αbm(r) = αbm δ(r−R−∆r) = αbm eiK(r−∆r) HereR denotes again the lattice vectors and ∆r the spac- ing between the electric and magnetic sublattices. The bare atomic polarizabilities αbe and αbm are assumed to be scalar for simplicity corresponding to an isotropic medium. The last expressions in eqn. (19) and (20) give the bare polarizabilities in reciprocal space, with K being the reciprocal lattice vectors. Due to the simultaneous presence of electric and mag- netic dipole lattices we now have to solve the coupled set of vector Helmholtz equations for the operators of the electric and magnetic fields ∇×∇× Ê− ω Ê = iωµ0∇× M̂+ µ0ω2P̂ (21) ∇×∇× Ĥ− ω M̂− iω∇× P̂. (22) In linear response the operator of the polarization P̂ and the magnetization M̂ are proportional to the elec- tric and magnetic fields respectively, P̂(r) = αbe(r)Ê(r) and M̂(r) = µ0αbm(r)Ĥ(r). In the following we will pursue a slightly different ap- proach to solve the coupled set of equations than used in the previous section. Taking into account the lattice symmetry we first write the field variables in the form Ê(r) = Ẽ(k−K)ei(k−K)r, (23) where the dependence on frequency ω was suppressed for notational simplicity. The subscript denotes integration over the first Brillouin zone. Substituting this and the corresponding expression for Ĥ into (21)-(22) gives the Helmholtz equations in reciprocal space. After some ele- mentary manipulations the following closed set of equa- tions is derived: ραbe/ε0 1− |k−K|2∆k−K Ẽ(k−K′) = µ0αbm eiK∆r(k−K)× 1− |k−K|2∆k−K H̃(k−K′)e−iK ρµ0αbm 1− |k−K|2∆k−K H̃(k−K′)e−iK ′∆r = − c ωµ0αbm e−iK∆r(k−K)× 1− |k−K|2∆k−K Ẽ(k−K′) where ρ = 1/a3 is the particle density. The sum in the brackets on the left hand sides of eqs. (24,25) can be rewritten as 1− |k−K|2∆k−K eikRG(0)(R) = G(0)(0) + R 6=0 eikRG(0)(R), where in the second line we have separated the singular contribution G(0)(0). One recognizes that this term can be added to the expressions containing the bare polariz- abilities in eqs.(24) and (25) yielding the dressed scatter- ing t-matrices for isolated electric and magnetic dipoles interacting with the free-space vacuum field: te(ω) + G(0)(0), (26) tm(ω) (ω2µ0 + G(0)(0). (27) The sum over the Greensfunction excludingR = 0 can be evaluated in a similar way as in the previous section. If we again assume a lattice constant a much smaller than the resonant wavelength, reciprocal K vectors different from zero can be disregarded. This leads to ρte(ω) + G̃(0)(k)− 1 3ω2/c2 Ê(k) = (28) µ0αbm 1− k2∆k Ĥ(k), ρtm(ω) + G̃(0)(k) − 1 3ω2/c2 Ĥ(k) = (29) c2αbe ωµ0αbm 1− k2∆k Ê(k). Since we are furthermore only interested in propagating, i.e. transversal modes, we can further simplify the calcu- lation by projecting onto transversal modes using ∆k ραe(ω)/ε0 ∆kÊ(k) = µ0αbm k×∆kĤ(k) (30) ρµ0αm(ω) k×∆kĤ(k) = c2αbe ωµ0αbm ∆kÊ(k). (31) Here we have substituted the dressed single parti- cle t-matrices by the free-space dressed polarizabilities αe(m)(ω) = te(m)(ω)c 2/ω2ε0 In order to find the dispersion k(ω) = n2ω2/c2 we have to determine the solution of the secular equation of the linear set of eqs. (30,31), which results in the condition ραe(ω)/ε0 × (32) ρµ0αm(ω) Solving for the refractive index of the transversal modes then gives n2 = εµ, where ε = 1 + ραe(ω)/ε0 1− ραe(ω)/3ε0 µ = 1 + ρµ0αm(ω) 1− ρµ0αm(ω)/3 are the relative dielectric permittivity and magnetic permeability, respectively, both satisfying the Clausius- Mossotti relations. Note that for longitudinal modes eqs. (28) and (29) de- couple. This can be seen by applying the corresponding projector to longitudinal waves k̂⊗k̂ which leads to a dis- appearance of the cross-coupling terms. The dispersion obtained in this way gives either ε = 0 corresponding to electric excitons [17, 18] or µ = 0 for magnetic excitons. NEGATIVE REFRACTION AND ABSORPTION REDUCTION DUE TO LOCAL FIELD EFFECTS IN MAGNETO-DIELECTRIC MEDIA It is interesting to consider the implications of the Clausius Mossotti relations for radiatively broadened me- dia in the large density limit. Let us first consider a purely dielectric medium and let us assume that the po- larizability αe(ω) = α e(ω) + i α e (ω) does not depend on the density, i.e. the medium is radiatively broadened. In this case one finds ρ→∞−→ −2 + i |αe|2 . (35) In the high-density limit and sufficiently close to reso- nance the response saturates at a value of −2 with an imaginary part that vanishes as 1/ρ. At this point the medium becomes totally opaque since the index of re- fraction attains an imaginary value n = i 2 indicating the emergence of a stopping band. This is illustrated in the left column of Fig 1 for a medium composed of either electric or magnetic dipole oscillators. For small densities (ρ|α0|/3 = 1/3) the resonance is centered at ω0 whereas for larger densities (ρ|α0|/3 = 3) the response shifts to smaller frequencies and is amplified. Eventu- ally (ρ|α0|/3 = 30) the refractive index becomes almost purely imaginary in which case light cannot propagate any longer. This behavior changes dramatically if we consider me- dia with overlapping electric and magnetic resonances described by both an electric polarizability αe(ω) and a magnetic polarizability αm(ω). Independent application of Clausius-Mossotti local-field corrections to the permit- tivity and the permeability leads in the high density limit n = −2 + i |αe|2 9α′′m µ0|αm|2 . (36) Thus in the spectral overlap region the real part of the index of refraction approaches the value −2, i.e. attains a constant negative value. Furthermore the imaginary part, responsible for absorption losses, approaches zero in that spectral region as 1/ρ. This rather peculiar behav- ior is illustrated in the right column of Fig.1. One clearly recognizes the emergence of a spectral region around the bare resonance frequency where the real part of the re- fractive index approaches −2 while the imaginary part is strongly suppressed. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (b)(a) ∆ [γ ] ∆ [γ ] FIG. 1: (color online) spectrum of the real (solid) and imagi- nary (dashed) part of the refractive index as well as the real (dotted) part of the response function(s) ε and/or µ as a func- tion of the detuning ∆ for a (a) pure dielectric or magnetic medium for ρ|α0|/3 at ∆ = 0 equal to = 1/3 (top), 3 (middle) and 30 (bottom) (b) magneto-dielectric medium for ρ|α0|/3 at ∆ = 0 equal to = 1/3 (top), 3 (middle) and 30 (bottom). Negative refraction of light is currently one of the most active research areas in photonics [19, 20, 21] due to fascinating potential applications such as superlens- ing [22] or electromagnetic cloaking [23, 24, 25]. In recent years substantial progress has been made in re- alizing negative refraction in so-called meta-materials [26, 27, 28, 29]. These are artificial periodic structures of electric and magnetic dipoles with a resonance wave- length much larger than the lattice constant which thus form a quasi-homogeneous magneto-dielectric medium. In order to achieve a large electromagnetic response, op- eration close to resonance is needed which is associated with rather substantial losses. The elimination of these losses represents one of the main challenges in the field [30]. We have shown here that in a radiatively broad- ened medium, i.e. a medium in which density-dependent broadening mechanism can still be disregarded for suf- ficiently large densities, local field effects can provide a negative index of refraction and at the same time effi- ciently suppress absorption losses. SUMMARY In the present paper we have given a rigorous micro- scopic derivation of Clausius-Mossotti relations for both the electric and magnetic response in an isotropic, radia- tively broadened magneto-dielectric medium formed by a simple bi-cubic lattice of electric and magnetic dipoles. As opposed to previous microscopic approaches we have taken into account possible modifications of the single- particle polarizabilities by the altered electromagnetic vacuum inside the medium in a self-consistent way. For a simple bi-cubic lattice it has been shown that the po- larizabilities entering the Clausius-Mossotti relations are those of single oscillators interacting with the free-space vacuum field. We showed that as a consequence of the local field corrections a radiatively broadened medium with overlapping electric and magnetic resonances be- comes lossless with a real part of the refractive index approaching the value −2 in the high-density limit. The latter could provide an interesting avenue to construct ar- tificial materials with negative refraction and low losses. This work was supported by the Alexander von Hum- boldt Foundation through the institutional collabora- tion grant between The Institute of Theoretical Physics and Astronomy of Vilnius University and the Techni- cal University of Kaiserslautern. J.K. acknowledges fi- nancial support by the Deutsche Forschungsgemeinschaft through the GRK 792 “Nichtlineare Optik und Ultra- kurzzeitphysik”. [1] J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, New York, 1999) [2] H.A. Lorentz, Wiedem. Ann. 9, 641 (1880); L. Lorenz ibid. 11, 70 (1881); L. Onsager, J. Am. Chem. Soc. 58, 1486 (1936); C.J.F. Böttcher, Theory of Electric Polar- ization, (Elsevier, Amsterdam, 1973) [3] D. M. Cook, The Theory of the Electromagnetic Field, (Prentice-Hall, Englewood Cliffs, N.J., 1975). [4] G. Nienhuis and C. Th. J. Alkemande, Physica C 81, 181 (1976). [5] J. Knoester and S. Mukamel, Phys. Rev. A 40, 7065 (1989). [6] R.J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991). [7] S.M. Barnett, B. Huttner, and R. Loudon, Phys. Rev. Lett. 68, 3698 (1992). [8] G. Juzeliunas, Phys. Rev. A 55 R4015 (1997). [9] S. Scheel, L. Knöll, and D.G. Welsch, Phys. Rev. A 60, 4094 (1999) [10] M. Fleischhauer, Phys. Rev. A 60, 2534 (1999). [11] H.T. Dung HT, S.Y. Buhmann, L. Knöll, D.G. Welsch, S. Scheel, and J. Kästel, Phys. Rev. A 68, 043816 (2003). [12] P. de Vries, D. V. van Coevorden, and A. Lagendijk, Rev. Mod. Phys. 70, 447 (1998). [13] J. Korringa, Physica 13, 392 (1947). [14] W. Kohn, N. Rostoker, Phys. Rev. 94, 1111 (1953). [15] J. M. Ziman, Proc. Phys. Soc. 86, 337 (1965). [16] P. de Vries, and A. Lagendijk, Phys. Rev. Lett 81, 1381 (1998). [17] A. S. Davydov, Theory of Molecular Excitons (Plenum, New York, 1971). [18] V. M. Agranovich, and M. D. Galanin, Electronic Exci- tation Energy Transfer in Condensed Matter, edited by V. M. Agranovich and A. A. Maradudin (North-Holland, Amsterdam, 1982). [19] V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968). [20] V. M. Agranovich, and Y. N. Gartstein, Physics Uspekhi 49, 1029 (2006) [21] V. M. Shalaev, Nature Photonics 1, 41 (2007) [22] J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000) [23] U. Leonhardt, Science 312, 1777 (2006). [24] J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006). [25] D. Schurig et al. , Science 314, 977 (2006). [26] J. B. Pendry et al. , IEEE Trans. Micro. Theory Tech. 47, 2075 (1999). [27] D. R. Smith et al. , Phys. Rev. Lett. 84, 4184 (2000); R. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001). [28] T. J. Yen et al. , Science 303, 1494 (2004). [29] S. Linden et al. , Science 306, 1351 (2004); C. Enkrich et al. , Phys. Rev. Lett. 95, 203901 (2005). [30] J. Kästel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, Phys. Rev. Lett. 99, 073602 (2007).

0704.0594

Search for a fourth generation b'-quark at LEP-II at sqrt{s}=196-209 GeV

arXiv:0704.0594v1 [hep-ex] 4 Apr 2007 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN–PH-EP/2006-023 20 June 2006 Search for a fourth generation b′-quark at LEP-II at√ s = 196 − 209 GeV DELPHI Collaboration Abstract A search for the pair production of fourth generation b′-quarks was performed using data taken by the DELPHI detector at LEP-II. The analysed data were collected at centre-of-mass energies ranging from 196 to 209 GeV, corresponding to an integrated luminosity of 420 pb−1. No evidence for a signal was found. Upper limits on BR(b′ → bZ) and BR(b′ → cW) were obtained for b′ masses ranging from 96 to 103 GeV/c2. These limits, together with the theoretical branching ratios predicted by a sequential four generations model, were used to constrain the value of RCKM = | Vcb′V tb′Vtb |, where Vcb′ , Vtb′ and Vtb are elements of the extended CKM matrix. (Accepted by Eur. Phys. J. C) http://arxiv.org/abs/0704.0594v1 J.Abdallah26, P.Abreu23, W.Adam55, P.Adzic12, T.Albrecht18, R.Alemany-Fernandez9, T.Allmendinger18, P.P.Allport24, U.Amaldi30, N.Amapane48, S.Amato52, E.Anashkin37, A.Andreazza29, S.Andringa23, N.Anjos23, P.Antilogus26, W-D.Apel18, Y.Arnoud15, S.Ask27, B.Asman47, J.E.Augustin26, A.Augustinus9, P.Baillon9, A.Ballestrero49, P.Bambade21, R.Barbier28, D.Bardin17, G.J.Barker57, A.Baroncelli40, M.Battaglia9, M.Baubillier26, K-H.Becks58, M.Begalli7, A.Behrmann58, E.Ben-Haim21, N.Benekos33, A.Benvenuti5, C.Berat15, M.Berggren26, L.Berntzon47, D.Bertrand2, M.Besancon41, N.Besson41, D.Bloch10, M.Blom32, M.Bluj56, M.Bonesini30, M.Boonekamp41, P.S.L.Booth†24, G.Borisov22, O.Botner53, B.Bouquet21, T.J.V.Bowcock24, I.Boyko17, M.Bracko44, R.Brenner53, E.Brodet36, P.Bruckman19, J.M.Brunet8, B.Buschbeck55, P.Buschmann58, M.Calvi30, T.Camporesi9, V.Canale39, F.Carena9, N.Castro23, F.Cavallo5, M.Chapkin43, Ph.Charpentier9, P.Checchia37, R.Chierici9, P.Chliapnikov43, J.Chudoba9, S.U.Chung9, K.Cieslik19, P.Collins9, R.Contri14, G.Cosme21, F.Cossutti50, M.J.Costa54, D.Crennell38, J.Cuevas35, J.D’Hondt2, J.Dalmau47, T.da Silva52, W.Da Silva26, G.Della Ricca50, A.De Angelis51, W.De Boer18, C.De Clercq2, B.De Lotto51 , N.De Maria48, A.De Min37, L.de Paula52, L.Di Ciaccio39, A.Di Simone40, K.Doroba56, J.Drees58,9, G.Eigen4, T.Ekelof53, M.Ellert53, M.Elsing9, M.C.Espirito Santo23, G.Fanourakis12, D.Fassouliotis12,3, M.Feindt18, J.Fernandez42 , A.Ferrer54, F.Ferro14, U.Flagmeyer58, H.Foeth9, E.Fokitis33, F.Fulda-Quenzer21, J.Fuster54, M.Gandelman52, C.Garcia54, Ph.Gavillet9, E.Gazis33, R.Gokieli9,56, B.Golob44,46, G.Gomez-Ceballos42, P.Goncalves23, E.Graziani40, G.Grosdidier21, K.Grzelak56, J.Guy38, C.Haag18, A.Hallgren53, K.Hamacher58, K.Hamilton36, S.Haug34, F.Hauler18, V.Hedberg27, M.Hennecke18, H.Herr†9, J.Hoffman56, S-O.Holmgren47, P.J.Holt9, M.A.Houlden24, J.N.Jackson24, G.Jarlskog27, P.Jarry41, D.Jeans36, E.K.Johansson47, P.D.Johansson47, P.Jonsson28, C.Joram9, L.Jungermann18, F.Kapusta26, S.Katsanevas28 , E.Katsoufis33, G.Kernel44, B.P.Kersevan44,46, U.Kerzel18, B.T.King24, N.J.Kjaer9, P.Kluit32, P.Kokkinias12, C.Kourkoumelis3, O.Kouznetsov17, Z.Krumstein17, M.Kucharczyk19, J.Lamsa1, G.Leder55, F.Ledroit15, L.Leinonen47, R.Leitner31, J.Lemonne2, V.Lepeltier21, T.Lesiak19, W.Liebig58, D.Liko55, A.Lipniacka47, J.H.Lopes52, J.M.Lopez35, D.Loukas12, P.Lutz41, L.Lyons36, J.MacNaughton55 , A.Malek58, S.Maltezos33, F.Mandl55, J.Marco42, R.Marco42, B.Marechal52, M.Margoni37, J-C.Marin9, C.Mariotti9, A.Markou12, C.Martinez-Rivero42, J.Masik13, N.Mastroyiannopoulos12, F.Matorras42, C.Matteuzzi30, F.Mazzucato37 , M.Mazzucato37, R.Mc Nulty24, C.Meroni29, E.Migliore48, W.Mitaroff55, U.Mjoernmark27, T.Moa47, M.Moch18, K.Moenig9,11, R.Monge14, J.Montenegro32 , D.Moraes52, S.Moreno23, P.Morettini14, U.Mueller58, K.Muenich58, M.Mulders32, L.Mundim7, W.Murray38, B.Muryn20, G.Myatt36, T.Myklebust34, M.Nassiakou12, F.Navarria5, K.Nawrocki56, R.Nicolaidou41, M.Nikolenko17,10, A.Oblakowska-Mucha20, V.Obraztsov43, O.Oliveira23, S.M.Oliveira23, A.Olshevski17, A.Onofre23, R.Orava16, K.Osterberg16, A.Ouraou41, A.Oyanguren54, M.Paganoni30, S.Paiano5, J.P.Palacios24, H.Palka19, Th.D.Papadopoulou33, L.Pape9, C.Parkes25, F.Parodi14, U.Parzefall9, A.Passeri40, O.Passon58, L.Peralta23, V.Perepelitsa54, A.Perrotta5, A.Petrolini14, J.Piedra42, L.Pieri40, F.Pierre41, M.Pimenta23, E.Piotto9, T.Podobnik44,46 , V.Poireau9, M.E.Pol6, G.Polok19, V.Pozdniakov17 , N.Pukhaeva17, A.Pullia30, J.Rames13, A.Read34, P.Rebecchi9, J.Rehn18, D.Reid32, R.Reinhardt58, P.Renton36, F.Richard21, J.Ridky13, M.Rivero42, D.Rodriguez42, A.Romero48, P.Ronchese37, P.Roudeau21, T.Rovelli5, V.Ruhlmann-Kleider41, D.Ryabtchikov43 , A.Sadovsky17, L.Salmi16, J.Salt54, C.Sander18, R.Santos23, A.Savoy-Navarro26, U.Schwickerath9, R.Sekulin38, M.Siebel58, A.Sisakian17, G.Smadja28, O.Smirnova27, A.Sokolov43, A.Sopczak22, R.Sosnowski56, T.Spassov9, M.Stanitzki18, A.Stocchi21, J.Strauss55, B.Stugu4, M.Szczekowski56, M.Szeptycka56 , T.Szumlak20, T.Tabarelli30, A.C.Taffard24, F.Tegenfeldt53, J.Timmermans32, L.Tkatchev17 , M.Tobin24, S.Todorovova13, B.Tome23, A.Tonazzo30, P.Tortosa54, P.Travnicek13, D.Treille9, G.Tristram8, M.Trochimczuk56, C.Troncon29, M-L.Turluer41, I.A.Tyapkin17, P.Tyapkin17, S.Tzamarias12, V.Uvarov43, G.Valenti5, P.Van Dam32, J.Van Eldik9, N.van Remortel16, I.Van Vulpen9, G.Vegni29, F.Veloso23, W.Venus38, P.Verdier28, V.Verzi39, D.Vilanova41, L.Vitale50, V.Vrba13, H.Wahlen58, A.J.Washbrook24, C.Weiser18, D.Wicke9, J.Wickens2, G.Wilkinson36, M.Winter10, M.Witek19, O.Yushchenko43, A.Zalewska19, P.Zalewski56, D.Zavrtanik45, V.Zhuravlov17, N.I.Zimin17, A.Zintchenko17 , M.Zupan12 1Department of Physics and Astronomy, Iowa State University, Ames IA 50011-3160, USA 2IIHE, ULB-VUB, Pleinlaan 2, B-1050 Brussels, Belgium 3Physics Laboratory, University of Athens, Solonos Str. 104, GR-10680 Athens, Greece 4Department of Physics, University of Bergen, Allégaten 55, NO-5007 Bergen, Norway 5Dipartimento di Fisica, Università di Bologna and INFN, Via Irnerio 46, IT-40126 Bologna, Italy 6Centro Brasileiro de Pesquisas F́ısicas, rua Xavier Sigaud 150, BR-22290 Rio de Janeiro, Brazil 7Inst. de F́ısica, Univ. Estadual do Rio de Janeiro, rua São Francisco Xavier 524, Rio de Janeiro, Brazil 8Collège de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, FR-75231 Paris Cedex 05, France 9CERN, CH-1211 Geneva 23, Switzerland 10Institut de Recherches Subatomiques, IN2P3 - CNRS/ULP - BP20, FR-67037 Strasbourg Cedex, France 11Now at DESY-Zeuthen, Platanenallee 6, D-15735 Zeuthen, Germany 12Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greece 13FZU, Inst. of Phys. of the C.A.S. High Energy Physics Division, Na Slovance 2, CZ-180 40, Praha 8, Czech Republic 14Dipartimento di Fisica, Università di Genova and INFN, Via Dodecaneso 33, IT-16146 Genova, Italy 15Institut des Sciences Nucléaires, IN2P3-CNRS, Université de Grenoble 1, FR-38026 Grenoble Cedex, France 16Helsinki Institute of Physics and Department of Physical Sciences, P.O. Box 64, FIN-00014 University of Helsinki, Finland 17Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, RU-101 000 Moscow, Russian Federation 18Institut für Experimentelle Kernphysik, Universität Karlsruhe, Postfach 6980, DE-76128 Karlsruhe, Germany 19Institute of Nuclear Physics PAN,Ul. Radzikowskiego 152, PL-31142 Krakow, Poland 20Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, PL-30055 Krakow, Poland 21Université de Paris-Sud, Lab. de l’Accélérateur Linéaire, IN2P3-CNRS, Bât. 200, FR-91405 Orsay Cedex, France 22School of Physics and Chemistry, University of Lancaster, Lancaster LA1 4YB, UK 23LIP, FCUL, IST, CFCUC - Av. Elias Garcia, 14-1o, PT-1000 Lisboa Codex, Portugal 24Department of Physics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK 25Dept. of Physics and Astronomy, Kelvin Building, University of Glasgow, Glasgow G12 8QQ 26LPNHE, IN2P3-CNRS, Univ. Paris VI et VII, Tour 33 (RdC), 4 place Jussieu, FR-75252 Paris Cedex 05, France 27Department of Physics, University of Lund, Sölvegatan 14, SE-223 63 Lund, Sweden 28Université Claude Bernard de Lyon, IPNL, IN2P3-CNRS, FR-69622 Villeurbanne Cedex, France 29Dipartimento di Fisica, Università di Milano and INFN-MILANO, Via Celoria 16, IT-20133 Milan, Italy 30Dipartimento di Fisica, Univ. di Milano-Bicocca and INFN-MILANO, Piazza della Scienza 3, IT-20126 Milan, Italy 31IPNP of MFF, Charles Univ., Areal MFF, V Holesovickach 2, CZ-180 00, Praha 8, Czech Republic 32NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlands 33National Technical University, Physics Department, Zografou Campus, GR-15773 Athens, Greece 34Physics Department, University of Oslo, Blindern, NO-0316 Oslo, Norway 35Dpto. Fisica, Univ. Oviedo, Avda. Calvo Sotelo s/n, ES-33007 Oviedo, Spain 36Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 37Dipartimento di Fisica, Università di Padova and INFN, Via Marzolo 8, IT-35131 Padua, Italy 38Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UK 39Dipartimento di Fisica, Università di Roma II and INFN, Tor Vergata, IT-00173 Rome, Italy 40Dipartimento di Fisica, Università di Roma III and INFN, Via della Vasca Navale 84, IT-00146 Rome, Italy 41DAPNIA/Service de Physique des Particules, CEA-Saclay, FR-91191 Gif-sur-Yvette Cedex, France 42Instituto de Fisica de Cantabria (CSIC-UC), Avda. los Castros s/n, ES-39006 Santander, Spain 43Inst. for High Energy Physics, Serpukov P.O. Box 35, Protvino, (Moscow Region), Russian Federation 44J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia 45Laboratory for Astroparticle Physics, University of Nova Gorica, Kostanjeviska 16a, SI-5000 Nova Gorica, Slovenia 46Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 47Fysikum, Stockholm University, Box 6730, SE-113 85 Stockholm, Sweden 48Dipartimento di Fisica Sperimentale, Università di Torino and INFN, Via P. Giuria 1, IT-10125 Turin, Italy 49INFN,Sezione di Torino and Dipartimento di Fisica Teorica, Università di Torino, Via Giuria 1, IT-10125 Turin, Italy 50Dipartimento di Fisica, Università di Trieste and INFN, Via A. Valerio 2, IT-34127 Trieste, Italy 51Istituto di Fisica, Università di Udine and INFN, IT-33100 Udine, Italy 52Univ. Federal do Rio de Janeiro, C.P. 68528 Cidade Univ., Ilha do Fundão BR-21945-970 Rio de Janeiro, Brazil 53Department of Radiation Sciences, University of Uppsala, P.O. Box 535, SE-751 21 Uppsala, Sweden 54IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, Avda. Dr. Moliner 50, ES-46100 Burjassot (Valencia), Spain 55Institut für Hochenergiephysik, Österr. Akad. d. Wissensch., Nikolsdorfergasse 18, AT-1050 Vienna, Austria 56Inst. Nuclear Studies and University of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Poland 57Now at University of Warwick, Coventry CV4 7AL, UK 58Fachbereich Physik, University of Wuppertal, Postfach 100 127, DE-42097 Wuppertal, Germany † deceased 1 Introduction The Standard Model (SM), although in agreement with the available experimental data [1], leaves several open questions. In particular, the number of fermion generations and their mass spectrum are not predicted. The measurement of the Z decay widths [1] established that the number of light neutrino species (m < mZ/2, where mZ is the Z boson mass) is equal to three. However, if a heavy neutrino or a neutrinoless extra generation exists, this bound does not exclude the possibility of extra generations of heavy quarks. Moreover the fit to the electroweak data [2] does not deteriorate with the inclusion of one extra heavy generation, if the new up and down-type quarks mass difference is not too large. It should be noticed however that in this fit no mixing of the extra families with the SM ones is assumed. The subject of this paper is the search for the pair production of a fourth generation b′-quark at LEP-II: b′ production and decay are discussed in section 2; in section 3, the data sets and the Monte Carlo (MC) simulation are described; the analysis is discussed in section 4; the results and their interpretation within a sequential model are presented in sections 5 and 6, respectively. 2 b′-quark production and decay Extra generations of fermions are predicted in several SM extensions [3,4]. In sequen- tial models [5–7], a fourth generation of fermions carrying the same quantum numbers as the SM families is considered. In the quark sector, an up-type quark, t′, and a down-type quark, b′, are included. The corresponding 4× 4 extended Cabibbo-Kobayashi-Maskawa (CKM) matrix is unitary, approximately symmetric and almost diagonal. As CP-violation is not considered in the model, all the CKM elements are assumed to be real. The b′-quark may decay via charged currents (CC) to UW, with U = t′, t, c, u, or via flavour-changing neutral currents (FCNC) to DX , where D = b, s, d and X = Z,H, γ, g (Fig. 1). As in the SM, FCNC are absent at tree level, but can appear at one-loop level, due to CKM mixing. If the b′ is lighter than t′ and t, the decays b′ → t′W and b′ → tW are kinematically forbidden and the one-loop FCNC decays can be as important as the CC decays [6]. The analysis of the electroweak data [1] shows that the mass difference |mt′ −mb′ | < 60 GeV/c2 is consistent with the measurement of the ρ parameter [3,5]. In particular, when mZ + mb < mb′ < mH + mb, either b ′ → cW or b′ → bZ decay tend to be domi- nant [5–7]. In this case, the partial widths of the CC and FCNC b′ decays depend mainly on mt′ , mb′ and RCKM = | Vcb′V tb′Vtb |, where Vcb′ , Vtb′ and Vtb are elements of the extended 4 × 4 CKM matrix [7]. Limits on the mass of the b′-quark have been set previously at various accelerators. At LEP-I, all the experiments searched for b′ pair production (e+e− → b′b̄′), yielding a lower limit on the b′ mass of about mZ/2 [8]. At the Tevatron, both the D0 [9] and CDF [10] experiments reported limits on σ(pp̄ → b′b̄′) × BR(b′ → bX)2, where BR is the branching ratio corresponding to the considered FCNC b′ decay mode and X = γ,Z. Assuming BR(b′ → bZ) = 1, CDF excluded the region 100 < mb′ < 199 GeV/c2. Although no dedicated analysis was performed for the b′ → cW decay, the D0 limits on σ(pp̄ → tt̄) × BR(t → cW)2 from Fig. 44 and Table XXXI of reference [11] can give a hint on the possible values for BR(b′ → cW) [12]. In the present analysis the on-shell FCNC (b′ → bZ) and CC (b′ → cW) decay modes were studied and consequently the mass range 96 GeV/c2 < mb′ < 103 GeV/c 2 was considered. This mass range is complementary to the one covered by CDF [10]. The mass range mW + mc < mb′ < mZ + mb was not considered because in this region the evaluation of the branching ratios for the different b′ decays is particularly difficult from the theoretical point of view [7]. In the present analysis no assumptions on the BR(b′ → bZ) and BR(b′ → cW) in order to derive mass limits were made. Different final states, corresponding to the different b′ decay modes and subsequent decays of the Z and W bosons, were analysed. 3 Data samples and Monte Carlo simulation The analysed data were collected with the DELPHI detector [13] during the years 1999 and 2000 in LEP-II runs at s = 196 − 209 GeV and correspond to an integrated luminosity of about 420 pb−1. The luminosity collected at each centre-of-mass energy is shown in Table 1. During the year 2000, an unrecoverable failure affected one sector of the central tracking detector (TPC), corresponding to 1/12 of its acceptance. The data collected during the year 2000 with the TPC fully operational were split into two energy bins, below and above s = 206 GeV, with 〈 s〉 = 204.8 GeV and 〈 s〉 = 206.6 GeV, respectively. The data collected with one sector of the TPC turned off were analysed separately and have 〈 s〉 = 206.3 GeV. s (GeV) 196 200 202 205 207 206∗ luminosity (pb−1) 76.0 82.7 40.2 80.0 81.9 59.2 Table 1: The luminosity collected with the DELPHI detector at each centre-of-mass energy is shown. The energy bin labelled 206∗ corresponds to the data collected with one sector of the TPC turned off. Signal samples were generated using a modified version of PYTHIA 6.200 [14]. Al- though PYTHIA does not provide FCNC decay channels for quarks, it was possible to activate them by modifying the decay products of an available channel. The angular distributions assumed for b′ pair production and decay were those predicted by the SM for any heavy down-type quark. Different samples, corresponding to b′ masses in the range between 96 and 103 GeV/c2 and with a spacing of 1 GeV/c2 were generated at each centre-of-mass energy. Specific Monte Carlo simulations (for both SM and signal processes) were produced for the period when one sector of the TPC was turned off. The most relevant background processes for the present analyses are those leading to WW or ZZ bosons in the final state, i.e. four-fermion backgrounds. Radiation in these events can mimic the six-fermion final states for the signal. Additionally qq̄(γ) and Bhabha events can not be neglected since for signal final states with missing energy these backgrounds can become important. SM background processes were simulated at each centre-of-mass energy using several Monte Carlo generators. All the four-fermion final states (both neutral and charged currents) were generated with WPHACT [15], while the particular phase space regions of e+e− → e+e−f f̄ referred to as γγ interactions were generated using PYTHIA [14]. The qq(γ) final state was generated with KK2F [16]. Bhabha events were generated with BHWIDE [17]. The generated signal and background events were passed through the detailed simu- lation of the DELPHI detector [13] and then processed with the same reconstruction and analysis programs as the data. 4 Description of the analyses Pair production of b′-quarks was searched for in both the FCNC (b′ → bZ) and CC (b′ → cW) decay modes. The b′ decay modes and the subsequent decays of the gauge bosons (Z or W) lead to several different final states (Fig. 2). The final states considered and their branching ratios are shown in Table 2. The choice of the considered final states was done taking into account their signatures and BR. About 81% and 90% of the branching ratio to the FCNC and CC channels were covered, respectively. All final states include two jets originating from the low energy b (c) quarks present in the FCNC (CC) b′ decay modes. A common preselection was adopted, followed by a specific analysis for each of the final states (Table 2). b′ decay boson decays BR (%) final states b′ → bZ (FCNC) ZZ → l+l−νν̄ 4.0 bb̄l+l−νν̄ ZZ → qq̄νν̄ 28.0 bb̄qq̄νν̄ ZZ → qq̄qq̄ 48.6 bb̄qq̄qq̄ b′ → cW (CC) WW → qq̄l+ν 43.7 cc̄qq̄l+ν WW → qq̄qq̄ 45.8 cc̄qq̄qq̄ Table 2: The final states considered in this analysis are shown. About 81% and 90% of the branching ratio to the FCNC and CC channels were covered, respectively. Events were preselected by requiring at least eight good charged-particle tracks and the visible energy measured at polar angles1 above 20◦, to be greater than 0.2 s. Good charged-particle tracks were defined as those with a momentum above 0.2 GeV/c and impact parameters in the transverse plane and along the beam direction below 4 cm and below 4 cm/ sin θ, respectively. The identification of muons relied on the association of charged particles to signals in the muon chambers and in the hadronic calorimeters and was provided by standard DELPHI algorithms [13]. The identification of electrons and photons was performed by combining information from the electromagnetic calorimeters and the tracking sys- tem. Radiation and interaction effects were taken into account by an angular clustering procedure around the main shower [18]. The search for isolated particles (charged leptons and photons) was done by construct- ing double cones oriented in the direction of charged-particle tracks or neutral energy deposits. The latter ones were defined as calorimetric energy deposits above 0.5 GeV, not matched to charged-particle tracks and identified as photon candidates by the stan- dard DELPHI algorithms [13,18]. For charged leptons (photons), the energy in the region between the two cones, which had half-opening angles of 5◦ and 25◦ (5◦ and 15◦), was required to be below 3 GeV (1 GeV), to ensure isolation. All the charged-particle tracks 1In the standard DELPHI coordinate system, the positive z axis is along the electron beam direction. The polar angle (θ) is defined with respect to the z axis. In this paper, polar angle ranges are always assumed to be symmetric with respect to the θ = 90◦ plane. final state assignment criteria bb̄l+l−νν̄ at least 1 isolated lepton bb̄qq̄νν̄ no isolated leptons Emissing > 50 GeV bb̄qq̄qq̄ no isolated leptons Emissing < 50 GeV cc̄qq̄l+ν only 1 isolated lepton cc̄qq̄qq̄ no isolated leptons Emissing < 50 GeV Table 3: Summary of the final state assignment criteria. and neutral energy deposits inside the inner cone were associated to the isolated particle. Its energy was then re-evaluated as the sum of the energies inside the inner cone and was required to be above 5 GeV. For well identified leptons or photons [13,18] the above requirements were weakened. In this case only the external cone was used (to ensure isolation) and its angle α was varied according to the energy of the lepton (photon) can- didate, down to 2◦ for Pℓ ≥ 70 GeV/c (3◦ for Pγ ≥ 90 GeV/c), with the allowed energy inside the cone reduced by sinα/ sin 25◦ (sinα/ sin 15◦). Isolated leptons were required to have a momentum greater than 10 GeV/c and a polar angle above 25◦. Events with isolated photons were rejected. All the events were clustered into two, four or six jets using the Durham jet algo- rithm [19], according to the number of jets expected in the signal in each of the final states, unless explicitly stated otherwise. Although two b jets are always present in the FCNC final states, they have a relatively low energy and b-tagging techniques [20] were not used. Events were assigned to the different final states according to the number of isolated leptons and to the missing energy in the event, as detailed in Table 3. Within the same b′ decay channel, the different selections were designed to be mutually exclusive. For the final states involving charged leptons (bb̄l+l−νν̄ and cc̄qq̄l+ν), events were divided into different samples according to the lepton flavour identification: e sample (well identified electrons), µ sample (well identified muons) and no-id sample (leptons with unidentified flavour or two leptons identified with different flavours). Specific analyses were then performed for each of the final states. The selection criteria for the bb̄qq̄qq̄ and cc̄qq̄qq̄ final states were the same. The bb̄l+l−νν̄ final state has a very clean signature (two leptons with ml+l− ∼ mZ, two low energy jets and missing mass close to mZ) and consequently a sequential cut analysis was adopted. For all the other final states, a sequential selection step was followed by a discriminant analysis. In this case, a signal likelihood (LS) and a background likelihood (LB) were assigned to each event, based on Probability Density Functions (PDF), built from the distributions of relevant physical variables. The discriminant variable was defined as ln(LS/LB). 4.1 The bb̄l+l−νν̄ final state The FCNC bb̄l+l−νν̄ final state events were preselected as described above, by re- quiring at least eight good charged-particle tracks, the visible energy measured at polar angles above 20◦, to be greater than 0.2 s and at least one isolated lepton. Distribu- tions of the relevant variables are shown in Fig. 3 for all the events assigned to this final state after the preselection. The event selection was performed in two levels. In the first one, events were required to have at least two leptons and an effective centre-of-mass energy [21], s′, below 0.95 s. The particles other than the two leptons in the events were clustered into two jets and the Durham resolution variable in the transition from two jets to one jet2 was required to be greater than 0.002. The number of data events and the SM expectation after the first selection level is shown in Table 4. The background composition and the signal efficiencies at this level of selection for mb′ = 100 GeV/c 2 and√ s = 205 GeV are given in Table 8. The efficiencies for the other relevant b′ masses and√ s values were found to be the same within errors. Data, SM expectation and signal distributions at this selection level are shown in Fig. 4. s (GeV) data (SM expectation ± statistical error) e sample µ sample no-id sample 196 2 (2.6±0.3) 1 (2.9±0.3) 47 (35.9±1.4) 200 3 (2.5±0.4) 4 (3.4±0.4) 30 (37.4±1.4) 202 2 (1.3±0.2) 1 (1.7±0.2) 20 (18.7±0.7) 205 5 (2.5±0.4) 3 (3.0±0.4) 35 (36.2±1.4) 207 3 (2.3±0.4) 3 (3.1±0.4) 45 (35.1±1.3) 206∗ 1 (1.9±0.3) 2 (2.6±0.2) 31 (27.6±1.0) total 16 (13.2±0.8) 14 (16.7±0.8) 208 (191.0±3.0) Table 4: First selection level of the bb̄l+l−νν̄ final state: the number of events selected in data and the SM expectations after the first selection level for each sample and cen- tre-of-mass energy are shown. In the final selection level the momentum of the more energetic (less energetic) jet was required to be below 30 GeV/c (12.5 GeV/c). Events in the e and no-id samples had to have a missing energy greater than 0.4 s. In the µ sample events were required to have an angle between the two muons greater than 125◦. In the no-id sample, the angle between the two charged leptons had to be greater than 140◦ and pmis/Emis < 0.4, where pmis and Emis are the missing momentum and energy, respectively. After the final selection, one data event was selected for an expected background of 1.5±0.7. This event belonged to the no-id sample and was collected at s = 200 GeV. The signal efficiencies for mb′ = 100 GeV/c 2 and s = 205 GeV are 30.6 ± 2.5% (e sample), 48.6 ± 2.7% (µ sample) and 7.2 ± 0.8% (no-id sample) and their variation with mb′ and s was found to be negligible in the relevant range. 4.2 The bb̄qq̄νν̄ final state The FCNC bb̄qq̄νν̄ final state is characterised by the presence of four jets and a missing mass close to mZ. At least 20 good charged-particle tracks and s′ > 0.5 were required. Events were clustered into four jets. Monojet-like events were rejected by requiring − log10(y2→1) < 0.7 (y2→1 is the Durham resolution variable in the two to one jet transition). Furthermore, − log10(y4→3) was required to be below 2.8 and the energy of the leading charged particle of the most energetic jet was required to be below 0.1 2The Durham resolution variable is the minimum value of the scaled transverse momentum obtained in the transition from n to n− 1 jets [19] and will be represented by yn→n−1. A kinematic fit imposing energy-momentum conservation and no missing energy was applied and the background-like events with χ2/n.d.f. < 6 were rejected. The data, SM expectation and signal distributions of this variable are shown in Fig. 5. Table 5 summarizes the number of selected data events and the SM expectation. The background composition and the signal efficiency at this level of selection for mb′ = 100 GeV/c 2 and√ s = 205 GeV are given in Table 8. The efficiencies for the other relevant b′ masses and√ s values were found to be the same within errors. s (GeV) data (SM expectation ± statistical error) 196 123 (106.3±4.0) 200 111 (104.8±4.0) 202 50 (49.8±1.9) 205 88 (94.2±3.7) 207 99 (91.2±3.6) 206∗ 62 (65.7±2.6) total 533 (511.7±8.3) Table 5: First selection level of the bb̄qq̄νν̄ final state: the number of events selected in data and the SM expectation for each centre-of-mass energy are shown. A discriminant selection was then performed using the following variables to build the PDFs: • the missing mass; • Aj1j2cop × min(sin θj1 , sin θj2), where Aj1j2cop is the acoplanarity3 and θj1,j2 are the polar angles of the jets when forcing the events into two jets4; • the acollinearity between the two most energetic jets5 with the event particles clus- tered into four jets; • the sum of the first and third Fox-Wolfram moments (h1 + h3) [22]; • the polar angle of the missing momentum. The data, SM expectation and signal distributions of these variables are shown in Fig. 6. 4.3 The bb̄qq̄qq̄ final state The FCNC bb̄qq̄qq̄ final state is characterised by the presence of six jets and a small missing energy. All the events were clustered into six jets and only those with at least 30 good charged-particle tracks were accepted. Moreover, events were required to have√ s′ > 0.6 s, − log10(y2→1) < 0.7 and − log10(y6→5) < 3.6. The number of selected data events and the expected background at this level are shown in Table 6. The background composition and the signal efficiency at this level of selection for mb′ = 100 GeV/c 2 and√ s = 205 GeV are given in Table 8. The efficiencies for the other relevant b′ masses and√ s values were found to be the same within errors. A discriminant selection was performed using the following variables to build the PDFs: 3The acoplanarity between two particles is defined as |180◦ − |φ1 −φ2||, where φ1,2 are the azimuthal angles of the two particles (in degrees). 4While the signal is characterised by the presence of four jets in the final state, the two jets configuration is used mainly for background rejection. 5The acollinearity between two particles is defined as 180◦ − α1,2, where α1,2 is the angle (in degrees) between those two particles. s (GeV) data (SM expectation ± statistical error) 196 349 (326.7±5.3) 200 347 (342.1±5.5) 202 165 (162.1±2.6) 205 322 (319.0±5.2) 207 287 (307.6±5.0) 206∗ 192 (215.8±3.6) total 1662 (1673.9±11.4) Table 6: First selection level of the bb̄qq̄qq̄ and cc̄qq̄qq̄ final states: the number of events selected in data and the SM expectations for each centre-of-mass energy are shown. • the Durham resolution variable, − log10(y4→3); • the Durham resolution variable, − log10(y5→4); • the acollinearity between the two most energetic jets, with the event forced into four jets; • the sum of the first and third Fox-Wolfram moments; • the momentum of the most energetic jet; • the angle between the two most energetic jets (with the events clustered into six jets). The distributions of these variables are shown in Fig. 7 for data, SM expectation and signal. 4.4 The cc̄qq̄l+ν final state The signature of this CC final state is the presence of four jets (two of them having low energy), one isolated lepton and missing energy (originating from the W → lν̄ decay). The events were accepted if they had at least 15 good charged-particle tracks. The event particles other than the identified lepton were clustered into four jets. Part of the qq̄ and γγ background was rejected by requiring − log10(y2→1) < 0.7. Furthermore, there should be only one charged-particle track associated to the isolated lepton, and the leading charged particle of the most energetic jet was required to have a momentum below 0.1 The number of selected data events and SM expectations at this level are summarized in Table 7. The background composition and the signal efficiencies at this level of selection for mb′ = 100 GeV/c 2 and s = 205 GeV are given in Table 8. The efficiencies for the other relevant b′ masses and s values were found to be the same within errors. The PDFs used to calculate the background and signal likelihoods were based on the following variables: • the sum of the first and third Fox-Wolfram moments; • the invariant mass of the two jets, with the event particles other than the identified lepton clustered into two jets; • the Durham resolution variable, − log10(y4→3); |~pi|/ s, where ~pi are the momenta of the charged particles (excluding the lepton) in the same hemisphere as the lepton (the hemisphere is defined with respect to the lepton); • the acollinearity between the two most energetic jets; s (GeV) data (SM expectation ± statistical error) e µ no-id 196 65 (51.1±1.4) 53 (56.1±1.5) 38 (34.4±1.4) 200 54 (58.1±1.7) 63 (59.9±1.6) 40 (35.0±1.4) 202 30 (27.8±0.8) 21 (28.4±0.8) 13 (16.9±0.7) 205 56 (50.8±1.5) 66 (53.6±1.5) 32 (33.3±1.4) 207 53 (53.8±1.6) 48 (57.2±1.6) 35 (33.8±1.4) 206∗ 31 (37.2±1.4) 42 (39.3±1.1) 21 (23.4±1.0) total 289 (278.8±3.5) 293 (294.5±3.4) 179 (176.8 ± 2.8) Table 7: First selection level of the cc̄qq̄l+ν final state: the number of events selected in data and the SM expectations for each sample and centre-of-mass energy are shown. • the angle between the lepton and the missing momentum. The data, SM expectation and signal distributions of these variables are shown in Fig. 8. In order to improve the efficiency, events with no leptons seen in the detector were kept in a fourth sample. For this sample, the selection criteria of the bb̄qq̄νν̄ final state were applied and the same variables as in section 4.2 were used to build the PDFs. The signal efficiency after the first selection level for mb′ = 100 GeV/c 2 and s = 205 GeV was 8.9±0.9%. The efficiencies for the other relevant b′ masses and s values were found to be the same within errors. 4.5 The cc̄qq̄qq̄ final state This final state is very similar to bb̄qq̄qq̄ (with slightly different kinematics due to the mass difference between the Z and the W). The analysis described in section 4.3 was thus adopted. The number of selected events and the SM expectations can be found in Table 6. At this level, the signal efficiency for mb′ = 100 GeV/c 2 and s = 205 GeV was 67.3±1.5%. The efficiencies for the other b′ masses and centre-of-mass energies were the same within errors. The PDFs were built using the same set of variables as in section 4.3. 5 Results For all final states, a good agreement between data and SM expectation was found. The summary of the total number of selected data events, SM expectations, the corresponding background composition and the signal efficiencies for the studied final states are shown in Table 8. In the bb̄l+l−νν̄ final state, one data event was retained after the final selection level, for a SM expectation of 1.5 ± 0.7 events. This event belonged to the no-id sample and was collected at s = 200 GeV. For all the other final states, discriminant analyses were used. In these cases, a discriminant variable, ln(LS/LB), was defined. The distributions of ln(LS/LB), for the different analysis channels are shown in Fig. 9. No evidence for a signal was found in any of the channels and the full information, i.e. event numbers and the shapes of the distributions of the discriminant variables were used to derive limits on BR(b′ → bZ) and BR(b′ → cW). data background signal final state (SM ± stat. error) composition (%) efficiency (%) qq̄ WW ZZ γγ bb̄l+l−νν̄ e sample 16 (13.2±0.8) 16 16 68 0 35.1±2.6 (first selection µ sample 14 (16.7±0.8) 0 10 90 0 53.4±2.7 level) no-id sample 208 (191.0±3.0) 8 80 12 0 12.3±1.0 bb̄qq̄νν̄ 533 (511.7±8.3) 76 17 2 5 57.6±1.7 bb̄qq̄qq̄ 1662 (1673.9±11.4) 35 65 0 0 66.0±1.5 e sample 289 (278.8±3.5) 7 82 11 0 45.3±2.7 cc̄qq̄l+ν µ sample 293 (294.5±3.4) 2 97 1 0 56.4±2.7 no-id sample 179 (176.8±2.8) 9 84 7 0 5.3±0.7 no lepton sample 533 (511.7±8.3) 76 17 2 5 8.9±0.9 cc̄qq̄qq̄ 1662 (1673.9±11.4) 35 65 0 0 67.3±1.5 Table 8: Summary of the total number of selected data events and SM expectations for the studied final states after the final selection (first selection level for bb̄l+l−νν̄). The corresponding background composition and signal efficiencies for mb′ = 100 GeV/c 2 and√ s = 205 GeV are also shown. 5.1 Limits on BR(b′ → bZ) and BR(b′ → cW) Upper limits on the product of the e+e− → b′b̄′ cross-section and the branching ratio as a function of the b′ mass were derived at 95% confidence level (CL) in each of the considered b′ decay modes (FCNC and CC), taking into account the values of the dis- criminant variables and their expected distributions for signal and background, the signal efficiencies and the data luminosities at the various centre-of-mass energies. Assuming the SM cross-section for the pair production of heavy quarks at LEP [7,14], these limits were converted into limits on the branching ratios corresponding to the b′ → bZ and b′ → cW decay modes. The modified frequentist likelihood ratio method [23] was used. The different final states and centre-of-mass energy bins were treated as inde- pendent channels. For each b′ mass only the channels with s > 2mb′ were considered. In order to avoid some non-physical fluctuations of the distributions of the discriminant variables due to the limited statistics of the generated events, a smoothing algorithm was used. The median expected limit, i.e. the limit obtained if the SM background was the only contribution in data, was also computed. In Fig. 10 the observed and expected limits on BR(b′ → bZ) and BR(b′ → cW) are shown as a function of the b′ mass. The 1σ and 2σ bands around the expected limit are also shown. The observed and expected limits are statistically compatible. At 95% CL and for mb′ = 96 GeV/c 2, the BR(b′ → bZ) and BR(b′ → cW) have to be below 51% and 43%, respectively. These limits were evaluated taking into account the systematic uncertainties, as explained in the next subsection. The limits obtained for BR(b′ → bZ) are compatible with those presented by CDF [10] for a b′ mass of 100 GeV/c2. Below this mass, the DELPHI result is more sensitive and the CDF limit degrades rapidly. For higher b′ masses, the LEP-II kinematical limit is reached and the present analysis looses sensitivity. 5.2 Systematic uncertainties The evaluation of the limits was performed taking into account systematic uncertain- ties, which affect the background estimation, the signal efficiency and the shape of the distributions used. The following systematic uncertainties were considered: • SM cross-sections: uncertainties on the SM cross-sections translate into uncertainties on the expected number of background events. The overall uncertainty on the most relevant SM background processes for the present analyses is typically less than 2% [24], which leads to relative changes on the branching ratio limits below 6%; • Signal generation: uncertainties on the final state quark hadronisation and fragmen- tation modelling were studied. The Lund symmetric fragmentation function was tested and compared with schemes where the b and c quark masses are taken into account [14]. This systematic error source was estimated to be of the order of 20% in the signal efficiency, by conservatively taking the maximum observed variation. The relative effect on the branching ratio limits is below 16%; • Smoothing: the uncertainty associated to the discriminant variables smoothing was estimated by applying different smoothing algorithms. The smoothing procedure does not change the number of SM expected events or the signal efficiency, but may lead to differences in the shape of the discriminant variables. The relative effect of this uncertainty on the limits evaluation was found to be below 9%. Further details on the evaluation of the systematic errors and the derivation of limits can be found in [25]. 6 Constraints on RCKM The branching ratios for the b′ decays can be computed within a four generations sequential model [5–7]. As discussed before, if the b′ is lighter than both the t and the t′ quarks and mZ < mb′ < mH, the main contributions to the b ′ width are BR(b′ → bZ) and BR(b′ → cW) [7]. Using the unitarity of the CKM matrix, its approximate diagonality (Vub′ Vub ≈ 0) and taking Vcb ≈ 10−2 [12], the branching fractions can be written as a function of three variables: RCKM = | Vcb′V tb′ Vtb |, mt′ and mb′ [5–7]. Fixing mt′ − mb′ , the limits on BR(b′ → bZ) and BR(b′ → cW) (Fig. 10) can be translated into 95% CL bounds on RCKM as a function of mb′ . Two extreme cases were considered: the almost degenerate case, with mt′−mb′ = 1 GeV/c2, and the case in which the mass difference is close to the largest possible value, mt′ − mb′ = 50 GeV/c2 [3,5]. The results are shown in Fig. 11 and Fig. 12. In the figures, the upper curve was obtained from the limit on BR(b′ → cW), while the lower curve was obtained from the limit on BR(b′ → bZ), which decreases with growing mt′ . This suppression is due to the GIM mechanism [26] as mt′ approaches mt. On the other hand, as the b ′ mass approaches the bZ threshold, the b′ → bg decay dominates over b′ → bZ [7] and the lower limit on RCKM becomes less stringent. The expected limits on BR(b ′ → bZ) did not allow to set exclusions for low values of RCKM and mt′ −mb′ = 1 GeV/c2 (see Fig. 11). 7 Conclusions The data collected with the DELPHI detector at s = 196−209 GeV show no evidence for the pair production of b′-quarks with masses ranging from 96 to 103 GeV/c2. Assuming the SM cross-section for the pair production of heavy quarks at LEP, 95% CL upper limits on BR(b′ → bZ) and BR(b′ → cW) were obtained. It was shown that, at 95% CL and for mb′ = 96 GeV/c 2, the BR(b′ → bZ) and BR(b′ → cW) have to be below 51% and 43%, respectively. The 95% CL upper limits on the branching ratios, combined with the predictions of the sequential fourth generation model, were used to exclude regions of the (RCKM , mb′) plane for two hypotheses of the mt′ − mb′ mass difference. It was shown that, for mt′ −mb′ = 1 (50) GeV/c2 and 96 GeV/c2 < mb′ < 102 GeV/c2, RCKM is bounded by an upper limit of 3.8×10−3 (1.2×10−3). For mb′ = 100 GeV/c2 and mt′ −mb′ = 50 GeV/c2, the CKM ratio was constrained to be in the range 4.6 × 10−4 < RCKM < 7.8 × 10−4. Acknowledgements We are greatly indebted to our technical collaborators, to the members of the CERN- SL Division for the excellent performance of the LEP collider, and to the funding agencies for their support in building and operating the DELPHI detector. We acknowledge in particular the support of Austrian Federal Ministry of Education, Science and Culture, GZ 616.364/2-III/2a/98, FNRS–FWO, Flanders Institute to encourage scientific and technological research in the industry (IWT) and Belgian Federal Office for Scientific, Technical and Cultural affairs (OSTC), Belgium, FINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil, Czech Ministry of Industry and Trade, GA CR 202/99/1362, Commission of the European Communities (DG XII), Direction des Sciences de la Matière, CEA, France, Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie, Germany, General Secretariat for Research and Technology, Greece, National Science Foundation (NWO) and Foundation for Research on Matter (FOM), The Netherlands, Norwegian Research Council, State Committee for Scientific Research, Poland, SPUB-M/CERN/PO3/DZ296/2000, SPUB-M/CERN/PO3/DZ297/2000, 2P03B 104 19 and 2P03B 69 23(2002-2004) FCT - Fundação para a Ciência e Tecnologia, Portugal, Vedecka grantova agentura MS SR, Slovakia, Nr. 95/5195/134, Ministry of Science and Technology of the Republic of Slovenia, CICYT, Spain, AEN99-0950 and AEN99-0761, The Swedish Research Council, Particle Physics and Astronomy Research Council, UK, Department of Energy, USA, DE-FG02-01ER41155, EEC RTN contract HPRN-CT-00292-2002. References [1] The LEP Collaborations ALEPH, DELPHI, L3, OPAL and the LEP Electroweak Working Group, A Combination of Preliminary Electroweak Measurements and Con- straints on the Standard Model (2005) CERN-PH-EP/2005-051, hep-ex/0511027; ALEPH, DELPHI, L3, OPAL and SLD Coll., LEP Electroweak Working Group, SLD Heavy Flavour Groups, Phys. Rept. 427 (2006) 257. [2] V.A. Novikov, L.B. Okun, A.N. Rozanov and M.I. Vysotsky, Phys. Lett. B529 (2002) [3] P.H. Frampton, P.Q. Hung and M. Sher, Phys. Rep. 330 (2000) 263. [4] A. Djouadi et al. in Electroweak symmetry breaking and new physics at the TeV scale, ed. Barklow, Timothy - World Scientific, Singapore (1997). [5] A. Arhrib and W.S. Hou, Phys. Rev. D64 (2001) 073016; A. Arhrib and W.S. Hou, JHEP 0607 (2006) 009. [6] W.S. Hou and R.G. Stuart, Phys. Rev. Lett. 62 (1989) 617; W.S. Hou and R.G. Stuart, Nucl. Phys. B320 (1989) 277; W.S. Hou and R.G. Stuart, Nucl. Phys. B349 (1991) 91. [7] S.M. Oliveira and R. Santos, Phys. Rev. D68 (2003) 093012; S.M. Oliveira and R. Santos, Acta Phys. Polon. B34 (2003) 5523. [8] ALEPH Coll., D. Decamp et al., Phys. Lett. B236 (1990) 511; DELPHI Coll., P. Abreu et al., Nucl. Phys. B367 (1991) 511; L3 Coll., O. Adriani et al., Phys. Rep. 236 (1993) 1; OPAL Coll., M.Z. Akrawy et al., Phys. Lett. B246 (1990) 285. [9] D0 Coll., S. Abachi et al., Phys. Rev. Lett. 78 (1997) 3818. [10] CDF Coll., T. Affolder et al., Phys. Rev. Lett. 84 (2000) 835. [11] D0 Coll., S. Abachi et al., Phys. Rev. D52 (1995) 4877. [12] Particle Data Group, W.-M. Yao et al., J. Phys. G33 (2006) 1. [13] DELPHI Coll., P. Aarnio et al., Nucl. Instr. Meth. A303 (1991) 233; DELPHI Coll., P. Abreu et al., Nucl. Instr. Meth. A378 (1996) 57. [14] T. Sjöstrand, Comp. Phys. Comm. 82 (1994) 74; T. Sjöstrand, PYTHIA 5.7 and JETSET 7.4, CERN-TH/7112-93; T. Sjöstrand et al., Comp. Phys. Comm. 135 (2001) 238. [15] E. Accomando and A. Ballestero, Comp. Phys. Comm. 99 (1997) 270; E. Accomando, A. Ballestrero and E. Maina, Comp. Phys. Comm. 150 (2003) 166; A. Ballestrero, R. Chierici, F. Cossutti and E. Migliore, Comp. Phys. Comm. 152 (2003) 175. [16] S. Jadach, B.F.L. Ward and Z. Was, Comp. Phys. Comm. 130 (2000) 260. [17] S. Jadach, W. P laczek and B.F.L. Ward, Phys. Lett. B390 (1997) 298. [18] F. Cossutti et al., REMCLU: a package for the Reconstruction of Elec- troMagnetic CLUsters at LEP200, DELPHI Note 2000-164 PROG 242, http://delphiwww.cern.ch/pubxx/delnote/public/2000 164 prog 242.ps.gz. [19] S. Catani et al., Phys. Lett. B269 (1991) 432. [20] DELPHI Coll., J. Abdallah et al., Eur. Phys. J. C32 (2004) 185. [21] P. Abreu et al., Nucl. Instr. Meth. A427 (1999) 487. [22] G. Fox and S. Wolfram, Phys. Lett. B82 (1979) 134. [23] A.L. Read, CERN report 2000-005 (2000) 81, “Workshop on Confidence Limits”, edited by F. James, L. Lyons and Y. Perrin. [24] S. Jadach et al., LEP2 Monte Carlo Workshop: Report of the Working Groups on Precision Calculations for LEP2 Physics, CERN report 2000-009 (2000); G. Altarelli et al., Physics at LEP2, CERN report 96-01 (1996). [25] N. Castro, Search for a fourth generation b′-quark at LEP-II. MSc. Thesis, Instituto Superior Técnico da Universidade Técnica de Lisboa (2004), CERN-THESIS-2005- [26] S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. Z / H / g / γ b / s / d t′ / t / c / u a) b) Figure 1: The Feynman diagrams corresponding to the b′ (a) FCNC and (b) CC decay modes are shown. Z/γ Z l−/ q̄ / q̄ l+/ q / q ν̄ / ν̄ / q̄ ν / ν / q Z/γ W q̄ / q̄ q / q ν / q̄ l+/ q a) b) Figure 2: The final states associated to the b′ (a) FCNC and (b) CC decay modes are shown. Only those states analysed here are indicated. DELPHI 0 50 100 150 αl1,tr (˚) a) (e sample) 0 20 40 60 80 100 pmis (GeV/c) b) (µ sample) 0 20 40 60 80 pjet 1 (GeV/c) c) (no-id sample) SM expectation signal (mb’=100 GeV/c Figure 3: Data and SM expectation after the preselection level for the bb̄l+l−νν̄ final state and centre-of-mass energies above 200 GeV. (a) The angle between the most energetic lepton and the closest charged-particle track (e sample), (b) the missing momentum (µ sample) and (c) the momentum of the most energetic jet (no-id sample) are shown. The signal distributions for mb′ = 100 GeV/c 2 and s = 205 GeV are also shown with arbitrary normalisation. The background composition is 11% of qq̄, 69% of WW, 15% of ZZ and 5% of γγ for the e sample, 6% of qq̄, 90% of WW and 4% of ZZ for the µ sample and 45% of qq̄, 48% of WW, 5% of ZZ and 2% of γγ for the no-id sample. DELPHI 0 20 40 60 80 pjet 1 (GeV/c) a) (e sample) 0 50 100 150 αll (˚) b) (µ sample) 0 0.25 0.5 0.75 1 pmis / Emis c) (no-id sample) SM expectation signal (mb’=100 GeV/c Figure 4: Data and SM expectation after the first selection level for the bb̄l+l−νν̄ final state and for centre-of-mass energies above 200 GeV. (a) The momentum of the most energetic jet (e sample), (b) the angle between the two leptons (µ sample) and (c) the ratio between the missing momentum and missing energy (no-id sample) are shown. The signal distributions for mb′ = 100 GeV/c 2 and s = 205 GeV are also shown with arbitrary normalisation. The arrows represent the cuts applied in the second selection level. DELPHI 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 χ2/n.d.f. SM expectation signal (mb’=100 GeV/c Figure 5: Comparison of data and SM expectation distributions of the χ2/n.d.f. of the fit imposing energy-momentum conservation and no missing energy for the bb̄qq̄νν̄ final state at centre-of-mass energies above 200 GeV. The arrow shows the applied cut. The signal for mb′ = 100 GeV/c 2 and s = 205 GeV is also shown with arbitrary normalisation. DELPHI 0 50 100 150 200 missing mass (GeV/c 0 10 20 30 40 scaled acoplanarity (˚) 0 50 100 150 acolj1j2 (˚) 0 0.5 1 1.5 h1+h3 0 50 100 150 θmis (˚) SM expectation signal (mb’=100 GeV/c Figure 6: Variables used in the discriminant analysis (bb̄qq̄νν̄ final state). The data and SM expectation distributions for centre-of-mass energies above 200 GeV are shown for (a) the missing mass, (b) Aj1j2cop ×min(sin θj1 , sin θj2), where Aj1j2cop is the acoplanarity and θj1,j2 are the polar angles of the jets when forcing the events into two jets, (c) the acollinearity between the two most energetic jets (with the event particles clustered into four jets), (d) the sum of the first and third Fox-Wolfram moments and (e) the polar angle of the missing momentum. The signal distributions for mb′ = 100 GeV/c 2 and s = 205 GeV are also shown with arbitrary normalisation. DELPHI 1 1.5 2 2.5 3 -log10(y4→3) 1 2 3 4 -log10(y5→4) 0 50 100 150 acolj1j2 (4 jets) (˚) 0 0.2 0.4 0.6 0.8 1 h1+h3 0 20 40 60 80 100 pj1 (GeV/c) /c e) 0 50 100 150 αj1j2 (˚) data SM expectation signal (mb’=100 GeV/c Figure 7: Variables used in the discriminant analysis (bb̄qq̄qq̄ final state). The data and SM expectation for centre-of-mass energies above 200 GeV are shown for (a) − log10(y4→3), (b) − log10(y5→4), (c) the acollinearity between the two most energetic jets, with the events clustered into four jets (see text for explanation), (d) the h1 + h3 Fox-Wolfram moments sum, (e) the momentum of the most energetic jet and (f) the angle between the two most energetic jets. The signal distributions for mb′ = 100 GeV/c 2 and√ s = 205 GeV are also shown with arbitrary normalisation. DELPHI 0 0.25 0.5 0.75 1 h1+h3 a) (e sample) 0 50 100 150 mj1j2 (2 jets) (GeV/c b) (e sample) 1 2 3 4 -log10(y4→3) c) (µ sample) 0 0.1 0.2 0.3 0.4 Σptracks lepton hem. / √s d) (µ sample) 0 50 100 150 acolj1j2 (˚) e) (no-id sample) 0 50 100 150 αlν (˚) f) (no-id sample) data SM expectation signal (mb’=100 GeV/c Figure 8: Variables used in the discriminant analysis (cc̄qq̄l+ν final state). The data events and background expectation for centre-of-mass energies above 200 GeV are shown for (a) the h1 + h3 Fox-Wolfram moments sum (e sample), (b) the invariant mass of the two jets with the events clustered into two jets (e sample), (c) − log10(y4→3) (µ sample), (d) |~pi|/ s, where ~pi are the momenta of the charged particles (excluding the lepton) in the same hemisphere as the lepton (µ sample), (e) the acollinearity between the two most energetic jets (no-id sample) and (f) the angle between the lepton and the missing momentum (no-id sample). The signal distributions for mb′ = 100 GeV/c 2 and√ s = 205 GeV are also shown with arbitrary normalisation. DELPHI -10 0 10 ln(LS/LB) -10 -5 0 5 ln(LS/LB) -20 -10 0 10 ln(LS/LB) -20 -10 0 10 ln(LS/LB) -20 -10 0 10 ln(LS/LB) -10 0 10 ln(LS/LB) -20 -10 0 10 ln(LS/LB) SM expectation Signal (mb’=100 GeV/c Figure 9: Discriminant variables ln(LS/LB) for data and SM simulation (centre-of– mass energies above 200 GeV). FCNC b′ decay mode: (a) bb̄qq̄νν̄ and (b) bb̄qq̄qq̄. CC b′ decay mode: (c) cc̄qq̄l+ν (e sample), (d) cc̄qq̄l+ν (µ sample), (e) cc̄qq̄l+ν (no-id sample) (f) cc̄qq̄l+ν (no lepton sample) and (g) cc̄qq̄qq̄. The signal distributions for mb′ = 100 GeV/c 2 and s = 205 GeV are also shown with arbitrary normalisation. DELPHI 96 97 98 99 100 101 102 103 mb’ (GeV/c a) b’→ bZ decay observed limit expected limit expected ± 1σ expected ± 2σ 96 97 98 99 100 101 102 103 96 97 98 99 100 101 102 103 mb’ (GeV/c 96 97 98 99 100 101 102 103 b) b’→ cW decay observed limit expected limit expected ± 1σ expected ± 2σ Figure 10: The observed and expected upper limits at 95% CL on (a) BR(b′ → bZ) and (b) BR(b′ → cW) are shown. The 1σ and 2σ bands around the expected limit are also presented. Systematic errors were taken into account in the limit evaluation. DELPHI Figure 11: The excluded region in the plane (RCKM , mb′) with mt′ −mb′ = 1 GeV/c2, obtained from the 95% CL upper limits on BR(b′ → bZ) (bottom) and BR(b′ → cW) (top) is shown. The light and dark shadings correspond to the observed and expected limits, respectively. The expected limits on BR(b′ → bZ) did not allow exclusions to be set for low values of RCKM . DELPHI Figure 12: The excluded region in the plane (RCKM , mb′) with mt′ −mb′ = 50 GeV/c2, obtained from the 95% CL upper limits on BR(b′ → bZ) (bottom) and BR(b′ → cW) (top) is shown. The light and dark shadings correspond to the observed and expected limits, respectively.

0704.0595

About curvature, conformal metrics and warped products

ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS FERNANDO DOBARRO BÜLENT ÜNAL Abstract. We consider the curvature of a family of warped prod- ucts of two pseduo-Riemannian manifolds (B, gB) and (F, gF ) furnished with metrics of the form c2gB ⊕ w 2gF and, in particular, of the type gB ⊕w gF , where c, w : B → (0,∞) are smooth functions and µ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B, gB) is Riemannian, the latter question involves nonlin- ear elliptic partial differential equations with concave-convex nonlinear- ities and singular partial differential equations of the Lichnerowicz-York type among others. 1. Introduction The main concern of this paper is the curvature of a special family of warped pseudo-metrics on product manifolds. We introduce a suitable form for the relations among the involved curvatures in such metrics and apply them to the existence and/or construction of Einstein and constant scalar curvature metrics in this family. Let B = (Bm, gB) and F = (Fk, gF ) be two pseudo-Riemannian manifolds of dimensions m ≥ 1 and k ≥ 0, respectively and also let B × F be the usual product manifold of B and F . For a given smooth function w ∈ C∞>0(B) = {v ∈ C∞(B) : v(x) > 0, ∀x ∈ B}, the warped product B ×w F = ((B ×w F )m+k, g = gB +w2gF ) was defined by Bishop and O’Neill in [19] in order to study manifolds of negative curvature. Date: November 4, 2018. 1991 Mathematics Subject Classification. Primary: 53C21, 53C25, 53C50 Secondary: 35Q75, 53C80, 83E15, 83E30. Key words and phrases. Warped products, conformal metrics, Ricci curvature, scalar curvature, semilinear equations, positive solutions, Lichnerowicz-York equation, concave- convex nonlinearities, Kaluza-Klein theory, string theory. http://arxiv.org/abs/0704.0595v1 2 FERNANDO DOBARRO & BÜLENT ÜNAL In this article, we deal with a particular class of warped products, i.e. when the pseudo-metric in the base is affected by a conformal change. Pre- cisely, for given smooth functions c, w ∈ C∞>0(B) we will call ((B × F )m+k, g = c2gB+w 2gF ) as a [c, w]-base conformal warped product (briefly [c, w]-bcwp), denoted by B ×[c,w] F . We will concentrate our attention on a special sub- class of this structure, namely when there is a relation between the conformal factor c and the warping function w of the form c = wµ, where µ is a real parameter and we will call the [ψµ, ψ]-bcwp as a (ψ, µ)-bcwp. Note that we generically called the latter case as special base conformal warped products, briefly sbcwp in [29]. As we will explain in §2, metrics of this type play a relevant role in several topics of differential geometry and theoretical physics (see also [29]). This article concerns curvature related questions of these metrics which are of interest not only in the applications, but also from the points of view of dif- ferential geometry and the type of the involved nonlinear partial differential equations (PDE), such as those with concave-convex nonlinearities and the Lichnerowicz-York equations. The article is organized in the following way: in §2 after a brief description of several fields where pseudo-metrics described as above are applied, we formulate the curvature problems that we deal within the next sections and give the statements of the main results. In §3, we state Theorems 2.2 and 2.3 in order to express the Ricci tensor and scalar curvature of a (ψ, µ)-bcwp and sketch their proofs (see [29, Section 3] for detailed computations). In §4 and 5, we establish our main results about the existence of (ψ, µ)-bcwp’s of constant scalar curvature with compact Riemannian base. 2. Motivations and Main results As we announced in the introduction, we firstly want to mention some of the major fields of differential geometry and theoretical physics where base conformal warped products are applied. i: In the construction of a large class of non trivial static anti de Sitter vacuum space-times • In the Schwarzschild solutions of the Einstein equations (see [10, 18, 41, 59, 69, 74]). • In the Riemannian Schwarzschild metric, namely (see [10]). • In the “generalized Riemannian anti de Sitter T2 black hole metrics” (see §3.2 of [10] for details). • In the Bañados-Teitelboim-Zanelli (BTZ) and de Sitter (dS) black holes (see [1, 15, 16, 28, 45] for details). Indeed, all of them can be generated by an approach of the fol- lowing type: let (F2, gF ) be a pseudo-Riemannian manifold and g be ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 3 a pseudo-metric on R+ × R× F2 defined by (2.1) g = u2(r) dr2 ± u2(r)dt2 + r2gF . After the change of variables s = r2, y = t, there results ds2 = 4r2dr2 and dy2 = dt2. Then (2.1) is equivalent to ds2 ± 4 s)dy2 + sgF 2 )2(− 2 ))2(−1)ds2 ± (2s 2 ))2dy2 2 )2gF . (2.2) Note that roughly speaking, g is a nested application of two (ψ, µ)- bcwp’s. That is, on R+ × R and taking (2.3) ψ1(s) = 2s 2 ) and µ1 = −1, the metric inside the brackets in the last member of (2.2) is a (ψ1, µ1)- bcwp, while the metric g on (R+ × R)× F2 is a (ψ2, µ2)-bcwp with (2.4) ψ2(s, y) = s 2 and µ2 = − In the last section of [29], through the application of Theorems 2.2 and 2.3 below and several standard computations, we generalized the latter approach to the case of an Einstein fiber (Fk, gF ) with dimension k ≥ 2. ii: In the study of the equivariant isometric embeddings of space-time slices in Minkowski spaces (see [39, 38]). iii: In the Kaluza-Klein theory (see [76, §7.6, Particle Physics and Ge- ometry], [60] and [77]) and in the Randall-Sundrum theory [30, 40, 63, 64, 65, 71] with µ as a free parameter. For example, in [46] the following metric is considered (2.5) e2A(y)gijdx idxj + e2B(y)dy2, with the notation {xi}, i = 0, 1, 2, 3 for the coordinates in the 4- dimensional space-time and x5 = y for the fifth coordinate on an extra dimension. In particular, Ito takes the ansatz (2.6) B = αA, which corresponds exactly to our sbcwp metrics, considering gB = dy2, gF = gijdx idxj , ψ(y) = e α = eA(y) and µ = α. iii: In String and Supergravity theories, for instance, in the Maldacena conjecture about the duality between compactifications of M/string 4 FERNANDO DOBARRO & BÜLENT ÜNAL theory on various Anti-de Sitter space-times and various confor- mal field theories (see [55, 62]) and in warped compactifications (see [40, 72] and references therein). Besides all of these, there are also frequent occurrences of this type of metrics in string topics (see [33, 34, 35, 36, 37, 53, 61, 71] and also [1, 12, 67] for some reviews about these topics). iv: In the derivation of effective theories for warped compactification of supergravity and the Hor̆ava-Witten model (see [50, 51]). For in- stance, in [51] the ansatz ds2 = hαds2(X4) + h βds2(Y ) is considered where X4 is a four-dimensional space-time with coordinates x is a Calabi-Yau manifold (the so called internal space) and h de- pends on the four-dimensional coordinates xµ, in order to study the dynamics of the four-dimensional effective theory. We note that in those articles, the structure of the expressions of the Ricci tensor and scalar curvature of the involved metrics result particularly use- ful. We observe that they correspond to very particular cases of the expressions obtained by us in [29], see also Theorems 2.2 and 2.3 and Proposition 2.4 stated below. v: In the discussion of Birkhoff-type theorems (generally speaking these are the theorems in which the gravitational vacuum solutions admit more symmetry than the inserted metric ansatz, (see [41, page 372] and [17, Chapter 3]) for rigorous statements), especially in Equation 6.1 of [66] where, H-J. Schmidt considers a special form of a bcwp and basically shows that if a bcwp of this form is Einstein, then it admits one Killing vector more than the fiber. In order to achieve that, the author considers for a specific value of µ, namely µ = (1 − k)/2, in the following problem: Does there exist a smooth function ψ ∈ C∞>0(B) such that the corresponding (ψ, µ)-bcwp (B2 × Fk, ψ2µgB + ψ2gF ) is an Einstein manifold? (see also (Pb-Eins.) below.) vi: In the study of bi-conformal transformations, bi-conformal vector fields and their applications (see [32, Remark in Section 7] and [31, Sections 7 and 8]). vii: In the study of the spectrum of the Laplace-Beltrami operator for p−forms. For instance in Equation (1.1) of [11], the author considers the structure that follows: let M be an n-dimensional compact, Rie- mannian manifold with boundary, and let y be a boundary-defining function; she endows the interior M of M with a Riemannian metric ds2 such that in a small tubular neighborhood of ∂M inM , ds2 takes the form (2.7) ds2 = e−2(a+1)tdt2 + e−2btdθ2∂M , ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 5 where t := − log y ∈ (c,+∞) and dθ2 is the Riemannian metric on ∂M (see [11, 56] and references therein for details). Notation 2.1. From now on, we will use the Einstein summation convention over repeated indices and consider only connected manifolds. Furthermore, we will denote the Laplace-Beltrami operator on a pseudo-Riemannian man- ifold (N,h) by ∆N (·), i.e., ∆N (·) = ∇N i∇Ni(·). Note that ∆N is elliptic if (N,h) is Riemannian and it is hyperbolic when (N,h) is Lorentzian. If (N,h) is neither Riemannian nor Lorentzian, then the operator is ultra-hyperbolic. Furthermore, we will consider the Hessian of a function v ∈ C∞(N), denoted by Hvh or H N , so that the second covariant differential of v is given by Hvh = ∇(∇v). Recall that the Hessian is a symmetric (0, 2) tensor field satisfying (2.8) Hvh(X,Y ) = XY v − (∇XY )v = h(∇X(grad v), Y ), for any smooth vector fields X,Y on N. For a given pseudo-Riemannian manifold N = (N,h) we will denote its Riemann curvature tensor, Ricci tensor and scalar curvature by RN , RicN and SN , respectively. We will denote the set of all lifts of all vector fields of B by L(B). Note that the lift of a vector field X on B denoted by X̃ is the vector field on B × F given by dπ(X̃) = X where π : B × F → B is the usual projection map. In Section 3, we will sketch the proofs of the following two theorems related to the Ricci tensor and the scalar curvature of a generic (ψ, µ)-bcwp. Theorem 2.2. Let B = (Bm, gB) and F = (Fk, gF ) be two pseudo-Rieman- nian manifolds with m ≥ 3 and k ≥ 1, respectively and also let µ ∈ R \ {0, 1, µ, µ±} be a real number with µ := − k m− 2 and µ± := µ± µ2 − µ. Suppose ψ ∈ C∞>0(B). Then the Ricci curvature tensor of the corresponding (ψ, µ)-bcwp, denoted by Ric verifies the relation Ric = RicB + β − β∆ 1 α∆ gB on L(B)× L(B), Ric = 0 on L(B)× L(F ), Ric = RicF − ψ2(µ−1) α∆ gF on L(F )× L(F ), (2.9) 6 FERNANDO DOBARRO & BÜLENT ÜNAL where (2.10) (m− 2)µ+ k , (m− 2)µ+ k , −[(m− 2)µ + k] µ[(m− 2)µ+ k] + k(µ− 1) , [(m− 2)µ + k]2 µ[(m− 2)µ+ k] + k(µ− 1) . Theorem 2.3. Let B = (Bm, gB) and F = (Fk, gF ) be two pseudo-Rieman- nian manifolds of dimensions m ≥ 2 and k ≥ 0, respectively. Suppose that SB and SF denote the scalar curvatures of B = (Bm, gB) and F = (Fk, gF ), respectively. If µ ∈ R and ψ ∈ C∞>0(B), then the scalar curvature S of the corresponding (ψ, µ)-bcwp verifies, (i) If µ 6= − k m− 1 , then (2.11) − β∆Bu+ SBu = Su2µα+1 − SFu2(µ−1)α+1 where (2.12) α = 2[k + (m− 1)µ] {[k + (m− 1)µ] + (1− µ)}k + (m− 2)µ[k + (m− 1)µ] , (2.13) β = α2[k + (m− 1)µ] > 0 and ψ = uα > 0. (ii) If µ = − k m− 1 , then (2.14) − k |∇Bψ|2B m−1 [S − SFψ−2]− SB. From the mathematical and physical points of view, there are several interesting questions about (ψ, µ)-bcwp’s. In [29] we began the study of existence and/or construction of Einstein (ψ, µ)-bcwp’s and those of constant scalar curvature. These questions are closely connected to Theorems 2.2 and In [29], by applying Theorem 2.2, we give suitable conditions that allow us to study some particular cases of the problem: (Pb-Eins.) Given µ ∈ R, does there exist a smooth function ψ ∈ C∞>0(B) such that the corresponding (ψ, µ)-bcwp is an Einstein manifold? ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 7 In particular, we obtain the following result as an immediate corollary of Theorem 2.2. Proposition 2.4. Let us assume the hypothesis of Theorem 2.2. Then the corresponding (ψ, µ)-bcwp is an Einstein manifold with λ if and only if (F, gF ) is Einstein with ν constant and the system that follows is verified λψ2µgB = RicB + β − β∆ 1 α∆ gB on L(B)× L(B) λψ2 = ν − 1 ψ2(µ−1) (2.15) where the coefficients are given by (2.10). Compare the system (2.15) with the well known one for a classical warped product in [18, 49, 59]. By studying (2.15), we have obtained the generaliza- tion of the construction exposed in the above motivational examples in i and v, among other related results. We suggest the interested reader consider the results about the problem (Pb-Eins.) stated in [29]. Now, we focus on the problems which we will deal in §4. Let B = (Bm, gB) and F = (Fk, gF ) be pseudo-Riemannian manifolds. There is an extensive number of publications about the well known Yamabe problem namely: (Ya) [79, 75, 68, 13] Does there exist a function ϕ ∈ C∞>0(B) such that (Bm, ϕ m−2 gB) has constant scalar curvature? Analogously, in several articles the following problem has been studied: (cscwp) [27] Is there a function w ∈ C∞>0(B) such that the warped product B ×w F has constant scalar curvature? In the sequel we will suppose that B = (Bm, gB) is a Riemannian manifold. Thus, both problems bring to the study of the existence of positive solutions for nonlinear elliptic equations on Riemannian manifolds. The involved non- linearities are powers with Sobolev critical exponent for the Yamabe problem and sub-linear (linear if the dimension k of the fiber is 3) for the problem of constant scalar curvature of a warped product. In Section 4, we deal with a mixed problem between (Ya) and (cscwp) which is already proposed in [29], namely: 8 FERNANDO DOBARRO & BÜLENT ÜNAL (Pb-sc) Given µ ∈ R, does there exist a function ψ ∈ C∞>0(B) such that the corresponding (ψ, µ)-bcwp has constant scalar curvature? Note that when µ = 0, (Pb-sc) corresponds to the problem (cscwp), whereas when the dimension of the fiber k = 0 and µ = 1, then (Pb-sc) corresponds to (Ya) for the base manifold. Finally (Pb-sc) corresponds to (Ya) for the usual product metric with a conformal factor in C∞>0(B) when µ = 1. Under the hypothesis of Theorem 2.3 i, the analysis of the problem (Pb- sc) brings to the study of the existence and multiplicity of solutions u ∈ C∞>0(B) of (2.16) − β∆Bu+ SBu = λu2µα+1 − SFu2(µ−1)α+1, where all the components of the equation are like in Theorem 2.3 i and λ (the conjectured constant scalar curvature of the corresponding (ψ, µ)-bcwp) is a real parameter. We observe that an easy argument of separation of variables, like in [24, §2] and [27], shows that there exists a positive solution of (2.16) only if the scalar curvature of the fiber SH is constant. Thus this will be a natural assumption in the study of (Pb-sc). Furthermore, note that the involved nonlinearities in the right hand side of (2.16) dramatically change with the choice of the parameters, an exhaustive analysis of these changes is the subject matter of [29, §6]. There are several partial results about semi-linear elliptic equations like (2.16) with different boundary conditions, see for instance [2, 5, 6, 9, 21, 23, 26, 73, 78] and references in [29]. In this article we will state our first results about the problem (Pb-sc) when the base B is a compact Riemannian manifold of dimension m ≥ 3 and the fiber F has non-positive constant scalar curvature SF . For brevity of our study, it will be useful to introduce the following notation: µsc := µsc(m,k) = − m− 1 and µpY = µpY (m,k) := − k + 1 m− 2 (sc as scalar curvature and Y as Yamabe). Notice that µpY < µsc < 0. We plan to study the case of µ = µsc in a preceding project, therefore the related results are not going to be presented here. We can synthesize our results about (Pb-sc) in the case of non-positive SF as follow. • The case of scalar flat fiber, i.e. SF = 0. Theorem 2.5. If µ ∈ (µpY , µsc) ∪ (µsc,+∞) the answer to (Pb-sc) is affirmative. ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 9 By assuming some additional restrictions on the scalar curvature of the base SB , we obtain existence results for the range µ ≤ µpY . • The case of fiber with negative constant scalar curvature, i.e. SF < 0. In order to describe the µ−ranges of validity of the results, we will apply the notations introduced in [29, §5] (see Appendix A for a brief introduction of these notations). Theorem 2.6. If “(m,k) ∈ D and µ ∈ (0, 1)” or “(m,k) ∈ CD and µ ∈ (0, 1) ∩ (µ−, µ+)” or “(m,k) ∈ CD and µ ∈ (0, 1) ∩ C[µ−, µ+]”, then the answer to (Pb-sc) is affirmative. Remark 2.7. The first two cases in Theorem 2.6 will be studied by adapting the ideas in [5] and the last case by applying the results in [73, p. 99]. In the former - Theorem 4.15, the involved nonlinearities correspond to the so called concave-convex whereas in the latter - Theorem 4.16, they are singular as in the Lichnerowicz-York equation about the constraints for the Einstein equations (see [22], [43], [58], [57, p. 542-543] and [73, Chp.18]). Similarly to the case of SF = 0, we obtain existence results for some remaining µ−ranges by assuming some additional restrictions for the scalar curvature of the base SB . Naturally the study of (Pb-sc) allows us to obtain partial results of the related question: Given µ ∈ R and λ ∈ R does there exist a function ψ ∈ C∞>0(B) such that the corresponding (ψ, µ)-bcwp has constant scalar curvature λ? These are stated in the several theorems and propositions in §4. 3. The curvature relations - Sketch of the proofs The proofs of Theorems 2.2 and 2.3 require long and yet standard com- putations of the Riemann and Ricci tensors and the scalar curvature of a general base conformal warped product. Here, we reproduce the results for the Ricci tensor and the scalar curvature, and we also suggest the reader see [29, §3] for the complete computations. Theorem 3.1. The Ricci tensor of [c, w]-bcwp, denoted by Ric satisfies (1) Ric = RicB − (m− 2)1 HcB + k +2(m− 2) 1 dc⊗ dc+ k 1 [dc⊗ dw + dw ⊗ dc] (m− 3)gB(∇ Bc,∇Bc) gB(∇Bw,∇Bc) on L(B)× L(B), 10 FERNANDO DOBARRO & BÜLENT ÜNAL (2) Ric = 0 on L(B)× L(F ), (3) Ric = RicF − (m− 2)gB(∇ Bw,∇Bc) +(k − 1)gB(∇ Bw,∇Bw) gF on L(F )× L(F ). Theorem 3.2. The scalar curvature S of a [c, w]-bcwp is given by c2S = SB + SF − 2(m− 1)∆Bc − 2k∆Bw − (m− 4)(m− 1)gB(∇ Bc,∇Bc) − 2k(m− 2)gB(∇ Bw,∇Bc) − k(k − 1)gB(∇ Bw,∇Bw) The following two lemmas (3.3 and 3.7) play a central role in the proof of Theorems 2.2 and 2.3. Indeed, it is sufficient to apply them in a suitable mode and make use of Theorems 3.1 and 3.2 several times, the reader can find all the details in [29, §2 and 4]. LetN = (Nn, h) be a pseudo-Riemannian manifold of dimension n, |∇(·)|2 = |∇N (·)|2N = h(∇N (·),∇N (·)) and ∆h = ∆N . Lemma 3.3. Let Lh be a differential operator on C >0(N) defined by (3.1) Lhv = where ri, ai ∈ R and ζ := riai, η := i . Then, (3.2) Lhv = (η − ζ) ‖grad hv‖2h (ii) If ζ 6= 0 and η 6= 0, for α = ζ and β = , then we have (3.3) Lhv = β Remark 3.4. We also applied the latter lemma in the study of curvature of multiply warped products (see [28]). ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 11 Corollary 3.5. Let Lh be a differential operator defined by (3.4) Lhv = r1 for v ∈ C∞>0(N), where r1a1 + r2a2 6= 0 and r1a21 + r2a22 6= 0. Then, by changing the variables v = uα with 0 < u ∈ C∞(N), α = r1a1 + r2a2 1 + r2a and β = (r1a1 + r2a2) 1 + r2a α(r1a1 + r2a2) there results (3.5) Lhv = β Remark 3.6. By the change of variables as in Corollary 3.5 equations of the (3.6) Lhv = r1 = H(v, x, s), transform into (3.7) β∆hu = uH(u α, x, s). Lemma 3.7. Let Hh be a differential operator on C∞>0(N) defined by (3.8) Hhv = riai and η := i , where the indices extend from 1 to l ∈ N and any ri, ai ∈ R. Hence, (3.9) Hhv = (η − ζ) dv ⊗ dv + ζ 1 where ⊗ is the usual tensorial product. If furthermore, ζ 6= 0 and η 6= 0, (3.10) Hhv = β where α = and β = 4. The problem (Pb-sc) - Existence of solutions Throughout this section, we will assume that B is not only a Riemannian manifold of dimension m ≥ 3, but also “compact” and connected. We further assume that F is a pseudo-Riemannian manifold of dimension k ≥ 0 with constant scalar curvature SF ≤ 0. Moreover, we will assume that µ 6= µsc. Hence, we will concentrate our attention on the relations (2.11), (2.12) and (2.13) by applying Theorem 2.3 (i). 12 FERNANDO DOBARRO & BÜLENT ÜNAL Let λ1 denote the principal eigenvalue of the operator (4.1) L(·) = −β∆B(·) + SB(·), and u1 ∈ C∞>0(B) be the corresponding positive eigenfunction with ‖u1‖∞ = 1, where β is as in Theorem 2.3. First of all, we will state some results about uniqueness and non-existence of positive solutions for Equation (2.16) under the latter hypothesis. About the former, we adapt Lemma 3.3 in [5, p. 525] to our situation (for a detailed proof see [5], [20, Method II, p. 103] and also [70]). Lemma 4.1. Let f ∈ C0(R>0) such that t−1f(t) is decreasing. If v and w satisfy (4.2) −β∆Bv + SBv ≤ f(v), v ∈ C∞>0(B), (4.3) −β∆Bw + SBw ≥ f(w), w ∈ C∞>0(B), then w ≥ v on B. Proof. Let θ(t) be a smooth nondecreasing function such that θ(t) ≡ 0 for t ≤ 0 and θ(t) ≡ 1 for t ≥ 1. Thus for all ǫ > 0, θǫ(t) := θ is smooth, nondecreasing, nonnegative and θ(t) ≡ 0 for t ≤ 0 and θ(t) ≡ 1 for t ≥ ǫ. Furthermore γǫ(t) := sθ′ǫ(s)ds satisfies 0 ≤ γǫ(t) ≤ ǫ, for any t ∈ R. On the other hand, since (B, gB) is a compact Riemannian manifold without boundary and β > 0, like in [5, Lemma 3.3, p. 526] there results (4.4) [−vβ∆Bw + wβ∆Bv]θǫ(v − w)dvgB ≤ [−β∆Bv]γǫ(v − w)dvgB . Hence, by the above considerations about θǫ and γǫ, (4.4) implies that (4.5) [−vβ∆Bw + wβ∆Bv]θǫ(v − w)dvgB ≤ ǫ [−β∆Bv≥0] [−β∆Bv]dvgB . Now, by applying (4.2) and (4.3) there results (4.6) − vβ∆Bw+wβ∆Bv = vLw−wLv ≥ vf(w)−wf(v) = vw − f(v) ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 13 Thus by combining (4.6) and (4.5), as ǫ→ 0+ we led to (4.7) [v>w] − f(v) dvgB ≤ 0 and conclude the proof like in [5, Lemma 3.3, p. 526-527]. But on [v > w] and hence meas[v > w] = 0; thus v ≤ w. 1 � Corollary 4.2. Let f ∈ C0(R>0) such that t−1f(t) is decreasing. Then (4.8) −β∆Bv + SBv = f(v), v ∈ C∞>0(B) has at most one solution. Proof. Assume that v and w are two solutions of (4.8). Then by applying Lemma 4.1 firstly with v and w, and conversely with w and v, the conclusion is proved. � Remark 4.3. Notice that Lemma 4.1 and Corollary 4.2 allow the function f ∈ C0(R>0) to be singular at 0. Related to the non-existence of smooth positive solutions for Equation (2.16), we will state an easy result under the general hypothesis of this section. Proposition 4.4. If either maxB SB ≤ infu∈R>0 u2µα(λ− SFu−2α) or minB SB ≥ supu∈R>0 u 2µα(λ − SFu−2α), then (2.16) has no solution in C∞>0(B). Proof. It is sufficient to apply the maximum principle with some easy ad- justments to the particular involved coefficients. � • The case of scalar flat fiber, i.e. SF = 0. In this case, the term containing the nonlinearity u2(µ−1)α+1 becomes non- influent in (2.16), thus (Pb-sc) equivalently results to the study of existence of solutions for the problem: (4.9) −β∆Bu+ SBu = λu2µα+1, u ∈ C∞>0(B), where λ is a real parameter (i.e., it is the searched constant scalar curvature) and ψ = uα. 1meas denotes the usual gB−measure on the compact Riemannian manifold (Bm, gB) 14 FERNANDO DOBARRO & BÜLENT ÜNAL Remark 4.5. 2 Let p ∈ R\{1} and (λ0, u0) ∈ (R\{0})×C∞>0(B) be a solution (4.10) −β∆Bu+ SBu = λup, u ∈ C∞>0(B). Hence, by the difference of homogeneity between both members of (4.9), it is easy to show that if λ ∈ R satisfies sign(λ) = sign(λ0), then (λ, uλ) is a solution of (4.10), where uλ = tλu0 and tλ = Thus by (4.9), we obtain geometrically: if the parameter µ is given in a way that p := 2µα + 1 6= 1 and B ×[ψµ0 ,ψ0] F has constant scalar curvature λ0 6= 0, then for any λ ∈ R verifying sign(λ) = sign(λ0), there results that B×[ψµ F is of scalar curvature λ, where ψλ = t ψ0 and tλ given as above. Theorem 4.6. (Case : µ = 0) The scalar curvature of a (ψ, 0)-bcwp of base B and fiber F (i.e., a singly warped product B ×ψ F ) is a constant λ if and only if λ = λ1 and ψ is a positive multiple of u 1 (i.e., ψ = tu 1 for some t ∈ R>0). Proof. First of all note that µ = 0 implies α = k + 1 . On the other hand, in this case, the problem (4.9) is linear, so it is sufficient to apply the well known results about the principal eigenvalue and its associated eigenfunctions of operators like (4.1) in a suitable setting. � Theorem 4.7. (Case : µsc < µ < 0) The scalar curvature of a (ψ, µ)-bcwp of base B and fiber F is a constant λ, only if sign(λ) = sign(λ1). Further- more, (1) if λ = 0 then there exists ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has constant scalar curvature 0 if and only if λ1 = 0. Moreover, such ψ’s are the positive multiples of uα1 , i.e. tu 1 , t ∈ R>0. (2) if λ > 0 then there exists ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has constant scalar curvature λ if and only if λ1 > 0. In this case, the solution ψ is unique. (3) if λ < 0 then there exists ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has constant scalar curvature λ when λ1 < 0 and is close enough to 0. Proof. The condition µsc < µ < 0 implies that 0 < p := 2µα + 1 < 1, i.e., the problem (4.9) is sublinear. Thus, to prove the theorem one can use variational arguments as in [24] (alternatively, degree theoretic arguments as in [7] or bifurcation theory as in [27]). 2Along this article we consider the sign function defined by sign = χ(0,+∞) − χ(−∞,0), where χA is the characteristic function of the set A. ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 15 We observe that in order to obtain the positivity of the solutions required in (4.9), one may apply the maximum principle for the case of λ > 0 and the antimaximum principle for the case of λ < 0. The uniqueness for λ > 0 is a consequence of Corollary 4.2. � Remark 4.8. In order to consider the next case we introduce the following notation. For a given p such that 1 < p ≤ pY , let (4.11) κp := inf |∇Bv|2 + SB dvgB , where Hp := v ∈ H1(B) : |v|p+1dvgB = 1 Now, we consider the following two cases. (1 < p < pY ): In this case by adapting [42, Theorem 1.3], there ex- ists up ∈ C∞>0(B) such that (βκp, up) is a solution of (4.10) and∫ up+1p dvgB = 1. (p = pY ): For this specific and important value, analogously to [42, §2], we distinguish three subcases along the study of our problem (4.10), in correspondence with the sign(κpY ). κpY = 0: in this case, there exists upY ∈ C∞>0(B) such that (0, upY ) is a solution of (4.10) and upY +1pY dvgB = 1. κpY < 0: here there exists upY ∈ C∞>0(B) such that (βκpY , upY ) is a solution of (4.10) and upY +1pY dvgB = 1. κpY > 0: this is a more difficult case, let Km be the sharp Eu- clidean Sobolev constant (4.12) Km = m(m− 2)ω where ωm is the volume of the unit m−sphere. Thus, if (4.13) κpY < then there exists upY ∈ C∞>0(B) such that (βκpY , upY ) is a solu- tion of (4.10) and upY +1pY dvgB = 1. Furthermore, the condi- (4.14) κpY ≤ 16 FERNANDO DOBARRO & BÜLENT ÜNAL is sharp by [42], so that this is independent of the underlying manifold and the potential considered. The equality case in (4.14) is discussed in [44]. This results allow to establish the following two theorems. Theorem 4.9. (Cases : µpY < µ < µsc or 0 < µ) There exists ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is a constant λ if and only if sign(λ) = sign(κp) where p := 2µα+1 and κp is given by (4.11). Furthermore if λ < 0, then the solution ψ is unique. Proof. The conditions (µpY < µ < µsc or 0 < µ) imply that 1 < p := 2µα+ 1 < pY , i.e. the problem (4.9) is superlinear but subcritical with respect to the Sobolev immersion theorem (see [29, Remark 5.5]). By recalling that ψ = uα, it is sufficient to prove that follows. Let up be defined as in the case of (1 < p < pY ) in Remark 4.8. If (λ, u) is a solution of (4.9), then multiplying (4.9) by up and integrating by parts there results (4.15) βκp upudvgB = λ pdvgB . Thus sign(λ) = sign(κp) since β, up and u are all positive. Conversely, if λ is a real constant such that sign(λ) = sign(κp) 6= 0, then by Remark 4.5, (λ, uλ) is a solution of (4.9), where uλ = tλup and On the other side, if λ = κp = 0, then (0, up) is a solution of (4.9). Since 1 < p, the uniqueness for λ < 0 is a consequence of Corollary 4.2. � Theorem 4.10. (Cases : µ = µpY ) If there exists ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµpY ,ψ] F is a constant λ, then sign(λ) = sign(κpY ). Furthermore, if λ ∈ R verifying sign(λ) = sign(κpY ) and (4.13), then there exists ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµpY ,ψ] F is λ. Besides, if λ ∈ R is negative, then there exists at most one ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµpY ,ψ] F is λ. Proof. The proof is similar to that of Theorem 4.9, but follows from the application of the case of (p = pY ) in Remark 4.8. Like above, the uniqueness of λ < 0 is a consequence of Corollary 4.2. � In the next proposition including the supercritical case, we will apply the following result (see also [73, p.99]). ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 17 Lemma 4.11. Let (Nn, gN ) be a compact connected Riemannian manifold without boundary of dimension n ≥ 2 and ∆gN be the corresponding Laplace- Beltrami operator. Consider the equation of the form (4.16) −∆gNu = f(·, u), u ∈ C∞>0(N) where f ∈ C∞(N × R>0). If there exist a0 and a1 ∈ R>0 such that (4.17) u < a0 ⇒ f(·, u) > 0 u > a1 ⇒ f(·, u) < 0, then (4.16) has a solution satisfying a0 ≤ u ≤ a1. Proposition 4.12. (Cases : −∞ < µ < µsc or 0 < µ) If maxSB < 0, then for all λ < 0 there exists ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is the constant λ. Furthermore, the solution ψ is unique. Proof. The conditions (−∞ < µ < µsc or 0 < µ) imply that 1 < p := 2µα+ 1. On the other hand, since B is compact, by taking f(., u) = −SB(·)u+ λup = (−SB + λup−1)u, we obtain that limu−→0+ f(·, u) = 0+ and limu−→+∞ f(·, u) = −∞. Thus (4.17) is verified. Hence, the proposition is proved by applying Lemma 4.11 on (Bm, gB). No- tice that a0 can take positive values and eventually gets close enough to 0 due to the condition of limu−→0+ f(·, u), and consequently the corresponding solution results positive. Again, since λ < 0 and 1 < p the uniqueness is a consequence of Corollary 4.2. � Proof. (of Theorem 2.5) This is an immediate consequence of the above results. � • The case of a fiber with negative constant scalar curvature, i.e. SF < 0. Here, the (Pb-sc) becomes equivalent to the study of the existence for the problem (4.18) −β∆Bu+ SBu = λup − SFuq, u ∈ C∞>0(B), where λ is a real parameter (i.e., the searched constant scalar curvature), ψ = uα, p = 2µα+ 1 and q = 2(µ − 1)α + 1. Remark 4.13. Let u be a solution of (4.18). 18 FERNANDO DOBARRO & BÜLENT ÜNAL (i) If λ1 ≤ 0, then λ < 0. Indeed, multiplying the equation in (4.18) by u1 and integrating by parts there results: (4.19) λ1 u1udvgB + SF qdvgB = λ pdvgB , where u1 and u are positive. (ii) If λ = 0, then λ1 > 0. (iii) If µ = 0 (the warped product case), then λ < λ1. These cases have been studied in [27, 24]. (iv) If µ = 1 (the Yamabe problem for the usual product with conformal factor in C∞>0(B)), there results sign(λ) = sign(λ1 + SF ). An immediate consequence of Remark 4.13 is the following lemma. Lemma 4.14. Let B and F be given like in Theorem 2.3(i). Suppose further that B is a compact connected Riemannian manifold and F is a pseudo- Riemannian manifold of constant scalar curvature SF < 0. If λ ≥ 0 and λ1 ≤ 0 (for instance when SB ≤ 0 on B), then there is no ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is λ. Theorem 4.15. [29, Rows 6 and 8 in Table 4] Under the hypothesis of Theorem 2.3(i), let B be a compact connected Riemannian manifold and F be a pseudo-Riemannian manifold of constant scalar curvature SF < 0. Suppose that “(m,k) ∈ D and µ ∈ (0, 1)” or “(m,k) ∈ CD and µ ∈ (0, 1) ∩ C[µ−, µ+]”. (1) If λ1 ≤ 0, then λ ∈ R is the scalar curvature of a B ×[ψµ,ψ] F if and only if λ < 0. (2) If λ1 > 0, then there exists Λ ∈ R>0 such that λ ∈ R \ {Λ} is the scalar curvature of a B ×[ψµ,ψ] F if and only if λ < Λ. Furthermore if λ ≤ 0, then there exists at most one ψ ∈ C∞>0(B) such that B ×[ψµ,ψ] F has scalar curvature λ. Proof. The proof of this theorem is the subject matter of §5. � Once again we make use of Lemma 4.11 for the next theorem about the singular case and the following propositions. Theorem 4.16. [29, Row 7 Table 4] Under the hypothesis of Theorem 2.3(i), let B be a compact connected Riemannian manifold and F be a pseudo- Riemannian manifold of constant scalar curvature SF < 0. Suppose that “(m,k) ∈ CD and µ ∈ (0, 1) ∩ (µ−, µ+)”, then for any λ < 0 there exists ψ ∈ C∞>0(B) such that the scalar curvature of B×[ψµ,ψ] F is λ. Furthermore the solution ψ is unique. ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 19 Proof. First of all note that the conditions “(m,k) ∈ CD and µ ∈ (0, 1) ∩ (µ−, µ+)” imply that q < 0 and 1 < p , i.e. the problem (4.18) is superlinear in p but singular in q. On the other hand, since B is compact, taking f(., u) = −SB(·)u+ λup − SFuq = [(−SB(·) + λup−1)u1−q − SF ]uq, there result limu−→0+ f(·, u) = +∞ and limu−→+∞ f(·, u) = −∞. Thus (4.17) is verified. Thus by an application of Lemma 4.11 for (Bm, gB), we conclude the proof for the existence part. The uniqueness part just follows from Corollary 4.2. � Remark 4.17. We observe that the arguments applied in the proof of The- orem 4.16 can be adjusted to the case of a compact connected Riemannian manifold B with 0 ≤ q < 1 < p, λ < 0 and SF < 0, so that some of the situations included in Theorem 4.15. However, both argumentations are compatible but different. Proof. (of Theorem 2.6) This is an immediate consequence of the above results. � The approach in the next propositions is similar to Proposition 4.12 and Theorem 4.16. Proposition 4.18. [29, Row 10 Table 4] Let 1 < µ < +∞. If maxSB < 0, then for all λ < 0 there exists ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is the constant λ. Proof. The condition 1 < µ < +∞ implies that 1 < q < p. On the other hand, since B is compact, taking f(., u) = −SB(·)u+ λup − SFuq = [−SB(·) + (λup−q − SF )uq−1]u, there result limu−→0+ f(·, u) = 0+ and limu−→+∞ f(·, u) = −∞. Thus (4.17) is satisfied. Thus an elementary application of Lemma 4.11 for (Bm, gB) proves the proposition. � Proposition 4.19. [29, Rows 2, 4 and 3 in Table 4] Let either “(m,k) ∈ D and µ ∈ (µsc, 0)” or “(m,k) ∈ CD and µ ∈ (µsc, 0) ∩ C[µ−, µ+]” or “(m,k) ∈ CD and µ ∈ (µsc, 0)∩ (µ−, µ+)”. If minSB > 0, then for all λ ≤ 0 there exists a smooth function ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is the constant λ. Proof. If either “(m,k) ∈ D and µ ∈ (µsc, 0)” or “(m,k) ∈ CD and µ ∈ (µsc, 0) ∩ C[µ−, µ+]”, then 0 < q < p < 1. 20 FERNANDO DOBARRO & BÜLENT ÜNAL On the other hand, since B is compact, taking f(., u) = −SB(·)u+ λup − SFuq = [−SB(·)u1−q + λup−q − SF ]uq, there result limu−→0+ f(·, u) = 0+ and limu−→+∞ f(·, u) = −∞. Thus (4.17) is verified and again we can apply Lemma 4.11 for (Bm, gB). If “(m,k) ∈ CD and µ ∈ (µsc, 0) ∩ (µ−, µ+)”, then q < 0 < p < 1. Con- sidering the limits as above, limu−→0+ f(·, u) = +∞ and limu−→+∞ f(·, u) = −∞. So, an application of Lemma 4.11 concludes the proof. � Remark 4.20. Notice that in Theorems 4.15 and 4.16 we do not assume hypothesis related to the sign of SB(·), unlike in Propositions 4.12, 4.18 and 4.19. Proposition 4.21. [29, Rows 5 and 9 in Table 4] Let (m,k) ∈ CD be. (1) If either “µ ∈ m− 1 , 0 ∩ {µ−, µ+} and minSB > 0” or “µ ∈ (0, 1) ∩ {µ−, µ+}”, then for all λ < 0 there exists a smooth function ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is the constant λ. In the second case, ψ is also unique . (2) If either “µ ∈ m− 1 , 0 ∩ {µ−, µ+}” or “µ ∈ (0, 1) ∩ {µ−, µ+}” and furthermore λ1 > 0, then there exists a smooth function ψ ∈ C∞>0(B) such that the scalar curvature of B ×[ψµ,ψ] F is 0. Proof. In both cases q = 0, so by considering f(., u) = −SB(·)u+ λup − SF , the proof of (1) follows as in the latter propositions, while that of (2) is a consequence of the linear theory and the maximum principle. � Remark 4.22. Finally, we observe a particular result about the cases studied in [27]. If µ = 0, then p = 1 and q = 1−2α = k − 3 k + 1 . When the dimension of the fiber is k = 2, the exponent q = −1 . So, writing the involved equation ∆Bu = f(., u) = −SB(·)u + λu− SFu− and by applying Lemma 4.11 as above, we obtain that if λ < minSB, then there exists a smooth function ψ ∈ C∞>0(B) such that the scalar curvature of B ×ψ F is the constant λ. Furthermore, by Corollary 4.2 such ψ is unique (see [27, 24] and [25]). ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 21 5. Proof of the Theorem 4.15 The subject matter of this section is the proof of the Theorem 4.15, so we naturally assume its hypothesis. Most of the time, we need to specify the dependence of λ of (4.18), we will do that by writing (4.18)λ. Furthermore, we will denote the right hand side of (4.18)λ by fλ(t) := λt p − SF tq. The conditions either “(m,k) ∈ D and µ ∈ (0, 1)” or “(m,k) ∈ CD and µ ∈ (0, 1) ∩ C[µ−, µ+]”, imply that 0 < q < 1 < p. But the type of nonlinearity in the right hand side of (4.18)λ changes with the signλ, i.e. it is purely concave for λ < 0 and concave-convex for λ > 0. The uniqueness for λ ≤ 0 is again a consequence of Corollary 4.2. In order to prove the existence of a solution for (4.18)λ with signλ 6= 0, we adapt the approach of sub and upper solutions in [5]. Thus, the proof of Theorem 4.15 will be an immediate consequence of the results that follows. Lemma 5.1. (4.18)0 has a solution if and only if λ1 > 0. Proof. This situation is included in the results of the second case of Theorem 4.7 by replacing −SF with λ (see [24, Proposition 3.1]). � Lemma 5.2. Let us assume that {λ : (4.18)λ has a solution} is non-empty and define (5.1) Λ = sup{λ : (4.18)λ has a solution}. (i) If λ1 ≤ 0, then Λ ≤ 0. (ii) If λ1 > 0, then there exists λ > 0 finite such that Λ ≤ λ. Proof. (i) It is sufficient to observe Remark 4.13 i. (ii) Like in [5], let λ > 0 such that (5.2) λ1t < λt p − SF tq,∀t ∈ R, t > 0. Thus, if (λ, u) is a solution of (4.18)λ, then p − SF λ1u1u < λ p − SF so λ < λ. Lemma 5.3. Let (5.3) Λ = sup{λ : (4.18)λ has a solution}. 22 FERNANDO DOBARRO & BÜLENT ÜNAL Figure 1. The nonlinearity fλ in Lemma 5.3, i.e. 0 < q < 1 < p, SF < 0, λ1 > 0, λ > 0. (i) Let E ∈ R>0. There exist 0 < λ0 = λ0(E) and 0 < M = M(E,λ0) such that ∀λ : 0 < λ ≤ λ0, so we have (5.4) 0 < E fλ(EM) (ii) If λ1 > 0, then {λ > 0 : (4.18)λ has a solution} 6= ∅. As a conse- quence of that, Λ is finite. (iii) If λ1 > 0, then for all 0 < λ < Λ there exists a solution of the problem (4.18)λ. Proof. (i) For any 0 < λ < λ0 0 < gλ(r) := E fλ(Er) = Erq−1(λEp−1rp−q − SFEq−1) < Erq−1(λ0E p−1rp−q − SFEq−1). It is easy to see that q − 1 p−q 1 ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 23 is a minimum point for gλ0 and gλ0(r0) = E q − 1 ) q−1 q − 1 p− 1 − 1 → 0+, as λ0 → 0+. Hence there exist 0 < λ0 = λ0(E) and 0 < M =M(E,λ0) such that (5.4) is verified. (ii) Since λ1 > 0, by the maximum principle, there exists a solution e ∈ C∞>0(B) of (5.5) LB(e) = −β∆Be+ SBe = 1. Then, applying item (i) above with E = ‖e‖∞ there exists 0 < λ0 = λ0(‖e‖∞) and 0 < M = M(‖e‖∞, λ0) such that ∀λ with 0 < λ ≤ λ0 we have that (5.6) LB(Me) =M ≥ fλ(Me), hence Me is a supersolution of (4.18)λ. On the other hand, since ǔ1 := inf u1 > 0, for all λ > 0 (5.7) ǫ−1fλ(ǫǔ1) = ǫ q−1[λǫp−qǔ 1 − SF ǔ 1] → +∞, as ǫ→ 0+. Furthermore, note that fλ is nondecreasing when λ > 0. Hence for any 0 < λ there exists a small enough 0 < ǫ verifying (5.8) LB(ǫu1) = ǫλ1u1 ≤ ǫλ1‖u1‖∞ ≤ fλ(ǫǔ1) ≤ fλ(ǫu1), thus ǫu1 is a subsolution of (4.18)λ. Then for any 0 < λ < λ0, (taking eventually 0 < ǫ smaller if necessary), we have that the above constructed couple sub super solution satisfies (5.9) ǫu1 < Me. Now, by applying the monotone iteration scheme, we have that {λ > 0 : (4.18)λ has a solution} 6= ∅. Furthermore by Lemma 5.2 (ii) there results Λ is finite. (iii) The proof of this item is completely analogous to Lemma 3.2 in [5]. We will rewrite this to be self contained. Given λ < Λ, let uν be a solution of (4.18)ν with λ < ν < Λ. Then uν is a supersolution of (4.18)λ and for small enough 0 < ǫ, the subsolution ǫu1 of (4.18)λ verifies ǫu1 < uν , then as above (4.18)λ has a solution. Lemma 5.4. For any λ < 0, there exists γλ > 0 such that ‖u‖∞ ≤ γλ for any solution u of (4.18)λ. Furthermore if SB is nonnegative, then positive zero of fλ can be choose as γλ. 24 FERNANDO DOBARRO & BÜLENT ÜNAL fΛHtL fΛHtL+Νt Figure 2. The nonlinearity in Lemma 5.5 , i.e. 0 < q < 1 < p, SF < 0, λ1 > 0, λ < 0. Proof. Define ŠB := minSB (recall that B is compact). There are two different situations, namely. • 0 ≤ ŠB: since there exists x1 ∈ B such that u(x1) = ‖u‖∞ and 0 ≤ −β∆Bu(x1) = −SB(x1)‖u‖∞ + λ‖u‖p∞ − SF‖u‖q∞, there results ‖u‖∞ ≤ γλ, where γλ is the strictly positive zero of fλ. • ŠB < 0: we consider f̃λ(t) := λtp − SF tq − ŠBt. Now, our problem (4.18)λ is equivalent to −β∆Bu+ (SB − ŠB)u = f̃λ(u), u ∈ C∞>0(B). But here the potential of (SB− ŠB) is non negative and the function f̃λ has the same behavior of fλ with a positive zero γ̃λ on the right side of the positive zero γλ of fλ. Thus, repeating the argument for the case of ŠB ≥ 0, we proved ‖u‖∞ ≤ γ̃λ. Lemma 5.5. Let λ1 > 0. Then for all λ < 0 there exists a solution of (4.18)λ. Proof. We will apply again the monotone iteration scheme. Define ŠB := minSB (note that B is compact). ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 25 • 0 ≤ ŠB: Clearly, the strictly positive zero γλ of fλ is a supersolution (5.10) − β∆Bu+ (SB + ν)u = fλ(u) + νu, for all ν ∈ R. On the other hand, for 0 < ǫ = ǫ(λ) small enough, (5.11) LB(ǫu1) = ǫλ1u1 ≤ fλ(ǫu1). Then ǫu1 is a subsolution of (5.10) for all ν ∈ R. By taking ε possibly smaller, we also have (5.12) 0 < ǫu1 < γλ. We note that for large enough values of ν ∈ R>0, the nonlinearity on the right hand side of (5.10), namely fλ(t) + νt, is an increasing function on [0, γλ]. Thus applying the monotone iteration scheme we obtain a strictly positive solution of (5.10), and hence a solution of (4.18)λ (see [3], [4], [54]). • ŠB < 0: In this case, like in Lemma 5.4 we consider f̃λ(t) := λtp − q − ŠBt. Then, the problem (4.18)λ is equivalent to (5.13) −β∆Bu+ (SB − ŠB)u = f̃λ(u), u ∈ C∞>0(B), where the potential is nonnegative and the function f̃λ has a similar behavior to fλ with a positive zero γ̃λ on the right side of the positive zero γλ of fλ. Here, it is clear that γ̃λ is a positive supersolution of (5.14) − β∆Bu+ (SB − ŠB + ν)u = f̃λ(u) + νu, for all ν ∈ R. Hence, we complete the proof similarly to the case of ŠB ≥ 0. Lemma 5.6. Let λ1 ≤ 0, λ < 0, ŠB := minSB and also let γλ be a positive zero of fλ and γ̃λ be a positive zero of f̃λ := fλ − ŠBidR≥0 . Then there exists a solution u of (4.18)λ. Furthermore any solution of (4.18)λ satisfies γλ ≤ ‖u‖∞ ≤ γ̃λ. Proof. First of all we observe that if SB ≡ 0 (so λ1 = 0), then u ≡ γλ is the searched solution of (4.18)λ. Now, we assume that SB 6≡ 0. Since λ1 ≤ 0, there results ŠB < 0. In this case, one can notice that 0 < γλ < γ̃λ. 26 FERNANDO DOBARRO & BÜLENT ÜNAL On the other hand, the problem (4.18)λ is equivalent to (5.15) −β∆Bu+ (SB − ŠB)u = f̃λ(u), u ∈ C∞>0(B). By the second part of the proof of Lemma 5.4, if u is a solution of (4.18)λ (or equivalently (5.15)), then ‖u‖∞ ≤ γ̃λ. Besides, since u1(fλ ◦ u) = λ1 u, u1 > 0 and λ1 ≤ 0 results γλ ≤ ‖u‖∞. From this point on, the proof of the existence of solutions for (5.15) follows the lines of the second part of Lemma 5.5. � 6. Conclusions and future directions Now, we would like to summarize the content of the paper and to propose our future plans on this topic. We remark to the reader that several computations and proofs, along with other complementary results mentioned in this article and references can be obtained in [29]. We have chosen this procedure to avoid the involved long computations. In brief, we introduced and studied curvature properties of a particular family of warped products of two pseudo-Riemannian manifolds which we called as a base conformal warped product. Roughly speaking the metric of such a product is a mixture of a conformal metric on the base and a warped metric. We concentrated our attention on a special subclass of this structure, where there is a specific relation between the conformal factor c and the warping function w, namely c = wµ with µ a real parameter. As we mentioned in §1 and the first part of §2, these kinds of metrics and considerations about their curvatures are very frequent in different physi- cal areas, for instance theory of general relativity, extra-dimension theories (Kaluza-Klein, Randall-Sundrum), string and super-gravity theories; also in global analysis for example in the study of the spectrum of Laplace-Beltrami operators on p-forms, etc. More precisely, in Theorems 3.1 and 3.2, we obtained the classical relations among the different involved Ricci tensors (respectively, scalar curvatures) for metrics of the form c2gB⊕w2gF . Then the study of particular families of either scalar or tensorial nonlinear partial differential operators on pseudo- Riemannian manifolds (see Lemmas 3.3 and 3.7) allowed us to find reduced expressions of the Ricci tensor and scalar curvature for metrics as above with c = wµ, where µ a real parameter (see Theorems 2.2 and 2.3). The operated reductions can be considered as generalizations of those used by Yamabe in [79] in order to obtain the transformation law of the scalar curvature under ABOUT CURVATURE, CONFORMAL METRICS AND WARPED PRODUCTS 27 a conformal change in the metric and those used in [27] with the aim to obtain a suitable relation among the involved scalar curvatures in a singly warped product (see also [52] for other particular application and our study on multiply warped products in [28]). In §4 and 5, under the hypothesis that (B, gB) be a “compact” and con- nected Riemannian manifold of dimension m ≥ 3 and (F, gF ) be a pseudo - Riemannian manifold of dimension k ≥ 0 with constant scalar curvature SF , we dealt with the problem (Pb-sc). This question leads us to ana- lyze the existence and uniqueness of solutions for nonlinear elliptic partial differential equations with several kinds of nonlinearities. The type of non- linearity changes with the value of the real parameter µ and the sign of SF . In this article, we concentrated our attention to the cases of constant scalar curvature SF ≤ 0 and accordingly the central results are Theorems 2.5 and 2.6. Although our results are partial so that there are more cases to study in forthcoming works, we obtained also other complementary results under more restricted hypothesis about the sign of the scalar curvature of the base. Throughout our study, we meet several types of partial differential equa- tions. Among them, most important ones are those with concave-convex nonlinearities and the one so called Lichnerowicz-York equation. About the former, we deal with the existence of solutions and leave the question of multiplicity of solutions to a forthcoming study. We observe that the previous problems as well as the study of the Ein- stein equation on base conformal warped products, (ψ, µ)-bcwp’s and their generalizations to multi-fiber cases, give rise to a reach family of interesting problems in differential geometry and physics (see for instance, the several recent works of R. Argurio, J. P. Gauntlett, M. O. Katanaev, H. Kodama, J. Maldacena, H. -J. Schmidt, A. Strominger, K. Uzawa, P. S. Wesson among many others) and in nonlinear analysis (see the different works of A. Am- brosetti, T. Aubin, I. Choquet-Bruat, J. Escobar, E. Hebey, J. Isenberg, A. Malchiodi, D. Pollack, R. Schoen, S. -T. Yau among others). Appendix A. Let us assume the hypothesis of Theorem 2.3 (i), the dimensions of the base m ≥ 2 and of the fiber k ≥ 1. In order to describe the classification of the type of nonlinearities involved in (2.11), we will introduce some notation (for a complete study of these nonlinearities see [29, Section 5]). The example in Figure 1 will help the reader to clarify the notation. Note that the denominator in (2.12) is (A.1) η := (m− 1)(m − 2)µ2 + 2(m− 2)kµ + (k + 1)k 28 FERNANDO DOBARRO & BÜLENT ÜNAL and verifies η > 0 for all µ ∈ R. Thus α in (2.12) is positive if and only if µ > − k m− 1 and by the hypothesis µ 6= − m− 1 in Theorem 2.3 (i), results α 6= 0. We now introduce the following notation: (A.2) p = p(m,k, µ) = 2µα+ 1 and q = q(m,k, µ) = 2(µ− 1)α + 1 = p− 2α, where α is defined by (2.12). Thus, for all m,k, µ given as above, p is positive. Indeed, by (A.1), p > 0 if and only if ̟ > 0, where ̟ := ̟(m,k, µ) := 4µ[k + (m− 1)µ] + (m− 1)(m− 2)µ2 + 2(m− 2)kµ + (k + 1)k = (m− 1)(m+ 2)µ2 + 2mkµ + (k + 1)k. But discr (̟) ≤ −4km2 ≤ −16 and m > 1, so ̟ > 0. Unlike p, q changes sign depending on m and k. Furthermore, it is im- portant to determine the position of p and q with respect to 1 as a function of m and k. In order to do that, we define (A.3) D := {(m,k) ∈ N≥2 × N≥1 : discr (̺(m,k, ·)) < 0}, where N≥l := {j ∈ N : j ≥ l} and ̺ := ̺(m,k, µ) := 4(µ − 1)[k + (m− 1)µ] + (m− 1)(m− 2)µ2 + 2(m− 2)kµ + (k + 1)k = (m− 1)(m + 2)µ2 + 2(mk − 2(m− 1))µ + (k − 3)k. Note that by (A.1), q > 0 if and only if ̺ > 0. Furthermore q = 0 if and only if ̺ = 0. But here discr (̺(m,k, ·)) changes its sign as a function of m and k. We adopt here the notation in [29, Table 4] below, namely CD = (N≥2× N≥1)\D if D ⊆ N≥2×N≥1 and CI = R\I if I ⊆ R. Thus, if (m,k) ∈ CD, let µ− and µ+ two roots (eventually one, see [29, Remark 5.3]) of q, µ− ≤ µ+. Besides, if discr (̺(m,k, ·)) > 0, then µ− < 0; whereas µ+ can take any sign. References [1] O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Large N Field Theories, String Theory and Gravity, Physics Reports 323 (2000), 183-386 [arXiv:hep-th/9905111]. [2] S. 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[74] K. Thorne, Warping spacetime, The Future of Theoretical Physics and Cosmology, Part 5, Cambridge University Press (2003), 74-104. [75] N. Trudinger, Remarks concerning the conformal deformation of Rieamnnian struc- tures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 (1968), 265-274. [76] P. S. Wesson, Space-Time-Matter, Modern Kaluza-Klein Theory, World Scientific (1999). [77] P. S. Wesson, On Higher-Dimensional Dynamics, arXiv:gr-qc/0105059. [78] M. Willem, Minimax Theorems, Birkhäuser, Boston (1996). [79] H. Yamabe On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37. (F. Dobarro) Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, Via Valerio 12/b, I-34127 Trieste, Italy E-mail address: [email protected] (B. Ünal) Department of Mathematics, Bilkent University, Bilkent, 06800 Ankara, Turkey E-mail address: [email protected] http://arxiv.org/abs/hep-th/9905221 http://arxiv.org/abs/hep-th/9906064 http://arxiv.org/abs/hep-th/0407176 http://arxiv.org/abs/gr-qc/9709071 http://arxiv.org/abs/gr-qc/0407095 http://arxiv.org/abs/physics/9905030 http://arxiv.org/abs/hep-th/0202016 http://arxiv.org/abs/gr-qc/0105059 1. Introduction 2. Motivations and Main results 3. The curvature relations - Sketch of the proofs 4. The problem (Pb-sc) - Existence of solutions 5. Proof of the Theorem ?? 6. Conclusions and future directions Appendix A. References

0704.0596

The local structure of conformally symmetric manifolds

THE LOCAL STRUCTURE OF CONFORMALLY SYMMETRIC MANIFOLDS ANDRZEJ DERDZINSKI AND WITOLD ROTER Abstract. This is a final step in a local classification of pseudo-Riemannian man- ifolds with parallel Weyl tensor that are not conformally flat or locally symmetric. Introduction The present paper provides a finishing touch in a local classification of essentially conformally symmetric pseudo-Riemannian metrics. A pseudo-Riemannian manifold of dimension n ≥ 4 is called essentially conformal- ly symmetric if it is conformally symmetric [2] (in the sense that its Weyl conformal tensor is parallel) without being conformally flat or locally symmetric. The metric of an essentially conformally symmetric manifold is always indefinite [4, Theorem 2]. Compact essentially conformally symmetric manifolds are known to exist in all dimensions n ≥ 5 with n ≡ 5 (mod 3), where they represent all in- definite metric signatures [8], while examples of essentially conformally symmetric pseudo-Riemannian metrics on open manifolds of all dimensions n ≥ 4 were first constructed in [16]. On every conformally symmetric manifold there is a naturally distinguished parallel distribution D, of some dimension d, which we call the Olszak distribution. As shown by Olszak [13], for an essentially conformally symmetric manifold d ∈ {1, 2}. In [7] we described the local structure of all conformally symmetric manifolds with d = 2. See also Section 3. This paper establishes an analogous result (Theorem 4.1) for the case d = 1. In both cases, some of the metrics in question are locally symmetric. In Remark 4.2 we explain why a similar classification result cannot be valid just for essentially con- formally symmetric manifolds. Essentially conformally symmetric manifolds with d = 1 are all Ricci-recurrent, in the sense that, for every tangent vector field v, the Ricci tensor ρ and the covariant derivative ∇vρ are linearly dependent at each point. The local structure of essentially conformally symmetric Ricci-recurrent manifolds at points with ρ ⊗∇ρ 6= 0 has already been determined by the second author [16]. Our new contribution settles the 2000 Mathematics Subject Classification. 53B30. Key words and phrases. Parallel Weyl tensor, conformally symmetric manifold. http://arxiv.org/abs/0704.0596v1 2 A. DERDZINSKI AND W. ROTER one case still left open in the local classification problem, namely, that of essentially conformally symmetric manifolds with d = 1 at points where ρ⊗∇ρ = 0. The literature dealing with conformally symmetric manifolds includes, among oth- ers, [9, 10, 12, 15, 17, 18] and the papers cited above. A local classification of homo- geneous essentially conformally symmetric manifolds can be found in [3]. 1. Preliminaries Throughout this paper, all manifolds and bundles, along with sections and connec- tions, are assumed to be of class C∞. A manifold is, by definition, connected. Unless stated otherwise, a mapping is always a C∞ mapping betweeen manifolds. Given a connection ∇ in a vector bundle E over a manifold M , a section ψ of E , and vector fields u, v tangent to M , we use the sign convention (1) R(u, v)ψ = ∇v∇uψ − ∇u∇vψ + ∇[u,v]ψ for the curvature tensor R = R∇. The Levi-Civita connection of a given pseudo-Riemannian manifold (M, g) is al- ways denoted by ∇. We also use the symbol ∇ for connections induced by ∇, in various ∇-parallel subbundles of TM and their quotients. The Schouten tensor σ and Weyl conformal tensor W of a pseudo-Riemannian manifold (M, g) of dimension n ≥ 4 are given by σ = ρ − (2n − 2)−1 sg, with ρ denoting the Ricci tensor, s = trgρ standing for the scalar curvature, and (2) W = R − (n− 2)−1g ∧ σ. Here ∧ is the exterior multiplication of 1-forms valued in 1-forms, which uses the ordinary ∧ as the valuewise multiplication; thus, g∧σ is a 2-form valued in 2-forms. Let (t, s) 7→ x(s, t) be a fixed variation of curves in a pseudo-Riemannian manifold (M, g), that is, an M-valued C∞ mapping from a rectangle (product of intervals) in the ts-plane. By a vector field w along the variation we mean, as usual, a section of the pullback of TM to the rectangle (so that w(t, s) ∈ Tx(t,s)M). Examples are xs and xt, which assign to (t, s) the velocity of the curve t 7→ x(t, s) or s 7→ x(t, s) at s or t. Further examples are provided by restrictions to the variation of vector fields on M . The partial covariant derivatives of a vector field w along the variation are the vector fields wt, ws along the variation, obtained by differentiating w covariantly along the curves t 7→ x(t, s) or s 7→ x(t, s). Skipping parentheses, we write wts, wstt, etc., rather than (wt)s, ((ws)t)t for higher-order derivatives, as well as xss, xst instead of (xs)s, (xs)t. One always has wts = wst + R(xt, xs)w, cf. [11, formula (5.29) on p. 460], and, since the Levi-Civita connection ∇ is torsionfree, xst = xts. Thus, whenever (t, s) 7→ x(s, t) is a variation of curves in M , (3) xtss = xsst + R(xt, xs)xs . CONFORMALLY SYMMETRIC MANIFOLDS 3 2. The Olszak distribution The Olszak distribution of a conformally symmetric manifold (M, g) is the parallel subbundle D of TM , the sections of which are the vector fields u with the property that ξ∧Ω = 0 for all vector fields v, v ′ and for the differential forms ξ = g(u, · ) and Ω =W (v, v ′, · , · ). The distribution D was introduced, in a more general situation, by Olszak [13], who also proved the following lemma. Lemma 2.1. The following conclusions hold for the dimension d of the Olszak distribution D in any conformally symmetric manifold (M, g) with dimM = n ≥ 4. (i) d ∈ {0, 1, 2, n}, and d = n if and only if (M, g) is conformally flat. (ii) d ∈ {1, 2} if (M, g) is essentially conformally symmetric. (iii) d = 2 if and only if rankW=1, in the sense that W, as an operator acting on exterior 2-forms, has rank 1 at each point. (iv) If d = 2, the distribution D is spanned by all vector fields of the form W (u, v)v′ for arbitrary vector fields u, v, v′ on M . Proof. See Appendix I. � In the next lemma, parts (a) and (d) are due to Olszak [13, 2o and 3o on p. 214]. Lemma 2.2. If d ∈ {1, 2}, where d is the dimension of the Olszak distribution D of a given conformally symmetric manifold (M, g) with dimM = n ≥ 4, then (a) D is a null parallel distribution, (b) at any x ∈M the space Dx contains the image of the Ricci tensor ρx treated, with the aid of gx, as an endomorphism of TxM, (c) the scalar curvature is identically zero and R = W + (n− 2)−1g ∧ ρ, (d) W (u, · , · , · ) = 0 whenever u is a section of D, (e) R(v, v ′, · , · ) = W (v, v ′, · , · ) = 0 for any sections v and v ′ of D⊥, (f) of the connections in D and E = D⊥/D, induced by the Levi-Civita connec- tion of g, the latter is always flat, and the former is flat if d = 1. Proof. Assertion (e) for W is immediate from the definition of D. Namely, at any point x ∈M , every 2-form Ωx in the image of Wx (for Wx acting on 2-forms at x) is ∧-divisible by ξ = gx(u, · ) for each u ∈ Dxr{0}, and so Ωx(v, v ′) = 0 if v, v ′ ∈ Dx We now proceed to prove (a), (b), (c) and (d). First, let d = 2. By Lemma 2.1(iii), this amounts to the condition rankW=1, so that (a), (b) and (c) follow from Lemma 2.1(iv) combined with [7, Lemma 17.1(ii) and Lemma 17.2]. Also, for a nonzero 2-form Ωx chosen as in the last paragraph, Dx is the image of Ωx, that is, Ωx equals the exterior product of two vectors in Dx (treated as 1-forms, with the aid of gx). Now (d) follows since, by (a), Ωx(ux, · ) = 0 if u is a section of D. 4 A. DERDZINSKI AND W. ROTER Next, suppose that d = 1. Replacing M by a neighborhood of any given point, we may assume that D is spanned by a vector field u. If u were not null, we would have W (u, v, u, v ′) = 0 for any sections v, v ′ of D⊥, as one sees contracting the twice-covariant tensor field W ( · , v, · , v ′) = 0, at any point x, in an orthogonal basis containing the vector ux. (We have already established (e) for W.) Combined with (e) for W and the symmetries of W, the relation W (u, v, u, v ′) = 0 for v, v ′ in D⊥ would then give W = 0, contrary to the assumption that d = 1. Thus, u is null, which yields (a). Now (4) we choose, locally, a null vector field u′ with g(u, u′) = 1. For any section v of D⊥ one sees that W (u, · , u′, v) = 0 by contracting the tensor field W ( · , · , · , v) = 0 in the first and third arguments, at any point x, in (5) a basis of TxM formed by ux, u x and n− 2 vectors orthogonal to them, and using (e) for W, along with the inclusion D ⊂ D⊥, cf. (a). Since u′ and D⊥ span TM , assertion (e) for W thus implies (d). To prove (b) and (c) when d = 1, we distinguish two cases: (M, g) is either es- sentially conformally symmetric, or locally symmetric. For (c), it suffices to establish vanishing of the scalar curvature s (cf. (2)). Now, in the former case, s = 0 accord- ing to [5, Theorem 7], while (b) follows since, as shown in [6, Theorem 7 on p. 18], for arbitrary vector fields v, v ′ and v ′′ on an essentially conformally symmetric pseu- do-Riemannian manifold, ξ ∧ Ω = 0, where ξ = ρ(v, · ) and Ω = W (v ′, v ′′, · , · ). In the case where g is locally symmetric, (b) and (c) are established in Appendix II. Assertion (e) for R is now obvious from (e) for W and (c), since, by (b), ρ(v, · ) = 0 for any section v of D⊥. The claim about E in (f) is in turn immediate from (1) and (e) for R, which states that R(w,w ′)v, for arbitrary vector fields w,w ′ and any section v of D⊥, is orthogonal to all sections of D⊥ (and hence must be a section of D). Finally, to prove (f) for D, with d = 1, let us fix a section u of D, a vector field v, and define a differential 2-form ζ by ζ(w,w ′) = (n−2)R(w,w ′, u, v) for any vector fields w,w ′. By (c) and (e), ζ = g(u, · )∧ ρ(v, · ), as D ⊂ D⊥ (cf. (a)), and so ρ(u, · ) = 0 in view of (b) and symmetry of ρ. However, by (b), both g(u, · ) and ρ(v, · ) are sections of the subbundle of T ∗M corresponding to D under the bundle isomorphism TM → T ∗M induced by g, so that ζ = 0 since the distribution D is one-dimensional. � 3. The case d = 2 For more details of the construction described below, we refer the reader to [7]. Let there be given a surface Σ, a projectively flat torsionfree connection D on Σ with a D-parallel area form α, an integer n ≥ 4, a sign factor ε = ±1, a real vector space V of dimension n− 4, and a pseudo-Euclidean inner product 〈 , 〉 on V . CONFORMALLY SYMMETRIC MANIFOLDS 5 We also assume the existence of a twice-contravariant symmetric tensor field T on Σ with divD(divDT ) + (ρD, T ) = ε (in coordinates: T jk,jk + T jkRjk = ε). Here divD denotes the D-divergence, ρD is the Ricci tensor of D, and ( , ) stands for the obvious pairing. Such T always exists locally in Σ. In fact, according to [7, Theorem 10.2(i)] combined with [7, Lemma 11.2], T exists whenever Σ is simply connected and noncompact. For T chosen as above, we define a twice-covariant symmetric tensor field τ on Σ, that is, a section of [T ∗Σ]⊙2, by requiring τ to correspond to the section T of [TΣ]⊙2 under the vector-bundle isomorphism TΣ → T ∗Σ which acts on vector fields v by v 7→ α(v, · ). In coordinates, τjk = αjlαkmT Next, we denote by hD the Patterson-Walker Riemann extension metric [14] on the total space T ∗Σ, obtained by requiring that all vertical and all D-horizontal vectors be hD-null, while hDx (ζ, w) = ζ(dπxw) for x ∈ T ∗Σ, any vector w ∈ TxT any vertical vector ζ ∈ Ker dπx = T Σ, and the bundle projection π : T ∗Σ → Σ. Finally, let γ and θ be the constant pseudo-Riemannian metric on V correspond- ing to the inner product 〈 , 〉, and the function V → R with θ(v) = 〈v, v〉. Our Σ,D, α, n, ε, V , 〈 , 〉 now give rise to the pseudo-Riemannian manifold (6) (T ∗Σ × V, hD− 2τ + γ − θρD) , of dimension n, with the metric hD− 2τ + γ − θρD, where the function θ and covariant tensor fields τ, ρD, hD, γ on Σ, T ∗Σ or V are identified with their pull- backs to T ∗Σ × V . (Thus, for instance, hD− 2τ + γ is a product metric.) We have the following local classification result, in which d stands for the dimen- sion of Olszak distribution D. Theorem 3.1. The pseudo-Riemannian manifold (6) obtained as above from any data Σ,D, α, n, ε, V , 〈 , 〉 with the stated properties is conformally symmetric and has d = 2. Conversely, in any conformally symmetric pseudo-Riemannian manifold such that d = 2, every point has a connected neighborhood isometric to an open subset of a manifold (6) constructed above from some data Σ, D, α, n, ε, V , 〈 , 〉. The manifold (6) is never conformally flat, and it is locally symmetric if and only if the Ricci tensor ρD is D-parallel. Proof. See [7, Section 22]. Note that, in view of Lemma 2.1(iii), the condition rankW=1 used in [7] is equivalent to d = 2. � The objects Σ,D, α, n, ε, V , 〈 , 〉 are treated as parameters of the above construc- tion, while T is merely assumed to exist, even though the metric g in (6) clearly depends on τ (and hence on T ). This is justified by the fact that, with fixed Σ,D, α, n, ε, V , 〈 , 〉, the metrics corresponding to two choices of T are, locally, iso- metric to each other, cf. [7, Remark 22.1]. 6 A. DERDZINSKI AND W. ROTER The metric signature of (6) is clearly given by −− . . .++, with the dots standing for the sign pattern of 〈 , 〉. 4. The case d = 1 Let there be given an open interval I, a C∞ function f : I → R, an integer n ≥ 4, a real vector space V of dimension n − 2 with a pseudo-Euclidean inner product 〈 , 〉, and a nonzero traceless linear operator A : V → V , self-adjoint relative to 〈 , 〉. As in [16], we then define an n-dimensional pseudo-Riemannian manifold (7) (I ×R× V, κ dt2 + dt ds + γ) , where products of differentials represent symmetric products, t, s denote the Carte- sian coordinates on the I × R factor, γ stands for the pullback to I × R × V of the flat pseudo-Riemannian metric on V that corresponds to the inner product 〈 , 〉, and the function κ : I ×R× V → R is given by κ(t, s, ψ) = f(t)〈ψ, ψ〉+ 〈Aψ, ψ〉. The manifolds (7) are characterized by the following local classification result, analogous to Theorem 3.1. As before, d is the dimension of the Olszak distribution. Theorem 4.1. For any I, f, n, V , 〈 , 〉, A as above, the pseudo-Riemannian man- ifold (7) is conformally symmetric and has d = 1. Conversely, in any conformally symmetric pseudo-Riemannian manifold such that d = 1, every point has a connected neighborhood isometric to an open subset of a manifold (7) constructed from some such I, f, n, V , 〈 , 〉, A. The manifold (7) is never conformally flat, and it is locally symmetric if and only if f is constant. A proof of Theorem 4.1 is given at the end of the next section. Obviously, the metric κ dt2+ dt ds + γ in (7) has the sign pattern − . . .+, where the dots stand for the sign pattern of 〈 , 〉. Remark 4.2. A classification result of the same format as Theorem 4.1 cannot be true just for essentially conformally symmetric manifolds with d = 1. Namely, such manifolds do not satisfy a principle of unique continuation: formula (7) with f which is nonconstant on I, but constant on some nonempty open subinterval I ′ of I, defines an essentially conformally symmetric manifold with a locally symmetric open submanifold U = I ′ ×R × V . At points of U, the local structure of (7) does not, therefore, arise from a construction that, locally, produces all essentially conformally symmetric manifolds and nothing else. As explained in [7, Section 24], an analogous situation arises when d = 2. 5. Proof of Theorem 4.1 The following assumptions will be used in Lemma 5.1. (a) (M, g) is a conformally symmetric manifold of dimension n ≥ 4 and y ∈M . CONFORMALLY SYMMETRIC MANIFOLDS 7 (b) The Olszak distribution D of (M, g) is one-dimensional. (c) u is a global parallel vector field spanning D. (d) t :M → R is a C∞ function with g(u, · ) = dt and t(y) = 0. (e) dim V = n− 2 for the space V of all parallel sections of E = D⊥/D. (f) ρ = (2−n)f(t) dt⊗dt for some C∞ function f : I ′ → R on an open interval I ′, where ρ is the Ricci tensor and f(t) denotes the composite f ◦ t. For local considerations, only (a) and (b) are essential. In fact, condition (e) (in which ‘parallel’ refers to the connection in E induced by the Levi-Civita connection of g), as well (c) and (d) for some u and t, follow from (a) – (b) if M is simply connected. See Lemma 2.2(f). On the other hand, (c) – (d), Lemma 2.2(b) and symmetry of ρ give ∇dt = 0 and ρ = χ dt⊗ dt for some function χ : M → R, so that ∇ρ = dχ⊗ dt⊗ dt. However, ∇ρ is totally symmetric (that is, ρ satisfies the Codazzi equation): our assumption ∇W = 0 implies the condition divW = 0, well known [11, formula (5.29) on p. 460] to be equivalent to the Codazzi equation for the Schouten tensor σ, while σ = ρ by Lemma 2.2(c). Thus, dχ equals a function times dt, and so χ is, locally, a function of t, which (locally) yields (f). For any section v of D⊥, we denote by v the image of v under the quotient-pro- jection morphism D⊥ → E = D⊥/D. The data required for the construction in Section 4 consist of I, f, n, V appearing in (a) – (f), along with the pseudo-Euclidean inner product 〈 , 〉 in V , induced in an obvious way by g (cf. Lemma 2.2(f)), and A : V → V characterized by 〈Aψ, ψ ′〉 = W (u′, v, v ′, u′), for ψ, ψ ′ ∈ V , with a vector field u′ and sections v, v ′ of D⊥ chosen, locally, so that g(u, u′) = 1, ψ = v and ψ ′ = v ′. (The resulting bilinear form (ψ, ψ ′) 7→ 〈Aψ, ψ ′〉 on V is well-defined, that is, unaffected by the choices of u′, v or v ′, as a consequence of Lemma 2.2(d),(e), while the function W (u′, v, v ′, u′) is in fact constant, by Lemma 2.2(d), as ones sees differentiating it via the Leibniz rule and noting that, since v and v ′ are parallel, the covariant derivatives of v and v ′ in the direction of any vector field are sections of D.) That A is traceless and self-adjoint is immediate from the symmetries of W. Finally, A 6= 0 since, otherwise, W would vanish. (Namely, in view of Lemma 2.2(d),(e), W would yield 0 when evaluated on any quadruple of vector fields, each of which is either u′ or a section of D⊥.) Under the assumptions (a) – (f), with f = f(t), we then have (8) R(u′, v)v ′ = [f g(v, v ′) + 〈Av, v ′〉]g(u′, u)u for any sections v, v ′ of D⊥ and any vector field u′. In fact, ρ(v, · ) = ρ(v ′, · ) = 0 from symmetry of ρ and Lemma 2.2(b), so that, by Lemma 2.2(c), R(u′, v)v ′ = W (u′, v)v ′ − (n − 2)−1g(v, v ′)ρu′, where ρu′ denotes the unique vector field with g(ρu′, · ) = ρ(u′, · ). Now (8) follows: due to (d), (f) and the definition of A, both sides have the same g-inner product with u′, and are orthogonal to u⊥ = D⊥ (with R(u′, v)v ′ orthogonal to D⊥ in view of Lemma 2.2(e)). 8 A. DERDZINSKI AND W. ROTER We fix an open subinterval I of I ′, containing 0, and a null geodesic I ∋ t 7→ x(t) in M with x(0) = y, parametrized by the function t (in the sense that the function t restricted to the geodesic coincides with the geodesic parameter). Namely, since ∇dt = 0, the restriction of t to any geodesic is an affine function of the parameter; thus, by (d), it suffices to prescribe the initial data formed by x(0) = y and a null vector ẋ(0) ∈ TyM with g(ẋ(0), uy) = 1. As g(ẋ(0), uy) = 1, the plane P in TyM , spanned by the null vectors ẋ(0) and uy (cf. Lemma 2.2(a)) is gy-nondegenerate, and so TyM = P ⊕ Ṽ , for Ṽ = P ⊥. Let pr : TyM → Ṽ be the orthogonal projection. Since pr(Dy) = {0}, the restriction of pr to Dy ⊥ descends to the quotient Ey = Dy ⊥/Dy, producing an isomorphism Ey → Ṽ , also denoted by pr. Finally, for ψ ∈ V , we let t 7→ ψ̃(t) ∈ Tx(t)M be the parallel field with ψ̃(0) = pr ψy, and set κ(t, s, ψ) = f(t)〈ψ, ψ〉+ 〈Aψ, ψ〉, as in Section 4. The formula F (t, s, ψ) = expx(t)(ψ̃(t) + sux(t)/2) now defines a C ∞ mapping F from an open subset of R2× V into M . Lemma 5.1. Under the above hypotheses, F ∗g = κ dt2+ dtds+ h. Proof. The F -images w,w ′, F∗ψ of the constant vector fields (1, 0, 0), (0, 1, 0) and (0, 0, ψ) in R2×V , for ψ ∈ V , are vector fields tangent to M along F (sections of F ∗TM). Since D⊥ is parallel, its leaves are totally geodesic and, by Lemma 2.2(e), the Levi-Civita connection of g induces on each leaf a flat torsionfree connection. Thus, w ′ and each F∗ψ are parallel along each leaf of D ⊥, as well as tangent to the leaf, and parallel along the geodesic t 7→ x(t). Therefore, w ′ = u/2, while the functions g(w ′, F∗ψ) and g(F∗ψ, F∗ψ ′), for ψ, ψ ′ ∈ V , are constant, and hence equal to their values at y, that is, 0 and 〈ψ, ψ ′〉. It now remains to be shown that g(w,w) = κ◦F , g(w, u/2) = 1/2 and g(w, F∗ψ) = 0. To this end, we consider the variation x(t, s) = F (t, sa, sψ) of curves in M , with any fixed a ∈ R and ψ ∈ V . Clearly, w = xt along the variation (notation of Section 1). Next, xts = xst is tangent to D ⊥, since so is xs, while D ⊥ is parallel. Consequently, [g(xt, u)]s = 0, as u is parallel and tangent to D. Thus, g(w, u) = g(xt, u) = 1. (Note that g(xt, u) = 1 at s = 0, due to (d), as the geodesic t 7→ x(t) is parametrized by the function t.) However, xss = 0 and xs is tangent to D ⊥, so that (3) and (8) now give xtss = [fg(xs, xs) + 〈Axs, xs〉]u, which is parallel in the s direction, while xts = xst = 0 at s = 0. Hence xts = s[fg(xs, xs) + 〈Axs, xs〉]u, and so g(xts, xts) = 0 (cf. (c) above and Lemma 2.2(a)). This further yields [g(xt, xt)]ss/2 = g(xt, xtss) = fg(xs, xs) + 〈Axs, xs〉. The last function is constant in the s direction, while g(xt, xt) = [g(xt, xt)]s = 0 at s = 0, and so g(w,w) = g(xt, xt) = s 2[fg(xs, xs) + 〈Axs, xs〉] = κ. Finally, being proportional to u at each point, xts is orthogonal to D ⊥, and hence to F∗ψ, which imples that [g(xt, F∗ψ)]s = 0, and, as g(w, F∗ψ) = g(xt, F∗ψ) = 0 at s = 0, we get g(w, F∗ψ) = 0 everywhere. � CONFORMALLY SYMMETRIC MANIFOLDS 9 We are now in a position to prove Theorem 4.1. First, (7) is conformally sym- metric and has d = 1, as one can verify by a direct calculation, cf. [16, Theorem 3]. Conversely, if conditions (a) and (b) above are satisfied, we may also assume (c) – (f). (See the comment following (f).) Our assertion is now immediate from Lemma 5.1. Appendix I: Proof of Lemma 2.1 We prove Lemma 2.1 here, since Olszak’s paper [13] may be difficult to obtain. The condition d = n is equivalent to conformal flatness of (M, g), since n > 2 and so Ω = 0 is the only 2-form ∧-divisible by all nonzero 1-forms ξ. At a fixed point x, the metric gx allows us to treat the Ricci tensor ρx and any 2-form Ωx as endomorphisms of TxM, so that we may consider their images (which are subspaces of TxM). If W 6= 0, fixing a nonzero 2-form Ωx in the image of Wx acting on 2-forms at x we see that, for every u ∈ Dx, our Ωx is ∧-divisible by ξ = gx(u, · ), and so the image of Ωx contains Dx. Thus, d ≤ 2, and (i) follows. (Being nonzero and decomposable, Ωx has rank 2.) As shown in [6, Theorem 7 on p. 18], if (M, g) is essentially conformally symmetric, the image of ρx is a subspace of Dx, so that (i) yields (ii), since g in (ii) cannot be Ricci-flat. Next, if d = 2, the image of our Ωx coincides with Dx (as rankΩx = 2). Every 2-form in the image of Wx thus is a multiple of Ωx, being the exterior product of two vectors in Dx, identified, via gx, with 1-forms. Hence rankW = 1. Conversely, if rankW = 1, all nonzero 2-forms Ωx in the image of Wx are of rank 2, as Wx, being self-adjoint, is a multiple of Ωx ⊗ Ωx, and so the Bianchi identity for W gives Ωx ∧ Ωx = 0. All such Ωx are therefore ∧-divisible by ξ = gx(u, · ), for every nonzero vector u in the common 2-dimensional image of such Ωx, which shows that d = 2. Finally, (iv) follows if one chooses Ωx 6= 0 equal to Wx(v, v ′, · , · ) for some v, v ′ ∈ TxM . Appendix II: Lemma 2.2(b),(c) in the locally symmetric case Parts (b) and (c) of Lemma 2.2 for locally symmetric manifolds with d = 1 could, in principle, be derived from Cahen and Parker’s classification [1] of pseudo-Riemann- ian symmetric manifolds. We prove them here directly, for the reader’s convenience. Our argument uses assertions (a), (d) in Lemma 2.2, along with (e) for W, which were established in the proof of Lemma 2.2 before Appendix II was mentioned. Suppose that ∇R = 0 and d = 1. Replacing M by an open subset, we also assume that the Olszak distribution D is spanned by a vector field u. By (1), (9) i) R( · , · )u = Ω ⊗ u or, in coordinates, ii) ulRjkl s = Ωjku for some differential 2-form Ω, which obviously does not depend on the choice of u. (It is also clear from (1) that Ω is the curvature form of the connection in the line bundle D, induced by the Levi-Civita connection of g.) Being unique, Ω is parallel, 10 A. DERDZINSKI AND W. ROTER and so are ρ and W, which implies the Ricci identities R · Ω = 0, R · ρ = 0, and R ·W = 0. In coordinates: Rmlj sτsk +Rmlk sτjs = 0, where τ = Ω or τ = ρ, and (10) Rqpj sWsklm + Rqpk sWjslm + Rqpl sWjksm + Rqpk sWjkls = 0. Summing Rmlj sΩsk + Rmlk sΩjs = 0 against u l, we obtain Ω ◦ Ω = 0, where the metric g is used to treat Ω as a bundle morphism TM → TM that sends each vector field v to the vector field Ωv with g(Ωv, v ′) = Ω(v, v ′) for all vector fields v ′. Lemma 2.2(d) and (9.i) give W ( · , · , u, v) = R( · , · , u, v) = 0 for our fixed vector field u, spanning D, and any section v of D⊥. Hence, by (2), g(u, · ) ∧ σ(v, · ) = g(v, · ) ∧ σ(u, · ). Thus, σu = cu for the Schouten tensor σ and some constant c, with σu defined analogously to Ωv. (Otherwise, choosing v such that u, σu and v are linearly independent at a given point x, we would obtain a contradiction with the equality between planes in TxM , corresponding to the above equality between exterior products.) Consequently, g(u, · ) ∧ (σ + cg)(v, · ) = 0, and so σv + cv is a section of D whenever v is a section of D⊥. Let us now fix u′ as in (4). Symmetry of σ gives g(σu′, u) = c. In a suitably ordered basis with (5), at any point x, the endomorphism of TxM corresponding to σx thus has an upper triangular matrix with the diagonal entries c,−c, . . . ,−c, c, so that trgσ = (4 − n)c. Consequently, (n − 2) s = 2(n − 1)(4 − n)c, for the scalar curvature s, and (n − 2)ρu = 2cu. However, contracting (9.ii) in k = s, we get ρu = −Ωu, and so (n− 2)Ωu = −2cu. The equality Ω ◦Ω = 0 that we derived from the Ricci identity R ·Ω = 0 now gives c = 0. Hence s = 0 (which yields Lemma 2.2(c)), and ρu = 0. As c = 0 and σ = ρ, the assertion about σv + cv obtained above means that ρv is a section of D whenever v is a section of D⊥. Let λ, µ, ξ be the 1-forms with λ = g(u, · ), µ = g(u′, · ), ξ(u′) = 0, and ρv = ξ(v)u for sections v of D⊥. Transvecting (9.ii) with µs, we get Ω = R( · , · , u, u ′) = (n − 2)−1λ ∧ ρ(u′, · ) from Lemma 2.2(c) with ρu = 0 and Lemma 2.2(d). However, evaluating ρ(u′, · ) on u′, u and sections v of D⊥, we see that ρ(u′, · ) = hλ+ ξ, with h = ρ(u′, u′). (Note that ξ(u) = 0 since ρu = 0, while D ⊂ D⊥ by Lemma 2.2(a).) Therefore, (11) i) (n− 2)Ω = λ ∧ ξ , ii) ρ = hλ⊗ λ + λ⊗ ξ + ξ ⊗ λ. In addition, if v ′ denotes the unique vector field with g(v ′, · ) = ξ, then u and v ′ are null and orthogonal, or, equivalently, (12) the 1-forms λ and ξ are null and mutually orthogonal. In fact, g(u, u) = 0 by Lemma 2.2(a), g(u, v ′) = 0 as ξ(u) = 0, and v ′ is null since (11) yields (n−2)[ρ(Ωu′)−Ω(ρu′)] = 2g(v ′, v ′)u, while, transvecting the Ricci identity Rmlj sRsk + Rmlk sRjs = 0 with u l and using (9.ii), we see that ρ and Ω commute as bundle morphisms TM → TM . Furthermore, transvecting with µkµm the coordinate form Rmlj sτsk+Rmlk sτjs = 0 of the Ricci identity R · τ = 0 for the parallel tensor field τ = (n − 2)Ω + ρ = CONFORMALLY SYMMETRIC MANIFOLDS 11 hλ⊗ λ + 2λ⊗ ξ (cf. (11)), we get 2λjblsξ s = 0, where b = W (u′, · , u′, · ). Namely, R = W + (n− 2)−1g ∧ ρ by Lemma 2.2(c), Wmlj sτsk = 0 in view of Lemma 2.2(d), µkµmWmlk sτjs = 2λjblsξ s since b(u, · ) = 0 (again from Lemma 2.2(d)), and the remaining terms, related to g ∧ ρ, add up to 0 as a consequence of (12), (11.ii) and the formula for τ . (Note that (12) gives Rj sτsk = Rj sτks = 0, and so four out of the eight remaining terms vanish individually.) However, u 6= 0, and so λ 6= 0, which gives b( · , v ′) = 0, where v ′ is the vector field with g(v ′, · ) = ξ. Thus, W (u′, · , u′, v ′) = 0. As a result, the 3-tensor W ( · , · , · , v ′) must vanish: it yields the value 0 whenever each of the three arguments is either u′ or a section of D⊥. (Lemma 2.2(e) for W is already established.) The relation W ( · , · , · , v ′) = 0 implies in turn that W ( · , · , · , ρv) = 0 (in coor- dinates: Wjkl sRsp = 0). In fact, by (11.ii), the image of ρ is spanned by u and v while W ( · , · , · , u) = 0 according to Lemma 2.2(d). As in [13, 1o on p. 214], we have W = (λ ⊗ λ) ∧ b (notation of (2)), where, again, b = W (u′, · , u′, · ). Namely, by Lemma 2.2(e) for W, both sides agree on any quadruple of vector fields, each of which is either u′ or a section of D⊥. Finally, transvecting (10) with µkµm and replacing R by W + (n− 2)−1g ∧ ρ, we obtain two contributions, one from W and one from g ∧ ρ, the sum of which is zero. Since W = (λ⊗λ)∧ b, the W contribution vanishes: its first two terms add up to 0, and so do its other two terms. (As we saw, b(u, · ) = 0, while, obviously, b(u′, · ) = 0.) Out of the sixteen terms forming the g∧ρ contribution, eight are separately equal to zero since Wjkl sRsp = 0, and so, in view of (11.ii) and the relation W = (λ⊗ λ) ∧ b, vanishing of the g∧ρ contribution gives λpSjlq = λqSjlp, for Sjlq = 2bjlξq−bqlξj−bqjξl. Thus, Sjlq = ηjlλq for some twice-covariant symmetric tensor field η, which, summed cyclically over j, l, q, yields 0 (due to the definition of Sjlq and symmetry of b). As λ 6= 0 and the symmetric product has no zero divisors, we get η = 0 and Sjlq = 0. The expression bjlξq− bqlξj is, therefore, skew-symmetric in j, l. As it is also, clearly, skew-symmetric in j, q, it must be totally skew-symmetric and hence equal to one- third of its cyclic sum over j, l, q. That cyclic sum, however, is 0 in view of symmetry of b, so that bjlξq = bqlξj. Thus, ξ = 0, for otherwise the last equality would yield b = ϕξ ⊗ ξ for some function ϕ, and hence W = (λ⊗ λ) ∧ b = ϕ(λ⊗ λ) ∧ (ξ ⊗ ξ), which would clearly imply that the vector field v ′ with g(v ′, · ) = ξ is a section of the Olszak distribution D, not equal to a function times u (as ξ(u′) = 0, while g(u, u′) = 1), contradicting one-dimensionality of D. Therefore, ρ = hλ ⊗ λ by (11.ii) with ξ = 0, which proves assertion (b) of Lemma 2.2 in our case. References [1] Cahen, M. & Parker, M., Pseudo-riemannian symmetric spaces. Mem. Amer. Math. Soc. 229 (1980), 1–108. 12 A. DERDZINSKI AND W. ROTER [2] Chaki, M. C. & Gupta, B., On conformally symmetric spaces. Indian J. Math. 5 (1963), 113–122. [3] Derdziński, A., On homogeneous conformally symmetric pseudo-Riemannian manifolds. Col- loq. Math. 40 (1978), 167–185. [4] Derdziński, A. & Roter, W., On conformally symmetric manifolds with metrics of indices 0 and 1. Tensor (N. S.) 31 (1977), 255–259. [5] Derdziński, A. & Roter, W., Some theorems on conformally symmetric manifolds. Tensor (N. S.) 32 (1978), 11–23. [6] Derdziński, A. & Roter, W., Some properties of conformally symmetric manifolds which are not Ricci-recurrent. Tensor (N. S.) 34 (1980), 11–20. [7] Derdzinski, A. & Roter, W., Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Preprint, math.DG/0604568. To appear in Tohoku Math. J. [8] Derdzinski, A. & Roter, W., Compact pseudo-Riemannian manifolds with parallel Weyl tensor. Preprint, http://arXiv.org/abs/math.DG/0702491. [9] Deszcz, R., On hypercylinders in conformally symmetric manifolds. Publ. Inst. Math. (Beograd) (N. S.) 51(65) (1992), 101–114. [10] Deszcz, R. & Hotloś, M., On a certain subclass of pseudosymmetric manifolds. Publ. Math. Debrecen 53 (1998), 29–48. [11] Dillen, F. J. E. & Verstraelen, L. C. A. (eds.), Handbook of Differential Geometry, Vol. I. North-Holland, Amsterdam, 2000. [12] Hotloś, M., On conformally symmetric warped products. Ann. Acad. Paedagog. Cracov. Stud. Math. 4 (2004), 75–85. [13] Olszak, Z., On conformally recurrent manifolds, I: Special distributions. Zesz. Nauk. Politech. Śl., Mat.-Fiz. 68 (1993), 213–225. [14] Patterson, E. M. & Walker, A. G., Riemann extensions. Quart. J. Math. Oxford Ser. (2) 3 (1952), 19–28. [15] Rong, J. P., On 2K∗ space. Tensor (N. S.) 49 (1990), 117–123. [16] Roter, W., On conformally symmetric Ricci-recurrent spaces. Colloq. Math. 31 (1974), 87– [17] Sharma, R., Proper conformal symmetries of conformal symmetric space-times. J. Math. Phys. 29 (1988), 2421–2422. [18] Simon, U., Compact conformally symmetric Riemannian spaces.Math. Z. 132 (1973), 173–177. Department of Mathematics The Ohio State University Columbus, OH 43210 E-mail address : [email protected] Institute of Mathematics and Computer Science Wroc law University of Technology Wybrzeże Wyspiańskiego 27, 50-370 Wroc law Poland E-mail address : [email protected] http://arxiv.org/abs/math/0604568 Introduction 1. Preliminaries 2. The Olszak distribution 3. The case d=2 4. The case d=1 5. Proof of Theorem ?? Appendix I: Proof of Lemma ?? Appendix II: Lemma ??(b),(c) in the locally symmetric case References

0704.0597

Investigation of Colour Reconnection in WW Events with the DELPHI detector at LEP-2

arXiv:0704.0597v1 [hep-ex] 4 Apr 2007 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN–PH-EP/2006-037 22 November 2006 Investigation of Colour Reconnection in WW Events with the DELPHI detector at LEP-2 DELPHI Collaboration Abstract In the reaction e+e− → WW → (q1q̄2)(q3q̄4) the usual hadronization models treat the colour singlets q1q̄2 and q3q̄4 coming from two W bosons indepen- dently. However, since the final state partons may coexist in space and time, cross-talk between the two evolving hadronic systems may be possible during fragmentation through soft gluon exchange. This effect is known as Colour Reconnection. In this article the results of the investigation of Colour Recon- nection effects in fully hadronic decays of W pairs in DELPHI at LEP are presented. Two complementary analyses were performed, studying the particle flow between jets and W mass estimators, with negligible correlation between them, and the results were combined and compared to models. In the frame- work of the SK-I model, the value for its κ parameter most compatible with the data was found to be: κSK−I = 2.2 corresponding to the probability of reconnection Preco to be in the range 0.31 < Preco < 0.68 at 68% confidence level with its best value at 0.52. (Accepted by Eur. Phys. J. C) http://arxiv.org/abs/0704.0597v1 J.Abdallah26, P.Abreu23, W.Adam55, P.Adzic12, T.Albrecht18, R.Alemany-Fernandez9, T.Allmendinger18, P.P.Allport24, U.Amaldi30, N.Amapane48, S.Amato52, E.Anashkin37, A.Andreazza29, S.Andringa23, N.Anjos23, P.Antilogus26, W-D.Apel18, Y.Arnoud15, S.Ask27, B.Asman47, J.E.Augustin26, A.Augustinus9, P.Baillon9, A.Ballestrero49, P.Bambade21, R.Barbier28, D.Bardin17, G.J.Barker57, A.Baroncelli40, M.Battaglia9, M.Baubillier26, K-H.Becks58, M.Begalli7, A.Behrmann58, E.Ben-Haim21, N.Benekos33, A.Benvenuti5, C.Berat15, M.Berggren26, L.Berntzon47, D.Bertrand2, M.Besancon41, N.Besson41, D.Bloch10, M.Blom32, M.Bluj56, M.Bonesini30, M.Boonekamp41, P.S.L.Booth†24, G.Borisov22, O.Botner53, B.Bouquet21, T.J.V.Bowcock24, I.Boyko17, M.Bracko44, R.Brenner53, E.Brodet36, P.Bruckman19, J.M.Brunet8, B.Buschbeck55, P.Buschmann58, M.Calvi30, T.Camporesi9, V.Canale39, F.Carena9, N.Castro23, F.Cavallo5, M.Chapkin43, Ph.Charpentier9, P.Checchia37, R.Chierici9, P.Chliapnikov43, J.Chudoba9, S.U.Chung9, K.Cieslik19, P.Collins9, R.Contri14, G.Cosme21, F.Cossutti50, M.J.Costa54, D.Crennell38, J.Cuevas35, J.D’Hondt2, J.Dalmau47, T.da Silva52, W.Da Silva26, G.Della Ricca50, A.De Angelis51, W.De Boer18, C.De Clercq2, B.De Lotto51 , N.De Maria48, A.De Min37, L.de Paula52, L.Di Ciaccio39, A.Di Simone40, K.Doroba56, J.Drees58,9, G.Eigen4, T.Ekelof53, M.Ellert53, M.Elsing9, M.C.Espirito Santo23, G.Fanourakis12, D.Fassouliotis12,3, M.Feindt18, J.Fernandez42 , A.Ferrer54, F.Ferro14, U.Flagmeyer58, H.Foeth9, E.Fokitis33, F.Fulda-Quenzer21, J.Fuster54, M.Gandelman52, C.Garcia54, Ph.Gavillet9, E.Gazis33, R.Gokieli9,56, B.Golob44,46, G.Gomez-Ceballos42, P.Goncalves23, E.Graziani40, G.Grosdidier21, K.Grzelak56, J.Guy38, C.Haag18, A.Hallgren53, K.Hamacher58, K.Hamilton36, S.Haug34, F.Hauler18, V.Hedberg27, M.Hennecke18, H.Herr†9, J.Hoffman56, S-O.Holmgren47, P.J.Holt9, M.A.Houlden24, J.N.Jackson24, G.Jarlskog27, P.Jarry41, D.Jeans36, E.K.Johansson47, P.D.Johansson47, P.Jonsson28, C.Joram9, L.Jungermann18, F.Kapusta26, S.Katsanevas28 , E.Katsoufis33, G.Kernel44, B.P.Kersevan44,46, U.Kerzel18, B.T.King24, N.J.Kjaer9, P.Kluit32, P.Kokkinias12, C.Kourkoumelis3, O.Kouznetsov17, Z.Krumstein17, M.Kucharczyk19, J.Lamsa1, G.Leder55, F.Ledroit15, L.Leinonen47, R.Leitner31, J.Lemonne2, V.Lepeltier21, T.Lesiak19, W.Liebig58, D.Liko55, A.Lipniacka47, J.H.Lopes52, J.M.Lopez35, D.Loukas12, P.Lutz41, L.Lyons36, J.MacNaughton55 , A.Malek58, S.Maltezos33, F.Mandl55, J.Marco42, R.Marco42, B.Marechal52, M.Margoni37, J-C.Marin9, C.Mariotti9, A.Markou12, C.Martinez-Rivero42, J.Masik13, N.Mastroyiannopoulos12, F.Matorras42, C.Matteuzzi30, F.Mazzucato37 , M.Mazzucato37, R.Mc Nulty24, C.Meroni29, E.Migliore48, W.Mitaroff55, U.Mjoernmark27, T.Moa47, M.Moch18, K.Moenig9,11, R.Monge14, J.Montenegro32 , D.Moraes52, S.Moreno23, P.Morettini14, U.Mueller58, K.Muenich58, M.Mulders32, L.Mundim7, W.Murray38, B.Muryn20, G.Myatt36, T.Myklebust34, M.Nassiakou12, F.Navarria5, K.Nawrocki56, R.Nicolaidou41, M.Nikolenko17,10, A.Oblakowska-Mucha20, V.Obraztsov43, A.Olshevski17, A.Onofre23, R.Orava16, K.Osterberg16, A.Ouraou41, A.Oyanguren54, M.Paganoni30, S.Paiano5, J.P.Palacios24, H.Palka19, Th.D.Papadopoulou33, L.Pape9, C.Parkes25, F.Parodi14, U.Parzefall9, A.Passeri40, O.Passon58, L.Peralta23, V.Perepelitsa54, A.Perrotta5, A.Petrolini14, J.Piedra42, L.Pieri40, F.Pierre41, M.Pimenta23, E.Piotto9, T.Podobnik44,46, V.Poireau9, M.E.Pol6, G.Polok19, V.Pozdniakov17, N.Pukhaeva17 , A.Pullia30, J.Rames13, A.Read34, P.Rebecchi9, J.Rehn18, D.Reid32, R.Reinhardt58, P.Renton36, F.Richard21, J.Ridky13, M.Rivero42, D.Rodriguez42, A.Romero48, P.Ronchese37, P.Roudeau21, T.Rovelli5, V.Ruhlmann-Kleider41, D.Ryabtchikov43 , A.Sadovsky17, L.Salmi16, J.Salt54, C.Sander18, A.Savoy-Navarro26, U.Schwickerath9, R.Sekulin38, M.Siebel58, A.Sisakian17, G.Smadja28, O.Smirnova27, A.Sokolov43, A.Sopczak22, R.Sosnowski56, T.Spassov9, M.Stanitzki18, A.Stocchi21, J.Strauss55, B.Stugu4, M.Szczekowski56, M.Szeptycka56 , T.Szumlak20, T.Tabarelli30, A.C.Taffard24, F.Tegenfeldt53 , J.Timmermans32, L.Tkatchev17 , M.Tobin24, S.Todorovova13, B.Tome23, A.Tonazzo30, P.Tortosa54, P.Travnicek13, D.Treille9, G.Tristram8, M.Trochimczuk56, C.Troncon29, M-L.Turluer41, I.A.Tyapkin17, P.Tyapkin17, S.Tzamarias12, V.Uvarov43, G.Valenti5, P.Van Dam32, J.Van Eldik9, N.van Remortel16, I.Van Vulpen9, G.Vegni29, F.Veloso23, W.Venus38, P.Verdier28, V.Verzi39, D.Vilanova41, L.Vitale50, V.Vrba13, H.Wahlen58, A.J.Washbrook24, C.Weiser18, D.Wicke9, J.Wickens2, G.Wilkinson36, M.Winter10, M.Witek19, O.Yushchenko43 , A.Zalewska19, P.Zalewski56, D.Zavrtanik45, V.Zhuravlov17, N.I.Zimin17, A.Zintchenko17 , M.Zupan12 1Department of Physics and Astronomy, Iowa State University, Ames IA 50011-3160, USA 2IIHE, ULB-VUB, Pleinlaan 2, B-1050 Brussels, Belgium 3Physics Laboratory, University of Athens, Solonos Str. 104, GR-10680 Athens, Greece 4Department of Physics, University of Bergen, Allégaten 55, NO-5007 Bergen, Norway 5Dipartimento di Fisica, Università di Bologna and INFN, Via Irnerio 46, IT-40126 Bologna, Italy 6Centro Brasileiro de Pesquisas F́ısicas, rua Xavier Sigaud 150, BR-22290 Rio de Janeiro, Brazil 7Inst. de F́ısica, Univ. Estadual do Rio de Janeiro, rua São Francisco Xavier 524, Rio de Janeiro, Brazil 8Collège de France, Lab. de Physique Corpusculaire, IN2P3-CNRS, FR-75231 Paris Cedex 05, France 9CERN, CH-1211 Geneva 23, Switzerland 10Institut de Recherches Subatomiques, IN2P3 - CNRS/ULP - BP20, FR-67037 Strasbourg Cedex, France 11Now at DESY-Zeuthen, Platanenallee 6, D-15735 Zeuthen, Germany 12Institute of Nuclear Physics, N.C.S.R. Demokritos, P.O. Box 60228, GR-15310 Athens, Greece 13FZU, Inst. of Phys. of the C.A.S. High Energy Physics Division, Na Slovance 2, CZ-180 40, Praha 8, Czech Republic 14Dipartimento di Fisica, Università di Genova and INFN, Via Dodecaneso 33, IT-16146 Genova, Italy 15Institut des Sciences Nucléaires, IN2P3-CNRS, Université de Grenoble 1, FR-38026 Grenoble Cedex, France 16Helsinki Institute of Physics and Department of Physical Sciences, P.O. Box 64, FIN-00014 University of Helsinki, Finland 17Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, RU-101 000 Moscow, Russian Federation 18Institut für Experimentelle Kernphysik, Universität Karlsruhe, Postfach 6980, DE-76128 Karlsruhe, Germany 19Institute of Nuclear Physics PAN,Ul. Radzikowskiego 152, PL-31142 Krakow, Poland 20Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy, PL-30055 Krakow, Poland 21Université de Paris-Sud, Lab. de l’Accélérateur Linéaire, IN2P3-CNRS, Bât. 200, FR-91405 Orsay Cedex, France 22School of Physics and Chemistry, University of Lancaster, Lancaster LA1 4YB, UK 23LIP, IST, FCUL - Av. Elias Garcia, 14-1o, PT-1000 Lisboa Codex, Portugal 24Department of Physics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK 25Dept. of Physics and Astronomy, Kelvin Building, University of Glasgow, Glasgow G12 8QQ 26LPNHE, IN2P3-CNRS, Univ. Paris VI et VII, Tour 33 (RdC), 4 place Jussieu, FR-75252 Paris Cedex 05, France 27Department of Physics, University of Lund, Sölvegatan 14, SE-223 63 Lund, Sweden 28Université Claude Bernard de Lyon, IPNL, IN2P3-CNRS, FR-69622 Villeurbanne Cedex, France 29Dipartimento di Fisica, Università di Milano and INFN-MILANO, Via Celoria 16, IT-20133 Milan, Italy 30Dipartimento di Fisica, Univ. di Milano-Bicocca and INFN-MILANO, Piazza della Scienza 3, IT-20126 Milan, Italy 31IPNP of MFF, Charles Univ., Areal MFF, V Holesovickach 2, CZ-180 00, Praha 8, Czech Republic 32NIKHEF, Postbus 41882, NL-1009 DB Amsterdam, The Netherlands 33National Technical University, Physics Department, Zografou Campus, GR-15773 Athens, Greece 34Physics Department, University of Oslo, Blindern, NO-0316 Oslo, Norway 35Dpto. Fisica, Univ. Oviedo, Avda. Calvo Sotelo s/n, ES-33007 Oviedo, Spain 36Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 37Dipartimento di Fisica, Università di Padova and INFN, Via Marzolo 8, IT-35131 Padua, Italy 38Rutherford Appleton Laboratory, Chilton, Didcot OX11 OQX, UK 39Dipartimento di Fisica, Università di Roma II and INFN, Tor Vergata, IT-00173 Rome, Italy 40Dipartimento di Fisica, Università di Roma III and INFN, Via della Vasca Navale 84, IT-00146 Rome, Italy 41DAPNIA/Service de Physique des Particules, CEA-Saclay, FR-91191 Gif-sur-Yvette Cedex, France 42Instituto de Fisica de Cantabria (CSIC-UC), Avda. los Castros s/n, ES-39006 Santander, Spain 43Inst. for High Energy Physics, Serpukov P.O. Box 35, Protvino, (Moscow Region), Russian Federation 44J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia 45Laboratory for Astroparticle Physics, University of Nova Gorica, Kostanjeviska 16a, SI-5000 Nova Gorica, Slovenia 46Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 47Fysikum, Stockholm University, Box 6730, SE-113 85 Stockholm, Sweden 48Dipartimento di Fisica Sperimentale, Università di Torino and INFN, Via P. Giuria 1, IT-10125 Turin, Italy 49INFN,Sezione di Torino and Dipartimento di Fisica Teorica, Università di Torino, Via Giuria 1, IT-10125 Turin, Italy 50Dipartimento di Fisica, Università di Trieste and INFN, Via A. Valerio 2, IT-34127 Trieste, Italy 51Istituto di Fisica, Università di Udine and INFN, IT-33100 Udine, Italy 52Univ. Federal do Rio de Janeiro, C.P. 68528 Cidade Univ., Ilha do Fundão BR-21945-970 Rio de Janeiro, Brazil 53Department of Radiation Sciences, University of Uppsala, P.O. Box 535, SE-751 21 Uppsala, Sweden 54IFIC, Valencia-CSIC, and D.F.A.M.N., U. de Valencia, Avda. Dr. Moliner 50, ES-46100 Burjassot (Valencia), Spain 55Institut für Hochenergiephysik, Österr. Akad. d. Wissensch., Nikolsdorfergasse 18, AT-1050 Vienna, Austria 56Inst. Nuclear Studies and University of Warsaw, Ul. Hoza 69, PL-00681 Warsaw, Poland 57Now at University of Warwick, Coventry CV4 7AL, UK 58Fachbereich Physik, University of Wuppertal, Postfach 100 127, DE-42097 Wuppertal, Germany † deceased 1 Introduction The space-time development of a hadronic system is still poorly understood, and models are necessary to transform a partonic system, governed by perturbative QCD, to final state hadrons observed in the detectors. WW events produced in e+e− collisions at LEP-2 constitute a unique laboratory to study and test the evolution of such hadronic systems, because of the clean environment and the well-defined initial energy in the process. Of particular interest is the possibility to study separately one single evolving hadronic system (one of the W bosons decaying semi-leptonically, the other decaying hadronically), and compare it with two hadronic systems evolving at the same time (both W bosons decaying hadronically). Interconnection effects between the products of the hadronic decays of the two W bosons (in the same event) are expected since the lifetime of the W bosons (τW ≃ ~/ΓW ≃ 0.1 fm/c) is an order of magnitude smaller than the typical hadronization times. These effects can happen at two levels: • in the evolution of the parton shower, between partons from different hadronic sys- tems by exchanging coloured gluons [1] (this effect is called Colour Reconnection (CR) for historical reasons); • between the final state hadrons, due to quantum-mechanical interference, mainly due to Bose-Einstein Correlations (BEC) between identical bosons (e.g. pions with the same charge). A detailed study by DELPHI of this second effect was recently published [2]. The first effect, the possible presence of colour flow between the two W hadronization systems, is the topic studied in this paper. This effect is worthy of study in its own right and for the possible effects induced on the W mass measurement in fully hadronic events (see for instance [3] for an introduction and [4] for an experimental review). The effects at the perturbative level are expected to be small [3], whereas they may be large at the hadronization level (many soft gluons sharing the space-time) for which models have to be used to compare with the data. The most tested model is the Sjöstrand-Khoze “Type 1” CR model SK-I [5]. This model of CR is based on the Lund string fragmentation phenomenology. The strings are considered as colour flux tubes with some volume, and reconnection occurs when these tubes overlap. The probability of reconnection in an event is parameterised by the value κ, set globally by the user, according to the space-time volume overlap of the two strings, Voverlap : Preco(κ) = 1− e−κVoverlap . (1) The parameter κ was introduced in the SK-I model to allow a variation of the percentage of reconnected events and facilitate studies of sensitivity to the effect. In this model only one string reconnection per event was allowed. The authors of the model propose the value of κ = 0.66 to give similar amounts of reconnection as other models of Colour Reconnection. By comparing the data with the model predictions evaluated at several κ values, it is possible to determine the value of κ most consistent with the data and extract the corresponding reconnection probability. Another model was proposed by the same authors, considering the colour flux tubes as infinitely thin, which allows for Colour Reconnection in the case the tubes cross each other and provided the total string length is reduced (SK-II′). This last model was not tested. Two further models are tested here, these are the models implemented in HERWIG [6] and ARIADNE [7] Monte Carlo programs. In HERWIG the partons are reconnected, with a reconnection probability of 1/9, if the reconnection results in a smaller total cluster mass. In ARIADNE, which implements an adapted version of the Gustafson-Häkkinen model [8], the model used [9] allows for reconnections between partons originating in the same W boson, or from different W bosons if they have an energy smaller than the width of the W boson (this model will be referred as ‘AR-2’). Colour Reconnection has been previously investigated in DELPHI by comparing in- clusive distributions of charged particles, such as the charged-particle multiplicity dis- tribution or the production of identified (heavy) particles, in fully hadronic WW events and the distributions in semi-leptonic WW events. The investigations did not show any effect as they were limited by statistical and systematic errors and excluded only the most extreme models of CR (see [10]). This article presents the results of the investigations of Colour Reconnection effects in hadronically decaying W pairs using two techniques. The first, proposed by L3 in [11], looks at the particle flow between the jets in a 4-jet WW event. The second, proposed by DELPHI in [12], takes into account the different sensitivity to Colour Reconnection of several W mass estimators. The first technique is more independent of the model and it can provide comparisons based on data. The second technique is more dependent on the model tested, but has a much larger sensitivity to the models SK-I and HERWIG. Since the particle flow and W mass estimator methods were found to be largely uncorrelated a combination of the results of these two methods is provided. The paper is organised as follows. In the next section, the LEP operation and the components of the DELPHI detector relevant to the analyses are briefly described. In section 3 data and simulation samples are explained. Then both of the analysis methods discussed above are described and their results presented in sections 4 and 5. The com- bination of the results is given in section 6 and conclusions are drawn in the seventh and final section. 2 LEP Operation and Detector Description At LEP-2, the second phase of the e+e− collider at CERN, the accelerator was operated at centre-of-mass energies above the threshold for double W boson production from 1996 to 2000. In this period, the DELPHI experiment collected about 12000 WW events corresponding to a total integrated luminosity of 661 pb−1. About 46% of the WW events are WW → q1q̄2q3q̄4 events (fully hadronic), and 44% are WW → q1q̄2ℓν̄, where ℓ is a lepton (semi-leptonic). The detailed description of the DELPHI detector and its performance is provided in [13,14]. A brief summary of the main characteristics of the detector important for the analyses follows. The tracking system of DELPHI consisted of a Time Projection Chamber (TPC), the main tracking device of DELPHI, and was complemented by a Vertex Detector (VD) closest to the beam pipe, the Inner and the Outer Detectors in the barrel region, and two Forward Chambers in the end caps. It was embedded in a 1.2 T magnetic field, aligned parallel to the beam axis. The electromagnetic calorimeter consisted of the High density Projection Chamber (HPC) in the barrel region, the Forward Electromagnetic Calorimeter (FEMC) and the Small angle Tile Calorimeter (STIC) in the forward regions, complemented by detectors to tag the passage of electron-positron pairs from photons converted in the regions between the FEMC and the HPC. The total depths of the calorimeters corresponded to about 18 radiation lengths. The hadronic calorimeter was composed of instrumented iron with a total depth along the shortest trajectory for a neutral particle of 6 interaction lengths, and covered 98% of the total solid angle. Embedded in the hadronic calorimeter were two planes of muon drift chambers to tag the passage of muons. The whole detector was surrounded by a further double plane of staggered muon drift chambers. For LEP-2, the DELPHI detector was upgraded as described in the following. Changes were made to some of the subdetectors, the trigger system [15], the run control and the algorithms used in the offline reconstruction of tracks, which improved the performance compared to the earlier LEP-1 period. The major changes were the extensions of the Vertex Detector (VD) and the Inner Detector (ID), and the inclusion of the Very Forward Tracker (VFT) [16], which increased the coverage of the silicon tracker to polar angles with respect to the z-axis1 of 11◦ < θ < 169◦. To further improve the track reconstruction efficiency in the forward regions of DELPHI, the tracking algorithms and the alignment and calibration procedures were optimised for LEP-2. Changes were also made to the electronics of the trigger and timing system which improved the stability of the running during data taking. The trigger conditions were optimised for LEP-2 running, to give high efficiency for 2- and 4-fermion processes in the Standard Model and also to give sensitivity to events which may have been signatures of new physics. In addition, improvements were made to the operation of the detector during the LEP operating states, to prepare the detector for data taking at the very start of stable collisions of the e+e− beams, and to respond to adverse background from LEP when it arose. These changes led to an overall improvement in the efficiency for collecting the delivered luminosity from about 85% in 1995, before the start of LEP-2, to about 95% at the end in 2000. During the operation of the DELPHI detector in 2000 one of the 12 sectors of the central tracking chamber, the TPC, failed. After 1st September it was not possible to detect the tracks left by charged particles inside the broken sector. The data affected corresponds to around 1/4 of the data collected in 2000. Nevertheless, the redundancy of the tracking system of DELPHI meant that tracks passing through the sector could still be reconstructed from signals in any of the other tracking detectors. As a result, the track reconstruction efficiency was only slightly reduced in the region covered by the broken sector, but the track parameter resolutions were degraded compared with the data taken prior to the failure of this sector. 3 Data and Simulation Samples The analyses presented here use the data collected by DELPHI in the years 1997 to 2000, at centre-of-mass energies s between 183 and 209 GeV. The data collected in the year 2000 with the TPC working in full, with centre-of-mass energies from 200 to 208 GeV and a integrated luminosity weighted average centre-of-mass energy of 206 GeV, were analysed together. Data acquired with the TPC with a broken sector, corresponding to a integrated luminosity weighted average centre-of-mass energy of 207 GeV, were analysed separately and included in the results presented here. The total integrated luminosity of the data sample is 660.8 pb−1, and the integrated luminosity weighted average centre-of-mass energy of the data is 197.1 GeV. To compare with the expected results from processes in the Standard Model including or not including CR, Monte Carlo (MC) simulation was used to generate events and 1The DELPHI coordinate system is a right-handed system with the z-axis collinear with the incoming electron beam, and the x axis pointing to the centre of the LEP accelerator. simulate the response of the DELPHI detector. These events were reconstructed and analysed with the same programs as used for the real data. The 4-fermion final states were generated with the code described in [17], based on WPHACT [18], for the WW signal (charged currents) and for the ZZ background (neutral currents), after which the events were fragmented with PYTHIA [19] tuned to DELPHI data [20]. The same WW events generated at 189, 200 and 206 GeV were also fragmented with PYTHIA implementing the SK-I model, with 100% reconnection probability. The systematic effects of fragmentation were studied using the above WW samples and WW samples generated with WPHACT and fragmented with either ARIADNE [7] or HERWIG [6] at 183, 189, 200 and 206 GeV. For systematic studies of Bose-Einstein Correlations (BEC), WW samples generated with WPHACT and fragmented with PYTHIA implementing the BE32 model [21] of BEC, were used at all energies, except at 207 GeV. The integrated luminosity of the simulated samples was at least 10 times that of the data of the corresponding year, and the majority corresponded to 100 times that of the data. To test the consistency of the SK-I model and measure the κ parameter, large WW samples were generated in an early stage of this work with EXCALIBUR [22] at 200 and 206 GeV, keeping only the fully hadronic decays. These samples were then fragmented with PYTHIA. It was verified for smaller subsets that the results using these large samples and the samples generated later with WPHACT are compatible. The qq̄(γ) background events were generated at all energies with KK2f [23] and frag- mented with PYTHIA. For systematic studies, similar KK2f samples fragmented with ARIADNE [7] were used at 183, 189, 200 and 206 GeV. These samples will be referred to as “DELPHI samples”. At 189 GeV, to compare with the other LEP experiments and with different CR mod- els, 6 samples generated with KORALW [24] for the 4-fermion final states were also used. These samples 2 will be referred to as “Cetraro samples”. The events in the different sam- ples have the final state quarks generated with the same kinematics, and differ only in the parton shower evolution and fragmentation. Three samples were fragmented respectively with PYTHIA, ARIADNE and HERWIG (using the tuning of the ALEPH collaboration), with no CR implementation. Three other samples were fragmented in the same manner but now implementing several CR models: the SK-I model with 100% reconnection proba- bility, the AR-2 model, and the HERWIG implementation of CR with 1/9 of reconnected events, respectively. 4 The Particle Flow Method The first of the two analyses presented in this paper is based on the so-called “particle flow method”. The particle flow algorithm is based on the selection of special event topologies, in order to obtain well defined regions between any two jets originating from the same W (called the Inside-W region) or from different Ws (called the Between-W region). It is expected that Colour Reconnection decreases (increases) particle production in the Inside-W (Between-W) region. Hence, by studying the particle production in the inter-jet regions it is possible to measure the effects of Colour Reconnection. However, this method requires a selection of events with a suitable topology (see below) which has a low efficiency (<∼25%). 2produced by ALEPH after the LEP-W Physics Workshop in Cetraro, Italy, October 2001 4.1 Event and Particle Selection Events with both Ws decaying into q1q̄2 are characterised by high multiplicity, large visible energy, and the tendency of the particles to be grouped in 4 jets. The background is dominated by qq̄(γ) events. Charged particles were required to have momentum p larger than 100 MeV/c and below 1.5 times the beam energy, a relative error on the momentum measurement ∆p/p < 1, and polar angle θ with respect to the beam axis between 20◦ and 160◦. To remove tracks from secondary interactions, the distance of closest approach of the extrapolated track to the interaction point was required to be less than 4 cm in the plane perpendicular to the beam axis and less than 4/sin θ cm along the beam axis, and the reconstructed track length was required to be larger than 30 cm. Clusters in the electromagnetic or hadronic calorimeters with energy larger than 0.5 GeV and polar angle in the interval 10◦ < θ < 170◦, not associated to charged particles, were considered as neutral particles. The events were pre-selected by requiring at least 12 charged particles, with a sum of the modulus of the momentum transverse to the beam axis, of charged and neutral particles, above 20% of the centre-of-mass energy. These cuts reduced the contributions from gamma-gamma processes and beam-gas interactions to a negligible amount. The momentum distribution of the charged particles for the pre-selected events is shown in Figure 1 and compared to the expected distribution from the simulation. A good agree- ment between data and simulation is observed. DELPHI 189 GeV WW WPHACT WW semileptonic p (GeV/c) 0 5 10 15 20 25 30 35 40 45 50 DELPHI 189 GeV WW WPHACT WW semileptonic p (GeV/c) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 1: Momentum distribution for charged particles (range 0-50 GeV/c (a) and 0-5 GeV/c (b)). Points represent the data and the histograms represent the contributions from simulation for the different processes (signal (white) and background contributions). About half of the e+e−→qq̄(γ) events at high-energy are associated with an energetic photon emitted by one of the beam electrons or positrons (radiative return events), thus reducing the energy available in the hadronic system to the Z mass. To remove these radiative return events, the effective centre-of-mass energy s′, computed as described in [25], was required to be above 110 GeV. It was verified that this cut does not affect the signal from W pairs, but reduces significantly the contribution from the qq̄(γ) process. In the WW fully hadronic decays four well separated energetic jets are expected which balance the momentum of the event and have a total energy near to the centre-of-mass energy. The charged and neutral particles in the event were thus clustered using the DURHAM algorithm [26], for a separation value of ycut = 0.005, and the events were kept if there were 4 and only 4 jets and a multiplicity (charged plus neutral) in each jet larger than 3. The combination of these two cuts removed most of the semi-leptonic WW decays and the 2-jet and 3-jet events of the qq̄(γ) background. The charged-particle multiplicity distribution for the selected events at 189 GeV is given in Figure 2, with data points compared to the histogram from simulation of signal and background processes. 0 10 20 30 40 50 60 70 80 90 DELPHI 189 GeV WW WPHACT WW semileptonic Figure 2: Uncorrected charged-particle multiplicity distribution at a centre-of-mass en- ergy of 189 GeV. Points represent the data and the histograms represent the contribution from simulation for the different processes. For the study of the charged-particle flow between jets, the initial quark configuration should be well reconstructed with a good quark-jet association. At 183 GeV and above, the produced W bosons are significantly boosted. This produces smaller angles in the laboratory frame of reference between the jets into which the W decays, when compared to these angles at threshold (back-to-back). Hence, this property tends to reduce the ambiguity in the definition of the Between-W and Inside-W regions. The selection criteria were designed in order to minimize the situation of one jet from one W boson appearing in the Inside-W region of the other W boson. The selection criteria are based on the event topology, with cuts in 4 of the 6 jet-jet angles. The smallest and the second smallest jet-jet angle should be below 100◦ and not adjacent (not have a common jet). Two other jet-jet angles should be between 100◦ and 140◦ and not adjacent (large angles). In the case that there are two different combinations of jets satisfying the above criteria for the large angles, the combination with the highest sum of large angles is chosen. This selection increases the probability to have a correct pairing of jets to the same W boson. s L Eff. Pur. Nsel MC tot. WW 4j qq̄(γ) ZZ W lep. εPAIR 183 52.7 22% 74% 127 114.2 84.4 22.3 0.7 7.0 69% 189 157.6 21% 75% 340 341.4 255.9 56.8 2.4 26.4 75% 192 25.9 21% 75% 61 56.1 41.9 9.4 0.4 4.4 77% 196 77.3 19% 74% 176 159.2 117.6 26.2 1.3 14.0 79% 200 83.4 18% 72% 173 165.0 119.5 27.8 1.3 16.4 82% 202 40.6 17% 72% 82 75.7 54.6 12.5 0.7 8.0 82% 206 163.9 15% 70% 282 274.7 193.1 47.8 2.7 31.1 79% 207 59.4 15% 70% 102 99.7 70.1 17.6 1.0 11.1 80% Table 1: Centre-of-mass energy ( s in GeV), integrated luminosity (L in pb−1), efficiency and purity of the data samples, number of selected events, number of expected events from 4-jet WW and background processes (total and separated by process), and efficiency of correct pairing of jets to the same W boson. The integrated luminosity, the efficiency to select 4-jet WW events and the purity of the selected data samples, estimated using simulation, and the number of selected events are summarised for each centre-of-mass energy in Table 1. The numbers of expected events are also given separately for the signal and the background processes, and were estimated using simulation. The efficiency to select the correct pairing of jets to the same W boson, estimated with simulation as the fraction of WW events for which the selected jets 1 and 2 (see later) correspond indeed to the same W boson, is given in the last column of the Table. The efficiency of the event selection criteria decreases with increasing centre-of-mass energy. This is primarily due to the ‘large’ angles being reduced as a result of the increased boost (becoming lower than the cut value of 100◦) and ‘small’ angles being increased due to the larger phase-space available (becoming higher than the cut value of 100◦). Much for the same reason, the efficiency to assign two jets to the same W boson in the selected events increases slightly with increasing centre-of-mass energy, in opposition to what would happen at threshold with the W boson decaying into two back-to-back jets, that would never be selected to come from the same W boson by the requirement that their interjet angle should be between 100◦ and 140◦. In the following analysis the jets and planar regions are labeled as shown in Figure 3: the planar region corresponding to the smallest jet-jet angle is region B in the plane made by jets 2 and 3; the second smallest jet-jet angle corresponds to the planar region D between jets 1 and 4 in the plane made by these two jets; the planar region corresponding to the greatest of the large jet-jet angles in this combination is region A and spans the angle between jets 1 and 2 in the plane made by these jets; and finally region C corresponds to the planar region spanned by the second large angle, between jets 3 and 4 in the plane made by these two jets. In general, the planar regions are not in the same plane, as the decay planes of the W bosons do not coincide, and the large angles in this combination are not necessarily the largest jet-jet angles in the event. The distribution of the reconstructed masses of the jet pairings (1,2) and (3,4), after applying a 4C kinematic fit requiring energy and momentum conservation, is shown in Figure 4 (two entries per event). In the figure, data at 189 GeV (points) are compared to the expected distribution from the 4-jet WW signal without CR, plus background processes, estimated using the simulation (histograms). The contribution from the 4-jet Inside-W region Inside-W region Between-W region Between-W region jet 2 jet 1 jet 3 jet 4 Figure 3: Schematic drawing of the angular selection. WW signal simulation is split between the case in which the two pairs of jets making the large angles actually come from their parent W bosons and the case in which the jets of a pair come from different W bosons (mismatch). 4.2 Particle Flow Distribution The particle flow analysis uses the number of particles in the Inside-W and the Between-W regions. An angular ordering of the jets is performed as in Figure 3. The two large jet-jet angles in the event are used to define the Inside-W regions, and the two smallest angles span the Between-W regions, the regions between the different Ws. In general, the two W bosons will not decay in the same plane, and this must be accounted for when comparing the particle production in the Inside-W and Between-W regions. So, for each region (A, B, C and D) the particle momenta of all charged particles are projected onto the plane spanned by the jets of that region: jets 1 and 2 for region A; jets 2 and 3 for region B; jets 3 and 4 for region C; jets 4 and 1 for region D. Then, for each particle the rescaled angle Φrescaled is determined as a ratio of two angles: Φrescaled = Φi/Φr , (2) when the particle momentum is projected onto the plane of the region r. The angle Φi is then the angle between the projected particle momentum and the first mentioned jet in the definition of the regions given above. The angle Φr is the full opening angle between the jets. Hence Φrescaled varies between 0 and 1 for the particles whose momenta are projected between the pair of jets defining the plane. However, due to the aplanarity of the event about 9% of the particles in the data and in the 4-jet WW simulation have projected angles outside all four regions. These particles were discarded from further analysis. In the case where a particle could be projected onto more than one region, with 0 < Φrescaled < 1, the solution with the lower momentum transverse to the region was used. This happened for about 13% of the particles in data, after background subtraction, and in the 4-jet WW simulation. This leads to the normalised particle flow distribution shown in Figure 5 at 189 GeV, where the rescaled angle of region A is plotted from 0 to 1, region B from 1 to 2, region 0 20 40 60 80 100 120 DELPHI 189 GeV WW WPHACT WW wrong pairing WW semileptonic Mass (GeV/c Figure 4: Reconstructed dijet masses (after a 4C kinematic fit) for the selected pairs at 189 GeV (2 entries per event)(see text). C from 2 to 3 and region D from 3 to 4. The statistical error on the bin contents (the average multiplicity per bin of Φrescaled divided by the bin width) was estimated using the Jackknife method [27], to correctly account for correlations between different bins. In this distribution the regions between the jets coming from the same W bosons (A and C), and from different W bosons (B and D), have the same scale and thus can be easily compared. After subtracting bin-by-bin the expected background from the observed distributions, we define the Inside-W (Between-W) particle flow as the bin-by-bin sum of regions A and C (B and D). These distributions are compared by performing the bin-by-bin ratio of the Inside-W particle flow to the Between-W particle flow. This ratio of distributions is shown for 189 GeV and 206 GeV in Figure 6. The data points are compared to several fully simulated WW MC samples with and without CR. A good agreement was found between the predictions using the WPHACT WW MC sam- ples and the predictions based on the KORALW WW MC samples, both for the scenario without CR and for the scenario with CR (SK-I model with 100% probability of recon- nection). For both sets of predictions the regions of greatest difference between the two scenarios span the rescaled variable Φrescaled from 0.2 to 0.8. 4.3 Particle Flow Ratio After summing the particle flow distributions for regions A and C, and regions B and D, the resulting distributions are integrated from 0.2 to 0.8. The ratio R of the Inside-W to the Between-W particle flow is then defined as (with Φ being the rescaled variable Φrescaled): 0 0.5 1 1.5 2 2.5 3 3.5 4 A B C D DELPHI 189 GeV WPH. SK-I 100% WW WPHACT WW semilept. Φrescaled Figure 5: Normalised charged-particle flow at 189 GeV. The lines correspond to the sum of the simulated 4-jet WW signal with the background contributions (estimated from DELPHI MC samples), normalised to the total number of expected events (Nevents). The dashed histogram corresponds to the sum with the simulated 4-jet WW signal generated by WPHACT with 100% SK-I. 0 0.2 0.4 0.6 0.8 1 Φrescaled DELPHI 189 GeV KW. SK-I 100%WPH. SK-I 100% WW KORALWWW WPHACT 0 0.2 0.4 0.6 0.8 1 Φrescaled DELPHI 206 GeV WPH. SK-I 100% WW WPHACT Figure 6: The ratio of the particle flow distributions (A+C)/(B+D) at 189 GeV (a) and at 206 GeV (b). The data (dots) are compared to WW MC samples generated with WPHACT (DELPHI samples) and KORALW (Cetraro samples), both without CR and implementing the SK-Imodel with 100% probability of reconnection. The lines corresponding to WPHACT are hardly distinguishable from the lines corresponding to KORALW in the same condition of implementation of CR. s (GeV) RData Rno CR RSK-I:100% 183 0.889 ± 0.084 0.928 ± 0.005 - 189 1.025 ± 0.063 0.966 ± 0.006 0.864 ± 0.005 192 1.008 ± 0.150 0.970 ± 0.006 - 196 1.041 ± 0.093 0.995 ± 0.006 - 200 0.922 ± 0.084 1.022 ± 0.007 0.889 ± 0.006 202 0.952 ± 0.126 1.015 ± 0.008 - 206 1.116 ± 0.088 1.012 ± 0.008 0.889 ± 0.006 207 1.039 ± 0.135 1.019 ± 0.008 - Table 2: Values of the ratio R for each energy (errors are statistical only), and expected values with errors due to limited statistics of the simulation, all from DELPHI WPHACT WW samples. MC Sample χ2/DF α,A β,B γ no CR 7.31/5 1.001± 0.003 (3.20± 0.36)× 10−3 (−1.35± 0.40)× 10−4 SK-I 100% 1.46/1 0.880± 0.003 (1.68± 0.44)× 10−3 - Table 3: Results of the fit to the evolution of R with ( s(GeV)− 197.5). ∫ 0.8 dnch/dΦ(A+ C)dΦ ∫ 0.8 dnch/dΦ(B +D)dΦ . (3) To take into account possible statistical correlations between particles in the Inside-W and Between-W regions, the statistical error on this ratio R was again estimated through the Jackknife method [27]. The values forR obtained for the different centre-of-mass energies are shown in Table 2, and compared to the expectations from the DELPHI WPHACT WW samples without CR and implementing the SK-I model with 100% reconnection probability. These values for data and MC are plotted as function of the centre-of-mass energy in Figure 7. The changes in the value of R for the MC samples are mainly due to the dif- ferent values of the boost of the W systems. In order to quantify this effect a linear function R( s− 197.5) = A + B · ( s− 197.5) was fitted to the MC points with CR (with s in GeV), while for the points without CR the quadratic func- tion R( s− 197.5) = α+ β · ( s− 197.5) + γ · ( s− 197.5)2 was assumed (with GeV), giving reasonable χ2/d.o.f. values. The fits yielded the results shown in Table 3. The MC without CR shows a stronger dependence on s. The function fitted to this sample was used to rescale the measured values of R for the data collected at different energies to the energy of 189 GeV, the centre-of-mass energy at which the combination of the results of the LEP experiments was proposed in [4]. All the rescaled values were combined with a statistical error-weighted average. The average of the R ratios rescaled to 189 GeV was found to be 〈R〉 = 0.979± 0.032(stat). (4) 180 185 190 195 200 205 210 WW no CR Fit to no CR WW SK-I 100% Fit to SK-I 100%Fit to SK-I 100% Combined Ratio √s (GeV) DELPHI Figure 7: The ratio R as function of s for data and MC (DELPHI WPHACTWW samples), and fits to the MC with and without CR, and the combined ratio after rescaling all values s = 189 GeV (see text). The value of the combined ratio at 189 GeV is shown at a displaced energy (upwards by 1 GeV) for better visibility, as well as all the values for the MC ‘WW no CR’ points and the corresponding fitted curve which are shown at centre-of-mass energies shifted downwards by 0.5 GeV. All errors for the MC values are smaller than the size of the markers. Performing the same weighted average when using for the rescaling the fit to the MC with CR, one obtains: 〈RCR rescale〉 = 0.987± 0.032(stat). (5) Repeating the procedure, but now without rescaling the R ratios, the result is: 〈Rno rescale〉 = 0.999± 0.033(stat). (6) 4.4 Study of the Systematic Errors in the Particle Flow The following effects were studied as sources of systematic uncertainties in this anal- ysis. 4.4.1 Fragmentation and Detector response A direct comparison between the particle flow ratios measured in fully hadronic data and MC samples, R4qData and R4qMC, respectively, is hampered by the uncertainties as- sociated with the modelling of the WW fragmentation and the detector response. These systematic uncertainties were estimated using mixed semi-leptonic events. In this tech- nique, two hadronically decaying W bosons from semi-leptonic events were mixed together to emulate a fully hadronic WW decay. Mixing Technique Semi-leptonic WW decays were selected from the data collected by DELPHI at centre- of-mass energies between 189 and 206 GeV, by requiring two hadronic jets, a well isolated identified muon or electron or, in case of a tau candidate, a well isolated particle, all associated with missing momentum (corresponding to the neutrino) pointing away from the beam pipe. A neural network selection, developed in [28], was used to select the events. The same procedure was applied to the WPHACT samples fragmented with PYTHIA and HERWIG at centre-of-mass energies of 189, 200 and 206 GeV and with ARIADNE at 189 and 206 GeV. The background to this selection was found to be of negligible importance in this analysis. Samples of mixed semi-leptonic events were built separately at each centre-of-mass energy for data and Monte Carlo semi-leptonic samples, following the mixing procedure developed in [2]. In each semi-leptonic event, the lepton (or tau-decay jet) was stripped off and the remaining particles constituted the hadronically decaying W boson. Two hadronically decaying W bosons were then mixed together to emulate a fully hadronic WW decay. The hadronic parts of W bosons were mixed in such a way as to have the parent W bosons back-to-back in the emulated fully hadronic WW decay. To increase the statistics of emulated events, and profiting from the cylindrical symmetry of the detector along the z axis, the hadronic parts of W bosons were rotated around the z axis, but were not moved from barrel to forward regions or vice-versa, as detailed in the following. When mixing the hadronic parts of different W events it was required that the two Ws had reconstructed polar angles back-to-back or equal within 10 degrees. In the latter case, when both Ws are on the same side of the detector, the z component of the momentum is sign flipped for all the particles in one of the Ws. The particles of one W event were then rotated around the beam axis, in order to have the two Ws also back-to-back in the transverse plane. Each semi-leptonic event was used in the mixing procedure between 4 and 9 times, to minimize the statistical error on the particle flow ratio R measured in the mixed semi-leptonic data sample. The mixed events were then subjected to the same event selection and particle flow analysis used for the fully hadronic events. The particle flow ratios Rmixed SLData and Rmixed SLMC were measured in the mixed semi-leptonic data and MC samples, respectively, and are plotted as function of the centre-of-mass energy in Figure 8. The values of Rmixed SL measured in MC show a dependence on s. This effect is quantified by performing linear fits to the points measured with PYTHIA, ARIADNE and HERWIG, respectively. The differences between the measured slopes were found to be small. The function fitted to the PYTHIA points was used to rescale the values of R measured in data at different energies to 189 GeV. The rescaled values were then combined using as weights the scaled statistical errors. The weighted average R at 189 GeV for the mixed semi-leptonic events built from data was found to be 〈Rmixed SLData〉 = 1.052± 0.027(stat). (7) For each MC sample, the ratio Rmixed SLData/Rmixed SLMC was used to calibrate the particle flow ratio measured in the corresponding fully hadronic sample, R4qMC, to compare it to the ratio measured in the data, 〈R4qData〉. The correction factor Rmixed SLData/Rmixed SLMC was computed from the values of R rescaled to 189 GeV, calcu- lated from the fits to the mixed semi-leptonic samples built from the data and the MC. 185 190 195 200 205 210 Data Mixed SL PYTHIA Fit to PYTHIA ARIADNE Fit to ARIADNE HERWIG Fit to HERWIG √s (GeV) DELPHI Figure 8: The ratio Rmixed SL as function of s for data and MC, and fits to the MC (see text). The ARIADNE points at 189 GeV and at 206 GeV have their centre-of-mass energy shifted and the error bars on data are tilted for readability. MC sample PYTHIA ARIADNE HERWIG Rmixed SLData/Rmixed SLMC 1.053 1.044 0.997 RCalibrated4qMC 1.018 1.011 1.004 Table 4: Ratio of data to MC fitted values of R in mixed semi-leptonic samples, used to calibrate the R4qMC values for different models (upper line), and calibrated values of R4qMC. All values were computed at s = 189 GeV. The values for Rmixed SLMC are presented in Table 4, for the different models, along with the calibrated values of R4q for the same models. The calibration factors differ from unity by less than 6%, and the largest difference of the calibrated R4qMC values when changing the fragmentation model, 0.014, was consid- ered as an estimate of the systematic error due to simulation of the fragmentation and of the detector response, and was added in quadrature to the systematic error. The error in the calibrated R4qMC values due to the statistical error on 〈Rmixed SLData〉 value used for the calibration, 0.026, was also added in quadrature to the systematic error. 4.4.2 Bose-Einstein Correlations Bose-Einstein correlations (BEC) between identical pions and kaons are known to exist and were established and studied in Z hadronic decays in [29]. They are expected to exist with a similar behaviour in the W hadronic decays, and this is studied in [2]. They are implemented in the MC simulation samples with BEC via the BE32 model of LUBOEI [21], which was tuned to describe the DELPHI data in [2]. However, the situation for the WW (ZZ) fully hadronic decays is not so clear, i.e. whether there are correlations only between pions and kaons coming from the same W(Z) boson or also between pions and kaons from different W(Z) bosons. The analyses of Bose-Einstein correlations between identical particles coming from the decay of different W bosons do not show a significant effect [30] for three of the LEP experiments, whereas for DELPHI, an effect was found at the level of 2.4 standard deviations [2]. Thus, a comparison was made between the WPHACT samples without CR and with BEC only between the identical pions coming from the same W boson (BEC only inside), to the samples without CR and with BEC allowed for all the particles stemming from both W bosons, implemented with the BE32 variant of the LUBOEI model (BEC all). The R values were obtained at each centre-of-mass energy, after which a linear fit was performed for each model to obtain a best prediction at 189 GeV. The fit values were found to be in agreement to the estimate at 189 GeV alone, and for simplicity this estimate was used. The measurement of BEC from DELPHI of 2.4 standard deviations above zero (corresponding to BEC only inside), was used to interpolate the range of 4.1 standard deviations of separation between BEC only inside and BEC all. To include the error on the measured BEC effect, one standard deviation was added to the effect before the interpolation. The difference in the estimated values of R at s = 189 GeV, between the model with BEC only inside and the model with partial BEC all (at the interpolated point of 3.4/4.1), -0.013, was added in quadrature to the systematic error. 4.4.3 qq̄(γ) Background Shape The fragmentation effects, in the shape of the qq̄(γ) background, were estimated by comparing the values of R obtained when the subtracted qq̄(γ) sample was fragmented with ARIADNE instead of PYTHIA at the centre-of-mass energy of 189 GeV, and the differ- ence, 0.003, was added in quadrature to the systematic error. 4.4.4 qq̄(γ) and ZZ Background Contribution At the centre-of-mass energy of 189 GeV, the qq̄(γ) cross-section in the 4-jet region is poorly known, due to the difficulty in isolating the qq̄(γ) → 4-jet signal from other 4-jet processes such as WW and ZZ. The study performed in [31] has shown that the maximal difference in the estimated qq̄(γ) background rate is 10% coming from changing from PYTHIA to HERWIG as the hadronization model, with the ARIADNE model giving intermediate results. Conservatively, at each centre-of-mass energy a variation of 10% on the qq̄(γ) cross-section was assumed, and the largest shift in R, 0.011, was added in quadrature to the systematic error. The other background process considered is the Z pair production. The Standard Model predicted cross-sections are in agreement with the data at an error level of 10% [32]. The cross-section was thus varied by ±10% at each energy and the effect in R was found to be negligible. 4.4.5 Evolution of R with Energy The R ratios were rescaled to s = 189 GeV using the fit to the MC without CR, however the correct behaviour might be given by the MC with CR. Hence, the difference of 0.009 between the R values obtained using the two rescaling methods, using MC without CR 〈R〉 and with CR 〈RCR rescale〉, was added in quadrature to the systematic error. 4.5 Results of the Particle Flow Analysis The final result for the average of the ratios R rescaled to 189 GeV is MC Sample R PYTHIA no CR 1.037± 0.004 PYTHIA SK-I 100% 0.917± 0.003 ARIADNE no CR 1.053± 0.004 ARIADNE AR2 1.021± 0.004 HERWIG no CR 1.059± 0.004 HERWIG 1/9 CR 1.040± 0.003 Table 5: R ratios for the Cetraro samples at 189 GeV, calibrated with the mixed semi-lep- tonic events. 〈R〉 = 0.979± 0.032(stat)± 0.035(syst). (8) In order to facilitate comparisons between the four LEP experiments, this value can be normalised by the one determined from simulation samples produced with the full detector simulation and analysed with the same method. The LEP experiments agreed to use for this purpose the Cetraro PYTHIA samples. These events were generated with the ALEPH fragmentation tuning but have been reconstructed with the DELPHI detector simulation and analysed with this analysis. The values of the R ratios obtained from the Cetraro samples at 189 GeV, calibrated using the mixed semi-leptonic events from these samples, are given in Table 5. The value of 〈R〉 measured from data is between the expected R ratios from PYTHIA without CR and with the SK-I model with 100% fraction of reconnection. The error of this measurement is larger than the difference between the values of R from ARIADNE samples without and with CR, and than the difference between values of R from the HERWIG samples without CR and with 1/9 of reconnected events. The following normalised ratios are obtained for the sample without CR and imple- menting the SK-I model with 100% CR probability, respectively: rdatano CR = 〈R〉data Rno CR = 0.944± 0.031(stat)± 0.034(syst), (9) rdataCR = 〈R〉data = 1.067± 0.035(stat)± 0.039(syst). (10) In the above expressions, the statistical errors in the MC predicted values were propagated and added quadratically to the systematic errors on the ratios. It is also possible to define the following quantity, taking the predictions for RCR and Rno CR at s = 189 GeV from the PYTHIA samples in Table 5, 〈Rdata〉 −Rno CR RCR − Rno CR = 0.49± 0.27(stat)± 0.29(syst) , (11) from which it can be concluded that the measured 〈Rdata〉 is compatible with intermediate probability of CR, and differs from the CR in the SK-I model at 100% at the level of 1.3 standard deviations. The ability to distinguish between these two models can be computed from the inverse of the sum in quadrature of the statistical and systematic errors; it amounts to be 2.5 standard deviations. In Figure 9 the result of δr is compared to the predicted values, in the scope of the SK-I model, as a funtion of the fraction of reconnected events. Fraction of reconnected events % DELPHI 0 20 40 60 80 100 Figure 9: Comparison of the measurement of the δr observable to the predictions from the SK-I model as a function of the fraction of reconnected events. The result for the value of 〈R〉 can also be used to test for consistency with the SK-I model as a function of κ and a log-likelihood curve was obtained. This also facilitates combination with the result obtained in the analysis in the following section, and for this reason the value of 〈R〉 is rescaled with PYTHIA without CR to a centre-of-mass energy of 200 GeV: the value obtained at 200 GeV is 〈R〉(200 GeV) = 1.024 ± 0.050. The values obtained for the predicted ratios RN at 200 GeV and the log-likelihood curve, as a function of κ, are shown in Figure 10. The value of κ most compatible with the data within one standard deviation is κSK-I = 4.13 +20.97 −3.46 . (12) 5 Different MW Estimators as Observables It has been shown [12] that the MW measurement inferred from hadronically decaying W+W− events at LEP-2, by the method of direct reconstruction, is influenced by CR effects, most visible when changing the value of κ in the SK-I model. For the MW(4q) estimator within DELPHI this is shown in [33]. Other published MW estimators in LEP experiments are equally sensitive to κ [34]. To probe this sensitivity to CR effects, alternative estimators for the MW measure- ment were designed which have different sensitivity to κ. In the following, the standard estimator and two alternative estimators, studied in this paper, are presented. The stan- dard estimator corresponds to that previously used in the measurement of the W mass by DELPHI [33]. Note that in the final DELPHI W mass analysis [35] results are given 0. 0. Fraction of reconnected events % DELPHI DELPHI Fraction of reconnected events % DELPHI DELPHI 0 20 40 60 80 100 0 20 40 60 80 100 Figure 10: a) Estimated ratio RN at 200 GeV plotted as a function of different κ values (top scale), or as function of the corresponding reconnection probabilities (bottom scale), compared to 〈R〉 measured from data after rescaling to 200 GeV (horizontal lines marked with R for the value and with 1σ(2σ) for the 〈R〉 value added/subtracted by one(two) standard deviations); the last three marks on the x axis, close to 100% of reconnection probability, correspond respectively to the values κ = 100, 300, 800; b) corresponding log-likelihood curve for the comparison of the estimated values (RN ) with the data (〈R〉). for the standard and hybrid cone estimators, with the hybrid cone estimator used to provide the primary result. The data samples, efficiencies and purities for the analysis corresponding to the standard estimator are provided in [33, 35]. • The standard MW estimator : This estimator is described in [33] and was optimised to obtain the smallest sta- tistical uncertainty for the W mass measurement. It results in an event-by-event likelihood Li(MW) for the parameter MW. • The momentum cut MW estimator : For this alternative MW estimator the event selection was performed in exactly the same way as for the standard MW estimator. The particle-jet association was also taken from this analysis. However, when reconstructing the event for the MW extraction a tighter track selection was applied. The momentum and energy of the jets were calculated only from those tracks having a momentum higher than a certain pcut value. An event-by-event likelihood L i (MW) was then calculated. • The hybrid cone MW estimator : In this second alternative MW estimator the reconstruction of the event is the same as for the standard analysis, except when calculating the jet momenta used for the MW extraction. coneR (cone) (std) Figure 11: Illustration of the iterative cone algorithm within a predefined jet as explained in the text. An iterative procedure was used within each jet (defined by the clustering algorithm used in the standard analysis) to find a stable direction of a cone excluding some particles in the calculation of the jet momentum, illustrated in Figure 11. Starting with the direction of the original jet ~p std , the jet direction was recalculated (direction (1) on the Figure) only from those particles which have an opening angle smaller than Rcone with this original jet. This process was iterated by constructing a second cone (of the same opening angle) around this new jet direction and the jet direction was recalculated again. The iteration was continued until a stable jet direction ~p jetcone was found. The jet momenta obtained, ~p jetcone, were rescaled to compensate for the lost energy of particles outside the stable cone, ~p jetcone → ~p jetcone · Ejetcone . (13) The energies of the jets were taken to be the same as those obtained with the standard clustering algorithm (E jetcone → E jet). This was done to increase the correlation of this estimator with the standard one. The rescaling was not done for the pcut estimator as it will be used in a cross-check observable with different systematic properties. Again the result is an event-by-event likelihood LRconei (MW). Each of these previously defined MW likelihoods had to be calibrated. The slope of the linear calibration curve for the MW estimators is tuned to be unity, therefore only a bias correction induced by the reconstruction method has to be applied. This bias is estimated with the nominal WPHACT Monte Carlo events and the dependence on the value of κ is estimated with the EXCALIBUR simulation. It was verified for smaller subsets that the results using these large EXCALIBUR samples and the samples generated with WPHACT are compatible. Neglecting the possible existence of Colour Reconnection (CR) in the Monte Carlo simulation results in event likelihoods Li(MW|event without CR), while Li(MW|event with CR) are the event likelihoods obtained when assuming the hypothesis that events do reconnect (100% CR in the scope of the SK-I model). To construct the event likelihoods for intermediate CR (values of κ larger than 0) the following weighting formula is used : Li(MW|κ) = [1−Pi(κ)]·Li(MW|event without CR)+Pi(κ)·Li(MW|event with CR) (14) where Pi(κ) is defined in Equation 1. The combined likelihood is produced for the event sample; the calibrated values for MW(κ) were obtained for different val- ues of κ using the maximum likelihood principle. In Figure 12 the difference dMW(κ) = MW(κ)−MW(κ = 0) or the influence of κ on the bias of the MW estimator is presented as function of κ. The uncertainty on this difference is estimated with the Jackknife method [27] to take the correlation between MW(κ) and MW(κ = 0) into account. It was observed from simulations that the estimators dependency on κ, for κ below about 5, was not signifi- cantly different in the centre-of-mass range between 189 and 207 GeV. Therefore in the determination of κ the dependency at 200 GeV was taken as default for all centre-of- mass energies. This value of centre-of-mass energy is close to the integrated luminosity weighted centre-of-mass energy of the complete data sample, which is 197.1 GeV. When neglecting the information content of low momentum particles or when using the hybrid cone algorithm, the influence of Colour Reconnection on the MW estimator is decreased. The dependence ∂MW of the estimator to κ is decreased when increasing the value of pcut or when working with smaller cone opening angles Rcone. 5.1 The Measurement of κ The observed difference ∆MW(std, i) = MW std − MWi in the event sample, where i is a certain alternative analysis, provides a measurement of κ. When both estimators std and MW i are calibrated in the same hypothesis of κ, the expectation values of ∆MW(std, i) will be invariant under a change of pcut or Rcone. When neglecting part of the information content of the events in these alternative MW analyses, by increasing pcut or decreasing Rcone, the statistical uncertainty on the value of 1 10 10 Standard MW analysis DELPHI pcut = 1 GeV/c pcut = 2 GeV/c pcut = 3 GeV/c Cone R=1.00 rad Cone R=0.75 rad Cone R=0.50 rad Cone R=0.25 rad SK-I Model parameter κ κ = 0.66 Figure 12: The difference dMW(κ) = MW(κ)−MW(κ = 0) is presented as a function of κ, for different MW estimators. The curve for the standard MW estimator is the curve at the top. The curves obtained with the hybrid cone analysis for different values of the cone opening angle, starting from the top with 1.00 rad down to 0.75 rad, 0.50 rad and 0.25 rad are indicated with dotted lines. The curves obtained with the momentum cut analysis for different values of pcut, starting from the top with 1 GeV/c, down to 2 GeV/c and 3 GeV/c are dashed. The vertical line indicates the value of κ preferred by the SK-I authors [5] and commonly used to estimate systematic uncertainties on measurements using e+e− → W+W− → q1q̄2q3q̄4 events. the MW estimator is increased. Therefore a balance must be found between the statistical precision on ∆MW(std, i) and the dependence of this difference to κ in order to obtain the largest sensitivity for a κ measurement. This optimum was found using the Monte Carlo simulated events and assuming that the data follow the κ = 0 hypothesis, resulting in the smallest expected uncertainty on the estimation of κ. For the pcut analysis an optimal sensitivity was found when using the difference ∆MW(std, pcut) with pcut equal to 2 GeV/c or 3 GeV/c. Even more information about κ could be extracted from the data, when using the difference ∆MW(std,Rcone), which was found to have an optimal sensitivity around Rcone = 0.5 rad. No significant im- provement in the sensitivity was found when combining the information from these two observables. Therefore the best measure of κ using this method is extracted from the ∆MW(std,Rcone = 0.5 rad) observable. Nevertheless, the ∆MW(std, pcut = 2GeV/c) ob- servable was studied as a cross-check. 5.2 Study of the Systematic Errors in the ∆MW Method The estimation of systematic uncertainties on the observables ∆MW(std, i) follows similar methods to those used within the MW analysis. Here the double difference is a measure of the systematic uncertainty between Monte Carlo simulation (‘MC’) and real data (‘DA’): ∆syst(MC,DA) = std(MC)−MWstd(DA)]− [MWi(MC)−MWi(DA)] ∣ (15) where i is one of the alternative MW estimators. The systematic error components are described below and summarised in Table 6. 5.2.1 Jet Reconstruction systematics with MLBZs A novel technique was proposed in [36] to study systematic uncertainties on jet recon- struction and fragmentation in W physics measurements with high statistical precision through the use of Mixed Lorentz Boosted Z events (MLBZs). The technique is similar to the one described in section 4.4.1. The main advantage of this method was that Monte Carlo simulated jet properties in W+W− events could be directly compared with the corresponding ones from real data using the large Z statistics. The main extension of the method beyond that described in [36] consisted in an improved mixing and boosting procedure of the Z events into MLBZs, demonstrated in Figure 13. The 4-momenta of the four primary quarks in WPHACT generated W+W− → q1q̄2q3q̄4 events were used as event templates. The Z events from data or simulation were chosen such that their thrust axis directions were close in polar angle to one of the primary quarks of the W+W− event template. Each template W was then boosted to its rest frame. The particles in the final state of a selected Z event were rotated so that the thrust axis matches the rest frame direction of the primary quarks in the W+W− template. After rescaling the kinematics of the Z events to match the W boson mass in the generated W+W− template, the two Z events were boosted to the lab frame of the W+W− template. All particles having an absolute polar angle with the beam direction smaller than 11◦ were removed from the event. The same generated WPHACT events were used for the construction of both the data MLBZs and Monte Carlo MLBZs in order to increase the correlation between both emulated samples to about 31%. This correlation was taken into account when boost lab frame WW "re−boost" rotate rest frame WW lab frame MLBZ boost Figure 13: Illustration of the mixing and boosting procedure within the MLBZ method (see text for details). quoting the statistical uncertainty on the systematic shift on the observables between data and Monte Carlo MLBZs. It was verified that when introducing a significant mass shift of 300 MeV/c2 on MW by using the cone rejection algorithm, it was reproduced within 15% by applying the MLBZ technique. Because the expected systematic uncertainties on the ∆MW(std, i) observables of interest are one order of magnitude smaller than 300 MeV/c2, this method is clearly justified. The double difference of Equation 15 was determined with the MLBZ method using Z events selected in the data sets collected during the 1998 calibration runs and Z events from the corresponding Monte Carlo samples. The following results were obtained for the ∆MW(std,Rcone = 0.5 rad) observable: ∆syst(ARIADNE ,DA) = −1.9 ± 3.9(stat)MeV/c2 ∆syst(PYTHIA ,DA) = −5.7 ± 3.9(stat)MeV/c2 ∆syst(HERWIG ,DA) = −10.6 ± 3.9(stat)MeV/c2 where the statistical uncertainty takes into account the correlation between the Monte Carlo and the data MLBZ events, together with the correlation between the two MW estimators. This indicates that most of the fragmentation, detector and Between-W Bose-Einstein Correlation systematics are small. The study was not performed for the ∆MW(std, pcut) observable. Other systematic sources on the reconstructed jets are not considered as the MW estimators used in the difference ∆MW(std, i) have a large correlation. 5.2.2 Additional Fragmentation systematic study The fragmentation of the primary partons is modelled in the Monte Carlo simulation used for the calibration of the MW i observables. The expected values on the MW estimators from simulation (in the κ = 0 hypothe- sis) are changed when using different fragmentation models [33], resulting in systematic uncertainties on the measured MW i observables and hence possibly also on our esti- mated κ. In Figure 14 the systematic shift δMW in the different MW i observables is shown when using HERWIG or ARIADNE rather than PYTHIA as the fragmentation model in the no Colour Reconnection hypothesis. When inferring κ from the data difference, ∆MW(std, i), the PYTHIA model is used to calibrate each MW i observable. This data difference for MW pcut=2GeV/c, ∆MW(std, pcut = 2GeV/c), changes 3 by (27 ± 12) MeV/c2 or (8 ± 12) MeV/c2 when replacing PYTHIA by respectively HERWIG or ARIADNE. Simi- larly, the observable ∆MW(std,Rcone = 0.5 rad) changes by (-4 ± 10) MeV/c2 or (-6 ± 10) MeV/c2 when replacing PYTHIA by respectively HERWIG or ARIADNE. The largest shift of the observable when changing fragmentation models (or the uncertainty on this shift if larger) is taken as systematic uncertainty on the value of the observable. Hence, systematic errors of 27 MeV/c2 for the ∆MW(std, pcut = 2GeV/c) observable and 10 MeV/c2 for the ∆MW(std,Rcone = 0.5 rad) observable were assumed as the contribution from fragmentation uncertainties. The MLBZ studies (see above) are compatible with these results, hence no additional systematic due to fragmentation was quoted for the ∆MW(std,Rcone = 0.5 rad) observable. 3This change, ∆MW(std, pcut = 2GeV/c) PYTHIA − ∆MW(std, pcut = 2GeV/c) HERWIG , is given by δMW(std ≡ pcut = 0.2GeV/c) PYTHIA−HERWIG − δMW(pcut = 2GeV/c) PYTHIA−HERWIG, and similar expressions for the ARIADNE and Rcone cases (for Rcone, std ≡ Rcone = π). 0 0.5 1 1.5 2 2.5 3 3.5 4 PYTHIA-ARIADNE DELPHI PYTHIA-HERWIG pcut / GeV/c 0 0.2 0.4 0.6 0.8 1 1.2 PYTHIA-ARIADNE DELPHI PYTHIA-HERWIG Rcone / rad Figure 14: Systematic shifts δMW, on MW observables, when applying different fragmen- tation models as a function of the pcut or Rcone values used in the construction of the MW observable. These Monte Carlo estimates were obtained at a centre-of-mass energy of 189 GeV. The uncertainties are determined with the Jackknife method. 5.2.3 Energy Dependence The biases of the different MW estimators have a different dependence on the centre- of-mass energy, hence the calibration of ∆MW(i, j) will be energy dependent. The energy dependence of each individual MW estimator was parameterised with a second order poly- nomial. Since WPHACT event samples were used at a range of centre-of-mass energies the uncertainty on the parameters describing these curves are small. Therefore a small systematic uncertainty of 3 MeV/c2 was quoted on the ∆MW(i, j) observables due to the calibration. 5.2.4 Background The same event selection criteria were applied for all the MW estimators, hence the same background contamination is present in all analyses. The influence of the qq̄(γ) background events on the individual MW estimators is small [33] and was taken into account when constructing the centre-of-mass energy dependent calibration curves of the individual MW estimators. The residual systematic uncertainty on both ∆MW(i, j) observables is 3 MeV/c2. 5.2.5 Bose-Einstein Correlations As for the particle flow method, the systematic uncertainties due to possible Bose- Einstein Correlations are estimated via Monte Carlo simulations. The relevant values for the systematic uncertainties on the observables are the differences between the ob- servables obtained from the Monte Carlo events with Bose-Einstein Correlations inside individual W’s (BEI) and those with, in addition, Bose-Einstein Correlations between identical particles from different W’s (BEA). The values were estimated to be (6.4 ± 9.3) MeV/c2 for the ∆MW(std, pcut = 2GeV/c) observable, and (7.2 ± 8.2) MeV/c2 for the ∆MW(std,Rcone = 0.5 rad) observable. As the uncertainties in the estimated contri- butions were larger than the contributions themselves, these uncertainties were added in quadrature to the systematic errors on the relevant observables. 5.2.6 Cross-check in the Semi-leptonic Channel Colour Reconnection between the decay products originating from different W boson decays can only occur in the W+W− → q1q̄2q3q̄4 channel. The semi-leptonic W+W− decay channel (i.e, qq̄′ℓνℓ) is by definition free of those effects. Therefore the determi- nation of Colour Reconnection sensitive observables, like ∆MW(std,Rcone = 0.5 rad), in this decay channel could indicate the possible presence of residual systematic effects. A study of the ∆MW(std,Rcone = 0.5 rad) observable was performed in the semi-leptonic decay channel. The semi-leptonic MW analysis in [33] was used and the cone algorithm was implemented in a similar way as for the fully hadronic decay channel. The same data sets have been used as presented throughout this paper and the following result was obtained: ∆MW(std,Rcone) = MW std − MWRcone = (8 ± 56(stat))MeV/c2 (17) where the statistical uncertainty was computed taking into account the correlation be- tween both measurements. Although the statistical significance of this cross-check is small, a good agreement was found for both MW estimators. 5.3 Results from the MW Estimators Analyses The observable ∆MW(std,Rcone) with Rcone equal to 0.5 rad (defined above), was found to be the most sensitive to the SK-I Colour Reconnection model, and the ∆MW(std, pcut = 2GeV/c) observable was measured as a cross-check. The analyses were calibrated with PYTHIA κ = 0 WPHACT generated simulation events. The values measured from the combined DELPHI data at centre-of-mass energies ranging between 183 and 208 GeV are: ∆MW(std,Rcone) = MW std − MWRcone = (59 ± 35(stat) ± 14(syst))MeV/c2 ∆MW(std, pcut) = MW std − MWpcut = (143 ± 61(stat) ± 29(syst))MeV/c2 where the first uncertainty numbers represent the statistical components and the sec- ond the combined systematic ones. The full breakdown of the uncertainties on both observables can be found in Table 6. Uncertainty contribution (MeV/c2) Source ∆MW(std,Rcone = 0.5 rad) ∆MW(std, pcut = 2GeV/c) Fragmentation 11 27 Calibration 3 3 Background 3 3 BEI-BEA 8 9 Total systematic 14 29 Statistical Error 35 61 Total 38 67 Table 6: Breakdown of the total uncertainty on both relevant observables. From these values estimates were made for the κ parameter by comparing them with the Monte Carlo expected values in different hypothesis of κ, shown in Figure 15 for the observable ∆MW(std,Rcone = 0.5 rad). The Gaussian uncertainty on the measured observables was used to construct a log- likelihood function L(κ) = −2 log L(κ) for κ. The log-likelihood function obtained is shown in Figure 16 for the first and in Figure 17 for the second observable. The result shown in Figure 16 is the primary result of this analysis, because of the larger sensitivity of the ∆MW(std,Rcone = 0.5 rad) observable to the value of κ (see sec- tion 5.1). The value of κ most compatible with the data within one standard deviation of the measurement is κSK-I = 1.75 +2.60 −1.30 . (19) The result on κ extracted from the cross-check ∆MW(std, pcut = 2GeV/c) observable is found not to differ significantly from the quoted result obtained with the more opti- mal ∆MW(std,Rcone = 0.5 rad) observable. The significance can be determined by the difference between both MW estimators : pcut − MWRcone = (−84 ± 59(stat))MeV/c2 . (20) Taking into account that the expectation of this difference depends on κ, we find a sta- tistical deviation of about 1 to 1.5σ between the measurements. No improved sensitivity is obtained by combining the information of both observables. 200 DELPHI at 188.6 GeV at 199.5 GeV at 206.5 GeV SK-I Model parameter κ κ = 0.66 Figure 15: The dependence of the observable ∆MW(std,Rcone = 0.5 rad) from simulation events on the value of the SK-I model parameter κ. The dependence is given at three centre-of-mass energies. 1 10 10 Likelihood of indirect measurement of κ SK-I DELPHI measured : std - R=0.50 (stat+syst) DELPHI measured : std - R=0.50 (stat) SK-I Model parameter κ Figure 16: The log-likelihood function −2 log L(κ) obtained from the DELPHI data mea- surement of ∆MW(std,Rcone = 0.5 rad). The bottom curve (full line) gives the final result including the statistical uncertainty on ∆MW(std,Rcone = 0.5 rad) and the investigated systematic uncertainty contributions. The top curve (dashed) is centred on the same min- imum and reflects the log-likelihood function obtained when only statistical uncertainties are taken into account. 1 10 10 Likelihood of indirect measurement of κ SK-I DELPHI measured : std - pcut=2 (stat+syst) DELPHI measured : std - pcut=2 (stat) SK-I Model parameter κ Figure 17: The log-likelihood function −2 log L(κ) obtained from the DELPHI data mea- surement of ∆MW(std, pcut = 2GeV/c). The bottom curve (full line) gives the final result including the statistical uncertainty on ∆MW(std, pcut = 2GeV/c) and the investigated systematic uncertainty contributions. The top curve (dashed) is centred on the same min- imum and reflects the log-likelihood function obtained when only statistical uncertainties are taken into account. In this paper the SK-I model for Colour Reconnection implemented in PYTHIA was studied because it parameterizes the effect as function of the model parameter κ. Other phenomenological models implemented in the ARIADNE [7,8] and HERWIG [6] Monte Carlo fragmentation schemes exist and are equally plausible. Unfortunately their effect in W+W− → q1q̄2q3q̄4 events cannot be scaled with a model parameter, analogous to κ in SK-I, without affecting the fragmentation model parameters. Despite this non- factorization property, the consistency of these models with the data can still be ex- amined. The Monte Carlo predictions of the observables in the hypothesis with Colour Reconnection (calibrated in the hypothesis of no Colour Reconnection) give the following values: ARIADNE → MWstd − MWRcone = (7.2 ± 4.1) MeV/c2 ARIADNE → MWstd − MWpcut = (9.4 ± 7.0) MeV/c2 HERWIG → MWstd − MWRcone = (19.7 ± 4.0) MeV/c2 HERWIG → MWstd − MWpcut = (22.8 ± 6.9) MeV/c2 . The small effects on the observables with the HERWIG implementation of Colour Reconnec- tion compared to those predicted by SK-I are due to the fact that the fraction of events that reconnect is smaller in HERWIG (1/9) compared to SK-I (& 25% at s = 200 GeV). After applying this scale factor between both models, their predicted effect on the W mass and on the ∆MW(i, j) observables becomes compatible. The ARIADNE implementa- tion of Colour Reconnection has a much smaller influence on the observables compared to those predicted with the SK-I and HERWIG Monte Carlo. 5.4 Correlation with Direct MW Measurement When using a data observable to estimate systematic uncertainties on some measur- and inferred from the same data sample, the correlation between the estimator used to measure the systematic bias and the estimator of the absolute value of the measurand should be taken into account. Therefore the correlation between the Colour Reconnection sensitive observables ∆MW(std,Rcone = 0.5 rad) and ∆MW(std, pcut = 2GeV/c) and the absolute MW(std) estimator was calculated. The correlation was determined from the Monte Carlo events and with κ = 0 or no Colour Reconnection. The values obtained were found to be stable as a function of κ within the statistical precision. The correlation between ∆MW(std,Rcone = 0.5 rad) and MW(std) was found to be 11%, while for the one between ∆MW(std, pcut = 2GeV/c) and MW(std) a value of 8% was obtained. Also the correlation between the different MW estimators was estimated and found to be stable with the value of κ. A value of 83% was obtained for the correlation between MW(std) and MW Rcone=0.5 rad, while 66% was obtained between MW(std) and MW pcut=2GeV/c. 6 Combination of the Results in the Scope of the SK-I Model The log-likelihood curve from the particle flow method was combined with the curve from the ∆MW method and the result is shown in Figure 18. The correlations between the analyses were neglected because the overlap between the samples is small and the nature of the analyses is very different. The total errors were used (statistical and systematic added in quadrature) in the combination. 1 10 10 Log-Likelihood of measurement of κ SK-I from ∆MW from particle flow ∆MW+part.flow combined SK-I Model parameter κ DELPHI Figure 18: The log-likelihood function −2 log L(κ) obtained from the combined DELPHI measurement via ∆MW(std,Rcone = 0.5 rad) and the particle flow. The full line gives the final result including the statistical and systematic uncertainties. The log-likelihood functions are combined in the hypothesis of no correlation between the statistical and systematic uncertainties of both measurements. The best value for κ from the minimum of the curve, with its error given by the width of the curve at the value −2 log L = (−2 log L)min + 1, is: κSK-I = 2.2 −1.3 . (22) 7 Conclusions Colour Reconnection (CR) effects in the fully hadronic decays of W pairs, produced in the DELPHI experiment at LEP, were investigated using the methods of the particle flow and the MW estimators, notably the ∆MW(std,Rcone = 0.5 rad) observable. The average of the ratios R of the integrals between 0.2 and 0.8 of the particle distri- bution in Inside-W regions to the Between-W regions was found to be 〈R〉 = 0.979± 0.032(stat)± 0.035(syst) . (23) The values used in this average were obtained after rescaling the value at each energy to the value at a centre-of-mass energy of 189 GeV using a fit to the MC without CR. The effects of CR on the values of the reconstructed mass of the W boson, as imple- mented in different Monte Carlo models, were studied with different estimators. From the estimator of the W mass with the strongest sensitivity to the SK-I model of CR, the ∆MW(std,Rcone = 0.5 rad) method, the difference in data was found to be ∆MW(std,Rcone) = MW std −MWRcone=0.5 rad = ( 59± 35(stat)± 14(syst) )MeV/c2 . (24) From the combination of the results from particle flow and MW estimators, corre- sponding to the curve in full line shown in Figure 18, the best value and total error for the κ parameter in the SK-I model was extracted to be: κSK-I = 2.2 −1.3 (25) which corresponds to a probability of reconnection of Preco = 52% and lies in the range 31% < Preco < 68% at 68% confidence level. The two analysis methods used in this paper are complementary: the method of parti- cle flow provides a model-independent measurement but has significantly less sensitivity towards the SK-I model of CR than the method of ∆MW estimators. The obtained value of κ in equation (25) can be compared with similar values obtained by other LEP experiments, and it was found to be compatible with, but higher than, the values obtained with the particle flow by L3 [37] and OPAL [38]. It is also compatible with, but higher than, the values obtained with the method of different MW estimators by OPAL [39] and ALEPH [40]. Acknowledgements We thank the ALEPH Collaboration for the production of the simulated “Cetraro Samples”. We are greatly indebted to our technical collaborators, to the members of the CERN- SL Division for the excellent performance of the LEP collider, and to the funding agencies for their support in building and operating the DELPHI detector. We acknowledge in particular the support of Austrian Federal Ministry of Education, Science and Culture, GZ 616.364/2-III/2a/98, FNRS–FWO, Flanders Institute to encourage scientific and technological research in the industry (IWT) and Belgian Federal Office for Scientific, Technical and Cultural affairs (OSTC), Belgium, FINEP, CNPq, CAPES, FUJB and FAPERJ, Brazil, Czech Ministry of Industry and Trade, GA CR 202/99/1362, Commission of the European Communities (DG XII), Direction des Sciences de la Matière, CEA, France, Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie, Germany, General Secretariat for Research and Technology, Greece, National Science Foundation (NWO) and Foundation for Research on Matter (FOM), The Netherlands, Norwegian Research Council, State Committee for Scientific Research, Poland, SPUB-M/CERN/PO3/DZ296/2000, SPUB-M/CERN/PO3/DZ297/2000, 2P03B 104 19 and 2P03B 69 23(2002-2004) FCT - Fundação para a Ciência e Tecnologia, Portugal, Vedecka grantova agentura MS SR, Slovakia, Nr. 95/5195/134, Ministry of Science and Technology of the Republic of Slovenia, CICYT, Spain, AEN99-0950 and AEN99-0761, The Swedish Research Council, Particle Physics and Astronomy Research Council, UK, Department of Energy, USA, DE-FG02-01ER41155, EEC RTN contract HPRN-CT-00292-2002. 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0704.0598

Evolutionary Neural Gas (ENG): A Model of Self Organizing Network from Input Categorization

Microsoft Word - ENG-EJTP.doc EVOLUTIONARY NEURAL GAS (ENG): A MODEL OF SELF ORGANIZING NETWORK FROM INPUT CATEGORIZATION. Ignazio Licata (a) ↑ , Luigi Lella (b) (a) Ixtucyber for Complex Systems, Marsala, TP and Institute for Scientific Methodology, Palermo, Italy; (b) A.R.C.H.I. - Advanced Research Center for Health Informatics, Ancona, Italy ABSTRACT Despite their claimed biological plausibility, most self organizing networks have strict topological constraints and consequently they cannot take into account a wide range of external stimuli. Furthermore their evolution is conditioned by deterministic laws which often are not correlated with the structural parameters and the global status of the network, as it should happen in a real biological system. In nature the environmental inputs are noise affected and “fuzzy”. Which thing sets the problem to investigate the possibility of emergent behaviour in a not strictly constrained net and subjected to different inputs. It is here presented a new model of Evolutionary Neural Gas (ENG) with any topological constraints, trained by probabilistic laws depending on the local distortion errors and the network dimension. The network is considered as a population of nodes that coexist in an ecosystem sharing local and global resources. Those particular features allow the network to quickly adapt to the environment, according to its dimensions. The ENG model analysis shows that the net evolves as a scale-free graph, and justifies in a deeply physical sense- the term “gas” here used. Key-words: Self-Organizing Networks; Neural Gas; Scale-Free Graph; Information in Network Functional Specialization. 1. INTRODUCTION Self organizing networks are systems widely used in categorization tasks. A network can be seen as a set A={c1, c2,… ,cn} of units with associated reference vectors wc∈R n where Rn is the same space where inputs are defined. Each unit (or node) can establish connections with the other ones, the units belonging to the same clusters are subjected to similar modification affecting their reference vectors. Self organizing networks can automatically adapt to input distributions without supervision by means of training algorithms that are simple sequences of deterministic rules. Competitive hebbian learning and neural gas are the most important strategies used for their training. Neural gas algorithm (Martinetz T.M. and Schulten K.J., 1991) sorts the network units according to the distance of their reference vector to each input. Then the reference vectors are adapted so that the ones related to the first nodes in the rank order are moved more close than the others to the considered input. Competitive hebbian learning (Martinetz and Schulten, 1991; Martinetz, 1993) consists in augmenting the weight of the link connecting the two units whose reference vectors are closest to the considered input (the two most activated units). Both strategies are examples of deterministic rules. As we know there are other rules that constrain the topology of the network which has a fixed dimensionality. That’s the case of Self Organizing Maps (Kohonen, 1982) and Growing Cell Structures (Fritzke, 1994). ↑ Corresponding author: [email protected] In other cases the network structures haven’t topological constraints, they take a well ordered distribution by exactly adapting to the manifold inputs. For example TRN (Martinetz and Schulten, 1994) and GNG are networks whose final structure is similar to a Delaunay Triangulation (Delaunay, 1934).We have tried to define a new self organizing network that is trained by probabilistic rules avoiding any topological constraints. According to Jefferson (1995) life and evolution are structured at least into four fundamental levels: molecular, cellular, organism and population. We propose a population level based on evolutionary algorithm where the network is seen as a population of units whose interactions are conditioned by the availability of resources in their ecosystem. The evolution of the population is driven by a selective process that favours the fittest units. This approach has a biological plausibility. As stated by recent theories (Edelman, 1987) human brain evolution is subjected to similar selective pressures. Obviously we are not interested in recreating the same structure as the human brain. Our work aims at finding innovative and effective solutions to the categorization problem adopting natural system strategies. So our system falls within the Artificial Life field (Langton, 1989). Our model is a complex system that shows emergent features. In particular its structure evolves as a scale free graph. In the training phase there arise clusters of units with a limited number of nodes that establish a great number of links with the others. Scale free graphs are a particular structure that is really common in natural systems. Human knowledge, for instance, seems to be structured as a scale free graph (Steyvers, Tenenbaum 2001). If we represent words and concepts as nodes, we’ll find that some of these are more connected than the others. Scale free graphs have three main features.The small world structure. It means there is a relatively short path between any couple of nodes (Watts, Strogatz, 1998).The inherent tendency to cluster that is quantified by a coefficient introduced by Watts and Strogatz. Given a node i of ki degree i.e. having ki edges which connect it to ki other nodes, if those make a cluster, they can establish ki(ki-1)/2 edges at best. The ratio between the actual number of edges and the maximum number gives the clustering coefficient of node i. The clustering coefficient of the whole network is the average of all the individual clustering coefficients. Scale free graphs are also characterized by a particular degree distribution that has a power-law tail P(k)~k n− . That’s why such networks are called “scale free” (Albert, Barabasi, 2000). The three previous features are quantified by three parameters: the average path length between any couple of nodes, the clustering coefficient and the exponent of the power law tail. We’ll show that the values of these parameters in our model seem to confirm its scale free nature. 2. AN OUTLINE ON SELF-ORGANIZATION AND EVOLUTIONARY SYSTEMS Natural selection mechanism has been successfully used for a lot of industrial applications spanning from projecting to real-time control and neural networks training. It was in the 60s that Genetic Algorithms based on the Evolution Theory’s three main mechanisms - reproduction, mutation and fitness – were first used in dealing with optimization problems. Although the solution is reached by a population of individuals, systems based on this approach are not considered self organizing because their dynamics depend on the external constraint of the fitness function. In the 80s a new approach to the study of living systems which mixed together self organization and evolutionary systems came out (Rocha, 1997). Its success was due to the studies on the way how biological systems work (metabolism, adaptability, autonomy, self repairing, growth, evolution etc.). The hybrid systems make us possible to get a better simulation both of the evolutionary optimization processes and the internal structure modification to reach a greater biological plausibility in the fitness Neuroevolutionary systems are an example of this approach. In classic neuroevolutionary models the network parameters are genetically set, whereas the connection weights are modified according to a training strategy. This solution follows the classic vision of cerebral development where genes control the formation of synaptic connections while their reinforcement depends on neural activity. More recent neuroevolutionary systems are characterized by different forms of self organizing processes which are cooperative coevolution (Paredis, 1995; Smith, Forrest and Perelson, 1993) and synaptic Darwinism (Edelman, 1987). Cooperative co evolutionary systems offer a promising alternative to classic evolutionary algorithms when we face complex dynamical problems. The main difference with respect to classic EA is the fact that each individual represents only a partial solution of the problem. Complete solutions are obtained by grouping several individuals. The goal of each individual is to optimize only a part of the solution, cooperating with other individuals that optimize other parts of the solution. It is so avoided the premature convergence towards a single group of individuals. An example of such approach is given by the Symbiotic Adaptive Neuroevolution System (Moriarty and Miikkulainen, 1998) that operates on populations of neural networks. While in most neuroevolutionary systems each individual represents a complete neural network, in SANE each individual represents a hidden unit of a two-layered network. Units are continuously combined and the resulting networks are evaluated on the basis of the performances shown in a given task. The global effect is equal to schemas promoting in standard EAs. In fact during the evolution of the population the neural schemas having the highest fitness values are favoured and the possible mutations in the copies of these schemas don’t affect the other copies in the population. Other recent strategies focus on the evolution of connection schemas in the network. In the human brain the number of synapses established by a single neuron is always much lower than the overall number of neurons. That gives the network a sparsely connected aspect. In the last years several models have been proposed to emulate the mechanism involved in the selection of links without referring to the physical and chemical properties of neurons. The Chialvo and Bak model (Chialvo and Bak, 1999) is based on two simple and biological inspired principles. First, the neural activity is kept low selecting the activated units by a winner takes all strategy. Second, the external environment gives a negative feedback that inhibits active synapses if the network behaviour is not satisfying. With these simple rules the model operates in a highly adaptive state and in critical conditions (extreme dynamics). The fundamental difference of this strategy based on the synaptic inhibition with respect to the classic one based on synaptic reinforcement is that the reinforcement-based learning is a continuative process by definition, while the inhibition-based learning stops when the training goal is achieved. The synaptic inhibition is also biologically plausible. According to Young (Young, 1964; Young, 1966) learning is the result of the elimination of synaptic connections (closing of unneeded channels). Dawkins (Dawkins R., 1971) stressed that pattern learning is achieved by synaptic inhibition. As stated by the neural groups’ selection theory developed by Edelman (Edelman, 1978; Edelman, 1987), brain development is characterized by generating a structural and dynamical variability within and between populations of neurons, by the interaction of the neural circuit with the environment and by the differential attenuation or amplification of synaptic connections. Research in neurobiology seems to confirm the validity of the negative feedback model and the fact that neural development follows the process of Darwinian evolution. The Chialvo and Bak model is a simple two-layered network. After the training each input pattern is associated with a single output unit leading to the formation of an associative map. When an input pattern is presented the most activated input unit i is selected. Then the neuron j from the hidden layer that establishes the most robust connection with i is selected. Finally the output neuron k that is the most strongly connected with j is selected. If k is not the desired output the two links connecting i with j and j with k are inhibited by a coefficient d that is the only parameter of the model. The iterative application of these rules leads to a rapid convergence towards any input- output mapping. This selective process followed by an inhibitory one is the essence of the natural selection in the evolutionary context. The fittest individual is selected on the basis of a strategy that doesn’t reward the best but punishes the worst. That’s the reason why this model has been considered a particular kind of synaptic Darwinism. Our neuroevolutionary model is also based on a selection strategy. The structural information of our network is not codified by genes. We directly consider the entire network as a population of nodes that can establish connections, generate other units or die. The probability of these events depends on the presence of local and global resources. If there are few resources the population falls, if there is a lot of resources the population grows. Like in the Chialvo and Bak model we don’t select the fittest nodes reinforcing their links, but we simply remove the worst nodes when the ecosystem resources are low. This generates a selective process that indirectly rewards the units which can better model the input patterns. Our evolutionary strategy can be seen as a selective retention process (Heylighen, 1992) that removes those units which cannot reach a stable state, remaining associated with several input patterns. Even if the stability of a unit is quantified by the minimum distortion error related to it, this information mustn’t be considered to be environmental information. The minimum distortion error simply quantifies the difficulty encountered by the unit during the modelling of input patterns. 3. THE EVOLUTIONARY ALGORITHM Research has confirmed (Roughgarden, 1979; Song and Yu, 1988) that in natural environments the population size along with competition and reproduction rates continuously changes according to some natural resources and the available space in the ecosystem. These mechanisms have been reproduced in some evolutionary algorithms, for example to optimize the evolution of a population of chromosomes in a genetic algorithm (Annunziato and Pizzuti, 2000). We have tried to use a similar strategy for the evolution of a population of units in a self organizing network without using the string representation of genetic programming. In our model each node is defined by a vector of neighbouring units connected to it, a reference vector and a variable D that is the smallest distance between its reference vector and the closest modelled input. The value of this variable quantifies the debility degree of the unit. The lower is D the higher are the chances for the unit to survive. At each presentation of the training input set, D is set to the maximum value. After the presentation of a given input x, if the reference vector w of the unit is modified, the resulting distance between the two vectors ||x-w|| is calculated. If this value is lower than D it becomes its new value. The training algorithm here used can be subdivided in three phases: 1) Winners are selected. For each input the unit having the closest reference vector is selected. 2) The reference vectors of the winners and their neighbours are updated according to the following formula : (3.1) ( ) ( ) ( )( )1w t w t x w tα+ = + − . So the reference vectors w of the selected units are moved towards the relative inputs x of a certain fraction of the distances that separate them. For winners this fraction is two or three orders of magnitude higher than the one used for their neighbours. So winners have the reference vectors moving more quickly towards the inputs. 3) The population of units evolves producing descendants, establishing new connections and eliminating the less performing units. All these events can occur with a well defined probability that depends on the availability of resources. These rules are iterated until a given goal is achieved. For example the minimization of the expected quantization error that is the mean of the distances between the winners and the K inputs they model: (3.2) D x wK = −� If this value falls below a certain threshold Dmin, training is stopped. The first two phases can be considered a kind of winner takes all strategy, where only the most activated units are selected and enabled to modify their reference vectors. The third phase is the evolutionary phase (fig. 3.1). Each unit i, i=[1…N(t)] where N(t) is the actual population size can meet the closest winner j with probability Pm: Fig. 3.1 – The evolutionary phase of the algorithm. If meeting occurs, the two units establish a link and they can interact by reproducing with probability Pr. In this case two new units are created. One is closer to the first parent, the other to the second parent: (3.3) If reproduction doesn’t take place due to the lack of resources the weaker unit of the population, i.e. the one with the highest debility degree, is removed. If unit i doesn’t meet any winner it can interact with the closest node k with probability Pr establishing a connection and producing a new unit whose reference vector is set between the parents reference vectors: (3.4) 1 22 2 p pw ww When we fix a maximum population size, the ratio between the actual size and the threshold N(t)/Nmax can be seen as a global resource of the ecosystem affecting the probabilities of the events. For example if the population size is low the reproduction rate should be high. So we can reasonably choose Pr = 1-N(t)/Nmax. If the population size is high, the chance for the units to meet each other will be higher, so we can set Pm = N(t)/Nmax. We can also consider a local resource that is the ratio between the threshold Dmin and the debility degree Di of the unit i. Each unit i should meet a winner with a probability Pm=(N(t)/Nmax)(1-Dmin/Di) and Pr = 1 – Pm. In this way winners are not encouraged to migrate to other groups of nodes and weaker units don’t participate in reproduction activities. We can estimate the population grow rate in the following way: (3.5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2min min 1 2 1 1 1 2 1 1 2 1 ( ) 2 1 1 1 2 1 1 ( ) m r m r m m d N t N t P P N t P P N t P N t P P N t N t P N t N t X t X t X t first model N t D D N t X t X t X t second model M D D + = + − − + − − − = = − = = − � + = −� � � � � �� � � �= − − � + = − −� � � �� � � �� �� �� � � �� �� � where X(t) is the normalized size N(t)/Nmax. This is the quadratic-logistic map of Annunziato and Pizzuti(Annunziato and Pizzuti, 2000): (3.6) ( ) ( ) ( )( )21 1X t aX t X t+ = − They proved that by varying the parameter different chaotic regimes arise. For a<1.7 the behaviour is not chaotic, for 1.7<a<2.1 we have chaotic regimes with simple attractors localized in a fixed part of the plane of the phases. Theoretically for the first model we expect to obtain a chaotic regime that is described by a simple attractor. In the second model the factor (1 – Dmin/D) might reduce the influence of the negative feedback in the final part of network training. It is possible to demonstrate that during the evolution the population size converges to N(t) = 0.72 Nmax. In this phase the probability that a unit establishes n connections with the other ones for the first model, considering only clusters of n units, is given by: (3.7) ( ) max0.72 1 max max 1max max 0.72 0.72 n i nN n P n n � � � � = − =� � � � � � � � It has to be pointed out we have subtracted the probability that such n links developed within a cluster of more than a n unit. The coefficients α and β of the power law are considered constant at the end of the training. To compute their values, we can take into consideration the cases n=1 and n=0.72N Nmax-1 which correspond to the minimum and maximum number of connection at the end of the training. (3.8) ( ) max0.72 2 1 0.72 0.72 1 P βα α = − = =� ( ) ( ) 0.72 2 0.72 1 1 max max 0.72 1 0.72 0.72 0.72 0.72 1 P N N − = = − −� � max max 0.72 2 0.72 1 0.72 1 0.72 0.72 � �� = The distribution tail of the degrees tends to stretch when the maximum size of the population increases, it means that in wider networks there are more hubs with a higher degree. For the second model we can consider that at the end of the training (1-Dmin/D) ∼ε So the probability that a unit establishes n links becomes: (3.9) max0.72 1 max max 1max max 0.72 0.72 n i nN n P n n βε ε α � � � � = − =� � � � � � � � max max 0.72 2 0.72 1 0.72 1 0.72 0.72 � �� = and the considerations made for the first model can be therefore extended to the second model. 4. TRAINING THE NET: SIMULATIONS We have compared the performances of our networks with those of a Growing Neural Gas in categorizing bidimensional inputs. GNG is a self organizing network which thanks to both the competitive hebbian learning strategy and the neural gas algorithm can categorize inputs without altering their exact dimensionality.For the GNG, the parameters of the model are �����α = 0.5, β = 0.0005 and at each λ = 300 steps a new unit is inserted. The maximum age of the links is set to 88. For the two different ENG models, the parameters are α = 0.05, β = 0.0006 and the maximum size is set to Nmax = 120. As stopping criterion for both the algorithms we have chosen the minimization of the expected quantization error that is the average distance between the winners and the corresponding inputs. We have considered two different input domains. In the first case inputs are localized within four square regions, in the second one inputs are uniformly distributed in a ring region. As shown in fig.4.1 after the training, GNG reference vectors are all positioned in the input domain. In the Evolutionary Self Organizing Networks (fig.4.2a and fig.4.2b) some units fall outside the input domain, but in this way the network remains fully connected. The nodes’ distribution statistical analysis confirms what appears to be intuitively patent: the emerging network structure is a typical scale-free one, i.e. a structure where few hubs manage the links. Fig. 4.1 – Growing Neural Gas simulations. Fig. 4.2 a – Evolutionary Self Organizing Network simulations (first model). Fig. 4.2 b – Evolutionary Self Organizing Network simulations (second model). We trained 30 networks of each type obtaining the average degree distributions reported in fig.4.3- 4.5. In tab. 4.1 – 4.2 are reported the average values of the structural parameters of the two networks. y = 37,074x -2,0371 1 2 3 4 5 degree y = 68,961x -2,9864 1 2 3 4 5 6 degree Fig. 4.3 – Average degree distribution in GNG (two different input domains) y = 23,75x-1,149 1 2 3 4 5 6 7 8 9 degree y = 22,069x -1,1143 1 2 3 4 5 6 7 8 degree Fig. 4.4 – Average degree distribution in ENG (first model, two different input domains) y = 24,415x-1,1411 1 2 3 4 5 6 7 8 9 degree y = 21,98x -1,1408 1 2 3 4 5 6 7 8 degree Fig. 4.5 – Average degree distribution in ENG (second model, two different input domains) While GNG have a high value for the average path length and a low clustering coefficient, ENG have a short average path length and a high clustering coefficient which along with the power law tail of the degree distribution confirm its scale free graph features. Average path length Clustering coefficient Power exponent GNG - 0.49 2.04 ESON (1st) 3.82 0.64 1.15 ESON(2nd) 3.92 0.63 1.14 Tab. 4.1 – Comparison of structural parameters (average values, first input domain) Average path length Clustering coefficient Power exponent GNG 6.4 0.42 2.98 ESON (1st) 3.61 0.58 1.11 ESON(2nd) 3.67 0.59 1.14 Tab. 4.2 – Comparison of structural parameters (average values, second input domain) Fig. 4.6 – 4.7 shows the population dynamics of the two ENG models. The structure shared by the two different ENG models is due to the fact that the winner units tend to establish the greatest number of connections. These are the favoured units with which each node try to establish a connection. If the probability depends also on the local distortion error as it happens in the second model, we obtain a final structure that is more similar to the GNG, which is to say more similar to a gas. In point of fact, the conditions to create a new link become more restrictive, reducing the interaction among each cluster and the whole network. The structure of connections seems to extend more uniformly in the regions where inputs are present as it can be seen in picture 4.2b (more evident in the circular distribution). Picture 4.7 shows the dynamics of the populations in the two different models of ENG. In the first model the population size seems to converge to the final value of 0.72Nmax, confirming the experimental results of Annunziato and Pizzuti. As it can be noticed in fig. 4.6, since the d value gradually diminishes during the training, the influence of the factor (1-Dmin/d) grows reducing the effects of the negative feedback which characterizes the quadratic logistic map. This justifies the sudden growth of the population at the end of the training in the second model. 1 3 5 7 9 1 epochs 1 2 3 4 5 6 7 8 9 10 11 epochs Fig. 4.6 – Network size evolution of the two ENG models (first input domains). At the end of training new units connect with the winner units which have a lower d, while the subgroups of units become more isolated. Considering the function (X(t),X(t+1)) the attractor becomes more marked in the second model. This means that the system tends to converge more toward a precise final state with a lower interaction among the groups of units. Fig. 4.7 – population dynamics (X(t),X(t+1)) of the two Esonet models (first input domain). 5. THE ROLE OF INFORMATION IN FUNCTIONAL SPECIALIZATION AND INTEGRATION We can classify a system as complex when it is made up of different parts heterogeneously interacting. In addition, its behaviour and its structure have to be neither completely casual (as it happens in a gas) nor too regular (as it happens in a crystal). In Nature we generally observe the co- existence of functionally highly specialized integrated areas. That’s what happens in the brain, where different areas and groups of neurons interact to give rise to an integrated and unitary cognitive scenario (G. M. Edelman, G. Tononi, 2000). Edelman has introduced the integration, reciprocal information and complexity concepts in order to mathematically define the functional organization of the cerebral structures. Within a complex system, a subset of elements can be defined an integrated process if – on a given temporal scale – the elements interact more strongly with each other than with the system. In a neural net or in a self-organizing one it means that the units of an integrated group will tend to simultaneously activate themselves. When the units in a subset are independent, the system’s entropy reaches its maximum value which is the sum of the entropies of the single elements (local entropies). On the contrary, when any kind of interaction occurs, the global entropy decreases so becoming lower than the sum of the local entropies. The integration measure is, therefore, a natural indicator of the system informational “capacity”. So the integration of a subset of network units can be calculated by deducting the sum of the entropies of each single component ( )ix from the entropy of the system considered as a whole. If each unit can only take two states (activated/not-activated), the amount of the possible activation patterns of a subset with N units is N2 . So the system maximum entropy is: (5.1) ( ) ( )max 2 2 2 1 2 1 log log 1 log 2 i i NN i i i H X H x p N � � � �= = = = =� �� � � �� � � � and the integration will be: (5.2) ( ) ( ) ( ) I X H x H X = −� for the self-organized net here considered, the integration of a sub-group of units takes the following expression: (5.3) ( ) I X N P � �� � = − � �� �+� � � � where Pi is the probability for a node to establish i connections. The overall number of the system’ states is equal to the total number of possible groups of i+1 units. Groups of units having the same dimension (groups of i+1 units) give the same contribution to the entropy of the system. If we choose the WTA strategy as activation modality, for each presented input only a single unit (the winner) and the 1<i<N-1 i units will activate themselves. All the other ones remain not- activated. The probability for a node to create connections is ruled by the power law βα −= kPi , with α and β depending on 1) the network dimension, 2) the local distortion errors (for the second model) and 3) the particular evolution of the network structure, i.e. the dynamic behaviour of ( )tα and ( )tβ . So the integration of the two self-organizing network here presented is: (5.4) ( ) ( ) I X N i � � � �= − � �� �+ � �� � The integration can be seen as a measure of the statistic dependency within a subset of units. The stronger their interactions are, the higher their integration. In order to measure the statistic dependency between a subset and the whole system, Edelman introduced the concept of mutual information. Given an n subset made up of k elements ( )knX and its complement in the system ( )knXX − , the mutual information is: (5.5) ( ) ( ) ( ) ( );k k k kn n n nIR X X X H X H X X H X− = + − − The mutual information is essential to evaluate the differentiation degree of a system, i.e. it is a significant index of the system’ “resolution” degree, calculated on the subdividable and distinct states. In order to measure the information of an integrated activation pattern, we calculate how the states of a given subset can differentiate them from the whole system ones. Which thing, following Edelman, is equivalent to considering the whole system as the observer of itself. In fact, if entropy measures the variability of a system according to an external observer evaluation, the mutual information measures the system variability according to an observer ideally placed within the system itself. The overall measure of the differentiation degree of a complex system is given by the mutual information average between each subset and the whole system: (5.6) ( ) ( ) C X IR X X X = −� Edelman defined such measure as complexity and its value is high if each subset can averagely take many different states which are statistically depending on the whole system’s ones, so it shows how the system is differentiated. High complexity values correspond to an optimal synthesis of functional specialization and functional integration. Systems whose elements are not integrated (such as a gas) or not specialized ( such as an homogeneous crystal) have a minimum complexity. In the evolutionary neural gas case, the WTA strategy limits the integration among the activation patterns. So the mutual information between any activation pattern and the other possible patterns is equal to zero. It justifies the use of the term “gas”, since the patterns behave like isles of information weakly interacting each other. If there were selected more winner units for the same input signal in the early training phase, we could get a given system status characterized by i + 1 activated units not only by the activation of just a single winner and its related i units, but also by the activation of more winners. therefore we should also take into consideration all the possible subgroups with j+1 elements. The mutual information formula between a subgroup with k activated units and the system is given by: (5.7) ( ) ( ) ( ) ( ) ( ) k ii jk n ii ji j H X i j = = − − � �+ � � � � � �= +� �� � � � +� �+ + � �� � � � � +� �� �+� �� � ( ) ( ) ( ) ( ) ( ) k ii jk n ii ji j H X X i j = = − − � �+ � � � � � �− = + +� �� � � � +� �+ + � �� � � � � +� �� �+� �� � ( )( ) ( ) ( )( ) ( ) 1 1 log kk jj −−− − −−−= − � � � � � � � �+ − − + +� �� �� � � � � �+ � �� � � �� � � − +� �� �+� �� � ( ) ( ) ( ) N ii j j N i i i j j i j k β β βα α α − − − + +� �� � � � � �� �+ + + − ⋅� �� � � � � �� �+ +� � � � � �� �� � � � ( ) ( ) ( ) ii j j i j j β β βα α α− − − + +� �� �� � � � + + −� �� �� � � �+� � � �� �� � ( ) ( ) ( ) ( ) ( ) N ii j ii ji j H X i j = = − − � �+ � � � � � �= +� �� � � � +� �+ + � �� � � � � +� �� �+� �� � To provide the system with a greater level of complexity, in order to favouring the integration among the network unit subgroups, it is, therefore, necessary adopting a strategy different from the WTA in the early training phases so as to select more winner units. 6) CONCLUSIONS AND FUTURE WORKS The here presented self-organizing network can be considered as an example of autopoietic system which evolves by means of a closed network of interactions and based upon the production of components (the categorization units). In the course of the reproductive dynamics, those ones produce other components, also belonging to the system (i.e. other categorization units) which maintain the system identity over time with respect to the experimental task. In particular, it has to be noticed that they are not just the environmental information to lead the evolution of the network of connections, but rather the network internal status, which is individuated globally by the size that the population has reached and locally by the values of the parameters of the units. The latter show the difficulties that the units encounter in modelling the presented input, such difficulty is directly proportional to the amount of variations their reference vectors are subjected to. Learning and the capability to model the system external inputs, therefore, emerges more by means of the population internal dynamics than by means of a learning algorithm. The appearing of a scale-free structure emerging from the choice of the population dynamics is peculiarly significant for the model’s biological plausibility. Which thing describes a quite phase- transition-like status where cluster “float” as informational “isles” in a “gaseous” configuration. It is worthy noticing that the WTA strategy and the environmental noise (probabilistic laws) suffice to create a kind of basic informational skeleton around which more interconnected functional structures can then aggregate. In the nervous system, it plausibly happens according to an essentially genetic design. Such kind of neural dynamics guarantees flexibility and redundancy to the informational nuclei which are ready to synchronize and connect through signals. Actually, what we tried here to describe is a proto-neural scenario with low integration of clusters which are specialized in easy categorization tasks. Developing the ENG model requires to investigate different synchronization scenarios among clusters and their ensuing functional integration to execute more complex tasks. In particular, it is necessary to modify the evolutive dynamics so as to mane the connections among units active. In this way, it should be possible to create a dynamic neural topology susceptible of hierarchical organization. Everything seems to confirm not only the deep reasons for the scale-free structures recurring in nature (Z. Toroczkai, K. E. Bassler, 2004), but also the fundamental lesson associating complexity with a thin border zone between integration and differentiation among the functional modules of a system. Acknowledgements: The authors thank Eliano Pessa and Graziano Terenzi for their precious suggestions. REFERENCES Albert R. and Barabasi A.(2000) Topology of evolving networks: Local events and universality. Physical Review Letters vol.85, p.5234. Annunziato M. and Pizzuti S.(2000), Adaptive Parametrization of Evolutionary Algorithms Driven by Reproduction and Competition, in Proceedings of ESIT 2000, Aachen, Germany. Chialvo D.R. and Bak P. (1999) Learning from mistakes, in Neuroscience Vol.90, No.4, pp.1137- 1148. Dawkins R. (1971), Selective neurone death as a possible memory mechanism, in Nature n.229, pp.118-119. Delaunay B. (1934), Bullettin of the Academy of Sciences USSR, vol.7, pp. 793-800. Edelman G.M. (1978), Group selection and phasic reentrant signaling: a theory of higher brain function, in The Mindful Brain (eds Edelman G.M. and Mountcastle V.), pp. 51-100, MIT, Cambridge. Edelman G.M. (1987), Neural Darwinism: The Theory of Neuronal Group Selection, Basic Books, New York. Edelman G.M., Tononi G. (2000), Un universo di coscienza, Biblioteca Einaudi, Torino. Fritzke B. (1994), Growing Cell Structures. A Self-Organizing Network for Unsupervised and Supervised Learning. in Neural Networks, 7(9), pp. 1441-1460. Heylighen F. (1992), Principles of Systems and Cybernetics: an evolutionary perspective, in: Cybernetics and Systems ’92, R. Trappl (ed.), World Science, Singapore, pp. 3-10. Langton C.G. (1989), Artificial Life: The Proceedings of an Interdisciplinary Workshop on the Synthesis and Simulation of Living Systems, Addison-Wesley. Jefferson D. and Taylor C. (1995), Artificial Life as a Tool for Biological Inquiry, in Artificial Life: an Overview, edited by C.G. Langton, MIT press, pp.1-10. Kohonen T. (1982), Self-Organized Formation of Topologically Correct Feature Maps, in Biological Cybernetics, n.43, pp.59-69. Martinetz T.M. (1993), Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving Maps, in ICANN’93, International Conference on Artificial Neural Networks, Springer, pp. 427- 434. Amsterdam. Martinetz T.M. and K.J. Schulten, (1991), A Neural Gas Network Learns Topologies, In Artificial Neural Networks, T.Kohonen, K. Makisara, O. Simula, and J. Kangas, eds, , pp. 397-402. North- Holland, Amsterdam. Martinetz T.M. and Schulten K.J. (1994), Topology Representing Networks, in Neural Networks, 7(3), pp. 507-522. Moriarty D.E. and R.Miikkulainen (1998), Forming Neural Networks Through Efficient and Adaptive Coevolution, in Evolutionary Computation, 5(4), pp. 373-399. Paredis J. (1995), Coevolutionary Computation, in Artificial Life, 2, pp.355-375. Rocha L.M. (1997) Evolutionary Systems and Artificial Life, Lecture Notes. Los Alamos, NM 87545 Roughgarden J.,(1979), Theory of Population Genetics and Evolutionary Ecology, Prentice-Hall. Smith R.E., Forrest S., Perelson A.S. (1993), Searching for Diverse Cooperative Populations with Genetic Algorithms, in Evolutionary Computation, 1(2), 127-149. Song J. and Yu J. (1988), Population System Control, Springer-Verlag. Steyvers M. and Tenenbaum J., 2001. The Large-Scale structure of Semantic Networks. Working draft submitted to Cognitive Science. Toroczkai,Z. and Bassler,K.E. (2004), Jamming is Limited in Scale-Free Systems, in Nature, 428 , p.716 Watts D.J., Strogatz S.H. , 1998. Collective dynamics of ‘small-world’ networks. Nature, vol. 393, pp. 440-442. Young J.Z. (1964), A Model of the Brain, Clarendon, Oxford. Young J.Z. , (1966), The Memory System of the Brain, University of California Press, Berkeley.

0704.0599

X-ray Dichroism and the Pseudogap Phase of Cuprates

X-ray Dichroism and the Pseudogap Phase of Cuprates S. Di Matteo1, 2 and M. R. Norman3 Laboratori Nazionali di Frascati INFN, via E. Fermi 40, C.P. 13, I-00044 Frascati, Italy Equipe de physique des surfaces et interfaces, UMR-CNRS 6627 PALMS, Université de Rennes 1, 35042 Rennes Cedex, France Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA (Dated: October 26, 2018) A recent polarized x-ray absorption experiment on the high temperature cuprate superconduc- tor Bi2Sr2CaCu2O8+x indicates the presence of broken parity symmetry below the temperature, T*, where a pseudogap appears in photoemission. We critically analyze the x-ray data, and con- clude that a parity-breaking signal of the kind suggested is unlikely based on the crystal structures reported in the literature. Possible other origins of the observed dichroism signal are discussed. We propose x-ray scattering experiments that can be done in order to determine whether such alternative interpretations are valid or not. PACS numbers: 78.70.Ck, 78.70.Dm, 75.25.+z, 74.72.Hs I. INTRODUCTION Twenty years since the discovery of high-temperature cuprate superconductivity, there is still no consensus on its origin. As the field has evolved, more and more at- tention has been directed to the pseudogap region of the phase diagram in underdoped compounds and the possible relation of this phase to the superconducting one.1 Time-reversal breaking has been predicted to occur in this pseudogap phase due to the presence of orbital currents2 and a subsequent experiment3 using angle- dependent dichroism in photoemission has claimed to de- tect this. However, this result has been challenged by others4 and an independent experimental verification of this would be highly desirable. Recently, Kubota et al.5 performed Cu K edge circu- lar and linear dichroism x-ray experiments for under- doped Bi2Sr2CaCu2O8+x (Bi2212), claiming no time- reversal breaking of the kind predicted in Ref. 2 exists, and that, on the contrary, a parity-breaking signal (but time-reversal even) is present with the same tempera- ture dependence as the photoemission dichroism signal, that was interpreted as x-ray natural circular dichroism (XNCD) as seen in other materials.6 The aim of the present paper is to critically examine the conclusions of Kubota et al.5 In particular, we find that the XNCD signal for the average7 Bi2212 crystal structure should be zero along all three crystallographic axes, therefore casting doubt on the original interpreta- tion of Ref. 5. To look into alternate explanations, we performed detailed numerical simulations aimed at ex- plaining the observed signal. At the basis of our study is the simple observation (see, e.g., Ref. 8) that circu- lar dichroism in absorption can be generated either by a non-magnetic effect in the electric dipole-quadrupole (E1-E2) channel (XNCD, a parity-breaking signal), or by a magnetic signal in the E1-E1 channel (parity-even). Alternately, such a signal can be due to contamination from x-ray linear dichroism (XNLD). We propose x-ray experiments that could be used to investigate these mat- ters further. The structure of the paper is as follows. In Sec. II we show how symmetry constrains any possible XNCD sig- nal that would be observed in Bi2212. We also perform numerical simulations for XNCD assuming an alignment displaced from the c-axis, using several crystal structure refinements proposed in the literature. We also calculate the XNLD signal and comment whether XNLD contam- ination could be responsible for the observed signal. In Sec. III we calculate the x-ray magnetic circular dichro- ism (XMCD) signal at both the Cu K and L edges as- suming magnetic order on either the copper or oxygen sites. Finally, in Sec. IV, we draw some general conclu- sions from our work. II. NON-MAGNETIC CIRCULAR DICHROISM IN BI2212 The structure of Bi2212 is strongly layered, with insu- lating BiO blocks intercalated between superconducting CuO2 planes. Crystal structure refinements reveal the presence of an incommensurate modulation whose origin has been the subject of much debate. The presence of this modulation has complicated the determination of the average crystal structure. Two different average struc- tures have been proposed in the literature for Bi2212: Bbmb9,10 and Bb2b11,12,13,14, where b is the modulation direction for the superstructure. We follow the general convention in the cuprate literature and use a rotated basis compared to those in the International Tables for Crystallography15 (respectively, Cccm, No. 66, and Ccc2, No. 37). In this way, the c-crystallographic direction is orthogonal to the CuO2 planes. The Bbmb structure is globally centrosymmetric and, as such, does not admit a non-zero value for the parity- odd operator ~L · (ǫ̂∗ × ǫ̂)(~Ω · k̂), whose expectation value determines the XNCD signal (~L, k̂, ǫ̂ and ~Ω are, re- spectively, the angular momentum operator, the direc- tion and polarization of the x-ray beam, and the toroidal http://arxiv.org/abs/0704.0599v2 momentum operator, see, e.g., Refs. 8,16). Therefore, if the signal measured by Kubota et al.5 were a true XNCD signal, this would imply a lower crystal symmetry than Bbmb. We note that most refinements in the literature suggest Bbmb,9,10 and this crystal structure is also con- sistent with recent photoemission data17 that indicate the presence of both a glide plane and a mirror plane based on dipole selection rules. The other average structure that has been suggested by x-ray and neutron diffraction is Bb2b.11,12,13,14 This space group is not centrosymmetric and, therefore, a par- ity breaking signal like that of XNCD is in principle al- lowed. However, not all wave vector directions are com- patible with the presence of a XNCD signal, as demon- strated below by symmetry considerations. In the last part of Section II, we numerically calculate the XNCD for a geometrical configuration allowing a signal - like ~k‖(101) - by means of the multiple-scattering subroutine in the FDMNES program.18 In the context of this program, atomic potentials are generated using a local density approximation with a Hedin-Lundqvist form for the exchange-correlation en- ergy. These potentials are then used in a muffin tin approximation to calculate the resulting XANES sig- nal by considering multiple scattering of the photo- electron about the absorbing site within a one-electron approximation.19 In the future, it would be desirable to repeat these calculations by using input from self- consistent band theory, as has recently been done for the Bbmb structure in regards to angle resolved photoemis- sion spectra.20 In the Bb2b setting, Cu ions belong to sites of Wyck- off multiplicity 8d. These eight equivalent copper sites can be partitioned in two groups of four sites that are related by the vector (1/2,0,1/2). Within each group the four sites are related by the symmetry operations {Ê, Ĉ2y, m̂x, m̂z}, where Ê is the identity, Ĉ2y is a two- fold axis around the b crystallographic axis, and m̂x(z) is a mirror-symmetry plane orthogonal to the a(c)-axis. The absorption at the Cu K edge, expressed in Mbarn, can be calculated through the equations: σ(±) = j , (1) j = 4π 2αh̄ω |〈Ψ(j)n |Ô (±)|Ψ 2δ[h̄ω − (En − E0)] The operator Ô(±) ≡ ǫ̂(±) ·~r(1+ i ~k ·~r) in Eq. (2) is the usual matter-radiation interaction operator expanded up to dipole (E1) and quadrupole (E2) terms, with the pho- ton polarization ǫ̂ and the wave vector ~k, where we label left- and right-handed polarization by the superscript ±. n ) is the ground (excited) state of the crystal, and E0 (En) its energy. The sum in Eq. (2) is extended over all the excited states of the system and h̄ω is the energy of the incoming photon, with α the fine-structure 9.029.019.008.998.98 Energy (keV) Pet (XNCD x 1000) Glady (XNCD x 400) Kan (XNCD x 100) FIG. 1: (Color online) XANES signal for ~k‖(001) and XNCD signal for ~k‖(101) at the Cu K edge for three Bb2b crystal refinements, with a cluster radius of 4.9 Å. The crystal struc- tures are Pet for Petricek et al.11, Glady for Gladyshevskii and Flükiger13 , and Kan for Kan and Moss12. The XNCD signals have been multiplied by the factors indicated. Each successive set of curves is displaced by 0.4 Mbarn. constant. Finally, the index j = 1, ..., 8 indicates the lattice site of the copper photoabsorbing atom in the unit cell. Eqs. (1) and (2) are the basis of the numer- ical calculations of the FDMNES program.18 The eight contributions can be written as the sum of two equal parts coming from the two subsets of four ions related by the (1/2,0,1/2) translation. Within each subset, the four absorption contributions are related to one another by the symmetry operations σ2 = Ĉ2yσ1, σ3 = m̂xσ1, and σ4 = m̂zσ1, implying that the total absorption is: σ = 2(1 + m̂z)(1 + m̂x)σ1 (3) The group from σ5 to σ8 is equivalent to the first group of four modulo a translation (this is the reason for the factor of two in Eq. (3)). Notice that the symmetry operators in Eq. (3) are meant to operate just on the electronic part of the operator Ô in Eq. (2). In the case of circular dichroism, the signal is given by σ = σ+−σ−. If we suppose that no net magnetization is present in the material (we shall analyze the possibility of magnetism in Sec. III), then the dichroism is natural, i.e., necessarily coming from the interference E1-E2 con- tribution in Eq. (2). In this case the signal is parity-odd, which implies that m̂z(x) ≡ ÎĈ2z(2x) → −Ĉ2z(2x) (Î is the inversion operator). Then Eq. (3) becomes σ = 2(1− Ĉ2z)(1 − Ĉ2x)σ1 (4) which implies that, of the possible five second-rank ten- sors involved in XNCD, only the term T 1 − T −1 sur- vives. To arrive at this result, we applied the usual op- erator rules on spherical tensors:21 Ĉ2xT m = T −m and Ĉ2zT m = (−) m . This in turn leads to a zero XNCD along the three crystallographic axes of the Bb2b crys- tal structure where, e.g., along the c-axis, the signal is proportional to T Therefore, even in the Bb2b crystal structure, the XNCD is exactly zero by symmetry when the wave vec- tor is directed along the c-axis (i.e., orthogonal to the CuO2 planes) as in the experiment of Ref. 5. This has been further checked by numerical calculations with clus- ter radii up to 6.5 Å, i.e., 93 atoms, centered on the Cu ion, based on the average crystal structures reported in Refs. 11,12,13,14. The only possibility to justify theo- retically the experimental evidence of circular dichroism of a non-magnetic nature is either by lowering of the or- thorhombic Bb2b symmetry, a misalignment of the k̂- direction with respect to the c-axis, or contamination from linear dichroism. We checked all of these possi- bilities. If we take into account the monoclinic supercell pro- posed in Ref. 13, with space group Cc, this implies a re- duction of the symmetry operations, with the loss of the two-fold screw axis. Nonetheless, the glide plane contain- ing the normal to the CuO2-planes is still present (m̂x in Eq. 3), which is responsible for the extinction rule of the quantity 〈Ψn|~L · (ǫ̂ ∗ × ǫ̂)(~Ω · k̂)|Ψn〉. Therefore the XNCD is again identically zero by symmetry, which we verified by direct numerical simulation of the supercell. Note that only a reduction to triclinic symmetry would allow for XNCD in the direction orthogonal to the CuO2 planes. We also checked for the possibility of misalignment, as shown in Fig. 1, by a direct calculation with ~k‖(101), corresponding to a tilting θ ∼ 10o with respect to the c-axis (note that the a lattice constant is ∼ 5.4 Å, and the c one ∼ 30.9 Å). We first remark that the calcu- lated XANES signal compares well to experiment (the XANES for ~k‖(001) and ~k‖(101) are identical on the scale of Fig. 1). Despite this, the energy profile of the XNCD signal is different from the one reported by Kub- ota et al.5 The main difference is the energy extension of the calculated dichroic signal, whose oscillations persist, though with decreased intensity, more than 50 eV above the edge itself. This characteristic is present for all four Bb2b refinements we have looked at (and as well for the monoclinic supercell, which we do not show) and is at variance with the experimental results of Ref. 5, where the dichroic signal is confined to a small energy range around the edge. Some comments on the calculations are in order. Each refinement gives rise to a different XNCD signal, and -0.05 9.029.019.008.998.98 Energy (keV) Pet (XNCD x 1000) Kan (XNCD x 100) Glady (XNCD x 400) FIG. 2: (Color online) XNCD signal, as in Fig. 1, but for a CuO5 cluster. 9.029.019.008.998.98 Energy (keV) R=2.1 R=3.1 R=4.9 R=4.1 R=5.5 R=6.5 XNCD x 1000 FIG. 3: (Color online) XNCD signal at the Cu K edge for ~k‖(101) as a function of the cluster radius (in Å) for the crys- tal structure of Petricek et al.11. The signal has been mul- tiplied by 1000. Each successive curve is displaced by 0.06 Mbarn. their intensities are quite different as well. They are found to be strongly dependent on the magnitude of the deviation of the atoms from their Bbmb positions, which differs significantly among the various Bb2b refinements. Moreover, the structure of Kan and Moss leads to a mani- festly different energy profile. This difference is seen even for CuO4 and CuO5 clusters (the results for a CuO5 clus- ter are shown in Fig. 2) and has to do with the large de- parture of this particular structure from the Bbmb one. Although there has been some criticism in the literature concerning this particular refinement,10 the point we wish to make is that each refinement has a different XNCD signal, showing how sensitive this signal is to the actual crystal structure. We note that Fig. 1 was done for a cluster radius of 4.9 Å, i.e., 37 atoms. In Fig. 3, we show results for the refinement of Petricek et al.11 up to a ra- dius of 6.5 Å (93 atoms), showing the development of ad- ditional structure in the energy profile as more and more atoms are included in the cluster. In the real system, the effective cluster radius is limited by the photoelectron escape depth, which is energy dependent.22 We also remark that the magnitude of the XNCD sig- nal we calculated for a 10 degree misalignment is com- parable to that measured in Ref. 5. On the other hand, the signal goes as sin(2θ), where θ measures the displace- ment from the c-axis. We note that Kubota et al.5 men- tion that their signal was insensitive to displacements from the c-axis of 5 degrees, which would argue against a misalignment given the strong angular dependence we predict. Moreover, we note that the size of the signal only depends on the projection of the k vector onto the a-c plane, i.e., a signal for ~k‖(111) is equivalent to that for ~k‖(101). The above leads us to suspect that neither misalign- ment, nor symmetry reduction, are the basis of the signal detected in Ref. 5. We now turn to the third possibility for a non-magnetic signal, that due to intermixing of lin- ear dichroism. All x-ray beams at a synchrotron have a linear polarization component (Kubota et al.5 mention the possibility of up to 5% of linear admixture). The resulting linear dichroism, which vanishes for a uniaxial crystal, can swamp the intrinsic XNCD signal for a biax- ial crystal where the a and b directions are inequivalent (like Bi2212). This was shown by Goulon et al. for a KTiOPO4 crystal with the same point group symmetry (mm2) as Bi2212.23 To analyze this further, we show the linear dichro- ism (the XANES signal for ~E‖(010) minus the one for ~E‖(100)), calculated for the Bb2b refinement of Glady- shevskii et al.,14 for various cluster radii, in Fig. 4. Simi- lar results have been obtained for the other crystal struc- tures, including the Bbmb refinement of Miles et al.10 The energy profile, with a positive peak followed by a negative peak, and its location at the absorption edge, is very reminiscent of the data. Moreover, the size of the XNLD signal is large, meaning that only a few percent admixture is necessary to explain the size of the signal seen in Ref. 5. One issue is that Kubota et al.5 did report the existence of an XNLD signal, but also claim that it is temperature independent. This is somewhat puzzling, as there are significant changes of the lattice constants with temperature.24 One obvious question would be why such an XNLD contamination would only appear below T*, though it should be remarked that there are anomalies in the superstructure periodicity near T*.24 A definitive test would be to rotate the sample under the beam, as any XNLD signal would vary as cos(2φ) where φ is the in- plane angle relative to the b-axis. Any circular dichroism (XNCD or XMCD) is instead φ-independent. 9.029.019.008.998.98 Energy (keV) R=3.0 R=4.1 R=5.1 XANES XNLD x 50 FIG. 4: (Color online) XNLD signal at the Cu K edge for ~k‖(001) as a function of the cluster radius (in Å) for the crys- tal structure of Gladyshevskii et al.14. The XNLD signal has been multiplied by 50. The XANES curve has been displaced by 0.05 Mbarn. A final possibility would be a small energy shift be- tween the left and right circularly polarized beams. Dif- ferentiating the absorption edge in Fig. 1 would indeed lead to a signal similar to that seen in Ref. 5 (but with an enhanced positive peak relative to the negative peak). But such an energy shift is difficult to imagine with the particular experimental set up used. The observed dichroism signal as a function of energy is also reminiscent of that typically seen for magnetic circular dichroism: in this case the signal would be of E1-E1 origin and its main features are indeed expected to be at the edge itself. In addition, the nature of the observed signal (a single sharp positive peak followed by a single sharp negative peak) is also much like a magnetic signal where the main features are expected to be more localized in energy starting from the rising edge of the absorption. Whether this possibility is realistic or not can only be determined by a quantitative calculation, which we offer in the next section. III. MAGNETIC DICHROISM IN BI2212 Over the years, there have been several claims of pos- sible magnetic order in the pseudogap phase of cuprate superconductors. Recently, a magnetic signal at a (101) Bragg vector has been observed below T* for several un- derdoped YBa2Cu3O6+x (YBCO) samples by polarized neutron diffraction.25 The signal, corresponding to a mo- ment of order 0.05-0.1 µB, is not simple ferromagnetism as it was not observed at the (002) Bragg vector. Even more recently, a Kerr rotation below T* has been de- tected in underdoped YBCO corresponding to a net fer- 9.029.019.008.998.98 Energy (keV) XANES Kan XMCD Pet XMCD Kan XMCD Glady XMCD x 10000 FIG. 5: (Color online) XMCD for ~k‖(001) at the Cu K edge with a ferromagnetic moment of 0.1 µB along the c-axis at each copper site. The cluster radius is 4.1 Å. The XMCD signal has been multiplied by 10000. The crystal structures are those of Fig. 1. romagnetic moment of 10−4 µB. This motivates us to consider the possibility of a mag- netic origin for the x-ray circular dichroism detected in Ref. 5. We restate that in absorption (see, e.g., Ref. 8), circular dichroism can be generated either by a non- magnetic effect in the E1-E2 channel (XNCD, a parity- breaking signal), or by a magnetic signal in the E1-E1 channel (XMCD, parity-even). The first possibility, an- alyzed in the previous section, does not seem to be com- patible with the experimental results. In order to analyze the second possibility, we need to provide the lattice with a magnetic structure that has a net magnetization (oth- erwise, the XMCD is zero). In what follows, we shall sup- pose two magnetic distributions, the first with the mag- netic moments on the Cu sites, the second on the planar O sites. The numerical calculations are performed with the relativistic extension of the multiple-scattering pro- gram in the FDMNES code,18 and provide results that are an extension of those of the previous section. The details of the calculations are as follows: we used again the average crystal structures discussed in Section II. For each magnetic configuration we employed clus- ters with radii ranging from 3.1 Å (a CuO5-cluster), to 4.9 Å (37 atoms) around the Cu photoabsorbing ion. In the first set of calculations, shown in Figs. 5 and 6, we built the input potential from a magnetic configuration of 4.55 3d↑ electrons and 4.45 3d↓ electrons (i.e., a moment of 0.1 µB per copper site). In the second set of calcula- tions, shown in Fig. 7, we built the input potential from a magnetic configuration of 2.05 2p↑ electrons and 1.95 2p↓ electrons (i.e., a moment of 0.1 µB per planar oxygen site). The following results are noteworthy: a) Differently from the XNCD calculations shown in -0.05 9.029.019.008.998.98 Energy (keV) XMCD Pet XMCD Kan XMCD Glady x 10000 FIG. 6: (Color online) XMCD as in Fig. 5, but with a cluster radius of 3.1 Å, i.e., a CuO5 cluster. 9.029.019.008.998.98 Energy (keV) XANES Kan XMCD Pet XMCD Kan XMCD Glady XMCD x 5000 FIG. 7: (Color online) XMCD for ~k‖(001) at the Cu K edge with a ferromagnetic moment of 0.1 µB at each planar oxygen site. The XMCD signal has been multiplied by 5000. The XANES signal has been displaced by 0.02 Mbarn. Fig. 1, all crystal structures give basically the same XMCD spectra. The reason for this behavior may be related to the fact that XMCD, when x-rays are orthog- onal to the CuO2 planes, mainly depends on the in-plane magnetisation density and on the in-plane crystal struc- ture, which is quite similar for the various refinements. b) There is a more marked dependence on the cluster radius compared to the XNCD, as shown by the compar- ison of Fig. 5 and Fig. 6. The calculations with a radius bigger than 4.1 Å (6 Cu, 9 O, 4 Sr, and 4 Ca), above the pre-edge energy, are basically equivalent to those shown in Fig. 5, with a positive peak at the edge energy, fol- 960950940930 Energy (eV) XANES Pet XMCD Pet XMCD Kan XMCD Glady XMCD x 10 FIG. 8: (Color online) XMCD for ~k‖(001) at the Cu L2,3 edges with a ferromagnetic moment of 0.1 µB along the c- axis at each copper site. The cluster radius is 4.1 Å. The XMCD signal has been multiplied by 10. lowed by a double negative peak, the latter at variance with the experimental results. On the contrary, the en- ergy shape obtained for a radius of 3.1 Å is very close to the experimental one, with a single negative peak after the sharp positive one, with relatively good agreement in the energy position and width. We could be tempted to suppose, therefore, that the virtual photoelectron has a very small mean free path before decaying and it is sensitive just to the nearest neighbor oxygens. Indeed we checked that an identical XMCD profile is obtained with just the in-plane CuO4-cluster. On the other hand, the size of the signal we calculate is about an order of magnitude smaller than that seen in Ref. 5. Since the XMCD signal is proportional to the moment, then we would need a moment of ∼1 Bohr magneton per copper to have a comparable signal. Such a huge moment would have been observed previously by neutron scattering if it existed. Of course we cannot exclude that spurious effects, such as strain fields, could have influenced the measurement. c) The energy profile in the case of magnetization at the oxygen sites is not much different from the cop- per case, except for the deeper negative peak around E ∼ 8.994 keV, as shown in Fig. 7. Also in this case the dif- ferent crystal structure refinements give basically equiv- alent results, as again the CuO2 planes are practically equivalent in the various cases. Note that the relative intensity is equivalent to the copper case, as 0.1 Bohr magnetons per planar oxygen corresponds to 0.2 Bohr magnetons per CuO2 cell (note we multiply by 5000 in Fig. 7 as compared to 10000 in Fig. 5). We also performed simulations for the Cu L2,3 edges for the magnetic configuration corresponding to Fig. 5, as shown in Fig. 8, which can be compared with future experimental investigations in order to confirm whether or not a net magnetization exists in this compound. Finally, we remark that the dependence of the XMCD signal on the tilting angle θ (i.e., the displacement of the photon wave vector from the c-axis) goes like cos(θ) and therefore the signal is not very sensitive to small displacements of 5 degrees, as noted by Kubota et al.5 This different angular dependence from the XNCD signal suggests a relatively easy way to unravel the question ex- perimentally: it is sufficient to measure the θ (azimuthal) dependence of the signal, noting that any XNLD contam- ination would be tested by the φ (polar) dependence of the signal. IV. CONCLUSIONS In our opinion, the experiments of Kubota et al.5 have raised more questions than they have answered. Al- though not treated in our paper, we believe that their results from the measurement of non-reciprocal linear dichroism are at this stage not conclusive, as only one di- rection for the toroidal moment has been investigated, of the two possible suggested by the orbital current pattern of Varma.2 The analysis performed in Section II showed, moreover, that the claimed XNCD signal is probably un- justified. In fact, even though the space group Bb2b is non-centrosymmetric, XNCD is absent by symmetry when the x-ray wave vector is chosen orthogonal to the CuO2 planes, as in the experimental measurement geom- etry of Ref. 5. The same extinction rule survives for the monoclinic supercell structure refined in Ref. 13. More- over, in both cases, it seems hard to mantain the hy- pothesis of misalignment, due to the experimental local- ization of the energy profile around the main absorption edge, which is absent in the calculations. We also note the difference of XNCD from the photoemission dichro- ism results of Kaminski et al.3 A direct comparison is however not immediate, as the former represents a q̂- integrated version of the latter (here q̂ is the solid angle in the space of the photoelectron wave-vector, see, e.g., Ref. 8). A more likely explanation is an XNLD contamination (Fig. 4), but then the challenge is to understand why such an effect would only exist below T*. We note that an op- tics experiment for an optimal doped Bi2212 sample has seen a change in linear birefrigence below Tc, which was accompanied by a non-zero circular birefringence.28 In addition, both Bi222329 and the Fe analogue of Bi221230 exhibit supercells with 222 space groups which would al- low for dichroism. So, it is conceivable that there is a subtle structural transition associated with T* which we suggest could be looked for by diffraction experiments. A final comment about the physical quantities detected by x-ray circular dichroism, either natural or magnetic, is in order. At the K edge of transition metal oxides, XMCD in the E1-E1 channel, at the excitation energy E = h̄ω − Eedge, gives information on the expectation value of L̂ · (ǫ̂∗× ǫ̂) for the excited states at the energy E . The orbital angular momentum is either induced from a spin moment via spin-orbit coupling (as calculated here) or directly by an orbital current (as in the scenario advo- cated in Ref. 231). No direct spin information is available at the K edge and therefore an XMCD measurement is not directly related to the ground state magnetic moment as for the L edge. Moreover, the observed states are those with p-like angular momentum projection on the photoabsorbing Cu ion, which are extended and there- fore mainly sensitive to the influence of the oxygen atoms surrounding the Cu site. In this case, the main contribu- tion to the XMCD energy profile is expected in an energy range of 10-20 eV from the main edge to the first shoul- der in the XANES spectrum, as found in Ref. 5 and in our own XMCD calculations. Although the results of Section III are in principle con- sistent with Ref. 5, the size of the ferromagnetic moment necessary to get a signal of the magnitude seen by exper- iment, ∼1 Bohr magneton, is excessive. If such a large ferromagnetic moment existed, it would have surely been seen by neutron scattering. From this point of view, ex- periments performed at the Cu L edge and O K edge would be desirable as they are more sensitive to the pres- ence of a magnetic moment. Finally, we would like to remark that XNCD along the c-axis is insensitive to orbital currents. These latter, confined to the CuO2 planes, develop a parity breaking characterized by a toroidal moment (~Ω) within the CuO2 planes. The XNCD experiment of Ref. 5 would only be sensitive to the projection of the toroidal moment out of this plane (i.e., along the direction of the x-ray wavevec- tor). Therefore, if performed as stated, it cannot tell us about any possible orbital current order. To conclude, we believe that the various interpreta- tions, XNCD, XNLD, or XMCD, have their drawbacks, and therefore the origin of the experimental signal of Ref. 5 is still open. In this sense, further experimen- tal checks of the energy extension of the dichroic signal would be highly desirable. Based on our results, the most stringent experimental test on the physical origin of the signal would come from the measurement of the depen- dence on the tilting (azimuthal) angle θ, due to the dif- ferent dependences of XNCD and XMCD, as well as the dependence on the in-plane (polar) angle φ, which would test for any possible XNLD contamination. V. ACKNOWLEDGMENTS The authors thank John Freeland, Zahir Islam, Stephan Rosenkranz, Daniel Haskel and Matti Lindroos for various discussions. This work is supported by the U.S. DOE, Office of Science, under Contract No. DE- AC02-06CH11357. SDM would like to thank the kind hospitality of the ID20 beamline staff at the ESRF. 1 M. R. Norman, D. Pines and C. Kallin, Adv. Phys. 54, 715 (2005). 2 C. M. Varma, Phys. Rev. B 55, 14554 (1997) and Phys. Rev. Lett. 83, 3538 (1999); M. E. Simon and C. M. Varma, ibid 89, 247003 (2002). 3 A. Kaminski, S. Rosenkranz, H. Fretwell, J. C. Cam- puzano, Z. Li, H. Raffy, W. G. Cullen, H. You, C. G. Olson, C. M. Varma and H. Hoechst, Nature 416, 610 (2002). 4 S. V. Borisenko, A. A. Kordyuk, A. Koitzsch, T. K. Kim, K. A. Nenkov, M. Knupfer, J. Fink, C. Grazioli, S. Turchini and H. Berger, Phys. Rev. Lett. 92, 207001 (2004). 5 M. Kubota, K. Ono, Y. Oohara and H. Eisaki, J. Phys. Soc. Jpn. 75, 053706 (2006). 6 L. Alagna, T. Prosperi, S. Turchini, J. Goulon, A. Ro- galev, C. Goulon-Ginet, C. R. Natoli, R. D. Peacock and B. Stewart, Phys. Rev. Lett. 80, 4799 (1998). 7 The average crystal structure is the base orthorhombic unit cell before the incommensurate superstructure is taken into account. As the latter involves a translation operator, it should not affect the symmetry arguments in this paper. 8 S. Di Matteo and C. R. Natoli, J. Synchr. Rad. 9, 9 (2002). 9 A. Yamamoto, M. Onoda, E. Takayama-Muromachi, F. Izumi, T. Ishigaki and H. Asano, Phys. Rev. B 42, 4228 (1990); D. Grebille, H. Leligny, A. Ruyter, Ph. Labbé and B. Raveau, Acta Cryst. B52, 628 (1996); N. Jakubowicz, D. Grebille, M. Hervieu and H. Leligny, Phys. Rev. B 63, 214511 (2001). 10 P. A. Miles, S. J. Kennedy, G. J. McIntyre, G. D. Gu, G. J. Russell and N. Koshizuka, Physica C 294, 275 (1998). 11 V. Petricek, Y. Gao, P. Lee and P. Coppens, Phys. Rev. B 42, 387 (1990). 12 X. B. Kan and S. C. Moss, Acta Cryst. B48, 122 (1992). 13 R. E. Gladyshevskii and R. Flükiger, Acta Cryst. B52, 38 (1996). 14 R. E. Gladyshevskii, N. Musolino and R. Flükiger, Phys. Rev. B 70, 184522 (2004). There is no superstructure for this Pb doped variant, so the crystal structure used is the actual one, not the average one. International Tables for Crystallography, 5th ed., ed. T. Hahn (Kluwer, Dordrecht, 2002). 16 P. Carra and R. Benoist, Phys. Rev. B 62, R7703 (2000). 17 A. Mans, I. Santoso, Y. Huang, W. K. Siu, S. Tavaddod, V. Apiainen, M. Lindroos, H. Berger, V. N. Strocov, M. Shi, L. Patthey and M. S. Golden, Phys. Rev. Lett. 96, 107007 (2006). 18 Y. Joly, Phys. Rev. B 63, 125120 (2001). This program can be downloaded at http://www-cristallo.grenoble.cnrs.fr/fdmnes. 19 C. R. Natoli, Ch. Brouder, Ph. Sainctavit, J. Goulon, Ch. Goulon-Ginet and A. Rogalev, Eur. Phys. J. B 4, 1 (1998). 20 V. Arpiainen and M. Lindroos, Phys. Rev. Lett. 97, 037601 (2006). 21 D. A. Varshalovich, A. N. Moskalev and V. K. Kersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988). 22 The results presented involve a convolution of the calcu- http://www-cristallo.grenoble.cnrs.fr/fdmnes lated spectrum with both a core hole (1.9 eV for the Cu K edge) and a photoelectron lifetime, with the latter having a strong energy dependence (ranging up to 15 eV with a midpoint value at 30 eV above the Fermi energy). But the calculation has a fixed cluster radius. 23 J. Goulon, C. Goulon-Ginet, A. Rogalev, G. Benayoun, C. Brouder and C. R. Natoli, J. Synchr. Rad. 7, 182 (2000). In addition, there is also an intrinsic non XNCD contribu- tion to the circular dichroism for a biaxial crystal, but this should vanish in Bi2212 for an x-ray beam aligned along the c-axis, see J. Goulon, C. Goulon-Ginet, A. Rogalev, V. Gotte, C. Brouder and C. Malgrange, Eur. Phys. J. B 12, 373 (1999). 24 P. A. Miles, S. J. Kennedy, A. R. Anderson, G. D. Gu, G. J. Russell and N. Koshizuka, Phys. Rev. B 55, 14632 (1997). 25 B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhes, C. T. Lin, X. Chaud and P. Bourges, Phys. Rev. Lett. 96, 197001 (2006). 26 A. Kapitulnik, unpublished results. 27 It should be remarked, though, that single-particle based approaches can miss some features of the spectral weight transfer between the L2 and L3 edges. 28 J. Kobayashi, T. Asahi, M. Sakurai, M. Takahashi, K. Okubo and Y. Enomoto, Phys. Rev. B 53, 11784 (1996). 29 E. Giannini, N. Clayton, N. Musolino, A. Piriou, R. Glady- shevskii and R. Flükiger, IEEE Trans. Appl. Supercond. 15, 3102 (2005). 30 Y. Le Page, W. R. McKinnon, J.-M. Tarascon and P. Bar- boux, Phys. Rev. B 40, 6810 (1989). 31 X-ray dichroism experiments in regards to the orbital cur- rent scenario of Varma have been discussed by S. Di Matteo and C. M. Varma, Phys. Rev. B 67, 134502 (2003).

0704.0600

Solvability of linear equations within weak mixing sets

9 Solvability of linear equations within weak mixing sets Alexander Fish November 4, 2018 Abstract We introduce a new class of “random” subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and suf- ficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that any partition-regular system of linear equations with integer coefficients is solvable in any WM set. 1 Introduction 1.1 Algebraic patterns within subsets of N We use extensively the notion of “algebraic pattern”. By an algebraic pattern we mean a solution of a diophantine system of equations. For example, an arithmetic progression of length k is an algebraic pattern corresponding to the following diophantine system: 2xi = xi−1 + xi+1, i = 2, 3, . . . , k − 1. We investigate the problem of finding linear algebraic patterns (these cor- respond to linear systems) within a family of subsets of natural numbers satisfying some asymptotic conditions. For instance, by Szemerédi theorem, subsets of positive upper Banach density (all S ⊂ N : d∗(S) > 0, where d∗(S) = lim supbn−an→∞ |S∩[an,bn]| bn−an+1 ) contain the pattern of an arithmetic progression of any finite length (see [12]). http://arxiv.org/abs/0704.0600v2 On the other hand, Schur patterns, namely triples of the form {x, y, x+ y}, which correspond to solutions of the so-called Schur equation, x+ y = z, do not necessarily occur in sets of positive upper density. For example, the odd numbers do not contain this pattern. But if we take a random subset of N by picking natural numbers with probability 1 independently, then this set contains the Schur pattern with probability 1. There is a deterministically defined analog of a random set - a normal set. To define a normal set we recall the notions of a normal infinite binary sequence and of a normal number. An infinite {0, 1}-valued sequence λ is called a normal sequence if every finite binary word w occurs in λ with frequency 1 , where |w| is the length of w. The more familiar notion is that of a normal number x ∈ [0, 1]. If to a number x ∈ [0, 1] we associate its dyadic expansion x = with xi ∈ {0, 1}, then x is called a normal number if the sequence (x1, x2, . . . , xn, . . .) is a normal sequence. Definition 1.1.1 A set S ⊂ N is called normal if the 0-1 sequence 1S (1S(n) = 1 ⇔ n ∈ S) is normal. Normal sets exhibit a non-periodic, “random” behavior. We notice that if S is a normal set then S − S contains N. Therefore, the equation z − y = x is solvable within every normal set. This implies that every normal set con- tains Schur patterns. Normal sets are related to a class of dynamical systems displaying maximal randomness; namely Bernoulli systems. In this work we investigate occur- rence of linear patterns in sets corresponding to dynamical systems with a lower degree of randomness, so called weakly mixing dynamical systems. The sets we obtain will be called WM sets. We will make this precise in the next section. In the present paper we treat the following problem: Give a complete characterization of the linear algebraic patterns which occur in all WM sets. Remark 1.1.1 It will follow from our definition of a WM set, that any normal set is a WM set. The problem of the solvability of a nonlinear equation or system of equa- tions is beyond the limits of the technique used in this paper. Nevertheless, some particular equations might be analyzed. In [3] it is shown that there exist normal sets in which the multiplicative Schur equation xy = z is not solvable. 1.2 Generic points and WM sets For a formal definition of WM sets we need the notions of measure preserving systems and of generic points. Definition 1.2.1 Let X be a compact metric space, B the Borel σ-algebra on X; let T : X → X be a continuous map and µ a probability measure on B. The quadruple (X,B, µ, T ) is called a measure preserving system if for every B ∈ B we have µ(T−1B) = µ(B). For a compact metric space X we denote by C(X) the space of continuous functions on X with the uniform norm. Definition 1.2.2 Let (X,B, µ, T ) be a measure preserving system. A point ξ ∈ X is called generic for the system (X,B, µ, T ) if for any f ∈ C(X) we f(T nξ) = f(x)dµ(x). (1.1) Example: Consider the Bernoulli system: (X = {0, 1}N0,B, µ, T ), where X is endowed with the Tychonoff topology, B is Borel σ-algebra on X , T is the shift to the left, µ is the product measure of µi’s where µi(0) = µi(1) = and N0 = N ∪ {0}. An alternative definition of a normal set which is purely dynamical is the following. A set S is normal if and only if the sequence 1S ∈ {0, 1} N0 is a generic point of the foregoing Bernoulli system. The notion of a WM set generalizes that of a normal set, where the role played by Bernoulli dynamical system is taken over by dynamical systems of more general character. Let ξ(n) be any {0, 1}−valued sequence. There is a natural dynamical system (Xξ, T ) connected to the sequence ξ: On the compact space Ω = {0, 1}N0 endowed with the Tychonoff topology, we define a continuous map T : Ω −→ Ω by (Tω)n = ωn+1. Now for any ξ in Ω we define Xξ = {T nξ}n∈N0 ⊂ Ω. Let A be a subset of N. Choose ξ = 1A and assume that for an appropriate measure µ, the point ξ is generic for (Xξ,B, µ, T ). We can attach to the set A dynamical properties associated with the system (Xξ,B, µ, T ). We recall the notions of ergodicity, total ergodicity and weak-mixing in er- godic theory: Definition 1.2.3 A measure preserving system (X,B, µ, T ) is called er- godic if every A ∈ B which is invariant under T , i.e. T−1(A) = A, satisfies µ(A) = 0 or 1. A measure preserving system (X,B, µ, T ) is called totally ergodic if for ev- ery n ∈ N the system (X,B, µ, T n) is ergodic. A measure preserving system (X,B, µ, T ) is called weakly mixing if the system (X ×X,BX×X , µ× µ, T × T ) is ergodic. In our discussion of WM sets corresponding to weakly mixing systems, we shall add the proviso that the dynamical system in question not be the trivial 1-point system supported on the point x ≡ 0. This implies that the “density” of the set in question be positive. Definition 1.2.4 Let S ⊂ N. If the limit of 1 n=1 1S(n) exists as N → ∞ we call it the density of S and denote by d(S). Definition 1.2.5 A subset S ⊂ N is called a WM set if 1S is a generic point of the weakly mixing system (X1S ,B, µ, T ) and d(S) > 0. 1.3 Solvability of linear diophantine systems within WM sets and normal sets Our main result is a complete characterization of linear systems of diophan- tine equations which are solvable within every WM set. The characterization is given by describing affine subspaces of Qk which intersect Ak, for any WM set A ⊂ N. Theorem 1.3.1 An affine subspace of Qk intersects Ak for every WM set A ⊂ N if and only if it contains a set of the form {n~a+m~b+ ~f |n,m ∈ N}, where ~a,~b, ~f have the following description: ~a = (a1, a2, . . . , ak) t, ~b = (b1, b2, . . . , bk) t ∈ Nk, ~f = (f1, f2, . . . , fk) t ∈ Zk and there exists a partition F1, . . . , Fl of {1, 2, . . . , k} such that: a) for every r ∈ {1, . . . , l} there exist c1,r, c2,r ∈ N, such that for every i ∈ Fr we have ai = c1,r , bi = c2,r and for every j ∈ {1, . . . , k} \ Fr we have aj bj c1,r c2,r 6= 0. ∀r ∈ {1, 2, . . . , l} ∃cr ∈ Z such that ∀i ∈ Fr : fi = cr. We also classify all affine subspaces of Qk which intersect Ak for any normal set A ⊂ N. Theorem 1.3.2 An affine subspace of Qk intersects Ak for every normal set A ⊂ N if and only if it contains a set of the form {n~a+m~b+ ~f |n,m ∈ N}, where ~a,~b, ~f have the following description: ~a = (a1, a2, . . . , ak) t, ~b = (b1, b2, . . . , bk) t ∈ Nk, ~f = (f1, f2, . . . , fk) t ∈ Zk and there exists a partition F1, . . . , Fl of {1, 2, . . . , k} such that for every r ∈ {1, . . . , l} there exist c1,r, c2,r ∈ N, such that for every i ∈ Fr we have ai = c1,r , bi = c2,r and for every j ∈ {1, . . . , k} \ Fr we have aj bj c1,r c2,r 6= 0. A family of linear algebraic patterns that has been studied previously are the “partition regular” patterns. These are patterns which for any finite partition of N: N = C1∪C2 ∪ . . .∪Cr, the pattern necessarily occurs in some Cj. (For example by van der Waerden’s theorem, arithmetic progressions are partition regular and by Schur’s theorem the Schur pattern is also partition regular). A theorem of Rado gives a complete characterization of such patterns. We will show in Proposition 4.1 that every linear algebraic pattern which is partition-regular occurs in every WM set. It is important to mention that if we weaken the requirement of weak mixing to total ergodicity, then in the resulting family of sets, Rado’s patterns need not necessarily occur. For example, for α 6∈ Q the set n ∈ N|nα (mod 1) ∈ is totally ergodic, i.e., 1S is a generic point for a totally ergodic system and the density of S is positive, but the equation x+ y = z is not solvable within In the separate paper [4] we will address the question of solvability of more general algebraic patterns, not necessarily linear, in totally ergodic and WM sets. The structure of the paper is the following. In Section 2 we prove the direction “⇐” of Theorems 1.3.1 and 1.3.2. In Section 3, by use of a probabilistic method, we prove the direction “⇒” of Theorems 1.3.1 and 1.3.2. In Section 4 we show that every linear system which is solvable in one of the cells of any finite partition of N is also solvable within every WM set. The paper ends with Appendix in which we collected proofs of technical statements which have been used in Sections 2 and 3. 1.4 Acknowledgments This paper is a part of the author’s Ph.D. thesis. I thank my advisor Prof. Hillel Furstenberg for introducing me to ergodic theory and for many useful ideas which I learned from him. I thank Prof. Vitaly Bergelson for fruitful discussions and valuable suggestions. Also, I would like to thank an anony- mous referee for numerous valuable remarks. 2 Proof of Sufficiency Notation: We introduce the scalar product of two vectors v, w of length N as follows: < v,w >N v(n)w(n). We denote by L2(N) the (finite-dimensional) Hilbert space of all real vectors of length N with the aforementioned scalar product. We define: ‖ w ‖2N =< w,w >N . First we state the following proposition which will prove useful in the proof of the sufficiency of the conditions of Theorem 1.3.1. Proposition 2.1 Let Ai ⊂ N (1 ≤ i ≤ k) be WM sets. Let ξi(n) = 1Ai(n)−d(Ai), where d(Ai) denotes density of Ai. Suppose there are (a1, b1), (a2, b2), . . . , (ak, bk) ∈ Z 2, such that ai > 0, 1 ≤ i ≤ k, and for every i 6= j ai bi aj bj 6= 0. Then for every ε > 0 there exists M(ε) ∈ N, such that for every M ≥ M(ε) there exists N(M, ε) ∈ N, such that for every N ≥ N(M, ε) where w(n) m=1 ξ1(a1n+b1m)ξ2(a2n+b2m) . . . ξk(akn+bkm) for every n = 1, 2, . . . , N . Since the proof of Proposition 2.1 involves many technical details, first we show how our main result follows from it. Afterwards we state and prove all the lemmas necessary for the proof of Proposition 2.1. We use an easy consequence of Proposition 2.1. Corollary 2.1 Let A be a WM set. Let k ∈ N, suppose (a1, b1), (a2, b2), . . . , (ak, bk) ∈ Z 2 satisfy all requirements of Proposition 2.1 and suppose f1, . . . , fk ∈ Z. Then for every δ > 0 there exists M(δ) such that ∀M ≥ M(δ) there exists N(M, δ) such that ∀N ≥ N(M, δ) we have ∣‖v‖N − d ∣ < δ, where v(n) m=1 1A(a1n + b1m + f1)1A(a2n + b2m + f2) . . . 1A(akn + bkm+ fk) for every n = 1, 2, . . . , N . Proof. We rewrite v(n) in the following form: v(n) = (ξ1(a1n+ b1m) + d(A)) . . . (ξk(akn + bkm) + d(A)), for every n = 1, 2, . . . , N . We introduce normalized WM sequences ξi(n) = ξ(n+ fi) (of zero average), where ξ(n) = 1A(n)− d(A). By use of triangular inequality and Proposition 2.1 it follows that for big enoughM and N (which depends on M) ‖v‖N is as close as we wish to d k(A). This finishes the proof. Proof. (of Theorem 1.3.1, ⇚) Let A ⊂ N be a WM set. Without loss of generality, we can assume that for every r : 1 ≤ r ≤ l we have r ∈ Fr. It follows from Corollary 2.1 that the vector v defined by 1A(a1n+ b1m+ f1)1A(a2n + b2m+ f2) . . . 1A(aln + blm+ fl) for every n = 1, 2, . . . , N , is not identically zero for big enough M and N . But this is possible only if for some n,m ∈ N we have (a1n+ b1m+ f1, a2n+ b2m+ f2, . . . , aln+ blm+ fl) ∈ A The latter implies that Ak intersects the affine subspace. Proof. (of Theorem 1.3.2, ⇚) For every r : 1 ≤ r ≤ l take all indices which comprise Fr. Denote this sequence of indices by Ir. Denote cr = mini∈Ir fi. Let Sr be the set of all non-zero shifts of fi, i ∈ Fr, centered at cr, i.e., Sr = {fi − cr | i ∈ Fr, fi > cr}. For example, if the sequence of fi’s where i ∈ F1 is (−5, 2, 3, 2,−5), then S1 = {7, 8}. Let A be a normal set. For every r : 1 ≤ r ≤ l we define sets Ar by Ar = {n ∈ N ∪ {0} |n ∈ A and n+ s ∈ A, ∀s ∈ Sr}. Then Ar is no longer a normal set provided that Sr 6= ∅ (d(A) = 21+|Sr | But, for all r : 1 ≤ r ≤ l the sets Ar’s are WM sets. Without loss of generality, assume that for every r : 1 ≤ r ≤ l we have r ∈ Fr. From Proposition 2.1 it follows that for big enough M and N 1A1(a1n+ b1m)1A(a2n+ b2m) . . . 1A(aln+ blm) ≈ d(Ar). The latter ensures that there exist m,n ∈ N such that (a1n+ b1m+ f1, . . . , akn+ bkm+ fk) ∈ A Now we state and prove all the claims that are required in order to prove Proposition 2.1. Definition 2.1 Let ξ be a WM-sequence (ξ is a generic point for a weakly mixing system (Xξ,BXξ , µ, T )) of zero average. The autocorrelation function of ξ of length j ∈ N with the shifts ~i = (i1, i2, . . . , ij) ∈ Z j and r ∈ Z is the sequence ψ which is defined by (n) = w∈{0,1}j ξ(n+ r + w ·~i), n ∈ N, where w ·~i is the usual scalar product in Qj, and (n) = 0, n ≤ 0. Lemma 2.1 Let ξ be a WM-sequence of zero average and suppose ε, δ > 0, b ∈ Z \ {0}. Then for every j ≥ 1, (c1, c2, . . . , cj) ∈ (Z \ {0}) j and (r1, r2, . . . , rj) ∈ Z j there exist I = I(ε, δ, c1, . . . , cn), a set S ⊂ [−I, I] density at least 1−δ and N(S, ε) ∈ N, such that for every N ≥ N(S, ε) there exists L(N, S, ε) such that for every L ≥ L(N, S, ε) r,(c1i1,...,cjij) (l + bn) for every (i1, i2, . . . , ij) ∈ S, where r = k=1 rk. Proof. We note that it is sufficient to prove the lemma in the case c1 = c2 = . . . = cj = 1, since if the average of nonnegative numbers over a whole lattice is small, then the average over a sublattice of a fixed positive density is also small. Recall that ξ ∈ Xξ = {T nξ}∞n=0 ⊂ supp(ξ) N0, where T is the usual shift to the left on the dynamical system supp(ξ)N0, and by the assumption that ξ is a WM-sequence of zero average it follows that ξ is a generic point of the weakly mixing system (Xξ,BXξ , µ, T ) and the function f : f(ω) = ω0 has zero integral. Denote ~i = (i1, . . . , ij). We define functions g on Xξ by T r+ǫ· ~i ◦ f, ǫ∈V ∗ T r+ǫ· ~i ◦ f, where Vj is the j-dimensional discrete cube {0, 1} j and V ∗j is the j-dimensional discrete cube except the zero point. Notice that (T nξ) = ψ We use the following theorem which is a special case of a multiparameter weakly mixing PET of Bergelson and McCutcheon (theorem A.1 in [2]; it is also a corollary of Theorem 13.1 of Host and Kra in [9]). Let (X, µ, T ) be a weakly mixing system. Given an integer k and 2k bounded functions fǫ on X, ǫ ∈ Vk , the functions Ni −Mi n∈[M1,N1)×...[Mk,Nk) ǫ∈V ∗ T ǫ1n1+...ǫknk ◦ fǫ converge in L2(µ) to the constant limit ǫ∈V ∗ when N1 −M1, . . . , Nk −Mk tend to +∞. From this theorem applied to the weakly mixing system Xξ × Xξ and the functions fǫ(x) = T r ◦ f ⊗ T r ◦ f for every ǫ ∈ Vj , we obtain for every Folner sequence {Fn} in N j that an average over the multi-index ~i = {i1, . . . , ij} of on Fn’s converges to zero in L 2(µ) (the integral of T r ◦ f ⊗ T r ◦ f is zero). Thus Xξ×Xξ Ni −Mi ~i∈[M1,N1)×...×[Mj,Nj) (y)dµ(x)dµ(y) = Ni −Mi ~i∈[M1,N1)×...×[Mj,Nj) (x)dµ(x) as N1 −M1, . . . , Nj −Mj → ∞. As a result we obtain the following statement: For every ε > 0, j ∈ N and every fixed (r1, r2, . . . , rj) ∈ N j, there exists a subset R ⊂ Nj of lower density equal to one, such that < ε (2.1) for every ~i ∈ R, where r = k=1 rj. Recall that lower density of a subset R ⊂ Nj is defined to be d∗(R) = lim inf N1−M1,...,Nj−Mj→∞ #{R ∩ [M1, N1)× . . .× [Mj , Nj)} k=1(Nk −Mk) Recall that ψ (l + bn) = g T l+bnξ The definition of the sequences ψj implies r1,~i (l + bn) = lim r2,(±i1,...,±ij) (l ± bn) for any r1, r2 ∈ Z, where ~i = (i1, . . . , ij). Therefore, in order to prove Lemma 2.1 it is sufficient to show the following: For every ε, δ > 0 and for any a priori chosen b ∈ N there exists I(ε, δ) ∈ N, such that for every I ≥ I(ε, δ) there exists a subset S ⊂ [1, I]j of density at least 1 − δ (namely, we have |S∩[1,I)j | ≥ 1 − δ) and N(S, ε) ∈ N, such that for every N ≥ N(S, ε) there exists L(N, S, ε) ∈ N such that for every L ≥ L(N, S, ε) the following holds for every ~i ∈ S: (l + bn) Let b ∈ N. Continuity of the function g0,~i and genericity of the point ξ ∈ Xξ yield (l + bn) = lim T bng0,~i T bng0,~i dµ. (2.2) By applying the von Neumann ergodic theorem to the ergodic system (Xξ,B, µ, T b) (ergodicity follows from weak-mixing of the original measure preserving system (Xξ,B, µ, T )) we have T bng0,~i → L2(Xξ) g0,~idµ. (2.3) From (2.1) there exists I(ε, δ) ∈ N big enough that for every I ≥ I(ε, δ) there exists a set S ⊂ [1, I]j of density at least 1− δ such that g0,~idµ for all ~i ∈ S. From equation (2.3) it follows that there exists N(S, ε) ∈ N, such that for every N ≥ N(S, ε) we have T bng0,~i for all ~i ∈ S. Finally, equation (2.2) implies that there exists L(N, S, ε) ∈ N, such that for every L ≥ L(N, S, ε) we have (l + bn) for all ~i ∈ S. The following lemma is a generalization of the previous lemma to a product of several autocorrelation functions. Lemma 2.2 Let ψ r1,~i , . . . , ψ rk,~i be autocorrelation functions of length j of WM-sequences ξ1, . . . , ξk of zero average, {c11, . . . , c j , . . . , c 1, . . . , c j} ∈ (Z \ {0}) jk and ε, δ > 0. Suppose (a1, b1), (a2, b2), . . . , (ak, bk) ∈ Z 2, such that ai > 0 for all i : 1 ≤ i ≤ k and for every i 6= j ai bi aj bj 6= 0. (If k = 1 assume that b1 6= 0.) Then there exists I(ε, δ) ∈ N, such that for every I ≥ I(ε, δ) there exist S ⊂ [−I, I]j of density at least 1− δ, M(S, ε) ∈ N, such that for every M ≥ M(S, ε) there exists X(M,S, ε) ∈ N, such that for every X ≥ X(M,S, ε) r1,(c i1,...,c (a1x+ b1m) . . . ψ rk,(c i1,...,c (akx+ bkm) for every (i1, i2, . . . , ij) ∈ S. Proof. The proof is by induction on k. THE CASE k = 1 (and arbitrary j): If a1 = 1 then the statement of the lemma follows from Lemma 2.1. If a1 > 1 then by Proposition 5.1 of Appendix for a given ~i = (i1, . . . , ij) ∈ S we have r1,(c i1,...,c (a1x+ b1m) r1,(c i1,...,c (x+ b1m) (2.4) (Limits exist by genericity of the point ξ.) By Lemma 2.1 the right hand side of (2.4) is small for large enough M . So, for large enough X (depending on M and (i1, . . . , ij)) the statement of the lemma is true. By finiteness of S we conclude that the statement of the lemma holds for k = 1. GENERAL CASE (k > 1): Suppose that the statement holds for k − 1. Denote vm(x) r1,(c i1,...,c (a1x+ b1m) . . . ψ rk,(c i1,...,c (akx+ bkm). Let ε, δ > 0. We show that there exists I(ε, δ) ∈ N such that for every I > I(ε, δ) a set S ⊂ [−I, I]j of density at least 1− δ can be chosen satisfying the following property: There exists I(ε, S) ∈ N such that for every I > I(ε, S) there exists M(I) ∈ N such that for all M > M(I) for a set of i’s in {1, 2, . . . , I} of density at least 1− ε we have < vm, vm+i >X (2.5) for all (i1, . . . , ij) ∈ S. The Van der Corput lemma (Lemma 5.1 of Appendix) finishes the proof. Note that the set of “good” i’s in the interval {1, 2, . . . , I} depends on (i1, . . . , ij) ∈ S. Denote < vm, vm+i >X 1,j+1 r1,(c i1,...c ij ,b1i) (a1x+ b1m) . . . ψ k,j+1 rk,(c i1,...,c ij ,bki) (akx+ bkm) Denote y = a1x+ b1m. Assume that (a1, b1) = d. Denote B̃y,m = ψ 1,j+1 r1,(c i1,...c ij ,b1i) (y) . . . ψ k,j+1 rk,(c i1,...,c ij ,bki) (a′ky + b where a′p = , b′p = bp − a pb1, 2 ≤ p ≤ k. We rewrite à as follows: y≡dl mod a1 m≡φ(l) mod B̃y,m + δX,M . (2.6) Here φ is a bijection of Za1 defined by the identity for every 0 ≤ l ≤ a1 − 1, Y = a1X , a p as above and δX,M accounts for the fact that for small y’s and y’s close to Y there is a difference between elements that are taken in the expression for à and in the expression on the right hand side of equation (2.6). Nevertheless, we have δX,M → 0 if Denote C̃y,m = ψ 2,j+1 r2,(c i1,...,c ij ,b2i) (a′2y + b 2m) . . . ψ k,j+1 rk,(c i1,...,c ij ,bki) (a′ky + b It will suffice to prove that there exists I(ε, δ) ∈ N such that for every I > I(ε, δ) we can find S ⊂ [−I, I]j of density at least 1 − δ with the following property: There exists I(ε, S) ∈ N such that for every I > I(ε, S) there exists M(I) ∈ N such that for every M > M(I) we can find X(M) ∈ N such that for every X > X(M) for a set of i’s in {1, 2, . . . , I} of density at least 1 − ε we have y≡dl mod a1 m≡φ(l) mod C̃y,m (2.7) for all 0 ≤ l ≤ a1 − 1, for all (i1, . . . , ij) ∈ S. Note that it is enough to prove the latter statement for every particular l : 0 ≤ l ≤ a1 Denote the left hand side of inequality (2.7) for a fixed l by D̃l. Introduce new variables z and n, such that y = za1+ dl and m = n +φ(l). We obtain D̃l = 2,j+1 t2n,z,l . . . ψ k,j+1 tkn,z,l 2,j+1 (a2z + c2n + q2) . . . ψ k,j+1 (akz + ckn+ qk) where shp = (rp, (c 1i1, . . . , c j ij , bpi)), n,z,l ap(a1z+dl)+(a1bp−apb1)( n+φ(l)) , qp = apld+(a1bp−apb1)φ(l) a1bp−apb1 6= 0, Z = Y and N = Md From the conditions on the function φ it follows that qp ∈ Z, 2 ≤ p ≤ k. From the conditions of the lemma we obtain for every p 6= q, p, q > 1, ap cp aq cq a1 det ap bp aq bq 6= 0. Therefore, D̃l can be rewritten as D̃l = φ2 (a2z + c2n) . . . φk (akz + ckn) where φℓ = ψ ℓ,j+1 rℓ+qℓ,(c i1,...,c ij ,bℓi) , 2 ≤ ℓ ≤ k. By the induction hypothesis the following is true. There exists Il(ε, δ ′) ∈ N big enough, such that for every Il ≥ Il(ε, δ ′) there exist a subset Sl ⊂ [−Il, Il] j+1 of density at least 1 − δ′2 and N(Sl, ε) ∈ N, such that for every N ≥ N(Sl, ε) there exists Z(N, Sl, ε) ∈ N, such that for every Z ≥ Z(N, Sl, ε) we have D̃l < (2.8) for all (i1, . . . , ij , i) ∈ Sl. For every (i1, . . . , ij) ∈ [−Il, Il] j we denote by Sli1,...,ij the fiber above (i1, . . . , ij): Sli1,...,ij = {i ∈ [−Il, Il] | (i1, . . . , ij , i) ∈ Sl}. Then there exists a set Tl ⊂ [−Il, Il] j of density at least 1− δ′, such that for every (i1, . . . , ij) ∈ Tl the density of S i1,...,ij is at least 1 − δ′. Let ε, δ > 0. Take δ′ < min ( ε , δ) and I > max (I ′(ε), Il(ε, δ ′)) (I ′(ε) is taken from the van der Corput lemma). Then it follows by (2.8) that there exists M(Tl, ε, δ) ∈ N, such that for every M ≥ M(Tl, ε, δ) there exists X(M,Tl, ε, δ) ∈ N, such that for every X ≥ X(M,Tl, ε, δ) the inequality (2.7) holds for every fixed (i1, . . . , ij) ∈ Tl for a set of i’s within the interval {1, . . . , I} of density at least 1 − ε . The lemma follows from the van der Corput lemma. Proof of Proposition 2.1. Denote vm(n) = ξ1(a1n+b1m) . . . ξk(akn+bkm). For every i ∈ N we introduce à defined by < vm, vm+i >N 0,(b1i) (a1n+ b1m) . . . ψ 0,(bki) (akn + bkm) where the functions ψp,j’s are autocorrelation functions of the ξp’s of length By Lemma 2.2 it follows that for every ε > 0 there exists I(ε) ∈ N such that for every I ≥ I(ε) there exist S ⊂ {1, 2, . . . , I} of density at least 1− ε M(S, ε) such that for every M ≥ M(S, ε) there exists N(M,S, ε) such that for every N ≥ N(M,S, ε) we have 0,(b2i) (a2n+ b2m) . . . ψ 0,(bki) (akn+ bkm) ≤ ε2. The proposition follows from the van der Corput Lemma 5.1. 3 Probabilistic constructions of WM sets The goal of this section is to prove the necessity of the conditions of Theorem 1.3.1. The following proposition is the main tool for this task. Proposition 3.1 Let a, b ∈ N, c ∈ Z such that a 6= b. Then there exists a normal set A within which the equation ax = by + c (3.1) is unsolvable, i.e., for every (x, y) ∈ A2 we have ax 6= by + c. Remark 3.1 The proposition is a particular case of Theorem 1.3.1. It is a crucial ingredient in proving the necessity direction of the theorem in general. Proof. Let S ⊂ N. We construct from S a new set AS within which the equation ax = by + c is unsolvable. Without loss of generality, suppose that a < b. Assume (a, b) = 1 (the general case follows easily). It follows from (a, b) = 1 that (3.1) is solvable. Any solution (x, y) of the equation ax = by + c has restrictions on x. Namely, x ≡ φ(a, b, c)(mod b), where φ(a, b, c) ∈ {0, 1, . . . , b − 1} is determined uniquely. Let us denote l0 = φ(a, b, c). We define inductively a sequence {li} ⊂ N ∪ {0}. If a pair (x, y) is a solution of equation (3.1) and y ∈ biN + li−1 then choose li ∈ {0, 1, . . . , b i+1 − 1} such that x ∈ bi+1N+ li. Note that from (a, b) = 1 it follows that (a, bi+1) = 1. It is clear that if u, v ∈ N satisfy (u, v) = 1 then for any w ∈ Z there exists a solution (x, y) ∈ N2 of the equation ux = vy + w. The latter implies that there exist x ∈ N, y ∈ biN+ li−1 such that ax = by+ c. Any such x should be a member of bi+1N+ li. Note that li and li−1 are connected by the identity ali ≡ bli−1 + c ( mod b i+1). (3.2) In addition, if x ∈ N is given then the equation ax ≡ by + c ( mod bi+1) has at most one solution y ∈ {0, 1, . . . , bi − 1}. We define sets Hi = biN+ li−1 ; i ∈ N. We prove that for every i ∈ N, Hi+1 ⊂ Hi. All elements of Hi+1 are in the same class modulo b i+1, therefore all elements of Hi+1 are in the same class modulo b i. So, if we show for some x ∈ Hi+1 that x ≡ li−1(mod b i) then we are done. For i = 1 we know that if y ∈ N then any x ∈ N such that (x, y) is a solution of the equation (3.1) has to be in H1. Take x ∈ H2 such that there exists y ∈ H1 with ax = by + c. Then x ∈ H1. Therefore, we have shown that H2 ⊂ H1. For i > 1 there exists x ∈ Hi+1 such that there exists y ∈ Hi with ax = by + c. By induction Hi ⊂ Hi−1. Therefore, the latter y is in Hi−1. Therefore, by construction of li’s we have that x ∈ Hi. This shows Hi+1 ⊂ Hi. We define sets Bi; 0 ≤ i <∞: B0 = N \H1, B1 = H1 \H2 . . . Bi = Hi \Hi+1 . . . Clearly we have Bi∩Bj = ∅ , ∀i 6= j and |N\ (∪ i=0Bi)| = | ∩ i=1Hi| ≤ 1. The latter is because for every i the second element (in the increasing order) of Hi is ≥ b We define AS = i=0Ai, where Ai’s are defined in the following manner: = S ∩ B0, C0 = B0 \ A0 = B1 \ {x | ax ∈ bB0 + c}, A1 = (B1 ∩ {x | ax ∈ bC0 + c}) ∪ (D1 ∩ S) , = B1 \A1 . . . = Bi \ {x | ax ∈ bBi−1 + c}, Ai = (Bi ∩ {x | ax ∈ bCi−1 + c}) ∪ (Di ∩ S) , = Bi \ Ai . . . Here it is worthwhile to remark that for every i, Bi = Ai ∪ Ci. Therefore AS ⊂ ∪ i=0Bi. If for some i ≥ 1 we have y ∈ Ai ⊂ Bi = Hi \ Hi+1, then any x with ax = by + c satisfies ax ≡ bli−1 + c ( mod b i+1). From (a, bi+1) = 1 it follows that there exists a unique solution x modulo bi+1. By identity (3.2) we have x ≡ li ( mod b i+1). Thus x ∈ Hi+1. If x ∈ Hi+2, then x ≡ li+1 ( mod b i+2). Thus we have ali+1 ≡ by + c ( mod b i+2). By uniqueness of a solution ( y ) modulo bi+1 we get y ≡ li ( mod b i+1). Thus y ∈ Hi+1. We have a contradiction, which shows that x ∈ Hi+1\Hi+2 = Bi+1. The same argument works for y ∈ A0 ⊂ B0 and it shows that any x with ax = by + c satisfies x ∈ B1. So, if y ∈ Ai (i ≥ 0) then any x with ax = by + c should satisfy x ∈ Bi+1. By construction of AS, x 6∈ AS. Thus equation (3.1) is not solvable in AS. We make the following claim: For almost every subset S of N the set AS is a normal set. (The probability measure on subsets of N considered here is the product on {0, 1}∞ of probability measures (1 The tool for proving the claim is the following easy lemma (for a proof see Appendix, Lemma 5.2). A subset A of natural numbers is a normal set if and only if for any k ∈ (N ∪ {0}) and any i1 < i2 < . . . < ik we have χA(n)χA(n + i1) . . . χA(n+ ik) = 0, (3.3) where χA(n) = 2 · 1A(n)− 1. First of all, we denote TN = n=1 χAS(n)χAS(n + i1) . . . χAS(n + ik). Be- cause of randomness of S, TN is a random variable. We will prove that N=1E(T ) <∞ and this will imply by Lemma 5.3 that TN →N→∞ 0 for almost every S ⊂ N. E(T 2N) = n,m=1 E(χAS(n)χAS(n+i1) . . . χAS(n+ik)χAS(m) . . . χAS(m+ik)). Adding (removing) of a finite set to (from) a normal set does not affect the normality of the set. The set ∪iBi might differ from N by at most one element (| ∩∞i=1 Hi| ≤ 1). This possible element does not affect the normality of AS and we assume without loss of generality that ∩∞i=1Hi = ∅, thus N = ∪ i=0Bi. For every number n ∈ N we define the chain of n, Ch(n), to be the following finite sequence: If n ∈ B0, then Ch(n) = (n). If n ∈ B1, then two situations are possible. In the first one there exists a unique y ∈ B0 such that an = by+c. We set Ch(n) = (n, y) = (n, Ch(y)). In the second situation we can not find such y from B0 and we set Ch(n) = (n). If n ∈ Bi+1, then again two situations are possible. In the first one there exists y ∈ Bi such that an = by + c. In this case we set Ch(n) = (n, Ch(y)). In the second situation there is no such y from Bi. In this case we set Ch(n) = (n). We define l(n) to be the length of Ch(n). For every n ∈ N we define the ancestor of n, a(n), to be the last element of the chain of n (of Ch(n)). To determine whether or not n ∈ AS will depend on whether a(n) ∈ S. The exact relationship depends on the i for which n ∈ Bi and on the j for which a(n) ∈ Bj or in other words on the length of Ch(n): χAS(n) = (−1) i−jχS(a(n)) = (−1) l(n)−1χS(a(n)). We say that n is a descendant of a(n). It is clear that E(χAS(n1) . . . χAS(nk)) 6= 0 (E(χAS(n1) . . . χAS(nk)) ∈ {0, 1}) if and only if every number a(ni) occurs an even number of times among numbers a(n1), a(n2), . . . , a(nk). We bound the number of n,m’s inside the square [1, N ] × [1, N ] such that E(χAS(n)χAS(n+ i1) . . . χAS(n+ ik)χAS(m)χAS(m+ i1) . . . χAS(m+ ik)) 6= 0. For a given n ∈ [1, N ] we count all m’s inside [1, N ] such that for the ancestor of n there will be a chance to have a twin among the ancestors of n+i1, . . . , n+ ik, m,m+ i1, . . . , m+ ik. First of all it is obvious that in the interval [1, N ] for a given ancestor there can be at most log b N + C1 descendants, where C1 is a constant. For all but a constant number of n’s it is impossible that among n + i1, . . . , n + ik there is the same ancestor as for n. Therefore we should focus on ancestors of the set {m,m + i1, . . . , m + ik}. For a given n we might have at most (k + 1)(log b N + C1) options for the number m to provide that one of the elements of {m,m + i1, . . . , m + ik} has the same ancestor as n. Therefore for most of n ∈ [1, N ] (except maybe a bounded number C2 of n’s which depends only on {i1, . . . , ik} and doesn’t depend on N) we have at most (k + 1)(log b N + C1) possibilities for m’s such that E(χAS(n)χAS(n+ i1) . . . χAS(n+ ik)χAS(m)χAS(m+ i1) . . . χAS(m+ ik)) 6= 0. Thus we have E(T 2N) ≤ (k + 1)(log b N + C1) + C2N ((k+1) log b N +C3), where C3 is a constant. This implies E(T 2N2) <∞. Therefore TN2 →N→∞ 0 for almost every S ⊂ N. By Lemma 5.3 it follows that TN →N→∞ 0 almost surely. In the general case, where a, b are not relatively prime, if c satisfies (3.1) then it should be divisible by (a, b). Therefore by dividing the equation (3.1) by (a, b) we reduce the problem to the previous case. We use the following notation: Let W be a subset ofQn. Then for any increasing subsequence I = (i1, . . . , ip) ⊂ {1, 2, . . . , n} we define ProjIW = WI = {(wi1, . . . , wip) | ∃w = (w1, w2, . . . , wn) ∈ W}. We recall the notion of a cone. Definition 3.1 A subset W ⊂ Qn is called a cone if (a) ∀w1, w2 ∈ W we have w1 + w2 ∈ W (b) ∀α ∈ Q : α ≥ 0 and ∀w ∈ W we have αw ∈ W . The next step involves an algebraic statement with a topological proof which we have to establish. Lemma 3.1 Let W be a non-trivial cone in Qn which has the property that for every two vectors ~a = {a1, a2, . . . , an} t,~b = {b1, b2, . . . , bn} t ∈ W there exist two coordinates 1 ≤ i < j ≤ n (depend on the choice of ~a,~b) such that ai bi aj bj There exist two coordinates i < j such that the projection of W on these two coordinates is of dimension ≤ 1 (dimQ SpanProj(i,j)W ≤ 1). Proof. First of all W has positive volume in V = SpanW (Volume is Haar measure which normalized by assigning measure one to a unit cube and W contains a parallelepiped). Fix an arbitrary non-zero element ~x ∈ W . For every i, j : 1 ≤ i < j ≤ n we define the subspace Ui,j = {~v ∈ V |Proj(i,j)~v ∈ SpanProj(i,j)~x}. From the assumptions of the lemma it follows that i,j;1≤i<j≤n (W ∩ Ui,j). For every i 6= j we obviously have that the volume of Ui,j is either zero or Ui,j = V . If we assume that the statement of the lemma does not hold then Ui,j 6= V, ∀i 6= j, and thus the volume of Ui,j , ∀i 6= j is zero. We get a contradiction because a finite union of sets with zero volume cannot be equal to a set with positive volume. Proof. (of Theorem 1.3.1, ⇛) Assume that an affine subspace A of Qk intersects Ak for any WM set A ⊂ N. First of all, we shift the affine space to obtain a vector subspace, denote it by U . The linear space U must contain vectors with all positive coordinates, since A ∩Ak must be infinite. Denote by W = {~v ∈ U | 〈~v, ~ei〉 ≥ 0 , ∀ i : 1 ≤ i ≤ k}. W is a non-trivial cone. Assume that for every ~a = (a1, . . . , ak) t,~b = (b1, . . . , bk) t ∈ W we have that ∃i, j : 1 ≤ i < j ≤ k such that ai bi aj bj Then by Lemma 3.1 we deduce that there exist maximal subsets of coordi- nates F1, . . . , Fl (one of them, assume F1, should have at least two coordi- nates) such that for every r ∈ {1, 2 . . . , l} we have VFr = SpanWFr is one dimensional. We fix r : 1 ≤ r ≤ l. We show that the projection on Fr of W + ~f is on a diagonal, where ~f ∈ Zk is such that U + ~f = A. If the projection of W on Fr is not on a diagonal then there exist two coordinates i < j from Fr such that W(i,j) = {(ax, bx) | x ∈ N} for some a 6= b natural numbers. Therefore the projection of A on (i, j) has the form {(ax+ f1, bx+ f2) | x ∈ N}, where f1, f2 are integers. From Proposition 3.1 it follows that for any a, b, c, where a 6= b, there exists a WM set A (even a normal set) such that the equation ax = by + c is not solvable within A. This proves the existence of a WM set A0 such that for every x ∈ Z we have (ax+ f1, bx+ f2) 6∈ A 0 (introduce the new variables z1, z2 by (z1, z2) = (ax1+ f1, bx+ f2) and take a normal set A0 such that the equation az2 = bz1 + (af2 − bf1) is unsolvable within A0). Thus ∀i, j ∈ Fr : W(i,j) = {(ax, ax) | x ∈ N}. To prove that a shift is the same for all coordinates in Fr we merely should know that for any natural number c there exists a WM set Ac such that inside Ac the equation x− y = c is not solvable. The last statement is easy to verify. Let jr ∈ Fr, ∀1 ≤ r ≤ l. Denote I = (j1, . . . , jl). We have proved that there exist g1, . . . , gl ∈ N, c1, . . . , cl ∈ Z such that (U + ~f)I = {(g1x1 + c1, . . . , glxl + cl) | x1, . . . , xl ∈ Q}. It is clear that we can find ~a,~b which satisfy all the requirements of Theorem 1.3.1. This completes the proof. Remark 3.2 We have proved that if an affine subspace A ⊂ Qk intersects Ak for any normal set A ⊂ N, then there exist ~a,~b ∈ Nk and a partition F1, . . . , Fl of {1, 2, . . . , k} such that: (a) ∀r : 1 ≤ r ≤ l and ∀i ∈ Fr, ∀j 6∈ Fr we have ai bi aj bj 6= 0. (b) ∃~f ∈ Zk such that the set {n~a +m~b+ ~f |n,m ∈ N} is in A. Thus, we have proved the direction “⇛” of Theorem 1.3.2. 4 Comparison with Rado’s Theorem We recall that the problem of solvability of a system of linear equations in one cell of any finite partition of N was solved by Rado in [10]. Such systems of linear equations are called partition-regular. We show that partition- regular systems are solvable within every WM set by use of Theorem 1.3.1. It is important to note that solvability of partition-regular linear systems of equations within WM sets can be shown directly (without use of Theorem 1.3.1) by use of the technique of Furstenberg and Weiss that was developed in their dynamical proof of Rado’s theorem (see [8]). First of all we describe Rado’s regular systems. Definition 4.1 A rational p × q matrix (aij) is said to be of level l if the index set {1, 2, . . . , q} can be divided into l disjoint subsets I1, I2, . . . , Il and rational numbers crj may be found for 1 ≤ r ≤ l and 1 ≤ j ≤ q such that the following relationships are satisfied: aij = 0 aij = c1jaij . . . aij = j∈I1∪I2∪...∪Il−1 cl−1j aij for i = 1, 2, . . . , p. Theorem 4.1 (Rado) A system of linear equations is partition-regular if and only if for some l the matrix (aij) is of level l and it is homogeneous, i.e. a system of the form aijxj = 0, i = 1, 2, . . . , p. The following claim is the main result of this section. Proposition 4.1 A partition-regular system is solvable in every WM set. Proof. Let a system j=1 aijxj = 0, i = 1, 2, . . . , p be partition-regular. We will use the fact that the system is solvable for any finite partition of N. First of all, the set of solutions of a partition-regular system is a subspace of Qq; denote it by V . It is obvious that V contains vectors with all positive components. If for some 1 ≤ i < j ≤ q we have Proj+i,jV (where Proj i,jV = {(x, y)|x, y ≥ 0 & ∃~v ∈ V :< ~v, ~ei >= x , < ~v, ~ej >= y}) is contained in a line, then Proj+i,jV is diagonal, i.e. it is contained in {(x, x)|x ∈ Q}. Otherwise, we can generate a partition of N into two disjoint sets S1, S2 such that no S 1 and no S 2 intersects V : This partition is constructed by an iterative process. Without loss of gener- ality we may assume that the line is x = ny, where n ∈ N. The general case is treated in the simillar way. We start with S1 = S2 = ∅. Let 1 ∈ S1. We “color” the infinite geometric progression {nm |m ∈ N} (adding elements to either S1 or S2) in such way that there is no (x, y) on the line from S 1 , S Then we take a minimal element from N which is still uncolored. Call it a. Add a to S1. Next, “color” {an m |m ∈ N}. Continuing in this fashion, we obtain the desired partition of N. This contradicts the assumption that the given system is partition-regular. Let F1, . . . , Fl be a partition of {1, 2, . . . , k} such that for every r ∈ {1, . . . , l} we have for every i 6= j , i, j ∈ Fr : dim QSpan(Proj i,jV ) = 1, and for every r : 1 ≤ r ≤ l, every i ∈ Fr .and for every j 6∈ Fr we have dim QSpan(Proj i,jV ) = 2. For every r : 1 ≤ r ≤ l we choose arbitrarily one representative index within Fr and denote it by jr (jr ∈ Fr). Then there exist g1, . . . , gl ∈ N such that VI = {(g1x1, . . . , glxl) | x1, . . . , xl ∈ Q}. The latter ensures that there exist vectors ~a,~b ∈ V which satisfy all the requirements of Theorem 1.3.1 and, therefore, the system is solvable in every WM set. 5 Appendix In this section we prove all technical lemmas and propositions that were used in the paper. We start with the key lemma which is a finite modification of Bergelson’s lemma in [1]. Its origin is in a lemma of van der Corput. Lemma 5.1 Suppose ε > 0 and {uj} j=1 is a family of vectors in Hilbert space, such that ‖uj‖ ≤ 1 (1 ≤ j ≤ ∞). Then there exists I ′(ε) ∈ N, such that for every I ≥ I ′(ε) there exists J ′(I, ε) ∈ N, such that the following holds: For J ≥ J ′(I, ε) for which we obtain 〈uj, uj+i〉 for a set of i’s in the interval {1, . . . , I} of density 1− ε we have Proof. For an arbitrary J define uk = 0 for every k < 1 or k > J . The following is an elementary identity: uj−i = I Therefore, the inequality i=1 ui i=1 ‖ui‖ yields ≤ (J + I) = (J + I) uj−p, uj−s〉 = (J+I) ‖uj−p‖ +2(J+I) r,s=1;s<r 〈uj−r, uj−s〉 = (J+I)(Σ1+2Σ2), where Σ1 = I by the aforementioned elementary identity and h=1(I−h) j=1〈uj, uj+h〉. The last expression is obtained by rewrit- ing Σ2, where h = r − s. By dividing the foregoing inequality by I 2J2 we obtain J + I J + I J + I Choose I ′(ε) ∈ N, such that 12 ≤ I ′(ε) ≤ 12 +1. Then for every I ≥ I ′(ε) we have 1 ≤ 11ε . There exists J ′(I, ε) ∈ N, such that for every J ≥ J ′(I, ε): . As a result, for every I ≥ I ′(ε) there exists J ′(I, ε), such that for every J ≥ J ′(I, ε) The next proposition was used in Section 2. Proposition 5.1 Let A ⊂ N be a WM-set. Then for every integer a > 0 and every integers b1, b2, . . . , bk ξ(n+ b1)ξ(n+ b2) . . . ξ(n+ bk) = ξ(an+ b1)ξ(an+ b2) . . . ξ(an+ bk), where ξ = 1A − d(A). Proof. Consider the weak-mixing measure preserving system (Xξ,B, µ, T ). The left side of the equation in the proposition is T b1fT b2f . . . T bkfdµ, where f(ω) = ω0 for every infinite sequence inside Xξ. We make use of the notion of disjointness of measure preserving systems. By [6] we know that every weak-mixing system is disjoint from any Kronecker system which is a compact monothethic group with Borel σ-algebra, the Haar probability measure, and the shift by a chosen element of the group. In particular, every weak-mixing system is disjoint from the measure preserving system (Za,BZa , S, ν), where Za = Z/aZ, S(n) = n+ 1( mod a). The measure and the σ-algebra of the last system are uniquely determined. Therefore, from Furstenberg’s theorem (see [6], Theorem I.6) it follows that the point (ξ, 0) ∈ Xξ×Za is a generic point of the product system (Xξ×Za,B×BZa , T×S, µ×ν). Thus, for every continuous function g on Xξ × Za we obtain Xξ×Za g(x,m)dµ(x)dν(m) = lim g(T nξ, Sn0). Let g(x,m) = f(x)10(m), which is obviously continuous on Xξ × Za. Then genericity of the point (ξ, 0) yields Xξ×Za f(x)10(m)dµ(x)dν(m) = f(x)dµ(x) = f(T nξ)10(n) = lim f(T anξ). Taking instead of the function f the continuous function T b1fT b2f . . . T bkf in the definition of g finishes the proof. The next two lemmas are very useful for constructing normal sets with specif- ical properties. Lemma 5.2 Let A ⊂ N. Let λ(n) = 21A(n)− 1. Then A is a normal set ⇔ for any k ∈ (N ∪ {0}) and any i1 < i2 < . . . < ik we have λ(n)λ(n+ i1) . . . λ(n+ ik) = 0. Proof. “⇒” If A is normal then any finite word w ∈ {−1, 1}∗ has the “right” frequency 1 inside wA. This guarantees that “half of the time” the function λ(n)λ(n + i1) . . . λ(n + ik) equals 1 and “half of the time” is equal to −1. Therefore we get the desired conclusion. “⇐” Let w be an arbitrary finite word of plus and minus ones: w = a1a2 . . . ak and we have to prove that w occurs in wA with the frequency 2 −k. For every n ∈ N the word w occurs in 1A and starting from n if and only if 1A(n) = a1 . . . 1A(n+ k − 1) = ak The latter is equivalent to the following λ(n) = 2a1 − 1 . . . λ(n+ k − 1) = 2ak − 1 The frequency of w within 1A is equal to λ(n)(2a1 − 1) + 1 . . . λ(n+ k − 1)(2ak − 1) + 1 The limit is equal to 1 Lemma 5.3 Let {an} be a bounded sequence. Let TN = n=1 an. Then TN converges to a limit t ⇔ there exists a sequence of increasing indices {Ni} such that Ni → 1 and TNi →i→∞ t. References [1] Bergelson, V. Weakly mixing PET. Ergodic Theory Dynam. Sys- tems 7 (1987), no. 3, 337–349. [2] Bergelson, V.; McCutcheon, R. An ergodic IP polynomial Sze- merédi theorem. Mem. Amer. Math. Soc. 146 (2000), no. 695. [3] Fish, A. Random Liouville functions and normal sets. Acta Arith. 120 (2005), no. 2, 191–196. [4] Fish, A. Polynomial largeness of sumsets and totally ergodic sets, see http://arxiv.org/abs/0711.3201. [5] Fish, A. Ph.D. thesis, Hebrew University, 2006. [6] Furstenberg, H. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1-49. [7] Furstenberg, H. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. d’ Analys Math. 31 (1977), 204–256. [8] Furstenberg, H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ. Press 1981. [9] Host, B.; Kra, B. Nonconventional ergodic averages and nilman- ifolds. Ann. of Math. (2) 161 (2005), no. 1, 397–488. [10] Rado, R. Note on combinatorial analysis. Proc. London Math. Soc. 48 (1943), 122–160. [11] Schur, I. Uber die Kongruenz xm+ym ≡ zm(modp). Jahresbericht der Deutschen Math.-Ver. 25 (1916), 114–117. [12] Szemerédi, E. On sets of integers containing no k elements in arithmetic progression. Collection of articles in memory of Jurǐi Vladimirovič Linnik. Acta Arith. 27 (1975), 199–245. Current Address: Department of Mathematics University of Wisconsin-Madison 480 Lincoln Dr. Madison, WI 53706-1388 E-mail: [email protected] http://arxiv.org/abs/0711.3201 Introduction Algebraic patterns within subsets of N Generic points and WM sets Solvability of linear diophantine systems within WM sets and normal sets Acknowledgments Proof of Sufficiency Probabilistic constructions of WM sets Comparison with Rado's Theorem Appendix

0704.0601

D-\bar D mixing and rare D decays in the Littlest Higgs model with non-unitarity matrix

arXiv:0704.0601v4 [hep-ph] 24 Oct 2007 D − D̄ mixing and rare D decays in the Littlest Higgs model with non-unitarity matrix Chuan-Hung Chen1,2∗, Chao-Qiang Geng3,4† and Tzu-Chiang Yuan3‡ 1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan 2National Center for Theoretical Sciences, Hsinchu 300, Taiwan 3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan 4Theory Group, TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C. V6T 2A3, Canada (Dated: October 27, 2018) Abstract We study the D − D̄ mixing and rare D decays in the Littlest Higgs model. As the new weak singlet quark with the electric charge of 2/3 is introduced to cancel the quadratic divergence induced by the top-quark, the standard unitary 3× 3 Cabibbo-Kobayashi-Maskawa matrix is extended to a non-unitary 4× 3 matrix in the quark charged currents and Z-mediated flavor changing neutral currents are generated at tree level. In this model, we show that theD−D̄ mixing parameter can be as large as the current experimental value and the decay branching ratio (BR) of D → Xuγ is small but its direct CP asymmetry could be O(10%). In addition, we find that the BRs of D → Xuℓ+ℓ−, D → Xuνν̄ and D → µ+µ− could be enhanced to be O(10−9), O(10−8) and O(10−9), respectively. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] http://arxiv.org/abs/0704.0601v4 I. INTRODUCTION As the observation of the Bs − B̄s mixing in 2006 by CDF [1], all neutral pseudoscalar- antipseudoscalar oscillations (P − P̄ ) in the down type quark systems have been seen. In the standard model (SM), the most impressive features of flavor physics are the Glashow- Iliopoulos-Maiani (GIM) mechanism [2] and the large top quark mass. The former results in the cancellation between the lowest order short-distance (SD) contributions of the first two generations to the mass difference ∆mK in the K 0 system, while the latter makes ∆mBq (q = d, s) in the Bq systems dominated by the SD effects [3]. In addition, these features also lead to sizable flavor changing neutral currents (FCNCs) from box and penguin diagrams, which contribute to the rare decays, such as K → πνν̄ and B → K(∗)ℓℓ̄. It is known that these processes could be good candidates to probe new physics effects [4–6]. However, it is clear that the new physics signals deviated from the SM predications for the P − P̄ mixings and rare FCNC decays have to wait for precision measurements on these processes. Unlike K and Bq systems, the SD contributions to charmed-meson FCNC processes, such as the D − D̄ mixing [7] and the decays of c → uℓ+ℓ− and D → ℓ+ℓ− [8], are highly suppressed due to the stronger GIM mechanism and weaker heavy quark mass enhancements in the loops. On the other hand, it is often claimed that the long-distance (LD) effect for the D−D̄ mixing should be the dominant contribution in the SM. Nevertheless, because the nonperturbative hadronic effects are hard to control, the result is still inconclusive [9–12]. Recently, BABAR [13] and BELLE [14, 15] collaborations have reported the evidence for the D − D̄ mixing with x′2 = (−0.22± 0.30± 0.21)× 10−3 , y′ = (9.7± 4.4± 3.1)× 10−3 , (1) x ≡ ∆mD = (0.80± 0.29± 0.17)% , y ≡ ∆ΓD = (0.33± 0.24± 0.15)% , yCP = (1.31± 0.32± 0.25)% , (2) respectively, where x′ = x cos δ + y sin δ and y′ = −x sin δ + y cos δ with the assumption of CP conservation and δ being the relative strong phase between the amplitudes for the doubly-Cabbibo-suppressed D → K+π− and Cabbibo-favored D → K−π+ decays [16, 17] and yCP = τ(D → K−π+)/τ(D → K+K−) − 1. Moreover, no evidence for CP violation is found. The combined results of Eqs. (1) and (2) at the 68% C.L. are [18] x = (5.5± 2.2)× 10−3 , y = (5.4± 2.0)× 10−3 , δ = (−38± 46)0 . (3) Note that the upper bound of x < 0.015 at 95%C.L. can be extracted from the BELLE data in Eq. (2) [14, 15]. The evidences of the mixing parameters by BABAR and BELLE collaborations reveal that the era of the rare charmed physics has arrived. The results in Eq. (3) can not only test the SU(3) breaking effects for the D− D̄ mixing [10, 12], but also examine new physics beyond the SM [17–21]. It is known that a straightforward way to enhance the rare D processes is to include some new heavy quarks within the framework of the SM. For instance, if a new heavy quark with the electric charge of −1/3 is introduced, it could affect the D system since the extra down type quark violates the GIM mechanism. However, the constraint on this heavy quark is quite strong as it could also lead to FCNCs for the down type quark sector at tree level, which are strictly limited by the well measured rare K and B decays, such as KL → µ+µ− and B → Xsγ [22]. On the other hand, if the charge of the new heavy quark is 2/3, it could generate interesting tree level FCNCs for the up type quark sector, for which the constraints are much weaker. In this paper, we will study D physics based on a new weak singlet upper quark. It has been known that in the framework of the Littlest Higgs model [23], there exists a new SU(2)L singlet vector-like up quark [24], hereafter denoted by T . Since the number of down type quarks is the same as that in the SM, the standard unitary 3 × 3 Cabibbo- Kobayashi-Maskawa (CKM) matrix is extended to a non-unitary 4×3 matrix in the charged currents. Moreover, Z-mediated FCNCs for the up quark sector are generated at tree level. In Ref. [25], it has been shown that the contributions of this new quark to the rare D processes are small and cannot reach the sensitivities of future experiments [25, 26]. In this paper, we will demonstrate that by adopting some plausible scenario, the effects could not only generate a large D − D̄ oscillation but also marginally reach the sensitivity proposed by BESIII for the rare D decays [27]. We note that the implication of the new data on the D − D̄ mixing in the Littlest Higgs model with T-parity has been recently studied in Ref. [21]. The paper is organized as follows. In Sec. II, we investigate that when a gauge singlet T - quark is introduced in the Littlest Higgs model, how the non-unitary matrix for the charged current and the tree level Z-mediated FCNC are formed. By using the leading perturbation, the mixing matrix elements related to the new parameters in the Littlest Higgs model are derived. In addition, we study how to get the small mixing matrix element for Vu(c)b, which describes the b → u(c) decays. In Sec. III, we discuss the implications of the non-unitarity on the D− D̄ mixing and rare D decays by presenting some numerical analysis. Finally, we summarize our results in Sec. IV. II. NON-UNITARY MIXING MATRIX IN THE LITTLEST MODEL To study the new flavor changing effects in the Littlest Higgs model, we start by writing the Yukawa interactions for the up quarks to be [24, 25] λabfǫijkǫxyχaiΣjxΣkyu b + λ0fTT c + h.c. , (4) where χT1 = (d1, u1, 0), χ 2 = (s2, c2, 0), χ 3 = (b3, t3, T ), u b is the weak singlet and Σ = eiΠ/fΣ0e iΠT /f with 112×2 112×2 , Π = 2 h∗/ φ hT/ . (5) The scale f denotes the global symmetry spontaneously breaking scale, which, as usual, could be around 1 TeV. Consequently, the 4× 4 up-quark mass matrix is given by [25] iλijv − − − − − 0 0 λ33f | λ0f . (6) We remark that the quadratic divergences for the Higgs mass from one-loop diagrams in- volving t and T get exactly cancelled as shown in Ref. [28]. Moreover, for other quarks other than the top quark, the one-loop quadratic divergent contributions do not necessitate fine-tuning the Higgs potential as the cutoff is around 10 TeV for f ∼ 1 TeV due to the small corresponding Yakawa couplings. That is why there is no need to introduce extra singlet states T [28, 29]. To obtain the quark mass hierarchy of mt ≫ mc ≫ mu, we can choose a basis such that the up-quark mass matrix is [30] m̂U 0 hf λ0f  (7) where m̂Uij = δijλiv/ 2 ≡ mi is diagonal matrix and h = (h1, h2, h3). The hi is related to λ33 by hi = Ṽ i3 λ33 and hh † = |λ33|2, in which Ṽ UR is the unitary transformation for the right-handed up quarks. We note that mi are not the physical masses and in principle their magnitudes could be as large as the weak scale. In order to preserve the hierarchy in the quark masses, one expects that m3 > m2 > m1. Furthermore, in terms of this basis, the charged and neutral currents, defined by LC = g√ J−µ W +µ − g√ 2 tan θ J−µ W H + h.c. , cos θW 3 − sin2 θWJµem tan θ 3 ZHµ + h.c. , (8) are expressed by J−µ = ŪLγµṼ 0aVDL , µṼ 0aV Ṽ 0†UL − µDL , (9) respectively, where UT = (u1, c2, t3, T ), D T = (d, s, b), aV = diag(1, 1, 1, 0) and Ṽ 0 = (V 0CKM)3×3 0  (10) with V 0CKMV CKM = 113×3. The null entry in aV denotes the new T -quark being a weak singlet; and without the new T -quark, V 0CKM is just the CKM matrix. Since the down quark sector is the same as that in the SM, we have set the unitary transformation UDL to be an identity matrix. For getting the physical eigenstates, the mass matrix in Eq. (7) can be diagonalized by unitary matrices V UL,R so that we have †diag U = V ULMUM L (11) m̂Um̂ f (|λ33|2 + |λ0|2)f 2  . (12) Since (|λ33|2 + |λ0|2)f 2 is much larger than other elements, we can take the leading order of the perturbation in himi/f (i = 1, 2, 3). According to Eq. (11), the leading expansion is given by †diag U = V ULMUM L ≈ (1 + ∆L)MUM †U (1−∆L) . (13) By looking at the off-diagonal terms (M †diag U )i4(4i), we can easily get ∆Li4 ≈ −∆L4i = − himif (|λ33|2 + |λ0|2)f 2 −m2i with i 6= 4. From the diagonal entries, if we set the light quark masses to be mu ≈ mc ≈ 0, we obtain 0 ≈ m2uj ≈ m j + 2∆Lj4(MUM U)4j , ∆Lj4 ≈ − with j = 1, 2. To be consistent with Eq. (14), at the leading expansion the relation 2h2j = (|λ33|2 + |λ0|2) (16) should be satisfied. We emphasize that the choice of Eq. (16) is somewhat fine-tuned in order to have Eqs. (14) and (15) simultaneously. Since the top-quark is much heavier than other ordinary quarks, we have 2h23 ≈ (1−m2t/m23)(|λ33|2+ |λ0|2) if f > m3 > mt. Similarly, one obtains the flavor mixing effects for i 6= j 6= 4 to be ∆Lij = hihjmimj m2i −m2j f 2[2(|λ33|2 + |λ0|2)f 2 − (m2i +m2j )] (|λ33|2 + |λ0|2)f 2 −m2j ((|λ33|2 + |λ0|2)f 2 −m2i ) . (17) After diagonalization, the currents become J−µ = ŪLγµV ULṼ 0aVDL = ŪLγµV ULV 0DL , 3 = ŪLγ µV ULṼ 0aV Ṽ 0†V †ULUL = ŪLγ µV ULV 0V 0†V †ULUL , (18) where UT = (u, c, t, T ), V 0 = Ṽ 0aV = (V 0CKM)3×3 0  (19) and diag(V 0V 0†) = aV . Since the 4-th component of aV is different from the first 3 ones, it is obvious that the matrix V ≡ V ULV 0 associated with the charged current does not satisfy unitarity. In addition, V ULaV V L, which is associated with the neutral current, is not the identity matrix. As a result, Z-mediated FCNCs at tree level are induced. According to Eq. (18), we see that V V † = V ULaV V L (20) which is just the same as the effects of the Z-mediated FCNCs. Due to V being a non-unitary matrix, one finds (V V †)ij = Vi4V j4 . (21) Consequently, the interesting phenomena arising from non-unitary matrix elements are al- ways related to Vi4V j4 = ∆Li4∆j4. We note that as we do not particularly address CP problem, in most cases, we set the parameters to be real numbers. It has been known that enormous data give strict bounds on the flavor changing effects. In particular, the pattern describing the charged current has been fixed quite well. Any new parametrization should respect these constraints. It should be interesting to see the relationship with and without the new vector-like T -quark. From Eq. (18), we know that the new flavor mixing matrix for the charged current is given by V = V ULV 0. At the leading order perturbation, one gets V = V ULV 0 ≈ (1 + ∆L)V 0 = V 0 +∆LV 0 . (22) If V 0tb ∼ 1 is taken, one finds that Vub ≈ V 0ub+∆L13 and Vcb ≈ V 0cb+∆L23. In terms of Eq. (17) and h1 = h2 ≈ h3, the relations ∆L13 ≈ −m1/m3 and ∆L23 ≈ −m2/m3 are obtained. Hence, in our approach, we have Vus ≈ V 0us − , Vub ≈ V 0ub − , Vcb ≈ V 0cb − . (23) From these results, it is clear that when the T -quark decouples from ordinary quarks, Vus → V 0us, Vub → V 0ub and Vcb → V 0cb, while m1/m2 → mu/mc, m1/m3 → mu/mt and m2/m3 → mc/mt, respectively. According to the observations in the decays of b → uℓν̄ℓ and b → cℓν̄ℓ, the corresponding values have been determined to be |Vub| = 3.96 ± 0.09 × 10−3 and |Vcb| = 42.21+0.10−0.80 × 10−3, respectively [16]. Since V 0ij and mi are free parameters, to satisfy the experimental limits with interesting phenomena in low energy physics, it is rea- sonable to set the orders of magnitude for m1/m3 and m2/m3 (m1/m2) to be O(10 −2) and O(10−1), respectively. Consequently, the non-unitary effects on the rare charmed meson decays governed by V14V 24 could be as large as ∆14∆24 ∼ O(10−4), which could be one order of magnitude larger than those in Ref. [25]. III. D − D̄ MIXING AND RARE D DECAYS A. D − D̄ mixing It is well known that the GIM mechanism has played an important role in the K − K̄ oscillation in the SM. In addition, due to the top-quark in the box and penguin diagrams, Bq − B̄q mixings are dominated by the SD effects, which are consistent with the data [16]. On the contrary, for the D− D̄ mixing the GIM cancellation further suppresses the mixing effect to be ∆mD ∼ O(m4s/m2Wm2c) [7] and the bottom quark contribution actually is a subleading effect due to the suppression of (VubV 2. In the SM, the SD contribution to the mixing parameter is O(10−7) [34]. However, the LD contribution to the mixing is believed to be dominant. Due to the nonperturbative hadronic effects, the result is still uncertain with the prediction on the mixing parameter ranging from O(10−3) [9] to O(10−2) [10–12]. Nonetheless, the mixing parameters shown in Eq. (3) could arise from the LD contribution. Thus, it is important to have a better understanding of the LD effect. On the other hand, it is also possible that the mixings in Eq. (3) could result from new physics. In the following, we will concentrate on the Littlest Higgs model. In the quark sector of the Littlest Higgs model due to the introduction of a new weak singlet, a direct impact on the low energy physics is the FCNCs at tree level. According to Eq. (18), the most attractive process with |∆C| = 2 via the Z-mediated c−u−Z interaction, illustrated in Fig. 1, is given by H(|∆C| = 2) = g (V14V ūγµPLc ūγ µPLc , (V14V ūγµPLc ūγ µPLc . (24) In terms of the hadronic matrix element, defined by 〈D̄|(ūc)V−A (ūc)V−A|D〉 = D , (25) FIG. 1: Z-mediated flavor diagram with |∆C| = 2. the mass difference for the D meson is [25] ∆mD ≈ DmDBD|(V14V ∗24)2| . (26) If we assume no cancellation between new physics and SM contributions, by taking τD = 1/ΓD = 6.232 × 1011 GeV−1, fD BD = 200 MeV [31, 32] and mD = 1.86 GeV and using Eq. (3), we obtain ζ0 ≡ |V14V ∗24| = |∆L14∆L24| = (1.47± 0.29)× 10−4 , (27) which is in the desirable range. In other words, the result in Eq. (27) demonstrates that the non-unitarity in the Littlest Higgs model could enhance the D − D̄ mixing at the observed level. We note that the limit of x < 0.015 (95%C.L.) leads to ζ0 < 2.5× 10−4. (28) In addition, we note that cancellation between the LD effect in the SM and the SD one from new physics could happen. In this case, the values in Eqs. (27) and (28) could be relaxed. B. D → Xuγ decay In the SM, the D-meson FCNC related processes are all suppressed since the internal fermions in the loops are all much lighter than mW . For the decay of D → Xuγ, without QCD corrections, the branching ratio is O(10−17); and it becomes O(10−12) when one-loop QCD corrections are included [8]. However, it is found that the two-loop QCD corrections can boost the BR to be as large as 3.5× 10−8 [33]. It should be interesting to see how large the non-unitarity effect on c → uγ is in the Littlest Higgs model. To study the radiative decay of c → uγ, we write the effective Lagrangian to be Lc→uγ = − mcūσµνPRcF µν , (29) where C7 = C 7 + C 7 and C 7 ≈ −(0.007 + i0.02) = 0.021eiδs with δs = 70.7◦ [33] being the strong phase induced by the two-loop QCD corrections. In the extension of the SM by including a weak singlet particle, the flavor mixing matrix in the charged current is not unitary and the Z-mediated FCNC at tree level is generated as well. For c → uγ, besides the QED-penguin diagrams induced by the W -boson displayed in Figs. 2a and 2b, the Z-mediated QED-penguin one in Fig. 2c will also give contributions. We note that the d, s, b W d, s, b u, c, T (a) (c)(b) FIG. 2: Flavor diagrams for c → uγ. contributions from WH and ZH can be ignored as m and m2Z/m are much less than one. At the first sight, due to the light quarks in the loops, the contributions from Figs. 2a and 2b could be negligible. However, due to the non-unitarity of (V V †)uc = V14V 24 6= 0, even in the limits of md, s, b → 0, the contributions from the mass independent terms do not vanish anymore and can be sizable. In terms of unitary gauge [22], we obtain CW7 = (V V †)12 VusV ∗cs VusV ∗cs . (30) Furthermore, if we set mu ≈ mc = 0, the contributions from Fig. 2c are given by CZ7 = (f u + f c + f T )/VusV fZc + f − eu sin2 θW [4ξ0(0)− 6ξ1(0) + 2ξ2(0)] +eu sin 2 θW [4ξ0(0)− 4ξ1(0)] fZT = eu [2ξ0(yT )− 3ξ1(yT ) + ξ2(yT )] , (31) where the functions ξn(x) are defined by ξn(x) ≡ zn+1dz 1 + (x− 1)z and yT = mT/mZ . Numerically, the total contribution in Fig. 2 is C7 = C 7 + C V ∗csVus 24 . (33) If we regard V14V 24 as an unknown complex parameter, i.e. V14V 24 = ζ0e iθ with θ being the CP violating phase, one can study the decay BR and direct CP asymmetry (CPA) of D → Xuγ defined by BR(D → Xuγ) = 6αem|C7|2 π|Vcd|2 BR(D → Xdeν̄e) , ACP = BR(c̄ → ūγ)− BR(c → uγ) BR(c̄ → ūγ) + BR(c → uγ) , (34) as functions of ζ0 and θ. In Fig. 3, the BR and CPA as functions of ζ0 are presented, where the solid, dotted, dashed and dash-dotted lines represent the CP violating phase at θ = 0, 45◦, 90◦ and 135◦, respectively. From these results, it is interesting to see that 0 1 2 3 4 5 6 0 1 2 3 4 5 6 (a) (b) FIG. 3: BR (in units of 10−8) and CPA (in units of 10−2) for D → Xuγ as functions of ζ0, where the solid, dotted, dashed and dash-dotted lines represent the CP violating phase at θ = 0, 45◦, 90◦ and 135◦, respectively. BR(D → Xuγ) is insensitive to the new physics effects, whereas the direct CPA could be as large as O(10%). Explicitly, if we take θ = 90◦ and ζ0 = 1.5 × 10−4, the CPA is about 3%. Note that this CPA vanishes in the SM. C. D → Xuℓℓ̄ and D0 → ℓ+ℓ− decays Because the current experimental measurements in K and Bq decays are all consistent with the SM predictions, it is inevitable that if we want to observe any deviations from the SM, we have to wait for precision measurements for K and Bq. SuperB factories or LHCb could provide a hope. However, the situation in D physics is straightforward. As stated before, unlike K and Bq systems, due to no heavy quark enhancement in the D system, the rare D-meson decays, such as D → Xuℓℓ̄ (ℓ = e, µ, ν), are always suppressed. Even by considering the long-distance effects, the related decays, such as D → µ+µ− and D → Xuνν̄, get small corrections to the SD predictions on the BRs [35]. Therefore, these rare decays definitely could be good candidates to probe the new physics effects. Since the values in the SM are hardly reachable at D factories [27], if any exotic event is found, it must be a strong evidence for new physics. In the following analysis, we are going to discuss the implication of the Littlest Higgs model on the rare D decays involving di-leptons. To study these decays, we first write the effective Hamiltonian for c → uℓ+ℓ− (ℓ = e, µ) H(c → uℓ+ℓ−) = − GFαem√ V ∗csVus Ceff9 O9 + C7O7 + C10O10 , (35) O7 = − ūiσµνq νPRcℓ̄γ O9 = ūγµPLc ℓ̄γ O10 = ūγµPLc ℓ̄γ µγ5ℓ , (36) where the effective Wilson coefficients are given by Ceff9 = (V V †)14 V ∗csVus cℓV + (h(zs, s)− h(zd, s)) (C2(mc) + 3C1(mc)) , C10 = − (V V †)14 V ∗csVus cℓA , (37) with s = q2/m2c , zi = mi/mc, c V = −1/2 + 2 sin2 θW , cℓA = −1/2 and h(z, s) = −4 ln z + (2 + x) |1− x| 1−x+1√ 1−x−1 − i π, for x ≡ 4z2/s < 1 , 2 arctan 1√ x−1 , for x ≡ 4z 2/s > 1 . (38) Here, we have neglected the small contributions from the penguin and box diagrams. We note that in the SM, the SD contributions are mainly from the term with h(z, s), induced by the insertion of O2 = ūLγµqLq̄Lγ µcL and mixing with O9 at one-loop level [35, 36]. We note that the resonant decays of D → XuV → Xuℓ+ℓ− (V = φ, ρ, ω) would have large corrections to c → uℓ+ℓ− at the resonant regions. However, in this paper we do not discuss these contributions as we only concentrate on the SD contributions. Moreover, these resonance contributions can be removed by imposing proper cuts in the phase space in dedicated searches. From Eq. (35), the decay rate for D → Xuℓ+ℓ− as a function of the invariant mass s = q2/m2c can be found to be 768π5 |VusV ∗cs|2(1− s)2R(s) , R(s) = |Ceff9 |2 + |C10|2 (1 + 2s) + 12Re(C∗7C 9 ) + 4 |C7|2 . (39) In addition, by utilizing the lepton angular distribution, we can also study the forward- backward asymmetry (FBA), given by −1 d cos θdΓ/dsd cos θ sgn(cos θ) −1 d cos θdΓ/dsd cos θ Ceff9 + , (40) where θ is the angle of ℓ+ related to the momentum of the D meson in the ℓ+ℓ− invariant mass frame. Since C10 is small in the SM, AFB is negligible. With mc = 1.4 GeV and the mixing parameter in Eq. (27), we get BR(D → Xue+e−) = (4.18± 0.91)× 10−10 , BR(D → Xuµ+µ−) = (2.51± 0.86)× 10−10 , (41) comparing with the SM predictions of BR(D → Xue+e−)SM = 2.1 × 10−10 and BR(D → +µ−)SM = 0.5 × 10−10, respectively. Clearly, if some cancellation occurs between new physics and SM contributions in the D − D̄ mixing, a larger value of ζ0 can be allowed. In Fig. 4, we show the tendency of the decay as a function of ζ0, where the negative horizontal values correspond to -ζ0. In addition, we present the differential decay BR [FBA] of D → Xue+e− as a function of s = q2/m2c in Fig. 5a [b], where the thick solid, dotted and dashed lines correspond to ζ0 = 1.5, 3.0 and 5.0, while the thin ones denote the cases for −ζ0 except ζ0 = 0 for the thin solid line in Fig. 5a. From Fig. 5b, we see that the FBA -6 -4 -2 0 2 4 6 FIG. 4: BR(in units of 10−9) for D → Xue+e− as a function of ζ0. 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 (a) (b) FIG. 5: (a)[(b)] Differential BR (in units of 10−9) [FBA] for D → Xue+e− as a function of s, where the thick solid, dotted and dashed lines correspond to ζ0 = 1.5, 3.0 and 5.0, while the thin ones denote the cases for −ζ0 except ζ0 = 0 for the thin solid line in (a). is only at percent level. In the Littlest Higgs model, this is because the Z coupling to the charged lepton cℓV = −1/2 + 2 sin2 θW appearing in Ceff9 is much smaller than one. This is quite different from that in b → sℓ+ℓ− where the dominant effect in the SM for the FBA is from the box and QED-penguin diagrams. Next, we discuss the decay of D → Xuνν̄. In the SM, the BR for D → Xuνν̄ is estimated to be O(10−16) − O(10−15) [35], which is vanishing small. In the Littlest Higgs model, by taking C7 = 0, C 9 = −C10 = −π(V V †)14/(αemVusV ∗cs), the effective Hamiltonian in Eq. (35) can be directly applied to c → uνν̄. The decay rate forD → Xuνν̄ as a function of s = q2/m2c can be obtained as 768π5 (1− s)2(1 + 2s) 2π2|(V V †)12|2 , (42) where the factor of 3 stands for the neutrino species. With ζ0 = 1.5×10−4, we get BR(D → Xuνν̄) = 1.31 × 10−9. However, if we relax the constraint on V14V †24, the BR as a function of ζ0 is shown in Fig. 6a. For a larger value of ζ0, BR(D → Xuνν̄) could be as large as O(10−8). 0 1 2 3 4 5 6 0 1 2 3 4 5 6 (a) (b) FIG. 6: (a) BR (in units of 10−8) for D → Xuνν̄ and (b) BR (in units of 10−9) for D → µ+µ−. Finally, we study the decays of D → ℓ+ℓ−. It has been known that, in the SM, the SD contributions to D → µ+µ− are O(10−18), while the LD ones are O(10−13) [35]. It is clear that any signal to be observed at the sensitivity of the proposed detector, such as BESIII, will indicate new physics effects. Since the effective interactions at quark level are the same as those in Eq. (35), one finds that BR(D → ℓ+ℓ−) = τDmDm |πV14V ∗24|2 . (43) Here we have used equation of motion for the charged lepton so that ℓ̄/pDℓ = 0. We also note that operators O7,9 make no contributions. With |V14V ∗24| = ζ0 = 1.5 × 10−4, the predicted BR for D → µ+µ− is 1.17× 10−10. In Fig. 6b, we present the BR as a function of ζ0. We see that BR(D → µ+µ−) in the Littlest Higgs model could be as large as O(10−9). IV. CONCLUSIONS We have studied the D − D̄ mixing and rare D decays in the Littlest Higgs model. In the model, as the new weak singlet vector-like quark T with the electric charge of 2/3 is introduced to cancel the quadratic divergence induced by the top-quark, the standard unitary 3× 3 CKM matrix is extended to a non-unitary 4× 3 matrix in the quark charged currents and Z-mediated flavor changing neutral currents are generated at tree level. We have shown that the effects on |∆C| = 2 and |∆C| = 1 processes are all related to V14V ∗24 in Eq. (21). To avoid the scenario adopted by Ref. [25], in which λ0 ∼ λ33 ≫ λij was assumed, we choose the basis such that the effective mass matrix for u1, c2 and t3 is diagonal, while the corresponding masses m1, m2 and m3 are free parameters and can be as large as the weak scale v. Since the global symmetry breaking scale f is larger than v, the mixing matrix relating physical and unphysical states could be extracted by taking the leading perturbative expansion. Accordingly, by using the approximation of mu ≈ mc ≈ 0, the explicit expressions for V14 and V24 have been obtained. In terms of the data for Vub and Vcb, we have found that the natural value for ζ0 ≡ |V14V ∗24| is O(10−4), which agrees with the observed parameter in the D − D̄ mixing but it is one order of magnitude larger than that in Ref. [25]. 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0704.0602

Alternative Large Nc Schemes and Chiral Dynamics

CERN–PH–TH/2007–068 SU–4252–8489 Alternative Large N Schemes and Chiral Dynamics Francesco Sannino∗ CERN Theory Division, CH-1211 Geneva 23, Switzerland. and University of Southern Denmark, Campusvej 55, DK-5230, Odense M., Denmark. Joseph Schechter† Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA. (Dated: March 2007) We compare the dependences on the number of colors of the leading ππ scattering amplitudes using the single index quark field and two index quark fields. These are seen to have different relationships to the scattering amplitudes suggested by chiral dynamics which can explain the long puzzling pion pion s wave scattering up to about 1 GeV. This may be interesting for getting a better understanding of the large Nc approach as well as for application to recently proposed technicolor models. BACKGROUND Gaining control of QCD in its strongly interacting (low energy) regime constitutes a real challenge. One very at- tractive approach is based on studying the theory in the large number of colors (Nc) limit [1, 2]. At the same time one may obtain more information by requiring the theory to model the (almost) spontaneous breakdown of chiral symmetry [3, 4]. A standard test case is pion pion scattering in the energy range up to about 1 GeV. Some time ago, an attempt was made [5, 6] to implement this combined scenario. Since the leading large Nc ampli- tude contains only tree diagrams involving mesons of the standard quark-antiquark type, it is expected that the required amplitude should be gotten by calculating just the chiral tree diagrams for rho meson exchange together with the four point pion contact diagram. There are no unknown parameters in this calculation. The crucial question is whether the scattering amplitude calculated in this way will satisfy unitarity. When one compares the result with experimental data up to about 1 GeV on the real part of the (most sensitive to unitarity violation) J=I=0 partial wave, one finds (see Fig.1 of [6]) that the result violates the partial wave unitarity bound by just a “little bit”. On the other hand, the pion contact term by itself violates unitarity much more drastically so one might argue that the large Nc approach, which suggests that the tree diagrams of all quark anti-quark resonances in the relevant energy range be included, is helping a lot. To make matters more quantitative one might ask the question: by how much should Nc be increased in or- der for the amplitude in question to remain within the unitarity bounds for energies below 1 GeV? This question was answered in a very simple way in [7], as we now briefly review. In terms of the con- ventional amplitude, A(s, t, u) the I = 0 amplitude is 3A(s, t, u) + A(t, s, u) + A(u, t, s). One gets the J = 0 channel by projecting out the correct partial wave. The current algebra (pion contact diagram) contribution to the conventional amplitude is Aca(s, t, u) = 2 s−m2π , (1) where the pion decay constant, Fπ depends on Nc as Fπ(Nc) = 131 3 so that Fπ(3) = 131 MeV. Fur- thermore mπ = 137 MeV is independent of Nc. The desired amplitude is obtained by adding to the current algebra term the following vector meson ρ(770) contribu- tion: Aρ(s, t, u) = g2ρππ (4m2π − 3s) g2ρππ (m2ρ − t)− imρΓρθ(t− 4m2π) (m2ρ − u)− imρΓρθ(u − 4m2π) , (2) where gρππ(Nc) = 8.56 Nc is the ρππ coupling con- stant. Also, mρ = 771 MeV is independent of Nc and Γρ(Nc) = g2ρππ (Nc) 12πm2ρ . (3) It should be noted that the first term in Eq. (2), which is an additional non-resonant contact interaction other than the current algebra contribution, is required when we include the ρ vector meson contribution in a chiral invariant manner. In Fig. 1 we show the real part of the I = J = 0 amplitude (denoted R0 ) due to current alge- bra plus the ρ contribution for increasing values of Nc. Since in this channel the vector meson is never on shell we suppress the contribution of the width in the vector http://arxiv.org/abs/0704.0602v1 meson propagator in Eq. (2). One observes that the uni- tarity bound (i.e., |R0 | ≤ 1/2) is satisfied for Nc ≥ 6 till well beyond the 1 GeV region. However unitarity is still a problem for 3, 4 and 5 colors. At energy scales larger than the one associated with the vector meson clearly other resonances are needed [5] but we shall not be concerned with that energy range here. It is also interesting to note that these considerations are essentially unchanged when the pion mass (i.e. explicit chiral symmetry breaking in the Lagrangian) is set to zero. FIG. 1: Real part of the I = J = 0 partial wave amplitude due to the current algebra +ρ terms, plotted for the follow- ing increasing values of Nc (from up to down), 3, 4, 5, 6, 7. The curve with largest violation of the unitarity bound corre- sponds to Nc = 3 while the ones within the unitarity bound are for Nc = 6, 7. Note that essentially we are just using the scaling, A(s, t, u) = Ã(s, t, u) . (4) where Ã(s, t, u) is defined replacing Fπ and gρππ with the Nc independent quantities F̃π = Fπ Nc and g̃ρππ = 3. Other authors [8] have found the same minimum value, Nc=6 for the practical consistency of the large Nc approximation, by using different methods. In order to explain low energy ππ scattering for the physical value Nc = 3 one must go beyond the large Nc approximation. It is attractive to keep the assumption of tree diagram dominance involving near by resonances, however. One easily sees that adding a scalar singlet res- onance (sigma) at the location where the unitarity bound on R0 (s) is first violated should restore unitarity. This is because the real part of a Breit Wigner resonance is zero at the pole location and negative just above it, so will bring R0 (s) below the bound, as required. In [7], the resonance mass was found to be around 550 MeV on this basis. Such a low value would make it different from a p-wave quark-antiquark state, which is expected to be in the 1000-1400 MeV range. We assume then that it is a four quark state (glueball states are expected to be in the 1.5 GeV range from lattice investigations). Four quark states of diquark-anti diquark type [9] and meson-meson type [10] have been discussed in the literature for many years. Accepting this picture, however, poses a problem for the accuracy of the large Nc inspired description of the scattering since four quark states are predicted not to exist in the large Nc limit of QCD. We shall take the point of view that a four quark type state is present since it allows a natural fit to the low energy data. Of course, it is still necessary to fine tune the parameters and shape of the sigma resonance to fit the experimental ππ scat- tering data in detail. In practice, since the parameters of the pion contact and rho exchange contributions are fixed, the sigma is the most important one for fitting and fits may even be achieved [11] if the vector meson piece is neglected. However the well established, presumably four quark type, f0(980) resonance must be included to achieve a fit in the region just around 1 GeV. There is by now a fairly large recent literature [12]- [44] on the effect of light “exotic” scalars in low energy meson meson scattering. There seems to be a consene- sus, arrived at using rather different approaches (keeping however, unitarity), that the sigma exists. TWO INDEX QUARK FIELDS Now, consider redefining the Nc = 3 quark field with color index A (and flavor index not written) as ǫABCq BC , qBC = −qCB, (5) so that, for example, q1 = q 23 and similarly for the ad- joint field, q̄1 = q̄23 etc. This is just a trivial change of variables and will not change anything for QCD. How- ever, if a continuation of the theory is made to Nc > 3 the resulting theory will be different since the two index anti- symmetric quark representation has Nc(Nc− 1)/2 rather than Nc color components. As was pointed out by Cor- rigan and Ramond [45], who were mainly interested in the problem of the baryons at large Nc, this shows that the extrapolation of QCD to higher Nc is not unique. Further investigation of the properties of the alternative extrapolation model introduced in [45] was carried out by Kiritsis and Papavassiliou [46]. Here, we shall discuss the consequences for the low energy ππ scattering dis- cussed above, of this alternative large Nc extrapolation, assuming for our purpose, that all the quarks extrapolate as antisymmetric two index objects. It may be worthwhile to remark that gauge theo- ries with two index quarks have recently gotten a great deal of attention. Armoni, Shifman and Veneziano [47, 48, 49, 50, 51] have proposed an interesting relation between certain sectors of the two index antisymmetric (and symmetric) theories at large number of colors and sectors of super Yang-Mills (SYM). Using a supersym- metry inspired effective Lagrangian approach 1/Nc cor- rections were investigated in [52]. Information on the su- per Yang-Mills spectrum has been obtained in [53]. On the validity of the large Nc equivalence between differ- ent theories and interesting new possible phases we refer the reader to [54, 55, 56]. The finite temperature phase transition and its relation with chiral symmetry has been investigated in [57] while the effects of a nonzero baryon chemical potential were studied in [58]. When adding flavors the phase diagram as a function of the number of flavors and colors has been provided in [59]. The complete phase diagram for fermions in ar- bitrary representations has been unveiled in [60]. The study of theories with fermions in a higher dimensional representation of the gauge group and the knowledge of the associated conformal window led to the construction of minimal models of technicolor [59, 61, 62] which are not ruled out by current precision measurements and lead to interesting dark matter candidates [63, 64, 65] as well as to a very high degree of unification of the standard model gauge couplings [66]. Besides these two limits a third one for massless one- flavor QCD, which is in between the ’t Hooft and Cor- rigan Ramond ones, has been very recently proposed in [67]. Here one first splits the QCD Dirac fermion into the two elementary Weyl fermions and afterwards assigns one of them to transform according to a rank-two antisym- metric tensor while the other remains in the fundamental representation of the gauge group. For three colors one reproduces one-flavor QCD and for a generic number of colors the theory is chiral. The generic Nc is a particular case of the generalized Georgi-Glashow model [68]. To illustrate the large Nc counting for the ππ scatter- ing amplitude when quarks are designated to transform according to the two index antisymmetric representation of color SU(3) one may employ [1] the mnemonic where each tensor index of this group is represented by a di- rected line. Then the quark-quark gluon interaction is pictured as in Fig. 2. The two index quark is pictured FIG. 2: Two index fermion - gluon vertex. as two lines with arrows pointing in the same direction, as opposed to the gluon which has two lines with arrows pointing in opposite directions. The coupling constant representing this vertex is taken to be gt/ Nc, where gt (the ’t Hooft coupling constant) is to be held constant. A “one point function”, like the pion decay constant, Fπ would have as it’s simplest diagram, Fig. 3. The X represents a pion insertion and is associated FIG. 3: Diagram for Fπ for the two index quark. with a normalization factor for the color part of the pion’s wavefunction, Nc(Nc − 1) , (6) which scales for large Nc as 1/Nc. The two circles each carry a quark index so their joint factor scales as N2c for large Nc; more precisely, taking the antisymmetry into account, the factor is Nc(Nc − 1) . (7) The product of Eqs. (6) and (7) yields the Nc scaling for F 2π (Nc) = Nc(Nc − 1) F 2π (3). (8) For largeNc, Fπ scales proportionately to Nc rather than Nc as in the case of the ’t Hooft extrapolation. Using this scaling together with Eq.(1) suggests that the ππ scatttering amplitude, A scales as, A(Nc) = Nc(Nc − 1) A(3), (9) which, for large Nc scales as 1/N c rather than as 1/Nc in the ’t Hooft extrapolation. This scaling law for large Nc may be verified from the mnemonic in Fig. 4, where there is an N2c factor from the two loops multiplied by four factors of 1/Nc from the X’s. FIG. 4: Diagram for the scattering amplitude, A with the 2 index quark. It is interesting to find the minimum value of Nc for which the tree amplitude due to the pion and rho meson terms (given in Eqs.(1) and (2) above) is unitary in this two antisymmetric index quark extrapolation scheme. Fig. 1 shows that the the peak value of the partial wave amplitude, R0 due to these two terms is numerically about 0.9. This is to be identified with Aca(3) + Aρ(3) in Eq.(9). Thus the condition that the extrapolated am- plitude be unitary is, Nc(Nc − 1) < 1/2. (10) Clearly, the extrapolated amplitude is unitary already for Nc = 4, which indicates better convergence in Nc than for the ’t Hooft case which became unitary at Nc = 6. There is still another different feature; consider the typical ππ scattering diagram with an extra internal (two index) quark loop, as shown in Fig. 5. FIG. 5: Diagram for the scattering amplitude, A including an internal 2 index quark loop. In this diagram there are four X’s (factor from Eq.(6)), five index loops (factor from Eq.(7)) and six gauge cou- pling constants. These combine to give a large Nc scaling behavior proportional to 1/N2c for the ππ scattering am- plitude. We see that diagrams with an extra internal 2 index quark loop are not suppressed compared to the leading diagrams. This is analogous, as pointed out in [46], to the behavior of diagrams with an extra gluon loop in the ’t Hooft extrapolation scheme. Now, Fig. 5 is a diagram which can describe a sigma particle exchange. Thus in the 2 index quark scheme, “exotic” four quark resonances can appear at the leading order in addition to the usual two quark resonances. Given the discussion we reviewed above, the possibility of a sigma appearing at leading order means that one can construct a unitary ππ amplitude already at Nc = 3 in the 2 antisymmetric index scheme. From the point of view of low energy ππ scattering, it seems to be unavoidable to say that the 2 index scheme is more realistic than the ’t Hooft scheme given the existence of a four quark type sigma. Of course, the usual ’t Hooft extrapolation has a num- ber of other things to recommend it. These include the fact that nearly all meson resonances seem to be of the quark- antiquark type, the OZI rule predicted holds to a good approximation and baryons emerge in an elegant way as solitons in the model. A fair statement would seem to be that each extrap- olation emphasizes different aspects of the true Nc = 3 QCD. In particular, the usual scheme is not really a re- placement for the true theory. That appears to be the meaning of the fact that the continuation to Nc > 3 is not unique. QUARKS IN TWO INDEX SYMMETRIC COLOR REPRESENTATION Clearly the assignment of quarks to the two index sym- metric representation of color SU(3) looks very similar. We may denote the quark fields as, AB = q BA , (11) in contrast to Eq.(5). There will be Nc(Nc + 1)/2 differ- ent color states for the two index symmetric quarks. This means that there is no value of Nc for which the sym- metric theory can be made to correspond to true QCD. For Nc = 3 there are 6 color states of the quarks and 8 color states of the gluon. If we choose Nc = 2, there are 3 color states of the quarks but unfortunately only three color states of the gluon. On the other hand, for large Nc it would seem reasonable to make approximations like, Asym(Nc) ≈ Aasym(Nc), (12) for the ππ scattering amplitude. As far as the large Nc counting goes, the mnemonics in Figs. 2-5 are still applicable to the case of quarks in the two index symmetric color representation. For not so large Nc, the scaling factor for the pion insertion would Nc(Nc + 1) , (13) and the pion decay constant would scale as F symπ (Nc) ∝ Nc(Nc + 1) . (14) With the identification AQCD = Aasym(3), the use of Eq.(12) enables us to estimate the large Nc scattering amplitude as, Asym(Nc) ≈ AQCD. (15) In applications to recently proposed minimal walking technicolor theories this formula is useful for making es- timates involving weak gauge bosons via the Goldstone boson equivalence theorem [69]. Finally we remark on the large Nc scaling rules for meson and glueball masses and decays in either the two index antisymmetric or two index symmetric schemes. Both meson and glueball masses scale as (Nc) 0. Further- more, all six reactions of the type a → b+ c, (16) where a,b and c can stand for either a meson or a glueball, scale as 1/Nc. This is illustrated in Fig.6 for the case of a meson decaying into two glueballs; note that the glueball insertion scales as 1/Nc and that two interaction vertices are involved. FIG. 6: Diagram for meson decay into two glueballs. SUMMARY We have investigated the dependences on the number of colors of the leading ππ scattering amplitudes using the single and the two index quark fields. We have seen that in the 2 index quark extension of QCD, exotic four quark resonances can appear at the leading order in addition to the usual two quark reso- nances. From the point of view of low energy ππ scatter- ing the 2 index scheme is more realistic than the ’t Hooft one given the existence of a four quark type sigma. This allows one to explain the long puzzling pion pion s wave scattering up to about 1 GeV. Of course, the usual ’t Hooft extrapolation has a num- ber of other important predictions to recommend it. A fair statement is that each largeNc extrapolation of QCD captures different aspects of the physical Nc = 3 case. We have also related the QCD scattering amplitude at large Nc with the one featuring two index symmetric quarks (Similar connections exist for adjoint fermions). The results are interesting for getting a better under- standing of the large Nc approach as well as for applica- tion to recently proposed technicolor models. Acknowledgments It is a pleasure to thank A. Abdel Rehim, D. Black, D.D. Dietrich, A. H. Fariborz, M.T. Frandsen, M. Harada, S. Moussa, S. Nasri and K. Tuominen for helpful discussions. The work of F.S. is supported by the Marie Curie Excellence Grant under contract MEXT-CT-2004- 013510 as well as the Danish Research Agency. The work of J.S. is supported in part by the U. S. DOE under Con- tract no. 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0704.0603

The height dependence of temperature - velocity correlation in the solar photosphere

Modern Solar Facilities – Advanced Solar Science, 139–142 F. Kneer, K. G. Puschmann, A. D. Wittmann (eds.) c© Universitätsverlag Göttingen 2007 The height dependence of temperature – velocity correlation in the solar photosphere J. Koza1,2,*, A. Kučera2, J. Rybák2, and H. Wöhl3 1Sterrekundig Instituut, Utrecht University, The Netherlands 2Astronomical Institute, Slovak Academy of Sciences, Tatranská Lomnica, Slovakia 3 Kiepenheuer-Institut für Sonnenphysik, Freiburg, Germany *Email: [email protected] Abstract. We derive correlation coefficients between temperature and line-of-sight velocity as a func- tion of optical depth throughout the solar photosphere for the non-magnetic photosphere and a small area of enhanced magnetic activity. The maximum anticorrelation of about −0.6 between temperature and line-of-sight velocity in the non-magnetic photosphere occurs at log τ5 = −0.4. The magnetic field is another decorrelating factor along with 5-min oscillations and seeing. 1 Introduction The correlative analysis proves to be an essential tool in disentangling of causal relations in the solar atmosphere. Recently, Rutten & Krijger (2003) and Rutten et al. (2004) quanti- fied the correlation of the reversed granulation observed in the low chromosphere and mid- photosphere with surface granulation in quest for the nature of internetwork background brightness patterns in these layers. In agreement with these studies Puschmann et al. (2003) demonstrated that filtering out of the p-modes is inevitable for studying the convective struc- tures in the solar photosphere because p-modes mostly reduce the correlation between var- ious line parameters. Odert et al. (2005) showed that correlation coefficients can fluctuate strongly in time with amplitudes of over 0.4 due to 5-min oscillations and the amplitudes are larger for higher formed lines. In case of weak lines the situation worsens even more, because correlations derived from them are influenced stronger by seeing. In this paper, we address the dissimilarity between non-magnetic and magnetic region seen in height variations of the correlation between temperature and line-of-sight velocity. We compare our results with a similar study by Rodrı́guez Hidalgo et al. (1999). Our analysis follows on the paper Koza et al. (2006, henceforth Paper I) and we invite the reader to have the paper at hand for further references. 2 Observational data and inversion procedure We use a time sequence of spectrograms obtained at the German Vacuum Tower Telescope at the Observatorio del Teide on April 28, 2000. The inversion code SIR (Ruiz Cobo & del Toro Iniesta 1992) was employed in the analysis of this observation. Thorough descriptions of the obser- vational data, inversion procedure, and spectral lines are given in Paper I. http://arxiv.org/abs/0704.0603v1 140 J. Koza et al.: Temperature – velocity correlation in the solar photosphere Figure 1. The height variation of the correlation between line-of-sight velocity and temperature for the results of Rodrı́guez Hidalgo et al. (1999) (solid) and for the non-magnetic (dashed) and magnetic (dotted) region defined in Paper I. 3 Results Figure 1 shows the height variations of the correlation between temperature and line-of-sight velocity for three different sets of data. The results of Rodrı́guez Hidalgo et al. (1999) in- dicate the significant anticorrelation between granules and intergranular lanes reaching the maximum of about −0.7 at log τ5 = −0.2. The subsequent weakening of this anticorrelation over the log τ5 ∈ 〈−0.2,−1.0〉 range is followed by a rise of correlation up to 0.28 at the middle photosphere at log τ5 = −1.75. No significant correlation exists in the upper photo- sphere. In the lower layers of the non-magnetic region (Paper I) the anticorrelation decreases to −0.63 at log τ5 = −0.4. However, in the middle photosphere temperature and line-of-sight velocity are almost uncorrelated with a local peak value of 0.08 at log τ5 = −1.7. Higher up at log τ5 = −2.9 the anticorrelation of about −0.42 is established again. In the sub-photospheric layers of the magnetic region the anticorrelation of −0.6 was found at log τ5 = 0.5. An approximately constant value of anticorrelation −0.2 was obtained in the low and middle photosphere. In the upper photosphere the anticorrelation reaches again −0.6. Figure 2 compares temperatures and line-of-sight velocities in the form of scatter corre- lation plots. Each plotted sample represents temperature and line-of-sight velocity specified along the x and y axes at a given pixel along the slit at a time within the interval of 15 min. From the top down, the row panels show correlations of the results of Rodrı́guez Hidalgo et al. (1999) and our results in the non-magnetic and magnetic region in three selected optical depths log τ5 = −0.3,−1.3, and −1.8. Plot saturation is avoided by showing sample density contours rather than individual points, except for the extreme outliers. The total distributions of temperatures and line-of-sight velocities are shown at the top and the left side of each panel, respectively. The first-moment curves are aligned at large correlation and become perpendicular in the absence of any correlation (Rutten & Krijger 2003). The first column in Fig. 2 shows good agreement of correlation coefficients and positions of maxima of ve- J. Koza et al.: Temperature – velocity correlation in the solar photosphere 141 Figure 2. Height-dependent scatter correlations of the line-of-sight velocity versus temperature. Top row: data from Rodrı́guez Hidalgo et al. (1999, see p. 315, Fig. 1). Middle and bottom row: our data for the non-magnetic and magnetic region (Paper I), respectively. The optical depths log τ5 and correlation coefficients cc are specified at each panel. Negative velocities indicate upflows. The rescaled total distributions of temperatures and line-of-sight velocities are shown as solid curves at the top and the left side of each panel, respectively. The dashed curves show the first moments of the sample density distributions over temperature and velocity bins. locity distributions in the non-magnetic region with the results of Rodrı́guez Hidalgo et al. (1999). However, the temperature distributions are dissimilar both in terms of asymme- try and also the positions of maxima. Our results indicate predominance of higher tem- peratures in the sample in contrast with lower temperatures prevailing in the results of Rodrı́guez Hidalgo et al. (1999). In the magnetic region, weak anticorrelation was found. The temperature distribution in this region is almost symmetric with maximum at higher temperatures than in the non-magnetic region. The second column of Fig. 2 corresponds to the layers where granulation is almost erased. While the temperature distributions in the non-magnetic region and in the results of Rodrı́guez Hidalgo et al. (1999) are symmetric, in the magnetic region the asymmetry indicates the abundant higher temperatures. The posi- tive correlation in the results of Rodrı́guez Hidalgo et al. (1999) shown in the third column suggests reversed granulation. However, this is not seen in our results. In the magnetic re- gion the asymmetries of temperature and velocity distributions indicate higher abundance of relatively hotter pixels with faster upflows. 142 J. Koza et al.: Temperature – velocity correlation in the solar photosphere 4 Discussion Figures 1 and 2 show dissimilarities both in height variations of correlation and distributions, although we and Rodrı́guez Hidalgo et al. (1999) used VTT observations and the SIR code. Because the maximum of anticorrelation found at sub-photospheric layers of the magnetic region is out of the range of sensitivity of the spectral lines (Paper I), we disregard this fea- ture. Very low anticorrelation found over log τ5 ∈ 〈0.0,−2.0〉 in the magnetic region (Fig. 1) suggests that magnetic field is another important decorrelating factor along with 5-min os- cillations and seeing (Puschmann et al. 2003; Odert et al. 2005). In our results, the middle layers of the non-magnetic and magnetic region lack signatures of reversed granulations (Fig. 1). The sinusoidal shape of the correlation coefficient in the non-magnetic region over the log τ5 ∈ 〈−1.2,−3.5〉 range can be explained as a sum of positive correlation typical for reversed granulation and negative anticorrelation characteristic for 5-min oscillations. 5 Summary Using a time sequence of high-resolution spectrograms and the SIR inversion code we have inferred height variation of correlation between the temperature and line-of-sight velocity throughout the quiet solar photosphere and a small magnetic region. The most important as- pect is comparison of the results with the akin study by Rodrı́guez Hidalgo et al. (1999). We found in agreement with Rodrı́guez Hidalgo et al. (1999) that the maximum anticorrelation −0.6 between the temperature and line-of-sight velocity in the non-magnetic region occurs at log τ5 = −0.4. The absence of signatures of reversed granulation in the middle layers of the non-magnetic region is likely to be due to 5-min oscillations, which negative anticorrelation prevails over weaker positive correlation typical for reversed granulation. Our results show that magnetic field is another decorrelating factor along with 5-min oscillations and seeing. Acknowledgements. The VTT is operated by the Kiepenheuer-Institut für Sonnenphysik, Freiburg, at the Observatorio del Teide of the Instituto de Astrofı́sica de Canarias. We are grateful to B. Ruiz Cobo (IAC) for kindly providing of the original data used in Figs. 1 and 2. This research is part of the European Solar Magnetism Network (EC/RTN contract HPRN-CT-2002-00313). This work was supported by the Slovak grant agency VEGA (2/6195/26) and by the Deutsche Forschungsgemein- schaft grant (DFG 436 SLK 113/7). J. Koza’s research is supported by a Marie Curie Intra-European Fellowships within the 6th European Community Framework Programme. References Koza, J., Kučera, A., Rybák, J., & Wöhl, H. 2006, A&A, 458, 941, (Paper I) Odert, P., Hanslmeier, A., Rybák, J., Kučera, A., & Wöhl, H. 2005, A&A, 444, 257 Puschmann, K., Vázquez, M., Bonet, J. A., Ruiz Cobo, B., & Hanslmeier, A. 2003, A&A, 408, 363 Rodrı́guez Hidalgo, I., Ruiz Cobo, B., Collados, M., & del Toro Iniesta, J. C. 1999, in ASP Conf. Ser. 173: Stellar Structure: Theory and Test of Connective Energy Transport, ed. A. Giménez, E. F. Guinan, & B. Montesinos, 313 Ruiz Cobo, B. & del Toro Iniesta, J. C. 1992, ApJ, 398, 375 Rutten, R. J., de Wijn, A. G., & Sütterlin, P. 2004, A&A, 416, 333 Rutten, R. J. & Krijger, J. M. 2003, A&A, 407, 735 Introduction Observational data and inversion procedure Results Discussion Summary

0704.0604

Magnetism in the high-Tc analogue Cs2AgF4 studied with muon-spin relaxation

Magnetism in the high-Tc analogue Cs2AgF4 studied with muon-spin relaxation T. Lancaster,1, ∗ S.J. Blundell,1 P.J. Baker,1 W. Hayes,1 S.R. Giblin,2 S.E. McLain,2 F.L. Pratt,2 Z. Salman,1, 2 E.A. Jacobs,3 J.F.C. Turner,3 and T. Barnes3 Clarendon Laboratory, Oxford University Department of Physics, Parks Road, Oxford, OX1 3PU, UK ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, UK Department of Chemistry and Neutron Sciences Consortium, University of Tennessee, Knoxville, Tennessee 37996, USA (Dated: November 4, 2018) We present the results of a muon-spin relaxation study of the high-Tc analogue material Cs2AgF4. We find unambiguous evidence for magnetic order, intrinsic to the material, below TC = 13.95(3) K. The ratio of inter- to intraplane coupling is estimated to be |J ′/J | = 1.9 × 10−2, while fits of the temperature dependence of the order parameter reveal a critical exponent β = 0.292(3), implying an intermediate character between pure two- and three- dimensional magnetism in the critical regime. Above TC we observe a signal characteristic of dipolar interactions due to linear F–µ +–F bonds, allowing the muon stopping sites in this compound to be characterized. PACS numbers: 74.25.Ha, 74.72.-h, 75.40.Cx, 76.75.+i Twenty years after its discovery, high-Tc superconduc- tivity remains one of the most pressing problems in con- densed matter physics. High-TC cuprates share a lay- ered structure of [CuO2] planes with strong antiferro- magnetic (AFM) interactions between S = 1/2 3d9 Cu2+ ions [1, 2]. However, analogous materials based upon 3d transition metal systems such as manganites [3] and nickelates [4] share neither the magnetic nor the super- conducting properties of the high-TC cuprates, leading to speculation that the spin- 1 character of Cu2+ is unique in this context. A natural extension to this line of in- quiry is to explore compounds based on the 4d analogue of Cu2+, namely S = 1 4d9 Ag2+ [5]; this motivated the synthesis of the layered fluoride Cs2AgF4 which contains silver in the unusual divalent oxidation state [6, 7]. This material possesses several structural similarities with the superconducting parent compound La2CuO4; it is com- prised of planes of [AgF2] instead of [CuO2] separated by FIG. 1: (Color online.) Structure of Cs2AgF4 showing a pos- sible magnetic structure. Candidate muon sites occur in both the [CsF] and [AgF2] planes. planes of [CsF] instead of [LaO] (Fig. 1). Magnetic measurements [7] suggest that in contrast to the antiferromagnetism of La2CuO4, Cs2AgF4 is well modelled as a two-dimensional (2D) Heisenberg ferro- magnet (described by the Hamiltonian H = J 〈ij〉 Si · Sj) with intralayer coupling J/kB = 44.0 K. The ob- servation of a magnetic transition below TC ≈ 15 K with no spontaneous magnetization in zero applied field (ZF) and a small saturation magnetization (∼ 40 mT), suggests the existence of a weak, AFM interlayer cou- pling. This behavior is reminiscent of the 2D ferromag- net K2CuF4 [8], where ferromagnetic (FM) exchange re- sults from orbital ordering driven by a Jahn-Teller distor- tion [9, 10]. On this basis, it has been suggested that in Cs2AgF4 a staggered ordering of dz2−x2 and dz2−y2 hole- containing orbitals on the Ag2+ ions gives rise to the FM superexchange [7]. An alternative scenario has also been advanced on the basis of density functional calcula- tions in which a d3z2−r2 − p− dx2−y2 orbital interaction through the Ag–F–Ag bridges causes spin polarization of the dx2−y2 band [11]. Although inelastic neutron scattering measurements have been carried out on this material [7], both Cs and Ag strongly absorb neutrons, resulting in limited resolu- tion and a poor signal-to-noise ratio. In contrast, spin- polarized muons, which are very sensitive probes of local magnetic fields, suffer no such impediments and, as we shall see, are ideally suited to investigations of the mag- netism in fluoride materials. In this paper we present the results of a ZF muon- spin relaxation (µ+SR) investigation of Cs2AgF4. We are able to confirm that the material is uniformly ordered throughout its bulk below TC and show that the critical behavior associated with the magnetic phase transition is intermediate in character between 2D and 3D. In ad- dition, strong coupling between the muon and F− ions allows us to characterise the muon stopping states in this http://arxiv.org/abs/0704.0604v1 compound. ZF µ+SR measurements were made on the MuSR in- strument at the ISIS facility, using an Oxford Instru- ments Variox 4He cryostat. In a µ+SR experiment spin- polarized positive muons are stopped in a target sample, where the muon usually occupies an interstitial position in the crystal. The observed property in the experiment is the time evolution of the muon spin polarization, the behavior of which depends on the local magnetic field B at the muon site [12]. Polycrystalline Cs2AgF4 was syn- thesised as previously reported [7]. Due to its chemical reactivity, the sample was mounted under an Ar atmo- sphere in a gold plated Ti sample holder with a cylindri- cal sample space of diameter 2.5 cm and depth 2 mm. A 25 µm thick window was screw-clamped onto a gold o-ring on the main body of the sample holder resulting in an airtight seal. Example ZF µ+SR spectra measured on Cs2AgF4 are shown in Fig. 2(a). Below TC (Fig. 2(c)) we observe oscillations in the time dependence of the muon polar- ization (the “asymmetry” A(t) [12]) which are character- istic of a quasi-static local magnetic field at the muon stopping site. This local field causes a coherent preces- sion of the spins of those muons for which a component of their spin polarization lies perpendicular to this local field (expected to be 2/3 of the total spin polarization for a powder sample). The frequency of the oscillations is given by νi = γµ|Bi|/2π, where γµ is the muon gyro- magnetic ratio (= 2π × 135.5 MHz T−1) and Bi is the average magnitude of the local magnetic field at the ith muon site. Any fluctuation in magnitude of these local fields will result in a relaxation of the oscillating signal [13], described by relaxation rates λi. Maximum entropy analysis (inset, Fig. 2(c)) reveals two separate frequencies in the spectra measured below TC, corresponding to two magnetically inequivalent muon stopping sites in the material. The precession frequen- cies, which are proportional to the internal magnetic field experienced by the muon, may be viewed as an effective order parameter for these systems [12]. In order to ex- tract the temperature dependence of the frequencies, the low temperature data were fitted to the function A(t) = Ai exp(−λit) cos(2πνit) (1) +A3 exp(−λ3t) +Abg, where A1 and A2 are the amplitudes of the precession signals and A3 accounts for the contribution from those muons with a spin component parallel to the local mag- netic field. The term Abg reflects the non-relaxing signal from those muons which stop in the sample holder or cryostat tail. The ratio of the two precession frequencies was found to be ν2/ν1 = 0.83 across the temperature range T < TC and this ratio was fixed in the fitting procedure. The FIG. 2: (Color online.) (a) Temperature evolution of ZF µ+SR spectra measured on Cs2AgF4 between 1.3 K and 59.7 K. (b) Above TC low frequency oscillations are observed due to the dipole-dipole coupling of F–µ+–F states. Inset: The energy level structure allows three transitions, leading to three observed frequencies. (c) Below TC higher frequency oscillations are observed due to quasistatic magnetic fields at the muon sites. Inset: Maximum entropy analysis reveal two magnetic frequencies corresponding to two magnetically inequivalent muon sites. amplitudes Ai were found to be constant across the tem- perature range and were fixed at values A1 = 1.66%, A2 = 3.74% and A3 = 5.54%. This shows that the prob- ability of a muon stopping in a site that gives rise to fre- quency ν1 is approximately half that of a muon stopping in a site that corresponds to ν2. We note also that A3 is in excess of the expected ratio of A3/(A1 +A2) = 1/2. The unambiguous assignment of amplitudes is made difficult by the resolution limitations that a pulsed muon source places on the measurement. The initial muon pulse at ISIS has FWHM τmp ∼ 80 ns, limiting the response for frequencies above ∼ τ−1mp [12]. We should expect, there- fore, slightly reduced amplitudes or increased relaxation (see below) for the oscillating components in our spectra for which ν1,2 & 5 MHz. The amplitudes of the oscilla- tions are large enough, however, for us to conclude that the magnetic order in this material is an intrinsic prop- erty of the bulk compound. Moreover, above TC there is a complete recovery of the total expected muon asymme- try. This observation, along with the constancy of A1,2,3 below TC, leads us to believe that Cs2AgF4 is completely ordered throughout its bulk below TC. Fig. 3(a) shows the evolution of the precession frequen- 11-T/TC 0 2 4 6 8 10 12 14 T (K) FIG. 3: Results of fitting data measured below TC to Eq. (1). (a) Evolution of the muon-spin precession frequencies ν1 (closed circles) and ν2 (open circles) with temperature. Solid lines are fits to the function νi(T ) = νi(0)(1 − T/TC) described in the text. Inset: Scaling plot of the precession frequencies with parameters TC = 13.95(3) K and β = 0.292. (b) Relaxation rates λ1 (closed circles), λ2 (open circles) and λ3 (closed triangles), as a function of temperature showing a rapid increase as TC is approached from below. cies νi, allowing us to investigate the critical behavior as- sociated with the phase transition. From fits of νi to the form νi(T ) = νi(0)(1−T/TC)β for T > 10 K, we estimate TC = 13.95(3) K and β = 0.292(3). In fact, good fits to νi(T ) = νi(0)(1−T/13.95)0.292 are achieved over the en- tire measured temperature range (that is, no spin wave related contribution was evident at low temperatures), yielding ν1(0) = 6.0(1) MHz and ν2(0) = 4.9(2) MHz corresponding to local magnetic fields at the two muon sites of B1 = 44(1) mT and B2 = 36(1) mT. A value of β = 0.292(3) is less than expected for three dimen- sional models (β = 0.367 for 3D Heisenberg), but larger than expected for 2D models (β = 0.23 for 2D XY or β = 0.125 for 2D Ising) [14, 15]. This suggests that in the critical regime the behavior is intermediate in char- acter between 2D and 3D; this contrasts with the mag- netic properties of K2CuF4 where β = 0.33, typical of a 3D system, is observed in the reduced temperature re- gion tr ≡ (TC − T )/TC > 7 × 10−2, with a crossover to more 2D-like behavior at tr < 7 × 10−2, where β = 0.22 [16, 17, 18]. Our measurements probe the behavior of Cs2AgF4 for tr ≥ 5.5 × 10−3, for which we do not ob- serve any crossover. A knowledge of TC and the intraplane coupling J , al- lows us to estimate the interplane coupling J ′. Recent studies of layered S = 1/2 Heisenberg ferromagnets us- ing the spin-rotation invariant Green’s function method [19], show that the interlayer coupling may be described by an empirical formula = exp with a = 2.414 and b = 2.506. Substituting our value of TC = 13.95 K and using |J |/kB = 44.0 K [7], we obtain |J ′|/kB = 0.266 K and |J ′/J | = 1.9× 10−2. The applica- tion of this procedure to K2CuF4 (for which TC = 6.25 K and |J |/kB = 20.0 K [8]) results in |J ′|/kB = 0.078 K and |J ′/J | = 3.9 × 10−3. This suggests that, although highly anisotropic, the interlayer coupling is stronger in Cs2AgF4 than in K2CuF4. This may account for the lack of dimensional crossover in Cs2AgF4 down to tr = 5.5× 10−3. Both transverse depolarization rates λ1 and λ2 are seen to decrease with increasing temperature (Fig. 3(b)) ex- cept close to TC where they rapidly increase. The large values of λ1,2 at low temperatures may reflect the re- duced frequency response of the signal due to the muon pulse width described above. The large upturn in the depolarization rate close to TC, which is also seen in the longitudinal relaxation rate λ3 (which is small and nearly constant except on approach to TC), may be at- tributed to the onset of critical fluctuations close to TC. The component in the spectra with the larger precession frequency ν1 has the smaller depolarization rate λ1 at all temperatures. These features provide further evidence for a magnetic phase transition at TC = 13.95 K. Above TC the character of the measured spectra changes considerably (Fig.2(a) and (b)) and we observe lower frequency oscillations characteristic of the dipole- dipole interaction of the muon and the 19F nucleus [20]. The Ag2+ electronic moments, which dominate the spec- tra for T < TC, are no longer ordered in the param- agnetic regime, and fluctuate very rapidly on the muon time scale. They are therefore motionally narrowed from the spectra, leaving the muon sensitive to the quasistatic nuclear magnetic moments. This interpretation is sup- ported by µ+SR measurements of K2CuF4 where simi- lar behavior was observed [21]. In many materials con- taining fluorine, the muon and two fluorine ions form a strong hydrogen bond usually separated by approxi- mately twice the F− ionic radius. The linear F–µ+–F spin system consists of four distinct energy levels with three allowed transitions between them (inset, Fig. 2(b)) giving rise to the distinctive three-frequency oscillations observed. The signal is described by a polarization function [20] D(ωdt) = uj cos(ωjt) , where u1 = 1, u2 = (1 + 1/ 3) and u3 = (1 − 1/ 3). The transition frequencies (shown in Fig. 2(b)) are given by ωj = 3ujωd/2 where ωd = µ0γµγF/4πr 3, γF is the nuclear gyromagnetic ratio and r is the µ+–19F separa- tion. This function accounts for the observed frequencies very well, leading us to conclude that the F–µ+–F bonds are highly linear. A successful fit of our data required the multiplication of D(ωdt) by an exponential function with a small re- laxation rate λ4, crudely modelling fluctuations close to TC. The addition of a further exponential component A5 exp(−λ5t) was also required in order to account for those muon sites not strongly dipole coupled to fluorine nuclei. The data were fitted with the resulting relaxation function A(t) = A4D(ωdt) exp(−λ4t) +A5 exp(−λ5t) +Abg, (3) The frequency ωd was found to be constant at all measured temperatures, taking the value ωd = 2π × 0.211(1) MHz, which corresponds to a constant F–µ+ separation of 1.19(1) Å, typical of linear bonds [20]. The relaxation rates only vary appreciably within 0.2 K of the magnetic transition, increasing as TC is approached from above, probably due to the onset of critical fluctua- tions. This provides further evidence for our assignment of TC = 13.95 K. Our determination of νi(0) and observation of the lin- ear F–µ+–F signal allow us to identify candidate muon sites in Cs2AgF4. Although the magnetic structure of the system is not known, magnetic measurements [7] suggest the existence of loosely coupled FM Ag2+ layers arranged antiferromagnetically along the c-direction. Dipole fields were calculated for such a candidate magnetic structure with Ag2+ moments in the ab planes oriented parallel (antiparallel) to the a direction for z = 0 (z = 1/2). The calculation was limited to a sphere containing ≈ 105 Ag ions with localized moments of 0.8 µB [7]. The above considerations suggest that the muon sites will be situ- ated midway between two F− ions. Two sets of candidate muon sites may be identified in the planes containing the fluorine ions. Magnetic fields corresponding to ν2(0) are found in the [CsF] planes (i.e. those with z = 0.145 and z = 0.355) at the positions (1/4, 1/4, z), (1/4, 3/4, z), (3/4, 1/4, z) and (3/4, 3/4, z). Sites corresponding to the frequency ν1(0) are more difficult to assign, but good candidates are found in the [AgF2] planes (at z = 0, 1/2) at positions (1/4, 1/2, z), (3/4, 1/2, z), (1/4, 0, z) and (3/4, 0, z). The candidate sites are shown in Fig. 1. We note that there are twice as many [CsF] planes in a unit cell than there are [AgF2] planes in agreement with our observation that components with frequency ν2 oc- cur with twice the amplitude of those with ν1. Such an assignment then implies that the presence of the muon distorts the surrounding F− ions such that their separa- tion is ∼ 2.38 Å. This contrasts with the in-plane F–F separation in the unperturbed material of 4.55 Å ([CsF] planes) and∼ 3.2 Å ([AgF2] planes) [7]. Thus the two ad- jacent F− ions in the magnetic [AgF2] planes each shift by ∼ 0.4 Å from their equilibrium positions towards the µ+, demonstrating that the muon introduces a non-negligible local distortion; however, the distortion in the Ag2+ ion positions is expected to be much less significant. In conclusion, we have shown unambiguous evi- dence for magnetic order in Cs2AgF4 with an exchange anisotropy of |J ′/J | ≈ 10−2 and critical behavior inter- mediate in character between 2D and 3D. The presence of coherent F–µ+–F states allows a determination of can- didate muon sites and an estimate of the perturbation of the system caused by the muon probe. This study demonstrates that µ+SR is an effective and useful probe of the Cs2AgF4 system. In order to further explore this system as an analogue to the high-TC materials it is desir- able to perform investigations of doped materials based on the Cs2AgF4 parent compound. Part of this work was carried out at the ISIS facility, Rutherford Appleton Laboratory, UK. This work is sup- ported by the EPSRC (UK). T.L. acknowledges support from the Royal Commission for the Exhibition of 1851. J.F.C.T and S.E.M acknowledge the U.S. National Sci- ence Foundation under awards CAREER-CHE 039010 and OISE 0404938, respectively. ∗ Electronic address: [email protected] [1] P.A. Lee, N. Nagaosa and X.-G. Wen, Rev. Mod. Phys. 78 17 (2006). [2] M.A. Kastner et al, Rev. Mod. Phys. 70 897 (1998). [3] E. Dagotta, T. Hotta and A Moreo, Phys. Rep. 344 1 (2001). [4] R.J. Cava et al., Phys. Rev. B 43 1229 (1991). [5] W. Grochala and R. Hoffmann, Angew. Chem. Int. Ed. 40 2742 (2001). [6] R.-H. Odenthal, D. Paus and R. Hoppe, Z. Anorg. Allg. Chem. 407 144 (1974). [7] S.E. McLain et al., Nature Mat. 5, 561 (2006). [8] I. Yamada, J. Phys. Soc. Jpn. 33, 979 (1972). [9] Y. Ito and J. Akimitsu, J. Phys. Soc. Jpn. 40, 1333 (1976). [10] D.I. Khomskii and K.I. Kugel, Solid State Commun. 13 763 (1973). [11] D. Dai et al., Chem. Mater. 18, 3281 (2006). [12] S.J. Blundell, Contemp. Phys. 40, 175 (1999). [13] R.S. Hayano et al., Phys. Rev. B 20, 850 (1979). [14] S.J. Blundell, Magnetism in Condensed Matter (Oxford University Press, 2001). [15] S.T. Bramwell and P.C.W. Holdsworth, J. Phys.: Con- dens. Matter 5 L53 (1993). [16] K. Hirakawa and H. Ikeda, J. Phys. Soc. Jpn. 35, 1328 (1973). [17] T. Hashimoto et al., J. Magn. Magn. Mater., 15-18, 1025 (1980). [18] M. Suzuki and H. Ikeda, J. Phys. Soc. Jpn. 50, 1133 (1981). [19] D. Schmalfuß, J. Richter and D. Ihle, Phys. Rev. B 72 224405 (2005). [20] J.H. Brewer et al., Phys. Rev. B 33, 7813 (1986). [21] C. Mazzoli et al., Physica B 326 427 (2003). mailto:[email protected]

0704.0605

Reconciling the X(3872) with the near-threshold enhancement in the D^0\bar{D}^{*0} final state

FZJ–IKP(TH)–2007–14 Reconciling the X(3872) with the near-threshold enhancement in the D0D̄∗0 final state. C. Hanhart Institut für Kernphysik, Forschungszentrum Jülich GmbH, D–52425 Jülich, Germany Yu. S. Kalashnikova, A. E. Kudryavtsev, and A. V. Nefediev Institute of Theoretical and Experimental Physics, 117218, B.Cheremushkinskaya 25, Moscow, Russia We investigate the enhancement in the D0D̄0π0 final state with the mass M = 3875.2 ± 0.7+0.3 −1.6 ± 0.8 MeV found recently by the Belle Collaboration in the B → KD0D̄0π0 decay and test the possibility that this is yet another manifestation of the well–established resonance X(3872). We perform a combined Flattè analysis of the data for the D0D̄0π0 mode, and for the π+π−J/ψ mode of the X(3872). Only if the X(3872) is a virtual state in the D0D̄∗0 channel, the data on the new enhancement comply with those on the X(3872). In our fits, the mass distribution in the D0D̄∗0 mode exhibits a peak at 2 ÷ 3 MeV above the D0D̄∗0 threshold, with a distinctive non-Breit–Wigner shape. PACS numbers: 14.40.Gx, 13.25.Gv, 12.39.Mk, 12.39.Pn I. INTRODUCTION The X(3872) state, discovered by Belle [1] in the B-meson decay, remains the most prominent member of the family of “homeless” charmonia, that is those mesons which definitely contain a c̄c pair but do not fit the standard charmonium assignment. The state was confirmed then by CDF [2], D0 [3], and BaBar [4]. The charmonium option for the X(3872) looks implausible as the state lies too high to be a 1D charmonium, and too low to be a 2P one [5]. This could, in principle, mean that we simply do not understand the spectra http://arxiv.org/abs/0704.0605v2 of higher charmonia. Indeed, most of the quark model predictions consider charmonia as cc̄ states in the quark potential model, with the potential parameters found from the description of lower charmonia, with uncertainties coming from proper treatment of relativistic effects. Another source of uncertainty is the role of open charm thresholds, the problem which is far from being resolved, though the attempts in this direction can be found in the literature — see, for example, Refs. [6, 7]. In any case, it looks premature to reject the cc̄ assignment for the X(3872) on basis of the mass only. However, the further development has revealed more surprises. The discovery mode of the X(3872) is π+π−J/ψ. The observation of the X(3872) in the γJ/ψ and π+π−π0J/ψ (ωJ/ψ) modes [8] implies that the X has positive C-parity, and the dipion in the π+π−J/ψ mode is C-odd, that is it originates from the ρ. Coexistence of the ρJ/ψ and ωJ/ψ modes points to a considerable isospin violation. Studies of the dipion mass spectrum in X(3872) → π+π−J/ψ decay establish that only the 1++ or 2−+ quantum number assignments are compatible with the data, while all other hypotheses are excluded by more than 3σ [9]. Both 1++ or 2−+ quantum numbers options for the X(3872) require drastic revisions of naive quark potential models, and no alternative explanation of the 2−+ state in this mass region was suggested. On the other hand, it was pointed out in Refs. [10, 11] that the DD̄∗ system with 1++ quantum numbers can be bound by pion exchange, forming a mesonic molecule (see also Ref. [12])1. As confirmed by actual calculations [13], large isospin mixing due to about 8 MeV difference between the D0D̄∗0 and D+D∗− thresholds can be generated in the molecular model in quite a natural way. This model was supplied, in Ref. [14], by quark–exchange kernels responsible for the transitions DD̄∗ → ρJ/ψ, ωJ/ψ, predicting the ωJ/ψ decay mode of the X(3872). Note, however, that one-pion-exchange as a binding mechanism in the DD̄∗ system should be taken with caution, as, in contrast to NN case, here the pion can be on-shell, as pointed in [15], where the ability to provide strong enough binding with one-pion exchange was questioned. For the most recent work on the implications of the nearby pion threshold see Refs. [16, 17]. For recent work for the X as quark state we refer to Ref. [18] and references therein. The molecular model has received additional support with the new data on the mass of 1An obvious shorthand notation is used here and in what follows: DD̄∗ ≡ 1√ (DD̄∗ + D̄D∗). the D0 meson [19] which yield a very weak binding, MX −M(D0D̄∗0) = −0.6± 0.6 MeV. (1) In the meantime, the Belle Collaboration has reported the first observation [20] of the near–threshold enhancement in the D0D̄0π0 mode in the decay B → KD0D̄0π0, with the branching fraction Br(B → KD0D̄0π0) = (1.22± 0.31+0.23 −0.30) · 10−4. (2) The peak mass of the enhancement is measured to be Mpeak = 3875.2± 0.7+0.3−1.6 ± 0.8 MeV. (3) Obviously it is tempting to relate this new state to the X(3872). However, the average value of the X(3872) mass is [21] MX = 3871.2± 0.5 MeV. (4) The central value (3) of the D0D̄0π0 peak mass enhancement is about 4 MeV higher than that, which obviously challenges attempts to relate this new state to the X(3872). Quite recently, the indication appeared that the Belle result [20] is likely to be con- firmed. Namely, the BaBar Collaboration has reported the preliminary data [22] on the B → KD0D̄∗0 decay, where the enhancement with the mass of M = 3875.6± 0.7+1.4 −1.5 ± 0.8 MeV, (5) was found, in a very good agreement with (3). BaBar observes the enhancement in the D0D̄0π0 and in the D0D̄0γ modes, which strongly supports the presence of the D0D̄∗0 intermediate state in the decay of the new X . If the new BaBar data persist, and the enhancement at 3875 MeV is indeed seen in two independent experiments, the possibility should be considered seriously of the presence of two charmonium-like states, X(3872) and X(3875), surprisingly close to each other and to the D0D̄∗0 threshold. However, there exists another, less exotic possibility. Namely, if the X(3872) is indeed strongly coupled to the D0D̄∗0 channel, and indeed has 1++ quantum numbers, one could expect the existence of a near–threshold peak in the D0D̄∗0 mass distribution. In the present paper we perform a phenomenological Flattè-like analysis of the data on the decay B → KD0D̄0π0 in the near–threshold region under the assumption of theX → D0D̄∗0 → D0D̄0π0 decay chain and 1++ quantum numbers for the X . The data on the π+π−J/ψ decay modes of the X(3872) are analyzed in the same framework, in order to investigate whether these data can accommodate the X(3875) state as a manifestation of the X(3872). II. FLATTÈ PARAMETRIZATION In this Section we introduce the Flattè-like parametrization of the near–threshold ob- servables. The relevant mass range is between the thresholds for the neutral and charged D-mesons. A natural generalization of the standard Flattè parametrization for the near– threshold resonance [23] of the D0D̄∗0 scattering amplitude reads F (E) = − 1 , (6) D(E) = E −Ef − − g2κ2 Γ (E) , E < 0 E −Ef − Γ (E) , 0 < E < δ E −Ef + i Γ (E) , E > δ δ =M(D+D∗−)−M(D0D̄∗0) = 7.6 MeV, 2µ1E, κ1 = −2µ1E, k2 = 2µ2(E − δ), κ2 = 2µ2(δ − E). Here µ1 and µ2 are the reduced masses in the D 0D̄∗0 and D+D∗− channels, respectively, and the energy E is defined relative to the D0D̄∗0 threshold. In what follows we assume isospin conservation for the coupling constants, g1 = g2 = g. The term iΓ/2 in Eq. (7) accounts for non-DD̄∗ modes. The X(3872) was observed in the π+π−J/ψ, π+π−π0J/ψ, and γJ/ψ modes, with Br(X → π+π−π0J/ψ) Br(X → π+π−J/ψ) = 1.0± 0.4± 0.3, (8) Br(X → γJ/ψ) Br(X → π+π−J/ψ) = 0.14± 0.05, (9) reported in Ref. [8]. Thus we assume that Γ (E) in Eq. (7) is saturated by the π+π−J/ψ and π+π−π0J/ψ modes and, in accordance with findings of Ref. [8], the dipion in the π+π−J/ψ mode comes from the ρ whereas the tripion in the π+π−π0J/ψ mode comes from the ω. The γJ/ψ channel is neglected due to its small branching fraction (9). The nominal thresholds for both ρJ/ψ and ωJ/ψ (3872 MeV and 3879 MeV, respectively) are close to the mass range under consideration, but both the ω meson and, especially, the ρ meson have finite widths, which are large in the scale under consideration. Thus Γ (E) is calculated as Γ (E) = Γπ+π−J/ψ(E) + Γπ+π−π0J/ψ(E), (10) Γπ+π−J/ψ(E) = fρ ∫ M−mJ/ψ q(m)Γρ (m−mρ)2 + Γ2ρ/4 , (11) Γπ+π−π0J/ψ(E) = fω ∫ M−mJ/ψ q(m)Γω (m−mω)2 + Γ2ω/4 , (12) with fρ and fω being effective couplings and q(m) = (M2 − (m+mJ/ψ)2)(M2 − (m−mJ/ψ)2) being the c.m. dipion/tripion momentum (M = E +M(D0D̄∗0)). Now we are in a position to write down the differential rates in the Flattè approximation. These are dBr(B → KD0D̄∗0) = B 1 |D(E)|2 , (14) dBr(B → Kπ+π−J/ψ) = B 1 Γπ+π−J/ψ(E) |D(E)|2 , (15) dBr(B → Kπ+π−π0J/ψ) Γπ+π−π0J/ψ(E) |D(E)|2 . (16) We assume the short–ranged dynamics of the weak B → K transition to be absorbed into the coefficient B. Obviously, the rate (14) is defined for E > 0 only, while the rates (15) and (16) are defined both above and below the D0D̄∗0 threshold. The formulae (14)–(16) are valid in the zero-width approximation for the D∗-mesons. In principle, one could include the finite width of the D∗-mesons either analogous to Eqs. (11) and (12) or in a more sophisticated way, as there are interference effects possible in the final state, as described in Ref. [24]. However, the widths of the D∗ mesons are small. Indeed, the total width of the D∗±-meson is measured to be 96 ± 22 keV [21]. There are no data on the D∗0 width, but one can estimate the D0π0 width of the D∗0 from the data [21] on charged D∗±, which gives Γ(D∗0 → D0π0) = 42 keV. The branching fractions of D∗0 are known [21]: Br(D∗0 → D0π0) = (61.9± 2.9)%, (17) Br(D∗0 → D0γ) = (38.1± 2.9)%, (18) so the total D∗0 width can be estimated to be only about 68 keV. The effect of such a small width was checked to be negligible in our studies, and we assume the D0D̄0π0 differential rate to be dBr(B → KD0D̄0π0) = 0.62B |D(E)|2 , (19) where the branching fraction (17) is taken into account. Analogously we have for the D0D̄0γ differential rate dBr(B → KD0D̄0γ) = 0.38B 1 |D(E)|2 . (20) Expressions (19) and (20) neglect final–state interactions; in particular, no DD̄ resonance within a few MeV above D0D̄0 threshold is assumed to exist, and π-rescattering is neglected. The latter is expected to be weak, as a consequence of chiral symmetry [17]. III. FLATTÈ ANALYSIS: PROCEDURE AND RESULTS Let us first specify the data used in our analysis. For the π+π−J/ψ mode we use the data from the B-meson decay. These are the ones reported by the Belle [1] and BaBar [25] Collaborations. The X(3872) is seen by Belle in the charged B-meson decay, with 35.7±6.8 signal events, and with the branching fraction [1] Br(B+ → K+X)Br(X → π+π−J/ψ) = (13.0± 2.9± 0.7) · 10−6. (21) The BaBar Collaboration [25] has observed the X(3872) both in the charged and neutral B-meson decays, with 61.2 ± 15.3 signal events for the charged mode, and only 8.3 ± 4.5 signal events for the neutral one. The branching fraction for the charged mode was found to be Br(B− → K−X)Br(X → π+π−J/ψ) = (10.1± 2.5± 1.0) · 10−6, (22) while the result for the neutral mode is much less certain: a 90% confidence interval was established as 1.34 · 10−6 < Br(B0 → K0X)Br(X → π+π−J/ψ) < 10.3 · 10−6. (23) Due to large errors and much smaller number of events, the X(3872) peak in the neutral mode looks much less convincing than the peak in the charged mode. A similar situation takes place for the D0D̄0π0 final state. The Belle data [20] include both B+ → K+D0D̄0π0 and B0 → K0D0D̄0π0 decays. There are 17.4± 5.2 signal events in the charged mode, with the branching fraction Br(B+ → K+D0D̄0π0) = (1.02± 0.31+0.21 −0.29) · 10−4, (24) and 6.5± 2.6 signal events in the neutral mode, with Br(B0 → K0D0D̄0π0) = (1.66± 0.70+0.32 −0.37) · 10−4. (25) Data on the B+ and B0 decays separately are presented in Ref. [26]. The D0D̄0π0 enhance- ment appears to be clearly seen in the data on charged B decays while, again, the neutral mode displays, within the errors, a much less pronounced peak. We conclude therefore that the data on charged and neutral B decays should be analyzed separately. The present analysis is performed for the charged mode only. Namely, with the Flattè formalism, we attempt to describe simultaneously the π+π−J/ψ mass spectrum from the charged mode and the D0D̄0π0 spectrum from the B+ mode, taken from Ref. [26]. The branching fractions (21) and (22) differ but, within the errors, are consistent with each other. In both sets of data, the fitted width of the signal is consistent with the resolu- tion, so only the upper limits on the X(3872) width were established: Γtot(Belle) < 2.3 MeV (26) Γtot(BaBar) < 4.1 MeV, (27) for the Belle and BaBar data, respectively. In view of this discrepancy we prefer to present two sets of fits, based on the two aforementioned sets of the π+π−J/ψ data. The π+π−J/ψ data are fitted in the interval −20 < E < 20 MeV (as before, E is the energy relative to the D0D̄∗0 threshold), after subtraction of the full background found in the corresponding analysis. The free parameters of the fit are the short–range factor B and the Flattè parameters Ef , g, and fρ. The parameter fω is constrained, in accordance with Eq. (8) through the condition RρJ/ψ RωJ/ψ = 1, (28) where RρJ/ψ = 20MeV −20MeV dBr(B → Kπ+π−J/ψ) dE, (29) RωJ/ψ = 20MeV −20MeV dBr(B → Kπ+π−π0J/ψ) dE. (30) The limits of integration in Eqs. (29) and (30) are somehow arbitrary but, as most of the support of the distributions (15) and (16) comes from within a few MeV around the D0D̄∗0 threshold, the uncertainty introduced by the limits of integration is much less than the experimental errors in Eq. (8). The D0D̄0π0 data are fitted in the energy region 0 < E < 20 MeV. Equation (19) describes the production of the D0D̄0π0 mode via the X-resonance, while the DD̄∗ pairs are known to be copiously produced in the B → K decay in a non-resonant way. Besides, the D0D̄0π0 final state could come from non-D0D̄∗0 modes like, for example, B → K∗D0D̄0. Therefore, we are to make assumptions on the background. The background in Refs. [20] and [26] is mostly combinatorial, and this part, given explic- itly in the publications, was subtracted prior to the analysis. For the rest of the background it is not possible to separate the contributions of the D0D̄∗0 and the D0D̄0π0 due to a limited phase space [20]. So we work under two extreme assumptions for the background. In Case A we consider the D0D̄0π0 background as unrelated to the D0D̄∗0 channel, while in Case B we assume that all the D0D̄0π0 events come from the D0D̄∗0 mode. The background was evaluated by fitting the Belle data off–peak (25 < E < 50 MeV). In Case A the background function is assumed to be proportional to the three–body D0D̄0π0 phase space R3 ∝ E2DDπ, where EDDπ = E +mD∗0 −mD0 −mπ0 . Then the total B → KD0D̄0π0 differential rate is dBrA(B → KD0D̄0π0) = 0.62 |D(E)|2 + cAE DDπ, (31) with cA as fitting constant. In Case B the background function is proportional to the two–body D0D̄∗0 phase space R2 ∝ k1 (see the definition below Eq. (7)). Then the signal– TABLE I: The set of the Flattè parameters for the best fits to the Belle data Ref. [1] and [20]. Fit g fρ fω Ef , MeV B · 104 φ ABelle 0.3 0.0070 0.036 -11.0 11.0 — BBelle 0.3 0.0086 0.046 -10.9 8.9 -144 background interference is to be taken into account: dBrB(B → KD0D̄0π0) = 0.62 + cB cosφ + cB sin φ with the relative phase φ and cB being fitting constants. The differential rates are translated into number-of-events distributions as follows. There are about 36 signal events in the Belle data, which corresponds to the branching fraction of about 1.3 · 10−5 (see Eq. (21)). Thus the number-of-events per 5 MeV distribution for the π+π−J/ψ mode is given by ππJ/ψ Belle (E) = 5 [MeV] 1.3 · 10−5 dBr(B → Kπ+π−J/ψ) . (33) For the BaBar data, with 61 events and the branching fraction of about 1.02 · 10−5 (see Eq. (22)), we have ππJ/ψ BaBar (E) = 5 [MeV] 1.02 · 10−5 dBr(B → Kπ+π−J/ψ) . (34) As to the D0D̄0π0 mode, the Belle Collaboration states to have 17.4 signal events in the charged mode [20], which corresponds to the branching fraction (24) of about 1.02 · 10−4. The number-of-events distributions per 4.25 MeV for the D0D̄0π0 mode is calculated as 0D̄0π0 A,B (E) = 4.25[MeV] 1.02 · 10−4 dBrA,B(B → KD0D̄0π0) . (35) The best fit to the π+π−J/ψ data alone requires a vanishing value of the DD̄∗ coupling constant, g = 0, so that such solution cannot accommodate the D0D̄0π0 enhancement as a related phenomenon. To describe both π+π−J/ψ and D0D̄∗0 modes we are to compromise on the π+π−J/ψ line–shape. It appears that a decent combined fit can be achieved only if the π+π−J/ψ distribution is peaked exactly at the D0D̄∗0 threshold, with the peak width (defined as the width at the TABLE II: The set of the Flattè parameters for the best fits to the BaBar data of Ref. [25] and the Belle data of Ref. [20]. Fit g fρ fω Ef , MeV B · 104 φ ABaBar 0.3 0.0042 0.021 -8.8 11.4 — BBaBar 0.3 0.0056 0.027 -8.8 8.9 -153 peak half–height) close to the upper limits given by Eq. (26) or (27). The values of the coupling g were found to be of the order of magnitude or larger than 0.3. Finally, the fits exhibit the scaling behaviour: they remain stable under the transformation g → λg, Ef → λEf , fρ → λfρ, fω → λfω, B → λB, (36) with tiny variations of the phase φ in the Case B. In Tables I, II we present the sets of the best fitting parameters — for both Case A and Case B and for g = 0.3 — for the Belle (Table I) and BaBar (Table II) data on the π+π−J/ψ mode and for the Belle data for the D0D̄0π0 mode. To assess the quality of the fits we calculate the π+π−J/ψ distributions integrated over the 5 MeV bins, as in Refs. [1] and [25], and the D0D̄0π0 distributions integrated over the 4.25 MeV bins, as in Refs. [20] and [26]. The results are shown at Fig. 1 together with the experimental data. The above–mentioned scaling behaviour does not allow one to perform a proper fit with the estimate of uncertainties in the parameters found. Indeed, the parameters of the best fits found for the values of coupling constant g larger than 0.3 differ only by a few % from the ones given by the scaling transformation (36), and the corresponding distributions are very similar to those given at Fig. 1. As seen from the figures, acceptable fits require the D0D̄∗0 differential rate to be peaked at around 2 ÷ 3 MeV above the D0D̄∗0 threshold. The scattering length in the D0D̄∗0 channel which follows from the expression (6) of the D0D̄∗0 scattering amplitude, is given by the expression a = − 2µ2δ + 2Ef/g + iΓ (0)/g 2µ2δ + 2Ef/g)2 + Γ (0)2/g2 , (37) and is calculated to be (−3.98− i0.46) fm, Case ABelle (−3.95− i0.55) fm, Case BBelle, -20 -10 0 10 20 E [MeV] -20 -10 0 10 20 E [MeV] -20 -10 0 10 20 E [MeV] -20 -10 0 10 20 E [MeV] 0 5 10 15 20 E [MeV] 0 5 10 15 20 E [MeV] 0 5 10 15 20 E [MeV] 0 5 10 15 20 E [MeV] ABelle BBelle ABaBar BBaBar FIG. 1: Upper plots: Our fits to the differential rates for the π+π−J/ψ channel measured by Belle [1] and BaBar [25] using prescription A and B (see Eqs. (31) and (32)). Lower plots: Cor- responding fits for the differential rates in the D0D̄0π0 channel measured by Belle [26]. The distributions integrated over the bins are shown in each panel as filled dots, experimental data as filled squares with error bars. (−3.10− i0.16) fm, Case ABaBar (−3.10− i0.22) fm, Case BBaBar. The real part of the scattering length for all the fits appears to be large and negative, and the imaginary part is much smaller. This, together with the beautiful cusp in the π+π−J/ψ mass distribution, signals the presence of a virtual state in the D0D̄∗0 channel. The cusp scenario for the π+π−J/ψ excitation curve in the X(3872) mass range was advocated in Ref. [27]. The X(3872) as a virtual DD̄∗ state was found in the coupled–channel microscopic quark model [7]. A large scattering length explains naturally the scaling behaviour of the Flattè param- eters. Such kind of scaling was described in Ref. [28] in the context of light scalar mesons properties: the scaling behaviour occurs if the scattering length approximation is operative. In the case of X the situation is more complicated, as there are two near–threshold channels, neutral and charged. Nevertheless, if it is possible to neglect the energy E in the expression (7) for the Flattè denominator D(E) then, as seen from the expression (6), scaling for the D0D̄∗0 scattering amplitude indeed takes place. If the factor B obeys the scaling transfor- mation, the differential rates (14)–(16) also exhibit the scaling behaviour. Note that, if the energy dependence of the charged D+D∗− and non-DD̄∗ channel contributions is neglected as well, this corresponds to the scattering length approximation, and neglect of the effective radius term. IV. DISCUSSION Our analysis shows that the large branching fraction (2) implies the X to be a virtual D0D̄∗0 state, and not a bound state. We illustrate this point by calculating the rates (14) and (15) for the set of the Flattè parameters (fit C) g = 0.3, Ef = −25.9 MeV, fρ = 0.007, fω = 0.036, B = 1.32 · 10−4. (40) The values of the coupling constants coincide with those of the fit ABelle, while the parameter Ef is chosen to yield the real part of the scattering length to be equal in magnitude to the one evaluated for the given fit ABelle, but positive: ã = (+3.98− i0.46) fm. The parameter B for this set yields the same value of the total branching fraction for the π+π−J/ψ mode as the fit ABelle. The π +π−J/ψ and D0D̄0π0 rates are shown in Fig. 2, together with the rates obtained for the case ABelle (without background). The new curve (dashed line in Fig. 2) displays a very narrow peak in the π+π−J/ψ distribution, corresponding to theD0D̄∗0 bound state, with binding energy of about 1 MeV (there is no corresponding peak in the D0D̄0π0 distribution as the finite width of the D∗0 is not taken into account in our analysis). Note that the π+π−J/ψ rates (Fig. 2) are normalized to give the branching ratio 1.3 · 10−5, which requires the coefficient B to be much larger for the virtual state than for the bound state. As a result, the D0D̄∗0 rate is much smaller for the bound state, as seen from Fig. 2. Obviously, the difference between the bound–state and virtual-state cases for the ratio Br(X → D0D̄0π0) Br(X → π+π−J/ψ) -3 -2 -1 0 1 2 3 E [MeV] 0 5 10 15 20 E [MeV] FIG. 2: The differential rates for the π+π−J/ψ (first plot) and D0D̄∗0 (second plot) for the fits ABelle (solid curves) and C (dashed curves). is driven by the strength of the bound–state peak, as discussed in Ref. [29], where the scattering length approximation was used to describe the X(3872). Following Ref. [29], let us write down the scattering length in the D0D̄∗0 channel as γre + iγim . (42) Then, in the scattering length approximation, the π+π−J/ψ differential rate is proportional to the factor γ2re + (k1 + γim) , E > 0, (γre − κ1)2 + γ2im , E < 0, while the D0D̄∗0 rate is proportional to γ2re + (k1 + γim) . (44) The line–shape for the D0D̄∗0 channel does not depend on the sign of γre. The same is true for the π+π−J/ψ line–shape above the D0D̄∗0 threshold while, below the threshold, the line–shapes differ drastically: in the bound–state case there is a narrow peak below threshold, and in the virtual-state case a threshold cusp appears. For γre > 0 and γim → 0 the expression (43) becomes a δ-function (see Ref. [29]): γreδ(E + γ re/(2µ1)). (45) Then the total rate does not depend on γim, if it is small enough. This simply means that, for γim = 0, we have a real bound state, which is not coupled to inelastic channels. In contrast to the bound–state case, for the virtual state, the rate (43) tends to zero with γim → 0, while the D0D̄∗0 rate does not vanish in such a limit. So it is possible, adjusting γim, to obtain large values of the ratio (41). Exactly the same situation is encountered in our fit: we need g & 0.3 for the fit to be reasonable and, in this scaling regime, as soon as we have a positive real part of the scattering length, the ratio (41) becomes small while, with a negative real part, we get a solution compatible with the data. The large branching fraction (2) was identified in Ref. [30] as a disaster for the molecular model of the X(3872). Indeed, the bound–state molecule decay into D0D̄0π0 is driven by the process D∗0 → D0π0 which gives the width of order 2Γ(D∗0 → D0π0) (up to the interference effects calculated in Ref. [24] which, for the bound–state case, cannot be neglected anymore and should be taken into account). The main decay mode of the X is π+π−J/ψ because the phase space available is large. This is confirmed by model calculations of Ref. [14] yielding Br(X → D0D̄0π0) Br(X → π+π−J/ψ) ≈ 0.08, (46) in a strong contradiction with data. The estimate (46) describes the decay of an isolated bound state. However, the suppres- sion is more moderate as, in B-decay, the continuum contribution is also to be considered. The bound–state contribution would be zero in the zero–width approximation for D∗0, while the D0D̄∗0 continuum contribution remains finite if the D∗0 width is neglected. However, if the X is a bound state, the continuum contribution is not large (see Fig. 2), Br(X → D0D̄0π0) Br(X → π+π−J/ψ) ≈ 0.62. (47) Such a small rate would remain unnoticed against the background. So, in practice, the bound–state X(3872) would reveal itself only as a narrow peak below threshold, with a very small rate (see Eq. (46)). In contrast to this we get for the virtual state Br(X → D0D̄0π0) Br(X → π+π−J/ψ) ≈ 9.9. (48) In our analysis, the X appears to be a virtual state in the D0D̄∗0 channel. This does not contradict the assumption g1 = g2 = g employed in the analysis. The latter means that the underlying strong interaction conserves isospin, and all the isospin violation comes from the mass difference between charged and neutral DD̄∗ thresholds. No charged partners of the X are observed, so it is reasonable to assume that the strong attractive interaction takes place in the isosinglet DD̄∗ channel. We do not specify the nature of this attractive force. It is known that in the one-pion- exchange model for the X , the force is attractive in the isosinglet channel, and is repulsive in the isotriplet one. However, as was already mentioned, the doubts were cast in [15] on the role of one-pion-exchange in the DD̄∗ binding, and it was advocated there that the X may fit the 23P1 charmonium assignment if the coupling to DD̄ ∗ channel is taken into account. In such a scenario the strong binding force obviously takes place in the isosinglet channel. We note, however, that, with the Flattè parameters found, one can make a definite statement: whatever the nature of the X(3872) is, the admixture of a compact cc̄ state in its wavefunction is small. Both large scattering length and the scaling behaviour of the DD̄∗ amplitude are consequences of the large value of the coupling constant of the state to the DD̄∗ channel. As shown in Ref. [31], this points to a large DD̄∗ component and a dynamical origin of the X . Although formulated for quasi–bound states in Ref. [31] the argument can also be generalized to virtual states. To clarify the connection between effective coupling and the nature of the state observe that the two–point function g(s) for the resonance can be written as g(s) = s−M2 − iΣ̄(s) , (49) whereM is the physical mass of the resonance and Σ̄(s) = Σ(s)−ReΣ(M2) is the self–energy responsible for the dressing through the mesonic channels. In the near–threshold region the momenta involved are much smaller than the inverse of the range of forces. As a result one may neglect the s–dependence of the real part of Σ̄ and replace its imaginary part by the leading terms g(s) ≃ 1 s−M2 + iM , (50) where the sum is over near–threshold channels, and the contributions of distant thresholds are absorbed into the renormalised mass M . Nonrelativistic reduction of Eq. (50) immedi- ately yields the Flattè formula (6). Thus the Flattè parameter Ef acquires clear physical meaning: the quantity M(D0D̄∗0) +Ef is the physical mass of the resonance, renormalised by the coupling to the decay channels. Now, if the couplings gi are small, the distribution for the resonance takes a standard Breit–Wigner form, and the scattering length is small. Correspondingly, the state is mostly cc̄, with a small admixture of the DD̄∗ component. If the couplings are large, the terms proportional to giki control the denominator in Eq. (50), the Breit–Wigner shape is severely distorted, the scattering length approximation is operative, and the mesonic component dominates the near–threshold wavefunction. Formulated differently: if the couplings are large, the properties of the resonance are given mainly by the continuum contribution — which is equivalent to saying it is mostly of molecular (dynamical) nature. It should be stressed that this kind of reasoning can only be used, if the resonance mass is very close to a threshold, for then the contribution of the continuum state is dominated by the unitarity cut piece which is unique and model independent. This argument is put into more quantitative terms in Ref. [31]. It is also important to note that our analysis does not allow for any conclusion on the mechanism that leads to the molecular structure. On the level of the phenomenological parametrisations used here a molecule formation due to t–channel exchanges and due to short–ranged s–channel forces (cc̄–DD∗ mixing) would necessarily lead to the same properties of the state, once the parameters are adjusted to the data. V. SUMMARY In this paper we present a Flattè analysis of the Belle data [20] on the near–threshold enhancement in the D0D̄0π0 mode. We constrain the Flattè parametrization with the data on the X(3872) seen in the π+π−J/ψ and π+π−π0J/ψ modes. With such constraints the new state can be understood as a manifestation of the well–established X(3872) resonance. We showed that the structure at 3875 MeV can only be related to the X(3872), if we assume the X to be of a dynamical origin, however, not as a bound state but as a virtual state. The situation is then similar to that of nucleon–nucleon scattering in the spin– singlet channel near threshold: in contrast to the spin–triplet channel, where there exists the deuteron as a bound state, the huge scattering length in the spin–singlet channel — about 20 fm — comes from a near–threshold virtual state. The attractive interaction is just not strong enough to form a bound state in this channel as well. The line–shape in the D0D̄∗0 mode appears to differ substantially from the one extracted previously from the Belle data directly. It peaks much closer to the D0D̄∗0 threshold, though the overall description of the data looks quite reasonable within the experimental errors. It is the π+π−J/ψ line–shape which, in our solutions, differ drastically from the one described by a simple Breit–Wigner form. We found a threshold cusp, with a width close to the limits imposed by the data analysis. While the data currently available allows for such a line–shape, a considerable improvement in the experimental resolution could confirm or rule out this possibility. In the meantime, we urge to perform an analysis of the data on the D0D̄0π0 final state with Flattè formulae given in Eqs. (14)–(16). Equally important is the Flattè analysis of the D0D̄0γ data [22]: if the structure in the D0D̄0π0 is indeed due to D0D̄∗0 and is indeed related to the X(3872) as a virtual state, one should observe an enhancement in D0D̄0γ similar to the one seen in the D0D̄0π0. The phase space available in this final state is larger than that in D0D̄0π0, so it is easier to separate the contributions of D0D̄∗0 and D0D̄0γ to the peak. The D0D̄0γ enhancement would be described with the Flattè formula (20) and, up to background and possible FSI effects, the ratio of branching fractions would be Br(X → D0D̄0π0) Br(X → D0D̄0γ) ≈ 1.6. (51) The most interesting situation would happen if, due to an improved resolution in the π+π−J/ψ mode, the combined Flattè analysis of the π+π−J/ψ, D0D̄0π0, and D0D̄0γ data fails to deliver a self-consistent result. Such a situation would point to the new X(3875) state being completely unrelated to the X(3872). Acknowledgments We would like to thank A. Dolgolenko for useful comments and suggestions and P. Pakhlov for illuminating discussions on various aspects of the Belle experiment. This re- search was supported by the Federal Agency for Atomic Energy of Russian Federation, by the grants RFFI-05-02-04012-NNIOa, DFG-436 RUS 113/820/0-1(R), NSh-843.2006.2, and NSh-5603.2006.2, and by the Federal Programme of the Russian Ministry of Industry, Science, and Technology No. 40.052.1.1.1112. 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B 579, 316 (2004). http://arxiv.org/abs/hep-ex/0505037 http://arxiv.org/abs/hep-ph/0308277 http://arxiv.org/abs/hep-ph/0402237 http://arxiv.org/abs/hep-ph/0702128 http://arxiv.org/abs/hep-ph/0703168 http://moriond.in2p3.fr/QCD/2007/SundayAfternoon/Grenier.pdf [25] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 73, 011101 (2006). [26] G. Majumder, ICHEP2006 talk, http://belle.kek.jp/belle/talks/ICHEP2006/Majumber.ppt. [27] D. V. Bugg, Phys. Lett. B 598, 8 (2004). [28] V. Baru, J. Haidenbauer, C. Hanhart, A. Kudryavtsev, and Ulf-G. Meißner, Eur. Phys. J. A 23, 523 (2005). [29] E. Braaten and M. Kusunoki, Phys. Rev. D 71, 074005 (2005). [30] E. S. Swanson, Phys. Rep. 429, 243 (2006). [31] V. Baru et al, Phys. Lett. B 586, 53 (2004). http://belle.kek.jp/belle/talks/ICHEP2006/Majumber.ppt Introduction Flattè parametrization Flattè analysis: procedure and results Discussion Summary Acknowledgments References

0704.0607

Plasmon Amplification through Stimulated Emission at Terahertz Frequencies in Graphene

Plasmon Amplification through Stimulated Emission at Terahertz Frequencies in Graphene Farhan Rana1, Faisal R. Ahmad2 1School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 2Department of Physics, Cornell University, Ithaca, NY 14853 Abstract We show that plasmons in two-dimensional graphene can have net gain at terahertz frequencies. The coupling of the plasmons to interband electron-hole transitions in population inverted graphene layers can lead to plasmon amplification through the process of stimulated emission. We calculate plasmon gain for different electron-hole densities and temperatures and show that the gain values can exceed 104 cm−1 in the 1-10 terahertz frequency range, for electron-hole densities in the 109-1011 cm−2 range, even when plasmon energy loss due to intraband scattering is considered. Plasmons are found to exhibit net gain for intraband scattering times shorter than 100 fs. Such high gain values could allow extremely compact terahertz amplifiers and oscillators that have dimensions in the 1-10 µm range. http://arxiv.org/abs/0704.0607v2 conduction valence plasmon plasmon plasmons Figure 1: (LEFT) Energy bands of graphene showing stimulated absorption of plasmons. (RIGHT) Population inverted graphene bands showing stimulated emission of plasmons. 1 Introduction Tremendous interest has been generated recently in the electronic properties of two dimensional (2D) graphene in both experimental and theoretical arenas [1, 2, 3, 5, 6, 7]. Graphene is a single atomic layer of carbon atoms forming a dense honeycomb crystal lattice [8]. The massless energy dispersion relation of electrons and holes with zero (or close to zero) bandgap results in novel behavior of both single-particle and collective excitations [1, 2, 3]. In addition, the high mobility of electrons in graphene has generated interest in developing novel high speed devices. Recently, it has been shown that the frequencies of plasma waves in graphene at moderate carrier densities ( 109−1011 cm−2) are in the terahertz range [3]. Electron-hole decay through plasmon emission has been recently experimentally observed in graphene [4]. The zero bandgap of graphene leads to strong damping of the plasma waves (plasmons) at finite temperatures as plasmons can decay by exciting interband electron-hole pairs [1, 2]. In this paper we show that plasmon amplification through stimulated emission is possible in population inverted graphene layers. This process is depicted in Fig.1. We show that plasmons in graphene can have a net gain at frequencies in the 1-10 THz range even if plasmon losses from electron and hole intraband scattering are considered. A net gain for the plasmons implies that terahertz amplifiers and oscillators based on plasmon amplification through stimulated emission are possible. The gain at terahertz frequencies is possible due to the (almost) zero bandgap of graphene. Although terahertz gain is also achievable in population inverted subbands in 2D quantum wells [9], intrasubband plasmons in quantum wells, being longitudinal collective modes, do not couple with intersubband transitions that require field polarization perpendicular to the plane of the quantum wells. The electromagnetic energy in the two-dimensional plasmon mode is confined within very small distances of the graphene layer and therefore waveguiding structures with large dimensions, such as those required in terahertz quantum cascade lasers [9], are not required for realizing plasmon based terahertz devices. We also present results for plasmon gain under different population inversion conditions taking into account both intraband and interband electronic transitions and carrier scattering. 2 Theoretical Model In this section we discuss the theoretical model used to obtain the values for the plasmon gain in graphene. In graphene, the valence and conduction bands resulting from the mixing of the pz-orbitals are degenerate at the inequivalent K and K ′ points of the Brillouin zone [8]. Near these points, the conduction and valence band dispersion relations can be written compactly as [2], Es,k = sh̄v|k| (1) where s = ±1 stand for conduction (+1) and valence (−1) bands, respectively, and v is the “light” velocity of the massless electrons and holes. The wavevector k is measured from the K(K ′) point. The frequencies ω(q) of the longitudinal plasmon modes of wavevector q are given by the equation,ǫ(q, ω) = 0, where ǫ(q, ω) is the longitudinal dielectric function of graphene [2]. In the random phase approximation (RPA) ǫ(q, ω) can be written as [10], ǫ(q, ω) = 1− V (q)Π(q, ω) (2) Here, V (q) is the bare 2D Coulomb interaction and equals e2/2ǫ∞q. ǫ∞ is the average of the dielectric constant of the media on either side of the graphene layer. Π(q, ω) is the electron-hole propagator including both intraband and interband processes and is given by the expression [1, 2], Π(q, ω) = 4 s s′ k | < ψs′,k+q|e iq.r|ψs,k > | f(Es,k − Efs)− f(Es′,k+q − Efs′) h̄ω + Es,k − Es′,k+q + iη The factor of 4 outside in the above equation comes from the degenerate two spins and the two valleys at K and K ′. f(E − Ef ) is the Fermi distribution function with Fermi energy Ef . |ψs,k) > are the Bloch functions for the conduction and valence bands near the K(K ′) point. The occupancy of electrons in the conduction and valence bands are described by different Fermi levels to allow for nonequilibrium population inversion. The Bloch functions have the following matrix elements [8], | < ψs′,k+q|e iq.r|ψs,k > | 1 + ss′ |k|+ |q| cos (θ) |k+ q| where θ is the angle between the vectors k and q. The condition v|q| < ω(q) must be satisfied in order to avoid direct intraband absorption of plasmons. Assuming v|q| < ω, and using the symmetry between conduction and valence bands, the intraband and interband contributions to the propagator can be approximated as follows, Πintra(q, ω) ≈ q2K T /πh̄2 ω(ω + i/τ)− v2q2/2 eEf+/KT + 1 e−Ef−/KT + 1 Πinter(q, ω) ≈ [f(h̄ω/2− Ef+)− f(−h̄ω/2− Ef−)] ω2 − ω2 [f(h̄ω/2− Ef+)− f(−h̄ω/2− Ef−)] (6) Here, q = |q|. In Equation (5), the intraband contribution to the propagator is written in the plasmon-pole approximation that satisfies the f-sum rule [10]. This approximation is not valid for large value of the wavevector q when ω(q) → vq. However, in this paper we will be concerned with small values of the wavevector for which the plasmons have net gain, and therefore the approximation used in Equation (5) is adequate. Plasmon energy loss due to intraband scattering has been included with a scattering time τ in the number-conserving relaxation-time approximation which assumes that as a result of scattering the carrier distribution relaxes to the local equilibrium distribution [11]. The real part of the interband contribution to the propagator modifies the effective dielectric constant and leads to a significant reduction in the plasmon frequency under population inversion conditions. The imaginary part of the interband contribution to the propagator incorporates plasmon loss or gain due to stimulated interband transitions. A necessary condition for plasmon gain from stimulated interband transitions is that the splitting of the Fermi levels of the conduction and valence electrons exceed the plasmon energy, i.e. Ef+−Ef− > h̄ω. But the plasmons will gave net gain only if the plasmon gain from stimulated interband transitions exceed the plasmon loss due to intraband scattering. The real and imaginary parts of the propagator in Equations (5) and (6) satisfy the Kramers-Kronig relations. Equations (5) and (6) can be used with Equation (2) to calculate the real and imaginary parts of the plasmon frequency ω(q) as a function of q. However, from the point of view of device design, it is more useful to assume that the frequency ω is real and the propagation vector q(ω), written as a function of ω, is complex. Since the charge density wave corresponding to plasmons has the form eiq.r−iωt, the imaginary part of the propagation vector corresponds to net gain or loss. We define the net plasmon energy gain g(ω) as −2Imag{q(ω)}. 3 Results and Discussion In simulations we use v = 108 cm/s and ǫ∞ = 4.0ǫo (assuming silicon-dioxide on both sides of the graphene layer) [1]. We assume a nonequilibrium situation, as in a semiconductor interband laser [12], in which the electron and hole densities are equal and Ef+ = −Ef−. Such a non-equilibrium situation can be realized experimentally by either carrier injection in an electrostatically defined graphene pn-junction or through optical pumping [13, 14]. The value of the scattering time τ (momentum relaxation time) is also critical for calculations of the net plasmon gain. Value of τ can be estimated from the experimentally reported values of mobility using the following expression for the graphene conductivity (assuming that only electrons are present) [17], e2 τ K T eEf+/KT + 1 Values of mobility between 20,000 and 60,000 cm2/V-s have been experimentally measured at low temperatures (T¡77K) in graphene [6, 7, 15]. Assuming a mobility value of 27,000 cm2/V-s , reported in Ref. [15] for an electron density of 3.4×1012 cm−2 at T=58K, the value of τ comes out to be approximately 0.6 ps. The phonon scattering time was experimentally determined to be close to 4 ps at T=300K [15]. Therefore, impurity or defect scattering is expected to be the 0 0.5 1 1.5 2 2.5 3 Wavevector (105 cm−1) T = 10K n=p=1, 3.5, 6, 8.5 × 109 cm−2 increasing density ω = v q Figure 2: Calculated plasmon dispersion relation in graphene at 10K is plotted for different electron-hole densities (n = p = 1, 3.5, 6, 8.5×109 cm−2). The condition ω(q) > h̄vq is satisfied for frequencies that have net gain in the terahertz range. The assumed values of v and τ are 108 cm/s and 0.5 ps, respectively. dominant momentum relaxation mechanism in graphene, and the scattering time is expected to be relatively independent of temperature [17]. In the results presented below, unless stated otherwise, we have used a temperature independent scattering time of 0.5 ps. Figs. 2-7 show the calculated dispersion relation of the plasmons and the net plasmon gain at T=10K, 77K, and 300K for different electron-hole densities. At very low frequencies the losses from intraband scattering dominate. At frequencies ranging from 1 to 15 THz, the plasmons can have net gain. The values of the net gain are found to be significantly large reaching 1−4×104 cm−1 for electron-hole densities in the 109 cm−2 range at low temperatures and 1011 cm−2 range at room temperature. The calculated plasmon dispersions indicate that ω(q) > vq at all frequencies for which the plasmons have net gain. Therefore, direct intraband absorption of plasmons is not possible at these frequencies and will not reduce the calculated gain values. Plasmons acquire net gain for smaller electron-hole densities at lower temperatures. At higher temperatures the distribution of electrons and holes in energy is broader and the 0 1 2 3 4 5 6 Frequency (THz) T = 10K n=p=1, 3.5, 6, 8.5 × 109 cm−2 increasing density Figure 3: Net plasmon gain in graphene at 10K is plotted for different electron-hole densities (n = p = 1, 3.5, 6, 8.5 × 109 cm−2). The assumed values of v and τ are 108 cm/s and 0.5 ps, respectively. gain at any particular frequency is therefore smaller. At T=10K, the plasmons have net gain for electron-hole densities as small as 2 × 109 cm−2. Almost an order of magnitude larger electron-hole densities are required to achieve the same net gain values at T=77K compared to T=10K. The linear energy dependence of the density of states associated with the massless dispersion relation of electrons and holes in graphene results in the maximum plasmon gain values to increase with the electron-hole density. The peak gain values shift to higher frequencies with the increase in the electron-hole density for the same reason. The fact that plasmons can acquire net gain for relatively small carrier densities suggests that plasmon gain is relatively robust with respect to intraband scattering losses. Fig. 8 shows the net gain at T=10K for n = p = 1010 cm−2 and values of the intraband scattering time τ varying from 0.1 to 0.5 ps. The net gain decreases as the plasmon losses increase with a decrease in the value of τ and the maximum gain value equals zero for τ = 0.15 ps. However, it should not be concluded from Fig. 8 that plasmons cannot have net gain for τ less than 0.15 ps since electron-hole density can always be increased to achieve net gain for smaller values of 0 2 4 6 8 Wavevector (105 cm−1) increasing density ω = v q T = 77K n=p=1, 2, 3, 4 × 1010 cm−2 Figure 4: Calculated plasmon dispersion relation in graphene at 77K is plotted for different electron-hole densities (n = p = 1, 2, 3, 4 × 1010 cm−2). The condition ω(q) > h̄vq is satisfied for frequencies that have net gain in the terahertz range. The assumed values of v and τ are 108 cm/s and 0.5 ps, respectively. 0 2 4 6 8 10 Frequency (THz) T = 77K n=p=1, 2, 3, 4 × 1010 cm−2 increasing density Figure 5: Net plasmon gain in graphene at 77K is plotted for different electron-hole densities (n = p = 1, 2, 3, 4 × 1010 cm−2). The assumed values of v and τ are 108 cm/s and 0.5 ps, respectively. 0 5 10 15 20 Wavevector (105 cm−1) T = 300K n=p=1, 1.5, 2, 2.5 × 1011 cm−2 increasing density ω = v q Figure 6: Calculated plasmon dispersion relation in graphene at 300K is plotted for different electron-hole densities (n = p = 1, 1.5, 2, 2.5×1011 cm−2). The condition ω(q) > h̄vq is satisfied for frequencies that have net gain in the terahertz range. The assumed values of v and τ are 108 cm/s and 0.5 ps, respectively. 0 5 10 15 20 Frequency (THz) T = 300K n=p=1, 1.5, 2, 2.5 × 1011 cm−2 increasing density Figure 7: Net plasmon gain in graphene at 300K is plotted for different electron-hole densities (n = p = 1, 1.5, 2, 2.5 × 1011 cm−2). The assumed values of v and τ are 108 cm/s and 0.5 ps, respectively. 0 2 4 6 8 Frequency (THz) T = 10K n = p = 1010 cm−2 τ = 0.5, 0.4, 0.3, 0.2, 0.15, 0.1 ps increasing τ Figure 8: Net plasmon gain in graphene at 10K is plotted for different intraband scattering times τ (τ = 0.5, 0.4, 0.3, 0.2, 0.15, 0.1 ps). The assumed value of v is 108 cm/s and the electron-hole density is 1010 cm−2. 0 2 4 6 8 10 12 Frequency (THz) increasing τ T = 10K n = p = 3 × 1010 cm−2 τ = 150, 125, 100, 75 fs Figure 9: Net plasmon gain in graphene at 10K is plotted for different scattering times τ (τ = 150, 125, 100, 75 fs). The assumed value of v is 108 cm/s and the electron-hole density is 3× 1010 cm−2. τ . Fig. 9 shows the net gain at T=10K for n = p = 3× 1010 cm−2 and values of the intraband scattering time τ varying from 75 to 150 fs. It can be seen that at these larger carrier densities plasmons have net gain for scattering times that are sub-100 fs. The exceedingly large values of the net plasmon gain (> 104 cm−1) in graphene implies that terahertz plasmon oscillators only a few microns long in length could have sufficient gain to overcome both intrinsic losses and losses associated with external radiation coupling. Plasmon fields with in-plane wavevector magnitude q decay as e−q |z| away from the graphene layer where |z| is the distance from the graphene layer. Figs. 2, 4, and 6 show that q has values exceeding 105 cm−1 at terahertz frequencies. Therefore, the electromagnetic energy associated with the terahertz plasmons is confined within 100 nm of the graphene layer. Strong field confinement and low plasmon losses at terahertz frequencies are both partly responsible for the high net gain values in graphene. Recent theoretical predictions for electron-hole recombination rates in graphene due to Auger scattering indicate that electron-hole recombination times can be much longer than 1 ps at temperatures ranging from 10K to 300K for electron-hole densities smaller than 1012 cm−2 [16]. This suggests that population inversion can be experimentally achieved in graphene via current injection in electrostatically defined pn-junctions or via optical pumping [13, 14]. It also needs to be pointed out here that graphene monolayers and multilayers produced from currently available experimental techniques are estimated to have defect/impurity densities anywhere between 1011 and 1012 cm−2 [17]. Therefore, at low electron-hole densities (less than 1011 cm−2) graphene is expected to exhibit localized electron and hole puddles rather than continuous electron and/or hole sheet charge densities [17]. This implies that with the currently available techniques graphene based terahertz plasmon oscillators might only be realizable with higher electron-hole densities (> 1011 cm−2) for operation at higher frequencies (> 5 THz). 4 conclusion In conclusion, we have shown that high gain values for plasmons are possible in population inverted graphene layers in the 1-10 THz frequency range. The plasmon gain remains positive even for carrier intraband scattering times shorter than 100 fs. The high gain values and the strong plasmon field confinement near the graphene layer could enable compact terahertz amplifiers and oscillators. The authors would like to thank Edwin Kan and Sandip Tiwari for helpful discussions. References [1] X. F. Wang, T. Chakraborty, Phys. Rev. B, 75, 033408 (2007). [2] E. H. Hwang, S. D. Darma, cond-mat/0610561. [3] V. Ryzhii, A. Satou, J. Appl. Phys., 101, 024509 (2007). [4] A. Bostwick, T. Ohta, T. Seyller, Karsten Horn AND E. Rotenberg, Nature, 3, 36, (2007). [5] K. S. Novoselov et. al., Nature, 438, 197 (2005). [6] K. S. Novoselov et. al., Science, 306, 666 (2004). [7] Y. Zhang et. al., Nature, 438, 201 (2005). [8] R. Saito, G. Dresselhaus, M. S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, UK (1999). [9] B. Williams, H. Callebaut, S. Kumar, Q. Hu, Appl. Phys. Letts., 82, 1015 (2003). [10] H. Huag, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Sientific, NJ (1994). [11] N. D. Mermin, Phys. Rev. B, 1, 2362 (1969). [12] L. A. Coldren, S. W. Corzine, Diode Lasers and Photonics Integrated Circuits, Wiley, NY (1995). [13] J. R. Williams, L. DiCarlo, C. M. Marcus, cond-mat/0704.3487 (2007). [14] B. zyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov, Philip Kim, cond-mat/0705.3044. http://arxiv.org/abs/cond-mat/0610561 [15] W. De Heer et. al., Science, 312, 1191 (2006). [16] F. Rana, cond-mat/0705.1204. [17] E. H. Hwang, S. Adam, S. Das Sarma, Phys. Rev. Letts., 98, 186806 (2007). Introduction Theoretical Model Results and Discussion conclusion

0704.0608

On the homology of two-dimensional elimination

On the Homology of Two-Dimensional Elimination Jooyoun Hong Department de Mathematics University of California, Riverside 900 Big Springs Drive Riverside, CA 92521 e-mail: [email protected] Aron Simis Departamento de Matemática Universidade Federal de Pernambuco 50740-540 Recife, PE, Brazil e-mail: [email protected] Wolmer V. Vasconcelos Department of Mathematics Rutgers University 110 Frelinghuysen Rd Piscataway, New Jersey 08854-8019 e-mail: [email protected] September 9, 2021 Abstract We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method of their calculation in terms of certain Hilbert coefficients. In dimension two the structure of the irreducible ideals leads naturally to the calculation of Sylvester determinants via a computer-assisted method. For degree at most 5 we produce the full set of defining equations of the base ideal. The results answer affirmatively some questions raised by D. Cox ([9]). AMS 2000 Mathematics Subject Classification. Primary 13H10; Secondary 13F20, 05C90. Partially supported by CNPq, Brazil. Partially supported by the NSF. http://arxiv.org/abs/0704.0608v1 1 Introduction Let R be a Noetherian ring and f = {f1, . . . , fm} a set of elements of R. Such sets are the ingredients of rational maps between affine and other spaces. At the cost of losing some definition, we choose to examine them in the setting of the ideal I they generate. Specifically, we consider the presentation of the Rees algebra of I 0 → M −→ S = R[T1, . . . , Tm] ϕ−→ R[It] → 0, Ti 7→ fit. The context of Rees algebra theory allows for the examination of the syzygies of the fi but also of the relations of all orders, which are carriers of analytic information. We set R = R[It] for the Rees algebra of I. The ideal M will be referred to as the equations of the fj, or by abuse of terminology, of the ideal I. If M is generated by forms of degree 1, I is said to be of linear type (this is independent of the set of generators). The Rees algebra R[It] is then the symmetric algebra S = Sym(I) of I. Such is the case when the fi form a regular sequence, M is then generated by the Koszul forms fiTj − fjTi, i < j. We will treat mainly almost complete intersections in a Cohen-Macaulay ring R, that is, ideals of codimension r generated by r + 1 elements. Almost exclusively, I will be an ideal of finite co-length in a local ring, or in a ring of polynomials over a field. Our focus on R is shaped by the following fact. The class of ideals I to be considered will have the property that both its symmetric algebra S and the normalization R′ of R have amenable properties, for instance, one of them (when not both) is Cohen-Macaulay. In such case, the diagram S ։ R ⊂ R′ gives a convenient dual platform from which to examine R. There are specific motivations for looking at (and for) these equations. In order to describe our results in some detail, let us indicate their contexts. (i) Ideals which are almost complete intersections occur in some of the more notable birational maps and in geometric modelling ([3], [4], [5], [6], [7], [8], [9], [10], [17], [18], [21]). (ii) It is possible interpret questions of birationality of certain maps as an interaction between the Rees algebra of the ideal and its special fiber. The mediation is carried by the first Chern coefficient of the associated graded ring of I. In the case of almost complete intersections the analysis is more tractable, including the construction of suitable algorithms. (iii) At a recent talk in Luminy ([9]), D. Cox raised several questions about the character of the equations of Rees algebras in polynomial rings in two variables. They are addressed in Section 4 as part of a general program of devising algorithms that produce all the equations of an ideal, or at least some distinguished polynomial (e.g. the ‘elimination equation’ in it) ([3], [13]). We now describe our results. Section 2 is an assemblage for the ideals treated here of basics on symmetric and Rees algebras, and on their Cohen-Macaulayness. We also introduce the general notion of a Sylvester form in terms of contents and coefficients in a polynomial ring over a base ring. This is concretely taken up in Section 4 when the base ring is a polynomial ring in 2 variables over a field. In Section 3 we examine the connection between typical algebraic invariants and the geometric background of rational maps and their images. Here, besides the dimension and the degree of the related algebras, we also consider the Chern number e1(I) of an ideal. In particular we explain a criterion for a rational map to be birational in terms of an equality of two such Chern numbers, provided the base locus of the map is empty and defined by an almost complete intersection ideal. In Section 4, we discuss the role of irreducible ideals in producing Sylvester forms. Of a general nature, we describe a method to obtain an irreducible decomposition of ideals of finite co-length. In rings such as k[s, t], due to a theorem of Serre, irreducible ideals are complete intersections, a fact that leads to Sylvester forms of low degree. Turning to the equations of almost complete intersections, we derive several Sylvester forms over a polynomial ring R = k[s, t], package them into ideals and examine the incident homological properties of these ideals and the associated algebras. It is a computer-assisted approach whose role is to produce a set of syzygies that afford hand computation: the required equations themselves are not generated by computation. Concretely, we model a generic class of ideals cases to define ‘super-generic’ ideals L in rings with several new variables L = (f, g, h1, . . . , hm) ⊂ A. Using Macaulay2 ([11]), we obtain the free resolutions of L. In degrees ≤ 5, the resolution has length ≤ 3 (2 when degree = 4) 0 → F3 d3−→ F2 d2−→ F1 −→ F0 −→ L → 0. It has the property that after specialization the ideals of maximal minors of d3 and d2 have codimension 5 and ≥ 4, respectively. Standard arguments of the theory of free resolutions will suffice to show that the specialization of L is a prime ideal. For ideals in R = k[s, t] generated by forms of degrees ≤ 5, the method succeeds in describing the full set of equations. In higher degree, in cases of special interest, it predicts the precise form of the elimination equation. For a technical reason–due to the character of irreducible ideals–the method is limited to dimension two. Nevertheless, it is supple enough to apply to non-homogeneous ideals. This may be exploited elsewhere, along with the treatment of ideals with larger numbers of generators in a two-dimensional ring. 2 Preliminaries on symmetric and Rees algebras We will introduce some basic material of Rees algebras ([2], [12], [22]). Since most of the questions we will consider have a local character, we pick local rings as our setting. Whenever required, the transition to graded rings will be direct. Throughout we will consider a Noetherian local ring (R,m) and I an m-primary ideal (or a graded algebra over a field k, R = n≥0Rn = R0[R1], R0 = k, and I a homogeneous ideal of finite colength λ(R/I) < ∞). We assume that I admits a minimal reduction J generated by n = dimR elements. This is always possible when k is infinite. The terminology means that for some integer r, Ir+1 = JIr. This condition in turn means that the inclusion of Rees algebras R[Jt] ⊂ R[It] is an integral birational extension (birational in the sense that the two algebras have the same total ring of fractions). The smallest such integer, rJ(I), is called the reduction number of I relative to J ; the infimum of these numbers over all minimal reductions of I is the (absolute) reduction number r(I) of I. For any ideal, not necessarily m-primary, the special fiber of R[It] – or of I by abuse of terminology – is the algebra F(I) = R[It]⊗R (R/m). The dimension of F(I) is called the analytic spread of I, and denoted ℓ(I). When I is m-primary, ℓ(I) = dimR. A minimal reduction J is generated by ℓ(I) elements, and F(J) is a Noether normalization of F(I). Hilbert polynomials The Hilbert polynomial of I by (m ≫ 0) is the function ([2]): λ(R/Im) = e0(I) m+ n− 1 − e1(I) m+ n− 2 + lower terms. e0(I) is the multiplicity of the ideal I. If R is Cohen-Macaulay, e0(I) = λ(R/J), where J is a minimal reduction of I (generated by a regular sequence). For such rings, e1(I) ≥ 0. For instance, if R = k[x1, . . . , xn], m = (x1, . . . , xn) and I = m λ(R/Im) = λ(R/mmd) = md+ n− 1 m+ n− 1 − e1(I) m+ n− 2 + lower terms where e1(I) = (dn − dn−1). Both coefficients will be the focus of our interest soon. Cohen-Macaulay Rees algebras There is broad array of criteria expressing the Cohen-Macaulayness of Rees algebra (see [1], [14], [19], [23, Chapter 3]). Our needs will be filled by single criterion whose proof is fairly straightforward. We briefly review its related contents. Let (R,m) be a Cohen-Macaulay local ring of dimension ≥ 1, and let I be an m-primary ideal with a minimal reduction J . The Rees algebra R[Jt] is Cohen-Macaulay and serves as an anchor to derive many properties of R[It]. Here is one that we shall make use of. Define the Sally module SJ(I) of I relative to J to be the cokernel of the natural inclusion of finite R[Jt]-modules I R[Jt] ⊂ I R[It]. Thus, SJ(I) = It/IJ t−1. It has a Hilbert function, unlike the algebra R[It], that gives information about the Hilbert function of I (see [22, Chapter 2]). The module on the left, I ·R[Jt], is a Cohen-Macaulay R[Jt]-module of depth dimR + 1. The Cohen-Macaulayness of I · R[It] is directly related to that of R[It]. These considerations lead to the criterion: Theorem 2.1 If dimR ≥ 2 and the reduction number of I is ≤ 1, that is I2 = JI, then R[It] is Cohen-Macaulay. The converse holds if dimR = 2. Symmetric algebras Throughout R is a Cohen-Macaulay ring and I is an almost complete intersection. The symmetric algebra Sym(I) will be denoted by S. Hopefully there will be no confusion between S and the rings of polynomials S = R[T1, . . . , Tn] that we use to give a presentation of either R or S. What keeps symmetric algebras of almost complete intersections fairly under control is the following: Proposition 2.2 Let (R,m) be a Cohen-Macaulay local ring. If I is an almost complete intersection and depth R/I ≥ dimR/I − 1, then S is Cohen-Macaulay. In particular, if I is m-primary then S is Cohen-Macaulay. Proof. The general assertion follows from [12, Proposition 10.3]; see also [16]. ✷ Let R be a Noetherian ring and let I be an R-ideal with a free presentation ϕ−→ Rn −→ I → 0. We assume that I has a regular element. If S = R[T1, . . . , Tn], the symmetric algebra S of I is defined by the ideal M1 ⊂ S of 1-forms, M1 = I1([T1, . . . , Tn] · ϕ). The ideal of definition of the Rees algebra R of I is the ideal M ⊂ S obtained by elimination (M1 : x t) = M1 : x where x is a regular element of I. Sylvester forms To get additional elements of M , evading the above calculation, we make use of general Sylvester forms. Recall how these are obtained. Let f = {f1, . . . , fn} be a set of polynomials in B = R[x1, . . . , xr] and let a = {a1, . . . , an} ⊂ R. If fi ∈ (a)B for all i, we can write f = [f1 · · · fn] = [a1 · · · an] ·A = a ·A, where A is a n×n matrix with entries in B. By an abuse of terminology, we refer to det(A) as a Sylvester form of f relative to a, in notation det(f)(a) = det(A). It is not difficult to show that det(f)(a) is well-defined mod (f). The classical Sylvester forms are defined relative to sets of monomials (see [9]). We will make use of them in Section 4. The structure of the matrix A may give rise to finer constructions (lower order Pfaffians, for example) in exceptional cases (see [20]). In our approach, the fi are elements of M1, or were obtained in a previous calculation, and the ideal (a) is derived from the matrix of syzygies ϕ. 3 Algebraic invariants in rational parametrizations Let f1, . . . , fn+1 ∈ R = k[x1, . . . , xn] be forms of the same degree. They define a rational Ψ : Pn−1 99K Pn p → (f1(p) : f2(p) : · · · : fn+1(p)). Rational maps are defined more generally with any number m of forms of the same degree, but in this work we only deal with the case where m = n+ 1. There are two basic ingredients to the algebraic side of rational map theory: the ideal theoretic and the algebra aspects, both relevant for the nature of Ψ. First the ideal I = (f1, . . . , fn+1) ⊂ R, which in this context is called the base ideal of the rational map. Then there is the k-subalgebra k[f1, . . . , fn+1] ⊂ R, which is homogeneous, hence a standard k- algebra up to degree renormalization. As such it gives the homogeneous coordinate ring of the (closed) image of Ψ. Finding the irreducible defining equation of the image is known as elimination or implicitization. We refer to [21] and [18] (also [20] for an even earlier overview) for the interplay between the ideal and the algebra, as well as its geometric consequences. In particular, the Rees algebra R = R[It] plays a fundamental role in the theory. A pleasant side of it is that, since I is generated by forms of the same degree, one has R⊗R k ≃ k[f1t, . . . , fn+1t] ⊂ R, which retro-explains the (closed) image of Pn−1 by Ψ as the image of the projection to Pn of the graph of Ψ. In particular, the fiber cone is reduced and irreducible. 3.1 Elimination degrees and birationality Although a rational map Pn−1 99K Pn has a unique set of defining forms f1, . . . , fn+1 of the same degree and unit gcd, two such maps may look “nearly” the same if they happen to be composite with a birational map of the target Pn - a so-called Cremona transformation. If this is the case the two maps have the same degree, in particular the final elimination degrees are the same. However, it may still be the case that the two maps are composite with a rational map of the target which is not birational, so that their degrees as maps do not coincide, yet the degrees of the respective images are the same. In such an event, one would like to pick among all such maps one with smallest possible degree. This leads us to he notion of improper and proper rational parametrizations. Definition 3.1 Let Ψ = (f1 : · · · : fn+1) : Pn−1 99K Pn be a rational map, where gcd(f1, . . . , fn+1) = 1. We will say that Ψ (or the parametrization defined by f1, . . . , fn+1) is improper if there exists a rational map Ψ′ = (f ′1 : · · · : f ′n+1) : Pn−1 99K Pn, with gcd(f ′1, . . . , f n+1) = 1, such that: 1. There is an inclusion of k-algebras k[f1, . . . , fn+1] ⊂ k[f ′1, . . . , f ′n+1]; 2. There is an isomorphism of k-algebras k[f1, . . . , fn+1] ≃ k[f ′1, . . . , f ′n+1]; 3. degΨ′ < degΨ. We note that if Ψ is improper and Ψ′ is as above then the rational map (P1 : · · · : Pn+1) : Pn 99K Pn is not birational, where fj = Pj(f 1, . . . , f n+1), for 1 ≤ j ≤ n + 1. Of course, the transition forms Pj = Pj(y1, . . . , yn+1) are not uniquely defined. Example 3.2 The parametrization given by f1 = x 1, f2 = x 2, f3 = x 2 is improper since it factors through the parametrization f ′1 = x 2 = x1x2, f 3 = x 2 through either one of the rational maps (y1 : y2 : y3) 7→ (y21 : y22 : y23) or (y1 : y2 : y3) 7→ (y21 : y1t3 : y23) neither of which is birational. Moreover, the forms x21, x1x2, x 2 define a birational map onto its image. We say that a rational map Ψ = (f1 : · · · : fn+1) : Pn−1 99K Pn is proper if it is not improper. The need for considering proper rational maps will become apparent in the context. It is also a basic assumption in elimination theory when one is looking for the elimination degrees (see [9]). Clearly, if Ψ is birational onto its image then it is proper. The converse does not hold and one seeks for precise conditions under which Ψ is birational onto its image. This is the object of the following parts of this subsection. When the ideal I = (f1, . . . , fn+1) has finite co-length – that is, I is (x1, . . . , xn)-primary – it is natural to consider another mapping, namely, the corresponding embedding of the Rees algebra R = R[It] into its integral closure R̃. We will explore the attached Hilbert functions into the determinations of various degrees, including the elimination degree of the mapping. Thus, assume that I has finite co-length. Then we may assume (k is infinite) that f1, . . . , fn is a regular sequence, hence the multiplicity of J = (f1, . . . , fn) is d n, the same as the multiplicity of md. This implies that J is a minimal reduction of I and of md. We will set up a comparison between R and R′ = R[md], where m = (x1, . . . , xn), through two relevant exact sequences: 0 → R −→ R′ −→ D → 0, (1) and its reduction mod m R̄ −→ R̄′ −→ D̄ → 0. (2) F = R̄ is the special fiber ofR (or, of I), and since I is generated by forms of the same degree, one has F ≃ k[f1, . . . , fn+1] as graded k-algebras. By the same token, F ′ = R̄′ ≃ k[md] – the d-th Veronese subring of R. In particular, since dimF = dimF ′, the leftmost map in the exact sequence (2) is injective. AlsoD is annihilated by a power of m, hence dimD = dim D̄. These are the degrees (multiplicities) deg(F) and deg(F ′) of the special fibers. Since F ′ is an integral extension of F , one has deg(F ′) = deg(F)[F ′ : F ], (3) where [F ′ : F ] = dimK(F ′ ⊗F K), where K denotes the fraction field of F (see, e.g., [21, Proposition 6.1 (b) and Theorem 6.6] for more general formulas). Since F ′ is besides integrally closed, the latter is also the field extension degree [ k(md) : K ]. Note that [F ′ : F ] = 1 means that the extension F ⊂ F ′ is birational (equivalently, the rational map Ψ maps Pn−1 birationally onto its image). As above, set L = md. We next characterize birationality in terms of both the coefficient e1 and the dimension of the R-module D. Proposition 3.3 The following conditions are equivalent: (i) [F ′ : F ] = 1, that is Ψ is birational onto its image; (ii) deg(F) = dn−1; (iii) dim D̄ ≤ n− 1; (iv) dimD ≤ n− 1 (v) e1(L) = e1(I). Proof. (i) ⇐⇒ (ii) This is clear from (3) since deg(F ′) = dn−1. (i) ⇐⇒ (iii) Since ℓ(I) = n and F ⊂ F ′ is integral, then F ⊂ F ′ is a birational extension if and only if its conductor F :F F ′ is nonzero, equivalently, if and only if dim D̄ ≤ n− 1. (iv) ⇐⇒ (iii) Clearly, dimD ≤ n and in the case of equality its multiplicity is e1(L) − e1(I) > 0. Therefore, the equivalence of the two statements follows suit. ✷ There is some advantage in examining D̄ since F is a hypersurface ring, F = k[T1, . . . , Tn+1]/(f) = R[T1, . . . , Tn+1]/(x1, . . . , xn, f) a complete intersection. Since F ′ is also Cohen-Macaulay, with a well-known presentation, it affords an understanding of D̄, and sometimes, of D. 3.2 Calculation of e1(I) of the base ideal of a rational map One objective here is to apply some general formulas for the Chern number e1(I) of an ideal I to the case of the base ideal of a rational map with source P1 = Proj(k[x1, x2]). Here is a method put together from scattered facts in the literature of Rees algebras (see [23, Chapter 2]). Proposition 3.4 Let (R,m) be a Cohen-Macaulay local ring of dimension d, let I be an m-primary ideal with a minimal reduction J = (a1, . . . , ad). Set R ′ = R/(a1, . . . , ad−1), I ′ = IR′. Then (i) e0(I) = e0(I ′) = λ(R/J), e1(I) = e1(I (ii) r(I ′) < degR′ ≤ e0(I); in particular, for n ≥ r = r(I ′), one has I ′n+1 = adI ′n (iii) λ(R′/I ′ r+1) = λ(R′/I ′ r) + λ(I ′ r/adI ′ r) = e0(I)(r + 1)− e1(I) (iv) e1(I) = −λ(R′/I ′ r) + e0(I)r It would be desirable to develop a direct method suitable for the ideal I = (a, b, c) generated by forms of R = k[s, t], of degree n. We may assume that a, b for a regular sequence (i.e. gcd(a, b) = 1). We already know that e0(I) = n 2. For regular rings, one knows ([15]) that e1(I) ≤ d−12 e0(I), d = dimR. Nevertheless the steps above already lead to an efficient calculation for two reasons: the multiplicity e0(I) is known at the outset and it does not really involve the powers of I. Forms of degree up to 10 are handled well by Maucalay2 ([11]). 4 Sylvester forms in dimension two We establish the basic notation to be used throughout. R = k[s, t] is a polynomial ring over the infinite field k, and I ⊂ R = k[s, t] is a codimension 2 ideal generated by 3 forms of the same degree n+ 1, with free graded resolution 0 −→ R(−n−1−µ)⊕R(2(−n−1)+µ) ϕ−→ R3(−n−1) −→ I −→ 0, ϕ = α1 β1 γ1 α2 β2 γ2 Then the symmetric algebra of I is S ≃ R[T1, T2, T3]/(f, g) with f = α1T1 + β1T2 + γ1T3 g = α2T1 + β2T2 + γ2T3. Starting out from these 2 forms, the defining equations of S, following [9], we obtain by elim- ination higher degrees forms in the defining ideal of R(I). It will make use of a computer- assisted methodology to show that these algorithmically specified sets generate the ideal of definition M of R(I) in several cases of interest–in particular answering some questions raised [9]. More precisely, the so-called ideal of moving forms M is given when I is gen- erated by forms of degree at most 5. In arbitrary degree, the algorithm will provide the elimination equation in significant cases. 4.1 Basic Sylvester forms in dimension 2 Let R = k[s, t] and let F,G ∈ B = R[s, t, T1, T2, T3]. If F,G ∈ (u, v)B, for some ideal (u, v) ⊂ R, the form derived from h = ad− bc = det(F,G)(u,v), will be called a basic Sylvester form. To explain their naturalness, even for ideals I not necessarily generated by forms, we give an approach to irreducible decomposition of certain ideals. Theorem 4.1 Let (R,m) be a Gorenstein local ring and let I be an m–primary ideal. Let J ⊂ I be an ideal generated by a system of parameters and let E = (J : I)/J be the canonical module of R/I. If E = (e1, . . . , er), ei 6= 0, and Ii = ann (ei), then Ii is an irreducible ideal The statement and its proof will apply to ideals of rings of polynomials over a field. Proof. The module E is the injective envelope of R/I, and therefore it is a faithful R/I– module (see [2, Section 3.2] for relevant notions). For each ei, Re1 is a nonzero submodule of E whose socle is contained in the socle of E (which is isomorphic to R/m) and therefore its annihilator Ii (as an R-ideal) is irreducible. Since the intersection of the Ii is the annihilator of E, the asserted equality follows. ✷ Corollary 4.2 Let (R,m) be a regular local ring of dimension two and let I be an m– primary ideal with a free resolution 0 → Rn−1 ϕ−→ Rn −→ I → 0,  an−1,1 · · · an−1,n−1 an,1 · · · an,n−1  and suppose that the last two maximal minors ∆n−1,∆n of ϕ form a regular sequence. If e1, . . . , en−1 are as above, then (∆n−1,∆n) : I = In−2(ξ ′) = (e1, . . . , en−1) and each ideal (∆n−1,∆n) : ei is a complete intersection of codimension 2. Proof. The assertion that the irreducible Ii is a complete intersection is a result of Serre, valid for all two-dimensional regular rings whose projective modules are free. ✷ Remark 4.3 In our applications, I = C(f, g), the content ideal of f, g. In some of these cases, C(f, g) = (s, t)n, for some n, an ideal which admits the irreducible decomposition (s, t)n = (si, tn+1−i). One can then process f, g through all the pairs {si, tn−i+1}, and collect the determinants for the next round of elimination. As in the classical Sylvester forms, the inclusion C(f, g) ⊂ (s, t)n may be used anyway to start the process, although without the measure of control of degrees afforded by the equality of ideals. 4.2 Cohen-Macaulay algebras We pointed out in Theorem 2.1 that the basic control of Cohen-Macaulayness of a Rees algebra of an ideal I ⊂ k[s, t] is that its reduction number be at most 1. We next give a mean of checking this property directly off a free presentation of I. Theorem 4.4 Let I ⊂ R be an ideal of codimension 2, minimally generated by 3 forms of the same degree. Let α1 α2 β1 β2 γ1 γ2 be the Hilbert-Burch presentation matrix of I. Then R is Cohen-Macaulay if and only if the equalities of ideals of R hold (α1, β1, γ1) = (α2, β2, γ2) = (u, v), where u, v are forms. Proof. Consider the presentation 0 → L −→ S = R[T1, T2, T3]/(f, g) −→ R → 0, where f, g are the 1-forms [ T1 T2 T3 If R is Cohen-Macaulay, the reduction number of I is 1 by Theorem 2.1, so there must be a nonzero quadratic form h with coefficients in k in the presentation ideal M of R. In addition to h, this ideal contains f, g, hence in order to produce such terms its Hilbert-Burch matrix must be of the form  p1 p2 q1 q2 where u, v are forms of k[s, t], and the other entries are 1-forms of k[T1, T2, T3]. Since p1, p2 are q1, q2 are pairs of linearly independent 1-forms, the assertion about the ideals defined by the columns of ϕ follow. 4.3 Base ideals generated in degree 4 This is the case treated by D. Cox in his Luminy lecture ([9]). We accordingly change the notation to R = k[s, t], I = (f1, f2, f3), forms of degree 4. The field k is infinite, and we further assume that f1, f2 form a regular sequence so that J = (f1, f2) is a reduction of I and of (s, t)4. Let 0 → R(−4− µ)⊕R(−8 + µ) ϕ−→ R3(−4) −→ R −→ R/I → 0, ϕ = α1 α2 β1 β2 γ1 γ2  (4) be the Hilbert-Burch presentation of I. We obtain the equations of f1, f2, f3 from this matrix. Note that µ is the degree of the first column of ϕ, 4 − µ the other degree. Let us first consider (as in [9]) the case µ = 2. Balanced case We shall now give a computer-assisted treatment of the balanced case, that is when the resolution (4) of the ideal I has µ = 2 and the content ideal of the syzygies is (s, t)2. Since k is infinite, it is easy to show that there is a change of variables, T1, T2, T3 → x, y, z, so that (s2, st, t2) is a syzygy of I. The forms f, g that define the symmetric algebra of I can then be written [f g] = [s2 st t2] where u, v, w are linear forms in x, y, z. Finally, we will assume that the ideal I2 x y z u v w has codimension two. Note that this is a generic condition. We introduce now the equations of I. • Linear equations f and g: [f g] = [x y z] ϕ = [x y z] α1 α2 β1 β2 γ1 γ2 = [s2 st t2] where u, v, w are linear forms in x, y, z. • Biforms h1 and h2: Write Γ1 and Γ2 such that [f g] = [x y z] ϕ = [ s t2 ] Γ1 = [ s 2 t ] Γ2. Then h1 = detΓ1 and h2 = detΓ2. • Implicit equation F = detΘ, where [h1 h2] = [s t] Θ. Using generic entries for ϕ, in place of the true k-linear forms in old variables x, y, z, we consider the ideal of k[s, t, x, y, z, u, v, w] defined by f = s2x+ sty + t2z g = s2u+ stv + t2w h1 = −syu− tzu+ sxv + txw h2 = −szu− tzv + sxw + tyw F = −z2u2 + yzuv − xzv2 − y2uw + 2xzuw + xyvw − x2w2 Proposition 4.5 If I2 x y z u v w specializes to a codimension two ideal of k[x, y, z], then L = (f, g, h1, h2, F ) ⊂ A = R[x, y, z, u, v, w] specializes to the defining ideal of R. Proof. Macaulay2 ([11]) gives a resolution 0 → A d2−→ A5 −→ A5 −→ L → 0 where  zv − yw zu− xw −yu+ xv  The assumption on I2 x y z u v w says that the entries of d2 generate an ideal of codimension four and thus implies that the specialization LS has projective dimension two and that it is unmixed. Since LS 6⊂ (s, t)S, there is an element q ∈ (s, t)R that is regular modulo S/LS. If LS = Q1 ∩ · · · ∩Qr is the primary decomposition of LS, the localization LSq has the corresponding decompo- sition since q is not contained in any of the Qi. But now Symq = Rq, so LSq = (f, g)u, as Iq = Rq. ✷ Non-balanced case We shall now give a similar computer-assisted treatment of the non-balanced case, that is when the resolution (4) of the ideal I has µ = 3. This implies that the content ideal of the syzygies is (s, t). Let us first indicate how the proposed algorithm would behave. • Write the forms f, g as f = as+ bt g = cs+ dt, where x y z u v w • The next form is the Jacobian of f, g with respect to (s, t) h1 = det(f, g)(s,t) = ad− bc = −bxs2 − byst− bzt2 + aus2 + avst+ awt2. • The next two generators h2 = det(f, h1)(s,t) = b 2xs+ b2yt− abzt− abus− abvt+ a2wt and the elimination equation h3 = det(f, h2)(s,t) = −b3x+ ab2y − a2bz + ab2u− a2bv + a3w. Proposition 4.6 L = (f, g, h1, h2, h3) ⊂ A = k[s, t, x, y, z, u, v, w] specializes to the defin- ing ideal of R. Proof. Macaulay2 ([11]) gives the following resolution of L 0 → A2 ϕ−→ A6 ψ−→ A5 −→ L → 0, x + abu −b y + abz + abv − a w −bsx− bty + asu + atv −btz + atw −s x− sty − t u− stv − t t −s 0 0 0 0 a b t −s 0 0 0 0 a b t −s 0 0 0 0 a b The ideal of 2 × 2 minors of ϕ has codimension 4, even after we specialize from A to S in the natural manner. Since LS has projective dimension two, it will be unmixed. As LS 6⊂ (s, t), there is an element u ∈ (s, t)R that is regular modulo S/LS. If LS = Q1 ∩ · · · ∩Qr is the primary decomposition of LS, the localization LSu has the corresponding decompo- sition since u is not contained in any of the Qi. But now Symu = Ru, so LSu = (f, g)u, as Iu = Ru. ✷ 4.4 Degree 5 and above It may be worthwhile to extend this to arbitrary degree, that is assume that I is defined by 3 forms of degree n+1 (for convenience in the notation to follow). We first consider the case µ = 1. Using the procedure above, we would obtain the sequence of polynomials in A = R[a, b, x1, . . . , xn, y1, . . . , yn] • Write the forms f, g as f = as+ bt g = cs+ dt, where x1 · · · xn y1 · · · yn  sn−2t stn−2  • The next form is the Jacobian of f, g with respect to (s, t) h1 = det(f, g)(s,t) = ad− bc • Successively we would set hi+1 = det(f, hi)(s,t), 1 < n. • The polynomial hn = det(f, hn−1)(s,t) is the elimination equation. Proposition 4.7 L = (f, g, h1, . . . , h5) ⊂ A specializes to the defining ideal of R. In Macaulay2, we checked the degrees 5 and 6 cases. In both cases, the ideal L (which has one more generator in degree 6) has a projective resolution of length 2 and the ideal of maximal minors of the last map has codimension four. Conjecture 4.8 For arbitrary n, L = (f, g, h1, . . . , hn) ⊂ A has projective dimension two and specializes to the defining ideal of R. In degree 5, the interesting case is when the Hilbert-Burch matrix φ has degrees 2 and 3. Let us describe the proposed generators. For simplicity, by a change of coordinates, we assume that the coordinates of the degree 2 column of ϕ are s2, st, t2 f = s2x+ sty + t2z g = (s3w1 + s 2tw2 + st 2w3 + t 3w4)x+ (s 3w5 + s 2tw6 + st 2w7 + t 3w8)y + (s3w9 + s 2tw10 + st 2w11 + t 3w12)z Let [ x y z sA sB + tC tD  = φ x ys+ zt sA+ tB stC + t2D xs+ yt z s2A+ stB sC + tD where A,B,C,D are k-linear forms in x, y, z. h1 = det(B1) = s2(−yA) + st(xC − yB − zA) + t2(xD − zB) = s2(−yA) + t(xCs− yBs− zAs+ xDt− zBt) = s(−yAs+ xCt− yBt− zAt) + t2(xD − zB), h2 = det(B2) = s2(xC − zA) + st(xD + yC − zB) + t2(yD) = s2(xC − zA) + t(xDs+ yCs− zBs+ yDt) = s(xCs− zAs + xDt+ yCt− zBt) + t2(yD). x ys+ zt −yA xCs− yBs− zAs + xDt− zBt xs+ yt z −yAs+ xCt− yBt− zAt xD − zB x ys+ zt xC − zA xDs+ yCs− zBs+ yDt xs+ yt z xCs− zAs+ xDt+ yCt− zBt yD c1 = det(C1) = x 2(Cs+Dt) + xy(−Bs) + xz(−As−Bt) + yz(At) + y2(As) c2 = det(C2) = x 2(Ds) + xy(Dt) + xz(−Bs−Ct) + yz(As) + z2(At) c3 = det(C3) = x 2(Ds) + xy(Dt) + xz(−Bs−Ct) + yz(As) + z2(At) c4 = det(C4) = xy(Ds) + xz(−Cs−Dt) + yz(−Ct) + z2(As +Bt) + y2(D) x y z −yA xC − yB − zA xD − zB xC − zA xD + yC − zB yD Then F = −x3D2+x2yCD+xy2(−BD)+x2z(2BD−C2)+xz2(2AC−B2)+xyz(BC− 3AD) + y2z(−AC)+ yz2(AB)+ y3(AD) + z3(−A2), an equation of degree 5. In particular, the parametrization is birational. Proposition 4.9 L = (f, g, h1, h2, c1, c2, c4, F ) specializes to the defining ideal of R. Proof. Using Macaulay2, the ideal L has a resolution: 0 −→ S1 d3−→ S6 d2−→ S12 d1−→ S8 −→ L −→ 0. d3 = [−z y x − t s 0]t y z 0 0 0 0 x 0 z 0 0 0 −v 0 0 z 0 x2w4 − xzw7 + xyw8 + xzw12 u 0 0 0 z −xzw3 + xyw4 + z 2w6 − yzw7 + y 2w8 − xzw8 − z 2w11 + yzw12 0 x −y 0 0 0 0 −v 0 −y 0 xzw1 − x 2w3 + yzw5 + z 2w9 − xzw11 0 u 0 0 −y xzw2 − x 2w4 + z 2w10 − xzw12 0 0 u 0 −x xzw1 + yzw5 − xzw6 + x 2w8 + z 0 0 0 u v 0 0 0 v x 0 −xyw1 + x 2w2 − y 2w5 + xyw6 − x 2w7 − yzw9 + xzw10 0 0 0 0 0 −t 0 0 0 0 0 s The ideals of maximal minors give codim I1(d3) = 5 and codim I5(d2) = 4 after special- ization. As we have been arguing, this suffices to show that the specialization is a prime ideal of codimension two. ✷ Elimination forms in higher degree In degrees greater than 5, the methods above are not very suitable. However, in several cases they are still supple enough to produce the elimination equation. We have already seen this when one of the syzygies is of degree 1. Let us describe two other cases. • Degree n = 2p, f and g both of degree p. We use the decomposition (s, t)p = (si, tp+1−i). For each 1 ≤ i ≤ p, let hi = det(f, g)(si,tp+1−i). These are quadratic polynomials with coefficients in (s, t)p−1. We set [h1, · · · , hp] = [sp−1, · · · , tp−1] ·A, where A is a p × p matrix whose entries are 2-forms in k[x, y, z]. The Sylvester form of degree n, F = det(A), is the required elimination equation. • Degree n = 2p+ 1, f of degree p. We use the decomposition (s, t)p = (si, tp+1−i). For each 1 ≤ i ≤ p, let hi = det(f, g)(si,tp+1−i). These are quadratic polynomials with coefficients in (s, t)p. We set [f, h1, · · · , hp] = [sp, · · · , tp] ·B, where A is a (p + 1) × (p + 1) matrix with one column whose entries are linear forms and the remaining columns with entries 2-forms in k[x, y, z]. The Sylvester form F = det(B) is the required elimination equation. References [1] I. M. Aberbach, C. Huneke and N. V. Trung, Reduction numbers, Briançon-Skoda theorems and depth of Rees algebras, Compositio Math. 97 (1995), 403–434. [2] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1993. [3] L. Busé and J.-P. Jouanolou, On the closed image of a rational map and the implicit- ization problem, J. Algebra 265 (2003), 312-357. [4] L. Busé, M. Chardin and J.-P. Jouanolou, Complement to the implicitization of ratio- nal hypersurfaces by means of approximation complexes, Arxiv, 2006. [5] L. Busé, D. Cox and C. DAndrea, Implicitization of surfaces in P3 in the presence of base points, J. Algebra Appl. 2 (2003), 189-214. [6] D. A. Cox, T. Sederberg, and F. Chen, The moving line ideal basis of planar rational curves, Comput. Aided Geom. Des. 15 (1998) 803–827. [7] D. A. Cox, R. N. Goldman, and M. Zhang, On the validity of implicitization by moving quadrics for rational surfaces with no base points, J. Symbolic Computation 29 (2000) 419–440. [8] D. A. Cox, Equations of parametric curves and surfaces via syzygies, Contemporary Mathematics 286 (2001) 1–20. [9] D. A. Cox, Four conjectures: Two for the moving curve ideal and two for the Bezoutian, Proceedings of “Commutative Algebra and its Interactions with Algebraic Geometry”, CIRM, Luminy, France, May 2006 (available in CD media). [10] C. D’Andrea, Resultants and moving surfaces, J. Symbolic Computation 31 (2001) 585–602. [11] D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/. [12] J. Herzog, A. Simis and W. V. Vasconcelos, Koszul homology and blowing-up rings, in Commutative Algebra, Proceedings: Trento 1981 (S. Greco and G. Valla, Eds.), Lecture Notes in Pure and Applied Mathematics 84, Marcel Dekker, New York, 1983, 79–169. [13] J. P. Jouanolou, Formes d’inertie et résultant: un formulaire, Adv. Math. 126 (1997), 119–250. [14] B. Johnson and D. Katz, Castelnuovo regularity and graded rings associated to an ideal, Proc. Amer. Math. Soc. 123 (1995), 727-734. [15] C. Polini, B. Ulrich and W. V. Vasconcelos, Normalization of ideals and Briançon- Skoda numbers, Math. Research Letters 12 (2005), 827–842. [16] M. E. Rossi, On symmetric algebras which are Cohen-Macaulay, Manuscripta Math. 34 (1981), 199-210. http://www.math.uiuc.edu/Macaulay2/ [17] T. Sederberg, R. Goldman and H. Du, Implicitizing rational curves by the method of moving algebraic curves, J. Symbolic Computation 23 (1997), 153–175. [18] A. Simis, Cremona transformations and some related algebras, J. Algebra 280 (1) (2004), 162–179. [19] A. Simis, B. Ulrich and W. V. Vasconcelos, Cohen-Macaulay Rees algebras and degrees of polynomial relations, Math. Annalen 301 (1995), 421–444. [20] A. Simis, B. Ulrich and W. V. Vasconcelos, Jacobian dual fibrations, Amer. J. Math. 115 (1993), 47–75. [21] A. Simis, B. Ulrich and W. V. Vasconcelos, Codimension, multiplicity and integral extensions, Math. Proc. Camb. Phil. Soc. 130 (2001), 237–257. [22] W. V. Vasconcelos, Arithmetic of Blowup Algebras, London Math. Soc., Lecture Note Series 195, Cambridge University Press, 1994. [23] W. V. Vasconcelos, Integral Closure, Springer Monographs in Mathematics, New York, 2005. Introduction Preliminaries on symmetric and Rees algebras Algebraic invariants in rational parametrizations Elimination degrees and birationality Calculation of e1(I) of the base ideal of a rational map Sylvester forms in dimension two Basic Sylvester forms in dimension 2 Cohen-Macaulay algebras Base ideals generated in degree 4 Degree 5 and above

0704.0609

The effectiveness of quantum operations for eavesdropping on sealed messages

7 The effectiveness of quantum operations for eavesdropping on sealed messages Paul A Lopata† and Thomas B Bahder∗1 †Sensors and Electronic Devices Directorate, Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland, 20783 USA ∗Charles M Bowden Research Facility, Aviation and Missile Research, Development and Engineering Center, US Army RDECOM, Redstone Arsenal, Alabama, 35898 USA E-mail: [email protected] Abstract. A quantum protocol is described which enables a user to send sealed messages and that allows for the detection of active eavesdroppers. We examine a class of eavesdropping strategies, those that make use of quantum operations, and we determine the information gain versus disturbance caused by these strategies. We demonstrate this tradeoff with an example and we compare this protocol to quantum key distribution, quantum direct communication, and quantum seal protocols. 1. Introduction We have all become accustomed to sending messages electronically, whether by fax machine, telephone, computer or other electronic media. Most of these messages contain data that is already publicly known or at least easily found. Other messages are things we would like to keep to ourselves, and it would be inconvenient if some third party came across the message. Still other messages are extremely private and resources, jobs, or even lives(!) might be lost if the message fell into the wrong hands. A great deal of effort is employed to encrypt the messages that fall in this last category, sending them with some sort of code in order to prevent any third party from understanding them even if the messages are intercepted.[1] However, when a message is sent electronically there is no commonly available technology to determine if someone has been trying to intercept the message. When sending typed letters, such a technology does exist, albeit in an imperfect form. We often seal our letters in envelopes. These envelopes are not secure, that is, they do not prevent anyone from opening the envelope and reading the letter inside. However, when an envelope is received intact, without any tears or other indication that it has been tampered with, we have a strong reason to believe that the message inside has not been seen by anyone since the earlier time when the sender sealed it. Yet a seal on an envelope is not to be wholly trusted for this task of detecting eavesdroppers. 1 Previous address: Army Research Laboratory, Adelphi, MD http://arxiv.org/abs/0704.0609v1 A skilled person might be able to examine the contents of the sealed envelope in any number of ways: by using x-rays or other similar non-destructive testing methods, by steaming the seal off and re-sealing, or by ripping open the envelope and then placing the letter in a new, forged envelope that matches the original in every detail. In this paper we introduce a quantum cryptographic protocol that allows two users to send and receive a message in a manner that is, in effect, quite similar to the use of a sealed envelope. The receiver of the message has the opportunity to check if there have been any active eavesdroppers trying to learn the contents of the message. And similar to a message sealed in an envelope, the message remains unknown to anyone who is not actively trying to learn the contents. This protocol has the advantage over sealing letters in envelopes because the limited types of interactions allowed by quantum mechanics prevent someone from eavesdropping on the message without leaving signs of the eavesdropping activities. It is important to make it clear that any messages sent using the protocol introduced here are not secure. That is, an eavesdropper can always choose to take some action in order to determine the content of a message sent using this protocol. (We give an example of one such effective eavesdropping strategy below.) The quality that makes this protocol distinct from other methods of message transmission is that any such active eavesdropping strategies will cause an appreciable amount of “noise” that is detectable by the message receiver. The analysis that a message receiver undertakes to place a bound on what an eavesdropper could have learned during a particular message transmission is not undertaken here. This analysis can be found elsewhere.[2] The goal of this manuscript is to examine a certain class of strategies for eavesdropping on these sealed messages, and it is divided into four parts: First, the quantum message sealing protocol is introduced. Following this, we examine a certain class of eavesdropping strategies and describe what an eavesdropper expects to learn by employing such strategies. Next, we describe the type and amount of disturbance the eavesdropper will cause by such an activity and work out the details of an example from this class of eavesdropping strategies. We conclude with a discussion of this protocol and its similarities and differences to other quantum cryptographic protocols. 2. Message sealing protocol We describe the protocol where the message sender named Alice transfers a message to the receiver named Bob. This message will be a single bit b which is either zero or one. The protocol utilizes a single quantum mechanical system which has two degrees of freedom. The standard notation for such a system is used, with |0〉 and |1〉 representing vectors that form an orthonormal basis. The protocol also involves a number of announcements made by the message sender. These announcements are to be considered as public announcements to which everyone is assumed to have access. A process, referred to as a single shot, will be repeated many times and goes as follows: Step 1 - Bob prepares a quantum system, which we refer to as a particle, in one of four pure states: |0〉, |1〉, |+〉 ≡ (|0〉+ |1〉)/ 2, or |−〉 ≡ (|0〉 − |1〉)/ 2. The decision as to which state to prepare is made at random with equal probability for each state. He records the state he has prepared and then he sends the particle to Alice. Step 2 - Alice makes one of two measurements with equal probability. She either makes a measurement corresponding to σ1 = |+〉〈+|−|−〉〈−| or she makes a measurement corresponding to σ3 = |0〉〈0| − |1〉〈1|. Each of these two measurements can be said to have a result m that is either m = +1 or m = −1. Step 3 - Alice announces whether her measurement corresponded to σ1 or σ3. Step 4 - Alice makes one of two possible announcements. With probability pa she makes a bit-announcement (described immediately below) and with probability (1 − pa) she makes a result-announcement. She also makes it known which of the two types of announcement she is making. Bit-Announcement: She announces a bit c that is determined by using the message bit b and the measurement result. If her measurement yielded the result m = +1 then her announced bit c will be the same as the message bit b and if her measurement yielded the result m = −1 her announced bit c will be the opposite of the message bit b. Result-Announcement: She announces the result of her measurement, m = +1 or m = −1. When Bob prepares the particle in the state |0〉 or |1〉 and Alice makes a σ3 measurement, or when Bob prepares the particle in the state |+〉 or |−〉 and Alice makes a σ1 measurement we say that Alice’s measurement and Bob’s state preparation have a matching basis. They will have a matching basis on half the shots performed. When this occurs then Bob knows the result of the measurement without Alice having to announce it, provided that the state of the particle did not change from when Bob prepared it to when Alice makes the measurement. The correlations between Bob’s state preparation and Alice’s measurement results allow Bob to both determine the message bit and check the channel for any disturbances. When Alice makes a measurement in the basis matching Bob’s state preparation, Bob determines the message by applying a controlled-bit-flip operation on the announced bit. When the state in which he prepared the particle is either |0〉 or |+〉 then the message bit b is the same as the announced bit c and if he prepared |1〉 or |−〉 then the message b is the opposite of the announced bit c. From an eavesdropper’s point of view, the probability that the message bit is one value or the other is determined from the coded bit-announcements. When both values of the measurement result are equally likely then both values of the message bit are equally likely (for either bit-announcement). The four possible initial states that Bob prepares and the two possible measurements were chosen so that either measurement result is equally likely. Moreover, the only opportunity that an eavesdropper has to change these probabilities is to change the state of the particle when it is traveling from Bob to Alice. The rules of quantum mechanics allow for the state of a quantum mechanical system to change in two different ways: by a unitary evolution or by a measurement. If we want to describe the effects of coupling the quantum system composed of the particle to another (auxiliary) quantum system and then letting the state of whole system (particle plus auxiliary) change via unitary evolution of measurement, the entire process can be described as a quantum operation or a generalized measurement on the state of the particle subsystem.[3] In the following sections we examine the case of when an eavesdropper chooses to change the state of the particle by applying a quantum operation. It is worthwhile to emphasize that while using this type of eavesdropping activity is not optimal,[2] it provides us with some intuition as to how this protocol can be expected to work. 3. Information gain from quantum operations In this section we quantify what an eavesdropper learns by applying a quantum operation to change the state of the particle as it travels from Bob to Alice. We first describe quantum operations[3] and then tackle the problem of quantifying an eavesdropper’s gain by using the Shannon mutual information.[4] A quantum operation E acting on states in Hilbert space H is described by a set of operators {E1, . . . , En} subject to the requirement that iEi = I where I is the identity operator acting on H. We say that the quantum operation E maps the initial state ρ to final state E(ρ) = i . A quantum operation is a convex linear map on the space of mixed states, which is to say that if ρ = pρ1 + (1 − p)ρ2 with 0 ≤ p ≤ 1, then E(ρ) = pE(ρ1) + (1 − p)E(ρ2). A special class of quantum operations are the unital quantum operations that map the chaotic state, which is 1 I where d = dim(H), to itself. We quantify the amount an eavesdropper learns by using the Shannon mutual information between two random variables: the random variable B which describes the possible values of the message and their probabilities, and the random variable C which describes the possible strings of bit-announcements and their probabilities. These strings result from the fact that there will be N shots, and an announcement will be made on each shot. On some of the shots only the result of the measurement will be announced, and this result does not depend on the message in any way. Therefore, only the bit-announcements will be of any concern to us in quantifying what the eavesdropper learns. The possible messages are b = 0 and b = 1 with one-half prior probability each. On each shot there are four possible bit-announcements — (σ1, 0), (σ1, 1), (σ3, 0), and (σ3, 1) — and when N shots are made, k of which result in bit-announcements (where 0 ≤ k ≤ N), there are 4k possible bit-announcement strings. Because of the probabilistic nature of the protocol, the number of bit-announcements is not fixed. The probability pk of making k bit-announcements is found using the binomial distribution p ka (1− pa)N−k . We use the symbol c to denote a bit-announcement string, and we use the symbol C(k) to describe the ensemble of all possible bit-announcement strings of length k. Given that there are k bit-announcements, the Shannon mutual information I(C(k) : B) is calculated using I(C(k) : B) = Pr(c) log Pr(c) Pr(c | b) log Pr(c | b) where the sum over c(k) indicates that this sum is taken over all 4k bit-announcement strings of length k. This can be used to determine the expected mutual information when taking the weighted sum over the various possible lengths of bit-announcement strings, I(C : B) = pkI(C (k) : B) . (2) This can be calculated once the probabilities Pr(c|b) are known for every c and both values of b. The remainder of this section is devoted to determining these probabilities, which will change depending upon which quantum operation is applied. For a given value of the message, the probabilities of the four bit-announcements depend upon the probability of Alice getting the m = +1 measurement result. That is, Pr(σi, c = b|b) = Pr(m = +1|σi) Pr(σi) = Pr(m = +1|σi)/2 , Pr(σi, c 6= b|b) = Pr(m = −1|σi) Pr(σi) = Pr(m = −1|σi)/2 , Table 1. The probabilities for the four results relevant to the bit-announcements, given that an eavesdropper acts with a quantum operation Eλv that maps the chaotic state to ρ(λv). Pr(m = +1|σ1, Eλv) = 12(1 + λv1) Pr(m = −1|σ1, Eλv) = 12(1− λv1) Pr(m = +1|σ3, Eλv) = 12(1 + λv3) Pr(m = −1|σ3, Eλv) = 12(1− λv3) where i = 1, 3 and b = 0, 1. The notation Pr(m = +1|σi), for example, is used to mean that this is the probability that the result m = +1 will be found when a measurement that corresponds to σi is made on the particle and Pr(σi) is the probability that the measurement corresponding to σi will be performed. Of course, the machinery of quantum mechanics requires us to specify the state of the particle in order to calculate a probability of a certain measurement result. From an eavesdropper’s point of view, if she does nothing to the particle then there are four possible states with equal probability. So Pr(m = ±1|σi) = 14 (Tr( (I±σi)|0〉〈0|)+Tr(12 (I±σi)|1〉〈1|)+Tr( (I±σi)|+〉〈+|)+ (I ± σi)|−〉〈−|)) where i = 1, 3. By the linearity of the Trace function, this is equivalent to Pr(m = ±1|σi) = Tr(12(I ± σi) I). In this way, it is quite reasonable to say that the state of the particle, to the eavesdropper’s best description, is the chaotic state ρ = 1 When an eavesdropper applies a quantum operation E to change the state of the particle, it will in general change each of the four possible initial states differently. By the linearity of the Trace function and the convex linearity of the quantum operation E , the probability of m = ±1 can be calculated for the state ρ′ = E(1 I). That is, Pr(m = ±1|σi) = Tr(12 (I ± σi)E( I)) for i = 1, 3. Every (generally mixed) state of a two-level quantum system can be described by ρ(λv) = (I + λ[v1σ1 + v2σ2 + v3σ3]) where v 1 + v 2 + v 3 = 1, σ2 = iσ1σ3, and 0 ≤ λ ≤ 1. This “Bloch sphere” description of the two-level state can be pictured as a vector λv in a real three dimensional space. When E(1 I) = 1 (I + λ[v1σ1 + v2σ2 + v3σ3]), the probabilities for the four possible announcements are shown in Table 1. If an eavesdropper applies the same quantum operation each time a particle is sent from Bob to Alice, the probabilities for each bit-announcement string is found by taking the product of the probabilities of each of the four announcements, with each of the probabilities appearing in the product the same number of times that that announcement appears in the string. We can now calculate the mutual information for any quantum operation by calculating the probabilities for each bit-announcement string and then using Equations (1) and (2). To summarize this section, we have described how to calculate the mutual information which quantifies what an eavesdropper expects to learn about the message given a particular quantum operation used as an eavesdropping strategy. In the next section, we determine the amount of “noise” that such eavesdropping strategies cause. 4. Disturbance caused by quantum operations In the previous section we focused on the bit-announcements and ignored the result- announcements. In this section we will do the opposite. The bit-announcements are used Table 2. The four events that correspond to mismatches. Bob prepares the state Alice measures measurement result |+〉 σ1 m = −1 |−〉 σ1 m = +1 |0〉 σ3 m = −1 |1〉 σ3 m = +1 by both Bob and any eavesdroppers to determine the message, but the result-announcements are of no use to the eavesdropper and serve Bob’s purpose to check the channel for “noise”. There are sixteen different event statistics that are kept by Bob relating to the measurement- announcements: four possible initial states, two possible measurement types, and two possible measurement results for each measurement. Out of these sixteen, there are four events that would be the most surprising to Bob, and would each indicate that the state of the particle, when Alice measured it, was not the same as the one he had prepared. These four types of events will be referred to as mismatches and are shown in Table 2. The probability of a mismatch, on a particular shot, is Pr(mismatch) = Pr(|+〉, σ1,−1) + Pr(|−〉, σ1,+1) + Pr(|0〉, σ3,−1) + Pr(|1〉, σ3,+1) Pr(σ1,−1||+〉) + Pr(σ1,+1||−〉) + Pr(σ3,−1||0〉) + Pr(σ3,+1||1〉) Pr(−1||+〉, σ1) + Pr(+1||−〉, σ1) + Pr(−1||0〉, σ3) + Pr(+1||1〉, σ3) Of course, when Bob analyzes the data, a mismatch can only occur on a particular shot if the bases are matched up. A factor of 1/2 disappears when we account for this to give the probability that there will be a mismatch error on a shot when the bases are matched. For a fixed quantum operation E employed by an eavesdropper, these probabilities are easily calculated. Note that these probabilities depend upon the final states E(|+〉〈+|), E(|−〉〈−|), E(|0〉〈0|), and E(|1〉〈1|), and not just on the evolution of the chaotic state. In general, there are many different quantum operations that have the same effect on the chaotic state. (The exception to this is when the chaotic state is mapped to a pure state, in which case it is easily seen by the convex linearity of quantum operations that every initial state must be mapped to that pure state.) 5. An Example Let us now examine a family of eavesdropping strategies that utilize the quantum operation Ex, where x is a parameter which falls in the range 0 ≤ x ≤ 1. When x = 0 the strategy corresponds to the eavesdropper doing nothing (and as we shall see, learning nothing), and when x = 1 it corresponds to a quantum operation eavesdropping strategy with the greatest mutual information. The quantum operation Ex can be achieved by coupling the initial state ρ (from Bob) to an auxiliary quantum system in the state |φ〉, letting the coupled system evolve unitarily (described by some unitary operator U that acts on the combined system) and then tracing over the auxiliary system. The unitary operator acts as follows: |0〉 ⊗ |φ〉 = |0〉 ⊗ |F 〉 ≡ |Γ0〉 Table 3. The probabilities, from the eavesdropper’s point of view, of the four possible bit- announcements for a given value of b when the quantum operation Ex, introduced in Section 5, is applied. b = 0 b = 1 Pr(σ1, c = 0|b) 1/4 1/4 Pr(σ1, c = 1|b) 1/4 1/4 Pr(σ3, c = 0|b) (1 + x)/4 (1− x)/4 Pr(σ3, c = 1|b) (1− x)/4 (1 + x)/4 |1〉 ⊗ |φ〉 x |0〉 ⊗ |G〉+ 1− x |1〉 ⊗ |F 〉 ≡ |Γ1〉 , where 〈F |G〉 = 0 and 〈F |F 〉 = 〈G|G〉 = 1. The fact that 〈0|1〉〈φ|φ〉 = 〈Γ0|Γ1〉 is sufficient to show that such a unitary operator U exists. The action of the quantum operation Ex on any initial pure state |η〉 is found by tracing over the auxiliary subsystem after performing the unitary transformation U : |η〉〈η| = Traux |η〉 ⊗ |φ〉 〈η| ⊗ 〈φ| By the convex linearity of quantum operations we also know the action of Ex on any mixed state as well. From the preceeding considerations, it is straightforward to show that Ex acts on the relevant initial states in the following way: |0〉〈0| = |0〉〈0| |1〉〈1| = x |0〉〈0| + (1− x) |1〉〈1| |+〉〈+| (1 + x) |0〉〈0| + (1− x) |1〉〈1| + |0〉〈1| + |1〉〈0| |−〉〈−| (1 + x) |0〉〈0| + (1− x) |1〉〈1| − |0〉〈1| + |1〉〈0| from which it is easy to see that (1 + x) |0〉〈0| + (1− x) |1〉〈1| (I + xσ3). The probability of a mismatch, calculated using Equation (3), for this quantum operations is (1 + x− 1− x). In order to calculate the mutual information for this quantum operation, we must be able to determine the values of Pr(c|b, Ex), that is, the probability of a every string of result- announcements c given each value of b. If a particular string of k result-announcements c(k, d1, d2, d3, d4) consists of (σ1, c = 0) announced d1 times, (σ1, c = 1) announced d2 times, (σ3, c=0) announced d3 times, and (σ3, c=1) announced d4 times — in any order — then the probability for this announcement to occur is c(k, d1, d2, d3, d4)|b=0, Ex (1 + x)d3(1− x)d4 ≡ px,k,d3,d4 c(k, d1, d2, d3, d4)|b=1, Ex (1− x)d3(1 + x)d4 ≡ qx,k,d3,d4 . 0.2 0.4 0.6 0.8 1 I HC : BL Figure 1. Mutual information as a function of x, describing the amount an eavesdropper learns about the message bit given that she uses the quantum operation Ex on each shot when Bob sends N = 119 particles and Alice has probability pa = 0.01 of making a bit- announcement. 0.2 0.4 0.6 0.8 1 Mismatch Probability Figure 2. Probability of a mismatch as a function of x when an eavesdropper uses the quantum operation Ex on each shot. This calculation utilizes the probabilities for the single announcements found in Table 3. There are k!/(d3!d4!(k − d3 − d4)!) different strings of k bit announcements that share this same probability (for each value of b). Using these results, we can now calculate the mutual information. I(C(k) : B) = − d3!d4!(k−d3−d4)! px,k,d,d3 + qx,k,d,d3 px,k,d,d3 + qx,k,d,d3 px,k,d,d3 log px,k,d,d3 − qx,k,d,d3 log qx,k,d,d3 If we choose some exemplary values of pa and N , this will give us some numerical results for the mutual information. Say that Alice sets pa = 0.05 and Bob agrees with Alice to send N = 119 particles in order to communicate the value of a single bit. This choice of pa and N gives them slightly more than a 95% chance of matching their bases on a shot when a result-announcement is made. The mutual information I(C : B), when an eavesdropper applies the quantum operation Ex on every shot, is plotted for all values of 0 ≤ x ≤ 1 in Figure 1. Compare this with the disturbance caused, quantified by the probability of a mismatch, by applying the same quantum operation, which is shown in Figure 2. As a final note, this example demonstrates that a passive eavesdropper learns nothing about the message. That is, if we describe a passive eavesdropper as someone who is only listening to the announcements that Alice makes but does not interfere with the particles in any way,[1] that person’s eavesdropping strategy would correspond to Ex when x = 0. It is easily seen from the Figures that this strategy causes the eavesdropper to learn nothing and also to causes no disturbance. 6. Discussion This protocol represents something new in the field of cryptography. It provides the message receiver with a way to check if an eavesdropper is attempting to access the message. The analysis shown here demonstrates both the amount learned by an eavesdropper and the disturbance caused, measured in the number of mismatches, when an eavesdropper employs a particular quantum operation. As shown in the example above, this protocol is not secure against active attacks in which an eavesdropper interacts with the particles as they travel from the message receiver to the message sender. However, this example also demonstrates that such attacks cause a disturbance in the system, which can be quantified by the number of mismatches found by the message receiver. A more general analysis a message receiver’s bound on the amount of information an eavesdropper could have learned during a particular transmission is taken up elsewhere.[2] The protocol discussed here has similarities to other quantum cryptography protocols that have been introduced and it is worthwhile to examine these similarities, as well as what makes this current protocol distinct. The three types of quantum cryptographic protocols that will be discussed here are quantum key distribution (QKD) protocols, quantum secure direct communication (QSDC) protocols, and quantum seal protocols. The main distinction between this new protocol and the QKD protocols is that the goal of QKD is to develop a shared private key between two parties while here it is important that a particular message gets transmitted. Said in a different way, each party in a QKD setting starts with nothing and ends up with a random string of bits, but neither one of them cares which string of bits results from the process, so long as they share the same one. Here, one party starts with a particular string of bits — the message — and when the process ends the other party will (hopefully) have the message as well. (There is a tunably small probability that the process will be unsuccessful.[2]) Of course, in QKD the random string of bits can later be used to encrypt a message (which can be sent on a classical channel), but the QKD process itself transfers no information. It is worthwhile mentioning that this current protocol is very similar, in some ways, to a specific QKD protocol, called BB84.[5] The two protocols use the same four initial possible states and the same two measurements. The difference between the two is the classical messages that are sent and how these messages are used. These two protocols are so similar that if two users have a system that implements BB84 then they should be able to implement this new protocol with only minor modifications to the system. The second type of quantum cryptographic protocol that we will discuss is the so-called “quantum secure direct communication” (QSDC).[6] The greatest similarity between the QSDC protocols and the one introduced here is that they both use quantum states of some transferred system to transmit a message from one party to another, rather than generating a key. Moreover, this is done without the use of any pre-shared key. However, the goal of QSDC is to transmit the messages securely (that is, to prevent any eavesdropper from understanding the message), while the goal of the protocol introduced here is to detect the activity of any active eavesdroppers. The final comparison we will make is with those quantum cryptographic protocols that have been called “quantum seal” protocols.[7] These quantum seal protocols are distinct from the current one. The goal of the quantum seal protocols is for a message sender to prepare a quantum mechanical system in some initial state so that someone else can determine the message by making a measurement on that quantum mechanical system. Moreover, the message preparer also creates correlations between the quantum mechanical system and a second quantum mechanical system so that a measurement can be made, by the message preparer, on the second system to determine if anyone has read the message. The major distinction between these quantum seal protocols and the protocol introduced here is that protocol introduced here has a preferred message receiver (the person who sends the particles to the message sender) who can check if anyone else has tried to read the message, while in these earlier quantum seal protocols[7] all receivers are on equal footing and it is the message sender who can check if someone has accessed the message. We conclude this discussion by emphasizing that the protocol introduced here is neither a QKD protocol, nor a QSDC protocol, nor a quantum seal protocol. It has distinct goals and the various security (or no-security) proofs that have been applied to these earlier protocols do not apply here. Acknowledgments This work was funded in part by the Disruptive Technology Office (DTO) and by the Army Research Office (ARO). This research was performed while Paul Lopata held a National Research Council Research Associateship Award at the Army Research Laboratory. References [1] Brassard G Modern Cryptology 1988 (Spring-Verlag New York, Inc.) [2] Lopata P and Bahder T, manuscript in preparation [3] Nielsen M and Chuang I 2000 Quantum Computation and Quantum Information (Cambridge University Press) [4] Shannon C 1993 Claude Elwood Shannon Collected Papers (IEEE Press) p 84 [5] Bennett C and Brassard G 1984 Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE Press) pp 175–179 [6] Boström K and Felbinger T 2002 Physical Review Letters 89 187902 Wójcik A 2003 Physical Review Letters 90 157901 Deng F-G, Long G L, and Liu X-S 2003 Physical Review A 68 042317 Deng F-G and Long G L 2004 Physical Review A 69 052319 Lucamarini M and Mancini S Physical Review Letters 94 140501 and others. [7] Bechmann-Pasquinucci H 2003 Quantum Seals Preprint quant-ph/0303173 Bechmann-Pasquinucci H, D’Ariano G M, and Macchiavello C 2005 Impossibility of Perfect Sealing of Classical Information Preprint quant-ph/0501073 Singh S K and Srikanth R 2005 Physica Scripta 71 pp 433–5 He G-P 2005 Physical Review A 71 054304 Chau H F 2006 Physics Letters A 354 pp 31–4 http://arxiv.org/abs/quant-ph/0303173 http://arxiv.org/abs/quant-ph/0501073

0704.0610

Shocks in nonlocal media

Shocks in nonlocal media Neda Ghofraniha,1 Claudio Conti,2,3 Giancarlo Ruocco,3,4 Stefano Trillo5∗ 1 Research Center SMC INFM-CNR, Università di Roma “La Sapienza”, P. A. Moro 2, 00185, Roma, Italy 2Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89/A, 00184 Rome, Italy 3Research center SOFT INFM-CNR Università di Roma “La Sapienza”, P. A. Moro 2, 00185, Roma, Italy 4Dipartimento di Fisica, Università di Roma “La Sapienza”, P. A. Moro 2, 00185, Roma, Italy 5 Dipartimento di Ingegneria, Università di Ferrara, Via Saragat 1, 44100 Ferrara, Italy (Dated: October 1, 2018) We investigate the formation of collisionless shocks along the spatial profile of a gaussian laser beam propagating in nonlocal nonlinear media. For defocusing nonlinearity the shock survives the smoothing effect of the nonlocal response, though its dynamics is qualitatively affected by the latter, whereas for focusing nonlinearity it dominates over filamentation. The patterns observed in a thermal defocusing medium are interpreted in the framework of our theory. Shock waves are a general phenomenon thoroughly in- vestigated in disparate area of physics (fluids and water waves, plasma physics, gas dynamics, sound propagation, physics of explosions, etc.), entailing the propagation of discontinuous solutions typical of hyperbolic PDE mod- els [1, 2]. They are also expected in (non-hyperbolic) universal models for dispersive nonlinear media, such as the Korteweg-De Vries (KdV) and nonlinear Schrödinger (NLS, or analogous Gross-Pitaevskii) equations, since hy- drodynamical approximations of such models hold true in certain regimes (typically, in the weakly dispersive or strongly nonlinear case) [3, 4, 5]. However, in the lat- ter cases, no true discontinuous solutions are permitted. The general scenario, first investigated by Gurevich and Pitaevskii [3], is that dispersion regularizes the shock, de- termining the onset of oscillations that appear near wave- breaking points and expand afterwards. This so-called collisionless shock has been observed for example in ion- acoustic waves [6], or wave-breaking of optical pulses in a normally dispersive fiber [7], and recently in a Bose- Einstein condensate with positive scattering length [8]. In this Letter we investigate how nonlocality of the nonlinear response affects the formation of a collisionless shock in a system ruled by a NLS model. In fact nonlocal- ity plays a key role in many physical systems due to trans- port phenomena and finite range interactions (e.g. as in Bose-Einstein condensation), and can be naively thought to smooth and eventually wipe out steep fronts character- istic of shocks. More specifically, we place this problem in the context of nonlinear optics where nonlocality arises quite naturally in different media [9, 10, 11, 12], study- ing the spatial propagation of a fundamental (gaussian TEM00) laser mode subject to diffraction and nonlocal focusing/defocusing action (Kerr effect). In a defocus- ing and ideal (local and lossless) medium, high intensity portions of the beam diffract more rapidly than the tails leading, at sufficiently high powers, to overtaking and os- cillatory wave-breaking similar (in 1D) to what observed in the temporal case [18]. We find that, while shock is not hampered by the presence of (even strong) nonlocal- ity, the mechanism of its formation as well as post-shock patterns are qualitatively affected by the nonlocality. Ex- perimental results obtained with a thermal defocusing nonlinearity are consistent with our theory and shed new light on the interpretation of the thermal lensing phe- nomenon. Importantly, our theory permits also to establish that nonlocality allows the shock to form also in the focusing regime where, contrary to the local case, it can prevails over filamentation or modulational instability (MI). Theory We start from the paraxial wave equation obeyed by the envelope A of a monochromatic field E = ( 2 )1/2A exp(ikZ − iωT ) (|A|2 is the intensity) + k0∆nA = −i A. (1) where k = k0n = n is the wave-number, and α0 the intensity loss rate. A sufficiently general nonlocal model can be obtained by coupling Eq. (1) to an equa- tion that rules the refractive index change ∆n of nonlin- ear origin. Introducing the scaled coordinates x, y, z = X/w0, Y/w0, Z/L, and complex variables ψ = A/ and θ = k0Lnl∆n, where Lnl = (k0|n2|I0)−1 is the non- linear length scale associated with peak intensity I0 and a local Kerr coefficient n2 (∆n = n2|A|2), Ld = kw20 is the characteristic diffraction length associated with the input spot-size w0, and L ≡ LnlLd, such model can be conveniently written as follows [12] ψ + χθψ = −iα εψ, (2) −σ2∇2 θ + θ = |ψ|2, (3) where α = α0L, ∇2⊥ = ∂2x + ∂2y , χ = n2/|n2| = ±1 is the sign of the nonlinearity, and σ2 is a free parameter that measures the degree of nonlocality. The peculiar dimensionless form of Eqs. (2-3) where ε ≡ Lnl/L = Lnl/Ld is a small quantity, highlights the fact that we will deal with the weakly diffracting (or strongly non- linear) regime, such that the local σ = 0 and lossless α = 0 limit yields a semiclassical Schrödinger equation with cubic potential (ε and z replace Planck constant http://arxiv.org/abs/0704.0610v1 and time, respectively). We study Eqs. (2-3) subject to the axi-symmetric gaussian input ψ0(r) = exp(−r2), x2 + y2, describing a fundamental laser mode at its waist. For ε ≪ 1, its evolution can be studied in the framework of the WKB trasformation ψ(r, z) = ρ(r, z) exp [iφ(r, z)/ ε] [4]. Substituting in Eqs. (2-3) and retaining only leading orders in ε, we obtain (D − 1) ρu+ (ρu)r = −αρ; uz + uur − χθr = 0, θrr + D − 1 + θ = ρ. (4) where u ≡ φr is the phase chirp, and D = 2 is the trans- verse dimensionality. The 1D case described by Eqs. (4) with D = 1 and r → x (∂y = 0) illustrates the ba- sic physics with least complexity. In the defocusing case (χ = −1) for an ideal medium (σ = α = 0, θ = ρ), Eqs. (4) are a well known hyperbolic system of conser- vation laws (Eulero and continuity equations) with real celerities (or eigenspeeds, i.e. velocities of Riemann in- variants) v± = u ± −χρ, which rules gas dynamics (u and ρ are velocity and mass density of a gas with pres- sure ∝ ρ2). A gaussian input is known to develop two symmetric shocks at finite z [4]. Importantly the diffrac- tion, which is initially of order ε2, starts to play a major role in the proximity of the overtaking point, and regu- larize the wave-breaking through the appearance of fast (wavelength ∼ ε) oscillations which connect the high and low sides of the front and expand outwards (far from the beam center) [3]. Such oscillations, characteristic of a col- lisionless shock, appear simultaneously in intensity and phase chirp (u) as clearly shown in Fig. 1(a,c). In the nonlocal case, the index change θ(x) can be wider than the gaussian mode (for large σ) and the shock dynamics is essentially driven by the chirp u with ρ adia- batically following. This can be seen by means of the fol- lowing approximate solution of Eqs. (4): considering that the equation for ρ is of lesser order [O(ǫ)], with respect to those for θ and u [O(1)], we assume ρ = exp(−2x2) unchanged in z and solve exactly the third of Eq. (4) for θ(x) (though derived easily, its full expression is quite cumbersome). Then, applying the theory of characteris- tics [1], the second of Eqs. (4) is reduced to the following ODEs, where dot stands for d/dz ẋ = u ; u̇ = χθx, (5) equivalent to the motion of a unit mass in the potential V (x) = −χθ with conserved energy E = u(z) + V (x). The solution of Eqs. (5) with initial condition x(0) = s, u(0) = 0 yields x(s, z), u(s, z) in parametric form, from which overtaking is found whenever u(x, z) (obtained by eliminating s) becomes a multivalued function of x at finite z = zs. The shock point corresponding to |du/dx| → ∞ is found from the solution u(x, z) displayed in Fig. 2(a) [ 2(b)], at positions x = ±xs 6= 0 (defocusing FIG. 1: (Color online) 1D spatial profiles of phase chirp u(x) (a-b) and amplitude |ψ(x)| = ρ(x) (c-d), as obtained from Eqs. (2-3) with ε = 10−3, α = 0, χ = −1 (defocusing), and ψ0 = exp(−x 2), for different z as indicated: (a-c) local case, σ2 = 0; (b-d) nonlocal case, σ2 = 5. FIG. 2: (Color online) (a) u(x) for different z and χ = 1 (focusing), σ = 1; (b) as in (a) for χ = −1 (defocusing); (c) shock distance zs (χ = −1 bold solid, χ = 1 thin solid) and shock position xs in the defocusing case (dashed line). case) or xs = 0 (focusing case). The shock distance zs increases with σ in both cases, as shown in Fig. 2(c). We have tested these predictions by integrating nu- merically Eqs. (2-3). Simulations with χ = −1 [see Fig. 1(b,d)] show indeed steepening and post-shock oscilla- tions in the spatial chirp u, which are accompanied by a steep front in ρ moving outward. The shock location in x and z is in good agreement with the results of our approximate analysis summarized in Fig. 2. Numerical simulations of Eqs. (2-3) validates also the focusing scenario. The field evolution displayed in Fig. 3(a) exhibits shock formation at the focus point (xs = 0, zs ≃ 8, for σ = 5) driven the phase whose chirp is shown in Fig. 3(b). This is remarkable because, in the local limit σ = 0, the celerities become imaginary (the equivalent gas would have pressure decreasing with increasing density ρ), and no shock could be claimed to exist. In this limit, the reduced problem (4) is elliptic and the initial value problem is ill-posed [13], an ultimate con- sequence of the onset of MI: modes with transverse (nor- FIG. 3: (Color online) Level plot of the intensity in the focusing case (χ = 1, ε = 0.01): (a) nonlocal case (σ2 = 25); (b) chirp profile for various z for (a); (c) quasi-local case (σ2 = 10−5). malized) wavenumber q < ∆q grow exponentially with z, with both gain g and bandwidth ∆q scaling as 1/ε. How- ever, the nonlocal response tends to frustrate MI (see also Refs. [9, 12]), as shown by standard linear stability anal- ysis which yields g = d(2χ− d)/ε2 (we set d ≡ ε2q2/2 and χ ≡ χ/(1 + σ2q2)), in turn implying a strong reduc- tion of both gain and bandwidth for large σ. In order to emphasize the difference between the local and non- local regime, we contrast Fig. 3(a) with the analogous evolution [see Fig. 3(c)] obtained in the quasi-local limit (σ2 = 10−5), which appears to be clearly dominated by filamentation. Thermal nonlinearity The physics of the defocusing case can be experimentally tested by exploiting ther- mal nonlinearities of strongly absorptive bulk samples, that we show below to fit our model. In this case, the system relaxes to a steady-state refractive index change ∆n = (dn/dT )∆T , where dn/dT is the thermal coef- ficient, and ∆T the local temperature change due to optical absorption. It is well known that this so-called thermal lens distorts a laser beam propagating in the medium [14, 15, 16]. However, only perturbative ap- proaches to the problem have been proposed (ray optics or Fresnel diffraction theory is applied after the lens pro- file is worked out from gaussian ansatz [14]), while the role of shock phenomena was completely overlooked. FIG. 4: 2D evolution according to Eqs. (2-3) with σ2 = 1, α = 1: (a) radial phase chirp at different z, as indicated, showing steepening and shock formation for ε = 10−2; (b) cor- responding intensity profile |ψ|2 (maximum scaled to unity) at z = 4.9; (c) transverse intensity profile vs. x (at y = 0) at z = 1/(4ε) and different values of ε (α0 = 62cm −1, σ = 0.3). We assume that the temperature field ∆T = ∆T⊥(X,Y ) obeys the following 2D heat equation (∂2X + ∂ Y )∆T⊥ − C∆T⊥ = −γ|A|2 (6) where the source term account for absorption pro- portional to intensity through the coefficient γ = α0/(ρ0cpDT ), where ρ0 the material density, cp the spe- cific heat at constant pressure, and DT is the thermal diffusivity (see e.g. [16]). Eq. (6) has been already em- ployed to model a refractive index of thermal origin in Ref. [10], and in Ref. [11] in the limit C = 0 which is equivalent to consider the range of nonlocality (mea- sured by 1/C, see below) to be infinite. Starting from the 3D heat equation ∇2∆T = −γ|A|2, the latter regime amounts to assume ∆T (X,Y, Z) = ∆T⊥(X,Y ), which is justified when longitudinal changes in intensity |A|2 are negligible as for solitary (invariant in Z) wave-packets in the presence of low absorption [11]. Viceversa, in the regime of strong absorption, we need to account for lon- gitudinal temperature profiles that are known from solu- tions of the 3D heat equations to be peaked at charac- teristic distance Ẑ in the middle of sample and decay to room temperature on the facets [14]. Since highly non- linear phenomena occurs in the neighborhood of Ẑ where the index change is maximum, we can use a (longitudi- nal) parabolic approximation with characteristic width Leff (∼ L) of the 3D temperature field ∆T (X,Y, Z) = 1− (Z−Ẑ) ∆T⊥(X,Y ) and consequently approximate ∇2∆T ≃ (∂2X + ∂2Y )∆T⊥ − L eff∆T⊥, so that the 3D heat equation reduces to Eq. (6) with C = 1/L2eff . Following this approach, Eq. (6) coupled to Eq. (1) can be casted in the form of Eqs. (2-3) by posing θ = k0Lnl|dn/dT |∆T⊥ and σ2 = 1/(Cw20) = L2eff/w20. The model reproduces the infinite range nonlocality for negligible losses (Leff → ∞); while for thin samples [|(∂2X + ∂2Y )∆T⊥| << |L eff∆T⊥|], Leff can be related to the Kerr coefficient n2 as Leff = γ|dn/dT | DTρ0cp|n2| α0|dn/dT | which establishes a link between the degree of nonlocal- ity and the strength of the nonlinear response (similarly to other nonlocal materials [12]). Having retrieved the model Eqs. (2-3), let us show next that the scenario illustrated previously applies sub- stantially unchanged in bulk (2D case) even on account for the optical power loss (α 6= 0). An example of the general dynamics is shown in Fig. 4, where we report a simulation of the full model (2-3), with σ2 = 1 and rela- tively large loss α = 1. In analogy to the 1D case, Fig. 4(a) clearly shows that the radial chirp u = φr steep- ens and then develop characteristic oscillations after the shock point (z ≃ 6, where |∂ru| → ∞). Correspond- ingly the intensity exhibits also an external front which is connected to a flat central region with a characteris- tic overshoot [see Fig. 4(b)] corresponding to a brighter ring [inset in Fig. 4(c)]. For larger distances this struc- ture moves outward following the motion of the shock. In the experiment such motion can be observed, at fixed physical lenght, by increasing the power, which amounts to decrease ε while scaling z and α accordingly (z ∝ 1/ε, α ∝ ε), as displayed in Fig. 4(c) for σ = 0.3. As a sample of a strongly absorbing medium we choose a 1 mm long cell filled with an acqueous solution of Rhodamine B (0.6 mM concentration). Our measure- ments of the linear and nonlinear properties of the sam- ple performed by means of the Z-scan technique gives data consistent with the literature [17], and allows us to extrapolate at the operating vacuum wavelength of 532 nm, a linear index n = 1.3, a defocusing nonlin- ear index n2 = 7 × 10−7 cm2W−1, and α0 = 62 cm−1. For our sample DT = 1.5 × 10−7 m2s−1, ρ0 = 103 kg m−3, cp = 4 × 103Jkg−1K−1 and |dn/dT | = 10−4 K−1 (γ ∼= 104 K W−1), and exploiting Eq. (7) we estimate Leff ∼= 10µm (Leff << L because of the strong ab- sorption that causes strong heating of our sample near the input facet), and correspondingly the degree of non- locality σ ∼= 0.3. We operate with an input gaussian beam with fixed intensity waist w0I = w0/ 2 = 20 µm (Ld ∼= 12 mm) focused onto the input face of the cell. With these numbers, an input power P = πw20II0 = 200 mW yields a nonlinear length Lnl ∼= 8 µm (L ∼= 0.3 mm), which allows us to work in the semiclassical regime with ε ∼= 0.025. The radial intensity profiles together with the 2D patterns imaged by means of a 40×microscope objec- tive and a recording CCD camera are reported in Fig. 5. As shown the beam exhibits the formation of the bright ring whose external front moves outward with increasing power, consistently with the reported simulations. We point out that, at higher powers, we observe (both ex- perimentally and numerically) that the moving intensity front leaves behind damped oscillations that correspond to inner rings of lesser brightness, as reported in litera- ture [15]. This, however, occurs well beyond the shock point that we have characterized so far. In summary, the evolution of a gaussian beam in the strong nonlinear regime is characterized by occurence of collisionless (i.e., regularized by diffraction) shocks that survive the smoothing effect of (even strong) nonlocality. While experimental results support the theoretical sce- nario in the defocusing case, the remarkable result that the nonlocality favours shock dynamics over filamenta- tion requires future investigation. ∗ Electronic address: [email protected] [1] G. B. Whitman, Linear and Nonlinear Waves (Wiley, New York, 1974); [2] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Perg- amon, 1995); M. A. Liberman and A. L. Yelikovich, Physics of Shock Waves in Gases and Plasmas (Springer, FIG. 5: Radial profiles of intensity observed in the thermal medium for different input powers. The insets show the cor- responding 2D output patterns. Heidelberg, 1986). [3] A.V. Gurevich and L.P. Pitaevskii, Sov. Phys. JETP 38, 291 (1973); A.V. Gurevich and A. L. Krylov, Sov. Phys. JETP 65, 944 (1987). [4] J. C. Bronski and D. W. McLaughlin, in Singular Limits of Dispersive Waves, NATO ASI Series, Ser. B 320, pp. 21-28 (1994); M. G. Forest and K. T. R. McLaughlin, J. Nonlinear Science 7, 43 (1998); Y. Kodama, SIAM J. Appl. Math. 59, 2162 (1999). M. G. Forest, J. N. Kutz, and K. T. R. McLaughlin, J. Opt. Soc. Am. B 16, 1856 (1999). [5] A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002). [6] R. J. Taylor, D.R. Baker, and H. Ikezi, Phys. Rev. Lett. 24, 206 (1970). [7] J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989). [8] M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006). V. M. Perez-Garcia, V.V. Konotop, V.A. Brazhnyi, Phys. Rev. Lett. 92, 220403 (2004); B. Damski, Phys. Rev. A 69, 043610 (2004); [9] J. Wyller, W. Krolikowski, O. Bang, J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002). [10] A. Yakimenko, Y. Zaliznyak and Y.S. Kivshar, Phys. Rev. E 71, 065603(R) (2005). [11] C. Rotschild, O. Cohen, O. Manela, M. Segev and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005). [12] C. Conti, M. Peccianti and G. Assanto, Phys. Rev. Lett. 91 073901 (2003); Phys. Rev. Lett. 92 113902 (2004); C. Conti, G. Ruocco and S. Trillo, Phys. Rev. Lett. 95 183902 (2005). [13] P. D. Miller and S. Kamvissis, Phys. Lett. A 247, 75 (1998); J. C. Bronski, Physica D 152, 163 (2001). [14] C. A. Carter and J. M. Harris, Appl. Opt. 23, 476 (1984); S. Wu and N. J. Dovichi, J. Appl. Phys. 67, 1170 (1990); F. Jürgensen and W. Schröer, Appl. Opt. 34 41 (1995). [15] C. J. Wetterer, L. P. Schelonka, and M. A. Kramer, Opt. Lett. 14, 874 (1989). mailto:[email protected] [16] P. Brochard, V. Grolier-Mazza and R. Cabanel, J. Opt. Soc. Am. B 14, 405 (1997). [17] S. Sinha, A. Ray, and K. Dasgupta, J. Appl. Phys. 87, 3222 (2000). [18] paraxial diffraction in defocusing media is well known to be isomorphus in 1D to propagation in a normally dispersive focusing medium as considered in Ref. [7]

0704.0611

Modeling the field of laser welding melt pool by RBFNN

Modeling the field of laser welding melt pool by RBFNN Anamarija Borštnik Bračič, Edvard Govekar, Igor Grabec Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, POB 394, SI-1001 Ljubljana, Slovenia Email: [email protected] Abstract— Efficient control of a laser welding process requires the reliable prediction of process behavior. A statistical method of field modeling, based on normalized RBFNN, can be successfully used to predict the spatiotemporal dynamics of surface optical activity in the laser welding process. In this article we demon- strate how to optimize RBFNN to maximize prediction quality. Special attention is paid to the structure of sample vectors, which represent the bridge between the field distributions in the past and future. I. INTRODUCTION Laser systems are efficiently applied in welding pro- cesses [1], where a laser beam is used to melt material. To maintain high performance in a welding process, efficient control should be established. The crucial task in planning the control system is to deter- mine representative variables which can effectively describe the welding process. For this purpose, the intensity and spatial distribution of reflected light, surface temperature values or properties of the emitted electron plasma are usually chosen. However, characteristic dynamic properties in space and time can also be obtained by recording surface optical activity in the heated zone, known as the melt pool [2]. After choosing the representative variables, extraction of evolution laws from temporal data becomes a crucial problem. To date, this problem has been extensively studied in relation to chaotic time series prediction [3], [4], [5], [6], [7]. The basis of these methods is to reconstruct a state-space from a recorded scalar time series by using an embedded technique, and then to estimate deterministic dynamic evolution from the reconstructed trajectory using statistical average estima- tors. We present a generalization of this approach, where the modeling of dynamic laws is extended from one dimen- sion (time) to multiple dimensions in spatiotemporal space. This generalization requires a new embedding method, which makes feasible a reconstruction of trajectory in the state-space from spatiotemporal data. The embedded technique, which was initially developed for time series analysis, can be simply generalized to spatially related data [8], [9], [10], [11], [12] and results in a good agreement between predicted and original chaotic fields over short time scales. Since, in a properly reconstructed state-space, the modeled dynamics must have similar statistical properties to the actual dynamics, we use a new state-space reconstruction method which also considers Manuscript received: January 31, 2007 statistical properties of a field structure. Such reconstruction results in an accurate short-term prediction as well as a sta- tistically proper long-term prediction of deterministic chaotic field evolution [13]. In this article, a statistical method of field generators, which is based on normalized radial basis function neural network (RBFNN), is used to model the spatiotemporal dynamics of laser welding melt pool images. The stochastic field evolution is modeled from sample state vectors reconstructed from recorded spatiotemporal data. The field evolution equation is estimated non-parametrically from the samples, using the conditional average estimator which determines the governing equation of RBFNN. The goal of this article is to find an optimal dimensionality of the neural network, i.e., to determine its optimal structure and an adequate number of sampling patterns, which will result in the best quality Q of field generator prediction. Accurate modeling of laser welding images, together with a criterion function specified by the operator of the laser system, provides the basis for optimal control of the laser welding process. II. DESCRIPTION OF RBFNN A. Non parametric statistical modeling Experimental analysis of process dynamics is based on a representative record of the field ϕ = ϕ(s), where the variable s represents space as well as time components s = s(r,t). Most commonly, the spatiotemporal field evolution of ϕ(s) is described analytically by a system of nonlinear partial differential equations or integrodifferential equations. An analytical form of the model can be estimated from the recorded data, based on spatial and temporal derivatives [15], [16], [17]. In the case of experimentally obtained data, it is difficult to estimate derivatives. Therefore, for a more general approach, a model of field evolution should be expressed in terms of recorded data only. In our model, field evolution is expressed in terms of data recorded at equally spaced discrete points in space and time. We assume that the dynamics of the field can be described in terms of the generator equation ϕ(s) = G (ϕ(s′ ∈ S(s)), σ) , (1) where ϕ(s′ ∈ S) represents the past distribution of the record, while ϕ(s) represents its future distribution. S represents http://arxiv.org/abs/0704.0611v1 Fig. 1. Illustration of point s and its surroundings s′ ∈ S . The future distribution of field ϕ (s) is located in plane t+1, while the surrounding points s′ ∈ S , which represent the past distribution of field, are located in planes t, t−1, t−2... the surroundings of point s. The field generator G provides for determination of the future field distribution from its past distribution. σ is a model parameter depending on the experimental setup and will be specified in greater detail later. An arbitrary point s and its surroundings s′ ∈ S are illustrated in Fig. 1. The source of information for modeling the field generator is a field record containing joint sample pairs ϕ(s) and ϕ(s′ ∈ S). These joint sample pairs form a sample vector Vi(s) = (ϕi(s), ϕi(s ′ ∈ S)). To make further derivation more transparent, the past field distribution ϕi(s ′) and the future field distribution ϕi(s) will be denoted by xi and yi, respectively. Hence Vi(s) = (yi,xi). The samples Vi are interpreted as random variables and can therefore be used to express the joint probability distribution function (PDF) by the kernel estimator [14] fN (V) = ψ(V −Vi, σ), (2) in which ψ denotes an acceptable kernel function such as the Gaussian function ψ(x − xi, σ) = 1/( 2πσ)exp(−(x − 2/2σ) and N is the number of sample pairs. Once the samples from the field record have been taken, the question of how to determine the optimal predictor becomes relevant. We consider as an optimal predictor of the future field distribution y from a given value x the value ŷ at which the mean square prediction error is minimal: E[(y − ŷ)2|x] = min(ŷ). (3) Here E[ ] denotes averaging over all points in a field record at a given time t. The solution of Eq. (3) yields together with PDF from Eq. (2) the conditional average estimator ŷ(x) = yiψ(x− xi, σ) ψ(x− xj , σ) yiCi(x), (4) where coefficients of the expansion Ci(x) represent basis functions that measure the similarity between the temporary vector x and vector xi from the field record. The conditional average estimator described by Eq. 4 represents a radial basis function neural network in which the recorded data xi,yi rep- resent the memorized contents of neurons, x and ŷ(x) are the input and the output of the network, while the basis functions Ci(x) correspond to activation functions of neurons. Since Ci(x) = 1, the conditional average estimator represents a normalized RBFNN. In this function, the parameter σ can be interpreted as the width of receptive fields of neurons. B. Quality of predictor Working towards optimal modeling of future field distribu- tions requires a quantitative estimation of modeling quality. We therefore introduce a testing field y and define the prediction quality Q, based upon the difference between the predicted field ŷ and the testing field y as: Q = 1− E[(ŷ − y) E[(ŷ − ˆ̄y)2] + E[(y − ȳ)2] . (5) Here ˆ̄y and ȳ stand for the average values of predicted field ŷ and testing field y, i.e., E[ŷ] = ˆ̄y and E[y] = ȳ. A perfect prediction ŷ = y yields Q = 1, while uncorrelated ŷ and y result in Q = 0. C. Prediction of field evolution The prediction process consists of three steps: 1) Learning, that corresponds to setting up the basis of joint sample pairs (ϕi(s), ϕi(s ′ ∈ S)) = (xi, yi) from the field record, 2) predicting the field ŷ by using the conditional average estimator from Eq. (4), 3) and, if the testing field exists, comparing predicted field with testing field and calculating prediction quality Q. In order to achieve the highest quality of prediction for the process, answers to the following crucial questions are needed: • How to find the surrounding S of a given point s, which gives the best prediction of field ϕ̂(s) at this point? • How to determine an optimal number of joint sample pairs (ϕi(s), ϕi(s ′ ∈ S)) = (xi, yi)? These questions will be addressed in the following chapters. III. TIME EVOLUTION OF MELT POOL Characteristic dynamic properties of laser welding process in space and time can be experimentally obtained by recording the surface optical activity of the melt pool. With respect to the energy supplied to the material, various dynamic regimes of the welding process can be distinguished. In Fig. 2 visual records of two different welding regimes are shown, a deep welding regime (a) and a heat conduction welding regime (b). In the following discussion, only the deep welding regime is considered. Dynamics of the welding regime are here represented by a record of 1000 images of size 32 × 32 points in space with sampling time 1/220 s. This experimental record forms a three-dimensional field of light intensity ϕ(s = r, t) in two- dimensional space {(rx,i, ry,i); i = 1, ..32, j = 1..32} and time {tk; k = 1..1000}. Due to local energy supply, the field is non-homogeneous in space. Consequently, we model its evolution locally at each spatial point separately. A model of field evolution, i.e., the learning sample is formed from the first 800 images. We then predict the time evolution of the field and compare it with the next 200 images, which represent 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 Fig. 2. Time series of laser welding records for two different welding regimes: (a) a deep welding regime , and (b) a heat conduction welding regime. The next time-step images denoted by ? are unknown and must be predicted. the testing sample. Based on the quality Q of these predicted images, we optimize our prediction procedure, i.e., and define the structure of the surroundings S, and the optimal number of joint sample pairs N and parameter σ. A. Optimal value of parameter σ Parameter σ in the conditional average estimator ŷ (Eq. (4)) was to this point left undetermined. However, as shown in Fig. 3, obtained for the deep welding regime, the quality of prediction depends on the value of σ. The learning sample consisted of 800 images and the surrounding of predicted field distribution in point s = (r, t) was taken to be just one neighboring point with the same space coordinate and the time coordinate being one step behind s′ = (r, t−1). Based on the learning sample, ten images from the testing interval have been predicted and compared with the corresponding images from the testing field. The value Q shown in Fig. 3 is the average quality of these ten images. As Fig. 3 shows, the quality exhibits a strong σ dependence at the beginning of the interval, and reaches its largest value for σ approximately equal to 4. For larger values, it becomes a weakly decreasing function of σ. Since the optimal quality is reached for σ ≈ 4, this value is used in our further calculations. As shown in Eq. (4), prediction of the field distribution at a given point is determined on the basis of similarity between the field distribution surrounding this point and the 5 10 15 20 25 30 Fig. 3. Dependence of prediction quality Q on the value of parameter σ from conditional average estimator ŷ. Surrounding set S is taken to be only one point with the same spatial position and a neighboring position in time. Number of learning images in the learning field is set to 800. field distribution in surrounding points taken from the learning field. If we keep in mind that the parameter σ defines the width of the Gaussian kernel function ψ (see comment to Eq. 2), we can conclude, that for very small σ, only those joint sample pairs from the learning field which have a field distribution in the surrounding points (xi) very similar to the field distribution in the surroundings of the point to be predicted (x) contribute to the predicted value of the field. On the other hand, for larger σ those joint sample pairs with larger difference x − xi also contribute to the prediction of ŷ. In the limit of large σ, almost all joint sample pairs contribute equally to ŷ. B. Optimal number of joint sample pairs If the prediction of welding pool images is to be part of a laser welding control system, the prediction operation has to be performed in the shortest time interval possible. Since the number of operations needed to predict a field distribution in a given point increases linearly with the number of joint sample pairs (see Eq. 4), it is necessary to find the smallest number of joint sample pairs which is still able to give predicted images of good quality. In Fig. 4 we show the dependence of the quality of predic- tion on the number of images N defining the learning field. As in the case of Fig. 3, the surrounding of the predicted field distribution in point s = (r, t) was taken to be just one neighboring point with the same space coordinate and the time coordinate being one step behind s′ = (r, t−1). Parameter σ is set to 4. Again Q is taken to be the average quality of ten predicted images, which were compared with the corresponding images from the testing field. As one can see from Fig. 4, Q increases rather strongly with small values of N (N < 400), while for N > 600 an increase in N does not result in a significant improvement of prediction quality. Therefore, in further calculations we apply N = 600. C. Choosing the surrounding S Our next goal is to find an optimal structure of RBFNN which yields the best quality of prediction in the shortest time 200 400 600 800 Fig. 4. Dependence of prediction quality Q on number of images defining the learning field. The surrounding set S is taken to be only one point with the same spatial position and a neighboring position in time. Parameter σ is set to 4. interval. The structure of surrounding set S plays an important role in this optimization process since each additional point in the surrounding increases the dimensionality of vectors xi and therefore the time needed to predict the field distribution in a given point. Our task is to find the smallest surrounding of point s, which results in high prediction quality. In Fig. 5 the prediction quality is presented for various selections of surrounding set S. Parameters N and σ are 600 and 4, respectively. As before, Q represents the average quality of ten predicted images which were compared with the corresponding images from the testing field. All the member points of the first six surrounding sets in the diagram lie in the plane t−1. Member points of other surrounding sets lie in several planes. For each of these, only those planes containing the member points are plotted.. As can be seen in Fig. 5, the smallest surrounding sets give the best quality of prediction - see sets Nr. 1-3 and 7-10. If more points belonging to the same time-plane are added to S, prediction quality is decreased- compare, for example sets Nr. 1 and Nr. 6 or Nr. 14 and Nr. 15. In contrast, surrounding sets containing points from two planes, t−1 and t−2, give a slightly better Q than sets containing only points from t−1 - compare for example sets Nr. 1 and Nr. 7. However, an addition of multiple time-planes reduces the quality (see set Nr. 12). As the best quality is obtained for set Nr. 7, this surrounding set is considered optimal in further calculations. We would like to stress, that in Fig. 5 only those surrounding sets which seemed to have the potential to give the best quality were taken into account. The optimal structure of S was chosen on the basis of selected sets. To be sure that the chosen structure was really optimal, it is necessary to calculate the prediction quality of all the subsets containing all combinations of neighboring points. Since the number of points in our learning set is 32×32×600, a calculation of Q for all sets would become a computationally prohibitive task. D. Optimal prediction of melt pool evolution After determining the optimal parameters of our RBFNN model, we next show the discrepancy between the predicted images of the laser welding melt pool and the corresponding 0 5 10 15 8 97 10 14 15 Fig. 5. Dependence of prediction quality Q (*) on the structure of the surrounding set S . Parameters are N = 600 and σ = 4. The netlike patterns (1-16) describe the position of surrounding points, while the netlike pattern denoted by P describes the position of prediction point s. In the netlike patterns, only those time planes which contain points from S are plotted. images from the testing field. In Fig. 6 we therefore present predicted laser welding images and corresponding images from the testing field for the optimal structure of RBFNN. Parameters are N = 600, and σ = 4, while the surrounding set S has only two member points, both having the same spatial position as the predicted point, but neighboring positions in time. Since the quality of prediction is 0.93 (see Fig. 5), a very good similarity between the predicted and corresponding image from the testing field is expected. Comparison of predicted images and images from the testing field in Fig. 6 indeed exhibits a good resemblance. However, we would like to draw attention to surface smoothness. As can be seen, the predicted surface is smoother than the original surface. This can be easily understood if the origin of prediction of images in the conditional average estimator (Eq. 4) is taken into account. Predicted ŷ is therefore a weighted average of all those yi, for which xi is similar to x. Consequently, the surface roughness is diminished due to conditional averaging. IV. CONCLUSION Time evolution of multi-dimensional fields is usually ob- tained by solving a system of partial differential equations. However, if the only source of information is a record of the field, a neural network can successfully replace differential equations by extracting field evolution properties from the recorded data. Neural-network-like structures are also expected to be the working algorithm of living organisms’ intelligence. In the same way as neural networks, living organisms predict the evolution of events in their surroundings solely on the basis a) predicted field b) testing field 51015202530 51015202530 51015202530 51015202530 Fig. 6. Comparison of predicted melt pool images (a) with corresponding images from the testing field (b) for two randomly chosen testing records. ϕ stands for field, rx and ry denote spatial coordinates of the record. Parameters are N = 600 and σ = 4. The surrounding set S has only two member points, both having the same spatial position (r) as the predicted point s=(r,t), but different neighboring positions in time, i.e., t−1 and t−2. of recorded data. It could be conjectured that this operation is probably performed by extracting simple evolution laws from recorded data. In this paper we show how to optimize a statistical modeling of a field generator performed by the normalized RBFNN, to efficiently learn spatiotemporal dynamics of multi-dimensional fields. In our experimental approach, all information about process dynamics is contained in a measured space-time record of the characteristic variable. To extract the model of field evolution from the corresponding discrete sample data, we employ a non-parametric approach, following a state-space re- construction technique. The basis of state-space reconstruction is the formation of sample vectors which are composed of past and future field distributions. We assume that the field distribution in a given spatiotemporal point s is correlated with the field distribution in the spatiotemporal surroundings of this point, S. The prediction of field distribution in s is then accomplished as a mapping relation between the field distribution in the surroundings S and field distribution in s. Since the optimization of the state-space reconstruction technique also requires a quantitative measure of the pre- diction quality, we introduce the quality estimator Q, which incorporates the difference between the predicted field and the corresponding testing field. We consider as a proper set of model parameters those values at which the prediction quality achieves a maximum. This strategy is used here to find a proper value of parameter σ and the structure of the surrounding S utilized in the prediction process. Generally, an estimation of the proper number of sample points must also consider the complexity of the experiments, which is numerically demanding in a multidimensional case [18]. Con- sequently, we also specify here the proper number N based upon the analysis of prediction quality. We demonstrate the proposed method of modeling of the properties of the laser-heated melt pool. For this purpose, we employ non-parametric statistical modeling of field evolution on a spatiotemporal record of the melt pool of the laser welding process. The major part of the field record is used for learning, while the minor part of the record serves for testing. We show how to construct the set of joint sample pairs containing past and future values of field distributions and pay special attention to the structure of these sample pairs. We also present the optimal structure of sample vectors, which gives the highest resemblance between predicted images and images from the testing field and has a small number of member points in order to make the prediction algorithm work quickly. ACKNOWLEDGMENT This work was supported by the Ministry of Higher Educa- tion, Science and Technology of the Republic of Slovenia and EU-COST. REFERENCES [1] E. Govekar, J. Gradišek, I. Grabec, M. Geisel, A. Otto, M. Geiger, ”On characterization of CO2 laser welding process by means of light emitted by plasma and images weld pool”, in Third Int. Symp: Investigation of Non-linear Dynamics Effects in Production Systems (Cottbus, Germany), 2000. [2] E. Govekar, J. Gradišek, I. Grabec, M. Geisel, A. Otto, M. Geiger, ”Influence of feed rate on dynamics of laser welding process” in Second Int. Symp: Investigation of Non-linear Dynamics Effects in Production Systems (Aachen, Germany), 1999. [3] M. Casdagli, S. Eubank, ”Nonlinear Modeling and Forcasting”, Santa Fe Institute: Addison-Wesley, 1992. [4] H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, L. S.Tsmiring, ”The analysis of observed chaotic data in physical systems”,Rev. Mod. Phys. Vol. 65, pp.1331-1392, 1993. [5] E. J. Kosterlich, T. Schreiber, ”Noise reduction in chaotic time-series data: A survey to common methods ”,Phys. Rev. E Vol. 48, pp.1752- 1763, 1993. [6] H. Kantz, T. Schreiber, ”Nonlinear Time Series Analysis”, Cambridge University Press, 1997. [7] S. Sigert, R. Friedrich, J. Peinke, ”Analysis of data sets of stochasitc systems”, Phys. Lett. A Vol. 243, pp.275-289, 1998. [8] D. M. Rubin, ”Use of forecasting signatures to hlep destinguish period- icity, randomness, and chaos in ripples and other spatial patterns”, Chaos Vol. 2, pp.525-535, 1992. [9] I. Grabec, S. Mandelj, ”Continuation of chaotic fields by RBFNN”, in Bilological and Artificial Computation: From Neuroscience to Technol- ogy: Proc., eds. J. Mira, R. Moreno-Diaz, J. Cebestany, Lecture Notes in Computer Science (Springer-Verlag, Berlin), Vol. 1240, pp.597-606, 1997. [10] S. Ørstavik, J. Stark, ”Reconstruction and cross-prediction in coupled map lattices using spatiotemporal embedding techniques”, Phys. Lett. A Vol. 247, pp.145-160, 1998. [11] U. Parlitz, C. Merkwirth, ”Prediction of spatiotemporal time series based on reconstructed local states”, Phys. Rev. Lett. Vol. 84, pp.1890-1893, 2000. [12] S. Mandelj, I. Grabec and E. Govekar, ”Statistical modeling of stochastic surface profiles”, CIRP-J. Manuf. Syst, Vol. 30, pp 281-287, 2000. [13] S. Mandelj, I. Grabec and E. Govekar, ”Nonparametric statistical model- ing of spatiotemporal dynamics based on recorded data”, Int. Jour. Bifur. Chaos, Vol. 14, No. 6, pp 2011-2025, 2004. [14] R. O. Duda, P. E. Hart, ”Pattern Clasification and Scene analysis”, J. Wiley and Sons, New York, Chap. 4. 1997. [15] H. U. Voss, M. J. Bünner, M. Abel, ”Identification of continuous, spatiotemporal systems”, Phys. Rev. E Vol. 57, pp.2820-2823, 1998. [16] M. Bär, R. Hegger, H.Kantz, ”Fitting partial differential equations to space-time dynamics”, Phys. Rev. E Vol. 59, pp.337-342, 1999. [17] H. U. Voss, P. Kolodner, M. Abel, J. Kurths, ”Amplitude equations from spatiotemporal binary-fluid convection data”, Phys. Rev. Lett Vol. 83, pp.3422-3425, 1999. [18] I. Grabec, ”Extraction of Physical Laws from Joint Experimental Data,” Eur. Phys. J. B, vol. 48, pp. 279-289, 2005. Introduction Description of RBFNN Non parametric statistical modeling Quality of predictor Prediction of field evolution Time evolution of melt pool Optimal value of parameter Optimal number of joint sample pairs Choosing the surrounding S Optimal prediction of melt pool evolution Conclusion References

0704.0612

Solving The High Energy Evolution Equation Including Running Coupling Corrections

Solving The High Energy Evolution Equation Including Running Coupling Corrections Javier L. Albacete∗ and Yuri V. Kovchegov† Department of Physics, The Ohio State University Columbus, OH 43210,USA April 2007 Abstract We study the solution of the nonlinear BK evolution equation with the recently calcu- lated running coupling corrections [1, 2]. Performing a numerical solution we confirm the earlier result of [3] (obtained by exploring several possible scales for the running coupling) that the high energy evolution with the running coupling leads to a universal scaling behavior for the dipole-nucleus scattering amplitude, which is independent of the initial conditions. It is important to stress that the running coupling corrections calculated recently significantly change the shape of the scaling function as compared to the fixed coupling case, in particular leading to a considerable increase in the anomalous dimension and to a slow-down of the evolution with rapidity. We then concentrate on elucidating the differences between the two recent calculations of the running coupling corrections. We explain that the difference is due to an extra contribution to the evolution kernel, referred to as the subtraction term, which arises when running coupling corrections are included. These subtraction terms were neglected in both recent calculations. We evalu- ate numerically the subtraction terms for both calculations, and demonstrate that when the subtraction terms are added back to the evolution kernels obtained in the two works the resulting dipole amplitudes agree with each other! We then use the complete running coupling kernel including the subtraction term to find the numerical solution of the re- sulting full non-linear evolution equation with the running coupling corrections. Again the scaling regime is recovered at very large rapidity with the scaling function unaltered by the subtraction term. ∗e-mail: [email protected] †e-mail: [email protected] http://arxiv.org/abs/0704.0612v2 1 Introduction Recently our understanding of the linear Balitsky–Fadin–Kuraev–Lipatov (BFKL) [4, 5] and non-linear Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) [6–13] and Balitsky-Kovchegov (BK) [14–18] small-x evolution equations in the Color Glass Condensate [6–29] has been improved due to the completion of the calculations determining the scale of the running coupling in the evolution kernel in [1,2,30,31]. The calculations in [1,2] proceeded by including αsNf corrections into the evolution kernel and by then completing Nf to the complete one-loop QCD beta-function by replacing Nf → −6πβ2. Calculation of the αsNf corrections is particularly easy in the s-channel light-cone perturbation theory formalism [32, 33] used to derive the BK and JIMWLK equations: there αs Nf corrections are solely due to chains of quark bubbles placed onto the s-channel gluon lines. The analytical results of [1,2] are not very concise and could not have been guessed without an explicit calculation. After finding αsNf corrections, the obtained contributions had to be divided into the running coupling part, which has a form of a running coupling correction to the leading-order (LO) JIMWLK or BK kernel, and into the ”subtraction piece”, which would bring in new structures into the kernel. Such separation had to be done both in [1] and in [2]. Unfortunately, there appears to be no unique way to perform this separation: it is not surprising, therefore, that it was done differently in both papers [1, 2]. This resulted in two different running coupling terms, shown below in Eqs. (35) and (36) along with Eqs. (7) and (8). Such a discrepancy has led to a misconception in the community that the calculations of [1] and of [2] disagree at some fundamental level. Indeed to compare the results of [1] and [2] one has to undo the separation into the running coupling and subtraction terms: combining both terms one should compare full kernels of the evolution equation obtained in [1, 2]. There is another more physical reason to perform such comparison: in principle, there is no small parameter making the subtraction term smaller than the running coupling term and thus justifying neglecting the former compared to the latter. Even the labeling of one term as “running coupling” piece is somewhat misleading, since it may give an impression that the neglected subtraction term has no running coupling corrections in it. As was shown in [31] both terms actually contribute to the running coupling corrections to the BFKL equation (if one uses the separation of [2] to define the terms). In this paper we perform numerical analysis of the BK evolution equation with the αsNf corrections resummed to all orders and with Nf completed to the QCD beta-function, Nf → −6πβ2, with β2 given in Eq. (20). We first solve the BK equation keeping the running coupling term only, with the kernels given by Eqs. (7) and (8). Indeed the solutions we find this way are different from each other. We then evaluate the subtraction terms for both cases and show that inclusion of subtraction terms puts the results of [1] and [2] in perfect agreement with each other! We complete our analysis by solving the BK equation with the full kernel including both the running coupling and subtraction terms. This work is structured as follows. Section 2 begins with Sect. 2.1 in which we review the αsNf corrections to the dipole scattering amplitude evolution equation recently derived in [1, 2] and the subtraction method employed in both works to separate the running coupling contributions from the subtraction terms. We discuss the scheme dependence of the running coupling terms introduced by this separation. We proceed in Sect. 2.2 by deriving the explicit expressions for the subtracted terms. The calculation is based on the results of [2]. Our analytical results are summarized in Sect. 2.3, where we give the explicit final expression for the kernel of the subtraction term in Eq. (39), which, combined with Eq. (38) gives us the subtraction terms (40) and (41) for the subtractions performed in [1] and in [2] correspondingly. In Sect. 3 we explain the numerical method we use to solve the evolution equations. We also list the initial conditions used, along with the definition of the saturation scale employed. Throughout the paper we will avoid the important question of the Landau pole and the contri- bution of renormalons to small-x evolution. As we explain in Sect. 3, we will simply “freeze” the running coupling at a constant value in the infrared. For a detailed study of the renormalon effects in the non-linear evolution we refer the readers to [30]. Our numerical results are presented in Sect. 4. By solving the evolution equations with the running coupling term only in Sect. 4.1 we show that the resulting dipole amplitude differs significantly from the fixed coupling case. We also observe that the amplitude obtained by solving the equation obtained in [2] is very close to the result of solving the BK evolution with a postulated parent-dipole running of the coupling constant. Both these amplitudes are quite different from the solution of the equation derived in [1], as one can see from Fig. 4. In spite of that, all three evolution equations studied (the ones derived in [1], [2] and the parent-dipole running coupling model) give approximately identical scaling function for the dipole amplitude at high rapidity, as demonstrated in Fig. 5 in Sect. 4.2. It is worth noting that, as can be seen from Fig. 6, the anomalous dimension we extracted from our solution is γ ≈ 0.85, which is different from the fixed coupling anomalous dimension of γ ≈ 0.64. The former anomalous dimension also appears to disagree with the predictions of analytical approximations to the behavior of the dipole amplitude with running coupling from [34–38]. In Sect. 4.3 we numerically evaluate the subtraction terms for both [1] and [2] and show that their contributions are important, as shown in Fig. 7. However, subtraction terms decrease with increasing rapidity, such that at high enough rapidities their relative contribution becomes small (see Fig. 8). In Fig. 9 we show that inclusion of subtraction terms makes the results of [1] and [2] agree with each other. Finally, the numerical solution of the full (all orders in αs β2) evolution equation including both the running coupling and subtraction terms is performed in Sect. 4.4. The results are shown in Fig. 10. All the main features of the evolution with the running coupling are preserved in the full solution: the growth of the dipole amplitude and of the saturation scale with rapidity is slowed down (for the latter see Fig. 11). The scaling function of Fig. 5 is unaltered by the subtraction term, as shown in Fig. 12. We summarize and discuss our main conclusions in Sect. 5. 2 Scheme dependence 2.1 Inclusion of running coupling corrections: general concepts The BK evolution equation for the dipole scattering matrix reads ∂S(x0, x1; Y ) d2z K(x0, x1, z) [S(x0, z; Y )S(z, x1; Y )− S(x0, x1; Y )] , (1) where K(x0, x1, z) = r21 r is the kernel of the evolution. Here transverse two-dimensional vectors x0 and x1 denote the transverse coordinates of the quark and the anti-quark in the parent dipole, while z is the position of the gluon produced in one step of evolution [39–42]. We have introduced the notation r = x0 − x1, r1 = x0 − z, r2 = z − x1 for the sizes of the parent and of the new (daughter) dipoles created by one step of the evolution. The notation r ≡ |r| for all the 2- dimensional vectors will be also employed throughout the rest of the paper. Eq. (1) admits a clear physical interpretation: the original parent dipole, when boosted to higher rapidities, may emit a new gluon which, in the large-Nc limit, is equivalent to a quark-antiquark pair. Thus, the original dipole splits into two new dipoles sharing a common transverse coordinate: the transverse position of the emitted gluon, z. The nonlinear term in the right hand side of Eq. (1) accounts for either one of the two new dipoles interacting with the target, along with the possibilities of only one dipole interacting or no interaction at all, while the subtracted linear term reflects virtual corrections. The kernel of the evolution is just the probability of one gluon emission calculated at leading logarithmic accuracy in αs ln(1/xB), where xB is the fraction of momentum carried by the emitted gluon [39–42]. Under the eikonal approximation the dipole scattering matrix off a hadronic target at a fixed rapidity is given by the average over the hadron field configurations of Wilson lines V calculated along fixed transverse coordinates (those of the quark and of the antiquark). More specifically S(x0, x1; Y ) = V (x0)V †(x1) 〉 . (3) Hence, the integrand of Eq. (1) can be regarded as a three point function in the sense that the gluon fields of the target are evaluated at three different transverse positions, those of the original quark and antiquark plus the one of the emitted gluon. However, the inclusion of higher order corrections to the evolution equation via all order resummation of αsNf contributions as recently derived in [1, 2] brings in new physical chan- nels that modify the three point structure of the leading-log equation. The dipole structure generated under evolution by diagrams like the one depicted in Fig. 1A (for more detailed dis- cussion of the diagrammatic content of the high order corrections see [2]) is identical to the one previously discussed for the leading order equation, the only novelty being that the propagator of the emitted gluon is now dressed with quark loops, modifying the emission probability but leaving untouched the interaction terms. On the contrary, diagrams like the one in Fig. 1B in which a quark-antiquark pair (rather than a gluon) is added to the evolved wave function modify the interaction structure of the evolution equation. The evolution of the parent dipole scattering matrix driven by these kind of terms is proportional to the scattering matrix of the two newly created dipoles (the one formed by the original quark and the new antiquark and vice versa), ∼ S(x0, z1)S(z2, x1). This term depends on four different transverse coordinates, i.e., it is a four point function and, therefore, its contribution to the evolution equation cannot be accounted for by a mere modification of the emission kernel of the leading order equation. To discuss in more detail the modifications introduced by the high order corrections, we find it useful to rewrite the evolution equation in the following, rather general way ∂S(x0, x1; Y ) = F [S(x0, x1; Y )] (4) where F is a functional of the dipole scattering matrix which for the original derivation of the equation is given by the right hand side of Eq. (1). In general it can be decomposed into two Figure 1: Schematic representation of the diagrams contributing to quark-NLO evolution. pieces F [S] = R [S]− S [S] . (5) The first term, R, which we will call the ’running coupling’ contribution, gathers all the higher order in αsNf corrections to the evolution that can be recast in a functional form that looks identical to the leading order one but with a modified kernel, K̃, which includes all the terms setting the scale for the running coupling: R [S(x0, x1; Y )] = d2z K̃(x0, x1, z) [S(x0, z; Y )S(z, x1; Y )− S(x0, x1; Y )] . (6) The second term, S, henceforth referred to as the ’subtraction’ contribution, encodes those contributions that depart from the three point structure of the leading-log equation. The explicit derivation and expressions for this term are presented in the next section. The relative minus sign between the two terms in Eq. (5) has been introduced for latter convenience. Importantly, the decomposition of F into running coupling and subtraction contributions, although constrained by unitarity arguments, is not unique. Two different separation schemes have been proposed in [1, 2]. They are both based on a similar strategy, sketched in Fig. 2, that can be summarized as follows. The newly created quark-antiquark pair added to the wave function in the diagrams Fig. 1B is shrunk to a point, called the subtraction point, by integrating out one of the coordinates in the dipole-qq̄ wave function, rendering the previously discussed four point nature of these contributions into a three point one. This integrated three point contribution is added to the running coupling contribution, whereas the original four point term minus its integrated version are assigned to the subtraction contribution. The divergence between the two approaches stems from the choice of the subtraction point. In the subtraction scheme proposed by Balitsky in [1] the subtraction point is chosen to be the transverse coordinate of either the quark, z2, or the antiquark, z1. The kernel for the running coupling functional, Eq. (6), obtained in this way is K̃Bal(r, r1, r2) = Nc αs(r r21 r . (7) On the other hand, in the subtraction procedure followed in [2] (which we will refer to as KW) the zero size quark-antiquark pair is fixed at the transverse coordinate of the gluon, x 0 x 0 Figure 2: Schematic representation of the subtraction procedure. z = αz1 + (1 − α)z2, where α is the fraction of the gluon’s longitudinal momentum carried by the quark, yielding the following expression for the kernel of the running coupling contribution: K̃KW(r, r1, r2) = − 2 αs(r 1)αs(r αs(R2) r1 · r2 r21 r + αs(r , (8) where R2(r, r1, r2) = r1 r2 r1·r2 . (9) As we shall discuss later, the scheme dependence originated by the choice of the subtraction point is substantial and has an important effect in the solutions of the evolution equation when only the running contribution is taken into account. In our numerical study we will also consider the following ad hoc prescription for the kernel of the running coupling functional in which the scale for the running of the coupling is set to be the size of the parent dipole K̃pd(r, r1, r2) = Nc αs(r r21 r . (10) This prescription is useful as a benchmark used to compare with previous numerical [3] and analytical works [34, 35, 43] where this ansatz was used. 2.2 Derivation of the subtraction term We begin by considering the NLO contribution to the kernel of the JIMWLK and BK evolution equations with the s-channel gluon splitting into a quark-antiquark pair, which then interacts with the target, as shown on the left hand side of Fig. 3. The contribution of this diagram has been calculated in [2]. The resulting JIMWLK kernel is [2] KNLO1 (x0, x1; z1, z2) = 4Nf (2π)2 (2π)2 (2π)2 (2π)2 e−iq·(z−x0)+iq ′·(z−x )−i(k−k′)·z q2q′2 (1− 2α)2q · k k′ · q′ + q · q′ k · k′ − q · k′ k · q′ k2 + q2α(1− α) k′2 + q′2α(1− α) 2α (1− α) (1− 2α) k2 + q2α(1− α) k′2 + q′2α(1− α) k · q k′ · q′ 4α2 (1− α)2 k2 + q2α(1− α) k′2 + q′2α(1− α)  . (11) The momentum labels in the above equation are explained on the left hand side of Fig. 3. If k1 and k2 are the transverse momenta of the quark and of the antiquark in the produced pair as shown in Fig. 3, then the transverse momentum of the gluon is q = k1 + k2. The other transverse momentum we use is k = k1(1 − α) − k2α, where α is the fraction the of gluon’s “plus” momentum carried by the quark, α ≡ k1+/(k1+ + k2+). The prime over the transverse momentum denotes the momentum of the same particle in the complex conjugate amplitude. For instance q′ is the momentum of the s-channel gluon in the complex conjugate amplitude. Finally, z1 and z2 denote the transverse coordinates of the quark and the antiquark. In Eq. (11) we use z12 = z1−z2 (the transverse separation between the quark and the antiquark) and z = α z1 + (1− α) z2 (the transverse coordinate of the gluon). Figure 3: A lowest order leading-Nf NLO correction which gives rise to the subtraction term is shown on the left. The same diagram with the gluon lines “dressed” by chains of fermion bubbles, as shown on the right, gives the full (resumming all powers of αµNf ) contribution to the subtraction term. Calculation of the subtraction term is pictured in Fig. 2. To obtain the BK kernel from Eq. (11) one should sum over all possible emissions of the gluon off the quark and antiquark lines in the incoming dipole both in the amplitude and in the complex conjugate amplitude, which is accomplished by KNLO1 (x0, x1; z1, z2) = CF m,n=0 (−1)m+n KNLO1 (xm, xn; z1, z2). (12) Below we will label the JIMWLK kernel by calligraphic letter K and the corresponding BK kernel by K. The contribution of the kernel from Eq. (12) to the right hand side of the NLO version of Eq. (1) is given by the following term d2z1 d 1 (x0, x1; z1, z2)S(x0, z1, Y )S(z2, x1, Y ) (13) with αµ the bare coupling. As shown in Fig. 2, at the NLO level, the subtraction term introduced in Eq. (5), is then defined by SNLO[S] = α2µ d2z1 d 2z2 K 1 (x0, x1; z1, z2) × [S(x0, w, Y )S(w, x1, Y )− S(x0, z1, Y )S(z2, x1, Y )], (14) where w is the point of subtraction in the transverse coordinate space. In [1] it was chosen to be equal to the transverse coordinate of either the quark or the antiquark, w = z1 or w = z2, (15) as both choices lead to the same subtraction term SBalNLO[S]: SBalNLO[S] = d2z1 d 2z2 K 1 (x0, x1; z1, z2) × [S(x0, z1, Y )S(z1, x1, Y )− S(x0, z1, Y )S(z2, x1, Y )] . (16) In [2] the subtraction point was chosen to be the transverse coordinate of the gluon z, w = z = α z1 + (1− α) z2. (17) This leads to the following subtraction term, which we denote SKWNLO[S]: SKWNLO[S] = d2z1 d 1 (x0, x1; z1, z2) × [S(x0, z, Y )S(z, x1, Y )− S(x0, z1, Y )S(z2, x1, Y )] . (18) Indeed the complete kernel in Eq. (5) is independent of the choice of w. However, since the subtraction term of Eq. (14) was neglected both in [1] and in [2], different choices of w led to different expressions for the remaining running coupling part R[S], i.e., to different answers as far as investigations in [1] and in [2] were concerned. Different choice of w is the main source of the discrepancy of final answers of [1] and [2], though it does not imply any disagreement in the full expression (5). Our goal in this Section is to evaluate KNLO1 (x0, x1; z1, z2) from Eq. (11) including the running coupling corrections. The s-channel light-cone perturbation theory formalism makes such inclusion simple [2]: all we have to do is include infinite chains of quark bubbles on the gluon lines in the amplitude and in the complex conjugate amplitude, as depicted on the right hand side of Fig. 3. Performing calculations similar to those done in [2] one arrives at K ❣1 (x0, x1; z1, z2) = 4Nf (2π)2 (2π)2 (2π)2 (2π)2 e−iq·(z−x0)+iq ′·(z−x )−i(k−k′)·z q2q′2 (1− 2α)2q · k k′ · q′ + q · q′ k · k′ − q · k′ k · q′ k2 + q2α(1− α) k′2 + q′2α(1− α) 2α (1− α) (1− 2α) k2 + q2α(1− α) k′2 + q′2α(1− α) k · q k′ · q′ 4α2 (1− α)2 k2 + q2α(1− α) k′2 + q′2α(1− α) 1 + αµβ2 ln q2 e−5/3 1 + αµβ2 ln q′2 e−5/3 ) (19) where K ❣1 denotes the kernel with the running coupling corrections resummed to all orders. Just like in [2, 31], here we will use MS renormalization scheme. Inclusion of fermion bubble chains generated two denominators at the end of Eq. (19), which is its only difference from Eq. (11). Here 11Nc − 2Nf . (20) Now we have to perform the transverse momentum integrals in Eq. (19). First we expand the denominators at the end of Eq. (19) into a power series and rewrite Eq. (19) as K ❣1 (x0, x1; z1, z2) = 4Nf n,m=0 (−αµβ2)n+m (2π)2 (2π)2 (2π)2 (2π)2 e−iq·(z−x0)+iq ′·(z−x )−i(k−k′)·z q2q′2 (1− 2α)2q · k k′ · q′ + q · q′ k · k′ − q · k′ k · q′ k2 + q2α(1− α) k′2 + q′2α(1− α) 2α (1− α) (1− 2α) k2 + q2α(1− α) k′2 + q′2α(1− α) k · q k′ · q′ 4α2 (1− α)2 k2 + q2α(1− α) k′2 + q′2α(1− α) λ=λ′=0 where we have defined µ2 = µ2 e5/3 to make the expressions more compact. Indeed we can not always expand the denominators of Eq. (19) into a geometric series employed in Eq. (21), but one has to remember that the summation of bubble chain diagrams shown on the right side of Fig. 3 gives one the geometric series. Hence the geometric series come first: later they are absorbed into the denominators shown in Eq. (19), which is an approximation not valid for all q and q′. Therefore, by keeping the geometric series in Eq. (21) we are not making any approximations. In general, in what follows we are not going to keep track of the issues of convergence of perturbation series. The contribution of renormalons to non-linear small-x evolution was thoroughly investigated in [30] and was found to be significant at low Q2. We refer the interested reader to [30] for more details on this issue. Using the following formulas (2π)2 e−ik·z k2 + q2 K0(q z) (22) (2π)2 e−ik·z k2 + q2 q K1(q z) (23) we can now perform the k- and k′-integrals in Eq. (21). Integrating over the angles of q and q′ as well yields K ❣1 (x0, x1; z1, z2) = (2π)4 n,m=0 (−αµβ2)n+m dq q dq′ q′ z212 |z − x0| |z − x1| − 4α ᾱ z12 · (z − x0) z12 · (z − x1) + z212 (z − x0) · (z − x1) × J1(q |z − x0|)K1(z12 q α ᾱ) J1(q ′ |z − x1|)K1(z12 q′ α ᾱ) + 2α ᾱ (α− ᾱ) z12 · (z − x0) z12 |z − x0| J1(q |z − x0|)K1(z12 q α ᾱ) J0(q ′ |z − x1|)K0(z12 q′ α ᾱ) z12 · (z − x1) z12 |z − x1| J0(q |z − x0|)K0(z12 q α ᾱ) J1(q ′ |z − x1|)K1(z12 q′ α ᾱ) + 4α2 ᾱ2 J0(q |z − x0|)K0(z12 q α ᾱ) J0(q ′ |z − x1|)K0(z12 q′ α ᾱ) λ=λ′=0 . (24) We have defined ᾱ = 1− α (25) for brevity. Now the integrals over q and q′ can be carried out to give K ❣1 (x0, x1; z1, z2) = (2π)4 n,m=0 (−αµβ2)n+m z212 µ 2 α ᾱ )λ+λ′ Γ2(1 + λ) Γ2(1 + λ′) − 4α ᾱ z12 · (z − x0) z12 · (z − x1) + z212 (z − x0) · (z − x1) (1 + λ) (1 + λ′) z812 (α ᾱ) 1 + λ, 2 + λ; 2;−|z − x0| α ᾱ z212 1 + λ′, 2 + λ′; 2;−|z − x1| α ᾱ z212 α− ᾱ z12 · (z − x0) (1 + λ)F 1 + λ, 2 + λ; 2;−|z − x0| α ᾱ z212 1 + λ′, 1 + λ′; 1;−|z − x1| α ᾱ z212 z12 · (z − x1) 1 + λ, 1 + λ; 1;− |z − x0|2 α ᾱ z212 (1 + λ′)F 1 + λ′, 2 + λ′; 2;− |z − x1|2 α ᾱ z212 1 + λ, 1 + λ; 1;− |z − x0|2 α ᾱ z212 1 + λ′, 1 + λ′; 1;− |z − x1|2 α ᾱ z212 λ=λ′=0 . (26) Unfortunately further simplification of the expression in Eq. (26) is impossible without approx- imations. The series resulting from summation over n and m are likely to be divergent due to renormalons. As we mentioned before, here we neglect the renormalon problem referring the reader to [30]. Similar to how it was done in [2] we are not going to attempt to resum the series exactly: instead we will calculate the next-to-leading order terms and assume that with a good accuracy they give us the scale(s) of the running coupling constant. This procedure is similar to the well-known prescription due to Brodsky, Lepage and Mackenzie [44]. Using the Taylor-expansions of hypergeometric functions F (1 + λ, 2 + λ; 2; z) = − λ 1 1 + ln(1− z) + 1 ln(1− z) + o(λ2). (27) F (1 + λ, 1 + λ; 1; z) = ln (1− z) + o(λ2) (28) after some algebra we obtain K ❣1 (x0, x1; z1, z2) = [α (z1 − x0)2 + ᾱ (z2 − x0)2] [α (z1 − x1)2 + ᾱ (z2 − x0)2] z412 − 4α ᾱ z12 · (z − x0) z12 · (z − x1) + z212 (z − x0) · (z − x1) 1− αµ β2 ln R2T (x0)µ + o(α2µ) 1− αµ β2 ln R2T (x1)µ + o(α2µ) + 2α ᾱ (α− ᾱ) z212 z12 · (z − x0) 1− αµ β2 ln R2T (x0)µ + o(α2µ) 1− αµ β2 ln R2L(x1)µ + o(α2µ) + z12 · (z − x1) 1− αµ β2 ln R2L(x0)µ + o(α2µ) 1− αµ β2 ln R2T (x1)µ + o(α2µ) +4α2 ᾱ2 z412 1− αµ β2 ln R2L(x0)µ + o(α2µ) 1− αµ β2 ln R2L(x1)µ + o(α2µ) In arriving at Eq. (29) we employed functions RT (x) and RL(x), which have dimensions of transverse coordinates and are defined by R2T (x)µ 4 e−2γ−5/3 [α (z1 − x)2 + ᾱ (z2 − x)2]µ2MS α ᾱ z212 (z − x)2 α (z1 − x)2 + ᾱ (z2 − x)2 α ᾱ z212 R2L(x)µ 4 e−2γ−5/3 [α (z1 − x)2 + ᾱ (z2 − x)2]µ2MS α (z1 − x)2 + ᾱ (z2 − x)2 α ᾱ z212 The subscripts T and L stand for transverse and longitudinal gluon polarizations which give rise to the two different functions under the logarithm. Recombining the series in Eq. (29) into physical running couplings finally yields α2µK ❣1 (x0, x1; z1, z2) = [α (z1 − x0)2 + ᾱ (z2 − x0)2] [α (z1 − x1)2 + ᾱ (z2 − x0)2] z412 − 4α ᾱ z12 · (z − x0) z12 · (z − x1) + z212 (z − x0) · (z − x1) R2T (x0) R2T (x1) + 2α ᾱ (α− ᾱ) z212 z12 · (z − x0) αs R2T (x0) R2L(x1) +z12 · (z − x1)αs R2L(x0) R2T (x1) + 4α2 ᾱ2 z412 αs R2L(x0) R2L(x1) with the physical running coupling in the MS scheme given by αs(1/R 1 + αµβ2 ln R2 µ2 ) . (33) Eq. (32) is the contribution to the JIMWLK evolution kernel of the resummed diagram on the right hand side of Fig. 3. 2.3 Brief summary of analytical results Let us briefly summarize our analytical results. The non-linear small-x evolution equation with the running coupling corrections included reads ∂S(x0, x1; Y ) = R [S]− S [S] . (34) The first term on the right hand side of Eq. (34) is referred to as the running coupling contribution. It was calculated independently in [1] and in [2]: the results of those calculations are given above in Eqs. (7) and (8) correspondingly, which have to be combined with Eq. (6) to obtain RBal [S] = d2z K̃Bal(x0, x1, z) [S(x0, z; Y )S(z, x1; Y )− S(x0, x1; Y )] (35) RKW [S] = d2z K̃KW(x0, x1, z) [S(x0, z; Y )S(z, x1; Y )− S(x0, x1; Y )] . (36) One notices immediately that RBal [S] calculated in [1] is different from RKW [S] calculated in [2] due to the difference in the kernels K̃Bal and K̃KW in Eqs. (7) and (8). However that does not imply disagreement between the calculations of [1] and [2]: after all, it is the full kernel on the right of Eq. (34), R [S]− S [S], that needs to be compared. To do that one has to calculate the second term on the right hand side of Eq. (34). The second term on the right hand side of Eq. (34) is referred to as the subtraction contri- bution. It is given by S[S] = α2µ d2z1 d 2z2K ❣1 (x0, x1; z1, z2) [S(x0, w, Y )S(w, x1, Y )− S(x0, z1, Y )S(z2, x1, Y )] with the resummed BK kernel K ❣1 (x0, x1; z1, z2) = CF m,n=0 (−1)m+n K ❣1 (xm, xn; z1, z2). (38) The resummed JIMWLK kernel K ❣1 (xm, xn; z1, z2) is given by Eq. (32), along with Eqs. (30) and (31) defining the scales of the running couplings. In the numerical solution below we will replace Nf → −6πβ2 in its prefactor, obtaining α2µK ❣1 (x0, x1; z1, z2) = − [α (z1 − x0)2 + ᾱ (z2 − x0)2] [α (z1 − x1)2 + ᾱ (z2 − x0)2] z412 − 4α ᾱ z12 · (z − x0) z12 · (z − x1) + z212 (z − x0) · (z − x1) R2T (x0) R2T (x1) + 2α ᾱ (α− ᾱ) z212 z12 · (z − x0) αs R2T (x0) R2L(x1) +z12 · (z − x1)αs R2L(x0) R2T (x1) + 4α2 ᾱ2 z412 αs R2L(x0) R2L(x1) This substitution is the same as for all other factors ofNf . The same substitution was performed in [2] to calculate the running coupling term. In fact, as was shown in [31], the linear part of the subtraction term (calculated using the prescription of [2]) contributes to the running coupling corrections to the BFKL equation. Therefore, in that case, the factor of Nf in front of Eq. (32) is definitely a part of the beta-function. Hence the replacement Nf → −6πβ2 is justified even in the subtraction term. Once again, in the numerical solution below we will use Eq. (39) along with Eq. (38) in Eq. (37) to calculate the subtraction term S[S]. Substituting w = z1 (or, equivalently, w = z2) in Eq. (37) would yield the subtraction term SBal[S] =α2µ d2z1 d 2z2K ❣1 (x0, x1; z1, z2) × [S(x0, z1, Y )S(z1, x1, Y )− S(x0, z1, Y )S(z2, x1, Y )] (40) which has to be subtracted from RBal [S] calculated in [1] and given by Eq. (35) to obtain the complete evolution equation resumming all orders of αsNf in the kernel. Substituting w = z = α z1 + (1− α) z2 in Eq. (37) yields SKW[S] =α2µ d2z1 d 2z2K ❣1 (x0, x1; z1, z2) × [S(x0, z, Y )S(z, x1, Y )− S(x0, z1, Y )S(z2, x1, Y )] (41) which has to be subtracted from RKW [S] calculated in [2] and given in Eq. (36) again to obtain the complete evolution equation resumming all orders of αsNf in the kernel. We checked explicitly by performing analytic calculations that the two evolution equations obtained this way agree at the NLO and NNLO. Below we will check the agreement of the two calculations to all orders by performing a numerical analysis of the solutions of these equations. The above discussion demonstrates that the separation of the evolution kernel into the running coupling and subtraction pieces, as done in Eq. (34), is somewhat artificial, and has no small parameter justifying one or another separation prescription. Therefore, the small-x evolution equation including all running coupling (or, more precisely, αsNf ) corrections should combine both terms in Eq. (34). Below we will solve such evolution equation numerically to obtain the full small-x evolution with the running coupling. 3 Numerical setup and initial conditions In our numerical study we consider the translational invariant approximation in which the scattering matrix is independent of the impact parameter of the collision, i.e., S = S(r, Y ). To solve the integro-differential equations, corresponding to the BK equation with running coupling we employ a second-order Runge-Kutta method with a step size in rapidity ∆Y = 0.1. We discretize the variable |r| into 800 points equally separated in logarithmic space between rmin = 10 −8 and rmax = 50. Throughout this paper, the units of r will be GeV −1, and those of Qs will be GeV. All the integrals have been performed using improved adaptative Gaussian quadrature methods. The accuracy of this numerical method has been checked in [3] to be better than a 4% in all the r range. We consider three different initial conditions for the dipole scattering amplitude, N(r, Y ) = 1− S(r, Y ). The first one is taken from the McLerran-Venugopalan (MV) model [22, 23]: NMV (r, Y = 0) = 1− exp . (42) where a constant term has been added to the argument of the logarithm in the exponent in order to regularize it for large values of r. The other two initial conditions are given by NAN (r) = 1− exp −(r Q , (43) with γ = 0.6 and γ = 0.8. These two last initial conditions will be referred hereinafter as AN06 and AN08 respectively. The interest in this ansatz, reminiscent of the Golec-Biernat–Wusthoff model [45], is that the small-r behavior NAN ∝ r2γ corresponds to an anomalous dimension 1− γ of the unintegrated gluon distribution at large transverse momentum. (AN labels initial conditions with anomalous dimension.) Our choices γ = 0.6 and γ = 0.8 can be motivated a posteriori by the observation that the anomalous dimension of the evolved BK solution for running coupling lies in between those two values and the one for the MV initial condition, γ ≈ 1 (see Section 4.2). Thus, the choice of distinct initial conditions allows us to better track the onset of the expected asymptotic universal behavior that is eventually reached at high energies and to study the influence of the pre-asymptotic, non-universal corrections to the solutions of the evolution equations. To completely determine our initial conditions, we set Q′s = 1 GeV at Y=0 in Eqs. (42) and (43) and put Λ = 0.2 GeV. Although Q s is normally identified with the saturation scale, our definition of the saturation scale through the rest of the paper will be purely pragmatical and given by the condition N(r = 1/Qs(Y ), Y ) = κ, (44) with κ = 0.5. We have checked that this choice of κ, albeit arbitrary, does not affect any of the major conclusions to be drawn in the rest of the paper. Finally, in order to avoid the Landau pole and to regularize the running coupling at large transverse sizes we stick to the following procedure: for small transverse distances r < rfr, with rfr defined by αs(1/r fr) = 0.5, the running coupling is given by the one loop expression αs(1/r β2 ln r2 Λ2 ) (45) with Nf = 3 and Λ = 0.2 GeV, whereas for larger sizes, r > rfr, we “freeze” the coupling at a fixed value αs = 0.5. A detailed study of the role of Landau pole in non-linear small-x evolution is given in [30]. 4 Results In this Section, we discuss our numerical results and how they compare to previous numerical work and analytical estimates. 4.1 Running coupling Fig. 4 shows the solutions of the evolution equation when only the running coupling contribution is taken into account, i.e., neglecting the subtraction term in Eq. (34), for different initial conditions and for the three schemes considered in this work: Balitsky’s, given by Eqs. (7) and (35), KW, given by Eqs. (8) and (36), and the the ad hoc parent dipole implementation of the running coupling, shown in Eq. (10). MV init. cond. Y=0,5,15,30KW Balitsky parent dipole init. cond. AN08 init. cond. Y=0,5,15,30 AN06 init. cond. Y=0,5,15,30 r )-1(GeV Figure 4: Solutions of the BK equation at rapidities Y=0, 5, 15 and 30 (curves are labeled from right to left) for the three running coupling schemes considered in this work: KW (solid line), Balitsky (dashed line) and parent dipole (dashed-dotted lines). The initial conditions are MV (top), AN08 (middle) and AN06 (bottom). As previously observed in [3, 46], the most relevant effect of including running coupling corrections in the evolution equation is a considerable reduction in the speed of the evolution with respect to the fixed coupling case. This is a common feature of the different running coupling schemes studied here and of other phenomenological ones considered in the literature (a detailed comparison between the solutions for fixed coupling evolution and for parent dipole running coupling can be found e.g. in [3]). This is not a surprising result, since a generic effect of the running of the coupling is to suppress the emission of small transverse size dipoles, which is the leading mechanism driving the evolution. However, despite this common feature of the running coupling solutions, significant dif- ferences are found between the solutions obtained under different schemes as we infer from Fig. 4. In particular, the evolution is much faster with the KW prescription than with that of Balitsky. Equivalently, the KW prescription yields a stronger growth of the saturation scale with rapidity/energy than Balitsky’s. Moreover, the solutions obtained when the parent dipole prescription is used lay much closer to those obtained within the KW scheme than to the ones obtained when Balitsky’s scheme is applied, contrary to what was suggested in [1]. As argued before, the differences observed in the solutions obtained using the two subtraction schemes are entirely due to neglecting the subtraction contribution and reflect the arbitrariness of the separation procedure. 4.2 Geometric scaling It has been found in previous analytical [34, 36, 47] and numerical studies on the solutions of the BK equation at leading order [3, 48–50] and for different heuristic implementations of next-to-leading order corrections [3,46], including the parent dipole prescription for the running coupling also considered in this work, that the solutions of the evolution equation at high enough rapidities are no longer a function of two separate variables r and Y , but rather they depend on a single scaling variable, τ = r Qs(Y ). This feature of the evolution, commonly referred to as geometric scaling, is an exact property of the solutions for fixed coupling evolution due to the conformal invariance of the leading-log kernel, and has become one of the key connections between the saturation based formalisms and the phenomenology of heavy ion collisions and deep inelastic scattering experiments [51–57]. It can be seen from Fig. 5 that the solutions of the BK equation with the running coupling terms discussed in the previous section also exhibit the property of scaling, in agreement with the analytical study carried out in [38], shown by the fact that the rescaled high rapidity solutions lay on a single curve which is independent of both the running coupling scheme and of the initial condition. The scaling behavior of the solution is observed in the whole τ range studied in this work, including the saturation region, τ > 1. The tiny deviations from a pure scaling behavior observed in Fig. 5 may be attributed to the fact that the full asymptotic behavior is reached at even larger rapidities (Y & 80, [3]) than those achieved by the numerical solution performed in this work. Remarkably, the scaling function for both KW and Balitsky’s scheme coincides with the one obtained with the parent dipole prescription, up to the above mentioned scaling violations. It has been observed in [3, 46, 48] that the scaling function differs significantly in the fixed and running coupling cases. Following that work, and to make a more quantitative study of the scaling property, we fitted our solutions to the functional form [34] f(τ) = a τ 2 γ ln τ 2 + b , (46) with a, b and γ free parameters, within a fixed window below the saturation region, τ ∈ [10−5, 0.1]. Noticeably, at large enough rapidities the whole fitting window lays within the geometric scaling window proposed in [36]: (Λ/Qs(Y )) < τ < 1, where Λ is some initial scale. -210 -110 1 MV init. cond. Y=0,40 Balitsky parent dipole fixed coupling init. cond. τ -110 1 10 AN06 init. cond. Y=0,40 Figure 5: Solutions of the BK equation at rapidities Y=0 and 40 for KW (solid line), Balitsky (dashed line) and parent dipole (dashed-dotted lines) schemes plotted versus the scaling variable τ = rQs(Y ). The asymptotic solution obtained with fixed coupling αs = 0.2 at Y = 40 in [3] is shown (black dashed-dotted line) for comparison. The initial conditions are MV (left) and AN06 (right). The value of γ extracted from the fits at rapidity Y = 40 lays in between γ ∼ 0.8 and γ ∼ 0.9. This conclusion holds for the three initial conditions used here: the anomalous dimension seems to converge to some intermediate value, in agreement with the value found in [3], for asymptotic running coupling solutions (γ ∼ 0.85 at Y = 70). This result for anomalous dimension is very far away from the value obtained in [3] for fixed coupling solutions (γ ∼ 0.64 at Y = 70) and from the predicted anomalous dimension for both running and fixed coupling solutions from analytical studies of the equation based on saddle point techniques [34–38], γc = χ(γc)/χ ′(γc) = 0.6275, where χ is the leading-log BFKL kernel. It might be argued that the numerical value of the anomalous dimension extracted from our fits is conditioned by the choice of the fitting function and by the fitting interval. Actually, it was shown in [3] that the solutions of the evolution could be well fitted by other functional forms, including the double-leading-log solution of BFKL, within a similar fitting region to the one considered in this work. On the other hand, several phenomenological parameterizations of the solution of the evolution have been proposed in [54–57] and have successfully confronted HERA 10 -410 10 -210 -110 1 2*0.6τ ~ Fixed coupling 2*0.85τ ~ Running coupling Figure 6: Asymptotic solutions (Y=40) of the evolution equation for running coupling (solid line) and fixed coupling with αs = 0.2 (dashed line). A fit to a power-law function aτ 2γ in the region τ ∈ [10−6, 10−2] yields γ ≈ 0.85 for the running coupling solution and γ ≈ 0.6 for the fixed coupling one. and RHIC experimental data. There, the dipole scattering amplitude at arbitrary rapidity is assumed to be given by a functional form analogous to our ansatz for the initial condition Eq. (43), but allowing for geometric scaling violations by replacing γ → γ(r, Y ). The value of the anomalous dimension at r = 1/Qs and/or for Y → ∞ is fixed to be the BFKL saddle point, γc ∼ 0.63 (the saddle point value considered in [57] is slightly different, γ ∼ 0.53), while the value γ = 1 is recovered in the limit r → ∞ at any finite rapidity. The success of these phenomenological works supports the claim that the anomalous dimension of the solution is given by the BFKL saddle point, in agreement with the above mentioned analytical predictions. However, the relevant values of momenta probed at current phenomenological applications are very distinct from the fitting region considered here. For example, the inclusive structure function measured in HERA is fitted in [54,56] within the region 0.045 GeV2 < Q2 < 45 GeV2, whereas charged hadron pt spectra in dAu collisions is well reproduced by [55–57] in the region 1 GeV < pt < 4.5 GeV. Note that, for both sets of data, the measured regions overlap with the deeply saturated domain of the solution. On the contrary, our fitting region 10−5 < τ < 1 corresponds to values of momenta ∼ 10Qs(Y ) < pt < 105Qs(Y ) (always well above the saturation scale), with Qs(Y = 40) ∼ 500÷1000 GeV for the different running coupling schemes considered and, therefore, has no overlap with the kinematic regions measured experimentally, since we scrutinize a momentum region strongly shifted to the ultraviolet compared to currently available data. Moreover, it should be noticed that the rapidity interval covered by both sets of experimental data is ∆Y < 4 in both cases, while we study the solutions of the evolution (Y=0)]MVSub[N -410 -310 -210 -110 1 (Y=30)]MVSub[N (Y=0)] Sub[N 10 -210 -110 1 (Y=30)] Sub[N (Y=0)] Sub[N 10 -210 -110 1 (Y=30)] Sub[N Figure 7: Subtraction contribution calculated in the KW scheme (triangles) and in Balitsky’s (stars). The trial functions correspond to the solutions of the evolution under Balitsky run- ning coupling scheme at rapidities Y = 0, 30 for MV (left), AN08 (center) and AN06 initial conditions. at asymptotic rapidities, Y ∼ 40. We have checked that shifting our fitting region to larger values of τ (smaller momentum) would bring the value of γ extracted from our fits closer to the saddle point BFKL one, since the transition from the ultraviolet region to the deeply saturated domain of the scaling solution is realized by a locally less steeper function (see Figs. (5) and (12)). Therefore, there is no contradiction at all between the success of the phenomenological parameterizations of the solutions and the results reported here. With the above clarifications we reach the following conclusion: the asymptotic scaling solutions corresponding to fixed and running coupling evolution are intrinsically different in the whole r-range. This is emphasized in Fig. 6, where we represent the scaling solutions in a log scale for τ < 1. It is clear that the tail of the distribution falls off with decreasing τ much steeper for the running coupling solution than for the fixed coupling one. A fit to a pure power-law function, f = a τ 2γ , in the region τ ∈ [10−6, 10−2] yields γ ∼ 0.85 for the running coupling and γ ∼ 0.61 for the fixed coupling solution. The differences between fixed and running coupling solutions at τ > 1 are evident from Fig. 5. This is a puzzling result that remains to be understood from purely analytical methods. (Y=0)]MV[N 10 -210 -110 1 (Y=30)]MV[N (Y=0)] 10 -210 -110 1 (Y=30)]AN08[N (Y=0)] 10 -210 -110 1 (Y=30)]AN06[N Figure 8: Ratio of the subtraction over the running terms, D(r, Y ) = S[N(r, Y )]/R[N(r, Y )], calculated in both KW (triangles) and Balitsky (stars) schemes for MV (left), AN08 (middle) and AN06 (right) initial conditions at rapidities Y=0 (top) and Y=30 (bottom). 4.3 Subtraction Term Before attempting to solve the complete evolution equation, and in order to gain insight in the nature and structure of the subtraction contribution, we first evaluate the subtraction functional for both Balitsky, Eq. (40), and KW, Eq. (41), schemes using a set of trial functions for S which we choose to consist of the solutions of the evolution equation with the running coupling in Balitsky’s scheme at different rapidities and of the three initial conditions considered above in this work. Two main remarks can be made about our results, shown in Fig. 7: i) For all the trial functions considered in this work, the subtraction contribution is much larger in the KW scheme than in Balitsky’s. A plausible explanation for this is that Balitsky’s subtraction contribution, Eq. (40), when expanded in terms of dipole scattering amplitudes, N = 1 − S, reduces to a sum of non-linear terms, since all the linear terms in the expansion cancel each other due to the z1 ↔ z2 symmetry of the kernel, whereas in the KW case no such cancellation happens and the subtraction contribution, Eq. (41), also includes linear terms, which are dominant over the non-linear ones in the non-saturated domain where N ≪ 1. ii) The subtraction contribution S has the same sign as the running coupling contribution R in the whole τ range which, together with the relative minus sign assigned to the subtraction term in Eq. (34), implies that the proper inclusion of the subtraction term reduces the value of the functional that governs the evolution, F . In other words: the subtraction contribution tends to systematically slow down the evolution, as we shall explicitly confirm in the next subsection. To better quantify the size of the subtraction contribution, we plot the ratio D(r, Y ) ≡ S[N(r, Y )]/R[N(r, Y )] in Fig. 8. At Y = 0, the relative weight of the subtraction contribution with respect to the running one within the KW scheme and for a MV initial condition goes from a D ∼ 0.4 at small τ to D ∼ 1 at τ ∼ 1. The same ratio for the Balitsky scheme takes significantly smaller values: it goes from D ∼ 0.1 at small τ to D ∼ 0.4 for τ ∼ 1. As the evolved solutions get closer to the scaling function, i.e. for larger rapidities, the r dependence of the ratio becomes flatter and its overall normalization goes down to an approximately constant value D ∼ 0.15 for the KW scheme and D ∼ 0.025 for that of Balitsky. This behavior remains unaltered when going from rapidity Y = 20 to Y = 30, which suggests that the ratio may saturate to a fixed value in the asymptotic region. -210 -110 1 KWRun BalRun KW(Run-Sub) (Run-Sub) (Y=0)MVN τ -210 -110 1 (Y=30)MVN Figure 9: Total kernel F = R−S calculated under Balitsky’s scheme, Eqs (7) and (40), (solid line) and under the KW scheme, Eqs (8) and (41), (dashed line). The overlap of the two lines shows the agreement between the two calculations. Triangles stand for the running coupling term calculated in the KW approach, RKW, while stars stand for the running coupling term under Balitsky’s scheme, RBal. The trial functions N(r, Y ) correspond to the solution of the evolution with only running coupling under Balitsky’s scheme at Y=0 (left) and Y=30 (right) for a MV initial condition. -110 1 1.2 MV i.c. Y=0,3,10 KWRun BalRun F=Run-Sub init. cond. r )-1(GeV -110 1 scaling function i.c. Y=0,3,10 r )-1(GeV Figure 10: Solutions of the complete (all orders in αs β2) evolution equation given in Eq. (34) (solid lines), and of the equation with Balitsky’s (dashed lines) and KW’s (dashed-dotted) running coupling schemes at rapidities Y = 0, 5 and 10. Left plot uses MV initial condition. The right plot employs the initial condition given by the dipole amplitude at rapidity Y = 35 evolved using Balitsky’s running coupling scheme and with r-dependence rescaled down such that Qs = Q s = 1 GeV. Finally, we have checked that combining the subtraction and running coupling contributions for both schemes adds up to the same result. This is shown in Fig. 9, where we plot the value of the total functional F = R − S calculated under the KW scheme (Eqs. (8) and (36) for the running coupling term, R, and Eq. (41) for the subtraction term, S) and under Balitsky’s scheme (Eqs. (7) and (35) for the running coupling term and Eq. (40) for the subtraction term). The two results coincide within the estimation of the numerical accuracy previously discussed. The agreement between the two results is better in the small-τ region, where the two curves lay almost on top of each other. In the saturation region, τ & 1, the agreement is slightly worse, although the differences between the values of F calculated in both schemes is still much less than the differences between the running coupling terms themselves. This slight remaining disagreement between the Balitsky’s and KW prescriptions may also be due to inaccuracies in a Fourier transform of a geometric series performed in arriving at Eq. (39). This result serves as a cross-check of our numerical method and as an additional confirmation of the agreement of the independent calculations derived in [1, 2]. 0 1 2 3 4 5 6 7 8 9 MV i.c.KWRun BalRun F=Run-Sub )2(GeV 1 2 3 4 5 6 7 8 9 scaling function i.c. Figure 11: Saturation scale corresponding to the solutions plotted in Fig. 10. 4.4 Complete running coupling BK equation In this section we calculate the solutions of the complete evolution equation, Eq. (34), including both the running and subtraction terms obtained by the all-orders αs Nf resummation and by the Nf → −6πβ2 replacement. Since the numerical evaluation of the subtraction contribution at each point of the grid and each step of the evolution would require an exceedingly large amount of CPU time consumption, the strategy followed to include it in the evolution equation consists of calculating such contribution only in a small set of grid points at each step of the evolution, which we fixed at n = 16, between the points r1 and r2, which are determined at each step of the evolution by the conditions N(Y, r1) = 10 −9, and N(Y, r2) = 0.99, and then using power-law interpolation and extrapolation to the other points of the grid. Both the running and subtracted terms are calculated within Balitsky scheme. This procedure is motivated by the fact that, as discussed in the previous section, the subtraction contribution can be regarded as a small perturbation with respect to the running coupling term within Balitsky’s scheme and by the fact that it is a rather smooth function that can be well fitted by a power-law function in most of the r-range. The accuracy of this procedure has been checked by doubling the number of points at which the subtraction contribution is calculated at each step of the evolution, i.e. by setting n=32. At Y=2, the differences between the solutions obtained with the two above mentioned choices for n were less than a 8% in the tail of the solution, r < r1, and less than a 3% for r > r1. The results of the evolution calculated in this way and using MV and rescaled asymptotic running coupling solution (Y=35) as initial conditions are plotted in Fig. 10. They confirm -210 -110 1 10 Y=10F=Run-Sub (Y=35) MV BALi.c.=N Figure 12: Rescaled solutions given by the complete αs β2-evolution equation (solid line) and for KW (dashed-dotted line) and Balitsky’s (dashed line) running coupling schemes at Y = 10. The initial condition corresponds to the dipole amplitude at rapidity Y = 35 evolved using Balitsky’s running coupling scheme and with r-dependence rescaled down such that Qs = Q s = 1 GeV. the expectations raised in the previous Subsection: the inclusion of the subtraction terms considerably slows down the evolution with respect to the sole consideration of the running coupling contributions. Moreover, the reduction in the speed of the wave front is much larger for the KW scheme than for that of Balitsky one for both initial conditions. However, the closer the initial condition is to the asymptotic running coupling scaling function, the smaller are the effects of the subtraction contribution. These features can be better quantified by inspecting the rapidity dependence of the saturation scale generated by the evolution, plotted in Fig. 11. At rapidity Y = 10 the ratio of the saturation scale Qs yielded by the KW scheme to Qs given by the complete αs β2-evolution equation is a factor of ∼ 2.5 for the MV initial condition and a factor of ∼ 2.1 for the asymptotic running coupling initial condition. At the same rapidity, the ratio of the saturation scale obtained under Balitsky’s scheme to Qs corresponding to the complete αs β2-evolution is ∼ 1.25 for the MV initial condition and ∼ 1.15 for the scaling function initial condition. Thus, in spite of the smallness of the ratio of the subtraction terms to the running coupling contributions at high rapidity, which is ∼ 0.025 for Balitsky’s and ∼ 0.15 for KW scheme at Y = 30 (see bottom plots in Fig. 8), the proper inclusion of the subtraction term results in fairly sizable effects in the solutions of the evolution equation. Finally, we notice that the scaling behavior of the solution is not affected by the subtraction term. This is seen in figure Fig. 12, where we evolve starting from an initial condition already close to the running coupling scaling function and plot the solutions of the evolution equation obtained with just running coupling terms (see Section 4.1) and the solution of the complete αs β2-evolution at rapidity Y = 10. It is clear that, within the numerical accuracy, no departure from the scaling behavior is observed. Therefore the main effect of a proper consideration of the subtraction term is the one of reducing the speed of the evolution. It does not violate or modify the geometric scaling property of the solutions established in Section 4.2. In our understanding geometric scaling appears to persist when the running coupling effects are included because, at high enough rapidity Qs(Y ) ≫ Λ, such that the new (from the LO standpoint) momentum scale Λ introduced by the running coupling can be safely neglected. Hence the dynamics is again characterized by a single momentum scale Qs(Y ). At the same time running coupling does modify the evolution kernel, leading to a different shape of the scaling function. 5 Conclusions In this paper we have taken into account all corrections to the kernels of the non-linear JIMWLK and BK evolution equations containing powers of αs Nf . We reiterated the fact that the sep- aration of the resulting kernel resumming all powers of αs Nf into the running coupling and subtraction parts, as done in the previous calculations of [1, 2], is not justified parametrically. We have then performed numerical analysis with the following conclusions. • First we solved the evolution equations derived in [1] and [2] keeping only the running coupling part or the evolution kernel and neglecting the subtraction term. Comparing to the results for fixed coupling obtained in [3] we confirmed the conclusion reached in [3] that the growth with rapidity is substantially reduced when running coupling corrections are included. The results for three different initial conditions are shown in Fig. 4. We observe that the solution of the equation derived in [2] differs significantly from that derived in [1], but agrees (with good numerical accuracy) with the solution of the BK evolution equation with the coupling running at the parent dipole size. (The latter is just a model of the running coupling not resulting from any calculations, which we plot for illustrative purposes.) We also observe that at sufficiently high rapidity both equations from [1] and from [2] give us the same scaling function for the dipole amplitude N(r, Y ) as a function of r Qs(Y ), which is also in agreement with the scaling function given by the parent dipole running, as shown in Fig. 5. The fact that the scaling is preserved when the running coupling corrections are included was previously established in [3], though for models of running coupling only. The shape of the scaling function is very different from that obtained from the fixed coupling evolution equations. In particular, we found that for dipole sizes below 0.1/Qs the anomalous dimension of the scaling function in the running coupling case becomes γ ≈ 0.85 (see Fig. 6). This is different from the result of several analytical estimates [34–38], which expect the anomalous dimension not to change when running coupling corrections are included and to remain at its fixed coupling value of γ ≈ 0.63. • We have then evaluated the subtraction term for both calculations performed in [1] and [2]. We demonstrated that subtracting the subtraction terms from the running coupling terms makes the full answer agree for both calculations of [1] and [2], as shown in Fig. 9 for the right hand side of the evolution equation. It turns out that the subtraction term SBal[S], which has to be subtracted from the result of [1], is systematically smaller than SKW[S], to be subtracted from the result of [2], over the whole rapidity range studied here. This implies that the result of [1] should have a smaller correction than the result of [2] and is thus closer to the full answer. The subtraction terms SBal[S] and SKW[S] are plotted in Fig. 7 as functions of the dipole size r for different values of rapidity. Their relative contributions to the evolution kernel are shown in Fig. 8, where we plotted the subtraction functional divided by the running coupling functional. From those figures we conclude that both the magnitude of these extra terms and their relative contribution to the evolution kernel decrease with increasing rapidity. Hence, while at ”moderate” rapidities (the ones closer to realistic experimental values) the subtraction term is important for both calculations [1, 2], it becomes increasingly less important at asymptotically large rapidities. The physics is easy to understand: the subtraction terms are o(α2s), while the running coupling part of the kernel is o(αs). Hence, if we suppose that the effective value of the coupling is given by its magnitude at the saturation scale Qs(Y ), then, as rapidity increases, the coupling would decrease, making the subtraction term much smaller than the running coupling term. Indeed, while at asymptotically high rapidities the assumption of [1,2] that the subtraction term could be neglected is justified, making the results of [1] and [2] agree with each other, for rapidities relevant to modern days experiments the subtraction term is numerically important. • With the last conclusion in mind we continued by numerically solving the full evolution equation resumming all powers of αs Nf in the evolution kernel, which now would combine both the running coupling and the subtraction terms. The five-dimensional integral in the subtraction term (37) made obtaining this solution rather difficult. The outcome of the calculation is shown in Fig. 10. All the main conclusions stated above were again confirmed by the solution of the full equation. At asymptotically high rapidity scaling regime is recovered, as can be seen from Fig. 12. As the subtraction term is less important in that regime, the scaling function appears to be the same as in the case of having only the running coupling term in the kernel. The anomalous dimension again turns out to be γ ≈ 0.85, in disagreement with the analytical expectations of [34–38]. However, the scaling of the saturation scale with rapidity appears to be in agreement with the expectations of analytical work of [34, 35, 38], as shown in Fig. 11. We conclude by observing that the knowledge of the non-linear small-x evolution equation with all the running coupling corrections included brings us to an unprecedented level of pre- cision allowing for a much more detailed comparison with experiments than was ever possible before. 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B599 (2004) 23–31, [hep-ph/0405045]. http://arxiv.org/abs/hep-ph/0307179 http://xxx.lanl.gov/abs/hep-ph/0307179 http://arxiv.org/abs/hep-ph/0106112 http://xxx.lanl.gov/abs/hep-ph/0106112 http://arxiv.org/abs/hep-ph/0104038 http://xxx.lanl.gov/abs/hep-ph/0104038 http://arxiv.org/abs/hep-ph/0007192 http://xxx.lanl.gov/abs/hep-ph/0007192 http://arxiv.org/abs/hep-ph/0407018 http://xxx.lanl.gov/abs/hep-ph/0407018 http://arxiv.org/abs/hep-ph/0502167 http://xxx.lanl.gov/abs/hep-ph/0502167 http://arxiv.org/abs/hep-ph/0310338 http://xxx.lanl.gov/abs/hep-ph/0310338 http://arxiv.org/abs/hep-ph/0512129 http://xxx.lanl.gov/abs/hep-ph/0512129 http://arxiv.org/abs/hep-ph/0608063 http://xxx.lanl.gov/abs/hep-ph/0608063 http://arxiv.org/abs/hep-ph/0405045 http://xxx.lanl.gov/abs/hep-ph/0405045 Introduction Scheme dependence Inclusion of running coupling corrections: general concepts Derivation of the subtraction term Brief summary of analytical results Numerical setup and initial conditions Results Running coupling Geometric scaling Subtraction Term Complete running coupling BK equation Conclusions

0704.0613

Anomalous c-axis transport in layered metals

Anomalous c-axis transport in layered metals D. B. Gutman and D. L. Maslov Department of Physics, University of Florida, Gainesville, FL 32611, USA (Dated: November 4, 2018) Transport in metals with strongly anisotropic single-particle spectrum is studied. Coherent band transport in all directions, described by the standard Boltzmann equation, is shown to withstand both elastic and inelastic scattering as long as EF τ ≫ 1. A model of phonon-assisted tunneling via resonant states located in between the layers is suggested to explain a non-monotonic temperature dependence of the c-axis resistivity observed in experiments. PACS numbers: 72.10.-d,72.10.Di Electron transport in layered materials exhibits a num- ber of unusual properties. The most striking example is a qualitatively different behavior of the in-plane (ρab) and out-of-plane (ρc) resistivities: whereas the temper- ature dependence of ρab is metallic-like, that of ρc is ei- ther insulating-like or even non-monotonic. At the level of non-interacting electrons, layered systems are metals with strongly anisotropic Fermi surfaces. A commonly used model is free motion along the planes and nearest- neighbor hopping between the planes: εk = k ||/2mab + 2J (1− cos k⊥d) , (1) where k|| and k⊥ are in the in-plane and c-axis com- ponents of momentum, respectively, mab is the in-plane mass, and d is lattice constant in the c-axis direction. For the strongly anisotropic case (J ≪ EF ), the equipoten- tial surfaces are “corrugated cylinders” (see Fig.1). If the Hamiltonian consists of the band motion with spectrum (1) and the interaction of electrons with poten- tial disorder as well as with inelastic degrees of freedom, e.g., phonons, the Boltzmann equation predicts that the conductivities are given by σBab = e 2ν〈vavbτtr〉, σBc = 4e2νJ2d2〈sin2 (k⊥d) τtr〉, (2) where 〈. . . 〉 denotes averaging over the Fermi surface and over the thermal (Fermi) distribution, ν = mab/πd is the density of states, and τtr is the transport time, resulting from all scattering processes (we set h̄ = kB = 1). If τtr decreases with the temperature, both σab and σc are expected to decrease with T as well. This is not what the experiment shows. The c-axis puzzle received a lot of attention in con- nection to the HTC materials [1], and a non-Fermi-liquid nature of these materials was suggested to be responsible for the anomalous c-axis transport [2]. However, other materials, such as graphite [3], TaS2 [4], Sr2RuO4 [5], or- ganic metals [6], etc., behave as canonical Fermi liquids in all aspects but the c-axis transport. This suggests that the origin of the effect is not related to the specific prop- erties of HTC compounds but common for all layered materials. A large number of models were proposed to explain the c-axis puzzle. Despite this variety, most au- thors seem to agree on that the coherent band transport FIG. 1: Fermi surface corresponding to Eq.(1) with Fermi velocity vectors at two different points. in the c-axis direction is destroyed. Although there is no agreement as to what replaces the band transport in the ”incoherent” regime, the most frequently discussed mech- anisms include incoherent tunneling between the layers, assisted by either out-of-plane impurities [8, 10, 11, 12] or by coupling to dissipative environment [13], and polarons [14, 15]. The message of this Letter is two-fold. First, we ob- serve that neither elastic or inelastic (electron-phonon) scattering can destroy band transport even in a strongly anisotropic metal as long as the familiar parameter EF τ is large. Nothing happens to the Boltzmann conductivi- ties in Eq.(2) except for σBc becoming very small at high temperatures so that other mechanisms, not included in Eq.(2), dominate transport. This observation is in agree- ment with recent experiment [7] where a coherent fea- ture (angle-dependent magnetoresistance) was observed in a supposedly incoherent regime. Second, we propose phonon-assisted tunneling through resonant impurities as the mechanism competing with the band transport. As such tunneling provides an additional channel for trans- http://arxiv.org/abs/0704.0613v1 port, the total conductivity is [8] σc = σ c + σres, (3) where σres is the resonant-impurity contribution. Be- cause σres increases with the temperature, the band chan- nel is short-circuited by the resonant one at high enough temperatures[9]. Accordingly, σc goes through a min- imum at a certain temperature (and ρc = σ c goes through a maximum). We consider phonon-assisted tun- neling through a wide band of resonant levels distributed uniformly in space. We show that the non-perturbative (in the electron-phonon coupling) version of this the- ory is in a quantitative agreement with the experiment on Sr2RuO4 [5]. Due to a similarity between phonon- assisted tunneling and other problems, in which inter- action leads to the formation of a cloud surrounding the electron (such as polaronic effect and zero bias anomaly), many ideas put forward earlier [8, 10, 11, 12, 13, 14, 15] agree with our picture. Nevertheless, we believe that only a combination of resonant impurities and electron- phonon interaction solves the puzzle of c-axis resistivity and provides a microscopic theory for some of the mech- anisms considered in prior work. We begin with the dis- cussion of the breakdown (or lack of it thereof) of the Boltzmann equation. One may wonder whether the band transport along the c-axis breaks down because the Anderson localization transition occurs in the c-direction whereas the in-plane transport remains metallic. This does not happen, how- ever, because an electron, encountering an obstacle for motion along the c-axis, moves quickly to another point in the plane, where such an obstacle is absent. More formally, it has been shown the Anderson transition oc- curs only simultaneously in all directions [16, 17, 18] and only if J is exponentially smaller than 1/τ . Therefore, localization cannot explain the observed behavior. Refs.[19, 20] suggested an idea of the “coherent- incoherent crossover”. It implies that the coherent band motion breaks down if electrons are scattered faster than they tunnel between adjacent layers, i.e., if Jτ ≪ 1. Con- sequently, the current in the c-direction is carried via in- coherent hops between conducting layers. It was noted by a number of authors that the assumption about inco- herent nature of the transport does not, by itself, explain the difference in temperature dependences of σab and σc [20, 21]: due to conservation of the in-plane momentum, σc is proportional to τ both in the coherent and inco- herent regimes. Nevertheless, an issue of the “coherent- incoherent crossover” poses a fundamentally important question: can scattering destroy band transport only in some directions, if the spectrum is anisotropic enough [22]? We argue here that this is not the case. Since we have already ruled out elastic scattering, this leaves inelastic one as a potential culprit. We focus on the case of the electron-phonon interaction as a source of inelastic scattering. For an isotropic metal, the quantum kinetic equation is derived from the Keldysh equations of motion for the Green’s function via the Prange-Kadanoff procedure [23] for any strength of the electron-phonon in- teraction. In this Letter, we apply the Prange-Kadanoff theory to metals with strongly anisotropic Fermi surfaces, such as the one in Fig. 1. We show that, exactly as in the isotropic case, the Boltzmann equation holds its stan- dard form as long as EF τe-ph ≫ 1. Since this form does not change between coherent (Jτe-ph ≫ 1) and incoher- ent (Jτe-ph ≪ 1) regimes, it means that the coherent- incoherent crossover is, in fact, absent. We adopt the standard Frölich Hamiltonian for the deformation-potential interaction with longitudinal acoustic phonons (ωq = sq) k+qak Since tunneling matrix elements are much more sensi- tive to the increase in the inter-plane distance than the elastic moduli, the anisotropy of phonon spectra in lay- ered materials, albeit significant, is still weaker than the anisotropy of electron spectra (see, e.g., Ref. [24]). Therefore, we treat phonons in the isotropic approxima- tion, and assume that the magnitude of the Fermi veloc- ity is larger than the speed of sound s. For a static and uniform electric field, the Keldysh component of the electron’s Green function satisfies the Dyson equation L̂GK + [ReΣR,⊗GK ]− + [ΣK ,⊗ReGK ]− [ΣK ,⊗A]+ − [Γ,⊗GK ]+ . (5) Here L̂ = (∂t + v · ∇R + eE · ∇k) is the Liouville op- erator, A = i(GR − GA) is the spectral function, Γ = ΣR − ΣA , and ⊗ denotes the convolution in space and time. Thanks to the Migdal theorem, the self-energy does not depend on electron’s dispersion ξk ≡ εk − EF , and Eq.(5) can be integrated over ξk. This results in an equa- L̂gK + [ReΣR, gK ]− = 2iΣ K − 1 [Γ, gK ]+ (6) for the “distribution function” gK(ǫ, n̂) = GK(ǫ, ξk, n̂)dξk , (7) where n̂ = vk/ |vk| is a local normal to the Fermi surface. We consider a linear dc response, when the self-energy is needed only at equilibrium. Within the Migdal theory, the Matsubara self-energy is given by a single diagram Σ(ǫ, n̂) = − g2 (q)G(ǫ− ω,k− q)D(ω, q) , where the dressed phonon propagator D−1 = D−10 − g is expressed through bare one D0(ω, q) = −s2q2/ ω2 + s2q2 and polarization operator Π which, for EF > 2J, is given by its 2D form Π(ω, q) = −ν 1− |ω|/ v2F q ‖ + ω We assume that the electron-phonon vertex decays on some scale kD shorter than Fermi momentum (kD ≪ kF ). This assumption allows one to linearize the dispersion ξk−q ≈ ξk − vk · q and simplifies the analysis without changing the results qualitatively. As long as J ≪ EF , we have |vk| ≈ kF /mab ≈ vF , where kF is the radius of the cylinder in Fig. 1 for J = 0. Despite the fact that the electron velocity does have a small component along the c-axis, its in-plane component is large (cf. Fig. 1). Since it is the magnitude of vk that controls the Migdal’s approximation, the problem reduces to the interaction of fast 2D electrons with slow 3D phonons. With these simplifications, we find ReΣR(ǫ, n̂) = −1 ǫ; (8a) ImΣR(ǫ, n̂) = − ζ 12(1− ζ)2 , (8b) where ζ = νg2 is a dimensionless coupling constant and ωD = skD. We see that, despite the strong anisotropy, the self-energy remains local, i.e., independent of ξk. Vertex renormalization leads to two types of correc- tions to the self-energy: those that are proportional to the Migdal’s parameter (s/vF ) and those that are pro- portional to ms2/ǫ. The second type of corrections inval- idates the Migdal’s theory for temperatures below ms2, which is about 1 K in a typical metal. For metals with anisotropic spectrum the existence of such a scale is po- tentially dangerous, since it is not obvious which of the masses (light or heavy) defines this scale. We find that the in-plane mass (mab) controls the vertex renormaliza- tion for the nearly cylindrical Fermi surface. This shows that the Migdal theory for layered metals has the same range of applicability as for isotropic metals [25]. The rest of the derivation proceeds in the same way as for the isotropic case [23], and the resulting Boltzmann equation assumes its standard form. Since no assump- tion about the relation between τe-ph and the dwell time (1/J) has been made, the conductivities obtained from the Boltzmann equation have the same form regardless of whether Jτe-ph is large or small. In other words, there is no coherent-incoherent crossover due to inelastic scat- tering in an anisotropic metal [29]. The situation changes qualitatively if resonant impu- rities are present in between the layers. Electrons that tunnel through such impurities are moving with the speed controlled by the broadening of a resonant level, i.e., much slower than speed of sound. For that reason they can not be treated within the formalism outlined above and require a separate study. To evaluate the resonant-impurity contribution to the conductivity, we assume that the impurities are randomly distributed in space with density nimp whereas their en- ergy levels uniformly distributed over an interval Eb. The tunneling conductance of a bilayer junction is G = −e2 dǫdǫ′Wǫ,ǫ′ (1 − n′ǫ) + , (9) where Wǫ,ǫ′ is a transition probability per unit time and nǫ is the Fermi function. To calculateWǫ,ǫ′ , we use the re- sults of Ref.[30, 31] for the probability of phonon-assisted tunneling through a single impurity Wǫ,ǫ′ = ΓLΓR it1(ǫ dt2dt3e i(t2−t3)(ǫ−ǭ0)−Γ(t2+t3) (10) × exp |αq|2 |1− e−it3 + eit1 e−it2 − 1 |2 coth e−it3 + eit2 + eit1(e−it2 − 1)(1− eit3)− c.c. where αq = −iΛq/ ρωq, Λ is the deformation-potential constant, ΓL and ΓR are tunneling widths of the resonant level, Γ = ΓL + ΓR, and ǭ0 is the energy of a resonant level renormalized by the electron-phonon interaction. In the limit of no electron-phonon interaction, Eq.(10) re- produces the well-known Breit-Wigner formula. From now on, we consider a wide band of resonant levels: Eb ≫ T ≫ Γ. Averaging Eq.(10) over spatial and en- ergy positions of resonant levels, one obtains σres=σel 1−coth sinh2 dteitǫ−λf(t) f(t)= (1−cos(ωt)) coth +i sin(ωt) .(11) Here σel is the conductivity due to elastic resonant tun- neling and λ ≡ Λ2ω2D/ρs5π2 is the dimensionless cou- pling constant for localized electrons. In the absence of electron-phonon interaction, σres is temperature inde- pendent and given by σel ≃ πe2Γ1nimpa0d/Eb[32], where a0 is the localization radius of a resonant state and Γ1 ≃ ǫ0e−d/a0 is its typical width. We note that the electron-phonon interaction is much stronger for localized electrons than for band ones: λ/ζ ∼ (kFd) (vF /s) ≫ 1. Since typically ζ ∼ 1, one needs to consider a non- perturbative regime of phonon-assisted tunneling. In that case, resonant tunneling is exponentially suppressed at T = 0: σres(T = 0) = σele −λ/2. At finite T , we find σres = σel e−λ/2 1 + π , T ≪ ωD√ , T ≫ λωD. As T increases, σres growth, resembling the zero-bias anomaly in disordered metals and Mössbauer effect. At high temperatures (T ≫ λωD) σres approaches the non- interacting value (σel). The asymptotic regimes in the interval ωD/ λ ≪ T ≪ λωD can also be studied but we will not pause for this here. Notice that, in contrast to the phenomenological model of Ref.[8], there is no simple relation between the T -dependences of σBc and σres. To compare our model with the experiment, we extract σBc from the low-temperature (between 10 and 50 K) c- axis resistivity of Sr2RuO4 and extrapolate it to higher temperatures [5]. The resonant part of the conductivity is calculated numerically using Eq.(11). The fit to the data for σel = 43 · 103Ω−1 cm−1, ωD = 41 K and λ = 16 is shown in Fig. 2. The agreement between the theory and experiment is quite good and the values of the fitting parameters are reasonable. An immediate consequence of our model is the sample-to-sample variation of the c-axis conductivity. Among the layered materials, the largest amount of data is collected for graphite [3]. Even within the group of samples with comparable in-plane mobili- ties, the temperature of the maximum in ρc varies from 40K to 300 K [3, 33]. 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Moses and R. H. McKenzie, Phys. Rev. B 60, 7998 (1999). [21] L. Ioffe, A. Larkin, A. Varlamov, and L. Yu, Phys. Rev. B 47, 8936 (1993). [22] D. G. Clarke, S. P. Strong, P. M. Chaikin, and E. I. Chashechkina, Science 279, 2071 (1998). [23] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). [24] J. Paglione, C. Lupien, W.A. MacFarlane, J.M. Perz, L. Taillefer, Z.Q. Mao, and Y. Maeno, Phys. Rev. B 65, 220506(R). [25] The self-energy in Eqs.(8a,8b) diverges at ζ = 1. This di- vergence –also present for the isotropic case – results from the renormalization of the sound velocity and is an arte- fact of the Frölich Hamiltonian. A divergence-free theory is obtained by applying the adiabatic approximation to the coupled system of electrons and ions [26, 27, 28]. [26] J. R. Schrieffer, Theory of Superconductivity, (Addison- Wesley, Redwood City, 1988). [27] E.G. Brovman and Yu. Kagan, Sov. Phys. JETP 25, 365 (1967). [28] B. T. Geilikman, Sov. Phys.-Usp. 18, 190 (1975). 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0704.0614

Proper holomorphic mappings of the spectral unit ball

arXiv:0704.0614v2 [math.CV] 14 Jun 2007 PROPER HOLOMORPHIC MAPPINGS OF THE SPECTRAL UNIT BALL W lodzimierz Zwonek Abstract. We prove an Alexander type theorem for the spectral unit ball Ωn show- ing that there are no non-trivial proper holomorphic mappings in Ωn, n ≥ 2. Let Mn denote the space of n× n complex matrices. In order to avoid some trivialities and ambiguities we assume in the whole paper that n ≥ 2. Let ρ(A) := max{|λ| : λ ∈ Spec(A)} be the spectral radius of A ∈ Mn. Denote also by Spec(A) := {λ ∈ C : det(A − λIn) = 0} the spectrum of A ∈ Mn, where the eigenvalues are counted with multiplicities (In denotes the identity matrix). We also denote the spectral unit ball by Ωn := {A ∈ Mn : ρ(A) < 1}. Note that Ωn is an unbounded pseudoconvex balanced domain in C with the continuous Minkowski functional equal to ρ. For A ∈ Mn denote PA(λ) := det(λIn − A) = j=1(−1) jσj(A)λ n−j , A ∈ Mn. Denote also σ := (σ1, . . . , σn). We put Gn := σ(Ωn). The domain Gn is called the symmetrized polydisc. Note that σ ∈ O(Mn,Gn). Denote also Jn := πn({(ζ1, . . . , ζn) : ζj = ζk for some j 6= k}), where πn,j(ζ1, . . . , ζn) := 1≤k1<...<kj≤n ζk1 · . . . · ζkj , ζl ∈ D, l = 1, . . . , n (D denotes the unit disc in C). Note that Gn \Jn is a domain and Gn \Jn is dense in Note that Ωn = Tz, where Tz := {A ∈ Ωn : σ(A) = z}, z ∈ C n. The sets Tz, z ∈ C n are pairwise disjoint analytic sets. Note that if the matrix A ∈ Tz is non- degoratory then A is a regular point of Tz – recall that in such a case rankσ ′(A) = n – it is the largest possible number. For definition and basic properties of non- derogatory matrices see [Nik-Tho-Zwo 2007] and references there. One of possible definitions of a non-derogatory matrix is that different blocks in the Jordan normal form correspond to different eigenvalues (or equivalently all eigenspaces are one- dimensional). We shall deliver some properties of the sets Tz (see Lemma 5, Lemma 6 and Corollary 7). It is also simple to see that T0 is a cone which contains at least n2 − n + 1 linearly independent vectors: for instance the ones consisting of one 1 lying not on the diagonal (and with other entries equal to 0) and the matrix A = (aj,k)j,k=1,... ,n such that a1,1 = 1, a1,2 = −1, a2,1 = 1, a2,2 = −1 (and with all other entries equal to 0). Consequently, we shall see that 0 is not a regular point of T0. On the other hand the sets Tπn(ζ1,... ,ζn), where the points ζ1, . . . , ζn ∈ D are The research was partially supported by the Research Grant No. 1 PO3A 005 28 of the Polish Ministry of Science and Higher Education. 2000 Mathematics Subject Classification. Primary: 32H35. Secondary: 15A18, 32C25, 47N99 keywords: spectral unit ball, proper holomorphic mappings, symmetrized polydisc Typeset by AMS-TEX http://arxiv.org/abs/0704.0614v2 2 W LODZIMIERZ ZWONEK pairwise different, are submanifolds – it follows from the fact that in this case all elements of Tπn(ζ1,... ,ζn) are non-derogatory. It is well-known that for a given mapping F ∈ O(Ωn,Ωn) there exists a mapping F̃ ∈ O(Gn,Gn) such that σ(F (A)) = F̃ (σ(A)) (see e.g. [Edi-Zwo 2005]). If f ∈ O(D,D) then one may well-define the following holomorphic mapping Ff : Ωn ∋ A 7→ f(A) := (j)(0) Aj ∈ Ωn. Note that F̃f (σ(A)) = σ(Ff (A)) = πn(f(λ1), . . . , f(λn)), where σ(A) = πn(λ1, . . . , λn). In particular, Fa ∈ Aut(Ωn) for any a ∈ AutD. On the other hand the function FB , where B(λ) = λ 2, λ ∈ D is a mapping of the form Ω2 ∋ A 7→ A 2 ∈ Ω2, which is not a proper holomorphic one – it maps T0 into 0. The structure of the group of automorphisms of Ωn has been been studied in several papers (see e.g. [Ran-Whi 1991] and [Ros 2003]). However, it is still not understood completely. Let us mention only that Aut(Ωn) is not transitive. Moti- vated by the results of the mentioned papers we are going to examine the structure of the class of proper holomorphic self-mappings of the spectral unit ball. It turns out that we get an analogue of the theorem of Alexander on proper holomorphic self mappings of the Euclidean ball in Cn stating that there are no non-trivial proper holomorphic self maps in the unit ball Bn, n ≥ 2 (see [Ale 1977]). In the paper we need some properties of proper holomorphic mappings between complex analytic sets that could be found in [Chi 1989] and [ Loj 1991]. The book [Rud 1980] may serve as another reference on proper holomorphic mappings (mostly between open sets in Cn). Theorem 1. Let F : Ωn 7→ Ωn be a proper holomorphic mapping, n ≥ 2. Then F is an automorphism. The following necessary form of proper holomorphic mappings of the spectral ball, which is a simple consequence of the description of the set of proper holomor- phic self-mappings of the symmetrized polydisc, will be crucial in our considerations and justifies the introducing of the condition (1) below. Proposition 2 (see Theorem 17 in [Edi-Zwo 2005]). Let F : Ωn 7→ Ωn be a proper holomorphic mapping. Then there is a non-constant finite Blaschke product B such that σ(F (A)) = πn(B(ζ1), . . . , B(ζn)), where A ∈ Ωn and σ(A) = πn(ζ1, . . . , ζn), ζ1, . . . , ζn ∈ D. In view of Proposition 2 it is natural that we study below the holomorphic mappings F : Ωn 7→ Ωn such that there is a function f ∈ O(D,D) with the property (1) σ(F (A)) = πn(f(ζ1), . . . , f(ζn)), A ∈ Ωn is such that σ(A) = πn(ζ1, . . . , ζn). We start with the following lemma. Lemma 3. Let F ∈ O(Ωn,Ωn) be such that F (0) = 0 and (1) is satisfied for f ∈ O(D,D) (then necessarily f(0) = 0) with f ′(0) 6= 0. Then F ′(0) is a linear isomorphism (of Mn). Proof. Put α := f ′(0). Fix V ∈ Mn. Let πn(µ) = σ(V ) for some µ = (µ1, . . . , µn) ∈ n. We first prove that F ′(0)(V ) ∈ σ(V ). PROPER HOLOMORPHIC MAPPINGS OF THE SPECTRAL UNIT BALL 3 Actually, σ(ζV ) = πn(ζµ), ζ ∈ C. Consequently, σ(F (ζV )) = πn(f(ζµ1), . . . , f(ζµn)) for sufficiently small ζ ∈ D and then F (ζV ) (f(ζµ1), . . . , f(ζµn)) Passing with ζ to 0 we get that F ′(0)(V ) ∈ Tπn(αµ). Therefore, Φ := F ′(0) : Mn 7→ Mn is a linear mapping such that (3) Φ(Tz) ⊂ Tz , z ∈ C To finish the proof of the lemma it is sufficient to show that Φ is a monomorphism. Suppose that it does not hold. Then there is an N ∈ Mn, N 6= 0 such that Φ(N) = 0. Because of (3) we get that N ∈ T0. But then there is an M ∈ T0 such that N + M 6∈ T0. In particular, T0 6∋ Φ(N + M) = Φ(N) + Φ(M) = Φ(M) ∈ T0 – contradiction. Lemma 4. Let F ∈ O(Ωn,Ωn) be such that (1) is satisfied with f(0) = 0 and f ′(0) 6= 0. Then F−1(0) ∩ T0 ⊂ {0}. Proof. Suppose that there is an A ∈ T0, A 6= 0 such that F (A) = 0. It follows from the Jordan decomposition theorem that there are linearly independent vectors v1, v2 ∈ Mn such that A(v2) = v1, A(v1) = 0 (at the moment it is essential that n ≥ 2). Let (v1, . . . , vn) be a vector base of C n. Define the linear mapping V : Cn 7→ n (equivalently an element from Mn) as follows V (v2) := v1, V (v1) := v2 and V (vj) := 0, j = 3, . . . , n. Then (A+ ζV ) 2(vj) = ζ(1 + ζ)vj , j = 1, 2. Consequently, the properties of the spectral radius imply that |ζ||1 + ζ| ≤ ρ((A + ζV )2) ≤ ρ2(A + ζV ). For any ζ ∈ C there are µj(ζ) ∈ C, j = 1, . . . , n such that πn(µ1(ζ), . . . , µn(ζ)) = σ(A + ζV ). Then σ(F (A + ζV )) = πn(f(µ1(ζ)), . . . , f(µn(ζ))) for ζ ∈ D small. We also know that max{|µj(ζ)| : j = 1, . . . , n} = ρ(A + ζV ) ≥ |1 + ζ| and ρ(A + ζV ) → 0 as ζ → 0. Note that ρ( F (A+ζV ) ) → ρ(F ′(A)(V )) as ζ → 0. But on the other hand F (A + ζV ) = max |f(µj(ζ))| : j = 1, . . . , n which tends to infinity as ζ → 0 because f ′(0) 6= 0 – a contradiction. � Note that the results proven so far referred to a larger class of mappings than only proper ones. It is possible that they may have application to the study of more general mappings than only the proper holomorphic ones. First we show simple results on the geometry of the sets Tz . 4 W LODZIMIERZ ZWONEK Lemma 5. The set of non-derogatory matrices is dense in Tz for any z ∈ Gn. Proof. Fix z ∈ Gn. Let A ∈ Tz. Without loss of generality assume that A is not non-derogatory. Choose a vector base B in which A has Jordan normal form. Let us study two different blocks corresponding to the same λ (and the corresponding vectors from B: v1, . . . , vk, w1, . . . , wl, k, l ≥ 1). Let Av1 = λv1, Avj = λvj + vj−1, j = 2, . . . , k, Aw1 = λw1, Awj = λwj + wj−1, j = 1, . . . , l. For ǫ > 0 define Bv1 := λv1 + ǫwl and for all other elements of the base B define Bv := Av, v ∈ B, v 6= v1. This easily gives an approximation of A with matrices still in Tz having one block corresponding to the eigenvalue λ less than in the original matrix. Repeating this procedure for all Jordan blocks having the same eigenvalues we easily construct a sequence of non-derogatory matrices in Tz tending to A. � Lemma 6. The set of non-derogatory matrices in Tz is connected and open in Tz for any z ∈ Gn. Proof. Fix z ∈ Gn. The non-trivial part of the lemma is the connectedness. Let us fix a system of numbers (ζ1, . . . , ζn) and the sequence of indices 1 = k1 < k2 < . . . < kl+1 = n + 1 where ζkj = ζkj+1 = . . . = ζkj+1−1, j = 1, . . . , l (and such that no other equalities between different ζj ’s hold) and πn(ζ1, . . . , ζn) = z. And now for any vector base (v1, . . . , vn) of C n we define the matrix A (more precisely, an element in Tz ⊂ Mn) as follows Avt = ζjvt + vt−1, j = 1, . . . , l, kj + 1 ≤ t < kj+1, Avkj = ζjvkj , j = 1, . . . , l. Note that the above mapping is continuous and its image equals the set of non-derogatory matrices in Tπn(ζ1,... ,ζn). This together with the fact that the set of all vector basis is connected in Mn completes the proof. � As a simple corollary of the results on the set of non-derogatory matrices in the sets Tz we get the following. Corollary 7. For any z ∈ Gn the set Tz is an analytic irreducible set of codimen- sion n. At the moment we are ready to move to the proof of our main result. Proof of Theorem 1. First recall that when F : Ωn 7→ Ωn is a proper holomor- phic mapping then there is a finite non-constant Blaschke product B such that σ(F (A)) = πn(B(ζ1), . . . , B(ζn)), where σ(A) = πn(ζ1, . . . , ζn). In particular, F (Tπn(ζ1,... ,ζn)) ⊂ Tπn(B(ζ1),... ,B(ζn)), ζj ∈ D, j = 1, . . . , n. But the proper- ness of F implies even that the equality F (Tπn(ζ1,... ,ζn)) = Tπn(B(ζ1),... ,B(ζn)), ζj ∈ D, holds – it is sufficient to note that Tz is always connected. Even more, F|Tπn(ζ1,... ,ζn) : Tπn(ζ1,... ,ζn) 7→ Tπn(B(ζ1),... ,B(ζn)) is open and proper for any ζj ∈ D, j = 1, . . . , n. We claim that for any λ0 ∈ D such that B ′(λ0) 6= 0 (note that such points exist) the function (4) F|Tπn(λ0,... ,λ0) is injective. Actually, making use of the automorphisms of Ωn and the properties of Blaschke products we may assume that λ0 = 0, B(0) = 0 and B ′(0) 6= 0. It follows from Lemma 4 that F−1(0) ∩ T0 = {0}. In particular, F (0) = 0. Now Lemma 3 applies and we get that F ′(0) is an isomorphism. Consequently, F is locally invertible near 0. Note that there is a neighborhood V of 0 such that #F−1(A) ∩ T0 = 1 for any A ∈ V ∩ T0. Otherwise there would exist T0 ∋ A ν , Ãν such that Aν 6= Ãν and PROPER HOLOMORPHIC MAPPINGS OF THE SPECTRAL UNIT BALL 5 F (Aν) = F (Ãν) → 0. But the properness of F implies that (taking if necessary a subsequence) either both sequences (Aν), (Ãν) converge to 0 or at least one of the sequences converges to a non-zero element à from T0 such that F (Ã) = 0. In the first case we contradict the local invertibility of F near 0 and in the second case we get two points in F−1(0) ∩ T0 – a contradiction, too. Now the analyticity of the set {A ∈ T0 : #F −1(A)∩T0 = 1} (see e.g. [ Loj 1991], Section V.7.1) (the mapping F|T0 : T0 7→ T0 is proper and open) and the fact that T0 is a cone shows that the mapping F|T0 : T0 7→ T0 is a one-to-one mapping. Now we prove the following property. (5) Let zν → z0 ∈ Gn, where z ν ∈ Gn be such that F|Tzν is not injective for any ν then F|T is not injective. Actually, to prove (5) note that because of the properties of proper holomorphic mappings we may assume that there are two sequences of non-derogatory matrices (Aν), (Ãν) with Aν 6= Ãν lying in Tzν , F (A ν) = F (Ãν) and tending to matrices A, à ∈ Tz0 such that A is non-derogatory and F|T is locally invertible in A. In the case A 6= à we are done, so assume that A = Ã. The local invertibility of F|T near A implies that there is a neighborhood U of A such that F|Tzν∩U is invertible for ν large enough, which contradicts the equality F (Aν) = F (Ãν). We claim that (6) for any z ∈ Gn the mapping F|Tz is injective. Put U := {z ∈ Gn : F|Tz is injective}. The fact that F|T0 is injective shows that U is not empty. The property (5) shows that U is open. To see that U is closed in Gn take a sequence U ∋ z ν → z ∈ Gn. Suppose that z 6∈ U . Then there are different non-derogatory matrices A1, . . . , Ak ∈ Tz , C ∈ Tw with k ≥ 2 such that F−1(C) ∩ Tz = {A1, . . . , Ak}. We may choose arbitrarily small open connected neighborhoods U1, . . . ,Uk,V of A1, . . . , Ak, C such that Ul ∩ Up = ∅ for l 6= p, Uj ∩ Tz̃ is connected, V ∩ Tw̃ is connected for any z̃, w̃ ∈ Gn, j = 1, . . . , k and F−1(V) ⊂ j=1 Uj . Consequently, for any ν there are pairwise disjoint sets F (Uj ∩ Tzν )∩V , j = 1, . . . , k that are open in V ∩ Twν and that are non-empty for ν large enough. Now the properness of F shows that for V sufficiently small the sets F (Uj ∩ Tzν ), j = 1, . . . , k cover the whole set V ∩ Twν for ν large enough; thus contradicting the connectedness of V ∩ Twν . Since Gn is connected we get that U = Gn, so (6) is satisfied. Let m denote the degree of B. We claim that m = 1. Suppose that m ≥ 2. Note that taking instead of F the composition of many F ’s we may assume that m ≥ n. There is a point ζ0 ∈ D such that B −1(ζ0) = {ζ1, . . . , ζm}. Composing, if necessary, with automorphisms of Ωn we may assume that ζ0 = 0. Recall that Tπn(ζ1,... ,ζn) is an n2 − n dimensional submanifold. Choose A ∈ Tπn(ζ1,... ,ζn) such that F (A) = 0. Let f := F|Tπn(ζ1,... ,ζn) : Tπn(ζ1,... ,ζn) 7→ T0. Then f is a holomorphic bijective mapping. Let us fix a regular point C in T0. Then the function ϕ : C ∋ λ 7→ f−1(λC) ∈ Tπn(ζ1,... ,ζn) is holomorphic on C \ {0} (the points λC, λ ∈ C \ {0}, are regular in T0) and continuous at 0 with ϕ(0) = A (use the properness and injectivity of f). Con- sequently, ϕ is holomorphic on C. Note that (F ◦ ϕ)(λ) = λC, λ ∈ C, so the tangent space to Tπn(ζ1,... ,ζn) at A i.e. TA(Tπn(ζ1,... ,ζn)) is mapped onto H := F ′(A)(TA(Tπn(ζ1,... ,ζn))), which contains the vector F ′(A)(ϕ′(0)) = (F ◦ϕ)′(0) = C. 6 W LODZIMIERZ ZWONEK Consequently, H contains all regular points of T0, so it contains the whole T0, which contains n2−n+ 1 linearly independent vectors contradicting the fact that H is at most n2 − n dimensional vector space. Consequently, we have proven that #F−1(C) = 1 for C ∈ Ωn showing that F is an automorphism. � Acknowledgment. The author wishes to express his gratitude to Witold Jarnicki for fruitful conversations on the properties of proper holomorphic mappings between analytic sets. References [Ale 1977] H. Alexander, Proper holomorphic mappings in Cn, Indiana Univ. Math. J. 26 (1977), 137–146. [Chi 1989] E. Chirka, Complex Analytic Sets, Kluwer, 1989. [Edi-Zwo 2005] A. Edigarian, W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), 364–374. [ Loj 1991] S. Lojasiewicz, Introduction to complex analytic geometry. Translated from the Polish by Maciej Klimek, Birkhäuser Verlag, Basel, 1991. [Nik-Tho-Zwo 2007] N. Nikolov, P. J. Thomas, W. Zwonek, Discontinuity of the Lempert function and the Kobayashi-Royden metric of the spectral ball, preprint. [Ran-Whi 1991] T. J. Ransford, M. C. White, Holomorphic self-maps of the spectral unit ball, Bull. London Math. Soc. 23 (1991), 256–262. [Ros 2003] J. Rostand, On the automorphisms of the spectral unit ball, Studia Math. 155 (2003), 207–230. [Rud 1980] W. Rudin, Function theory in the unit ball of Cn (1980), Grundlehren der Mathematischen Wissenschaften 241 Springer-Verlag, New York-Berlin. Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland E-mail address: [email protected]

0704.0615

Parsimony via concensus

Title: Parsimony via consensus Trevor C. Bruen and David Bryant Running Head: Parsimony via consensus Key Words: maximum parsimony, compatibility, supertree, matrix representation with parsimony, homoplasy Corresponding Author: David Bryant Abstract The parsimony score of a character on a tree equals the number of state changes required to fit that character onto the tree. We show that for unordered, reversible characters this score equals the number of tree rearrangements required to fit the tree onto the character. We discuss implications of this connection for the debate over the use of consensus trees or total evidence, and show how it provides a link between incongruence of characters and recombination. Introduction The (Fitch) parsimony length of a character on a tree equals the minimum number of state changes (substitutions) required to fit the character onto a tree (Fitch, 1971). We turn this definition on its head and show how the parsimony length of a character equals the minimum number of changes in the tree required to fit the tree onto the character. This may be a back-to-front way to look at parsimony, but it is also a useful one. We detail two applications of the result. The first application is that this reformulation of parsimony provides a closer link between parsimony based analysis and supertree methods. We demonstrate that the maximum parsi- mony tree can be viewed as a type of median consensus tree, where the median is computed with respect to the SPR distance (see below). As well, the result shows how to conduct a parsimony based analysis not just on characters but on trees, without having to recode the trees as binary character matrices. This opens the way to a hybrid between the consensus approach and the total evidence approach, where the data is a mix characters, trees, and subtrees. The second application of our observation on parsimony is to the analysis of pairs of characters. We show that the score of the maximum parsimony tree for two characters is a simple function of the smallest number of recombinations required to explain the incongru- ence between the characters without homoplasy. This result provides the basis of a highly efficient test for recombination (Bruen et al., 2006). Here and throughout the paper we assume that all phylogenetic trees are fully resolved (bifurcating) and that by ‘parsimony’ we refer to Fitch parsimony, where the character states are unordered and reversible. Some of the results presented here can be extended to other forms of parsimony, and possibly to incompletely resolved trees (Bruen, 2006), lie beyond the scope of this paper. Note that in this paper we are dealing with unrooted SPR rearrangements, which are those used in tree searches. There is a related, but distinct, concept of rooted SPR rearrangements, where the rearrangements are restricted to obey a type of temporal constraint Song (2003). It is this latter class of rooted SPR rearrangments that are used to model lateral gene transfers and recombination. It would be a worthwhile, but challenging, goal to investigate whether any of the results on unrooted SPR rearrangements in this paper can be extended to rooted SPR rearrangements. Linking Parsimony with SPR A subtree-prune and regraft (SPR) rearrangement is an operation on phylogenetic trees whereby a subtree is removed from one part of the tree and regrafted to another part of the tree, see Figure 1, (Felsenstein, 2004; Swofford et al., 1996). These SPR rearrangements are widely used by tree searching software packages like PAUP (Swofford, 1998) and Garli (Zwickl, 2006). The SPR distance between two trees can be defined as the minimal num- ber of SPR rearrangements required to transform one tree into the other (Hein, 1990; Allen and Steel, 2001; Goloboff, 2007). For example, the two trees T1 and T3 in Figure 1 can be transformed into each other using a minimum of two SPR rearrangements, via the tree T2, so their SPR distance is two. Figure 1: Two trees, T1 and T3, separated by two SPR rearrangements via the intermediate tree T2. A binary character of parsimony length 3 is indicated on tree T1 by the node colours. The character is compatible with a tree (T3) within SPR distance two, illustrating Theorem 1.. The parsimony length of a character on a tree is the minimum number of steps required to fit that character on the tree, as computed by the algorithm of (Fitch, 1971). We will always assume unordered reversible characters The length of a character Xi on a tree T is denoted `(Xi, T ). A character with ri states therefore has parsimony length at least (ri−1), as every state not at the root has to arise at least once. A character is compatible with a tree if it requires at most (ri − 1) changes on that tree (Felsenstein, 2004). So far, one thinks of fitting a character onto a tree; we could just as well fit the tree onto the character. If the character and the tree are compatible then we have a perfect fit. When there is not a perfect fit we can measure how many SPR rearrangements are required to give a tree that does make a perfect fit. It turns out that this measure gives an equivalent score to parsimony length. More formally: Theorem 1. Let Xi be a character with ri states and let T be a fully resolved phylogenetic tree. It takes exactly `(Xi, T ) − (ri − 1) SPR rearrangements to transform T into a tree compatible with Xi. The result still holds if Xi has some missing states. As an example, consider the character X1 mapping taxa A,C,D,F to one and B,E,G to zero. The length of this character on tree T1 of Figure 1 is three, and the number of SPR rearrangements needed to transform T1 onto some tree T3 compatible with with X1 is two. Note that there could be other trees compatible with X1 are are further than two SPR rearrangements away: the result only gives the number of rearrangements required to obtain the closest tree. Once stated, the theorem is not too difficult to prove. First show that performing an SPR rearrangement decreases the length by at most one step. Hence it takes at least `(Xi, T )− (ri − 1) SPR rearrangements to transform T into a tree compatible with the character Xi. Then show that this is the minimum required. A formal proof is presented in the Appendix. A restricted (binary character) version of this theorem was proved in (Bryant, 2003). The theorem captures an issue that is central to the interpretation of incongruence: is an observed incongruence to be explained by positing homoplasy or by modifying the tree. Define the SPR distance from a tree T to a character Xi to be the SPR distance from T to the closest tree T ′ that is compatible with Xi. Theorem 1 then tells us that the SPR distance from T is equal to the difference between the length `(Xi, T ) of Xi on T and the minimum possible length of Xi on any tree. Consensus trees, supertrees and parsimony In their insightful overview of supertree methods Thorley and Wilkinson (2003) characterise a family of supertree methods that all minimise a sum of the form d(T, ti) = d(T, t1) + d(T, t2) + ...+ d(T, tn). (1) Here t1, t2, . . . , tn are the input trees and d(T, ti) is a measure of the distance between the input tree ti with the supertree T . There are many choice for the distance measure d, and it need not be the case that the distance measure satisfies the symmetry condition d(T, ti) = d(ti, T ). Gordon (1986) was the first to propose this description of supertrees. Many supertree methods can be described in these terms, including Matrix representation with parsimony (MRP) (Baum, 1992; Ragan, 1992); Minimum Flip supertrees Chen et al. (2006); the Median Supertree (Bryant, 1997), Majority Rule Supertree (Cotton and Wilkin- son, 2007) and the Average Consensus Supertree (Lapointe and Cucumel, 1997). Let ds(T,Xi) denote the SPR distance from T to the closest fully resolved tree Ti that is compatible with Xi. By Theorem 1, a maximum parsimony tree for X1, . . . , Xm is one that minimises the expression ds(T,Xi) = ds(T,X1) + ds(T,X2) + ...+ ds(T,Xm). (2) In this way, maximum parsimony is a form of median consensus. The significance of this observation doesn’t come from the fact that we can write the the parsimony score of T in the form (2); it is from the close connection with SPR distances, and from the way we will now use this connection to combine different kinds of data in the same theoretical framework. An SPR median tree for fully resolved trees t1, . . . , tn on the same leaf set is a tree T that minimises ds(T, ti) = ds(T, t1) + ds(T, t2) + ...+ ds(T, tn), where here d(T, ti) denotes the SPR distance from T to ti (Hill, 2007). We extend this directly to a supertree method by mimicking the situation for characters. Suppose that ti is a phylogenetic tree, not necessarily fully resolved, on a subset of the set of leaves. We say that a fully resolved tree T on the full set of leaves is compatible with ti (equivalently, T displays ti) if we can obtain ti from T by pruning off leaves and contracting edges. In this general situation, we let ds(T, ti) denote the SPR distance from T to the closest fully resolved tree Ti that is compatible with ti. This is equivalent to the more traditional definition whereby we first prune leaves off T then compute the distance from this pruned tree to ti. Now suppose that we have both characters and trees in the input. Both types of phylo- genetic data can be into an SPR median tree T , chosen to minimise the sum ds(T, ti) + ds(T,Xi). We have, then, a way to bring together both the supertree/consensus methodology and the total evidence methodology. In the case that the data comprises only trees, the tree is a median supertree; in the case that the data comprises only character data, the tree is the maximum parsimony tree. It is important to note the difference between this approach and the MRP method (Baum, 1992; Ragan, 1992), which could be used to combine trees and characters. In MRP, the trees are broken down into multiple independent characters. This is a problem, since the characters encoding a tree are nowhere near independent. In contrast, the SPR median tree approach treats a tree as a single indivisible unit of information. There is one critical issue that has been side-stepped: computation time. At present, computational limitations make the construction of SPR median trees infeasible for all but the smallest data sets: just computing the SPR distance between two trees is an NP-hard problem (Hickey et al., 2006). In contrast, Total evidence and MRP approaches are possible for at least 100 taxa. However there are now good heuristics for unrooted SPR distance Goloboff (2007) and exact special case algorithms Hickey et al. (2006) that could be applied to the problem. Below we describe a lower bound method for the SPR distance that should also aid construction of these SPR median trees. Parsimony on pairs of characters Another valuable application of Theorem 1 follows when we consider parsimony analysis of just two unordered and reversible characters. The concept of pairwise character compatibil- ity was introduced by Le Quesne (1969) (see also Felsenstein (2004)). Two binary characters with states 0 and 1 are incompatible if and only if all four combinations of 00, 01, 10, and 11 are present as combination of states for the two characters (Le Quesne, 1969). In a standard setting, character incompatibility is interpreted as implying that at least one of the charac- ters has undergone convergent or recurrent mutation (homoplasy). In other words, for every possible phylogeny describing the history of the two characters, at least one homoplasy is posited for one of the characters. Another interpretation of incompatibility of two characters is that characters evolved without homoplasy on two different phylogenies, where the phylo- genies differ by one or more SPR rearrangement (Sneath et al., 1975; Hudson and Kaplan, 1985). Define the total incongruence score i(X1, X2) for two multi-state unordered characters X1 and X2 (with r1 and r2 states respectively) as i(X1, X2) = min `(X1, T ) + `(X2, T ) − (r1 − 1)− (r2 − 1). (3) This is the maximum parsimony score of the two characters X1, X2 minus the minimum number of changes required for each character. Equation (3) generalises the incompatibility notion for two binary characters. It is also equivalent to the incongruence length difference statistic applied to only two characters (Farris et al., 1995). Importantly, the total incongru- ence score can be computed rapidly (Bruen and Bryant, 2006). The following consequence of Theorem 1 strengthens the connection between incongruence and SPR rearrangements. Theorem 2. The total incongruence score i(X1, X2) for two characters equals the minimum SPR distance between a tree T1 and T2 such that X1 is compatible with T1 and X2 is compatible with T2. Although the notion of total incongruence for two characters has been considered before in the context of character selection and weighting (Penny and Hendy, 1986), it has not been considered in the context of genealogical similarity. Essentially, Theorem 2 shows that the total incongruence score equals the minimum possible number of SPR rearrangements that could have occurred between the phylogenetic histories for both characters, assuming that the characters have different histories with which they are each perfectly compatible. Indeed, Theorem 2 suggests a natural way to interpret genealogical similarity between two characters, which we have used to develop a powerful test for recombination (Bruen et al., 2006). Choosing two characters from two different genes (which have possibly different histories) gives a simple approach to identify the distinctiveness of the histories of the genes. We can also apply Theorem 2 to obtain a lower bound on an SPR distance between two trees. Suppose that we have two trees T1 and T2 and we wish to obtain a lower bound on the SPR distance d(T1, T2) between the two trees. If we choose any character X1 convex on T1 and any character X2 convex on T2 then, by Theorem 2, we have that i(X1, X2) ≤ d(T1, T2). By carefully choosing X1 and X2 we can obtain tighter bounds. One natural starting point for X1 and X2 is the four or five character encodings described by (Semple and Steel, 2002; Huber et al., 2005). Discussion and extensions We have presented a reformulation of parsimony that is, in some way, dual to the standard definitions. Instead of measuring how well a character fits onto a tree we look at how well the tree fits onto the character. A consequence of this new perspective is that we can combine trees and character data using one general SPR framework, and we also obtain new results connecting incongruence measures and recombination. Nevertheless, it is not immediately clear how the new reformulation can be interpreted in itself. Trees compatible with X1 Trees compatible with X2 Trees compatible with X3 Trees compatible with X4 Trees compatible with X(m-1)Trees compatible with Xm d(T,t1) Figure 2: Cartoon representation of parsimony in terms of tree rearrangements. Each characterXi gives a ‘cloud’ of trees containing those trees compatible withXi. The maximum parsimony tree is then the tree closest to these clouds under the SPR distance. One aid in this direction is to consider the information a single character, or tree, rep- resents. Given a single character, we can imagine a cloud of trees comprising exactly those trees compatible with the character (Figure 2). If we are told that this character evolved without homoplasy, then we know that the true evolutionary tree must be contained some- where within the cloud. However as there is only one character there is a lot of uncertainty regarding the tree, so there are a lot of trees in the clouds. Now suppose we have multiple characters, each with its own cloud. There may not be a single tree contained in the inter- section of all of these clouds. Instead, we search for a tree that is close as possible to all of the clouds. The distance from T to the cloud associated to character Xi is exactly ds(T,Xi), so by Theorem 1 a tree closest to all of the clouds is a maximum parsimony tree. Each cloud represents the uncertainty around each piece of data (tree or character). We note that several of the results in this article can be extended, for details. Firstly, both Theorems 1 and 2 are both valid if we replace the SPR distance with the tree bisection and reconnection (TBR) distance. In a TBR rearrangement, a subtree is removed from the tree and then reattached elsewhere in a tree, the difference with SPR being that we can reattach using any of the nodes in the subtree (Allen and Steel, 2001; Felsenstein, 2004). The TBR distance between two trees is the minimum number of TBR rearrangements required to transform one tree into the other. That Theorems 1 and 2 hold for the TBR distance might seem surprising, since the TBR distance between two trees is always less than, or equal to, the SPR distance between the trees. However the extension follows by a tiny change to the proof of Theorem 1, noting that a TBR move can still only reduce the parsimony score of a character by at most one. We have also explored extensions of the result to other distances between trees, notably the Robinson-Foulds or partition distance and the Nearest Neighbor Interchange distance, though the connections are not so clear. See Bruen (2006) for details. Acknowledgements We would like to thank Mike Steel, Sebastien Böcker, Olaf Bininda Emonds, Pablo Golloboff, Mark Wilkinson and an anonymous referee for their valuable suggestions. This research was partially supported by the New Zealand Marsden Fund. References Allen, B. and M. Steel. 2001. Subtree transfer operations and their induced metrics on evolutionary trees. Annals of Combinatorics 5:1–13. Baum, B. 1992. Combining trees as a way of combining datasets for phylogenetic inference, and the desirability of combining gene trees. Taxon 41:3–10. Bruen, T. 2006. Discrete and statistical approaches to genetics. Ph.D. thesis McGill Univer- sity School of Computer Science. Bruen, T. and D. Bryant. 2006. A subdivision approach to maximum parsimony. Annals of Combinatorics In Press. Bruen, T., H. Philippe, and D. Bryant. 2006. A simple and robust statistical test to detect the presence of recombination. Genetics 172:1–17. Bryant, D. 1997. Building trees, hunting for trees and comparing trees. Ph.D. thesis Dept. Mathematics, University of Canterbury. Bryant, D. 2003. A classification of consensus methods for phylogenetics. Pages 163–184 in Bioconsensus vol. 61 of DIMACS. American Math Society, Providence, RI. Bryant, D. 2004. The splits in the neighborhood of a tree. Annals of Combinatorics 8:1–11. Chen, D., O. Eulenstein, D. Fernandez-Baca, and M. Sanderson. 2006. Minimum-flip su- pertrees: Complexity and algorithms. IEEE/ACM Trans. Comput. Biol. Bioinformatics 3:165–173. Cotton, J. and M. Wilkinson. 2007. Majority-rule supertrees. Systematic Biology 56:445– Farris, J. S., M. Källersjö, A. G. Kluge, and C. Bult. 1995. Constructing a significance test for incongruence. Systematic Biology 44:570–572. Felsenstein, J. 2004. Inferring Phylogenies. Sinauer Associates. Fitch, W. M. 1971. Towards defining the course of evolution: Minimum change for a specific tree topology. Systematic Zoology 20:406–416. Goloboff, P. 2007. Calculating SPR distances between trees. Cladistics Online early access. Gordon, A. D. 1986. Consensus supertrees: the synthesis of rooted trees containing overlap- ping sets of labeled leaves. Journal of Classification 3:335–348. Hein, J. 1990. Reconstructing evolution of sequences subject to recombination using parsi- mony. Mathematical Biosciences 98:185–200. Hickey, G., F. Dehne, A. Rau-Chaplin, and C. Blouin. 2006. The computational complexity of the unrooted subtree prune and regraft distance. Tech. Rep. CS-2006-06 Faculty of Computer Science, Dalhousie University. Hill, T. 2007. Development of New Methods for Inferring and Evaluating Phylogenetic Trees. Ph.D. thesis Uppsala Universitet. Huber, K. T., V. Moulton, and M. A. Steel. 2005. Four characters suffice to convexly define a phylogenetic tree. SIAM Journal on Discrete Mathematics 18:835–843. Hudson, R. R. and N. L. Kaplan. 1985. Statistical properties of the number of recombination events in the history of a sample of dna sequences. Genetics 111:147–64. Lapointe, F.-J. and G. Cucumel. 1997. The average consensus procedure: combination of weighted taxa containing identical or overlapping sets of taxa. Systematic Biology 46:306– Le Quesne, W. J. 1969. A method of selection of characters in numerical taxonomy. System- atic Zoology 18:201–205. Penny, D. and M. Hendy. 1986. Estimating the reliability of evolutionary trees. Molecular Biology and Evolution 3:403–17. Ragan, M. A. 1992. Phylogenetic inference based on matrix representations of trees. Molec- ular Phylogenetics and Evolution 1:53–58. Semple, C. and M. Steel. 2002. Tree reconstruction from multi-state characters. Advances in Applied Mathematics 28:169–84. Semple, C. and M. Steel. 2003. Phylogenetics. Oxford University Press. Sneath, P., M. Sackin, and R. Ambler. 1975. Detecting evolutionary incompatibilities from protein sequences. Systematic Zoology 24:311–332. Song, Y. S. 2003. On the combinatorics of rooted binary phylogenetic trees. Ann. Comb. 7:365–379. Swofford, D., G. Olsen, P. Waddell, and D. Hillis. 1996. Molecular sytematics chap. Phylo- genetic Inference, Pages 407–514. Sinauer Associates, Inc. Swofford, D. L. 1998. PAUP*. Phylogenetic Analysis using Parsimony (*and other methods). Sinauer Associates, Sunderland, Massachusetts. Thorley, J. L. and M. Wilkinson. 2003. A view of supertree methods. Pages 185–194 in Bioconsensus (F. Roberts, ed.) vol. 61 of DIMACS series in discrete mathematics and theoretical computer science The American Mathematical Society, New York. Zwickl, D. 2006. Genetic algorithm approaches for the phylogenetic analysis of large biological sequence datasets under the maximum likelihood criterion. Ph.D. thesis University of Texas at Austin. Appendix Refer to (Semple and Steel, 2003) for a detailed description of the notation. The first observation is that an TBR rearrangement of a tree increases the length of a character by at most one. As SPR rearrangements are a special case of TBR rearrangements, the same result holds for SPR. Lemma 1. Let T be a fully resolved phylogenetic tree and Xi an unordered reversible charac- ter. Let T ′ be a phylogenetic tree that differs from T by a single TBR rearrangement. Then `(χ, T ′) ≤ `(χ, T ) + 1. Proof. The proof of Lemma 5.1 in (Bryant, 2004) for binary characters applies directly to the multistate case. Let dSPR(T, T ′) denote the unrooted SPR distance between two phylogenetic trees T and Theorem 1 LetXi be a character with ri states and let T be a fully resolved phylogenetic tree. It takes exactly `(Xi, T ) − (ri − 1) SPR rearrangements to transform T into a tree compatible with Xi. The result still holds if Xi has some missing states. Proof. Let T ′ be a fully resolved phylogenetic tree compatible with Xi for which dSPR(T, T is minimized and let m = dSPR(T, T ′). Then there exists a sequence of trees T ′ = T0, ..., Tm = T such that every adjacent pair of trees in the sequence differ by exactly one SPR rear- rangement. By Lemma 1 the existence of this sequence implies that `(T,Xi) − `(T ′, Xi) ≤ dSPR(T, T ′) and since Xi is compatible with Xi we have `(T ′, Xi) = ri − 1, giving `(T,Xi)− (ri − 1) ≤ dSPR(T, T ′). For the other direction, we show that we can construct a sequence of `(T,Xi) − (ri − 1) SPR rearrangements that transform T into a tree T ′ compatible with Xi. Firstly, if `(T,Xi) − (ri − 1) = 0, then T is compatible with Xi so the proof is finished. Otherwise, let X̂i be an assignment of states to internal nodes that minimises the number of state changes (that is, a minimum extension of Xi). Then since Xi is not convex on T there exist three vertices u, v and w, where {u, v} ∈ E(T ), v lies on the path from u to w and X̂i(u) = X̂i(w) 6= X̂i(v). Perform an SPR rearrangement by removing edge {u, v}, supressing the v vertex and creating a new edge {u, t} where t is a new vertex on an edge adjacent to w. Furthermore, set X̂i(t) = X̂i(w). Then the number of edges on which a change has occurred has decreased by 1 thereby decreasing the parsimony length by 1. This procedure can be repeated until the parsimony length equals ri − 1, constructing the desired sequence of trees and completing the proof. Let T be a maximum parsimony phylogenetic tree for X1 and X2 and let Theorem 2 The total incongruence score i(X1, X2) for two characters equals the mini- mum SPR distance between a tree T1 and T2 such that X1 is compatible with T1 and X2 is compatible with T2. Proof. Let T1 and T2 be any two trees compatible with X1 and X2 respectively. Then `(X1, T1) = r1−1 and by Theorem 1, `(X2, T1)− (r2−1) ≤ dSPR(T1, T2). We have then that i(X1, X2) ≤ `(X1, T1) + `(X2, T1)− (r1 − 1)− (r2 − 1) ≤ dSPR(T1, T2) and so i(X1, X2) is a lower bound for dSPR(T1, T2). We show that this bound can be achieved. Let T be a maximum parsimony tree for the pair of characters X1, X2. By Theorem 1 there exist two trees T1 and T2 such that T1 is compatible with X1, T2 is compatible with X2 and dSPR(T1, T ) + dSPR(T2, T ) = i(X1, X2), implying that dSPR(T1, T2) ≤ dSPR(T1, T ) + dSPR(T2, T ) ≤ i(X1, X2) and hence dSPR(T1, T2) = i(X1, X2).

0704.0617

Spectropolarimetric observations of the Ca II 8498 A and 8542 A lines in the quiet Sun

Draft version June 8, 2021 Preprint typeset using LATEX style emulateapj v. 08/22/09 SPECTROPOLARIMETRIC OBSERVATIONS OF THE CA II 8498 Å AND 8542 Å LINES IN THE QUIET SUN A. Pietarila , H. Socas-Navarro High Altitude Observatory, National Center for Atmospheric Research2, 3080 Center Green, Boulder, CO 80301, USA T. Bogdan Space Environment Center, National Oceanic and Atmospheric Administration, 325 Broadway, Boulder, CO 80305, USA Draft version June 8, 2021 ABSTRACT The Ca II infrared triplet is one of the few magnetically sensitive chromospheric lines available for ground-based observations. We present spectropolarimetric observations of the 8498 Å and 8542 Å lines in a quiet Sun region near a decaying active region and compare the results with a simulation of the lines in a high plasma-β regime. Cluster analysis of Stokes V profile pairs shows that the two lines, despite arguably being formed fairly close, often do not have similar shapes. In the network, the local magnetic topology is more important in determining the shapes of the Stokes V profiles than the phase of the wave, contrary to what our simulations show. We also find that Stokes V asymmetries are very common in the network, and the histograms of the observed amplitude and area asymmetries differ significantly from the simulation. Both the network and internetwork show oscillatory behavior in the Ca II lines. It is stronger in the network, where shocking waves, similar to those in the high-β simulation, are seen and large self-reversals in the intensity profiles are common. Subject headings: polarization, Sun: chromosphere, waves 1. INTRODUCTION Our understanding of solar magnetic fields outside active regions has increased signifi- cantly during the last years. This is due to new and better instrumentation (e.g., THEMIS, Paletou & Molodij 2001; VSM on SOLIS, Keller & The Solis Team 2001; Swedish Solar Telescope, Scharmer, Bjelksjo, Korhonen, Lindberg, & Petterson 2003; Solar Optical Telescope on Hinode, Shimizu 2004; and SPINOR, Socas-Navarro et al. 2006), better diagnostic techniques (see for example Bellot Rubio 2006 for a review on inversion techniques) and advanced numerical simulations (Stein & Nordlund 2006 and references therein). A large portion of the work has focused on photospheric magnetic fields. Only now we are starting to have adequate tools for investigating chromospheric magnetism in more detail. (For a review of chromospheric magnetic fields see Lagg (2005)). This is not surprising considering the numerous difficulties in observing chromospheric magnetic fields, interpreting the data, and performing realistic MHD simulations. There are two different sets of lines that are often used for chromospheric spectropolarimetry, the He I infrared (IR) triplet at 10830 Å, and the Ca II IR triplet at 8500 Å. Both line sets have their advantages and dis- advantages. The He I lines are formed over a relatively thin layer, and therefore observations can be inverted us- ing a simple Milne-Eddington model. The drawback is that while the formation range is fairly narrow, the pre- cise formation height remains uncertain, and the Milne- Eddington inversions do not give any information on 1 Institute of Theoretical Astrophysics, University of Oslo, P.O.Box 1029 Blindern, N-0315 Oslo, Norway 2 The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation. the atmospheric gradients. The lines are also sensitive to the Paschen-Back effect, which must be included in the inversion code (Socas-Navarro et al. 2004). Further- more, simulating the He I lines is difficult since coronal irradiation has a non-negligible effect on their formation (Andretta & Jones 1997). In contrast, the formation of the Ca II IR lines is fairly well understood (Lites et al. 1982). The broad Ca II lines sample a large region of the atmosphere, from the photosphere to the lower chromo- sphere. However, the Ca II lines are formed in nonLTE, making inversions considerably more cumbersome. Several investigations using the Ca II IR lines have studied intensity and velocity oscillations in the quiet Sun (e.g. Lites et al. 1982; Deubner & Fleck 1990) or, alternatively, magnetic fields in active regions (e.g. Socas-Navarro et al. 2000a). In both cases the lines have proven useful as diagnostics of the solar chromosphere. In this paper we present results of spectropolarimetric observations of two of the lines in an enhanced network region. We have both spatial maps and time series data. The observations show that the Ca II lines are formed in a very interesting region, namely the region where the atmosphere is transforming from a plasma dominated (β >> 1) to a magnetic field dominated (β << 1) regime in terms of dynamic force balance. Wave propagation is clearly seen in the highly dynamic magnetic regions, whereas the weakly magnetic internetwork is found to be less variable. Interestingly, the two Ca II lines ex- hibit significant differences even though in calculations they are formed fairly close together. The importance of gradients in the chromospheric network is clearly demon- strated by the prevalence of asymmetric Stokes V profiles in the data. The paper is arranged as follows: in § 2 the data and their reduction are addressed. Results of analyzing the data using different approaches are presented in § 3. We http://arxiv.org/abs/0704.0617v1 performed cluster analyses on the Stokes V profiles to classify them and to describe spatial patterns seen in the data. Statistics, such as profile amplitudes and asym- metries, are presented. The time dependent behavior of the lines in different network and internetwork regions is also discussed. In § 4 the observations are compared to simulations of the lines in a high plasma-β regime (Pietarila et al. 2006, hereafter P06). Finally, in § 5 the main results are summarized and discussed. 2. OBSERVATIONS AND DATA REDUCTION The Spectro-Polarimeter for INfrared and Optical Re- gions (SPINOR, Socas-Navarro et al. 2006) at the Dunn Solar telescope, Sacramento Peak Observatory, was used to observe two of the Ca II infrared triplet lines at 8498 Å and 8542 Å, as well as two photospheric Fe I lines at 8497 Å and 8538 Å. The setup included several other lines but because of computer problems only data from the two Ca lines which used the ASP TI TC245 cameras were recorded fully. The data have 256 points in both the wavelength and spatial position with a typical noise level of 6 × 10−4 Ic (1 σ deviation from the mean) and a spectral sampling of 25 mÅ. The pixel height corre- sponds to ≈ 0.38 arcseconds on the solar surface along the slit. We observed a quiet Sun region near disk center at S17.3 W32.1 on May 19, 2005 at 14:14 UT. An MDI- magnetogram of the region is shown in Figure 1. The slit was positioned in the vicinity of a decaying active re- gion, AR10763, but avoided flux concentrations from the active region (i.e., plages). A time series consisting of 99 time steps of short scans (3 slit positions), with a spacing of 0.375 arcseconds each, was acquired during variable seeing conditions. The cadence is ≈ 10 seconds (i.e., a given slit position was repeated every 30 s). The time series was followed by a 63 step raster centered around the position where the slit was during the time series. The raster step size was 0.375 arcseconds. Adaptive op- tics (AO, Rimmele 2000) were used during the observ- ing sequence but the compromised seeing conditions did not allow for continuous locking onto granulation. This caused the slit to jump occasionally, making the longest period with a stationary slit in the time series 17 time steps (8.5 min). The spatial resolution varied during the sequences being at best less than an arc second, but on average a factor of two worse. Standard procedures for flat field and bias were used for the data reduction. Instrumental polarization was re- moved using the available calibration data, as explained in Socas-Navarro et al. (2006). No absolute wavelength calibration was attempted because no suitable telluric lines are present. Instead a wavelength calibration using spatial pixels devoid of magnetic field was done by fitting the average spectrum to the Kitt Peak FTS-spectral at- las (Neckel & Labs 1984). The FTS atlas was also used to find the normalization factor for the intensities to the quiet Sun continuum intensities. Because of detector flat- field residuals and prefilter shape, the continua in the raw data from both detectors are tilted. The tilts were re- moved a posteriori by subtracting a linear fit (y = a+bλ) obtained by matching the continuum intensity levels to those of the FTS atlas. The data were analyzed using both the raster and time series for statistical purposes. The period when the slit was stationary on the solar disk was used to study the time-dependent behavior of the lines. Because of the short length of this period, we do not present any Fourier analysis of the data. To make a classification of Stokes V profile morphologies, we did cluster analyses based on a Principal Component Analysis (PCA) in a similar manner as the work of Sánchez Almeida & Lites (2000) and Khomenko et al. (2003) for photospheric lines. We computed amplitudes for Stokes I and V profiles. Be- cause the Ca line intensity profiles often exhibit strong self-reversals, no proxies for atmospheric velocities, such as lines’ centers of gravity or bisectors, are adequate. For those Stokes V profiles with amplitudes greater than 7 × 10−3 Ic, (i.e. ≥ 10σ), amplitude and area asymme- tries were also calculated. The amplitude asymmetry of a Stokes V profile is de- fined by (Mart́ınez Pillet 1997): ab − ar ab + ar , (1) where ab and ar are the unsigned extrema of the blue and red lobes of the Stokes V profile. The area asymmetry of a Stokes V profile is defined by (Mart́ınez Pillet 1997): σA = s V (λ)dλ |V (λ)|dλ , (2) where s is the sign of the blue lobe. Because of the broad, deep lines and large velocities (compared with the pho- tosphere) present in the chromosphere, the choice of the integration range for the area asymmetries is non-trivial for the Ca lines. We followed the same procedure as in P06. In the weak field regime the Stokes V profile is proportional to dI/dλ (strictly true only in the ab- sence of atmospheric velocity and magnetic gradients). Inspection of the data showed that most of the observed Stokes V profiles have roughly the same structures as the dI/dλ profiles. The intensity in the blue wing (λ0) of the line profile was matched with a point in the red wing (λ1) with the same intensity. The signal-to-noise in the intensity profiles is much higher than in the Stokes V profiles and also the slope is much steeper. This makes matching points with the same value more accurate in the intensity than in the Stokes V profiles. The selec- tion of a wavelength to start the integration range was made by choosing a wavelength point that is far enough from the line core so that self-reversals are not an issue. In our data this point, λ1, is at 600 mÅ from reference wavelength of line center. The same value was used in Magnetograms made from the 63 step scan are shown in Figure 2. The panels are in order of increasing for- mation height: Fe I 8497 Å, Fe I 8538 Å, Ca II 8498 Å and Ca II 8542 Å. The lower part of the slit was located above a flux concentration along the enhanced network and the upper part over an internetwork region with very little magnetic activity. The network becomes wider and more diffuse with increasing line formation height as de- scribed by Giovanelli (1980). Not all magnetic flux seen in the photosphere can be identified in the chromosphere Fig. 1.— MDI magnetogram showing the position of the slit for the time series and the map (rectangular region). The observed region was close to the decaying active region, AR10763. and vice versa. However, interpreting the chromospheric magnetograms is difficult due to the self-reversed features in the cores of the Ca line Stokes V profiles. 3. RESULTS In Figure 3, Stokes I and V spectra of the solar sur- face under the slit are shown for both Ca II lines as well as the two photospheric Fe I lines in the Ca lines’ wings (marked by arrows). Since the Fe I 8497 Å line is blended in the Ca line’s wing and the Fe I 8538 Å line is very close to the edge of the detector, no quantitative analysis is done for them. No signal above the noise was recorded in Stokes Q and U so they will not be addressed in what follows. Residual vertical fringing caused by the polar- ization modulator is visible in the Stokes V images. We chose not to try to remove the fringing since its’ ampli- tude is of the same order of magnitude as the noise. The network, present in the lower part of the slit, is associated with less absorption in the intensity profiles. Both Ca lines often show self reversals, which are usually stronger on the blue side of the line than in the red. The Stokes V profiles of both Ca lines have large, extended wings. At times, the profiles may have both polarities present on the blue side of the core but in almost all cases the far blue wing of the profile has the same po- larity (i.e., opposite sign) as the red wing. The Stokes I and V profiles of the chromospheric lines look distinc- tively different from the photospheric lines: the Ca lines have more structure, they are wider and exhibit more spatial variation than the photospheric Fe lines. Some differences are seen between the two Ca lines: the 8542 Å line is slightly broader, has more structure in the spec- Fig. 2.— Magnetograms of the map deduced by using the weak field method (Landi Degl’Innocenti 1992). The Stokes V signal is measured in units of Gauss. Vertical lines show the position of slit during the time series. First panel: Fe I 8497 Å, second panel: Fe I8538 Å, third panel: Ca II 8498 Å, third panel: Ca II 8542 Å. Location on the solar disk: S32.1, W17.3. The orientation of the magnetograms is 180 degrees from the MDI magnetogram in Figure 1. The plotted symbols (∗, ✸ and △) on the images show where the pixels discussed later in the text are located. tra and also stronger absorption than the 8498 Å line. The internetwork region, present in the upper part of the slit, is mostly devoid of Stokes V signal, and Stokes I is more homogeneous than in the network. Self reversals are usually not seen in the profiles. A small portion of the internetwork region has structures in Stokes I that are similar to those seen in the magnetic region: Stokes I is brighter than in the surrounding areas and the profiles show some self reversals. Closer inspection of the images reveals a visible, albeit a very small amplitude, Stokes V signal. The spatial patterns of Stokes I and V amplitudes and asymmetries in the two Ca lines (Figure 4) are fairly similar to one another. The network is clearly visible in the Stokes I and V amplitudes, though it is more diffuse in the 8542 Å line. There is a structure in the upper part of the map that is seen best in the 8498 Å intensity image. Parts of this structure appear also in both lines’ Stokes V amplitude and asymmetry images. The edges of the network have more asymmetric Stokes V profiles. This is seen clearly in the 8542 Å amplitude asymmetry. Photospheric velocities can be estimated from the lo- cations of the iron lines’ intensity minima. Except for a nearly constant offset caused by the convective down flows in the network, the internetwork and network re- gions have very similar spatial and temporal patterns. 3.1. Classification of the Stokes V profiles To classify the shapes of the 8498 Å and 8542 Å Stokes V profile pairs we used PCA, (Rees et al. 2000) and clus- ter analysis. The cluster analyses were performed sepa- rately for the map and the time series. Here we present a summary of the PCA procedure and cluster analysis for completeness. With the PCA we are able to reduce the number of parameters needed to describe a given profile. Each pro- file, S(λj), j = 1, ..., Nλ (Nλ is the number of wavelength points in the profile) is composed of a linear combination of eigenvectors ei(λj), i = 1, ..., n: S(λj) = Σ i=1ciei(λj), (3) where the ci are appropriate constants. The eigenvectors and constants for a given set of profiles are obtained from a singular value decomposition (SVD, Rees et al. 2000, Socas-Navarro et al. 2001) and form an orthonormal ba- sis with Nλ eigenvectors: j=1ei(λj)ek(λj) = δik. (4) Not all eigenvectors contain the critical information needed to reproduce the profiles, some of the eigenvec- tors carry information about the noise pattern. We can therefore truncate the series expansion and use only a small number of eigenvectors and corresponding coeffi- cients to reproduce a given profile. The PCA guarantees that when expansion of Eq. 3 is truncated at a given order m, the amount of information in the lower orders is maximized. We performed the SVD for the two 8498 Å and 8542 Å Stokes V profiles separately. The resulting orthonor- mal bases, and also the cluster analysis, depend on the subset of profiles used to construct it. Because of this we included all Stokes V profiles from pixels where the 8498 Å Stokes V amplitude is above 7 × 10−3Ic, alto- gether 13671 profiles. Visual inspection of the eigenvec- tors shows that the first 11 eigenvectors (approximately) Fig. 3.— Dispersed images of the slit. The arrows mark the locations of the two photospheric iron lines and the horizontal lines in the intensity images are the hairlines used to spatially coalign the two detectors. Wavelengths are measured from 8498 Å (left) and 8542 Å (right). contain relevant information about the actual shape of the profiles whereas the remainder are associated with the noise patterns. The Stokes V profile pairs, now described with 11 × 2 coefficients corresponding to the 11 × 2 eigenvectors in- stead of 102 (51 for each profile) wavelength points, were organized into a predefined number of clusters. Before doing this the vectors consisting of the 22 coefficients were standardized, i.e., no information of the absolute Stokes V amplitudes is left, only the relative amplitudes of the 8498 Å and 8542 Å profiles. Based on the values of the coefficients, 6 cluster centers were identified using the k-means method (MacQueen 1967). It starts with k random clusters, which through iterations are changed to minimize the variability within a cluster and maximize it between clusters. Each profile pair is then assigned to the nearest cluster center in the 22-dimensional Eu- clidean space. The choice of number of clusters used for the cluster analysis is non-trivial. Since each data point is described by 22 numbers we cannot visually distinguish patterns in the spatial distribution of the points. Instead the number of clusters was defined by trial and error, i.e. so that each profile type in the time series or map is rep- resented and each cluster is still clearly distinct from one another. For each cluster a profile was constructed using the eigenvectors and the averaged 2 × 11 coefficients of all profiles belonging to that cluster. Cluster analysis of the map shows the shapes of Stokes V profiles in network regions with different magnetic topologies, whereas the time series analysis describes how a set of profiles from a certain magnetic topology changes with time. The results for the map are shown in Figure 5. Above each profile is the percentage of all profiles belonging to the cluster, the mean distance in the Euclidean space Fig. 4.— Maps of the 8498 Å and 8542 Å lines’ Stokes I amplitudes, Stokes V amplitudes, area asymmetries and amplitude asymmetries in the raster scan. The horizontal line seen in the amplitude images is a hairline used to spatially coalign the detectors. The vertical line shows the position of the slit during the time series. Note that x-axis is stretched compared with y-axis. of the profiles to the cluster center, and the standard deviation of the mean. The smaller the distance to the cluster center, the more compact the cluster is and the better the cluster describes the profiles. The standard deviation is proportional to the spread of the distances in each cluster. In general, clusters with the least number of profiles belonging to them have larger mean distances. Three points can be deduced from the figure. First, asymmetric profiles should be common. In fact, they ap- pear to be more common than symmetric ones. Second, even though the two Ca lines are formed fairly close to one another (the 8498 Å line core optical depth is unity at about 1 Mm and the 8542 Å 0.2 Mm higher up in the ra- diation hydrodynamic simulations by Carlsson and Stein 1997), the 8498 Å and 8542 Å profiles in a given cluster are often clearly different from one another. Third, in all cluster profiles the far-red wings have the same polar- ity as the far-blue wings, indicating that the lower parts along the line-of-sight of the atmosphere, where the wings are formed, are dominated by a single magnetic polarity. The clusters differ from one another in several differ- ent ways: the degree of asymmetry, and distinct relation- ships between the 8498 Å and 8542 Å line profiles, rel- ative amplitudes, etc. However, quantitative measures, such as profile asymmetries, of the clusters do not nec- essarily represent the members of a given cluster very well. For example, the variation of Stokes V amplitude asymmetries within a cluster is large and the mean is not necessarily the same as that of the cluster profile. The cluster analysis retrieves qualitative similarities and gives a basis for morphological classification, rather than representing quantitative similarities within the data. To illustrate this point, Figure 6 displays histograms of the clusters showing the Stokes V amplitudes and asymme- tries for all profiles belonging to a given cluster. Shown in Figure 7 is the spatial distribution of the clusters. The smallest network patches often consist of only cluster 1 and cluster 2 profiles. The middle of the largest network patch is a mixture of different clusters. In most cases, the profiles at the edges of the network patches belong to cluster 1. This is the most common cluster consisting of 35.6 % of all the profile pairs in the map. The cluster 1 profiles are asymmetric, 8542 Å more so than 8498 Å, and they also have opposite signs of amplitude and area asymmetries. The amplitude histograms of profiles belonging to this cluster show that they have in general low amplitudes, as one might expect from profiles located at the edges of the network. The large amplitude asymmetry in the 8542 Å cluster profile is not seen in the observed profiles. In fact, only very few profiles exhibit such large asymmetries and there is only a slight tendency of the profiles having more often negative than positive amplitude asymmetries. The cluster area asymmetries are in better agreement with the observed profiles belonging to this cluster. Regions of cluster 2 profiles are often located adjacent to patches of cluster 1 profiles. The cluster 2 profiles account for 20.0 % of all profile pairs in the map. The cluster profiles are fairly antisymmetric. This is seen in the observed profiles as well: the asymmetry histograms tend to be narrow and only slightly offset from zero. The relative amplitudes of the two cluster profiles are very different: the 8498 Å amplitude is a factor 3 larger. The disproportionality is not as large in the observed profiles though the amplitude histograms show that in general 8498 Å has a larger amplitude than 8542 Å. The range of observed amplitudes is considerably larger than in cluster Of the profile pairs in the map 14.1 % belong to cluster 3. Also, these profiles are often found in regions close to the network edges by the patches of cluster 1 profiles. Both cluster profiles have multiple lobes and are asym- metric, 8498 Å more in amplitude and 8542 Å in area. This is also seen in the histograms of the observed asym- Fig. 5.— Results of cluster analysis of the Stokes V profile pairs in the map. Line on left is 8498 Å and on right 8542 Å. Shown are the percentages of profile pairs belonging to each cluster, and the mean distance and its standard deviation of the profiles to the cluster center. metries. There is a strong emission feature on the blue side of the line in the 8498 Å cluster profile. It is weaker in the 8542 Å profile. The histograms for cluster 3 are nearly identical to those of cluster 1. This illustrates how cluster analysis based on PCA is captures the qualitative differences in the line profiles. Cluster 4 consists of 13.5 % of the profile pairs. Most of the observed profiles belonging to this cluster are near to the middle of the largest network patches. The 8498 Å cluster profile is dominated by a strong emission feature in the blue lobe. This feature is not visible in the 8542 Å cluster profile. The overlap between the two lines’ ampli- tude histograms is fairly small. Also the cluster profiles show this difference in the relative amplitudes: 8542 Å has a significantly lower amplitude than 8498 Å. Except for the 8542 Å area asymmetry histogram, all histograms are centered around zero. The range of area asymmetries in the 8542 Å line is large and the distribution is skewed towards negative values. This trend in the 8542 Å area asymmetries is seen in several of the clusters. The patches of profiles belonging to the fifth cluster (9.6 %) are also found in the less homogeneous middle regions of the network elements. The 8498 Å cluster Fig. 6.— Stokes V statistics of the map clusters. The histograms are for all profiles belonging to the given cluster and the dotted vertical lines show the area and amplitude asymmetries for the cluster profiles. profile has a factor 2 lower amplitude than 8542 Å. This is not seen in the amplitude histograms but there is a large overlap between the two histograms. The cluster profiles are fairly antisymmetric and also the histograms of observed profile asymmetries are centered around zero. The 8542 Å area asymmetry is again the exception: it is centered around a negative value. Cluster 6 is the smallest cluster with 7.2 % of the pro- files. Patches of cluster 6 profiles are located in regions with cluster 4 and 5 profiles. The 8498 Å cluster profile is very similar to that of cluster 5. Like cluster 5, the 8542 Å cluster 6 profile has a factor 2 larger amplitude and the amplitude histograms overlap nearly entirely. All the cluster 6 histograms are very similar to cluster 5. The major difference between the two is that there is very little structure in the 8542 Å line profile. 3.2. Time-dependent behavior The cluster analysis results of the time series are shown in Figure 8, and the spatio-temporal distribution is cap- tured in Figure 9. The clusters consist of profiles at rest with varying degrees of structure, and profiles where the blue side is in emission. While there are temporal changes in the clusters, there are no clear periodic pat- terns visible. Most slit positions have a preferred cluster or in some cases the slit position is dominated by two clusters. Positions where more than 2 clusters are domi- nant are rare. Because the slit moved occasionally during the time series, no meaningful power spectra can be made from this data set. The time series data do however allow for a qualitative analysis of the time-dependent behav- ior. Comparing network and internetwork pixels reveals some interesting features: the network, especially in the intermediate flux regions, is very dynamic with propagat- ing shock-like features and large self-reversals appearing frequently in both Stokes I and V . In comparison, the internetwork is less dynamic, intensity oscillations are present but they are much weaker than in the network. No structures indicating the presence of shocks, are seen in the internetwork profiles. In agreement with prior ob- servations of chromospheric lines (e.g., Noyes 1967), any oscillation periods in the network appear to have a longer period than in the internetwork. We now examine three different regions, namely an internetwork pixel, an intermediate flux network pixel, and a strong network pixel. 3.2.1. Internetwork In Figure 10 the time evolution of a typical internet- work pixel is shown. The location of the pixel is marked by an asterisk in Figure 2. The data were taken when the slit was stationary. No Stokes V signal above the noise level is seen in the pixel. The Stokes I profiles of both Ca lines change periodically in width and position of the line center, but no self-reversals are seen. Also, the line-wing intensity shows some oscillations. 3.2.2. Intermediate flux network The difference between the internetwork and network regions with intermediate flux (Fig. 11) is dramatic: the Fig. 7.— Spatial distribution of the clusters in the map. The black areas (0 cluster) correspond to regions where the Stokes V amplitudes are below 7× 10−3Ic and where no cluster analysis was performed. network region is much more dynamic, and highly asym- metric profiles, in both lines Stokes I and V , are seen. The time dependent behavior of the photospheric iron line is quite similar to what is seen in the internetwork. The Stokes I in both lines has a clearly oscillating be- havior with bright, very asymmetric episodes followed by a darker, more symmetric episodes. The period for the oscillation is about 4 minutes, i.e. below that as- sociated with the acoustic cutoff frequency (about 5.3 mHz). This may be caused by the presence of inclined magnetic fields can lower effectively the acoustic cutoff frequency (Bel & Leroy 1977). The time evolution of the 8542 Å Stokes I has a diagonal structure moving from blue to red. This indicates the presence of propagating compressible waves (Carlsson & Stein 1997). The bright part, which corresponds to a large self-reversal, is clearly shifted towards the blue. This is seen in the 8498 Å line profiles as well, although these profiles tend to be more flat-bottomed. In general, the self-reversals and over all variation is larger in the blue wing than in the red. This is true for all slit positions which exhibit strong time- dependent behavior. The Stokes V image of the 8498 Å line also shows strong diagonal structures that coincide in time with the dark phases of Stokes I. Inspection of individual pro- files (Fig. 12) reveals a pattern of multiple lobes in the Stokes V profiles. These lobes are on the blue side of the line core and their amplitudes and positions vary period- ically in time resulting in the diagonal structure seen in the image. The lobes can be identified with the emission features seen in the Stokes I profiles. The 8542 Å line Stokes V image shows a pattern of a multi-lobed pro- files whose amplitudes vary strongly in time. The large Stokes V amplitude phase coincides with the bright, very asymmetric phase seen in the intensity profiles. The red wings always exhibit less structure and variation than the blue wings. 3.2.3. Strong network Stokes I and V profiles seen in the strong network re- gions (Figs. 13 and 14) would appear at first glance to be a mixture of the less dynamic internetwork and the highly dynamic intermediate flux region. The Stokes I profiles exhibit the same pattern of bright (more asym- metric) and dark (less asymmetric) phases as seen in the intermediate flux region. The difference between the two phases is however not as large: the amplitude of the self- reversals, especially in the 8542 Å intensity profiles, is much smaller than in the intermediate flux case. The Stokes V images resemble those of the intermedi- ate flux region: some diagonal structures are seen, but they are weaker. The 8542 Å line Stokes V profiles have a time varying amplitude but the profiles are not as asym- metric and they are not necessarily multi-lobed. The dif- Fig. 8.— As Fig. 5 but for the time series. ference between the time-dependent behavior of the red and blue lobes of the profile, i.e. the red lobe varies less in time, is even more clear here than in the intermediate flux region. 3.3. Statistics Histograms of the Stokes I amplitude integrated over 250 mÅ around the line core for the two Ca lines are shown in the top-left panel of Figure 15. These his- tograms include both the map and time-series profiles. Because there are almost five times as many profiles in the time-series as there are in the map, the histograms are dominated by the time-series profiles. Both lines ex- hibit a wide range of values. Except for the peaks at low intensities, the histograms are fairly flat. The dark- est (i.e. lowest core intensity or most absorption) am- plitudes, are associated with the internetwork, and the brightest with the network. Histograms of the Stokes V amplitudes (top right panel of Fig. 15) peak at the same value in both lines, 0.003 Ic, but the 8498 Å histogram tail decays more slowly. Since the 8498 Å line is formed slightly lower of the two and the lines are roughly equally sensitive to magnetic fields (effective Landé g factors are 1.07 and 1.10 for the 8498 Å and the 8542 Å lines, respectively), it is not surprising that the 8498 Å histogram has the longer tail. Fig. 9.— The spatio-temporal distributions of the clusters for the first slit position in the time series. The black areas correspond to regions where the Stokes V amplitudes is ≤ 7×10−3Ic. The vertical lines show the period with the best seeing when the slit was stationary. Both lines’ Stokes V amplitude asymmetry histograms (bottom left panel of Fig. 15) have very similar shapes and similar widths. There are more positive asymme- tries in both lines: 56% in 8498 Å and 64 % in 8542 Å (Table 1). The mean amplitude asymmetries are also positive, and the 8542 Å mean asymmetry is two times larger. There are more negative amplitude asymmetries in the 8542 Å map than in the time series. Non-zero am- plitude asymmetries indicate at least one of two things: the spatial pixels consist in most cases of at least two atmospheric components that are shifted relative to one another or that there are velocity and/or magnetic field gradients present in the atmosphere. The area asymmetry histograms (bottom right panel of Fig. 15) of the two calcium lines repeat the pattern al- ready seen in the cluster profiles: the 8542 Å histograms is centered around a negative value and the 8498 Å is centered at roughly zero, though the mean is slightly pos- itive. The 8542 Å histogram is significantly wider than the 8498 Å histogram. A multi-component atmosphere alone cannot produce area asymmetries, so the existence of non-zero area asymmetries indicates the presence of velocity and possibly magnetic gradients in the atmo- sphere. In the 8542 Å line 66 % of the profiles have negative area asymmetries whereas in the 8498 Å line the majority of the profiles, 64 %, have positive area asymmetries (Ta- ble 1). To better understand why the area asymmetry histograms of the lines are so different, we need to look at the components of the area asymmetry separately i.e. the sign of the blue lobe and the total area of the Stokes V profile. One possible cause for the difference in the histograms might be that the distribution of signs of the blue lobe is different in the two lines. Closer inspection reveals that this is not the explanation. The vast major- ity of both lines, over 80 %, have a negative sign. (Here the sign is defined to be the sign of the local maximum or minimum amplitude of the blue lobe). A second pos- sible explanation is that the V (λ)dλ is different in the two lines. This is found to be the case. The 8542 Å line has more profiles with a positive area and the 8498 Å has slightly more profiles with a negative area. (Note that the sign of the area asymmetry is the product of the sign of the blue lobe and the area; eq. 2.) The area of the Stokes V profile is strongly affected by the emis- sion features. These features, and their amplitudes, are related to the self-reversals seen in the Stokes I profiles. The self-reversals are stronger on the blue side of the line core than on the red. In general, the blue lobes of the Stokes V profiles have negative amplitudes and the effect of the emission features is then to reduce the amplitude, and in some cases, make it positive and this way reduce the overall negative area. The effect of the emission features on the amplitude asymmetries is not as large because the amplitude will be affected only if the emission feature is located at the Fig. 10.— Time dependent behavior of Stokes I in an internetwork pixel. Location of the pixel is marked with an asterisk in Figure 2. same wavelength as the maximum absolute amplitude. Also if the profile has a wide blue lobe, i.e., the wings contribute significantly, a local reduction in peak ampli- tude is counterbalanced by a comparable signal in the other parts of the blue lobe. The resulting profile will have nearly the same amplitude in the blue lobe as be- fore, but the area will be reduced leading to a smaller, or even negative, area asymmetry. Since the self-reversals are larger in the 8542 Å line, this scenario is more likely to apply to it than the 8498 Å line. Both lines’ area and amplitude asymmetries are found to be inversely proportional to the Stokes V amplitudes. The scatter, especially in the 8542 Å line, is fairly large. PCA also allows us to ensure that the determination of Stokes V asymmetries is not dominated by noise. Recon- structing the profiles using only the 11 first eigenvectors (i.e., essentially noise-free profiles) and then computing the asymmetries reproduces the Stokes V amplitude and asymmetry histograms. To test if the negative histogram peak in the 8542 Å line is an artifact caused by data re- duction, we computed area asymmetries for the datasets, but after first removing the fringe pattern caused by the optics. This did not alter the area asymmetry histogram. Another artifact that could cause the offset is an incor- rect subtraction of the tilt caused by the detector in the continuum intensity. To remove the offset in the his- tograms by means of changing the tilt causes a clearly visible lopsidedness in the Stokes I profiles. Lastly, to make sure that the choice of the integration range is not the cause of the offset, we used a constant bandwidth for area asymmetries and it also reproduces the 8542 Å area histogram offset. (Besides these issues, there are no other obvious artifacts that would cause the offset.) We therefore conclude that the offset is not caused by the fringing or incorrect subtraction of the tilt in the contin- uum intensity. 4. COMPARISON OF OBSERVATIONS WITH A HIGH-β SIMULATION In P06 we synthesized Stokes profiles for the Ca IR triplet lines in the high-β regime. This was done by combining a radiation hydrodynamic code (see for exam- ple Carlsson & Stein 1997) with a weak magnetic field and using a nLTE Stokes inversion and synthesis code (Socas-Navarro et al. 2000b) to produce, based on snap- shots of the simulation, a time series of the lines’ Stokes vectors. The simulation is driven by a photospheric ve- locity piston and its dynamics are dominated by upward propagating acoustic waves in a simple magnetic field topology. The simulation shows that the radiative trans- fer is very similar in all the Ca IR triplet lines. The differences between the line behaviors in the simulation are mainly due to the lines having slightly different for- mation heights and thus experiencing a difference in the amplitudes of the shocking waves: the higher the line is formed, the larger the amplitude of the passing wave is. In the simulation there is no feedback from the mag- netic fields on the dynamics and the waves are purely acoustic. The observations have limited spatial and tem- poral resolutions whereas the simulation is much better resolved. 4.1. Comparison of time dependent behavior As the acoustic waves in the simulation propagate up- wards and eventually form shocks, a time-varying pat- Fig. 11.— Time dependent behavior of Stokes I and V in an intermediate flux pixel. Location of the pixel is marked with a diamond in Figure 2. Fig. 12.— Time evolution of individual Stokes I and V profiles in an intermediate flux pixel. Location of the pixel is marked with a diamond in Figure 2. Fig. 13.— Time evolution of Stokes I and V in a network pixel. Location of the pixel is marked with a triangle in Figure 2. Fig. 14.— Time evolution of individual Stokes I and V profiles in a network pixel. Location of the pixel is marked with a triangle in Figure 2. Fig. 15.— Histograms of Stokes I and V amplitudes, and Stokes V amplitude and area asymmetries of the map and time-series. tern of disappearing and reappearing Stokes V lobes is seen (Fig. 16). The pattern is strongest in the highest forming line, i.e. 8542 Å. Wave propagation is also seen in the Stokes I profiles. There are no large self reversals or brightenings, instead the position of the line minimum changes periodically and forms a saw-tooth like pattern where the red shift takes more time than the blue shift phase. If we first compare the simulated profiles to the in- ternetwork observations (Fig. 10), we see that the strong signatures of shocks seen in the simulation are not present in the observations. In the simulation the Ca IR triplet is formed in a region where the waves are just be- ginning to shock. If the formation height of the lines or the shocks in the simulation is off, compared to the real Sun, by a small amount, even 50 km, the lines’ temporal evolution may look very different. Another possible ex- planation to why we see no strong indications of shocks is the temporal and/or spatial resolution: there may be several components oscillating out of phase relative to one another in a given resolution element. However, the photospheric velocities are very similar in the internet- work and network, but the network profiles show strong self-reversals. This suggests that spatial and temporal resolution alone cannot explain the lack of strong signa- tures of shocks in the internetwork. Observations of the quiet Sun show varying degrees of oscillatory power (compare for example Lites et al. (1993) [Ca II H and K] or UV data of Judge et al. 2003, McIntosh & Judge 2001 and Wikstøl et al. 2000). This variation may be related to the local magnetic topology, especially to the possible existence of a magnetic canopy (McIntosh et al. 2003; Vecchio et al. 2006). The region observed here was less oscillatory than average but still not exceptionally quiet. Both the simulated profiles and observed network pro- files (Fig. 13) show time varying patterns where the Stokes I and V amplitudes change periodically. In the simulation the wave propagation manifests itself in the Stokes I profiles most clearly as a shift of the line core and the saw-tooth shape of the time series. In the ob- servations, waves cause the lines’ periodically varying self-reversals that result in alternating bright and dark phases. There are indications of diagonal structures in the observed Stokes I images, but they are not nearly as clear as in the simulation. In the simulation the up- ward propagating waves cause the blue and red lobes of the Stokes V profiles to disappear alternately. In con- trast, the observed time varying pattern in Stokes V looks more complicated: there is much more structure in the observed profiles, especially in the line cores, than in the simulation. This is related to the simulated pro- files not exhibiting strong self-reversals as seen in the observations. In the simulation, because of radiative cooling and ex- pansion of the falling material, the down flows are in general cooler than the up flows. In the synthesized pro- files this manifests itself by the red wings of the Stokes I profiles showing less variations, though the difference with the blue wing is quite small. Similar behavior is also seen in the observations: the self-reversals are in general larger in the blue wing of the Stokes I profiles and the red lobes of the Stokes V profiles show clearly less variation. 4.2. Comparison of statistics and Stokes V morphologies In the simulation the magnetic field decays exponen- tially with height and therefore the Ca II Stokes V am- plitudes are significantly lower than the Fe I 8497 Å am- plitude. In the observations the Ca and Fe line Stokes V profiles have roughly the same amplitudes. This may be explained by the field decaying much slower with height in the observations, or by the filling factor in the ob- servations being smaller in the photosphere than in the chromosphere. Both Ca II lines’ observed Stokes V profiles have a sig- nificant amount of signal in the wings. In the simulations only the 8498 Å line Stokes V has extended wings with large amplitudes (Fig. 4 in P06). The amount of signal in the wings depends on the atmospheric magnetic field Fig. 16.— Time evolution of Stokes I and V profiles in the high-β simulation (P06). The Stokes V signal in wavelength range -1.2 to -0.6 Å in the 8498 Å image is scaled down with factor 7.5 in order to display both the Ca II 8498 Å and Fe I 8497 Å lines in the same panel. gradient. If there is no gradient the wings of all three Ca lines have very little signal. Whereas a model atmo- sphere with a constant field gradient produces profiles where all lines, 8498 Å the most, have some signal in the line wings and an exponential field produces profiles with the largest wings. Depending on where the gradi- ent is located and how strong the field is, the Ca lines may or may not have similar Stokes V profiles. Based on the profile shapes and relative amplitudes, it is ob- vious that the magnetic topology in the observations is different from the simulation. Formation of area and amplitude asymmetries in the simulation is coupled. The correlation is especially strong in the 8542 Å line (upper row of Fig. 17). In the 8498 Å Stokes V profiles the strong wings affect the asymmetries, and the correlation is weaker. The observed area and amplitude asymmetries of both lines show less correlation. This is at least partly because the observed profiles have more complex shapes than in the simulation. The lower panels in Figure 17 show the Stokes V asym- metry histograms for the simulation. The observed his- tograms are re-plotted to enable direct comparison. In the simulation both lines’ amplitude and asymmetry his- tograms are centered roughly around zero (percentage- wise there are a couple of percent more negative than positive asymmetries). This was not the case in the ob- servations where all the asymmetries, except the 8542 Å area asymmetry, have clearly more positive than nega- tive values, i.e. the blue lobe is larger in area/amplitude than the red lobe. The observed 8498 Å profiles are more dynamic than the simulated ones. Consequently the observed 8498 Å asymmetry histograms are clearly wider than the simu- lated. Because there is very little signal in the simulated 8542 Å Stokes V profile wings, when an upward propa- gating wave causes a Stokes V lobe to disappear, there is no signal in the line wing to contribute to the amplitude. This leads to the extreme amplitude asymmetries in the simulations and in the additional lobes at large values in the simulated 8542 Å line area asymmetry histogram. Since the observed profiles have a significant amount of signal in the wings, the extreme amplitude asymmetries are moderated, and no lobes at large values are seen in the histogram. 5. CONCLUSIONS AND DISCUSSION So far most spectropolarimetric studies using the Ca II IR triplet lines have focused on active regions (e.g., Socas-Navarro, Trujillo Bueno, & Ruiz Cobo 2000a; López Ariste, Socas-Navarro, & Molodij 2001; Socas-Navarro 2005; Uitenbroek, Balasubramaniam, & Tritschler 2006). The observations presented here show that these lines are also promising candidates for studying the magnetic chromosphere outside of active regions. Interpreting the observations, however, is not straight forward. The main results of the analysis presented here are: • Classification of Stokes V profile shapes. Asymmetric line profiles are very common and that the two lines, despite being formed fairly close in a geometrical sense, often do not have similar shapes. Furthermore, the edges of the network patches ex- hibit profile shapes different from those seen in the center of the patches. The cluster analysis results, as expected, in a qualitative, not quantitative, de- scription of the profile shapes. • Statistics of the line profiles. The 8542 Å area asymmetry is predominantly neg- ative; while the 8498 Å area asymmetry and the amplitude asymmetries are usually positive. • Time dependent behavior. The enhanced network has very different dynamic behavior compared with the internetwork. It is more dynamic and the oscillation period, as seen in both Stokes I and V , is greater than in the in- ternetwork. • Comparison with high-β simulation. Oscillations are present in both the observations and the simulation. The simulated profiles are more dynamic than the observed internetwork pro- files. The opposite is true for network profiles. In the simulation, the formation of asymmetries is more tightly coupled than what is seen in the ob- servations. Except for the 8542 Å amplitude asym- metry the observed profiles show a wider range of asymmetries. And lastly, the peculiar negative area asymmetries seen in the observed 8542 Å line and the tendency of the other asymmetries to be posi- tive are not reproduced by the simulation. The tendency of large Stokes V asymmetries to de- crease with an increasing signal amplitude has also been observed in photospheric lines (Grossmann-Doerth et al. 1996). In the photosphere a magnetic canopy is one pos- sible explanation: the canopy gives rise to asymmetries in the lines, and as a flux tube diameter increases, the relative contribution from the canopy to the Stokes V signal decreases. In the photosphere the scatter in an amplitude vs. asymmetry plot is significantly larger in the area than in the amplitude. No large difference is seen in the area and amplitude asymmetry scatters of the Ca II lines. In the quiet Sun photosphere, more positive than negative Stokes V asymmetries are found (Grossmann-Doerth et al. 1996). In contrast with 8498 Å line (where there is no large difference in the mean area and amplitude asymmetries) the photospheric mean area asymmetries are significantly smaller (4 % in the Fe I 6302 Å line) than the mean amplitude asymmetries (15 % in the Fe I 6302 Å line). The photospheric asymme- tries are often attributed to multiple atmospheric compo- nents within a resolution element. In the chromosphere, however, gradients have to play a dominant role since the formation of area asymmetries require them. An- other piece of evidence of the importance of gradients in the chromosphere is that Milne-Eddington inversions, which include the Paschen-Back effect of the He I 10830 Å triplet, are not able to reproduce the observed area asymmetries (Sasso & Solanki 2006). Fig. 17.— Stokes V asymmetries of the simulated and observed profiles. Upper 4 panels show the correlation of amplitude and area asymmetries in the simulated and observed Ca lines. The Pearson correlation coefficient for each case is given. The asterisk symbols show the mean for each 0.1 wide bin and the error bars show the standard deviation. The lower panels are histograms of observed and simulated amplitude and area asymmetries. Khomenko et al. (2005) used a 3-dimensional magne- toconvection model to synthesize photospheric magneti- cally sensitive lines in the visible and IR. There are more positive than negative Stokes V asymmetries in their syn- thetic profiles. They found that reducing the spatial res- olution increases the number of irregular stokes V pro- files (though the number of strongly asymmetric profiles decreases). They conclude that the asymmetries reflect more inhomogeneities in the horizontal direction than in the vertical. In the chromosphere large velocity gradients are more common and variation in the vertical direction are likely to be more important than variation in the hor- izontal direction. When these two factors are combined with the observed area asymmetries, one concludes that the chromospheric asymmetries mainly reflect the line-of- sight inhomogeneities, and not variations in the horizon- tal direction. Despite the apparent similarities between the photospheric and chromospheric Stokes V profiles, the underlying mechanism causing the asymmetries does not appear to be the same. Drawing parallels between the chromosphere and photosphere is problematic since the two regions exist in very different physical regimes. The discrepancy between the Stokes V asymmetry his- tograms of the observations and the simulation may be related to the self-reversals. The simulated profiles ex- hibit only small self-reversals. The observations show large self-reversals in the Stokes I profiles and accompa- nying emission features in the Stokes V profiles. These features are stronger on the blue side of the line cores. Another effect that contributes to the imbalance is that that the down flow phase lasts longer. Our observations, especially with a 5 second exposure time, sample more profiles with red-shifts and positive asymmetries (since there will be more emission on the blue side). However, inspection of Fig. 16 shows the same to be true of the simulations. If this is the case, why are there not more positive than negative asymmetries in the simulation as well? The sample of these observations is limited because the majority of the profiles are drawn from the same three slit positions which sample the same local magnetic field configuration. It would not be surprising if histograms made of profiles from a variety of quiet-Sun magnetic field topologies would have somewhat different shapes. The complexity of the observed profiles makes the inter- pretation of the area and amplitude asymmetries diffi- cult. Because of multiple lobes and the strong signal in the line wings, the asymmetries are not necessarily good proxies for the overall complexity of the Stokes V pro- files. This is especially true if the two asymmetries are viewed separately. It is a well known result that the network intensity oscillations have a longer period than the internetwork (e.g. Orrall 1966, Lites et al. 1993, Banerjee et al. 2001). This has also been observed before in the Ca II IR lines (Deubner & Fleck 1990). Why do the intermedi- ate flux regions in our observations appear to be more dynamic than the stronger flux regions? It may be re- lated to a more complex magnetic topology at the edges of the network patches. The observations show no sig- nal above the noise in Stokes Q and U , so we can- not draw any conclusions of possible horizontal fields. Any signal would be affected by atomic polarization (Manso Sainz & Trujillo Bueno 2003) making the inter- pretation exceedingly complex. The filling factor in the network is not likely to be very large, and is likely smaller at the edges than in the center of the network patch. In- versions by Bellot Rubio et al. (2000) of average Stokes profiles in a plage region gave a filling factor of 0.5 a z = 0 km. The filling factor in the photospheric net- work can safely be assumed to be lower than this. In fact, in recent inversions by Domı́nguez Cerdeña et al. (2006), which included a small patch of network, the pho- tospheric filling factor in the patch center was as small as 0.1. The network magnetic fields must expand with height and consequently the chromospheric filling fac- tor must exceed photospheric values. Results of com- paring photospheric and chromospheric magnetograms, however, Zhang & Zhang (2000) suggest that the sizes of the network magnetic elements are not very different at the two heights . The chromospheric magnetograms in the comparison are based on the Hβ line. Its interpreta- tion is complicated by the magnetically sensitive blends close to the line core, and the line may suffer from same problems as the Hα line when used as a proxy for chro- mospheric magnetic fields, namely that the photospheric contribution to the polarization signal is not insignificant (Socas-Navarro & Uitenbroek 2004). Lastly, the size of network patches is not directly linked with the filling fac- tor. We see some expansion of the network with height in the magnetograms of the map (Fig. 2), especially when comparing the Ca II 8498 Å and 8542 Å magnetograms. But since the magnetograms were constructed by using the weak field formula, and the network fields have gra- dients and are not necessarily weak, the magnetograms are not accurate. Also the choice of color scaling of the images affects the comparison. However, the apparent expansion is not necessarily an artifact, since expansion of network seen in magnetograms has also been reported by Giovanelli (1980). Obviously we need to understand better the topology of the network magnetic fields. To do this we plan to perform nLTE inversions of these data in the near fu- ture. The inversions will help further in understanding the formation dynamics of the Ca II IR lines in the quiet Sun, and hopefully reveal how the underlying atmosphere differs from that used in the simulation. An important question to answer is why the two Ca lines behave as differently as they do. Having a time series taken during good seeing would be helpful. Also in order to expand the analysis to internetwork regions, better spatial resolution is required. Another interesting question is how much variation there is in dynamics in different internetwork regions, and how well the differences can be explained in terms of the surrounding magnetic fields as has been sug- gested by Vecchio et al. (2006) based on imaging data of Ca II 8542 Å Stokes I. To fully investigate this in detail high quality data of the full Stokes vector are needed. TABLE 1 Observed Stokes V asymmetries 8498 Å 8498 Å 8498 Å 8542 Å 8542 Å 8542 Å < 0 (%) > 0 (%) mean (%) < 0 (%) > 0 (%) mean (%) σa 43.2 55.7 3.1 36.6 61.4 6.3 σA 35.5 64.5 3.3 69.7 30.3 -6.8 Note. — Percentages of observed Ca II 8498 Å and 8542 Å Stokes V amplitude and area asymmetries with negative (i.e. red lobe larger) and positive (i.e. blue lobe larger) signs. Thanks to Doug Gilliam, Joe Elrod and Mike Bradford for all their invaluable help during the observing run. 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0704.0618

Number of moduli of irreducible families of plane curves with nodes and cusps

NUMBER OF MODULI OF IRREDUCIBLE FAMILIES OF PLANE CURVES WITH NODES AND CUSPS. CONCETTINA GALATI Abstract. Let Σn n(n+3) 2 be the family of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let Σ ⊂ Σn be an irreducible component. We consider the natural rational map ΠΣ : Σ 99K Mg, from Σ to the moduli space of curves of genus g = −d−k. We define the number of moduli of Σ as the dimension dim(ΠΣ(Σ)). If Σ has the expected dimension equal to 3n+ g − 1− k, then (1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ− k), where ρ := ρ(2, g, n) = 3n − 2g − 6 is the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that Σ has the expected number of moduli if the equality holds in (1). In this paper we construct examples of families of irreducible plane curves with nodes and cusps as singularities having expected number of moduli and with non-positive Brill-Noether number. 1. Introduction In this paper we compute the number of moduli of certain families of irreducible plane curves with nodes and cusps as singularities. Let Σnk,d ⊂ P(H 0(P2,OP2(n))) := PN , with N = n(n+3) , be the closure, in the Zariski’s topology, of the locally closed set of reduced and irreducible plane curves of degree n with k cusps and d nodes. Let Σ ⊂ Σnk,d be an irreducible component of the variety Σ k,d. We denote by Σ0 the open set of Σ of points [Γ] ∈ Σ such that Σ is smooth at [Γ] and such that [Γ] corresponds to a reduced and irreducible plane curve of degree n with d nodes, k cusps and no further singularities. Since the tautological family S0 → Σ0, parametrized by Σ0, is an equigeneric family of curves, by normalizing the total space, we get a family // S0 // P2 × Σ0 of smooth curves of genus g = −k−d. Because of the functorial properties of the moduli space Mg of smooth curves of genus g, we get a regular map Σ0 → Mg, sending every point [Γ] ∈ Σ0 to the isomorphism class of the normalization of the plane curve Γ corresponding to the point [Γ]. This map extends to a rational map ΠΣ : Σ 99K Mg. We say that ΠΣ is the moduli map of Σ and we set number of moduli of Σ := dim(ΠΣ(Σ)). Date: 06 September 2005. 1991 Mathematics Subject Classification. 14H15; 14H10; 14B05. Key words and phrases. families of plane curves, number of moduli, nodes and cusps. http://arxiv.org/abs/0704.0618v1 2 CONCETTINA GALATI Notice that, when Σnk,d is reducible, two different irreducible components of Σ can have different number of moduli. We say that Σ has general moduli if ΠΣ is dominant. Otherwise, we say that Σ has special moduli. Definition 1.1. When Σ has the expected dimension equal to 3n+ g − 1 − k and g ≥ 2, we say that Σ has the expected number of moduli if dim(ΠΣ(Σ)) = min(dim(Mg), dim(Mg) + ρ− k), where ρ := ρ(2, g, n) = 3n−2g−6 is the number of Brill-Noether of the linear series of degree n and dimension 2 on a smooth curve of genus g. As we shall see in the next section, when g ≥ 2 and when Σ has the expected dimension equal to 3n+ g − 1 − k, the number of moduli of Σ is at most equal to the expected one. This happens in particular if k < 3n. If k ≥ 3n, in general we have not an upper-bound for the dimension of Σ and we cannot provide an upper bound for the number of moduli of Σ, (see lemma 2.2 and remark 2.3). Moreover, by classical Brill-Neother theory when ρ is positive and by a well know result of Sernesi when ρ ≤ 0 (see [18]), we have that Σn0,d, (which is irreducible by [8]), has the expected number of moduli for every d ≤ . When k > 0 there are known results giving sufficient conditions for the existence of irreducible components Σ of Σnk,d with general moduli, (see propositions 2.5 and 2.6 and corollary 2.7). In this article we construct examples of families of irreducible plane curves with nodes and cusps with finite and expected number of moduli. A large part of this paper is obtained working out the main ideas and techniques that Sernesi uses in [18]. In section 2.1 we introduce the varieties Σnk,d and we recall their main properties. In section 2.2 we discuss on definition 1.1 and we summarize known results on the number of moduli of families of irreducible plane curves with nodes and cusps. In theorem 3.5 we prove the existence of plane curves with nodes and cusps as singular- ities whose singular points are in sufficiently general position to impose independent linear conditions to a linear system of plane curves of a certain degree. This result is related to the moduli problem by lemma 3.2, remark 3.4 and proposition 4.1, where we find sufficient conditions in order that an irreducible component Σ ⊂ Σnk,d has the expected number of moduli. If Σ verifies the hypotheses of proposition 4.1, then the Brill-Neother number ρ is not positive and Σ has finite number of moduli. Moreover, by lemma 4.6 and corollary 4.7, for every k′ ≤ k and d′ ≤ d+k−k′, there is at least an irreducible component Σ′ ⊂ Σnk′,d′ , such that Σ ⊂ Σ ′ and the general element [D] ∈ Σ′ corresponds to a plane curve D verifying hypotheses of propo- sition 4.1 and so having the expected number of moduli. Finally, the main result of this paper is contained in theorem 4.9, where, by using induction on the degree n and on the genus g of the general curve of the family, we construct examples of families of irreducible plane curves with nodes and cusps verifying the hypotheses of proposition 4.1. In particular, we prove that, if k ≤ 6 and ρ ≤ 0, then Σnk,d has at least an irreducible component which is not empty and which has the expected number of moduli. This result may be improved and examples of families of curves showing that the condition k ≤ 6 is not sharp are given in remark 4.10. Notice that the previous theorem provides only examples of families of plane curves with nodes and cusps with expected number of moduli, when ρ is not positive. When the number of cusps k is very small, we expect it is possible to prove the existence of irreducible components of Σnk,d with expected number of moduli, for every value of ρ. For example, from a result of Eisenbud and Harris, it follows that Σn1,d, (which is irreducible by [16]), has general moduli if ρ ≥ 2, (see corollary 2.7). In theorem 4.11, by using induction on n we find that Σn1,d has general moduli also when ρ = 1. By recalling that, by theorem 4.9, Σn1,d has expected number of moduli when ρ ≤ 0, we conclude that Σn1,d has the expected number of moduli for every ρ or, equivalently, NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 3 for every d ≤ − 1. We still don’t know examples of irreducible components of Σnk,d having number of moduli smaller that the expected. 2. Preliminaries 2.1. On Severi-Enriques varieties. We shall denote by PN = P n(n+3) 2 the Hilbert scheme of plane curves of degree n, by [Γ] ∈ PN the point parametrizing a plane curve Γ ⊂ P2 and by Σnk,d ⊂ P N the closure, in the Zariski topology, of the locally closed set parametrizing reduced and irreducible plane curves of degree n with d nodes and k cusps as singularities. These varieties have been introduced at the beginning of the last century by Severi and Enriques. In particular, the case k = 0 has been studied first by Severi and for this reason the varieties Σn0,d are usually called Severi varieties, while for k > 0 the varieties Σnk,d are called Severi-Enriques varieties. We recall that every irreducible component Σ of Σnk,d has dimension at least equal to N − d− 2k = 3n+ g − 1− k, where g = −k−d. When the equality holds we say that Σ has expected dimen- sion. Moreover, it is well known that if k < 3n then every irreducible component Σ of Σnk,d has expected dimension, (see for example [23] or [25]). On the contrary, when k ≥ 3n, there exist examples of irreducible components of Σnk,d having di- mension greater than the expected, (see [25]). Moreover, we recall that Σn0,d is not empty for every d ≤ and it contains in its closure all points parameterizing irreducible plane curves of degree n and genus g = − d, (see [24], [25] and [1]). Often, we shall denote Σn0,d by Vn,g. While the proof of the existence of Vn,g is quite elementary and it is due to Severi, the irreducibility of Vn,g remained an open problem for a long time and it has been proved by Harris only in 1986. Later, by using the same techniques of Harris, Kang has proved the irreducibility of Σnk,d with k ≤ 3, see [8] and [16]. However, in general, Σnk,d is reducible and there exist values of n, d and k such that Σnk,d is empty, (see [25], [12], [20], [11] or chapter 2 of [7] and related references). Finally, we recall that, if Σ ⊂ Σnk,d is a non-empty irreducible component of the expected dimension equal to 3n + g − 1 − k, then, for every k′ ≤ k and d′ ≤ d + k − k′, there exists a non-empty irreducible compo- nent Σ′ ⊂ Σnk′,d′ such that Σ ⊂ Σ ′. This happens in particular if k < 3n. More precisely, it is true that, if Γ ⊂ P2 is a reduced (possibly reducible) plane curve of degree n with k < 3n cusps at points q1, . . . , qk, nodes at points p1, . . . , pd and no further singularities, then, chosen arbitrarily k1 cusps, say q1, . . . , qk1 among the k cusps of Γ, k2 cusps qk1+1, . . . , qk2 among qk1+1, . . . , qk and d1 nodes p1, . . . , pd1 among the nodes of Γ, there exists a family of reduced plane curves D → B ⊂ PN of degree n, whose special fibre is D0 = Γ and whose general fibre Dt = D has a node in a neighborhood of every marked node of Γ, a cusp in a neighborhood of each point q1, . . . , qk1 , a node in a neighborhood of each point qk1+1, . . . , qk2 and no further singularities, (see [25], corollary 6.3 of [11] or lemma 3.17 of chapter 2 of [7]). To save space, we shall say that the family D → B is obtained from Γ by preserving the singularities q1, . . . , qk1 and p1, . . . , pd1 , by deforming in a node each cusp qk1+1, . . . , qk2 and by smoothing the other singularities. 2.2. Known results on the number of moduli of Σnk,d. In order to explain the definition 1.1, we need to recall some basics of Brill-Noether theory. Given a smooth curve C of genus g, the set G2n(C) of linear series g n on C of dimension 2 and degree n, is a projective variety which verifies the following properties: (1) G2n(C) is not empty of dimension at least ρ, if ρ(2, n, g) = 3n− 2g − 6 ≥ 0, (see theorem V.1.1 and proposition IV.4.1 of [4]). 4 CONCETTINA GALATI (2) Let g2n be a given linear series, letH ∈ g n be a divisor and letW ⊂ H 0(C,H) be the three dimensional vector space corresponding to g2n. Denoting by ωC = OC(KC) the canonical sheaf of C and by µo,C :W ⊗H 0(C, ωC(−H)) → H 0(C, ωC) the natural multiplication map, also called the Brill-Noether map of the pair (C,W ), we have that the dimension of the tangent space to G2n(C) at the point [g2n],corresponding to g n, is equal to dim(T[g2 n(C)) = ρ+ dim(ker(µ0,C)), (see [2] or proposition IV.4.1 of [4] for a proof). (3) Moreover, if C is a curve with general moduli (i.e. if [C] varies in an open set of Mg), the variety G n(C) is empty if ρ < 0, it consists of a finite number of points if ρ = 0 and it is reduced, irreducible, smooth and not empty variety of dimension exactly ρ, when ρ ≥ 1, (see theorem V.1.5 and theorem V.1.6 of [4]). In the latter case, the general g2n on C defines a local embedding on C and it maps C to P2 as a nodal curve, (see theorem 3.1 of [1] or lemma 3.43 of [9]). From (3), we deduce that, the Severi variety Σn0,d = Vn,g of irreducible plane curves of genus g = − d, has general moduli when ρ ≥ 0 and it has special moduli when ρ < 0. When ρ < 0, and then g ≥ 3, by definition 1.1, we expect that the image of Vn,g into Mg has codimension exactly −ρ. Equivalently, recalling that, in this case, dim(Vn,g) = 3n+ g − 1 = 3g − 3 + ρ+ 8 = dim(Mg) + ρ+ dim(Aut(P we expect that on the smooth curve C, obtained by normalizing the plane curve corresponding to the general element of Vn,g, there is only a finite number of g mapping C to the plane as a nodal curve. This is a well known result proved by Sernesi in [18]. Theorem 2.1 (Sernesi, [18]). The Severi variety Vn,g = Σ 0,d of irreducible plane curves of degree n and genus g = − d has number of moduli equal to min(dim(Mg), dim(Mg) + ρ). What can we say about the number of moduli of an irreducible component Σ of Σnk,d, when k > 0? In this case we need to distinguish the two cases k < 3n and k ≥ 3n. In the first case we have the following result. Lemma 2.2. For every not empty irreducible component Σ of Σnk,d, with k < 3n and g = − k − d ≥ 2, the number of moduli of Σ is at most equal to min(dim(Mg), dim(Mg) + ρ− k), where ρ = 3n − 2g − 6 is the Brill-Neother number of moduli of linear series of dimension 2 and degree n on a smooth curve of genus g. Proof. We recall that an ordinary cusp P of a plane curve Γ corresponds to a simple ramification point p of the normalization map φ : C → Γ, i.e. to a simple zero of the differential map dφ. If we denote by G2n,k(C) ⊂ G n(C) the set of g n on C defining a birational morphism with k simple ramification points, then G2n,k(C) is a locally closed subset of G2n(C) and every irreducible component G of G n,k(C) has dimension at least equal to ρ− k, if it is not empty. In particular, if F 2n,k(C) is the variety whose points correspond to the pairs ([g2n], {s0, s1, s2}) where [g n] ∈ G n,k(C) and {s0, s1, s2} is a frame of the three dimensional space associated to the linear series g2n, then every irreducible component of F n,k(C) has dimension at least equal min(8, ρ− k + 8). NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 5 Now, let Σ be one of the irreducible components of Σnk,d and let [Γ] be a general point of Σ. Then, if Γ ⊂ P2 is the corresponding plane curve and φ : C → Γ is the normalization map, then the fibre over the point [C] ∈ Mg of the moduli map ΠΣ : Σ 99K Mg consists of an open set in one or more irreducible components of F 2n,k(C). In partic- ular, every irreducible component of the general fibre of ΠΣ has dimension at least equal to min(8, ρ− k + 8). Moreover, if k < 3n then Σ has the expected dimension equal to N−d−2k = 3n+g−1−k, (see [25] or [23]). Finally, if g = −k−d ≥ 2, dim(Σ) = 3n+ g − 1− k = 3g − 3 + ρ− k + 8. This proves the statement. � Remark 2.3. The proof of the previous lemma still holds if k ≥ 3n but Σ has the expected dimension. However in general, when k ≥ 3n, we don’t have a bound for dim(ΠΣ(Σ)). Indeed, in this case the dimension of the general fibre of the moduli map of Σ is still at least equal to ρ− k + 8, but Σ may have dimension larger than 3n+ g − 1 − k. Anyhow, by the following proposition, every not empty irreducible component of Σnk,d has special moduli if k ≥ 3n. Proposition 2.4 (Arbarello-Cornalba, [1]). Let C be a general curve of genus g ≥ 2 and let φ : C → P2 be a birational morphism, then the degree of the zero divisor of the differential map of φ is smaller than ρ. In particular, every irreducible component of Σnk,d has special moduli if ρ = 3n− 2g − 6 < k. A sufficient condition for the existence of irreducible families of plane curves with nodes and cusps with general moduli is given by the following result. Proposition 2.5 (Kang, [15]). Σnk,d is irreducible, not empty and with general moduli if n > 2g − 1 + 2k, where g = − d− k. Actually, in [15], Kang proves that if n > 2g−1+2k, then Σnk,d is not empty and irreducible. But from his proof it follows that, under the hypothesis of proposition 2.5, Σnk,d has general moduli because the general element of Σ k,d corresponds to a curve which is a projection of an arbitrary smooth curve C of genus g in Pn−g, from a general (n − 3)-plane intersecting the tangent variety of C in k different points. Another result which may be used to find examples of families of plane curves with nodes and cusps having general moduli is the following. Let grn be a linear series on C associated to a (r+1)-spaceW ⊂ H0(C,L), where L is an invertible sheaf on C, and let {s0, . . . , sr} be a basis of W , then the ramification sequence of the g n at p is the sequence b = (b0, . . . , br) with bi = ordpsi − i. Choosing another basis of W , the ramification sequence of grn at p doesn’t change. We say that the ramification sequence of the grn at p is at least equal to b = (b0, . . . , br) if bi ≤ ordpsi − i, for every i, and we write (ordps0, . . . , ordpsr − r) ≥ (b0, . . . , br). Proposition 2.6 (Proposition 1.2 of [10]). Let C be a general curve of genus g, let p be a general point on C and let b = (b0, . . . , br) be any ramification sequence. There exists a grn on C having ramification at least b at p if and only if (bi + g − n+ r)+ ≤ g, where (−)+ := max(−, 0). From proposition 2.6, we easily deduce the following result. Corollary 2.7. Suppose that k ≤ 3 and ρ = 3n− 2g − 6 ≥ 2k. Then Σnk,d is not empty, irreducible and it has general moduli. 6 CONCETTINA GALATI Proof. By [16], the variety Σnk,d is irreducible for every k ≤ 3 and d ≤ Moreover, by using classical arguments, one can prove that Σnk,d is not empty if k ≤ 4 and d ≤ − 4, (see, for example, corollary 3.18 of chapter two of [7]). Finally, by theorem 1.1 of [21], by using the terminology of proposition 2.6, under the hypothesis k ≤ 3n − 4, in particular if k ≤ 3, the variety Σnk,d contains every point of PN corresponding to a plane curve Γ of genus g = − k − d such that the normalization morphism of Γ has at least a ramification point with ramification sequence (b0, b1, b2) ≥ (0, k, k). Then, by proposition 2.6, if ρ ≥ 2k and k ≤ 3, the moduli map of Σnk,d is surjective. � 3. On the existence of certain families of plane curves with nodes and cusps in sufficiently general position As we already observed, we don’t have a complete answer for the existence prob- lem of Σnk,d. In this section we are interested in a little more specific existence problem. We shall prove the existence of plane curves with nodes and cusps as singularities whose singular points are in sufficiently general position to impose independent linear conditions to a linear system of plane curves of a certain degree. Definition 3.1. A projective curve C ⊂ Pr is said to be geometrically t-normal if the linear series cut out on the normalization curve C̃ of C by the pull-back to C̃ of the linear system of hypersurfaces of Pr of degree t is complete. From a geometric point of view, a projective curve C ⊂ Pr is geometrically t- normal if and only if the image curve νt,r(C) of C by the Veronese embedding νt,r : P r → P( t ) of degree t, is not a projection of a non-degenerate curve living in a higher dimensional projective space. We shall say that a curve is geometrically linearly normal (g.l.n. for short) if it is geometrically 1-normal. Every such a curve C is not a projection of a curve lying in a projective space of larger dimension. The following result is proved under more general hypotheses in [5], theorem 2.1. Lemma 3.2. Let Γ ⊂ P2 be an irreducible and reduced plane curve of degree n and genus g with at most nodes and cusps as singularities. Let t be an integer such that n− 3 − t < 0, then Γ is geometrically t-normal if and only if it is smooth. On the contrary, if n − 3 − t ≥ 0, the plane curve Γ is geometrically t-normal if and only if its singular points impose independent linear conditions to plane curves of degree n− 3− t. We recall the following classical definition. Definition 3.3. Let Γ ⊂ P2 be a plane curve of degree n with d nodes at p1, . . . , pd and k cusps at q1, . . . , qk as singularities. Let φ : C → Γ be the normalization of Γ. The adjoint divisor ∆ of φ is the divisor on C defined by ∆ = i=1 φ −1(pi) + j=1 2φ −1(qj). Proof of lemma 3.2. Let Γ be a plane curve as in the statement of the lemma. Then, Γ is geometrically t-normal if and only if, by definition, h0(C,OC(t)) = h 0(P2,OP2(t))− h 0(P2, IΓ(t)) where IΓ is the ideal sheaf of Γ in P 2 and OC(t) := OC(tφ ∗(H)), where H is the general line of P2. By Riemann-Roch theorem, Γ is geometrically t-normal if and only if (2) h0(C, ωC(−t))) = −nt+ g − 1 + (t+ 1)(t+ 2) − h0(P2, IΓ(t)), where g is the geometric genus of Γ and ωC is the canonical sheaf of C. On the other hand, it is well known that H0(C, ωC(−t)) = H 0(C,OC(n− 3− t)(−∆)), where ∆ NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 7 is the adjoint divisor of φ, (see definition 3.3 and [4], appendix A). If n− 3− t < 0 then h0(C,OC(n− 3− t)) = 0 and Γ is geometrically t-normal if and only if h0(P2,OP2(t)) − h 0(P2, IΓ(t)) = nt− n2 − 3n where δ = − g = deg(∆)/2. This equality is verified if and only if δ = 0, i.e. Γ is smooth. If n− 3 ≥ t, h0(P2, IΓ(t)) = 0 and (2) is verified if and only if h0(C,OC(n− 3− t)(−∆)) = h 0(P2,OP2(n− 3− t))− δ. On the other hand, if ψ : S → P2 is the blowing-up of the plane at the singular locus of Γ, denoting by Ei the pullback of the singular locus of Γ with respect to ψ and by OS(r) the sheaf OS(rψ ∗(H)), we have that h0(C,OC(n−3−t)(−∆)) = h 0(S,OS(n−3−t)(− Ei)) = h 0(P2,OP2(n−3−t)⊗A) where A is the ideal sheaf of singular points of Γ. � Remark 3.4. Notice that, if an irreducible and reduced plane curve Γ of degree n with only nodes and cusps as singularities is geometrically t-normal, with t ≤ n−3, then it is geometrically r-normal for every r ≤ t. Indeed, if a set of points imposes independent linear conditions to a linear system S, then it imposes independent linear conditions to every linear system S′ containing S. Theorem 3.5. Let Σnk,d be the variety of irreducible and reduced plane curves of degree n with d nodes and k cusps. Suppose that d, k, n and t are such that d+ k ≤ n2 − (3 + 2t)n+ 2 + t2 + 3t = h0(OP2(n− t− 3))(3) t ≤ n− 3 if k = 0,(4) k ≤ 6 if t = 1, 2 and(5) k ≤ 6 + [ ], if t = 3,(6) where [−] is the integer part of −. Then the variety Σnk,d is not empty and there ex- ists at least an irreducible component W ⊂ Σnk,d whose general element corresponds to a geometrically t-normal plane curve. Remark 3.6. As we shall see in the next section, (see proposition 4.1), the geomet- ric linear normality of the plane curve corresponding to the general element of an irreducible component Σ of Σnk,d, is related with the number of moduli of Σ. Another motivation for the previous theorem has been the family of irreducible plane sextics with six cusps. By [25], we know that Σ66,0 contains at least two irreducible compo- nents Σ1 and Σ2. The general point of Σ1 corresponds to a sextic with six cusps on a conic, whereas the general element of Σ2 corresponds to a sextic with six cusps not on a conic. Note that, by the previous lemma the general element of Σ2 param- eterizes a geometric linearly normal sextic, unlike the general element of Σ1, which corresponds to a projection of a canonical curve of genus four. Theorem 3.5, proves in particular that, under a suitable restriction, (see inequality (3)), on the genus of the curve corresponding to the general element of the family and, if the number of the cusps is small, the variety Σnk,d contains a not empty irreducible component whose general element corresponds to a curve which is not a projection of an other curve, lying in a projective space of larger dimension. We notice that the inequality (3) of the previous theorem can’t be improved. Indeed, if g = − k − d, then k + d > h0(P2,OP2(n− 3 − t)) if and only if g < 2tn−t2−3t . On the other hand, by using the same notation as in theorem (3.5), if g < 2tn−t , then, by Riemann- Roch theorem, we have that h0(C,OC(t)) ≥ tn− g+1 > +1 = h0(P2,OP2(t)). On the contrary, inequalities (5) and (6) are not sharp, (see example 3.7). 8 CONCETTINA GALATI In the case of k = 0 and t = 1, theorem 3.5 has been proved by Sernesi in [18], section 4. The case k = 0 and t ≤ n−3 is already contained in [5]. To show theorem 3.5, we proceed by induction on the degree n and on the number of nodes and cusps of the curve. The geometric idea at the base of the induction on the degree of the curve is, mutatis mutandis, the same as that of Sernesi. Proof of theorem 3.5. Let t be a positive integer such that n−3−t ≥ 0 and letW ⊂ Σnk,d be an irreducible component of Σ k,d. By standard semicontinuity arguments it follows that, if there exists a point [C] ∈ W corresponding to a geometrically t-normal curve with only k cusps and d nodes as singularities, then the general element of W corresponds to a geometrically t-normal plane curve. Moreover, if the theorem is true for fixed n, t ≤ n − 3, k as in (5) or in (6) and k + d as in (3), then the theorem is true for n, t and any k′ ≤ k and d′ ≤ d+ k − k′. Indeed, from the hypotheses (3), (5) and (6), it follows in particular that k < 3n. By section 2.1, under this hypothesis, for every k′ ≤ k and for every d′ ≤ d + k − k′, there exists a family of plane curves C → ∆ of degree n, parametrized by a curve ∆ ⊂ Σnk′,d′ , whose special fibre is C0 = C and whose general fibre Cz has d ′ nodes and k′ cusps as singularities. The statement follows by applying the semicontinuity theorem to the family C̃ → ∆̃, obtained by normalizing the total space of the pull-back family of C → ∆ to the normalization curve ∆̃ of ∆. Finally, it’s enough to show the theorem when the equality holds in (5), (6) and (3). First of all we consider the case k = 0. We will show the statement for any fixed t and by induction on n. Let, then t ≥ 1 and n = t+3. In this case the equality holds in (3) if d = 1 = h0(P2,OP2). Since one point imposes independent linear conditions to regular functions, by using lemma 3.2, we find that every irreducible plane curve of degree n = t + 3 with one node and no further singularities is geometrically t- normal. So, the first step of the induction is proved. Suppose, now, that the theorem is true for n = t+3+a and let [Γ] ∈ Vn,g be a point corresponding to a geometrically t-normal curve with a 2+3a+2 nodes. Let D be a line which intersects transversally Γ and let P1, ..., Pt+1 be t + 1 marked points of Γ ∩ D. If Γ ′ = Γ ∪ D ⊂ P2, then P1, ..., Pt+1 are nodes for Γ ′. Let C → Γ be the normalization of Γ and C′ → Γ′ the partial normalization of Γ′, obtained by smoothing all singular points of Γ′, except P1, ..., Pt+1. We have the following exact sequence of sheaves on C (7) 0 → OD(t)(−P1 − ...− Pt+1) → OC′(t) → OC(t) → 0, where OC′(t) := OC′(tH) and H is the pull-back with respect to C ′ → Γ′ of general line of P2. Since deg(OD(t)(−P1 − ...− Pt+1)) < 0, we get that h0(D,OD(t)(−P1 − ...− Pt+1)) = 0 and so (8) h0(C′,OC′(t)) = h 0(C,OC(t)) = h 0(P2,OP2(t)). Now, by section 2.1, we can obtain Γ′ as the limit of a 1-parameter family of irreducible plane curves ψ : C → ∆ ⊂ P (n+1)(n+4) of degree n+ 1 = t+ a+ 4 with a2 + 3a+ 2 + n− t− 1 = (a+ 1)2 + 3(a+ 1) + 2 = h0(P2,OP2(n+ 1− t− 3)) nodes specializing to nodes of Γ′ different from the marked points P1, ..., Pt+1. More- over, one can prove that ∆ is smooth, (see [24] or [25]). Normalizing C, we obtain a family whose general fibre is smooth and whose special fibre is exactly C′, and we conclude the inductive step by (8) and by semicontinuity theorem. Now we consider the case t = 1, 2 or 3 and k as in (5) and in (6). Suppose the theorem is true for n and let [Γ] ∈ Σnk,d be a general point in one of the irreducible NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 9 components of Σnk,d. Then, let D be a smooth plane curve of degree t if t = 1, 2 or an irreducible cubic with a cusp if t = 3. By the generality of Γ, we may suppose that D intersects Γ transversally. Let P1, ..., Pt2+1 be t 2+1 fixed points of Γ∩D. If Γ′ = Γ ∪D, then P1, ..., Pt2+1 are nodes for Γ ′. Let C → Γ be the normalization of Γ and C′ → Γ′ the partial normalization of Γ′, obtained by smoothing all singular points except P1, ..., Pt2+1. By using the same notation and by arguing as before, from the following exact sequence of sheaves on C′ 0 → OD(t)(−P1 − ...− Pt2+1) → OC′(t) → OC(t) → 0, we deduce that (9) h0(C′,OC′(t)) = h 0(C,OC(t)) = h 0(P2,OP2(t)). Now, by section 2.1, we can obtain Γ′ as limit of a family of irreducible plane curves φ : C → ∆ of degree n+ t with d+ nt− t2 − 1 = (n+t)2+(3+2t)(n+t)+t2+3t+2 nodes specializing to nodes of Γ′ different to P1, ..., Pt2+1, and k + 2−3t+2 cusps specializing to cusps of Γ. We conclude by (9) and by semicontinuity, as before. Now we have to show the first step of the induction. For t = 1 the induction begins with the cases (n, k) = (4, 1), (5, 3), (6, 6). Trivially, if n = 4 and k = 1 one point imposes independent conditions to the linear system of regular functions. If n = 5 and k = 3 we have to show that there are irreducible quintics with three cusps not on a line. A quintic with three cusps is a projection of the rational normal quintic C5 ⊂ P 5 from a plane generated by three points lying on three different tangent lines to C5. By Bezout theorem the three cusps of such a plane curve can’t be aligned. If n = k = 6, one can repeat the classical argument used by Zariski, see [24] or example 3.20 of chapter 2 of [7]. For t = 2 we have to show the theorem for (n, k) = (5, 1), (6, 3), (7, 6), (8, 6), while for t = 3 we have to show the theorem for (n, k) = (6, 1), (7, 3), (8, 6), (9, 6), (10, 6). The case t = 2 and (n, k) = (5, 1) is trivial. When t = 2, n = 6 and k = 3 we have that n − 3 − t = 1. To show that there exists an irreducible sextic with three cusps not on a line, consider a rational quartic C4 with three cusps, (see corollary 3.18 of chapter 2 of [7] for the existence). By Bezout theorem, the three double points of C4 can’t be aligned. Then consider a sextic C6 which is union of C4 and a conic C2 which intersects C4 transversally. By section 2.1, one can smooth the intersection points of C4 and C2 obtaining a family of sextics with three cusps not on a line. For t = 2, n = 7 and k = 6 we argue as in the previous case, by using a sextic C6 with six cusps not on a conic and a line R with intersects C6 transversally. Similarly for t = 2 , n = 8 and k = 6 and t = 3 and (n, k) = (6, 1), (7, 3), (8, 6), (9, 6), (10, 6). � Example 3.7. Inequalities (5) and (6) are not sharp. To see this, we can consider the example of curves of degree 10. We recall that we say that a plane curve is geometrically linearly normal (g.l.n. for short) if it is geometrically 1-normal. The- orem 3.5 ensures the existence of g.l.n. irreducible plane curves of degree 10 with k ≤ 6 cusps and nodes as singularities. But, by using the same ideas as we used in theorem 3.5, one can prove the existence of g.l.n. plane curves of degree 10 with nodes and k ≤ 9 cusps. It is enough to consider a sextic Γ6 with six cusps not on a conic and a rational quartic Γ4 with three cusps intersecting Γ6 transversally. We choose five points P1, . . . , P5 of Γ4 ∩ Γ6. If Γ 6 and Γ 4 are the normalization curves of Γ6 and Γ4 respectively and C ′ is the partial normalization of Γ6 ∪Γ4 obtained by normalizing all its singular points except P1, . . . , P5, by considering the following exact sequence 0 → OΓ′4(1)(−P1 − · · · − P5) → OC′(1) → OΓ′6(1) → 0 10 CONCETTINA GALATI we find that h0(C′,OC′(1)) = 3. By using terminology of section 2.1, the statement follows by smoothing the singular points P1, . . . , P5 of Γ6 ∪Γ4, and by semicontinu- ity, as in the proof of theorem 3.5. The bound on the number of cusps of theorem 3.5 can be improved also for t = 2 or t = 3. For example, theorem 3.5 ensures the existence of geometrically 3-normal curves of degree 12 with k ≤ 6 and nodes as further singularities. But, by considering a geometrically 3-normal curve of de- gree 8 with six cusps and a quartic with 3 cusps and arguing as before, we can find geometrically 3-normal irreducible plane curves of degree 12 with nodes and k ≤ 9 cusps. 4. Families of plane curves with nodes and cusps with finite and expected number of moduli. Let Σ ⊂ Σnk,d be an irreducible component of Σ k,d. We want to give sufficient conditions for Σ to have the expected number of moduli. Let [Γ] ∈ Σ be a general element, corresponding to a plane curve Γ with normalization map φ : C → Γ. We shall denote by ωC the canonical sheaf of C and by OC(1) the sheaf associated to the pullback to C of the divisor cut out on Γ from the general line of P2. Proposition 4.1. Let Σ ⊂ Σnk,d be an irreducible component of Σ k,d and let [Γ] ∈ Σ be a general element, corresponding to a plane curve Γ with normalization map φ : C → Γ. Suppose that Σ is smooth of the expected dimension equal to 3n+g−1−k at [Γ]. Moreover, suppose that: (1) Γ is geometrically linearly normal, i.e. h0(C,OC(1)) = 3, (2) the Brill-Noether map µo,C : H 0(C,OC(1))⊗H 0(C, ωC(−1)) → H 0(C, ωC) is surjective. Then Σ has the expected number of moduli equal to 3g − 3 + ρ− k. Proof. The case k = 0 has been proved by Sernesi in [18], section 4. We shall assume k > 0. Let Γ be a plane curve verifying the hypotheses of the proposition. By lemma 1.5.(b) of [22], the hypothesis that Σ is smooth of the expected dimension at [Γ] implies the vanishing H1(C,Nφ) = 0, where Nφ if the normal sheaf of φ. We recall that, denoting by ΘC and ΘP2 the tangent sheaf of C and P 2 respectively, then the normal sheaf of φ is defined as the cokernel of the differential map φ∗ of φ (10) 0 → ΘC → φ∗ΘP2 → Nφ → 0 By theorem 3.1 of [13], the vanishing H1(C,Nφ) = 0 is a sufficient condition for the existence of a universal deformation family of the normalization map φ, whose parameter space B is smooth at the point 0 corresponding to φ, with tangent space at 0 equal to H0(C, Nφ). On the contrary, by [3], p. 487, the Severi variety Vn,g = Σ 0,k+d of irreducible plane curves of genus − d − k is singular at the point [Γ] and the universal deformation space B of φ is a desingularization of Vn,g at [Γ]. Moreover, by corollary 6.11 of [2], if Bk = F −1(Σ) is the locus of points of B corresponding to a morphism with k ramification points, then the tangent space to Bk at 0 is a subspaceW of H 0(C,Nφ) NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 11 of codimension k such that W ∩H0(C,Kφ) = 0, where Kφ is the torsion subsheaf of Nφ. By [3], p.487, it follows that, if F : B → Vn,g is the natural (1 : 1)-map from B to Vn,g, then the differential map dF : H0(C, Nφ) → T[Γ]Vn,g restricts to an isomorphism between W and the tangent space T[Γ]Σ to Σ at [Γ]. We can now go back to the number of moduli of Σ. From the exact sequence (10), by using that H1(C, Nφ) = 0, we get the following long exact sequence 0 → H0(C,ΘC) → H 0(C, φ∗ΘP2) → H 0(C,Nφ) → H1(C,ΘC) → H 1(C, φ∗ΘP2) → 0 Recalling that the space H1(C,ΘC) is canonically identified with the tangent space T[C]Mg to Mg at the point associated to the normalization C of Γ, the coboundary map δC : H 0(C,Nφ) → H 1(C,ΘC) sends the Horikawa class of an infinitesimal deformation of φ to the Kodaira-Spencer class of the corresponding infinitesimal deformation of C. So, δC |W is the differential map at the point 0 ∈ B of the moduli map ΠΣ ◦ F : Bk = F −1(Σ) 99K Mg. Since the point [Γ] is general in Σ, and recalling the isomorphism dF :W → T[Γ]Σ, we have that the number of moduli of Σ = dim(δC(W )). Now, from the exact sequence (10), we have that dim(δC(H 0(C,Nφ)) = 3g − 3− h 1(C, φ∗ΘP2). Moreover, from the pull-back to C of the Euler exact sequence, we deduce the well known isomorphism H1(C, φ∗ΘP2) ≃ coker(µ 0,C) ≃ (ker(µ0,C)) and we conclude that (11) dim(δC(H 0(C,Nφ))) = 3g − 3− dim(ker(µ0,C)). Notice that the previous equality is always true, even if Γ doesn’t verify the hypoth- esis (1) or (2) of the statement. Moreover, if Γ is geometrically linearly normal, i.e. if h0(C,OC(1)) = 3, we have that ρ = 3n− 2g − 6 = dim(coker(µo,C))− dim(ker(µo,C)). When µo,C is surjective, ρ = − dim(ker(µo,C)) and (12) dim(δC(H 0(C,Nφ)) = 3g − 3 + ρ = dim(B)− 8 = dim(Vn,g)− 8. Since the dimension of the fibre of the moduli map ΠVn,g ◦ F : B 99K Mg has dimension at least equal to 8 = dim(Aut(P2)), from (12) we deduce that the differential map of ΠVn,g ◦F has maximal rank at 0 and, in particular, we have that dim((ΠVn,g ◦ F ) −1([C])) = 8. Equivalently, there exist only finitely many g2n on C. It follows that there are only finitely many g2n on C mapping C to the plane as a curve with k cusps and d nodes. Then, dim(δc(W )) = dim(ΠΣ(Σ)) = 3g − 3 + ρ− k. Remark 4.2. Arguing as in the proof of the previous proposition, it has been proved in [18] that, if Γ is a geometrically linearly normal plane curve with only d nodes as singularities and the Brill-Noether map µo,C of the normalization morphism of Γ is injective, then Σ = Σn0,d has general moduli. If Σ ⊂ Σ k,d and [Γ] ∈ Σ verify the hypotheses of proposition 4.1 but we assume that µo,C is injective, we may only conclude that ΠVn,g ◦ F is dominant with surjective differential map at [Γ]. 12 CONCETTINA GALATI So dim(Π−1Vn,g ([C])) = ρ + 8. But this is not useful to compute the dimension of δC(W ) = δC(T[Γ]Σ). However, in this case we get that δC(T[Γ]Σ) + δC(H 0(C,Kφ)) = δC(H 0(C,Nφ)) = H 1(C,ΘC). Then, by using that dim(δC(H 0(C,Kφ))) ≤ k and by recalling that if Σ has the expected dimension then the number of moduli of Σnk,d is at most the expected one (see lemma 2.2 and remark 2.3), we find that 3g − 3− k ≤ number of moduli of Σ ≤ 3g − 3 + ρ− k. Remark 4.3. Notice that, if a plane curve Γ of genus g verifies the hypotheses (1) and (2) of the previous proposition, then the Brill-Noether number ρ(2, g, n) is not positive and, in particular, g ≥ 3. We don’t know examples of complete irreducible families Σ ⊂ Σnk,d with the expected number of moduli whose general element [Γ] corresponds to a curve Γ of genus g, with ρ(2, g, n) ≤ 0, which doesn’t verify properties (1) and (2). Lemma 4.4 ([5], Corollary 3.4). Let Γ be an irreducible plane curve of degree n with only nodes and cusps as singularities and let φ : C → Γ be the normalization morphism of Γ. Suppose that Γ is geometrically 2-normal, i.e. h0(C,OC(2)) = 6. Then the Brill-Noether map µo,C : H 0(C,OC(1))⊗H 0(C, ωC(−1)) → H 0(C, ωC) is surjective. Proof. By lemma 3.2, the curve Γ is geometrically 2-normal if and only if the scheme N of the singular points of Γ imposes independent linear conditions to the linear systemH0(P2,OP2(n−5)) of plane curves of degree n−5. SinceH 0(P2,OP2(n−5)) ⊂ H0(P2,OP2(n−4)), N imposes independent linear conditions plane curves of degree n−4, and, by using lemma 3.2, we get that h0(C,OC(1)) = 3, i.e. Γ is geometrically linearly normal. Now, denote by IN |P2 the ideal sheaf of N . Notice that the curve Γ is geometrically 2-normal if and only if the ideal sheaf IN |P2(n− 4) is 0-regular, (in the sense of Castelnuovo-Mumford). Indeed, since h2(P2, IN |P2(n− 6)) = 0, the ideal sheaf IN |P2(n− 4) is 0-regular if and only if h 1(P2, IN |P2(n− 5)) = 0. Because of the 0-regularity of IN |P2(n− 4), we have the surjectivity of the natural map H0(P2, IN |P2(n− 4))⊗H 0(P2,OP2(1)) → H 0(P2, IN |P2(n− 3)), (see [17]). Finally, by the geometric linear normality of Γ, the vertical maps of the following commutative diagram H0(P2,OP2(1))⊗H 0(P2, IN |P2(n− 4)) // H0(P2, IN |P2(n− 3)) H0(C,OC(1))⊗H 0(C, ωC(−1)) // H0(C, ωC) are surjective and, hence, the Brill-Noether map µo,C is surjective too. � Corollary 4.5. Let Σ ⊂ Σnk,d be an irreducible component of Σ k,d of dimension equal to 3n + g − 1 − k, such that the general point [Γ] ∈ Σ corresponds to a geometrically 2-normal plane curve. Then Σ has the expected number of moduli equal to 3g − 3 + ρ− k. Proof. It follows from proposition 4.1 and lemma 4.4. � In order to produce examples of families of irreducible plane curves with nodes and cusps with the expected number of moduli, we study how increases the rank of the Brill-Noether map by smoothing a node or a cusp of the general curve of the family, (in the sense of section 2.1). NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 13 Let Σ ⊂ Σnk,d, with n ≥ 5, be an irreducible component of Σ k,d, let [Γ] ∈ Σ be a general point of Σ and let φ : C → Γ be the normalization of Γ. Choose a singular point P ∈ Γ and denote by φ′ : C′ → Γ the partial normalization of Γ obtained by smoothing all singular points of Γ, except the point P . If ωC′ is the dualizing sheaf of C′ and µo,C′ : H 0(C′,OC′(1))⊗H 0(C′, ωC′(−1)) → H 0(C′, ωC′), is the natural multiplication map, we have the following result. Lemma 4.6. If h0(C,OC(1)) = 3 and the geometric genus g of C is such that g > n−2, with n ≥ 5, then rk(µo,C′) ≥ rk(µo,C)+1. In particular, if h 0(C,OC(1)) = 3, n ≥ 5 and µo,C is surjective, then µo,C′ is also surjective. Proof. Let ψ : C → C′ be the normalization map. We recall that, if we set φ∗(P ) := p1 + p2 when P is a node and φ ∗(P ) = 2φ−1(P ) when P is a cusp, then the dualizing sheaf of C′ is a subsheaf of ψ∗(ωC(φ ∗(P ))), (see for example [10], p.80). In particular we have the following exact sequence (13) 0 → ωC′ → ψ∗ωC(φ ∗(P )) → CP → 0 where CP is the skyscraper sheaf on C with support at P . From this exact sequence, we deduce that H0(C′, ωC′) ≃ H 0(C, ωC(φ ∗(P ))). Moreover, tensoring (13) by OC′(−1), we find the exact sequence (14) 0 → ωC′(−1) → ψ∗ωC(φ ∗(P ))(−1) → CP → 0 from which we get an injective map H0(C′, ωC′(−1)) → H 0(C, ωC(φ ∗(P ))(−1)). On the other hand (15) h0(C′, ωC′(−1)) = h 0(C, ωC(φ ∗(P ))(−1)) = g − n+ 3 and so H0(C′, ωC′(−1)) ≃ H 0(C, ωC(φ ∗(P ))(−1)). Moreover, from the hypothesis h0(C,OC(1)) = 3, we have that H 0(C,OC(1)) ≃ H0(C′,OC′(1)) ≃ H 0(P2,OP2(1)). Therefore, in the following commutative dia- H0(C′,OC′(1))⊗H 0(C′, ωC′(−1)) µo,C′ H0(C′, ωC′) H0(C,OC(1))⊗H 0(C, ωC(−1)(φ ∗(P ))) // H0(C, ωC(φ ∗(P ))) where we denoted by µ′o,C the natural multiplication map, the vertical maps are isomorphisms. In particular, rk(µo,C′) = rk(µ o,C). In order to compute the rank of µ′o,C , we consider the following commutative dia- 14 CONCETTINA GALATI H0(C,OC(1))⊗H 0(C, ωC(−1)) H0(C, ωC) H0(C,OC(1))⊗H 0(C, ωC(−1)(φ ∗(P ))) // H0(C, ωC(φ ∗(P ))) where the vertical maps are injections. Notice that, since we supposed n ≥ 5, h0(C,OC(1)) = 3 and g > n−2 ≥ 3, the sheaf OC(1) is special. We deduce that C is not hyperelliptic and, chosen a basis of H0(C, ωC), the associated map C → P g−1 is an embedding. On the contrary, the sheaf ωC(φ ∗(P )) does not define an embedding on C. Choosing a basis of H0(C, ωC(φ ∗(P )) and denoting by Φ : C → Pg the associated map, this will be an embedding outside φ∗(P ). If P is a node of C and φ∗(P ) = p1 + p2, the image of C to P g, with respect to Φ, will have a node at the image point Q of p1 and p2. If P ∈ Γ is a cusp, then Φ(C) will have a cusp at the image point Q of φ−1(P ). The hyperplanes of Pg passing through Q cut out on C the canonical linear series |ωC |. Moreover, if we denote by B ⊂ P the subspace which is the base locus of the hyperplanes of Pg corresponding to Im(µ′o,C), then Q /∈ B. Indeed, B intersects the curve C in the image of the base locus of |OC(1)|+ |ωC(φ ∗(P ))(−1)| := P(Im(µ′0,C)), which coincides with the base locus of |ωC(φ ∗(P ))(−1)|, since |OC(1)| is base point free. Now, by (15), h0(ωC(φ ∗(P ))(−1)) = 3 + g − n = h0(C, ωC(−1)) + 1. Then φ∗(P ) does not belong to the base locus of |ωC(φ ∗(P ))(−1)|, and so dim(< Q,B >Pg) = dim(B) + 1. Finally, we find that rk(µo,C) = rk(Gµo,C) ≤ dim(Im(G) ∩ Im(µ o,C)) ≤ g + 1− dim(< B,Q >Pg)− 1 = g − 1− dim(B) = rk(µ′o,C)− 1. Corollary 4.7. Let Σ ⊂ Σnk,d be a non-empty irreducible component of the expected dimension of Σnk,d, with n ≥ 5. Suppose that Σ has the expected number of moduli and that the general element [Γ] ∈ Σ corresponds to a g.l.n. plane curve Γ of geometric genus g such that, if C → Γ is the normalization of Γ, then the map µo,C is surjective. Then, for every k ′ ≤ k and d′ ≤ d + k − k′, there is at least an irreducible component Σ′ ⊂ Σnk′,d′ , such that Σ ⊂ Σ ′, the general element [D] ∈ Σ′ corresponds to a g.l.n. plane curve D of geometric genus g′ with normalization Dν → D and the Brill-Noether map µ0,Dν surjective. In particular, Σ ′ has the expected number of moduli. Proof. Let Γ be the curve corresponding to the general element [Γ] of Σ ⊂ Σnk,d. Since by hypothesis Σ is smooth of the expected dimension at [Γ], by section 2.1, for every k′ ≤ k and for every d′ ≤ d+ k− k′ there exists an irreducible component Σ′ of Σnk′,d′ containing Σ. In order to prove the statement, it is enough to show it under the hypotheses k′ = k− 1 and d′ = d+1, k = k′ and d′ = d− 1 or d = d′ and k′ = k − 1. If k′ = k − 1 and d′ = d + 1, then the statement follows by standard semiconinuity arguments. If k = k′ and d′ = d − 1 or d = d′ and k′ = k − 1, the statement follows by lemma 4.6 and by standard semicontinuity arguments. � NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 15 The following lemma has been stated and proved by Sernesi in [18]. Actually, Sernesi supposes that Γ has only nodes as singularities. But, since his proof works for plane curves Γ with any type of singularities and, since we need it for curves with nodes and cusps, we state the lemma in a more general form. Lemma 4.8. ([18], lemma 2.3) Let Γ be an irreducible and reduced plane curve of degree n ≥ 5 with any type of singularities. Denote by C the normalization of Γ. Suppose that h0(C,OC(1)) = 3 and the Brill-Noether map µo,C : H 0(C,OC(1))⊗H 0(C, ωC(−1)) → H 0(C, ωC), has maximal rank. Let R be a general line and let P1, P2 and P3 be three fixed points of Γ ∩ R. We denote by C′ the partial normalization of Γ′ = Γ ∪ R, obtained by smoothing all the singular points, except P1, P2 and P3. Then h 0(C′,OC′(1)) = 3 and, denoting by ωC′ the dualizing sheaf of C ′, the multiplication map µo,C′ : H 0(C′,OC′(1))⊗H 0(C′, ωC′(−1)) → H 0(C′, ωC′), has maximal rank. Theorem 4.9. Let Σnk,d be the algebraic system of irreducible plane curves of degree n ≥ 4 with k cusps, d nodes and geometric genus g = − k − d. Suppose that: (16) n− 2 ≤ g equivalently k + d ≤ h0(P2,OP2(n− 4)) (17) k ≤ 6 + if 3n− 9 ≤ g and n ≥ 6, (18) k ≤ 6 otherwise. Then Σnk,d has at least an irreducible component Σ which is not empty and such that, if Γ ⊂ P2 is the curve corresponding to the general element of Σ and C is the normalization curve of Γ, then h0(C,OC(1)) = 3 and the map µo,C has maximal rank. In particular, when ρ ≤ 0, the algebraic system Σ has the expected number of moduli equal to 3g − 3 + ρ− k. Proof. Suppose that (17) holds. Then, by observing that g ≥ 3n− 9 if and only if k + d ≤ h0(P2,OP2(n− 6)) and by using theorem 3.5 for t = 3, we have that there exists an irreducible com- ponent Σ of Σnk,d whose general element is a geometrically 3-normal plane curve Γ. By remark 3.4, it follows that also the linear systems cut out on C by the conics and the lines are complete. The statement follows from corollary 4.5. In order to prove the theorem under the hypothesis (18), we consider the following subcases: (1) 2n− 5 ≤ g ≤ 3n− 9, i.e. h0(OP2(n− 6)) ≤ k+ d ≤ h 0(OP2(n− 5)) and n ≥ 5, (2) n− 2 ≤ g ≤ 2n− 7 and n ≥ 5, (3) g = 2n− 6 and n ≥ 4. Suppose that (1) holds. By theorem 3.5 for t = 2, we know that, under this hypothesis, there exists a nonempty component Σ ⊂ Σnk,d, whose general element is geometrically 2-normal. We conclude as in the previous case, by corollary 4.5. Now, suppose that g and n verify (2). We shall prove the theorem by induction on n and g. Set g = 2n − 7 − a, with a ≥ 0 fixed. Suppose that the theorem is true for the pair (n, g), with n ≥ 7. We shall prove the theorem for (n + 1, g + 2), observing that g + 2 = 2(n + 1) − 7 − a. Let Γ be a g.l.n. irreducible plane curve 16 CONCETTINA GALATI of degree n and genus g = 2n − 7 − a with k ≤ 6 cusps, d nodes and no more singularities. Let C be the normalization of Γ. Suppose that the Brill-Noether map µo,C has maximal rank. Let R ⊂ P 2 be a general line and let P1, P2 and P3 be three fixed points of Γ∩R. By section 2.1, since k ≤ 6 < 3n, one can smooth the singular points P1, P2, P3 and preserve the other singularities of Γ ∪ R ⊂ P 2, obtaining a family of plane curves C → ∆ whose general fibre is irreducible, has degree n+1 and genus g+2. We conclude by lemma 4.8 and by standard semicontinuity arguments. Now we prove the first step of the induction for n ≥ 7. If n = 7, we get 0 ≤ a ≤ 2. Let a = 0, i.e. g = 2n−7−a= 7. Let Γ be a g.l.n. irreducible plane curve of degree n = 7, of genus g = n = 7 with k ≤ 6 cusps and nodes as singularities, such that no seven singular points of Γ lie on an irreducible conic. To prove that there exists such a plane curve, notice that, by applying theorem 3.5 for t = 1, we get that, for any fixed k ≤ 6, there exists a g.l.n. irreducible sextic D of genus four with k cusps and d = 6 − k nodes. Let R1, . . . , R6 be the singular points of D. Since the points R1, . . . , R6 of D impose independent linear conditions to the conics, however we choose five singular points Ri1 , . . . , Ri5 of D, with I = (i1, . . . , i5) ⊂ (1, . . . , 6), there exists only one conic CI , passing through these points. Let us set S = CI ∩D and let R be a line intersecting D transversally at six points out of S. By Bezout theorem, no seven singular points of Γ′ = D ∪ R belong to an irreducible conic. Moreover, if D̃ is the normalization of D, if Q1, . . . , Q4 are four fixed points of D∩R and D′ is the partial normalization of Γ′ obtained by smoothing the singular points except Q1, . . . , Q4, then, by the following exact sequence (19) 0 → OR(1)(−Q1 − · · · −Q4) → OD′(1) → OD̃(1) → 0 we find that h0(D′, OD′(1)) = 3. By section 2.1, one can smooth the singulari- ties Q1, . . . , Q4 and preserve the other singularities of D ∪ R, getting a family of irreducible septics G → ∆ whose general fibre Γ is a geometrically linearly normal irreducible septic with k cusps and 8 − k nodes such that no seven singular point of Γ belong to an irreducible conic. Let, now, C be the normalization of Γ and let ∆ ⊂ C be the adjoint divisor of the normalization map φ : C → Γ. We shall prove that ker(µo,C) = 0. Since Γ is geometrically linearly normal, we have that h0(C, ωC(−1)) = h 0(C,OC(3)(−∆))) = g − n+ 2 = 2. Then, by the base point free pencil trick, we find that ker(µo,C) = H 0(C, ω∗C(B)⊗OC(2)), where B is the base locus of |ωC(−1) = OC(3)(−∆)|. Let F be the pencil of plane cubics passing through the eight double points P1, . . . , P8 of Γ and let BF be the base locus of the pencil F . Let Γ3 be the general element of F . Suppose that BF has dimension one. If BF contains a line l, then, by Bezout theorem, at most three points among P1, . . . , P8, say P1, . . . , P3 can lie on l and the other points have to be contained in the base locus of a pencil of conics F ′. Using again Bezout theorem, we find that the curves of F ′ are reducible and the base locus of F ′ contains a line l′. But also l′ contains at most three points of P4, . . . , P6. It follows that there is only one cubic through P1, . . . , P8. This is not possible by construction. Suppose that BF contains an irreducible conic Γ2. By Bezout theorem, at most seven points among P1, . . . , P8 may lie on Γ2. On the other hand, since dim(F) = 1, there are exactly seven points of P1, . . . , P8, say P1, . . . , P7, on Γ2 and the general cubic Γ3 of F is union of Γ2 and a line passing through P8. Since, by construction, no seven singular points of Γ lie on an irreducible conic, also in this case we get a contradiction. So the general element Γ3 of F is irreducible. Using again Bezout theorem, we find that Γ3 is smooth and F has only one more base point Q. We consider the following cases: a) Q doesn’t lie on Γ; NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 17 b) Q lies on Γ, but Q 6= P1, . . . , P8; c) Q is infinitely near to one of the points P1, . . . , P8, say Pî, i.e. the cubics of F have at P the same tangent line l, but l is not contained in the tangent cone to Γ d) Q is like in the case c), but l is contained in the tangent cone to Γ at P Suppose that the case a) or c) holds. Thus B = 0 and ker(µo,C) = H 0(C, ω∗C ⊗OC(2)) = H 0(C,OC(−2)(∆)). By Riemann-Roch theorem, h0(C,OC(−2)(∆)) = h 0(C,OC(6)(−2∆)) − 4. One sees that h0(C,OC(6)(−2∆)) = 4, by blowing-up the plane at P1, . . . , P8 and by using some standard exact sequences. Suppose now that the case b) holds. Thus B = Q and dim(ker(µo,C)) = h 0(C,OC(−2)(∆ +Q)) = h 0(C,OC(6)(−2∆−Q))− 3. Also in this case one sees that h0(C,OC(6)(−2∆ − Q)) = 3 by blowing-up at P1, . . . , P8 and Q and by using standard exact sequences. Finally, we analyze the case d). Let Φ : S → P2 be the blow-up of the plane at P1, . . . , P8 with exceptional divisors E1, . . . , E8. Let Q ∈ Eî be the intersection point of Eî and the strict transform C3 of the general cubic Γ3 of the pencil F . We denote by Φ̃ : S̃ → S the blow-up of S at Q and by Ψ : S̃ → P2 the composition map of the maps Φ and Φ̃. We still denote by E1, . . . , E8 their strict transforms on S̃, by C and C3 the strict transforms of Γ and Γ3 and by EQ the new exceptional divisor of S̃. In this case we have that Ψ∗(Γ) = C+2 Ei+3EQ, Ψ ∗(Γ3) = C3+ Ei+2EQ. Moreover, the divisor ∆ is cut out on C by iEi + EQ and the base locus B of the linear series |ωC(−1)| coincides with the intersection point of EQ and C. So, we have that dim(ker(µo,C)) = h 0(C,OC(−2)( Ei+2EQ)) = h 0(C,OC(6)(−2 Ei−3EQ))−3. Moreover, from the following exact sequence 0 → O (−1) → O (6)(−2 Ei − 3EQ) → OC(6)(−2 Ei − 3EQ) → 0 we find that H0(C,OC(6)(−2 Ei− 3EQ)) = H 0(S̃,O (6)(−2 Ei− 3EQ)). In order to show that h0(S̃,O (6)(−2 Ei − 3EQ)) = 3, we consider the following exact sequence 0 → O (3)(− Ei − EQ) → OS̃(6)(−2 Ei − 3EQ) →(20) → OC3(6)(−2 Ei − 3EQ) → 0 By Riemann-Roch theorem, we have that h0(C3,OC3(6)(−2 Ei − 3EQ)) = 1 and h 1(C3,OC3(6)(−2 Ei − 3EQ) = 0. Moreover, by Serre duality we have that H1(S̃,O (3)(− Ei − EQ))) = H 1(S̃,O (−6)(2 Ei + 3EQ))). From the exact sequence (21) 0 → O (−6)(+2 Ei + 3EQ)) → OS̃(1) → OC(1) → 0 18 CONCETTINA GALATI by using that the map H0(S̃,O (1)) → H0(C,OC(1)) is surjective and that h1(S̃,O (1)) = 0, we find that H1(S̃,O (−6)(+2 Ei + 3EQ))) = H 1(S̃,O (3)(− Ei − EQ))) = 0. Then, by (20), h0(S̃,O (6)(−2 Ei − 3EQ)) = h 0(S̃,O (3)(− Ei − EQ)) + h0(C3,OC3(6)(−2 Ei−3EQ)) = 3 and ker(µo,C) = 0. The first step of induction for g = n = 7 and k ≤ 6 is proved. We complete the proof of the first step of the induction, for n and g verifying (2). When n = 7 and 1 ≤ a ≤ 2, the existence of a g.l.n. plane curve Γ follows from theorem 3.5. Using the above notation, h0(C, ωC(−1)) = 1 if a = 1 and h0(C, ωC(−1)) = 0 if a = 2. In any case µo,C is injective. When n ≥ 8 and a ≤ n − 6 the theorem follows by induction from the case n = 7. For n ≥ 8 and a = n− 5, we find that g = n− 2, or, equivalently, k + d = h0(P2,OP2(n− 4)). In theorem 3.5, we proved the existence of geometrically linearly normal plane curves of degree n ≥ 8 and genus g = n− 2, with nodes and k ≤ 6 cusps. For every such plane curve Γ, using the above notation, the Brill-Noether map µo,C is injective since h0(C, ωC(−1)) = 0. The cases n = 5 and n = 6 are similar. Suppose now that n and g verify (3). First of all we prove the theorem for (n, g) = (4, 2), (5, 4), (6, 6). For n = 4 and g = 2, we find n = g + 2 and we argue as in the case n ≥ 8 and g = n − 2. Similarly, for (n, g) = (5, 4). For n = 6 and g = 6 in theorem 3.5 we proved the existence of geometrically linearly normal plane curves Γ with k ≤ 4 cusps and nodes as singularities. For every such a plane curve Γ, denoting by C its normalization, we get that h0(C, ωC(−1)) = 2, i.e. the linear system F of conics passing through the four singular points P1, . . . , P4 of Γ is a pencil which cuts out on C the complete linear series |ωC(−1)|. We have two possibilities: either the general element of this pencil is irreducible or it consists of a line containing exactly three singular points P1, P2, P3 of Γ and a line passing through P4. In any case the base locus of F intersects Γ only at P1, . . . , P4 and the linear series |ωC(−1)| has no base points. Then, by the base point free pencil trick , we find that ker(µo,C) = H 0(C, ω∗C ⊗ O(2)) = H 0(C,OC(−1)(∆)), where ∆ ⊂ C is the adjoint divisor of the normalization map C → Γ. By Riemann-Roch theorem, we have that h0(C,OC(−1)(∆)) = h 0(C,OC(4)(−2∆))−3. By blowing-up at P1, . . . , P4, one can see that h 0(C,OC(4)(−2∆)) = 3, as we wanted. Finally, we show the theorem under the hypothesis (3) for n ≥ 7, by using induction on n. In order to prove the inductive step we may use lemma 4.8, exactly as we did in the case (2). We prove the first step of induction. If n = 7 we have that g = 8. On pages 15 and 16 we proved the existence of geometrically linearly normal plane curves Γ of degree 7 and genus 7 with k ≤ 6, such that, if P1, . . . , P8 are the singular points of Γ, then no seven points among P1, . . . , P8 lie on a conic. In particular, we proved that, for every such a plane curve Γ, the general element of the pencil of cubics passing through P1, . . . , P8 is irreducible and, if φ : C → Γ is the normalization of Γ, then the Brill-Noether map µo,C is injective. Let C the partial normalization of Γ which we get by smoothing all the singular points of Γ except a node, say P8. By using the same notation and by arguing exactly as in the proof of lemma 4.6, we get the following commutative diagram H0(C′,OC′(1))⊗H 0(C′, ωC′(−1)) µo,C′ H0(C′, ωC′) H0(C,OC(1))⊗H 0(C, ωC(−1)(φ ∗(P8))) // H0(C, ωC(φ ∗(P8))) where µ′o,C is the multiplication map and the vertical maps are isomorphisms. We want to prove that the map µo,C′ is surjective. By the previous diagram it is enough NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 19 to prove that µ′o,C is surjective. Since h 0(C, ωC(φ ∗(P8))) = 8 and h0(C,OC(1))h 0(C, ωC(−1)(φ ∗(P8))) = 3(7− 7 + 3) = 9, we have that dim(ker(µo,C′)) ≥ 1 and µo,C′ is surjective if dim(ker(µo,C′)) = 1. By recalling that Γ is geometrically linearly normal, we have that, if Z is the scheme of the points P1, . . . , P7 and IZ|P2 is the ideal sheaf of Z in P 2, then in the following commutative diagram H0(C,OC(1))⊗H 0(C, ωC(−1)(φ ∗(P8))) H0(C, ωC(φ ∗(P8))) H0(P2,OP2(1))⊗H 0(P2, IZ|P2(3)) // H0(P2, IZ|P2(4)) the vertical maps are isomorphisms. Hence, it is enough to prove that the kernel of the multiplication map µ has dimension one. Let {f0, f1, f2} be a basis of the vector space H0(P2, IZ|P2(3)). Since the general cubic passing through P1, . . . , P8 is irreducible, we may assume that f0, f1 and f2 are irreducible. Suppose, by contradiction, that there exist at least two linearly independent vectors in the kernel of µ. Then, there exist sections u0, u1, u2 and v0, v1, v2 of H 0(P2,OP2(1)) such that the sections ui⊗ fi and vi⊗ fi are linearly independent in H 0(P2,OP2(1))⊗ H0(P2, IZ|P2(3)) and i=0 uifi = 0 i=0 vifi = 0. We can look at (22) as a linear system in the variables f0, f1, f2. The space of solutions of (22) is generated by the vector (u1v2 − u2v1, u3v0 − u0v3, u0v1 − u1v0). In particular, if we set qi = (−1) 1+iuivj − viuj, we find that fjqi = fiqj , for every i 6= j. But this is not possible since f1, f2 and f3 are irreducible. We deduce that dim(ker(µ)) = dim(ker(µo,C′)) = 1 and µo,C′ is surjective. The existence of a plane septic of genus 8 with k ≤ 6 cusps and nodes as singularities, with injective Brill-Neother map, follows now by smoothing the node P8 (in the sense of section 2.1) and by standard semicontinuity arguments. � Remark 4.10. Notice that the conditions which we found in theorem 4.9 in order that Σnk,d has at least an irreducible component with the expected number of moduli, are not sharp, even if we suppose ρ ≤ 0. To see this, notice that in remark 3.6 we proved the existence of an irreducible component Σ of Σ129,0 whose general element corresponds to a 3-normal plane curve. By remark 3.4 and corollary 4.5, we have that Σ has the expected number of moduli. Theorem 4.11. Σn1,d has the expected number of moduli, for every d ≤ Proof. First of all, we recall that, by [16], Σn1,d is irreducible for every d ≤ Moreover, from theorem 4.9 and from corollary 2.7, we know that Σn1,d is not empty and it has the expected number of moduli if either ρ ≤ 0 or ρ ≥ 2. Next we shall prove that, if ρ = 1, then the algebraic system Σn1,d = Σ (n−3)2 has general moduli. Equivalently, we will show that, if [Γ] ∈ Σn1,d is a general point and g = − 1 − d = 3n−7 , then, on the normalization curve C of Γ there are only finitely many linear series g2n with at least a ramification point. Notice that, 20 CONCETTINA GALATI if g = − 1 − d = 3n−7 , then n is odd and n ≥ 5. We prove the statement by induction on n. If n = 5 then g = 4. Let C ⊂ P3 be the canonical model of a general curve of genus four and let 2P + Q, with P 6= Q be a divisor in a g13 on C. This divisor is cut out on C by the tangent line to C at P . The projection of C from Q is a plane quintic of genus four with a cusp. This proves that Σ51,1 has general moduli. Now we suppose that the theorem is true for n and we prove the theorem for n+2. Let Γ ⊂ P2 be the plane curve with a cusp and (n−3)2 −1 nodes corresponding to a general point [Γ] ∈ Σn (n−3)2 and let C2 be an irreducible conic intersecting Γ transversally. By section 2.1, the point [C2 ∪ Γ] belongs to Σ (n+2−3)2 particular, however we choose four points P1, . . . , P4 of intersection between Γ and C2, there exists an analytic branch SP1,..., P4 of Σ (n−1)2 , passing through [C2∪Γ] and whose general point corresponds to an irreducible plane curve of degree n + 2 with a cusp in a neighborhood of the cusp of Γ and a node at a neighborhood of every node of C2 ∪ Γ different from P1, . . . , P4. Moreover, S := SP1,..., P4 is smooth at the point [C2 ∪ Γ], (see [7], chapter 2). Let Π : Σn+2 (n−1)2 99K M 3(n+2)−7 be the moduli map of Σn+2 (n−1)2 . In order to prove that Π is dominant it is sufficient to show that Π(S) = M 3n−1 . By section 2.1, there exist an analytic open sets Si ⊂ Σn+2 (n−3)2 −1+2n−i , with i = 1, 2, 3, such that S0 := S ∩ (P5 × Σn (n−3)2 ) ⊂ S1 ⊂ S2 ⊂ S3 ⊂ S. Every Si, with i = 1, 2, 3, has irreducible components, passing through [C2∪Γ] and intersecting transversally at [C2 ∪Γ], (see [7], chapter 2 or [25]). Moreover, the general point of every irreducible component of Si, with i = 1, 2, 3, corresponds to an irreducible plane curve Γi of degree n + 2 with a cusp in a neighborhood of the cusp of Γ, a node in a neighborhood of every node of C2 ∪ Γ different from P1, . . . , P4 and 4 − i nodes specializing to 4 − i fixed points among P1, . . . , P4, as Γi specializes to C2 ∪ Γ. Now, notice that the moduli map Π is not defined at the point [C2 ∪ Γ], but, if S is sufficiently small, then the restriction of Π to S extends to a regular function on S. More precisely, let C → ∆ be any family of curves, parametrized by a projective curve ∆ ⊂ S, passing through the point [C2 ∪ Γ] and whose general point corresponds to an irreducible plane curve of degree n+2 of genus 3(n+2)−7 with a cusp and nodes as singularities. If we denote by C′ → ∆ the family of curves obtained from C → ∆ by normalizing the total space, we have that the general fibre of C′ → ∆ is a smooth curve of genus 3n−1 , corresponding to the normalization of the general fibre of C → ∆, whereas the special fibre C′0 is the partial normalization of C2∪Γ, obtained by normalizing all the singular points, except P1, . . . , P4. Then, the map Π|S is defined at [C2 ∪Γ] and it associates to the point [C2∪Γ] the isomorphism class of C 0. Similarly, if [Γi] is a general point in one of the irreducible components of Si, with i = 1, 2, 3, then Π|S ([Γi]) is the partial normalization of Γi obtained by smoothing all the singular points except for the 4−i nodes of Γi tending to 4−i fixed points among P1, . . . , P4 as Γi specializes to C2∪Γ. It follows that, if we denote by M the locus of M 3n−1 parametrizing j-nodal curves, then ΠS(S i) ⊆ M4−i3n−1 , for every i = 0, . . . , 4, and ΠS(S i) ΠS(S i+1). In particular, we find that dim(Π|S (S)) ≥ dim(Π|S (S 0)) + 4. NUMBER OF MODULI OF IRREDUCIBLE FAMILIES... 21 In order to compute the dimension of Π|S (S 0) we consider the rational map F : Π|S (S 0) 99K M 3n−7 forgetting the rational tail. By the hypothesis that Σn (n−3)2 has general moduli and hence F is dominant. Moreover, if C is the normalization curve of Γ, by the generality of [Γ] in Σn (n−3)2 , we may assume that C is general in M 3n−7 want to show that dim(F−1([C])) = 5. In order to see this, we recall that, by the hypothesis that Σn (n−3)2 has general moduli, on C there exist only finitely many linear series of degree n and dimension two, mapping C to the plane as curve with a cusp and nodes as singularities. Let g2n be one of these linear series, let {s0, s1, s2} be a basis of g2n and φ ′ : C → Γ′ ⊂ P2 the associated morphism. If Q1, . . . , Q4 are four general points of Γ′, then the linear system of conics through Q1, . . . , Q4 is a pencil F(Q1, . . . , Q4). Let C2 and D2 be two general conics of F(Q1, . . . , Q4). We claim that, if η : P1 → C2 and β : P 1 → D2 are isomorphisms between P 1 and C2 and D2 respectively, then the points η −1(Q1), . . . , η −1(Q4) are not projectively equivalent to the points β−1(Q1), . . . , β −1(Q4). In order to prove this, it is enough to prove that there are at least two conics in the pencil F(Q1, . . . , Q4) which verify the claim. LetD ⊂ P2 be a conic. If we choose two sets of points p1, . . . , p4 and q1, . . . , q4 ofD not projectively equivalent onD, we may always find projective automorphisms A : P2 → P2 and A′ : P2 → P2 such that A(pi) = Qi and A ′(qi) = (Qi), for every i. By construction, the conics C2 = A(D) and D2 = A ′(D) belong to the pencil F (Q1, . . . , Q4) and verify the claim. This implies that the partial normalizations C and D′ of Γ′∪C2 and Γ ′∪D2, obtained by smoothing all the singular points except Q1, . . . , Q4, are not isomorphic. Now, let C 2 be a general conic of F(Q1, . . . , Q4) and let R1, . . . , R4 be four general points of Γ ′, different from Q1, . . . , Q4. If D is a general conic of the pencil F(R1, . . . , R4), then the partial normalization C and D′ of Γ′ ∪C′2 and Γ ′ ∪D′2 obtained, respectively, by smoothing all the singular points except Q1, . . . , Q4 and R1, . . . , R4, are not isomorphic. Indeed, since C is a general curve of genus 3n−7 ≥ 7, the only automorphism of C is the identity. This proves that dim(F−1([C])) = 5. In particular, we deduce that dim(Π|S (S 0)) = 3 3n− 7 − 3 + 5 dim(Π|S (S) ≥ 3 3n− 7 − 3 + 9 = 3 3(n+ 2)− 7 Remark 4.12. We expect that it is possible to prove that Σnk,d has expected number of moduli for every ρ also when k = 2 or k = 3. By corollary 2.7 and theorem 4.9, Σnk,d is not empty, irreducible and it has expected number of moduli for ρ ≤ 0 and ρ ≥ 2k. In order to extend theorem 4.11 to the case k = 2 and k = 3 one needs to consider a finite number of cases. Acknowledgment. The results of this paper are part of my PhD-thesis. I would like to express my gratitude to my advisor Prof. C. Ciliberto who initiated me into the subject of algebraic geometry and who provided me many invaluable suggestions. I have also enjoyed and benefited from conversation with many people including F. Flamini, E. Sernesi, L. Chiantini, L. Caporaso and G. Pareschi. Finally, I would like to thank the referee for useful remarks which allowed me to improve the finale version of this paper. 22 CONCETTINA GALATI References [1] E. Arbarello and M. Cornalba: Su una proprietà notevole dei morfismi di una curva a moduli generali in uno spazio proiettivo, Rend. Sem. Mat. Univ. Politec. Torino, vol. 38 (1980), no. 2, 87–99 (1981). [2] E. Arbarello and M. Cornalba: Su una congettura di Petri., Comment. Math. Helv., vol. 56 (1981), no. 1, 1–38. [3] E. Arbarello and M. Cornalba: A few remarks about the variety of irreducible plane curves of given degree and genus., Ann. Sci. École Norm. Sup. (4), vol. 16 (1983), 467–488 (1984). [4] E. Arbarello and M. Cornalba, P.A. Griffiths, J. Harris: Geometry of algebraic curves., vol. 1, Springer-Verlag. [5] A. Arsie and C. Galati: Geometric k-normality of curves and applications, Le Matematiche, Vol. LVIII (2003), Fasc. II, 179–199. [6] S. Diaz and J. Harris: Ideals associated to deformations of singular plane curves, Transactions of the American Mathematical Society, vol. 309, n. 2, 433–468 (1988). [7] C. Galati: Number of moduli of plane curves with nodes and cusps., PhD thesis, Università degli Studi di Tor Vergata, 2004-2005. [8] J. Harris: On the Severi problem, Invent. Math., vol. 84 (1986), no. 3, 445–461. [9] J. Harris and I. Morrison: Moduli of curves, Graduate texts in mathematics, vol. 187, Springer, New York, 1988. [10] D. Eisenbud and J. Harris: The Kodaira dimension of the moduli space of curves of genus ≥ 23. Invent. Math. vol. 90 (1987), no. 2, 359–387. [11] G.M Greuel and U. Karras: Families of varieties with prescribed singularities, Compositio Math. vol. 69 (1989), no. 1, 83–110. [12] G.M. Greuel, C. Lossen, and E. Shustin: Castelnuovo function, zero-dimensional schemes and singular plane curves. J. Algebraic Geom. vol. 9 (2000), no. 4, 663–710. [13] E. Horikawa: On the deformations of the holomorphic maps I, J. Math. Soc. Japan, vol. 25 (1973), 372–396. [14] E. Horikawa: On the deformations of the holomorphic maps II, J. Math. Soc. Japan, vol. 26 (1974), 647–667. [15] P. Kang: A note on the variety of plane curves with nodes and cusps, Proc. Amer. Math. Soc. vol. 106 (1989), no. 2, 309–312. [16] P. Kang: On the variety of plane curves of degree d with δ nodes and k cusps, Trans. Amer. Math. Soc. vol. 316 (1989), no. 1, 165–192. [17] D. Mumford: Lectures on curves on an algebraic surface. , Princeton University Press, 1966. [18] E. Sernesi: On the existence of certain families of curves, curves. Invent. Math. vol. 75 (1984), no. 1, 25–57. [19] F. Severi: Vorlesungen über algebraische Geometrie, Teuner, Leipzig, 1921. [20] E. Shustin: Smoothness and irreducibility of varieties of plane curves with nodes and cusps, Bull. Soc. Math. France vol. 122 (1994), no. 2, 235–253. [21] E. Shustin: Equiclassical deformations of plane algebraic curves, Singularities (Oberwolfach, 1996), 195–204, Progr. Math., vol. 162, Birkhuser, Basel, 1998. [22] A. Tannenbaum: On the classical characteristic linear series of plane curves with nodes and cuspidal points: two examples of Beniamino Segre, Compositio Mathematica vol. 51 (1984), 169–183. [23] J. Wahl: Deformations of plane curves with nodes and cusps, Amer. J. Math. vol. 96 (1974), 529–577. [24] O. Zariski: Dimension theoretic characterization of maximal irreducible sistems of plane nodal curves, Amer. J. Math. vol. 104 (1982), no. 1, 209–226. [25] O. Zariski: Algebraic surfaces, Classics in mathematics, Springer. Dipartimento di Matematica, Università degli Studi della Calabria, via P. Bucci, cubo 30B, Arcavacata di Rende (CS) E-mail address: [email protected] 1. Introduction 2. Preliminaries 2.1. On Severi-Enriques varieties 2.2. Known results on the number of moduli of nk,d 3. On the existence of certain families of plane curves with nodes and cusps in sufficiently general position 4. Families of plane curves with nodes and cusps with finite and expected number of moduli. Acknowledgment References

0704.0619

Search for Heavy Neutral MSSM Higgs Bosons with CMS: Reach and Higgs-Mass Precision

DCPT/07/12 IPPP/07/06 arXiv:0704.0619 Search for Heavy Neutral MSSM Higgs Bosons with CMS: Reach and Higgs-Mass Precision S. Gennai 1∗, S. Heinemeyer 2†, A. Kalinowski 3‡, R. Kinnunen 4§, S. Lehti 4¶, A. Nikitenko 5‖ and G. Weiglein 6∗∗ 1 Centro Studi Enrico Fermi, Rome and INFN Pisa, Italy 2 Instituto de Fisica de Cantabria (CSIC-UC), Santander, Spain 3 Institute of Experimental Physics, Warsaw, Poland 4 Helsinki Institute of Physics, Helsinki, Finland 5 Imperial College, London, UK; on leave from ITEP, Moscow, Russia 6 IPPP, University of Durham, Durham DH1 3LE, UK Abstract The search for MSSM Higgs bosons will be an important goal at the LHC. We analyze the search reach of the CMS experiment for the heavy neutral MSSM Higgs bosons with an integrated luminosity of 30 or 60 fb−1. This is done by combining the latest results for the CMS experimental sensitivities based on full simulation studies with state-of-the-art theoretical predictions of MSSM Higgs-boson properties. The results are interpreted in MSSM benchmark scenarios in terms of the parameters tan β and the Higgs-boson mass scale, MA. We study the dependence of the 5 σ discovery contours in the MA–tan β plane on variations of the other supersymmetric parameters. The largest effects arise from a change in the higgsino mass parameter µ, which enters both via higher-order radiative corrections and via the kinematics of Higgs decays into supersymmetric particles. While the variation of µ can shift the prospective discovery reach (and correspondingly the “LHC wedge” region) by about ∆ tan β = 10, we find that the discovery reach is rather stable with respect to the impact of other supersymmetric parameters. Within the discovery region we analyze the accuracy with which the masses of the heavy neutral Higgs bosons can be determined. We find that an accuracy of 1–4% should be achievable, which could make it possible in favourable regions of the MSSM parameter space to experimentally resolve the signals of the two heavy MSSM Higgs bosons at the LHC. ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] § email: [email protected] ¶ email: [email protected] ‖ email: [email protected] ∗∗ email: [email protected] http://arxiv.org/abs/0704.0619v1 http://arxiv.org/abs/0704.0619 1 Introduction Identifying the mechanism of electroweak symmetry breaking will be one of the main goals of the LHC. Many possibilities have been studied in the literature, of which the most popular ones are the Higgs mechanism within the Standard Model (SM) and within the Minimal Supersymmetric Standard Model (MSSM) [1]. Contrary to the case of the SM, in the MSSM two Higgs doublets are required. This results in five physical Higgs bosons instead of the single Higgs boson of the SM. These are the light and heavy CP-even Higgs bosons, h and H , the CP-odd Higgs boson, A, and the charged Higgs boson, H±.1 The Higgs sector of the MSSM can be specified at lowest order in terms of the gauge couplings, the ratio of the two Higgs vacuum expectation values, tan β ≡ v2/v1, and the mass of the CP-odd Higgs boson, MA. Consequently, the masses of the CP-even neutral Higgs bosons and the charged Higgs boson are dependent quantities that can be predicted in terms of the Higgs- sector parameters. Higgs-phenomenology in the MSSM is strongly affected by higher-order corrections, in particular from the sector of the third generation quarks and squarks, so that the dependencies on various other MSSM parameters can be important. After the termination of LEP in the year 2000 (the final LEP results can be found in Refs. [2, 3]), and the (ongoing) Higgs boson search at the Tevatron [4–6], the search will be continued at the LHC [7–9] (see also Refs. [10, 11] for recent reviews). The current exclusion bounds within the MSSM [3–5] and the prospective sensitivities at the LHC are usually dis- played in terms of the parameters MA and tan β that characterize the MSSM Higgs sector at lowest order. The other MSSM parameters are conventionally fixed according to certain benchmark scenarios [12–14]. The most prominent one is the “mmaxh scenario”, which in the search for the light CP-even Higgs boson allows to obtain conservative bounds on tan β for fixed values of the top-quark mass and the scale of the supersymmetric particles [15]. Besides the “no-mixing scenario”, which is similar to the mmaxh scenario, but assumes vanishing mix- ing in the stop sector, other CP-conserving scenarios that have been studied in LHC analyses (see e.g. Ref. [11]) are the “gluophobic Higgs scenario” and the “small αeff” scenario [13]. For the interpretation of the exclusion bounds and prospective discovery contours in the benchmark scenarios it is important to assess how sensitively the results depend on those parameters that have been fixed according to the benchmark prescriptions. While in the decoupling limit, which is the region of MSSM parameter space with MA ≫ MZ , the couplings of the light CP-even Higgs boson approach those of a SM Higgs boson with the same mass, the couplings of the heavy Higgs bosons of the MSSM can be sizably affected by higher-order contributions even for large values of MA. The kinematics of the heavy Higgs-boson production processes, on the other hand, is governed by the parameter MA, since in the region of large MA the heavy MSSM Higgs bosons are nearly mass-degenerate, MA ≈ MH ≈ MH±. In Ref. [14] it has been shown that higher-order contributions to the relation between the bottom-quark mass and the bottom-Yukawa coupling have a dramatic effect on the exclusion bounds in the MA–tanβ plane obtained from the bb̄φ, φ → bb̄ channel at the Tevatron. In this article we investigate how the 5 σ discovery regions in the MA–tanβ plane for the heavy neutral MSSM Higgs bosons (a corresponding analysis for the charged Higgs-boson 1We focus in this paper on the case without explicit CP-violation in the soft supersymmetry-breaking terms. search will be presented elsewhere) obtainable with the CMS experiment at the LHC depend on the other MSSM parameters. For the experimental sensitivities achievable with CMS we use up-to-date results based on full simulation studies for 30 or 60 fb−1(depending on the channel) [9]. This information is combined with precise theory predictions for the Higgs- boson masses and the involved production and decay processes incorporating higher-order corrections at the one-loop and two-loop level. In our analysis we investigate the impact on the discovery reach arising both from higher-order corrections and from possible decays of the heavy Higgs bosons into supersymmetric particles.2 The search for the heavy neutral MSSM Higgs bosons at the LHC will mainly be pursued in the b quark associated production with a subsequent decay to τ leptons [7–9]. In the region of large tanβ this production process benefits from an enhancement factor of tan2 β compared to the SM case. The main search channels are3 (here and in the following φ denotes the two heavy neutral MSSM Higgs bosons, φ = H,A): bb̄φ, φ → τ+τ− → 2 jets (1) bb̄φ, φ → τ+τ− → µ+ jet (2) bb̄φ, φ → τ+τ− → e+ jet (3) bb̄φ, φ → τ+τ− → e+ µ . (4) For our numerical analysis we use the program FeynHiggs [19–22]. We study in particular the dependence of the “LHC wedge” region, i.e. the region in which only the light CP-even MSSM Higgs boson can be detected at the LHC at the 5 σ level, on the variation of the higgsino mass parameter µ. The dependence on µ enters in two different ways, on the one hand via higher-order corrections affecting the relation between the bottom mass and the bottom Yukawa coupling, and on the other hand via the kinematics of Higgs decays into supersymmetric particles. We analyze both effects separately and discuss the possible impact of other supersymmetric parameters. Our results for the discovery reach of the heavy neutral MSSM Higgs bosons extend the known results in the literature in various ways. In comparison with Refs. [23, 24], where the prospective 5σ discovery contours for CMS in the MA–tanβ plane of the m h benchmark scenario were given for three different values of µ, the results in the present paper are based on full simulation studies and make use of the most up-to-date CMS tools for triggering and event reconstruction. Furthermore, in the analysis of Refs. [23, 24] relevant higher- order corrections, in particular those depending on ∆b (see Sect. 2.2 below), have been neglected. The effects induced by the ∆b corrections have been investigated in Ref. [14], where the results were obtained by a simple rescaling of the experimental results given in Refs. [7, 23–25]. Our present analysis, on the other hand, makes use of the latest CMS studies and provides a separate treatment of the different τ final states, channels (1)–(4). As a second step of our analysis we investigate the experimental precision that can be achieved for the determination of the heavy Higgs-boson masses in the discovery channels (1)– 2We restrict our analysis to the impact of supersymmetric contributions. For a discussion of uncertainties related to parton distribution functions, see e.g. Ref. [16]. 3In our analysis we do not consider diffractive Higgs production, pp → p ⊕ H ⊕ p [17]. For a detailed discussion of the search reach for the heavy neutral MSSM Higgs bosons in diffractive Higgs production we refer to Ref. [18]. (4). We discuss the prospective accuracy of the mass measurement in view of the possibility to experimentally resolve the signals of the heavy neutral MSSM Higgs bosons. The paper is organized as follows: Sect. 2 introduces our notation and gives a brief sum- mary of the most relevant supersymmetric radiative corrections to the Higgs-boson masses, production cross sections and decay widths at the LHC. The relevant benchmark scenarios are briefly reviewed. In Sect. 3 the experimental analysis is described. The results for the variation of the 5 σ discovery contours, obtainable at CMS with 30 or 60 fb−1 are given in Sect. 4, where we also discuss the achievable experimental precision in the Higgs mass determination. The conclusions can be found in Sect. 5. 2 Phenomenology of the MSSM Higgs sector 2.1 Notation The MSSM Higgs sector at lowest order is described in terms of two independent parameters (besides the SM gauge couplings): tan β ≡ v2/v1, the ratio of the two vacuum expectation values, and MA, the mass of the CP-odd Higgs boson A. Beyond the tree-level, large radiative corrections can occur from the t/t̃ sector, and for large values of tanβ also from the b/b̃ sector. Our notations for the scalar top and scalar bottom sector of the MSSM are as follows: the mass matrices in the basis of the current eigenstates t̃L, t̃R and b̃L, b̃R are given by +m2t + cos 2β ( s2w)M Z mtXt mtXt M +m2t + cos 2β s2wM , (5) +m2b + cos 2β (−12 + s2w)M Z mbXb mbXb M +m2b − 13 cos 2β s , (6) where mtXt = mt(At − µ cotβ ), mb Xb = mb (Ab − µ tanβ ). (7) Here MQ̃, Mt̃R and Mb̃R are the diagonal soft SUSY-breaking parameters, At denotes the trilinear Higgs–stop coupling, Ab denotes the Higgs–sbottom coupling, and µ is the higgsino mass parameter. For the numerical evaluation, it is often convenient to choose MQ̃ = Mt̃R = Mb̃R =: MSUSY. (8) Concerning analyses for the case where Mt̃R 6= MQ̃ 6= Mb̃R , see e.g. Refs. [20, 26, 27]. It has been shown that the upper bound on the mass of the light CP-even Higgs boson, Mh, obtained using eq. (8) is the same as for the more general case, provided that MSUSY is identified with the heaviest mass of MQ̃,Mt̃R ,Mb̃R [20]. Accordingly, the most important parameters entering the Higgs-sector predictions via higher-order corrections are mt, MSUSY, Xt, Xb and µ (see also the discussion in Sect. 2.2.2 below). The Higgs-sector observables furthermore depend on the SU(2) gaugino mass param- eter, M2, the U(1) parameter M1 and the gluino mass, mg̃ (the latter enters the predictions for the Higgs-boson masses only from two-loop order on). In numerical analyses the U(1) gaugino mass parameter, M1, is often fixed via the GUT relation M2. (9) We will briefly comment below on the possible impact of complex phases entering the Higgs- sector predictions via higher-order contributions. 2.2 Higher-order corrections in the Higgs sector In the following we briefly summarize the most important higher-order corrections affecting the observables in the MSSM Higgs-boson sector. As mentioned above, we focus on the MSSM with real parameters. For our numerical analysis we use the program FeynHiggs [19– 22]4, which incorporates a comprehensive set of higher-order results obtained in the Feynman- diagrammatic approach [20–22, 28–30]. 2.2.1 Higgs-boson propagator corrections Higher-order corrections to the Higgs-boson masses and the wave function normalization factors of processes with external Higgs bosons arise from Higgs-boson propagator-type con- tributions. These corrections furthermore contribute in a universal way to all Higgs-boson couplings. For the propagator-type corrections in the MSSM the complete one-loop re- sults [31–34], the bulk of the two-loop contributions [20, 27–29, 35–39] and even leading three-loop corrections [40] are known. The remaining theoretical uncertainty on the light CP-even Higgs-boson mass has been estimated to be below ∼ 3 GeV [21, 41]. The by far dominant contribution is the O(αt) term due to top and stop loops (αt ≡ h2t/(4π), where ht denotes the top-quark Yukawa coupling). Effects of O(αb) can be important for large values of tan β. 2.2.2 Corrections to the relation between the bottom-quark mass and the bot- tom Yukawa coupling Concerning the corrections from the bottom/sbottom sector, large higher-order effects can in particular occur in the relation between the bottom-quark mass and the bottom Yukawa coupling (which controls the interaction between the Higgs bosons and bottom quarks as well as between the Higgs and scalar bottoms), hb, for large values of tanβ. At lowest order the relation reads mb = hbv1. Beyond the tree level large radiative corrections proportional to hbv2 are induced, giving rise to tanβ-enhanced contributions [36–38,42]. At the one-loop level the leading terms proportional to v2 are generated either by gluino–sbottom one-loop diagrams of O(αs) or by chargino–stop loops of O(αt). The leading one-loop contribution ∆b in the limit of MSUSY ≫ mt and tanβ ≫ 1 takes the simple form [36] mg̃ µ tanβ × I(mb̃1 , mb̃2 , mg̃) + At µ tan β × I(mt̃1 , mt̃2 , µ) , (10) 4 The code can be obtained from www.feynhiggs.de . where the function I is given by I(a, b, c) = (a2 − b2)(b2 − c2)(a2 − c2) a2b2 log + b2c2 log + c2a2 log max(a2, b2, c2) The leading contribution can be resummed to all orders in the perturbative expansion [36– 38]. This leads in particular to the replacement 1 + ∆b , (12) where mb denotes the running bottom quark mass including SM QCD corrections. For the numerical evaluations in this paper we choose mb = mb(mt) ≈ 2.97 GeV. The ∆b corrections are numerically sizable for large tan β in combination with large values of the ratios of µmg̃/M SUSY or µAt/M SUSY. Negative values of ∆b lead to an enhancement of the bottom Yukawa coupling as a consequence of eq. (12) (for extreme values of µ and tanβ the bottom Yukawa coupling can even acquire non-perturbative values when ∆b → −1), while positive values of ∆b give rise to a suppression of the Yukawa coupling. Since a change in the sign of µ reverses the sign of ∆b, the bottom Yukawa coupling can exhibit a very pronounced dependence on the parameter µ. For large values of tanβ the correction to the production cross sections of the Higgs bosons H and A induced by ∆b enters approximately like tan 2 β/(1 + ∆b) 2, giving rise to potentially large numerical effects. In the case of the subsequent Higgs-boson decay φ → τ+τ−, however, the ∆b corrections in the production and the decay process cancel each other to a large extent. The residual ∆b dependence of σ(bb̄φ) × BR(φ → τ+τ−) is approximately given by tan2 β/((1+∆b) 2+9), which has a much weaker ∆b dependence (see Ref. [14] for a more detailed discussion). In the numerical analysis below the ∆b corrections, which have been discussed in this section in terms of simple approximation formulae, will be supplemented by other higher- order corrections as implemented in the program FeynHiggs (and possible decay modes into supersymmetric particles are taken into account). Higher-order corrections to Higgs decays into τ+τ− within the SM and MSSM have been evaluated in Refs. [34, 43]. 2.2.3 Corrections to the Higgs production cross sections For the prediction of Higgs-boson production processes at hadron colliders SM-type QCD corrections in general play an important role. The SM predictions for the process bb̄ → φ+X at the LHC are far advanced. In the five-flavor scheme the SM cross section is known at NNLO in QCD [44]. The cross section in the four-flavor scheme is known at NLO [45, 46]. Results obtained in the two schemes have been shown to be consistent [47–49] (see also Refs. [48, 50] and Refs. [45, 46] for results with one and two final-state b-quarks at high-pT , respectively). The predictions for the bb̄ → φ + X cross sections in the MSSM have been obtained with FeynHiggs [19–22]. The FeynHiggs implementation5 is based on the state-of-the-art 5The inclusion of the charged Higgs production cross sections is planned for the near future. SM prediction, namely the NNLO result in the five-flavor scheme [44] using MRST2002 parton distributions at NNLO [51], with the renormalization scale set equal to MHSM and the factorization scale set equal to MHSM/4. In order to obtain the MSSM prediction the SM cross section is rescaled with the ratio of the partial widths in the MSSM and the SM, Γ(φ → bb̄)MSSM Γ(φ → bb̄)SM . (13) The evaluation of the partial widths incorporates one-loop SM QCD and SUSY QCD correc- tions, as well as (in the SUSY case) the resummation of all terms of O((αs tanβ)n) [34,37,43] and the proper normalization of the external Higgs bosons as discussed in Refs. [22, 52]. Since the approximation of rescaling the SM cross section with the ratio of partial widths does not take into account the MSSM-specific dynamics of the production processes, the theoretical uncertainty in the predictions for the cross sections will in general be somewhat larger than for the decay widths. It should be noted that in comparison with other approaches for treat- ing the SM and SUSY contributions, for instance the program HQQ [53], sizable deviations can occur as a consequence of differences in the scale choices and the inclusion of higher-order corrections. 2.3 The mmaxh and no-mixing benchmark scenarios While the phenomenology of the production and decay processes of the heavy neutral MSSM Higgs bosons at the LHC is mainly characterised by the parametersMA and tanβ that govern the Higgs sector at lowest order, other MSSM parameters enter via higher-order contribu- tions, as discussed above, and via the kinematics of Higgs-boson decays into supersymmetric particles. The other MSSM parameters are usually fixed in terms of benchmark scenarios. The most commonly used scenarios are the “mmaxh ” and “no-mixing” benchmark scenar- ios [12–14]. According to the definition of Ref. [13] the mmaxh scenario is given by mmaxh : MSUSY = 1000 GeV, Xt = 2MSUSY, Ab = At, µ = 200 GeV, M2 = 200 GeV, mg̃ = 0.8MSUSY . (14) The no-mixing scenario differs from the mmaxh scenario only in that it has vanishing mixing in the stop sector and a larger value of MSUSY no-mixing: MSUSY = 2000 GeV, Xt = 0, Ab = At, µ = 200 GeV, M2 = 200 GeV, mg̃ = 0.8MSUSY . (15) The value of the top-quark mass in Ref. [13] was chosen according to the experimental central value at that time. For our numerical analysis below, we use the value, mt = 171.4 GeV [54] In Ref. [14] it was suggested that in the search for heavy MSSM Higgs bosons the mmaxh and no-mixing scenarios, which originally were mainly designed for the search for the light CP-even Higgs boson h, should be extended by several discrete values of µ, µ = ±200,±500,±1000 GeV . (16) 6 Most recently the central experimental value has shifted to mt = 170.9± 1.8 GeV [55]. This shift has a negligible impact on our analysis. As discussed above, the variation of µ in particular has an impact on the correction ∆b, modifying in this way the bottom Yukawa coupling. For very large values of tan β and large negative values of µ the bottom Yukawa coupling can be so much enhanced that a perturbative treatment is no longer possible. We have checked that in our analysis of the LHC discovery contours the bottom Yukawa coupling stays in the perturbative regime, so that all values of µ down to µ = −1000 GeV can safely be inserted. The variation of the parameter µ also modifies the mass spectrum and the couplings in the chargino and neutralino sector of the MSSM. Besides the small higher-order corrections induced by loop diagrams involving charginos and neutralinos, a change in the mass spectrum of the chargino and neutralino sector can have an important effect on Higgs phenomenology because decay modes of the heavy neutral MSSM Higgs bosons into charginos and neutralinos open up if the supersymmetric particles are sufficiently light (the mass spectrum in the mmaxh and no-mixing scenarios respects the limits from direct searches for charginos at LEP [56] for all values of µ specified in eq. (16)). Differences between the mmaxh and no-mixing scenarios in the searches for heavy neutral MSSM Higgs bosons are induced in particular by a difference in the ∆b correction. While in the mmaxh scenario both the O(αs) and O(αt) contributions to ∆b can be sizable, see eq. (10), in the no-mixing scenario the O(αt) contribution is very small because At is close to zero in this case. The larger value of MSUSY in the no-mixing scenario gives rise to an additional suppression of |∆b| compared to the mmaxh scenario. 3 Experimental analysis In this section we briefly review the recent CMS analysis of the φ → τ+τ− channel, see Ref. [9], yielding the number of events needed for a 5 σ discovery (depending on the mass of the Higgs boson). The analysis was performed with full CMS detector simulation and reconstruction for the following four final states of di-τ -lepton decays: τ+τ− → jets [57], τ+τ− → e+ jet [58], τ+τ− → µ+ jet [59] and τ+τ− → e + µ [60]. The Higgs-boson production in association with b quarks, pp → bb̄φ, has been selected using single b-jet tagging in the experimental analysis. The kinematics of the gg → bb̄φ production process (2 → 3) was generated with PYTHIA [61]. It has been shown that in this way the NLO kinematics is better reproduced than using the PYTHIA gb → bφ process (2 → 2) [62]. The backgrounds considered in the analysis were QCD muli-jet events (for the ττ → jets mode), tt̄, bb̄, Drell-Yan production of Z, γ∗, W+jet, Wt and ττbb̄. All background processes were generated using PYTHIA, except for τ+τ−bb̄, which was generated using CompHEP [63]. The results for the various channels, eqs. (1) – (4), are given in Tabs. 1 – 4. For every Higgs-boson mass point studied we show the number of signal events needed for 5 σ discovery, NS, the total experimental selection efficiency, εexp, and the ratio of the di-τ mass resolution to the Higgs-boson mass, RMφ . The last row in Tabs. 1 – 4 shows the expected precision of the Higgs-boson mass measurement, evaluated as explained below, for parameter points on the 5 σ discovery contour. Detector effects, experimental systematics and uncertainties of the background determination were taken into account in the evaluation of the NS. These effects reduce the discovery region in the MA–tanβ plane as shown in previous analyses [9] φ → τ+τ− → jets, 60 fb−1 MA [GeV] 200 500 800 NS 63 35 17 εexp 2.5× 10−4 2.4× 10−3 3.6× 10−3 RMφ 0.176 0.171 0.187 ∆Mφ/Mφ [%] 2.2 2.8 4.5 Table 1: Required number of signal events, NS, with L = 60 fb−1 for a 5 σ discovery in the channel φ → τ+τ− → jets. Furthermore given are the total experimental selection efficiency, εexp, the ratio of the di-τ mass resolution to the Higgs-boson mass, RMφ, and the expected precision of the Higgs-boson mass measurement, ∆Mφ/Mφ, obtainable from NS signal events. φ → τ+τ− → e+ jet, 30 fb−1 MA [GeV] 200 300 500 NS 72.9 45.5 32.8 εexp 3.0× 10−3 6.4× 10−3 1.0× 10−2 RMφ 0.216 0.214 0.230 ∆Mφ/Mφ [%] 2.5 3.2 4.0 Table 2: Required number of signal events, NS, with L = 30 fb−1 for a 5 σ discovery in the channel φ → τ+τ− → e+ jet. The other quantities are defined as in Tab. 1. φ → τ+τ− → µ+ jet, 30 fb−1 MA [GeV] 200 500 NS 79 57 εexp 7.0× 10−3 2.0× 10−2 RMφ 0.210 0.200 ∆Mφ/Mφ [%] 2.4 2.6 Table 3: Required number of signal events, NS, with L = 30 fb−1 for a 5 σ discovery in the channel φ → τ+τ− → µ+ jet. The other quantities are defined as in Tab. 1. φ → τ+τ− → e + µ, 30 fb−1 MA [GeV] 200 250 NS 87.8 136.7 εexp 6.4× 10−3 1.1× 10−2 RMφ 0.262 0.412 ∆Mφ/Mφ [%] 2.8 3.5 Table 4: Required number of signal events, NS, with L = 30 fb−1 for a 5 σ discovery in the channel φ → τ+τ− → e+ µ. The other quantities are defined as in Tab. 1. (see in particular Fig. 5.6 of Ref. [9] for the τ+τ− → µ+ jet mode). Now we turn to the evaluation of the expected precision of the Higgs-boson mass mea- surement. In spite of the escaping neutrinos, the Higgs-boson mass can be reconstructed in the H,A → ττ channel from the visible τ momenta (τ jets) and the missing transverse energy, EmissT , using the collinearity approximation for neutrinos from highly boosted τ ’s. In the investigated region of MA and tanβ the two states A and H are nearly mass-degenerate. For most values of the other MSSM parameters the mass difference of A and H is much smaller than the achievable mass resolution. In this case the difference in reconstructing the A or the H will have no relevant effect on the achievable accuracy in the mass determina- tion. In some regions of the MSSM parameter space, however, a sizable splitting between MA and MH can occur even for MA ≫ MZ . We will discuss below the prospects in scenarios where the splitting between MA and MH is relatively large. The precision ∆Mφ/Mφ shown in Tabs. 1 – 4 is derived for the border of the parameter space in which a 5 σ discovery can be claimed, i.e. with NS observed Higgs events. The statistical accuracy of the mass measurement has been evaluated via . (17) A higher precision can be achieved if more than NS events are observed. The corresponding estimate for the precision is obtained by replacing NS in eq. (17) by the number of observed signal events, Nev. It should be noted that the prospective accuracy obtained from eq. (17) does not take into account the uncertainties of the jet and missing ET energy scales. In the τ+τ− → jets mode these effects can lead to an additional 3% uncertainty in the mass measurement [57]. A more dedicated procedure of the mass measurement from the signal plus background data still has to be developed in the experimental analysis. However, we do not expect that the additional uncertainties will considerably degrade the accuracy of the Higgs boson mass measurement as calculated with eq. (17). 4 Results The results quoted in Sect. 3 for the required number of signal events depend only on the Higgs-boson mass, i.e. the event kinematics, but are independent of any specific MSSM scenario. In order to determine the 5 σ discovery contours in the MA–tan β plane these results have to be confronted with the MSSM predictions. The number of signal events, Nev, for a given parameter point is evaluated via Nev = L × σbb̄φ × BR(φ → τ+τ−)× BRττ × εexp . (18) Here L denotes the luminosity collected with the CMS detector, σbb̄φ is the Higgs-boson pro- duction cross section, BR(φ → τ+τ−) is the branching ratio of the Higgs boson to τ leptons, BRττ is the product of the branching ratios of the two τ leptons into their respective final state, BR(τ → jet +X) ≈ 0.65 , (19) BR(τ → µ+X) ≈ BR(τ → e+X) ≈ 0.175 , (20) and εexp denotes the total experimental selection efficiency for the respective process (as given in Tabs. 1 – 4). The Higgs-boson production cross sections and decay branching ratios have been evaluated with FeynHiggs as described in Sect. 2.2. 4.1 Discovery reach for heavy neutral MSSM Higgs bosons The number of signal events, Nev, in the MSSM depends besides the parameters MA and tan β, which govern the MSSM Higgs sector at lowest order, in principle also on all other MSSM parameters. In the following we analyze how stable the results for the 5σ discovery contours in theMA–tanβ plane are with respect to variations of the other MSSM parameters. We take into account both effects from higher-order corrections, as discussed in Sect. 2.2, and from decays of the heavy Higgs bosons into supersymmetric particles. As starting point of our analysis we use the mmaxh and no-mixing benchmark scenarios, where we investigate in detail the sensitivity of the discovery contours with respect to variations of the parameter µ. We then discuss the possible impact of varying other MSSM parameters. We have evaluated Nev in the two benchmark scenarios as a function of MA and tan β. For fixed MA we have varied tan β such that Nev = NS (as given in Tabs. 1 – 4). This tanβ value is then identified as the point on the 5 σ discovery contour corresponding to the chosen value of MA. In this way we have determined the 5 σ discovery contours for the m h and the no-mixing scenarios for µ = ±200,±1000 GeV. In Figs. 1 – 3 we show the 5σ discovery contours obtained from the process bb̄φ, φ → τ+τ− for the final states τ+τ− → jets, τ+τ− → e + jet and τ+τ− → µ + jet. As can be seen from Tab. 4, the fourth channel discussed above, τ+τ− → e + µ, contributes for 30 fb−1 only in the region of relatively small MA values and has a lower sensitivity than the other three channels. We therefore omit this channel in the following discussion. The discovery contours in Figs. 1 – 3 are given for the mmaxh and no-mixing benchmark scenarios with µ = ±200,±1000 GeV. As explained above, the 5 σ discovery contours are affected by a change in µ in two ways. Higher-order contributions, in particular the ones associated with ∆b, ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 60 fb j+j→ ττ → φ bb→pp scenariomaxhm 2 = 1 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom SUSY = 2 MtStop mix: X ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 60 fb j+j→ ττ → φ bb→pp no mixing scenario 2 = 2 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom = 0tStop mix: X Figure 1: Variation of the 5σ discovery contours obtained from the channel bb̄φ, φ → τ+τ− → jets in the mmaxh (left) and no-mixing (right) benchmark scenarios for different values of µ. ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 30 fb e+j→ ττ → φ bb→pp scenariomaxhm 2 = 1 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom SUSY = 2 MtStop mix: X ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 30 fb e+j→ ττ → φ bb→pp no mixing scenario 2 = 2 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom = 0tStop mix: X Figure 2: Variation of the 5σ discovery contours obtained from the channel bb̄φ, φ → τ+τ− → e+ jet in the mmaxh (left) and no-mixing (right) benchmark scenarios for different values of µ. ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 30 fb +jµ → ττ → φ bb→pp scenariomaxhm 2 = 1 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom SUSY = 2 MtStop mix: X ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 30 fb +jµ → ττ → φ bb→pp no mixing scenario 2 = 2 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom = 0tStop mix: X Figure 3: Variation of the 5σ discovery contours obtained from the channel bb̄φ, φ → τ+τ− → µ+ jet in the mmaxh (left) and no-mixing (right) benchmark scenarios for different values of µ. modify the Higgs-boson production cross sections and decay branching ratios. Furthermore the mass eigenvalues of the charginos and neutralinos vary with µ, possibly opening up the decay channels of the Higgs bosons to supersymmetric particles, which reduces the branching ratio to τ leptons. The results for the 5 σ discovery contours for the final state τ+τ− → jets are shown in Fig. 1 for themmaxh (left) and the no-mixing (right) scenario. As expected from the discussion of the ∆b corrections in Sect. 2.2, the variation of the 5 σ discovery contours with µ is more pronounced in the mmaxh scenario, where a shift up to ∆ tanβ = 12 can be observed for MA = 800 GeV. For lowMA values (corresponding also to lower tanβ values on the discovery contours) the variation stays below ∆ tanβ = 3. In the no-mixing scenario the variation does not exceed ∆ tan β = 5. The τ+τ− → jets channel has also been discussed in Ref. [14]. Our results, which are based on the latest CMS studies using full simulation [57], are qualitatively in good agreement with Ref. [14], in which the earlier CMS studies of Refs. [23, 24] had beed used. The 5 σ discovery regions are largest for µ = −1000 GeV and pushed to highest tanβ values for µ = +200 GeV. In the low MA region our discovery contours are very similar to those obtained in Ref. [14]. In the high MA region, MA ∼ 800 GeV, corresponding to larger values of tan β on the discovery contours, our improved evaluation of the 5 σ discovery contours gives rise to a shift towards higher tan β values compared to Ref. [14] of about ∆ tanβ = 8 (mostly due to the up-to-date experimental input). Accordingly, we find a smaller discovery region compared to Ref. [14] and therefore an enlarged “LHC wedge” region where only the light CP-even MSSM Higgs boson can be detected at the 5 σ level. The results for the channel τ+τ− → e+ jet are shown in Fig. 2. Again the mmaxh scenario shows a stronger variation than the no-mixing scenario. The resulting shift in tan β reaches up to ∆ tan β = 8 for MA = 500 GeV in the m h scenario, but stays below ∆ tanβ = 4 for the no-mixing scenario. Finally in Fig. 3 the results for the channel τ+τ− → µ+ jet are depicted. The level of variation of the 5 σ discovery contours is the same as for the e + jet final state.7 ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 60 fb j+j→ ττ → φ bb→pp )=0χ χ → φ, BR(maxhm 2 = 1 TeV/cSUSYM 2 = 200 GeV/c2M SUSY = 0.8 Mgluinom SUSY = 2 MtStop mix: X ,GeV/cAM 100 200 300 400 500 600 700 800 2 = -1000 GeV/cµ 2 = -200 GeV/cµ 2 = 200 GeV/cµ 2 = 1000 GeV/cµ CMS, 60 fb j+j→ ττ → φ bb→pp )=0χ χ → φno mixing, BR( 2 = 2 TeV/c 2 = 200 GeV/c2M = 0.8 M gluino = 0tStop mix: X Figure 4: Variation of the 5σ discovery contours obtained from the channel bb̄φ, φ → τ+τ− → jets in the mmaxh (left) and no-mixing (right) benchmark scenarios for different values of µ in the case where no decays of the heavy Higgs bosons into supersymmetric particles are taken into account (see text). In order to gain a better understanding of how sensitively the discovery contours in the MA–tan β plane depend on the chosen SUSY scenario, it is useful to separately investigate the different effects caused by varying the parameter µ. For simplicity, we restrict the following discussion to the bb̄φ, φ → τ+τ− → jets channel. In Fig. 4 we show the same results as in Fig. 1, but for the case where no decays of the heavy Higgs bosons into supersymmetric particles are taken into account. As a consequence, the variation of the 5 σ discovery contours with µ shown in Fig. 4 is purely an effect of higher-order corrections, predominantly those entering via ∆b. The difference between Fig. 1 and Fig. 4, on the other hand, is purely an effect of the change in BR(φ → τ+τ−) caused by the variation of the partial Higgs-boson decay widths into supersymmetric particles arising from a shift in the masses of the charginos and neutralinos. In Fig. 4 the dependence of the 5 σ discovery contours on µ significantly differs from the case of Fig. 1. While in Fig. 1 the inclusion of decays into supersymmetric particles gives 7Since the results of the experimental simulation for this channel are available only for two MA values, the interpolation is a straight line. This may result in a slightly larger uncertainty of the results shown in Fig. 3 compared to the other figures. rise to the fact that the smallest discovery region is found for small µ values, µ = +200 GeV (with the exception of the region of very small MA), in Fig. 4 the 5 σ discovery contours are ordered monotonously in µ: the largest (smallest) 5 σ discovery regions are obtained for µ = −(+)1000 GeV, i.e. for the largest (smallest) values of the bottom Yukawa coupling. As expected, the effect of the higher-order corrections is largest in the high tanβ-region (corresponding to large values of MA on the discovery contours). In this region the variation of µ shifts the discovery contours by up to ∆ tanβ = 11 for the case of the mmaxh scenario (left plot of Fig. 4), i.e. the effect is about the same as for the case where decays into supersymmetric particles are included. For lower values of tanβ (corresponding to smaller values of MA on the discovery contours), on the other hand, the modification of the Higgs branching ratio as a consequence of decays into supersymmetric particles yields the dominant effect on the 5 σ discovery contours. Accordingly, the observed variation with µ in this region is significantly smaller in Fig. 4 as compared to the full result of Fig. 1. The reduced sensitivity of the discovery contours on µ can also clearly be seen for the case of the no- mixing scenario (right plot), where as discussed above the ∆b correction is smaller than in the mmaxh scenario. ,GeV/cAM 100 200 300 400 500 600 700 800 2 = 200 GeV/c gluino 2 = 500 GeV/c gluino 2 = 1000 GeV/c gluino 2 = 2000 GeV/c gluino CMS, 60 fb j+j→ ττ → φ bb→pp scenariomaxhm 2 = 1 TeV/cSUSYM 2 = 200 GeV/c2M 2 = 1000 GeV/cµ SUSY = 2 MtStop mix: X ,GeV/cAM 100 200 300 400 500 600 700 800 2 = 200 GeV/c gluino 2 = 500 GeV/c gluino 2 = 1000 GeV/c gluino 2 = 2000 GeV/c gluino CMS, 60 fb j+j→ ττ → φ bb→pp no mixing scenario 2 = 2 TeV/cSUSYM 2 = 200 GeV/c2M 2 = 1000 GeV/cµ = 0tStop mix: X Figure 5: Variation of the 5σ discovery contours obtained from the channel bb̄φ, φ → τ+τ− → jets in the mmaxh (left) and no-mixing (right) benchmark scenarios with µ = +1000 GeV for different values of mg̃. A parameter affecting the ∆b corrections, see eq. (10), but not the kinematics of the Higgs-boson decays is the gluino mass, mg̃. We now investigate the impact of varying this parameter, which is normally fixed to the values mg̃ = 800, 1600 GeV in the m h and no-mixing benchmark scenarios, respectively. The results for four different values of the gluino mass, mg̃ = 200, 500, 1000, 2000 GeV, are shown in Fig. 5. The µ parameter has been set to µ = +1000 GeV in Fig. 5, such that the Higgs decay channels into charginos and neutralinos are suppressed. As one can see from eq. (10), the change of mg̃ affects the O(αs) part of ∆b and corresponds to a monotonous increase of ∆b. As an example, this yields for µ = 1000 GeV, tan β = 50 in the two scenarios: mmaxh , mg̃ = 200 GeV : ∆b = 0.50 mmaxh , mg̃ = 2000 GeV : ∆b = 0.94 no-mixing, mg̃ = 200 GeV : ∆b = 0.06 no-mixing, mg̃ = 2000 GeV : ∆b = 0.29 . (21) In the no-mixing scenario the At value is close to zero, suppressing the mg̃-independent contribution to ∆b, while the higher SUSY mass scale results in an overall reduction of ∆b in this scenario. The value of ∆b in the no-mixing scenario would slightly increase if mg̃ were raised to even larger values, but this effect would not change the qualitative behaviour. Fig. 5 shows that the results for the discovery reach in the MA–tanβ plane are relatively stable with respect to variations of the gluino mass. The shift in the discovery contours remains below about ∆ tanβ = 4 for the mmaxh scenario (left plot) and ∆ tanβ = 1 for the no-mixing scenario (right plot). For the positive sign of µ chosen in Fig. 5, where the ∆b correction yields a suppression of the bottom Yukawa coupling, the largest discovery reach is obtained for small mg̃, while the smallest discovery reach is obtained for large mg̃. This behaviour would be reversed by a change of sign of µ. We have also investigated the possible impact of other MSSM parameters (besides µ and mg̃) on the 5 σ discovery contours in the MA–tan β plane. The ∆b corrections depend also on the parameters in the stop and sbottom sector, see eq. (10). While the formulas in Sect. 2.2.2 have been given for the region where MSUSY ≫ mt, the qualitative effect of reducing the stop and sbottom masses can nevertheless be inferred. Sizable ∆b corrections require relative large values of µ and mg̃. If these parameters are kept large while the stop and sbottom masses are reduced, the ∆b corrections tend to decrease. It is obvious from eq. (10) that reducing the absolute value of At decreases the electroweak part of the ∆b correction. As discussed above, this effect of the ∆b corrections manifests itself in the comparison of the mmaxh and no-mixing scenarios, see Figs. 1–5. Concerning the possible impact of the ∆b corrections on the 5 σ discovery contours for the bb̄φ, φ → τ+τ− channel in the MA–tanβ plane we conclude that larger effects than those shown in Figs. 1–5 (where we have displayed the discovery contours up to tan β = 50) would only arise if the variation of µ were extended over an even wider interval than −1000 GeV ≤ µ ≤ +1000 GeV as done in our analysis above. We now turn to the possible effects of other higher-order corrections beyond those entering via ∆b on the 5 σ discovery contours for the bb̄φ, φ → τ+τ− channel. These effects are in general non-negligible, see the discussions in Sect. 2.2 and in Sect. 4.2 below, but smaller than those induced by ∆b. As a consequence, the impact on the 5 σ discovery contours in the MA–tan β plane of other supersymmetric parameters entering via higher-order corrections is in general much smaller than the effect of varying µ in the high-tanβ region of Fig. 4. As an example, the difference observed in Figs. 1–5 between the mmaxh and no-mixing scenarios arising from the different values of At and MSUSY in the two scenarios (see eqs. (14), (15)) is mainly an effect of the ∆b corrections, while the impact of other higher-order corrections involving At and MSUSY is found to be small. Also the decays of the heavy neutral MSSM Higgs bosons into supersymmetric particles are in general affected by other supersymmetric parameters in addition to the dependence on µ, MA and tan β. The resulting effects on BR(φ → τ+τ−) turn out to be rather small, however. We find that sizable deviations from the values of BR(φ → τ+τ−) occurring in the mmaxh and no-mixing scenarios for −1000 GeV ≤ µ ≤ +1000 GeV are only possible in quite extreme regions of the MSSM parameter space that are already highly constrained by existing experimental data. Our discussion above has been given in the context of the MSSM with real parameters. Since the sensitivity of the 5 σ discovery contours in the MA–tan β plane on the other super- symmetric parameters can mainly be understood as an effect of higher-order corrections to the bottom Yukawa coupling and of the kinematics of Higgs-boson decays into supersymmet- ric particles, no qualitative changes of our results are expected for the case where complex phases are taken into account. 4.2 Higgs-boson mass precision The discussion in the previous section shows that the prospective discovery reach of the bb̄φ, φ → τ+τ− channel in theMA–tanβ plane is rather stable with respect to variations of the other MSSM parameters. We now turn to the second part of our analysis and investigate the expected statistical precision of the Higgs-boson mass measurement. The expected statistical precision is evaluated as described in Sect. 3, see eq. (17). In Figs. 6 – 7 we show the expected precision for the mass measurement achievable from the channel bb̄φ, φ → τ+τ− using the final states τ+τ− → jets and τ+τ− → e + jet. Within the 5 σ discovery region we have indicated contour lines corresponding to different values of the expected precision, ∆M/M . The results are shown in the mmaxh benchmark scenario for µ = −200 GeV (left plots) and µ = +200 GeV (right plots). We find that experimental precisions of ∆Mφ/Mφ of 1–4% are reachable within the dis- covery region. A better precision is reached for larger tanβ and smaller MA as a consequence of the higher number of signal events in this region. The other scenarios and other values of µ discussed above yield qualitatively similar results to those shown in Figs. 6, 7. As discussed above, for large values of MA the heavy neutral MSSM Higgs bosons are nearly mass-degenerate, MH ≈ MA. The experimental separation of the two states H and A (or the corresponding mass eigenstates in the CP-violating case) will therefore be challenging. The results shown in Figs. 6 – 7 have been obtained using the combined sample of H and A events. It is important to note, however, that even in the region of large MA the mass splitting between MH and MH can reach the level of a few %. An example of such a scenario is (as above, we consider the CP-conserving case, i.e. the MSSM with real parameters; the corresponding scenario in the case of non-vanishing complex phases has been discussed in Ref. [22]) MSUSY = 500 GeV, At = Ab = 1000 GeV, µ = 1000 GeV, M2 = 500 GeV, M1 = 250 GeV, mg̃ = 500 GeV . (22) In Fig. 8 the mass splitting |MH −MA| min(MH ,MA) is given as a function of Xt for tanβ = 40 and two MA values, MA = 300 GeV (solid line) and MA = 500 GeV (dashed line). The dot-dashed and dotted parts of the contours for Figure 6: The statistical precision of the Higgs-boson mass measurement achievable from the channel bb̄φ, φ → τ+τ− → jets in the mmaxh benchmark scenario for µ = −200 GeV (left) and µ = +200 GeV (right) is shown together with the 5 σ discovery contour. Figure 7: The statistical precision of the Higgs-boson mass measurement achievable from the channel bb̄φ, φ → τ+τ− → e + jet in the mmaxh benchmark scenario for µ = −200 GeV (left) and µ = +200 GeV (right) is shown together with the 5 σ discovery contour. -1500 -1000 -500 0 500 1000 1500 [GeV] = 500 GeV, tanβ = 40 = 300 GeV = 300 GeV, LEP excl. = 500 GeV = 500 GeV, LEP excl. Figure 8: The mass splitting between the heavy neutral MSSM Higgs bosons, ∆MHA/M ≡ |MH −MA| /min(MH ,MA), is shown as a function ofXt forMA = 300, 500 GeV in a scenario with MSUSY = 500 GeV, µ = 1000 GeV and tanβ = 40. The other parameters are given in eq. (22). The dot-dashed (dotted) parts of the contours forMA = 300 GeV (MA = 500 GeV) indicate parameter combinations that are excluded by the search for the light CP-even Higgs boson of the MSSM at LEP [3]. MA = 300, 500 GeV, respectively, in the region of small |Xt| indicate parameter combinations that result in relatively low Mh values that are excluded by the search for the light CP-even Higgs boson of the MSSM at LEP [3]. One can see in Fig. 8 that the mass splitting between MH and MA shows a pronounced dependence on Xt in this scenario. Mass differences of up to 5% are possible for large Xt (while the widths of the Higgs bosons are at the 1–1.5% level in this parameter region). The example of Fig. 8 shows that a precise mass measurement at the LHC may in favourable regions of the MSSM parameter space open the exciting possibility to distin- guish between the signals of H and A production. In confronting Fig. 8 with the expected accuracies obtained in Figs. 6 – 7 one of course needs to take into account that a separate treatment of the H and A channels in Figs. 6 – 7 would reduce the number of signal events by a factor of 2, resulting in a degradation of the expected accuracies (for the same luminosity) by a factor of 2. A more detailed analysis of the potential for experimentally resolving two mass peaks would furthermore have to include effects arising from overlapping Higgs signals. Such an analysis goes beyond the scope of the present paper. 5 Conclusions We have analyzed the reach of the CMS experiment with 30 or 60 fb−1 for the heavy neutral MSSM Higgs bosons, depending on tanβ and the Higgs-boson mass scale, MA. We have focused on the channel bb̄H/A,H/A → τ+τ− with the τ ’s subsequently decaying to jets and/or leptons. The experimental analysis, yielding the number of events needed for a 5 σ discovery (depending on the mass of the Higgs boson) was performed with full CMS detector simulation and reconstruction for the final states of di-τ -lepton decays. The events were generated with PYTHIA. The experimental analysis has been combined with predictions for the Higgs-boson masses, production processes and decay channels obtained with the code FeynHiggs, taking into ac- count all relevant higher-order corrections as well as possible decays of the heavy Higgs bosons into supersymmetric particles. We have analyzed the sensitivity of the 5 σ discov- ery contours in the MA–tanβ plane to variations of the other supersymmetric parameters. We have shown that the discovery contours are relatively stable with respect to the im- pact of additional parameters. The biggest effects, resulting from higher-order corrections to the bottom Yukawa coupling and from the kinematics of Higgs decays into charginos and neutralinos, are caused by varying the absolute value and the sign of the higgsino mass parameter µ. The corresponding shift in the 5 σ discovery contours amounts up to about ∆ tanβ = 10. The effects of other contributions to the relation between the bottom-quark mass and the bottom Yukawa coupling, arising from the gluino mass and the parameters in the stop and sbottom sector, are in general smaller than the shifts induced by a variation of µ. The same holds for the impact of higher-order contributions beyond the corrections to the bottom Yukawa coupling and for the possible effects of other decay modes of the heavy Higgs bosons into supersymmetric particles. The results of our analysis, which was carried out in the framework of the CP-conserving MSSM, should not be substantially affected by the inclusion of complex phases of the soft-breaking parameters. We have analyzed the prospective accuracy of the mass measurement of the heavy neu- tral MSSM Higgs bosons in the channel bb̄H/A,H/A → τ+τ−. We find that statistical experimental precisions of 1–4% are reachable within the discovery region. These results, obtained from a simple estimate of the prospective accuracies, are not expected to consid- erably degrade if further uncertainties related to background effects and jet and missing ET scales are taken into account. We have pointed out that a %-level precision of the mass measurements could in favourable regions of the MSSM parameter allow to experimentally resolve the signals of the two heavy MSSM Higgs bosons. Acknowledgements S.H. and G.W. thank M. Carena and C.E.M. 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Buttar et al., Les Houches Physics at TeV Colliders 2005, “Standard Model and Higgs working group: Summary report”, hep-ph/0604120. [63] E. Boos et al. [CompHEP Collaboration], Nucl. Instrum. Meth. A 534 (2004) 250, hep-ph/0403113. http://arxiv.org/abs/hep-ph/0010017 http://arxiv.org/abs/hep-ph/0604120 http://arxiv.org/abs/hep-ph/0403113 Introduction Phenomenology of the MSSM Higgs sector Notation Higher-order corrections in the Higgs sector Higgs-boson propagator corrections Corrections to the relation between the bottom-quark mass and the bottom Yukawa coupling Corrections to the Higgs production cross sections The mhmax and no-mixing benchmark scenarios Experimental analysis Results Discovery reach for heavy neutral MSSM Higgs bosons Higgs-boson mass precision Conclusions

0704.0620

White dwarf masses derived from planetary nebulae modelling

Astronomy & Astrophysics manuscript no. gesicki c© ESO 2021 December 4, 2021 Letter to the Editor White dwarf masses derived from planetary nebulae modelling K. Gesicki1 and A.A. Zijlstra2 1 Centrum Astronomii UMK, ul. Gagarina 11, PL-87-100 Torun, Poland e-mail: [email protected] 2 School of Physics and Astronomy, University of Manchester, P.O. Box 88, Manchester M60 1QD, UK e-mail: [email protected] Received ; accepted ABSTRACT Aims. We compare the mass distribution of central stars of planetary nebulae (CSPN) with those of their progeny, white dwarfs (WD). Methods. We use a dynamical method to measure masses with an uncertainty of 0.02 M�. Results. The CSPN mass distribution is sharply peaked at 0.61 M�. The WD distribution peaks at lower masses (0.58 M�) and shows a much broader range of masses. Some of the difference can be explained if the early post-AGB evolution is faster than predicted by the Blöcker tracks. Between 30 and 50 per cent of WD may avoid the PN phase because of too low mass. However, the discrepancy cannot be fully resolved and WD mass distributions may have been broadened by observational or model uncertainties. Key words. Planetary nebulae: general – Stars: evolution – Stars: white dwarfs 1. Introduction White Dwarf (WD) mass distributions have been determined us- ing a variety of different methods. Discrepancies exist between the different determinations in particular between the photo- metric and spectroscopic WD masses. Boudreault & Bergeron (2005) compared the masses derived by fitting the observed Balmer lines with masses derived from trigonometric parallaxes and photometry. They found differences of ∼ 50 per cent for cool (6 500–14 000 K) DA white dwarfs. Spectroscopic masses are believed to be more accurate, especially for WDs in the temper- ature range between 15 000 and 40 000 K (Liebert et al. 2005). Atmospheric models are less well established for stars outside this range. For hotter WDs the atmospheric structure is modi- fied by an (often unknown) amount of metals and by non-LTE effects. For cooler WDs the convection has to be considered and the models are sensitive to the mixing length and the amount of helium convected to the surface (Boudreault & Bergeron 2005). Central stars of planetary nebulae (CSPN) provide a way to test the mass distributions. CSPNe evolve directly into WDs, with only very minor mass changes, allowing one to measure masses of currently forming white dwarfs. However, CSPN mass distributions have also been uncertain. For example, Napiwotzki (2006) shows that the very high CSPN masses (close to the Chandrasekhar limit) derived spectroscopically with state-of- the-art model atmospheres by Pauldrach et al. (2004) are physi- cally implausible and masses close to the peak of the CSPN/WD mass distribution are more likely. CSPN masses are normally obtained from the luminosi- ties. But more accurate masses can be derived using the age– temperature diagram, obtainable from the surrounding planetary nebula (PN). Gesicki et al. (2006) applied this to a sample of 101 PNe. In this Letter we discuss the resulting mass distribu- tions for hydrogen-rich and hydrogen-poor CSPNe and compare with published WD masses. 2. Methods and results 2.1. Models The method requires the age of the nebula and the temperature of the central star to be determined. Together these provide the heating time scale for the star. We derive the age of the PNe using a combination of line ratios, diameters (taken from the literature), and new high res- olution spectra (Gesicki et al. 2006). The diameters and line ratios are used to fit a spherically symmetric photo-ionization model. The model assumes a density distribution and finds a stellar black-body temperature. For each ion, the model finds a radial emissivity distribution. The observed line profiles for each ion represent the convolution of the thermal broadening and the expansion velocity at each radius. Thus, the line profiles for dif- ferent ions are used to fit a velocity field. An iterative procedure is used to improve the ionization model. The emissivity distribu- tions of different ions overlap, and this gives a strong constraint on the shape of the wings of the line profiles. A genetic algo- rithm, PIKAIA, is used to arrive at the optimum solution for ionization model and velocity field. A turbulent component is added if needed: turbulence is indicated by a Gaussian shape of the line profiles. The expansion velocities are found to increase with radius, due to the overpressure of the ionized region. ¿From the velocity field v(r), we derive the mass-weighted average over the nebula, vav. This parameter has been shown to be robust against the simplifications. Different models which provide comparable quality fits give the same vav to within 2 km s−1 (Gesicki et al 2006). Applying this to a radius of 0.8 times the outer radius (equivalent to the mass-averaged radius) allows us to define a kinematic age t to the nebula. A linear ac- celeration is assumed to have occurred from the AGB expansion velocity (10–15 km s−1) to the PN velocity vav (20–25 km s−1). The derived nebular age and stellar temperature are com- pared to the the H-burning tracks of Blöcker (1995), which pro- vide the largest and most uniform collection available. We inter- 2 K. Gesicki and A.A. Zijlstra: White dwarf masses from planetary nebulae 0.565 0.605 0.625 0.696 0.836 2 4 8 Fig. 1. Comparison of the 101 modelled PNe with the evolu- tionary tracks in the HR diagram. The model black-body tem- peratures are plotted against the luminosities interpolated from tracks. Filled circles indicate [WR] stars, open circles are wels and pluses indicate non-emission-line stars. The dotted lines show H-burning evolutionary models of Blöcker (1995), labeled by mass in units of M�. The solid lines are isochrones, labeled by the time after the nebula ejection, in units of 103 yr. polate between different tracks to find for each (t,Teff), the CSPN luminosity and mass. 2.2. Different CSPN types The CSPNe fall into two broad categories: the hydrogen-rich O-type stars and the emission-line central stars which are gen- erally hydrogen-deficient. The second group consists of [WR]- type stars with strong emission lines and wels (weak emission line stars). The [WR] are subdivided into hot [WO] and cool [WC]. [WR] stars are in most cases hydrogen-free (three possi- ble exceptions are mentioned by Werner & Herwig 2006). The wels may contain some hydrogen. Gesicki et al. (2006) show that one group of wels is located in the temperature gap between [WC] and [WO] stars. The other wels stars form a non-uniform group, including higher-mass objects where the high luminosity drives a wind but the star is not necessarily hydrogen-poor. The hydrogen-rich stars are believed to be related to the DA white dwarfs, while the [WR] may evolved into DB’s. 2.3. The HR diagram The full analyzed sample contains 101 PNe, of which about 60 are in the direction of the Galactic Bulge and the remainder are in the Galactic disk. Foreground confusion among the Bulge PNe is estimated at 20%. The sample contains 23 [WR]-type, 21 wels and 57 non-emission-line central stars1. The CSPN classification was adopted from literature. The last group contains also objects without any information about their spectrum. In Fig.1 we show the photoionization temperatures and inter- polated luminosities, plotted on the HR diagram. The H-burning tracks of Blöcker (1995) are also shown: the luminosities and masses of CSPNe fall into a rather restricted range of values. Isochrones of 1,2,4, and 8 × 103 yr are also shown. A previous HR diagram of CSPNe presented by Stanghellini et al. (2002) shows a much broader range of luminosities and, in consequence, masses. They use Zanstra temperatures and luminosities. The Zanstra method of locating a CSPN in the HR diagram was criticized by Schönberner & Tylenda (1990). 1 The data file is available from web page www.astri.uni.torun.pl/∼gesicki/modelled pne.dat Table 1. Comparison between our dynamical masses and spec- troscopic masses from Kudritzki et al. (2006). Observed mass- loss rates from the same paper are also listed and compared to values from the model tracks of Blöcker (1995). He 2-108 is classified as wels, the other three are non-emission-line stars. Object M [M�] Teff [103 K] log Ṁ [M� yr−1] dyn. spec. dyn. spec. spec. evol. tracks Tc 1 0.59 0.81 32 34 −7.46 −7.91 He 2-108 0.57 0.63 32 34 −6.85 −8.16 IC 418 0.61 0.92 37 36 −7.43 −7.82 NGC 3242 0.61 0.63 79 75 −8.08 −7.86 Observationally, the accuracy of the luminosity determinations is about a factor of 2. On the Schönberner tracks, a CSPN mass change from, e.g., 0.57 to 0.7 M� corresponds to a factor of 3 in luminosity. The masses determined directly from luminosity are thus accurate to only 0.1 M�. This is less than the typical dispersion of masses. In contrast, for the same mass range, the dynamical time scales differ by a factor of 60. Even for a factor of 2 uncertainty in the nebular age, the mass changes by only 0.02 M�. Therefore, the dynamical method improves the accu- racy. Schönberner & Tylenda (1990) also developed a method to improve the CSPN mass determination. This method (Tylenda et al. 1991) results in masses similar to ours. Table 1 compares, for four objects in common, our dynami- cal masses with the spectroscopic masses derived by Kudritzki et al. (2006). The spectroscopic masses are larger, in two cases very much larger. The lower masses are supported by the kinematical properties of Tc 1 and He 2-108 (see Fig. 5 of Napiwotzki 2006), which favour an old thin disk population. Kudritzki et al. also derive Teff : our photo-ionization values are in good agreement. Pauldrach et al. (2004) find from a spectroscopic analysis, five CSPNe with masses close to the Chandrasekhar limit. This result is implausible, as argued by Napiwotzki (2006). Three of their objects are also in our sample, and all are found to have regular masses. 2.4. The mass distributions In Fig.2 the upper panel presents the mass distribution of our whole sample of 101 PNe. All CSPNe masses fall into a narrow range, 0.55 − 0.66 M�, with a mean mass of 0.61 M�. The range of masses is almost identical to that of Tylenda et al. (1991) but they obtained a smaller mean mass of 0.593 M� and their distri- bution peaks at 0.58 M�. The lower panel of Fig.2 presents masses for the same types of CSPNe as shown in Fig. 1. The non-emission-line stars show a Gaussian mass distribution. The hydrogen-deficient emission- line stars seem to consist of two populations: one sharply peaked, containing [WR] stars, and the other showing a wider spread, composed of [WR] and wels. The sharp peak consists, with a single exception, of hot [WO] stars only. The presented histograms seem to suggest that hot [WO] stars form a different group from the combined cooler [WC] and wels CSPNe. K. Gesicki and A.A. Zijlstra: White dwarf masses from planetary nebulae 3 Fig. 2. The CSPN mass histograms. Upper panel: the histogram of all modelled PNe. Lower panel: the histogram of different subgroups of the 101 PNe. The dashed line indicates [WR] stars, the dotted line wels and the solid line non-emission-line stars. 3. Comparing CSPNe and WDs 3.1. The histograms The comparable birth rates of PNe and WDs suggests that most white dwarfs go through the PN phase (e.g. Liebert et al. 2005). The mass distribution in both samples should therefore be simi- Fig.3 presents the histograms of our interpolated O-type CSPN masses and the masses of DA white dwarfs from recent surveys. The WD data of Madej et al. (2004), kindly provided by the authors, contain 1175 new DA WDs extracted from the Sloan Digital Sky Survey. The data of Liebert et al. (2005) taken from the electronic version of their article, contain 347 DA WDs from the Palomar Green Survey. For Fig.3 we selected the ob- jects with temperatures between 15 000 K and 40 000 K. The two WDs histograms are not identical, but both peak at similar val- ues and show extended low- and high-mass tails. We plot the histograms using narrower bins than usually done for WDs, op- timized to the mass resolution of our CSPN data. The difference between the WD and CSPN distributions is striking. First, the obtained CSPN masses are restricted to a much nar- rower range of values than WDs, and are also much more sharply peaked. At face value, this implies that only some of the WDs have gone through the PN phase, in contrast to the conclusion from their similar birth rates (Liebert et al. 2005). Second, the two distributions peak at different masses. Here a systematic er- ror cannot be excluded, as discussed below. 3.2. Hydrogen-rich vs. hydrogen-deficient Hansen & Liebert (2003) point to a variety of WD mass distri- butions with clear differences between hydrogen- and helium- rich cool stars. Beauchamp et al. (1996) found for hot helium- atmosphere DB stars a sharp peak lacking almost entirely of low- and high-mass components. They also found that the DBA stars, which exhibit traces of atmospheric hydrogen, show a distinctly different, broad and flat distribution. The CSPN show an apparent difference between hydrogen- rich and hydrogen-deficient mass distributions. The hydrogen- deficient stars show a very narrow mass distribution; it is tempt- ing to relate this to the helium-rich DB and DBA populations. We use hydrogen-burning tracks to derive these masses. The Fig. 3. The mass distribution of non-emission-line O-type CSPNe (shaded area) is compared to two DA white dwarf distri- butions of intermediate temperatures: thin line: data from Liebert et al. (2005); dotted line: data from Madej et al. (2004) which are more numerous, and are rescaled. evolution after the thermal pulse leading to helium burners is very complicated and not well understood (Werner & Herwig 2006). This may not affect the derived masses too much: the ef- fect of a thermal pulse is to change the temperature of the star, but as shown in Fig. 1, the isochrones have only a weak depen- dence on temperature. The resulting offset in time (still very un- certain) when accounted for can shift those CSPN masses to- wards higher values. 4. Discussion 4.1. Uncertainties in mass determinations When comparing the CSPNe and WDs we have to remem- ber that we compare different spatial distributions. Because of their faintness the WD observations are restricted to our near- est neighbourhood while PNe are observed across the whole Galaxy. Nevertheless we didn’t obtain significantly different dis- tributions for PNe at different distances. Our mass determination relies on a single set of evolutionary tracks. There are two possible sources of errors in the Blöcker tracks. The first is the early post-AGB evolution where the time scales depend on how and when the AGB wind terminates. The Blöcker tracks end this at Teff ∼ 6000 K, (pulsation period of 50 days) to agree with the observations of detached shells around hotter stars but not around cooler stars. A later termination would lead to an earlier start of the ionization: in this case we would systematically overestimate the masses. For a reduction of the post-AGB transition time by 103 yr, the typical mass would re- duce by 0.01 M�. The second uncertainty is the mass-loss rate during the post- AGB phase. For M ∼ 0.6 M�, the post-AGB mass-loss rate in the Blöcker models is 0.1 times the nuclear burning rate, but for high-mass models the mass loss accelerates the evolution by 50% (Blöcker 1995). A higher post-AGB mass loss than as- sumed would reduce our masses, but for the typical masses we find a very large increase would be required. Table 1 compares the Blöcker mass-loss rates with observed values, where we used the dynamical mass to calculate the Blöcker rate. For the three non-emission-line stars, observed rates are higher by up to a fac- tor of 3. This appears to be in part related to the high luminos- ity derived by Kudritzki et al: if we compare their rates with Blöcker tracks at similar luminosity, then the Blöcker rates tend to be higher. The nuclear burning rate of ṀH ∼ −6.8 exceeds the observed wind by a factor of four (more for NGC 3242). For 4 K. Gesicki and A.A. Zijlstra: White dwarf masses from planetary nebulae Table 2. Blöcker track time scales: PN visibility is defined as between log Teff = 4.4 and either a nebular age t = 104 yr or a stellar luminosity log L = 3.0, whichever occurs earlier Mass [M�] tstart [yr] tend [yr] tvisibility [yr] 0.546 90 103 - - 0.565 4 103 10 103 6 103 0.605 1.5 103 7.4 103 5.9 103 0.625 660 3.6 103 2.9 103 0.696 100 880 780 0.836 100 840 740 0.940 12 90 78 this factor, the Blöcker tracks would underestimate the speed of evolution by only 10 per cent. We conclude that the post-AGB mass-loss rates have little effect on the derived masses. The ex- ception is the wels star in the sample, where the wind mass loss rate is comparable to the nuclear burning rate. There is also an uncertainty in the dynamical age estimate. A later acceleration would increase the ages by up to 50 per cent and shift the mass peak from 0.61 to 0.60 M�. The WD mass determinations also suffer from simplifica- tions and model assumptions, in addition to the uncertainties concerning cool and hot WDs as described in the Introduction. One uncertainty is in contemporary plasma physics, concerning the pressure broadening in a very high density plasma (Madej et al. 2004). The mass-radius relations used depend on the assumed mass of the hydrogen layer. Napiwotzki et al. (1999) compared estimates from different studies and concluded that the gravities obtained from spectroscopic method suffer from systematic er- rors of up to 0.1 dex in log g. This corresponds to an offset in masses of about 0.02 M� and could, in principle, explain the dif- ference in peak masses between WDs and CSPNe. The width of the peak may also be narrower than derived from the models. Nevertheless, the wide tails of the mass distribution are not in doubt. 4.2. Time scales, birth rates and binarity The derived CSPN mass distribution combines the effects of the birth rate as function of mass, and the observable life time of the PN. The latter depends on mass as indicated in Table 2. The period of visibility is defined here as beginning when the star reaches Teff = 25 103 K, and ending either when the star enters the cooling track (defines as log L = 3.00) or when the age of the nebula is 104 yr, whichever comes earlier. Our histogram should be corrected for the difference in visibility time. This increases the number at high CSPN mass only by a factor of up to 10, and brings the high mass tail in somewhat better agreement. We may also have a sample bias against high masses, as these are not expected in the Bulge objects. The de-selection of bipolar objects may have removed a few higher-mass nebulae in the disk. CSPNe with M < 0.56 M� would not produce a visible PN, as the post-AGB transition time becomes too long (’lazy PNe’). In the sample of Liebert et al. (2005), 30 per cent of white dwarfs have masses in this range, and 50 per cent in the sample of Madej et al. (2004). However, the sharp drop in the CSPN mass distri- bution below 0.60 M� occurs at too high mass to be affected. Hansen & Liebert (2003) argue that both the high- and low- mass tails in WDs distribution can be a result of binary evolu- tion. Merging leads to high-mass WDs while a close compan- ion stripping the envelope can cause an early termination of the evolution and produce a low-mass helium WD. Both channels together may account for some 10 per cent of all WDs (Moe & de Marco 2006). Therefore the histogram for single WDs could be narrower. Close binary evolution can affect the PN phase as well, leading to strongly non-spherical nebulae. Our model anal- ysis assumes spherical symmetry, and we did not analyze bipo- lar nebulae. Our selection therefore favours single CSPNe and rejects low-mass CSPNe in interacting binaries. Thus, the CSPN histogram (Fig. 3) is biased toward single-star evolution, while the WD histogram includes binary broadening. This may affect the tails of the WD histogram but is not expected to affect the main peak. Moe & de Marco (2006) predict a number of PNe in the Galaxy of around 46000. Based on local column densities, Zijlstra & Pottasch (1991) derive an actual number of 23000, suggesting that only about half the stars which could produce a PN, do so. This comparison is limited by our knowledge on the time a PN remains observable. Moe & de Marco (2006) pre- dict a birth rate of PNe of 1.1× 10−12 PNe yr−1 pc−3, comparable to the current, local WD birth rate of 1.0 × 10−12 PNe yr−1 pc−3. Again assuming only half their predicted number of PNe is actu- ally observed, the expectation is that half of all WDs have passed through the PN phase. 5. Conclusions We show that the mass distribution of CSPNe is sharply peaked at M = 0.61 M�. The published WD mass distributions show a much broader distribution peaking at a lower mass of M = 0.59 M�. Part of the difference in the peak may indicate faster evolution during the early post-AGB phase than assumed in the Blöcker tracks. CSPN mass-loss rates cannot explain the dif- ference. However considering the uncertainty of 0.02 M� in the WD mass estimations both peaks are in reasonable agreement. About 30 per cent of WDs have too low masses to have passed through the PN phase. Acknowledgements. We thank our referee Ralf Napiwotzki for important com- ments. This project was financially supported by the “Polish State Committee for Scientific Research” through the grant No. 2.P03D.002.025 and by a NATO collaborative program grant No. PST.CLG.979726. AAZ and KG gratefully ac- knowledge hospitality from the SAAO. References Beauchamp, A., Wesemael, F. & Bergeron, P. 1996, in: C. S. Jeffery and U. Heber (eds.) Hydrogen-Deficient Stars, ASP Conference Series, Vol. 96, p.295 Blöcker, T. 1995, A&A 299, 755 Boudreault, S. & Bergeron, P. 2005, ASP Conf. Series, Vol. 334, p.249 Gesicki, K., Zijlstra, A. A., Acker, A., Gorny, S. K., Gozdziewski & K., Walsh, J. R. 2006, A&A 451, 925 Hansen, B.,M.,S. & Liebert, J. 2003, ARA&A 41, 465 Kudritzki, R. P., Urbaneja, M. A., & Puls, J. 2006, IAU Symposium 234, Planetary Nebulae in our Galaxy and Beyond, M.J. Barlow and R.H. Méndez, Eds., (CUP Cambridge). , p. 119 Liebert, J., Bergeron, P. & Holberg, J. B. 2005, ApJS 156, 47 Madej, J., Nalezyty M. & Althaus, L. G. 2004, A&A 419, L5 Moe, M. & de Marco, O. 2006, ApJ, 650, 916 Napiwotzki, R., Green, P. J. & Saffer, R. A., 1999, ApJ, 517, 399 Napiwotzki, R. 2006, A&A, 451, L27 Pauldrach, A. W. A., Hoffmann, T. L. & Mendez, R. H. 2004, A&A 419, 1111 Schönberner, D. & Tylenda, R. 1990, A&A, 234, 439 Stanghellini, L., Villaver, E., Manchado, A. & Guerrero, M. A. 2002, ApJ, 576, Tylenda, R., Stasińska, G., Acker, A. & Stenholm, B. 1991, A&A, 246, 221 Werner, K. & Herwig, F. 2006, PASP 118, 183 Zijlstra A.A., & Pottasch S.R. 1991, A&A, 243, 478 Introduction Methods and results Models Different CSPN types The HR diagram The mass distributions Comparing CSPNe and WDs The histograms Hydrogen-rich vs. hydrogen-deficient Discussion Uncertainties in mass determinations Time scales, birth rates and binarity Conclusions

0704.0621

Uniqueness theorems for Cauchy integrals

arXiv:0704.0621v1 [math.CV] 4 Apr 2007 UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG Abstract. If µ is a finite complex measure in the complex plane C we denote by Cµ its Cauchy integral defined in the sense of principal value. The measure µ is called reflectionless if it is continuous (has no atoms) and Cµ = 0 at µ-almost every point. We show that if µ is reflectionless and its Cauchy maximal function C ∗ is summable with respect to |µ| then µ is trivial. An example of a reflectionless measure whose maximal function belongs to the ”weak” L1 is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi. 1. Introduction This article discusses uniqueness theorems for Cauchy integrals of complex measures in the plane. We consider the spaceM =M(C) of finite complex measures µ in C. The Cauchy integral of a measure fromM is defined in the sense of principal value. First, for any µ ∈M , ε > 0 and any z ∈ C consider Cµε (z) := ζ:|ζ−z|>ε dµ(ζ) ζ − z Consequently, the Cauchy integral of µ can be defined as Cµ(z) := lim Cµε (z) , if the limit exists. Unlike the Cauchy transform on the line, Cµ can vanish on a set of positive Lebesgue measure: consider for example µ = dz on a closed curve, whose Cauchy transform is zero at all points outside the curve. It is natural to ask if Cµ can also vanish on large sets with respect to µ. If µ = δz is a single point mass, its Cauchy transform will be zero µ-a.e. due to the above definition of Cµ in the sense of principal value. Examples of infinite discrete measures with vanishing Cauchy transforms can also be constructed with little effort. After that one arrives at the following corrected version of the question: Is it true that any continuous µ ∈ M , such that Cµ(z) = 0 at µ-a.e. point, is trivial? As usual, we call a measure continuous if it has no point masses. We denote the space of all finite complex continuous measures by Mc(C). This problem can also be interpreted in terms of uniqueness. Namely, if f and g are two functions from L1(|µ|) such that C(f−g)µ = 0, µ-a.e., does it imply that f = g, µ-a.e.? This way it becomes a problem of injectivity of the planar Cauchy transform. The first author is supported by grants No. MTM2004-00519 and 2001SGR00431. The second author is supported by N.S.F. Grant No. 0500852. The third author is supported by N.S.F. Grant No. 0501067 . http://arxiv.org/abs/0704.0621v1 2 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG First significant progress towards the solution of this problem was achieved by X. Tolsa and J. Verdera in [14]. It was established that the answer is positive in two important particular cases: when µ is absolutely continuous with respect to Lebesgue measure m2 in C and when µ is a measure of linear growth with finite Menger curvature. The latter class of measures is one of the main objects in the study of the planar Cauchy transform, see for instance [11], [12] or [13]. As to the complete solution to the problem, it seemed for a while that the answer could be positive for any µ ∈Mc, see for example [14]. However, in Section 5 of the present paper we show that there exists a large set of continuous measures µ satisfying Cµ(z) = 0, µ-a.e. Following [2], we call such measures reflectionless. This class seems to be an intriguing new object in the theory. On the positive side, we prove that if the maximal function associated with the Cauchy transform is summable with respect to |µ| then µ cannot be reflectionless, see Theorem 2.1. This result is sharp in its scale because the simplest examples of reflectionless measures produce maximal functions that lie in the ”weak” L1(|µ|). We prove this result in Section 2 In view of this fact, we believe that the class of continuous measures with summable Cauchy maximal functions also deserves attention. A full description of this class and the (disjoint) class of reflectionless measures remains an open problem. Let us mention that if µ is a measure with linear growth and finite Menger curvature then its Cauchy maximal function belongs to L2(|µ|), see [12, 13], and therefore is summable. This fact relates Theorem 2.1 to the beforementioned result from [14]. The latter can also be deduced in a different way, see Section 2. From the point of view of uniqueness, our results imply that any bounded planar Cauchy transform is injective, see corollary 2.5. This property is a clear analogue of the uniqueness results for the Cauchy integral on the line or the unit circle. In Section 3 we discuss other applications of Theorem 2.2. They involve structural theo- rems of De Giorgi and his notion of a set of finite perimeter, see [5]. In Section 4 we study asymptotic behavior of the Cauchy transform near its zero set. The results of this section imply that the Radon derivative of µ with respect to Lebesgue measure m2 vanishes a.e. on the set {Cµ = 0}. In particular the set {Cµ = 0} must be a zero set with respect to the variation of the absolutely continuous part of µ which is a slight generalization of the first result of [14]. It is interesting to note that the most direct analogue of this corollary on the real line is false: it is easy to construct an absolutely continuous (with respect to m1 = dx) measure µ ∈M(R) such that |µ|({Cµ = 0}) > 0. Finally, in Section 5 we attempt a geometric description of the set of reflectionless mea- sures. We give a partial description of reflectionless measures on the line in terms of so-called comb-like domains. We also provide tools for the construction of various examples of such measures. In particular, we show that the harmonic measure on any compact subset (of positive Lebesgue measure) of R is reflectionless. Acknowledgments. The authors are grateful to Fedja Nazarov for his invaluable comments and insights. The second author would also like to thank the administration and staff of Centre de Recerca Matemática in Barcelona for the hospitality during his visit in the Spring of 2006. UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 3 2. Measures with summable maximal functions If µ ∈M we denote by Cµ∗ (z) its Cauchy maximal function Cµ∗ (z) := sup |Cµε (z)|. Our first result is the following uniqueness theorem. Theorem 2.1. Let µ ∈Mc. Assume that Cµ∗ (z) ∈ L1(|µ|) and that Cµ(z) exists and vanishes µ-a.e. Then µ ≡ 0. We first prove Theorem 2.2. If Cµ∗ ∈ L1(|µ|) and Cµ(z) exists µ-a.e. then µdµ(z) = 2 Cµ(t)dµ(t) = [Cµ(z)] for m2-a.e. point z ∈ C . (1) Proof. Put F := {z ∈ C : d|µ|(t) |t− z| <∞} . As |µ| is a finite measure, m2(C \ F ) = 0 . (2) Let z ∈ F . Then the integral |t−ζ|>ε dµ(t)dµ(ζ) ζ − z is absolutely convergent for any ε > 0. Using the identity (t− z)(z − ζ) (z − ζ)(ζ − t) (ζ − t)(t− z) we obtain |t−ζ|>ε z − ζ ζ − t ζ − t dµ(t)dµ(ζ) = dµ(ζ) ζ − z |t−ζ|>ε dµ(t) dµ(t) |ζ−t|>ε dµ(ζ) ζ − t dµ(t) · Cµε (t) · dµ(ζ) · Cµε (ζ) · ζ − z Cµε (t)dµ(t) E := {z ∈ C : Cµ∗ (t)d|µ|(t) |t− z| <∞} . By assumption, the numerator Cµ∗ (t)d|µ|(t) is a finite measure. Therefore m2(C \ E) = 0 . (3) If z ∈ E then Cµε (t)dµ(t) Cµ(t)dµ(t) . (4) 4 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG This formula is true as long as Cµ∗ ∈ L1(|µ|) and the principal value Cµ exists µ-a.e. by the dominated convergence theorem. Thus I = 2CC µdµ(z) if z ∈ E . (5) It is left to show that, since z ∈ F , I = [Cµ(z)]2 . (6) Since z ∈ F , the following integral converges absolutely: φε(t, z) := ζ∈C,|ζ−t|>ε dµ(ζ) ζ − z φε(t, z) dµ(t) . Since the point z is fixed in F , we have that 1|ζ−z| ∈ L 1(|µ|), and therefore |ζ−z|d|µ|(ζ) is small if |µ|(A) is small. Denoting the disc centered at t and of radius ε by B(t, ε) we notice 1) φε(t, z) = dµ(ζ) ζ − z B(t,ε) dµ(ζ) ζ − z 2) lim |µ|(B(t, ε)) = 0. uniformly in t. Otherwise µ would have an atom. We conclude that, as ε → 0, the functions φε(t, z) converge uniformly in t ∈ C to φ(z) =∫ ζ−z . Hence for any z ∈ F and any t ∈ C \ z φε(t, z) → φ(z) , as ε → 0 . Since φε(t, z) converge uniformly and z ∈ F , dµ(t)φε(t, z) → φ(z) dµ(t) = [Cµ(z)]2 . We have verified (6). Combining (5) and (6) we conclude that for z ∈ E ∩ F (so for m2-a.e. z ∈ C) we have µdµ(z) = 2 Cµ(t)dµ(t) = lim I = [Cµ(z)]2 for m2-a.e. point z ∈ C . (7) This formula is true as long as Cµ∗ ∈ L1(|µ|) and the principal value Cµ exists µ-a.e. To deduce Theorem 2.1 suppose that Cµ vanishes µ-a.e. Then the left-hand side in (7) is zero form2-a.e. point z. The same must hold for [C µ(z)]2. But if Cµ(z) = 0 for Lebesgue-a.e. point z ∈ C then µ = 0, see for example [6]. Theorem 2.1 is completely proved. UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 5 Remark. In the statement of Theorem 2.2 the condition Cµ∗ ∈ L1(|µ|) can be replaced with the condition that Cµε converge in L 1(|µ|). The proof would have to be changed as follows. Like in the above proof one can show that at Lebesgue-a.e. point z I = [Cµ(z)]2 . (8) The relation I = 2 Cµε (t)dµ(t) for a.e. z can also be established as before. Since Cµε converge in L 1(|µ|), the last integral converges to CC µdµ(z) in the ”weak” L2(dxdy), which concludes the proof. Hence we arrive at the following version of Theorem 2.1: Theorem 2.3. Let µ ∈Mc. Assume that Cµε → 0 in L1(|µ|). Then µ ≡ 0. This version has the following corollary: corollary 2.4 ([14]). Let µ ∈M be a measure of linear growth and finite Menger curvature. If Cµ = 0 at µ-a.e. point then µ ≡ 0. Proof. The conditions on µ imply that the L2(|µ|)-norms of the functions Cµε are uniformly bounded, see for instance [11]. Since Cµε also converge µ-a.e., they must converge in L 1(|µ|). Remark As was mentioned in the introduction, Corollary 2.4 also follows from Theo- rem 2.1. However, the above version of the argument allows one to obtain it without the additional results of [12, 13] on the maximal function. We also obtain the following statement on the injectivity of any bounded planar Cauchy transform. As usual, we say that the Cauchy transform is bounded in L2(µ) if the functions Cfdµε are uniformly bounded in L 2(µ)-norm for any f ∈ L2(µ). If Cµ is bounded, then Cfdµε converge µ-a.e as ε → 0 and the image Cfdµ exists in a regular sense as a function in L2(µ), see [13]. corollary 2.5. Let µ ∈ M be a positive measure. If Cµ is bounded in L2(µ) then it is injective (has a trivial kernel). Proof. Suppose that there is f ∈ L2(µ) such that Cfdµ = 0 at µ-a.e. point. Since both f and Cfdµ∗ are in L 2(µ), Cfdµ∗ is in L 1(|f |dµ). Hence f is a zero-function by Theorem 2.1 � Remark We have actually obtained a slightly stronger statement: If Cµ is bounded in L2(µ) then for any f ∈ L2(µ) the functions f and Cfdµ cannot have disjoint essential supports, i.e. the product fCfdµ cannot equal to 0 at µ-a.e. point. In the rest of this section we will discuss what other kernels could replace the Cauchy kernel in the statement of Theorem 2.1. If K(x) is a complex-valued function in Rn, bounded outside of any neighborhood of the origin, and µ is a finite measure on Rn, one can define Kµ and Kµ∗ in the same way as C and Cµ∗ were defined in the introduction. 6 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG The proof of Theorem 2.2 relied on the fact that the Cauchy kernel K(z) = 1/z is odd, satisfies the symmetry condition (3), i.e. K(x− y)K(y − z) +K(y − z)K(z − x) +K(z − x)K(x− y) ≡ 0, (9) and is summable as a function of z for any t with respect to Lebesgue measure. Any K(x) having these three properties could be used in Theorem 2.1. Out of these three conditions the symmetry condition (9) seems to be most unique. However, other symmetry conditions may result in formulas similar to Theorem 2.2 that could still yield Theorem 2.1. Here is a different example. It shows that much less symmetry can be required from the kernel if the measure is positive. Theorem 2.6. Let µ be a positive measure in Rn. Suppose that the real kernel K(x) satisfies the following properties: 1) K(−x) = −K(x) for any x ∈ Rn; 2) K(x) > 0 for any x from the half-space Rn+ = {x = (x1, x2, ..., xn) | x1 > 0}. If Kµ∗ ∈ L1(µ) and Kµ(x) = 0 for µ-a.e. x then µ ≡ 0. Note that real and imaginary parts of the Cauchy kernel, Riesz kernels in Rn, as well as many other standard kernels satisfy the conditions of the theorem. We will need the following Lemma 2.7. Let K be an odd kernel. and let µ, ν ∈ M . Then Kµε (z)dν(z) = − Kνε (z)dµ(z) (10) for any ε > 0. Suppose that Kµ∗ ∈ L1(|ν|). If Kµ(z) exists ν-a.e. then Kµ(z)dν(z) = − lim Kνε (z)dµ(z). In particular, suppose that both Kµ∗ ∈ L1(|ν|) and Kν∗ ∈ L1(|µ|). If Kµ(z) exists ν-a.e. and Kν(z) exists µ-a.e. then Kµ(z)dν(z) = Kν(z)dµ(z). Proof. Since K is odd, the first equation can be obtained simply by changing the order of integration. The second and third equations now follow from the dominated convergence theorem. � Proof of Theorem 2.6. There exists a half-plane {x1 = c} in Rn such that µ({x1 = c}) = 0 but both µ({x1 > c}) and µ({x1 < c}) are non-zero. Denote by ν and η the restrictions of µ onto {x1 > c} and {x1 < c} respectively. Then∫ Kνε (z)dµ(z) = Kνε (z)dν(z) + Kνε (z)dη(z). The first integral on the right-hand side is 0 because of the oddness of K (apply the first equation in the last lemma with µ = ν). The second condition on K and the positivity of the measure imply that the second integral is positive and increases as ε → 0. Therefore∫ Kνε (z)dµ(z) cannot tend to zero. This contradicts the fact that K µ = 0, ν-a.e. and the second equation from the last lemma. � UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 7 3. Sets of finite perimeter In this section we give another example of an application of Theorem 2.2. It involves the notion of a set of finite perimeter introduced by De Giorgi in the 50’s, see [5]. We say that a set G ⊂ R2 has finite perimeter (in the sense of De Giorgi) if the distributional partial derivatives of its characteristic function χG are finite measures. Such sets have structural theorems. For example, if G is such a set then the measure ∇χG is carried by a set E, rectifiable in the sense of Besicovitch, i. e. a subset of a countable union of C1 curves and an H1-null set, where H1 is the one-dimensional Hausdorff measure. Also the measure ∇χG is absolutely continuous with respect to H1 restricted to E and its Radon-Nikodym derivative is a unit normal vector H1-a.e. (notice that ∇χG is a vector measure). At H1-almost all points of E the function χG has approximate “one-sided”’ limit. For more details we refer the reader to [5]. The general question we consider can be formulated as follows: What can be said about µ if Cµ coincides at µ-a.e. point with a ”good” function f? To avoid certain technical details, all measures in this section are compactly supported. Furthermore, we will only discuss the two simplest choices of f . As we will see, even in such elementary situations Theorem 2.2 yields interesting consequences. As usual, when we say that Cµ = f at µ-a.e. point, we imply that the principal value exists µ-almost everywhere. Theorem 3.1. Let µ ∈ Mc be compactly supported. Assume that Cµ(z) = 1, µ-almost everywhere and Cµ∗ ∈ L1(|µ|). Then µ = ∂̄χG, where G is a set of finite perimeter. In particular, µ is carried by a set E, H1(E) < ∞, rectifiable in the sense of Besicovitch, and µ is absolutely continuous with respect to the restriction of H1 to E. Remark. The most natural example of such a measure is dz on a C1 closed curve. The theorem says that, by the structural results of De Giorgi, this is basically the full answer. Proof. By Theorem 2.2 we get that for Lebesgue-almost every point in C [Cµ(z)]2 = 2Cµ(z) . (11) In other words for m2-a.e. point z we have C µ(z) = 0 or = 2. Let G denote the set where Cµ(z) = 2. Since the Cauchy transform of any compactly supported finite measure must tend to zero at infinity, this set is bounded. Consider the following equality χG = C understood in the sense that the two functions are equal as distributions. Taking distribu- tional derivatives on both sides we obtain ∂̄χG = µ/2 and ∂χ̄G = µ̄/2. Hence G has finite perimeter and the rest of the statement follows from the results of [5]. � We say that a set G has locally finite perimeter (in the sense of De Giorgi) if the distribu- tional derivatives of χG are locally finite measures. Our second application is the following 8 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG Theorem 3.2. Let µ ∈ Mc be compactly supported. Assume that Cµ(z) = z, µ-almost everywhere and Cµ∗ ∈ L1(|µ|). If µ(C) = 0 then µ = 2z∂̄χG, where G is a set with locally finite perimeter. Whether µ(C) = 0 or not, µ is carried by a set E, H1(E) <∞, which is a rectifiable set in the sense of Besicovitch, and µ is absolutely continuous with respect to the restriction of H1 to E. Remark. The most natural example of such a measure is zdz on a C1 closed curve. Our statement shows that this is basically one-half of the answer. The other half is given by√ z2 − cdz as will be seen from the proof. Proof. Again, from Theorem 2.2 we get that for Lebesgue-almost every point in C [Cµ(z)]2 = 2Cζdµ(ζ)(z) . (12) Notice that Cζdµ(ζ)(z) = ζ − z dµ(ζ) = µ(C) + zCµ(z) and we get a quadratic equation [Cµ(z)]2 = 2zCµ(z)− p , where p := −2µ(C). First case p = 0. Here we get [Cµ(z)]2 = 2zCµ(z) . We conclude that Cµ(z) = 0 or z for Lebesgue-a.e. point z ∈ C. Again a bounded set G appears on which Cµ = 2zχG(z) in terms of distributions. Therefore ∂̄χG = dµ/2z , and the right hand side is a finite measure on any compact set avoiding the origin. Therefore, G is a (locally) De Giorgi set. Let us consider the case p 6= 0. For simplicity we assume p = 1, other p’s are treated in the same way. Then we have to solve the quadratic equation Cµ(z)2 − 2zCµ(z) + 1 = 0 for Lebesgue-a.e. point in C. Let us make the slit [−1, 1] and consider two holomorphic functions in C \ [−1, 1] r1(z) = z − z2 − 1, r2(z) = z + z2 − 1 , where the branch of the square root is chosen so that r1(z) → 0, z → ∞ . In other words we have the sets E1 and E2 such that m2(C \ E1 ∪ E2) = 0 and z ∈ E1 ⇒ Cµ(z) = r1(z) , z ∈ E2 ⇒ Cµ(z) = r2(z) . UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 9 Obviously it is E1 that contains a neighborhood of infinity. The function z − z2 − 1 outside of [−1, 1] can be written as Cµ0(z) where dµ0(x) = 1π 1− x2dx. Consider ν = µ−µ0. z ∈ E1 ⇒ Cν(z) = 0 , z ∈ E2 ⇒ Cν(z) = 2 z2 − 1 := R(z) . Therefore, Cν(z) = R(z)χE2 . (13) Notice that if R was analytic in an open domain compactly containing E2 we would conclude from the previous equality that ν = R(z)∂̄χE2 . If, in addition, |R| was bounded away from zero on E2, we would obtain that ∂̄χE2 and ∂χE2 are measures of finite variation, and hence E2 is a set of finite perimeter. Notice that our R(z) = 2 z2 − 1 is analytic in O := C \ [−1, 1] and is nowhere zero. We will conclude that E2 is a set of locally finite perimeter. More precisely we will establish the following claim: For every open disk V ⊂ O the set O ∩ E2 has finite perimeter. Indeed, let W be a disk compactly containing V , W ⊂ O. Let ψ be a smooth function, supported in W , ψ|V = 1. Multiply (13) by ψ and take a distributional derivative (against smooth functions supported in V ). Then we get (using the fact that R is holomorphic on V ) ν|V = ∂̄(ψRχE2∩V )|V = ∂̄(RχE2∩V )|V = R∂̄(χE2∩V )|V . We conclude immediately that E2 ∩ V is a set of finite perimeter. Therefore, E2 ∩D is a set of finite perimeter, where D is a domain whose closure is contained compactly in O. Recalling that µ = ν + µ0 we finish the proof. � Remark 3.3. In is interesting to note that, as follows from the proof, if µ is the measure from the statement of the theorem then one of the connected components of supp µ must contain both roots of the equation z2 + 2µ(C) = 0. We conclude this section with the following examples of measures µ whose Cauchy trans- form coincides with z at µ-a.e. point Examples. 1. Let Ω be an open domain with smooth boundary Γ. Suppose that [−1, 1] ⊂ Ω. Let {Dj}∞j=1 be smoothly bounded disjoint domains in O := Ω \ [−1, 1], γj = ∂Dj . Assume H1(γj) <∞ . (14) LetR(z) be an analytic branch of 2 z2 − 1 inO. Consider the measure ν on Γ∪(∪γj)∪[−1, 1] defined as ν = R(z)dz|Γ − R(z)dz|∪γj − 1− x2dx|[−1,1]. 10 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG Cν(z) = 0 if z ∈ C \ Ō , 0 if z ∈ ∪jDj , R(z) if z ∈ O \ ∪jD̄j . Recall that R(z) = z + z2 − 1 − (z − z2 − 1) and that Cµ0(z) = z − z2 − 1 for µ0 = 1− x2dx|[−1,1]. We conclude that for µ = ν + µ0 one has Cµ(z) = z2 − 1 if z ∈ C \ Ō , z2 − 1 if z ∈ ∪jDj , z2 − 1 if z ∈ O \ ∪jD̄j . 2. The second example is exactly the same as the first one but Dj,k = B(xj,k, ), xj,k = j , 1 ≤ k ≤ j, j = 1, 2, 3.... Here the assumption (14) fails. But ν, defined as above, will still be a measure of finite variation (and so will be µ): |ν|(C) ≤ C In both examples Cµ(z) = z for µ-a.e. z. 4. Asymptotic behavior near the zero-set of Cµ In this section we take a slightly different approach. We study asymptotic properties of measures near the sets where the Cauchy transform vanishes. Theorem 4.2 below shows that near the density points of such sets the measure must display a certain ”irregular” asymptotic behavior. As was mentioned in the introduction, one of the results of [14] says that an absolutely continuous planar measure cannot be reflectionless. This result is not implied by our Theorem 2.1 because an absolutely continuous measure may not have a summable Cauchy maximal function. It is, however, implied by Theorem 4.2, see Corollary 4.4 below. When estimating Cauchy integrals one often uses an elementary observation that the difference of any two Cauchy kernels 1/(z − a) − 1/(z − b) can be estimated as O(|z|−2) near infinity. To obtain higher order of decay one may consider higher order differences. Here we will utilize the following estimate of that kind, which can be verified through simple calculations. Lemma 4.1. If a, b, c ∈ B(0, r) be different points, |a − b| > r. Then there exist constants A,B ∈ C such that |A|, |B| < 2 z − a z − b z − c ∣∣∣∣ < outside of B(0, 2r). (Namely, A = b−c b−a , B = a−b .) If µ ∈M consider one of its Riesz transforms in R3, R1µ(x, y, z), defined as R1µ(x, y, z) = |(u, v, 0)− (x, y, z)|3 dµ(u+ iv). UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 11 This transform is the planar analogue of the Poisson transform. In particular, R1µ(x, y, z) = (x+ iy) for all points w = x+ iy ∈ C where the Radon derivative (w) = lim µ(B(w, r)) |B(w, r)| exists. For measures on the line or on the circle their Poisson integrals and Radon derivatives (with respect to the one-dimensional Lebesgue measure) are very much related but not always equivalent. When the asymptotics of the Poisson integral and the ratio from the definition of the Radon derivative are different near a certain point it usually means that the measure is ”irregular” near that point. It is not difficult to show that if µ is absolutely continuous then at a Lebesgue point of its density function the Radon derivative of µ and the Poisson integral of |µ| (or R1|µ| if n > 1) behave equivalently. Even for singular measures on the circle, if a measure possesses a certain symmetry near a point, then the same equivalent behavior takes place, as follows for instance from [1], Lemma 4.1. In fact, it is not easy to construct a measure so that its Poisson integral and Radon derivative behaved differently near a large set of points. The same can be said about the Riesz transform and the Radon derivative. Thus one may interpret our next result as an evidence that, for a planar measure µ, most points where Cµ = 0 are ”irregular.” Theorem 4.2. Let µ ∈ M and let w = x + iy be a point of density (with respect to m2) of the set E = {Cµ = 0}. Then µ(B(w, r)) = o (R1|µ|(x, y, r)) as r → 0 + . In view of the above discussion this implies corollary 4.3. If w is a point of density of the set E = {Cµ = 0}, such that there exists the Radon derivative d|µ|/dm2(w) 6= 0, then µ(B(w, r)) = o (|µ|(B(w, r))) as r → 0+ (16) and dµ/dm2(w) = 0. Since m2-almost every point of a set is its density point, we also obtain the following version of the result from [14]: corollary 4.4. The set E = {Cµ = 0} has measure zero with respect to the absolutely continuous component of µ. Proof of Theorem 4.2. without loss of generality w = 0. Choose a C∞0 test-function φ sup- ported in B := B(0, r), and such that 0 ≤ φ ≤ D/r2, |∇φ| ≤ A/r3 and φ dm2 = 1. Denote the complement of E by Ec. Then φdµ = 〈φ, ∂̄Cµ〉 = 〈∂̄φ, Cµ〉 = 〈χEc∂̄φ, Cµ〉 = χEc ∂̄φ dm2(z) ζ − z dµ(ζ) (17) All we need is to show that the last integral is small. Then, since the first integral in (17) is similar to the right-hand side of (16) we will complete the proof. The main idea for 12 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG the rest of the proof is to make the function F (ζ) = χEc ∂̄φ dm2(z) ζ−z ”small” by subtracting a linear combination of Cauchy kernels corresponding to points from E, which will not change its integral with respect to µ. Namely, let a, b ∈ B(0, r) ∩ E be any two points such that |a − b| > r. By the previous lemma for any z ∈ B(0, r) there exist constants A = A(z), B = B(z), of modulus at most 2, such that (15) holds with c = z. Integrating (15) with respect to χEc ∂̄φ dm2(z) we obtain that ∣∣∣∣ χEc ∂̄φ dm2(z) ζ − z ζ − a ζ − b ∣∣∣∣ < C ε(r)r |ζ |3 outside of B(0, 2r) for some constants A∗, B∗, where ε(r) = |B(0, r)∩Ec|/r2 = o(1) as r → 0. The constants satisfy |A∗|, |B∗| < 2 ε(r) Notice that if w ∈ E then ζ−wdµ = 0 by the definition of the set E. Hence, since a, b ∈ E, χEc ∂̄φ dm2(z) ζ − z dµ(ζ) = χEc ∂̄φ dm2(z) ζ − z ζ − a ζ − b dµ(ζ) B(0,2r) C\B(0,2r) = I1 + I2. For I2 we now have C\B(0,2r) χEc ∂̄φ dm2(z) ζ − z ζ − a ζ − b dµ(ζ) C\B(0,2r) ε(r)r |ζ |3 d|µ|(ζ) ≤ Cε(r)R1|µ|(0, 0, r). In I1 we estimate each summand separately. First, B(0,2r) χEc ∂̄φ dm2(z) ζ − z dµ(ζ) ∣∣∣∣ ≤ B(0,2r) |ζ − z| χEcdm2(z)d|µ|(ζ) |µ|(B(0, 2r)) ≤ C ε(r)R1|µ|(0, 0, r). To estimate the second and third summands of I1, recall that the only restriction on the choice of a, b ∈ B(0, r) ∩ E was that |a − b| > r. This condition will be satisfied, for instance, if a ∈ B1 = B(−56r, r) and b ∈ B2 = B(56r, r). If we average the modulus of the second summand over all choices of a ∈ B1 ∩ E, recalling that A∗ = A∗(a) always satisfies |A∗| ≤ 2 ε(r) , we get |B1 ∩ E| B(0,2r) A∗(a) ζ − a dµ(ζ) ∣∣∣∣dm2(a) ≤ |B1 ∩ E| B(0,2r) |A∗(a)| |ζ − a| dm2(a)d|µ|(ζ) r|µ|(B(0, 2r)) ≤ Cε(r)R1|µ|(0, 0, r). It is left to choose a ∈ B1 ∩ E for which the modulus is no greater than its average. The same can be done for b. The proof is finished. � UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 13 5. Reflectionless measures and Combs As was mentioned in the introduction, following [2], we will call a non-trivial continuous finite measure µ ∈M(C) reflectionless if Cµ(z) = 0 at µ-a.e. point z. Perhaps the simplest example of a reflectionless measure is the measure µ = 1 (1−x2)−1/2dx on [−1, 1], the harmonic measure of C \ [−1, 1] corresponding to infinity. The fact that µ is reflectionless can be verified through routine calculations or via the conformal map interpretation of the harmonic measure. It will also follow from a more general Theorem 5.4 below. At the same time, since Cµ∗ ≍ (1 − x2)−1/2 on [−1, 1], this simple example complements the statement of Theorem 2.1. Since the function (1−x2)−1/2 belongs to the ”weak” L1(|µ|), the summability condition for the Cauchy maximal function proves to be exact in its scale. In the rest of this section we discuss further examples and properties of positive reflec- tionless measures on the line. Let us recall that functions holomorphic in the upper half plane C+ and mapping it to itself (having non-negative imaginary part) are called Nevanlinna functions. Let M+(R) denote the class of finite positive measures compactly supported on R. The function f is a Nevanlinna function if and only if it has a form f(z) = az + b+ t2 + 1 ]dρ(t) , where ρ is a positive measure on R such that dρ(t) < ∞, a > 0, b ∈ R are constants. If the representing measure is from M+(R) and f(∞) = 0, the formula becomes simpler: f(z) = dµ(x) x−z . Definition. A simply connected domain O is comb-like if it is a subset of a half-strip {w : ℑw ∈ (0, π),ℜw > q}, for some q ∈ R, contains another half-strip {w : ℑw ∈ (0, π),ℜw > r} for some r ∈ R and has the property that for any w0 = u0 + iv0 ∈ O the whole ray {w = u+ iv0, u ≥ u0} lies in O . (18) If in addition H1(∂O∩B(0, R)) <∞ for all finite R, we say that O is a rectifiable comb-like domain. Let O be a rectifiable comb-like domain, Γ = ∂O. Then by the Besicovitch theory we know that for H1-a.e. pont w ∈ Γ there exists an approximate tangent line to Γ, see [3] for details. We wish to consider rectifiable comb-like domains satisfying the following geometric property: for a.e. w ∈ Γ approximate tangent line is either vertical or horizontal. (19) It is not difficult to verify that for any conformal map F : C+ → O, O is comblike if and only if F ′ is a Cauchy potential of µ ∈ M+(R): F ′(z) = dµ(x) x−z . It is, therefore, natural to ask the following Question. Which comb-like domains correspond to reflectionless measures µ ∈M+(R)? An answer would give a geometric description of reflectionless measures from M+(R). If, in addition, a comb-like domain is rectifiable, then the answer is given by 14 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG Theorem 5.1. 1) Rectifiable comb-like domains correspond exactly to those measures µ ∈M+(R) that are absolutely continuous with respect to dx and satisfy dµ(x) ∈ H1loc(C+). (20) 2) An absolutely continuous measure satisfying (20) is reflectionless if and only if the corre- sponding comb-like domain has the property (19). Remarks. 1) Of course not every comb-like domain gives rise to a reflectionless measure fromM+(R). Just take any comb-like domain which appears as F (C+), where F = ∫ z ∫ dµ(x) x−z for a singular µ ∈M+(R). By a result from [9] singular measures cannot be reflectionless. 2) On the other hand, even if µ = g(x)dx is a reflectionless absolutely continuous measure, the corresponding conformal map F = ∫ z ∫ dµ(x) x−z : C+ → O can be onto a non-rectifiable domain. 3) For non-rectifiable domains we have no criteria to recognize which ones correspond to reflectionless measures. 4) It is well known, and not difficult to prove, that the antiderivative of a Nevanlinna function is a conformal map, see for instance [4]. If F = ∫ z ∫ dµ(x) x−z , µ ∈ M+(R) then ℑF (x) is an increasing function on R whose derivative in the sense of distributions is µ. The image F (C+) lies in the strip {ℑw ∈ (0, π‖µ‖)}. Theorem 5.1 will follow from Theorems 5.2 and 5.3 below. Theorem 5.2. Let F be a conformal map of C+ on a rectifiable comb-like domain O. Then F (z) = ∫ z ∫ dµ(x) x−z , µ ∈ M+(R), µ << dx. Also dµ(x) x−z ∈ H loc(C+). If in addition O satisfies (19) then µ is reflectionless. Proof. without loss of generality O ⊂ {ℜz > 0}. Put Φ = eF . Then the image Φ(O) is the subdomain of the complement of the unit half-disk in C+ which is the union of rays (R(θ)eiθ,∞). Consider the subdomain of the upper half-disk D := {z : 1/z ∈ Φ(O)}. Define G as the smallest open domain containing D and its reflection D := {z̄ : z ∈ D}. Then G is a star-like domain inside the unit disk. The preimage of G ∩ R under Φ is the union of two Infinite rays R1 = [−∞, a), R2 = (b,∞], a < b. Therefore, by reflection principle C \ [a, b] is mapped conformally (by the extension of Φ which we will also denote by Φ) onto star-like Since Φ : C+ → G, where G is star-like, it is well-known that argΦ(x+ iδ) is an increasing function of x, see [7]. We conclude that the argument of Φ is monotone. Therefore, ℑF (x + iδ) is monotone, and so ℑf(x+ iδ) is positive, where f = F ′. We see that f = F ′ is a Nevanlinna function. From the structure of our comb-like domain, we conclude immediately that its representing measure µ has compact support, so we are in M+(R). Also, let us prove that µ << dx. The boundary of our comb is locally rectifiable. So f = F ′ belongs locally to the Hardy class H1(C+), [16]. Since ℑf is the Poisson integral of µ, ℑf = Pµ = (x− t)2 + y2 dµ(t), UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 15 and f is in H1(C+) locally, we conclude that µ = ℑfdx,ℑf ≥ 0 a.e., [16]. Now suppose that, in addition, O = F (C+) has the property (19). Let us recall that for a simply connected domain with rectifiable boundary Γ the restriction of the Hausdorff measure H1|Γ is equivalent to the harmonic measure ν on O. Therefore the tangent lines to Γ are either vertical or horizontal a.e. with respect to ν. The measure ν is the image of the harmonic measure λ of C+ which is equivalent to the Lebesgue measure on the line. We have a conformal map F (a continuous function up to the boundary of C+ because it is an anti-derivative of an H1loc-function) which pushes forward λ to ν. Call a point w0 ∈ Γ accessible from O if there exists a ray x0 + iy, 0 < y < 1, such that w0 = limy→0 F (x0 + iy). Almost every point of Γ (w.r. to ν) is accessible from O. For ν-a.e. accessible w0 ∈ Γ where the tangent line is vertical (horizontal) we can say that ℜF ′(x0) = 0 (ℑF ′(x) = 0). So R = E1 ∪ E2 ∪ E3, where |E3| = 0, |E1 ∩ E2| = 0, and E1 = {x ∈ R : ℜF ′(x) = 0}, E2 = {x ∈ R : ℑF ′(x) = 0}. We already know that the measure µ = ℑF ′(x)dx represents f(z) = F ′(z) = dµ(t) t−z . Notice that R\E2 · = ·. But we also know that boundary values exist dx-almost everywhere, i.e. dµ(t) t− x− iy = ℜF ′(x) = 0 for a.e. x ∈ E1 and therefore for µ-a.e. x ∈ E1. This means (see [16]) that dµ(x) = 0 µ-a.e. Definition. A simply connected rectifiable comb-like domain O is called a comb if its “left” boundary consists of countably many horizontal and vertical segments. A comb is called a straight comb if O = {w : ℑw ∈ (0, π),ℜw > 0} \ S, where the set S is relatively closed with respect to the strip {w : ℑw ∈ (0, π),ℜw > 0} and is the union of countably many horizontal intervals Rn = (iyn, ln + iyn]. We require also that ln <∞ . Example. Let F be a conformal map of C+ on a comb O. By our last theorem F ′(z) =∫ dµ(x) x−z , where µ ∈M+(R) is reflectionless: C µ(x) = 0 for µ-a.e. x. Definition. Let E be a compact subset of the real line. Let E have positive logarithmic capacity, so Green’s function G of C \E exists. The domain C \E is called Widom domain G(c) <∞ , where the summation goes over all critical points of G (we assume that G is a Green’s function with pole at infinity. Example. Let E be a compact subset of the real line of the positive length. We assume that every point of E is regular in the sense of Dirichlet for the domain C \ E, and we also assume that C \ E is not a Widom domain. Such E exist in abundance. We will see below, that the harmonic measure ω of C \ E (with pole at infinity) is reflectionless. Consider F (z) = ∫ z ∫ dω(x) z−x for z ∈ C+. It is easy to see that F (z) = G(z) + iG̃(z) + const, 16 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG where G̃ is the harmonic conjugate of G. This F is a conformal map (see [4]) of C= onto a domain D lying in the strip {w : ℑw ∈ (0, π)}. It is easy to see that complementary intervals of E will be mapped by F onto straight horizontal segments on the boundary of D. Each finite complementary interval contains exactly one critical point of G, and clearly the length of the corresponding straight horizontal segment is G(c) (this follows from the formula F (z) = G(z) + iG̃(z) + const). As the domain C \ E was not a Widom domain, we have that the sum of lengths of abovementioned straight horizontal segment is infinite. So domain D is not rectifiable. Therefore the reflectionless property of µ alone does not say anything about the rectifiability of the domain, which is the target domain of the conformal map F (z) = ∫ z ∫ dµ(x) z−x . Theorem 5.3. Let µ be absolutely continuous positive measure on R and let Cµ ∈ H1loc(C+). Then F (z) = ∫ z ∫ dµ(x) x−z is a conformal map of C+ onto a rectifiable comb-like domain O. If µ is reflectionless then O has the property (19). Proof. Consider F (z) = ∫ z ∫ dµ(x) x−z . Since µ is positive, it is a conformal map. If µ is such that f(z) = Cµ ∈ H1loc(C+) then F (z) = f maps C+ onto a domain with locally rectifiable boundary (see [16]). If, in addition, µ = ℑfdx is reflectionless, then for a.e. point of P := {x ∈ R : ℑf(x) > 0} we have ℜf(x) = 0. Conformal map F (z) is continuous up to the boundary of C+ and its boundary values F (x) form a (locally) absolutely continuous function, F ′(x) = f(x) a.e. As at almost every point we have either ℑF ′(x) = 0 or ℜF ′(x) = 0 we conclude that O = F (C+) has the property (19). We also need the following definition. Definition. A compact subset E in R is called homogeneous if there exist r, δ > 0 such that for all x ∈ E, |E ∩ (x− h, x+ h)| ≥ δh for all h ∈ (0, r). Example. Let E ⊂ R be a compact set of positive length. Let µ be a reflectionless measure supported on E, µ = g(x)dx. Let in addition E be a homogeneous set. Then F (z) = ∫ z ∫ dµ(x) x−z is a conformal map from C+ on a rectifiable comb-like domain satisfying (19). Proof. The Cauchy integral Cgdx considered in C\E will be in the Hardy class H1(C\E). In fact the reflectionless property of gdx implies that its limits from C± will be both integrable with respect to dx|E . Now we use homogenuity of E and Zinsmeister’s theorem [15] to conclude that f(z) = Cgdx(z) is in the usual H1loc(C). Then the conformal map F (z) = f maps C+ onto a rectifiable subdomain of a strip. We use Theorem 5.3 to get the rest of our example’s claims. � The simple example of a reflectionless measure mentioned at the beginning of this section, as well as many other explicit examples, are given by our next statement. UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 17 Theorem 5.4. Let E be a compact set of positive lenght, E ⊂ R. Let ω be a harmonic measure of C \ E with pole at infinity. Then ω is reflectionless. Example. The simplest comb is a strip {w : ℑw ∈ (0, π),ℜw > 0}. Consider F (z) = log(z + z2 − 1). It maps conformally C+ onto the strip. Its derivative f(z) = 1√z2−1 is x−z and dµ = 1−x2 is the harmonic measure of C \ [−1, 1]. Proof of Theorem 5.4. We need to show that Cω = 0 at ω-a.e point. From our definitions it can be seen, that Cω on the line coincides with the Hilbert transform of ω, which in its turn is asymptotically equivalent to the conjugate Poisson transform Qω. Thus all we need to establish is that Qω(x+ ih) = (x− y)2 + h2 dω(y) = ℜ dω(y) x− ih− y → 0 as h→ 0+ (21) for almost every x. Instead, we have that the Green’s function F (x) defined as F (x) = log |x− y|dω(y) + C∞, where C∞ is a real constant (Robin’s constant), is equal to 0 at every density point of E, see for example [8]. The idea of the proof is to show that Qω(x + iε) behaves like (F (x+ ε) + F (x− ε))/ε near almost every x. The technical details are as follows. Introduce φ(y) := |1− y| |1 + y| y2 + 1 , (22) φx,h(y) := y − x The function φ(y) decreases as 1/y2 at infinity, hence it is in L1(R, dx) and so are φx,h(y) with a uniform bound on the norm. However, these functions are not bounded, which makes it difficult to use them in our estimates. To finish the proof we will first obtain a bounded version of φx,h(y) through the following averaging procedure. Let ω = g(x)dx. Choose x to be a Lebesgue point of g and a density point of E. Fixing sufficiently small h > 0 we can find the set A(x, h) ⊂ (x−h, x−h/2)∪ (x+h/2, x+h) such • A(x, h) consists of density points of E, • |A(x, h)| ≥ h/2, • A(x, h) is symmetric with respect to x. Let Tx,h := T := {t ∈ (0, h) : x+ t ∈ A(x, h)}. Then |T | ≥ h/4. Now put ψx,h(y) := φx,t(y) dt . By (22) one can see immediately that |ψx,h| ≤ for some M > 0 and |ψx,h(y)| ≤ C , for |y| > h . (23) 18 MARK MELNIKOV, ALEXEI POLTORATSKI, AND ALEXANDER VOLBERG Also, since ∫ φ dy = 0 . we have that ∫ ψx,h dy = 0 . Therefore, g(y)ψx,h(y) dy| = | (g(y)− g(x))ψx,h(y) dy| ≤ |g(y)− g(x)||ψx,h|(y) dy. Now notice that (23) implies that |ψx,h| is majorated by an approximate unity (for instance, by a constant multiple of the Poisson kernel corresponding to z = x + ih). Since x is a Lebesgue point for g(x), this means that the last integral tends to 0 as h→ 0. Looking at the definitions of Tx,h and ψx,h(y) we can see that g(y)ψx,h(y) dy = |Tx,h| (F (x+ t)− F (x− t))−ℜ g(y)dy x− it− y where F (x) is the Green’s function. As we mentioned before, F is zero at the density points of E. We conclude that |Tx,h| g(y)dy x− it− y → 0, h→ 0 + . for a.e. x on the Borel support of g. Since the Cauchy integral of g has a limit a.e. we obtain g(y)dy x− ih− y → 0, h→ 0 + . Remark. All reflectionless measures on R discussed in this section, including those provided by Theorem 5.4 are absolutely continuous with respect to Lebesgue measure. One may wonder if there exist singular reflectionless measures. The answer is negative. More generally, as follows from a theorem from [9], if principal values of the Hilbert transform exist µ-a.e. for a continuous µ ∈M(R) then µ << dx . References [1] A. B. Alexandrov, J. M. Anderson, A. Nicolau. Inner functions, Bloch spaces and symmetric measures, Proc. London Math. Soc. (3) 79 (1999), no. 2, 318–352. [2] E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its and V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994). [3] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. xii+343 pp. [4] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Prin- ciples of Mathematical Sciences], 259. Springer-Verlag, New York, 1983. [5] L.C. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Math- ematics. CRC Press, Boca Raton, FL, 1992. [6] T. W. Gamelin, Uniform Algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. UNIQUENESS THEOREMS FOR CAUCHY INTEGRALS 19 [7] G. M. Golusin, Geometric theory of functions. Hochschulbcher fr Mathematik, Bd. 31. VEB Deutscher Verlag der Wissenschaften, Berlin, 1957. xii+438 pp. 30.0X [8] W. K. Hayman, P. B. Kennedy, Subharmonic Functions, vol. 1, Academic Press, London-New York, 1976. [9] P. Jones, A. Poltoratski, Asymptotic growth of Cauchy transforms, Ann. Acad. Sci. Fenn. Math, 2004 [10] M. Krein, A. Nudelman The Markov moment problem and extremal problems. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development. Translated from the Russian by D. Louvish. Translations of Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977. v+417 pp. [11] M. Melnikov, J. Verdera, A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs, Internat. Math. Res. Notices 1995, no. 7, 325–331. [12] F. Nazarov, S. Treil, A. Volberg , Cauchy integral and Calder-Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Not. 15 (1997) 703726. [13] X. Tolsa, L2 -boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), 269-304. [14] X. Tolsa, J. Verdera, May the Cauchy transform of a non-trivial finite measure vanish on the support of the measure? Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 479–494. [15] M. Zinsmeister, Espaces de Hardy et domaines de Denjoy. (French) [Hardy spaces and Denjoy domains] Ark. Mat. 27 (1989), no. 2, 363–378. [16] I. Privalov, Boundary properties of analytic functions, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow- Leningrad, 1950. 336 pp. Department de Matematiques, Uneversitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain E-mail address : [email protected] Texas A& M University, Department of Mathematics, College Station, TX 77843, USA E-mail address : [email protected] Dept. Math., Michigan State Univ., East Lansing MI 48823, USA, and, School of Math., University of Edinburgh, Edinburgh UK EH9 EJ6 E-mail address : [email protected]

0704.0622

On the number of moduli of plane sextics with six cusps

ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS CONCETTINA GALATI Abstract. Let Σ6 be the variety of irreducible sextics with six cusps as singularities. Let Σ ⊂ Σ6 be one of irreducible components of Σ6 . Denoting by M4 the space of moduli of smooth curves of genus 4, we consider the rational map Π : Σ 99K M4 sending the general point [Γ] of Σ, corresponding to a plane curve Γ ⊂ P2, to the point of M4 parametrizing the normalization curve of Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We know that dim(Π(Σ)) ≤ dim(M4) + ρ(2, 4, 6) − 6 = 7, where ρ(2, 4, 6) is the Brill-Neother number of linear series of dimension 2 and degree 6 on a curve of genus 4. We prove that both irreducible components of Σ6 have number of moduli equal to seven. 1. Introduction Let Σnk,d ⊂ P(H0(P2,OP2(n))) := PN , with N = n(n+3) , be the closure, in the Zariski’s topology, of the locally closed set of reduced and irreducible plane curves of degree n with k cusps and d nodes. Let Σ ⊂ Σnk,d be an irreducible component of the variety Σnk,d. Denoting by Mg the moduli space of smooth curves of genus − k − d, it is naturally defined a rational map ΠΣ : Σ 99K Mg, sending the general point [Γ] ∈ Σ0 to the isomorphism class of the normalization of the curve Γ ⊂ P2 corresponding to [Γ]. We say that ΠΣ is the moduli map of Σ and we set number of moduli of Σ := dim(ΠΣ(Σ)). We say that Σ has general moduli if ΠΣ is dominant. Otherwise, we say that Σ has special moduli or that Σ has finite number of moduli. By lemma 2.2 of [4], we know that the dimension of the general fibre of ΠΣ is at least equal to max(8, 8 + ρ− k), where ρ := ρ(2, g, n) = 3n− 2g− 6 is the number of Brill-Noether of linear series of degree n and dimension 2 on a smooth curve of genus g. It follows that, if Σ has the expected dimension equal to 3n+ g − 1− k and g ≥ 2, then (1) dim(ΠΣ(Σ)) ≤ min(dim(Mg), dim(Mg) + ρ− k). Definition 1.1. We say that Σ has the expected number of moduli if equality holds in (1). In particular, we expect that, if ρ − k ≤ 0, then on the normalization curve C of the curve Γ ⊂ P2 corresponding to the general point [Γ] ∈ Σ, there exists only a finite number of linear series of degree n and dimension 2 mapping C to a plane curve with nodes and k cusps as singularities and corresponding to a point of Σ, (see the proof of lemma 2.2 of [4]). For a deeper discussion and a list of known results about the moduli problem of Σnk,d we refer to sections 1 and 2 of [4] and related references. In particular, in [4] we have found sufficient conditions in order Key words and phrases. number of moduli, sextics with six cusps, plane curves, Zariski pairs. http://arxiv.org/abs/0704.0622v1 2 CONCETTINA GALATI that an irreducible component Σ of Σnk,d has finite and expected number of moduli. If Σ verifies these conditions then ρ(2, n, g) ≤ 0. Finally in [4] we constructed examples of families of plane curves with nodes and cusps with finite and expected number of moduli. In this paper we consider the particular case of the variety Σ66,0 of irreducible sextics with six cusps. It was proved by Zariski (see [8]) that Σ66,0 has at least two irreducible com- ponents. One of them is the parameter space Σ1 of the family of plane curves of equation f32 (x0, x1, x2) + f 3 (x0, x1, x2) = 0, where f2 and f3 are homogeneous polynomials of degree two and three respectively. The general point of Σ1 corresponds to an irreducible sextic with six cusps on a conic as singularities. Moreover, Σ66,0 contains at least one irreducible component Σ2 whose general element corresponds to a sextic with six cusps not on a conic as singularities and containing in its closure the variety Σ69,0 of elliptic sextics with nine cusps. Recently, A. Degtyarev has proved that Σ1 and Σ2 are the unique irreducible components of Σ66,0, (see [1]). The moduli number of Σ1 and Σ2 can not be calculated by using the result of [4]. Indeed, in this case ρ(2, 4, 6) = 4 > 0 and then the general element of every irreducible component of Σ66,0 does not verify the hypotheses of proposition 4.1 of [4]. On the contrary, it is easy to verify that, if Γ ⊂ P2 is the plane curve corresponding to the general element of one of the irreducible components of Σ66,0 and C is the normalization curve of Γ, then the map µo,C is injective. But, in contrast with the nodal case, this information is not useful in order to study the moduli problem of Σnk,d, (see [6] and remark 4.2 of [4]). In the proposition 2.2 and corollary 2.4, we prove that Σ2 has the expected number of moduli equal to seven. Moreover, we show that there exists a stratification Σ69,0 ⊂ Σ′ ⊂ Σ̃ ⊂ Σ2, where Σ′ and Σ̃ are respectively irreducible components of Σ68,0 and Σ 7,0 with ex- pected number of moduli. Finally, in the corollary 2.8, we prove that also Σ1 has the expected number of moduli by using that every element of Σ1 is the branch locus of a triple plane. 2. On the number of moduli of complete irreducible families of plane sextics with six cusps First of all we want to find sufficient conditions in order that, if an irreducible component Σ of Σnk,d has the expected number of moduli, then every irreducible component Σ′ of Σnk′,d′ , containing Σ, has the expected number of moduli. In the corollary 4.7 of [4] we considered this problem under the hypothesis that Σ has the expected dimension and ρ(2, n, g) ≤ 0. Now we are interested to the case ρ > 0. We need the following local result. = {(a, b, x, y)| y2 = x3 + ax+ b} ⊂ C2 × A2 be the versal deformation family of an ordinary cusp (see [3] for the definition and properties of the versal deformation family of a plane singularity). We recall that the general curve of this family is smooth. The locus ∆ of C2 of the pairs (a, b) such that the corresponding curve is singular, has equation 27b2 = 4a3. For (a, b) ∈ ∆ and (a, b) 6= (0, 0), the corresponding curve has a node and no other singularities, whereas (0, 0) is the only point parametrizing a cuspidal curve. ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 3 Lemma 2.1 ([3], page 129.). Let G → C2 be a two parameter family of curves of genus g ≥ 2, whose general fibre is stable and which is locally given by y2 = x3 + ax + b, with (a, b) ∈ C2 and let D ⊂ C2 be a curve passing through (0, 0) and not tangent to the axis b = 0 at (0, 0). Then the j-invariant of the elliptic tail of the curve which corresponds to the stable limit of G(0,0), with respect to the curve D, doesn’t depend on D. Otherwise, for every j0 ∈ C, there exists a curve Dj0 ⊂ C2 passing through (0, 0) and tangent to the axis b = 0 at this point, such that the elliptic tail of the stable reduction of G(0,0) with respect to Dj0 , has j-invariant equal to j0. Proposition 2.2. Let Σ ⊂ Σnk,d, with k < 3n, be an irreducible component of Σnk,d. Let g be the geometric genus of the plane curve corresponding to the general element of Σ. Suppose that g ≥ 2, ρ(2, g, n)− k ≤ 0 and Σ has the expected number of moduli equal to 3g − 3 + ρ− k. Then, every irreducible component Σ′ of Σnk′,d′ , with k′ = k− 1 and d = d′ or k = k′ and d′ = d− 1, such that Σ ⊂ Σ′, has expected number of moduli. Proof. First we consider the case k′ = k − 1 and d = d′. Let q1, . . . , qk be the cusps of Γ. It is well known that, since k < 3n then [Γ] ∈ Σnk−1,d. In particular, for every fixed cusp qi of Γ there exists an irreducible analytic branch Si of Σnk−1,d passing through the point [Γ] and whose general point corresponds to a plane curve Γ′ of degree n with d nodes and k − 1 cusps specializing to the singular points of Γ different from qi, as Γ ′ specializes to Γ. Moreover, it is possible to prove that every Si is smooth at the point [Γ], see [7] or chapter 2 of [5]. Let Σ′ be one of the irreducible components of Σnk−1,d containing Σ. Notice that the general element of Σ′ corresponds to a curves of genus g′ = g + 1. Since ρ(2, g′, n) − k′ = 3n− 2g − 2− 6− k + 1 = ρ(2, g, n)− k − 1 < 0, in order to prove the theorem it is enough to show that the general fibre of the moduli map ΠΣ′ : Σ 99K Mg+1 has dimension equal to eight. Let us notice that the map ΠΣ′ is not defined at the general element [Γ] of Σ. More precisely, let γ ⊂ Si ⊂ Σ′ be a curve passing through [Γ] and not contained in Σ. Let C → γ be the tautological family of plane curves parametrized by γ. Let C′ → γ be the family obtained from C → γ by normalizing the total space. The general fibre of C′ → γ is a smooth curve of genus g+1, while the special fibre C′0 := Γ′ is the partial normalization of Γ obtained by smoothing all the singular points of Γ, except the marked cusp qi. If we restrict the moduli map ΠΣ′ to γ, we get a regular map which associates to [Γ] the point corresponding to the stable reduction of Γ with respect to the family C′ → γ, which is the union of the normalization curve C of Γ and an elliptic curve, intersecting at the point q ∈ C which maps to the cusp qi ∈ Γ. Now, let G ⊂ Σ′ × Mg+1 be the graph of ΠΣ′ , let π1 : G → Σ′ and π2 : G → Mg+1 be the natural projections and let U ⊂ Σ be the open set parametrizing curves of degree n and genus g with exactly k cusps and d nodes as singularities. From what we observed before, if we denote by ΠΣ′(Σ) the Zariski closure in Mg+1 of π2π−11 (U), then ΠΣ′(Σ) is contained in the divisor ∆1 ⊂ Mg+1, whose points are isomorphism classes of reducible curves which are union of a smooth curve of genus g and an elliptic curve, meeting at a point. Denoting by ΠΣ : Σ → Mg the moduli map of Σ, the rational map ∆1 99K Mg which forgets the elliptic tail, restricts to a rational dominant map q : ΠΣ′(Σ) 99K ΠΣ(Σ). The dimension of the general fibre of q is at most two. Since, by hypothesis, the dimension of the fibre of the moduli map ΠΣ is eight, there exists only a finite 4 CONCETTINA GALATI number of g2n on C, ramified at k points, which maps C to a plane curve D such that the associated point [D] ∈ P n(n+3) 2 belongs to Σ. In particular, the set of points x of C such that there is a g2n with k simple ramification points, one of which at x, is finite. So, the dimension of the general fibre of q is at most one. In order to decide if the general fibre of q has dimension zero or one, we have to understand how the j-invariant of the elliptic tail of the stable reduction of Γ′ with respect the family C′ → γ, depends on γ. If C → C2 is the étale versal deformation family of the cusp. By versality, for every fixed cusp pi of Γ, there exist étale neighborhoods n(n+3) 2 of [Γ] in P n(n+3) 2 , V v→ C2 of (0, 0) in C2 and Ui of pi in the tautological family U → P n(n+3) 2 with a morphism f : U → V such that the family Ui → U is the pullback, with respect to f , of the restriction to V of the versal family. By the properties of the étale versal deformation family of a plane singularity, (see [2]), we have that f−1((0, 0)) is an étale neighborhood of [Γ] in the (smooth) analytic branch Σn1,0 whose general element corresponds to an irreducible plane curves with only one cusp at a neighborhood of the cusp qi of Γ. So, dim(f −1((0, 0))) = n(n+3) − 2 and the map f is surjective. Moreover, if g is the restriction of f at u−1(Σ′), then also g is surjective. Indeed, g−1((0, 0)) = f−1((0, 0)) ∩ u−1(Σ′) = u−1((Σ)) and, since k < 3n, then dim(Σ) = 3n + g − 1 − k = dim(Σ′) − 2 and g is sur- jective. By using lemma 2.1, it follows that the general fibre of the natural map ΠΣ′(Σ) → ΠΣ(Σ) has dimension exactly equal to one. Therefore, dim(ΠΣ′ (Σ)) = dim(ΠΣ(Σ)) + 1 = 3g − 3 + ρ(2, g, n)− k + 1 = 3(g + 1)− 3 + ρ(2, g + 1, n)− k By using that dim(ΠΣ′ (Σ ′)) ≥ dim(ΠΣ′(Σ)) + 1 = 3(g + 1)− 3 + ρ(2, g + 1, n)− k + 1. and by recalling that, by lemma 2.2 of [4], it is always true that dim(ΠΣ′(Σ ′)) ≤ 3(g + 1)− 3 + ρ(2, g + 1, n)− k + 1, the statement is proved in the case k′ = k − 1 and d′ = d. Suppose, now, that k = k′ and d′ = d − 1. Also in this case Σ is not contained in the regularity domain of ΠΣ′ . More precisely, if [Γ] ∈ Σ is general, then ΠΣ′([Γ]) consists of a finite number of points, corresponding to the isomorphism classes of the partial normalizations of Γ obtained by smoothing all the singular points of Γ, except for a node. Then ΠΣ′(Σ) is contained in the divisor ∆0 of Mg+1 parametrizing the isomorphism classes of the analytic curves of arithmetic genus g + 1 with a node and no more singularities. The natural map ∆0 99K Mg sending the general point [C′] of ∆0 to the isomorphism class of the normalization of C restricts to a rational dominant map q : ΠΣ′(Σ) 99K ΠΣ(Σ). Since we suppose that Σ has the expected number of moduli and ρ(2, g, n)−k ≤ 0, if C is the normalization of the plane curve corresponding to the general element of Σ, then the set S of the linear series of dimension 2 and degree n on C with k simple ramification points, mapping C to a plane curve D such that the associated point [D] in the Hilbert Scheme belongs to Σ, is finite. We deduce that also the set S′ of the pairs of points (p1, p2) of C, such that there is a g n ∈ S such that the associated morphism maps p1 and p2 to the same point of the plane, is finite. So, also q −1([C]) is finite and dim(ΠΣ′ (Σ)) = dim(ΠΣ(Σ)). It follows that dim(ΠΣ′ (Σ ′)) ≥ 3g− 3+ 3n− 2g− 6− k+1 = 3(g+1)− 3+ 3n− 2(g+1)− 6− k. Remark 2.3. Notice that, the arguments used before to prove lemma 2.2 don’t work if the dimension of the general fibre of the moduli map of Σ has dimension bigger than eight. Indeed, in this case, the dimension of the general fibre of the map ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 5 ΠΣ′(Σ) 99K ΠΣ(Σ) could be bigger than one if k ′ = k − 1 and d = d′, or than zero if k′ = k and d = d′ − 1. Corollary 2.4. There exists at least one irreducible component Σ2 of Σ 6,0 having the expected number of moduli equal to dim(M4) − 2 and whose general element corresponds to a sextic with six cusps not on a conic. Remark 2.5. As we already observed in the previous section, Σ2 is the only com- ponent of Σ66,0 parametrizing sextics with six cusps not on a conic by [1]. Proof. Let Σ69,0 be the variety of elliptic plane curves of degree six with nine cusps and no more singularities. It is not empty and irreducible, because, by the Plücker formulas, the family of dual curves is Σ30,0 ≃ P9, which is irreducible and not empty. Moreover, if we compose an holomorphic map φ : C → P2 from a complex torus C to a smooth plane cubic with the natural map φ(C) → φ(C)∗, where we denoted by the dual curve of φ(C), we get a morphism from C to a plane sextic with nine cusps. Therefore, the number of moduli of Σ69,0 is equal of the number of moduli of Σ30,0, equal to one. Since 6 < 3n = 18, there is at least one irreducible component Σ′ of Σ68,0 containing Σ 9,0. Let ΠΣ′ : Σ 99K M2 be the moduli map of Σ′ and let G ⊂ Σ′ × M2 be its graph. If we denote by π1 : G → Σ′ and π2 : G → M2 the natural projection, by U the open set of Σ69,0 parametrizing cubics of genus one with nine cusps and by ΠΣ′(Σ 9,0) the Zariski closure in M2 of π2π−11 (U), then, by arguing as in the first part of the proof of the lemma 2.2, we have a dominant map ΠΣ′(Σ 9,0) 99K M1, whose general fibre has dimension one. We conclude that dim(πΣ′ (Σ ′)) ≥ dim(πΣ′(Σ69,0)) + 1 = 3 and so, the moduli map of Σ′ is dominant, as one expects, because ρ(2, 2, 6)− 8 = 18− 4− 6− 8 = 0. Let D be the plane sextic corresponding to the general point of Σ′. By Bezout theorem, the height cusps P1, . . . , P8 of D don’t belong to a conic and, however we choose five cusps of D, no four of them lie on a line. Then, let C2 be the unique conic containing P1, . . . , P5. There exists at least a cusp, say P6, which does not belong to C2. Since 8 < 3n = 18, there exists a family of plane sextics D → ∆, whose special fibre is D and whose general fibre has a cusp at a neighborhood of every cusp of D different from P7 and no further singularities. By lemma 2.2, the curve ∆ is contained in an irreducible component of Σ67,0 with expected number of moduli. By repeating the same argument for the general fibre of the family D → ∆ we get an irreducible component Σ2 of Σ66,0 with the expected number of moduli and whose general element parametrizes a sextic with six cusps not on a conic. � Now we consider the irreducible component Σ1 of Σ 6,0 parametrizing plane curves of equation f23 (x0, x1, x2) + f 2 (x0, x1, x2) = 0, where f2 is an homogeneous poly- nomial of degree two and f3 is an homogeneous polynomial of degree three. The general element of Σ1 corresponds to an irreducible plane curve of degree six with six cusps on a conic. We want to show that Σ1 has the expected number of moduli equal to 12− 3 + ρ(2, 4, 6)− 6 = 7 = dim(M4)− 2. Equivalently, we want to show that the general fibre of the moduli map Σ1 99K M4 has dimension equal to eight. Lemma 2.6. Let Γ2 : f2(x0, x1, x2) = 0 and Γ3 : f3(x0, x1, x2) = 0 be a smooth conic and a smooth cubic intersecting transversally. Then, the plane curve Γ : f23 (x0, x1, x2)− f32 (x0, x1, x2) = 0 is an irreducible sextic of genus four with six cusps at the intersection points of Γ2 and Γ3 as singularities. The curve Γ is projection of a canonical curve C ⊂ P3 from 6 CONCETTINA GALATI a point p ∈ P3 which is contained in six tangent lines to C. Moreover, for every point q ∈ P3 −C such that the projection plane curve πq(C) of C from q is a sextic with six cusps on a conic of equation g23(x0, x1, x2) − g32(x0, x1, x2) = 0, where g3 and g2 are two homogeneous polynomials of degree three and two respectively, there exists a cubic surface S3 ∈ |IC|P3(3)|, containing C, such that the plane curve πq(C) is the branch locus of the projection πq : S3 → P2. Remark 2.7. Notice that, by [1], every irreducible sextic with six cusps on a conic as singularities has equation given by (f2(x0, x1, x2)) 3 + (f3(x0, x1, x2)) 2 = 0, with f2 and f3 homogeneous polynomials of degree two and three. In order words, all the sextics with six cusps on a conic as singularities are parametrized by points of Σ1. An other proof of this result as been provided to us by G. Pareschi. Proof of lemma 2.6. Let f(x0, x1, x2) = f 3 (x0, x1, x2) − f32 (x0, x1, x2) = 0 be the equation of Γ. From the relation f3(x) = ±f2(x) f2(x), we deduce that (x) = ±2∂f2 f2(x) and hence (x) = 2 (x)f3(x)− 3f2(x)2 (x) = −f2(x)2 By using that the conic Γ2 : f2 = 0 is smooth, it follows that, if a point x ∈ Γ is singular, then x ∈ Γ2 and hence x ∈ Γ3∩Γ2. On the other hand, always from (2), if x ∈ Γ2∩Γ3, then x is a singular point of Γ. Hence, the singular locus of Γ coincides with Γ3 ∩ Γ2. Let x be a singular point of Γ. If p1(x, y) + terms of degree two = 0 q1(x, y) + terms of degree ≥ two = 0 are respectively affine equations of Γ2 and Γ3 at x, then, the affine equation of Γ at x is given by q1(x, y) 2 − p1(x, y)3 + terms of degree ≥ four = 0. Since Γ2 and Γ3 intersect transversally, we have that q1(x, y) does not divide p1(x, y) and hence Γ has an ordinary cusp at x. Let now φ : C → Γ be the normalization of Γ. We recall that the cubics passing through the six cusps of Γ cut out on C the complete canonical series |ωC |. Since the cusps of Γ is contained in the conic Γ2 ⊂ P2 of equation f2 = 0, the lines of P 2 cut out on C a subseries g ⊂ |ωC | of dimension two of the canonical series. Moreover, if we still denote by C a canonical model of C in P3, then the linear series g is cut out on C in P3 from a two dimensional family of hyperplanes passing through a point p ∈ P3−C. If we project C from p we get a plane curve projectively equivalent to Γ. Since Γ has six cusps as singularities, we deduce that there are six tangent lines to C passing through p. To see that Γ is the branch locus of a triple plane, let S3 ⊂ P3 be the cubic surface of equation F3(x0, . . . , x3) = x 3 − 3f2(x0, x1, x2)x3 + 2f3(x0, x1, x2) = 0. If p = [0, 0, 0, 1], then, by using Implicit Function Theorem, the ramification locus of the projection πp : S3 → P2, is given by the intersection of S3 with the quadric S2 of equation = x23 − f2(x0, x1, x2) = 0. Now, if x = [x0, x1, x2] ∈ S3 ∩ S2, then x3 = ± f2(x0, x1, x2). By substituting in the equation of S3, we find that the branch locus of the projection πp : S3 → P2 coincides with the plane curve Γ. From what we proved before, it follows that the ramification locus of the projection map πp : S3 → P2 is the normalization curve C of Γ. Finally, if q ∈ P3 − C is an other point such that the plane projection πq(C) is an irreducible sextic with six cusps on a conic parametrized by a point xq ∈ Σ1 ⊂ P27, then, up to projective motion, we may always assume that q = [0 : 0 : 0 : 1] and hence, if g23(x0, x1, x2)− g32(x0, x1, x2) = 0 ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 7 is the equation of the plane curve πq(C), then C is the locus of ramification of the projection from q to the plane of the cubic surface of equation x33 − 3g2(x0, x1, x2)x3 + 2g3(x0, x1, x2) = 0. Corollary 2.8. The irreducible component Σ1 of Σ 6,0 parametrizing plane curves of equation f23 (x0, x1, x2)+ f 2 (x0, x1, x2) = 0, where f2 is an homogeneous polynomial of degree two and f3 is an homogeneous polynomial of degree three, has the expected number of moduli equal to 7 = dim(M4) + ρ(2, 4, 6)− 6. Proof. Let [Γ] ⊂ P2 be a plane sextic of equation f23 (x0, x1, x2) − f32 (x0, x1, x2) = 0, where the conic f2 = 0 and the cubic f3 = 0 are smooth and they intersect transversally. Let C ⊂ P3 be the normalization curve of Γ and let SC be the set of points v = [v0 : · · · : v3] ∈ P3 such that there exists a cubic surface S3 ∈ |IC|P3(3)|, containing C, such that the curve C is the ramification locus of the projection πv : S3 → P2. By the former lemma, in order to prove that Σ1 has the expected number of moduli, it is enough to find a point [Γ] of Σ1 corresponding to an irreducible plane sextic Γ ⊂ P2 with six cusps of a conic such that the set SC is finite. Let Γ2 be the smooth conic of equation f2(x0, x1, x2) = x 1 − x22 = 0 and let Γ3 be the smooth cubic of equation f3(x0, x1, x2) = x 0+x0x 2−x21x2 = 0. If a1, a2 and a3 are the three different solutions of the polynomial x3 + x2 + x− 1 = 0, then Γ2 and Γ3 intersect transversally at the points [ai, ai, 1], [ai,− ai, 1], with i = 1, 2, 3. By the former lemma, the plane sextic Γ of equation f32 − f23 = 0 is irreducible and it has six cusps at the intersection points of Γ2 and Γ3 as singularities. Moreover, the normalization curve C of Γ is the canonical curve of genus 4 in P3 which is intersection of the cubic surface S3 ⊂ P3 of equation F3(x0, x1, x2, x3) = x 3 + (x 0 + x 1 − x22)x3 + x30 + x0x22 − x21x2 = 0 and the quadric S2 of equation = 3x23 + x 0 + x 1 − x22 = 0. We want to show that SC is finite. To see this we observe that, since h0(P3, IC|P3(2)) = 1 and h0(P3, IC|P3(3)) = 5, the equation of every cubic surface containing C and which is not the union of S2 and an hyperplane is given by G(x0, . . . , x3;β0, . . . , β3) = F3(x0, x1, x2, x3) + ∂F3(x0, x1, x2, x3) with βj ∈ C, for i = 0, . . . , 3. Now, a point [v] = [v0, . . . , v3] ∈ SC if and only if there exist β0, . . . , β3 such that C is contained in the intersection of G(x0, . . . , x3;β0, . . . , β3) = 0 and ∂G(x0, . . . , x3;β0, . . . , β3) Still using that h0(P3, IC|P3(2)) = 1, a point [v] ∈ P3 belongs to SC if and only if ∂G(x0, . . . , x3;β0, . . . , β3) ∂F3(x0, . . . , x3) for some λ ∈ R− 0, or, equivalently, βjxj) ∂x3∂xi = (λ− viβi − v3) 8 CONCETTINA GALATI The previous equality of polynomials is equivalent to the following bilinear system of ten equations in the variables v0, . . . , v3 and β0, . . . , β3 (1 + β3)v0 + 3β0v3 = 0 (x0x3) (1 + β3)v1 + 3β1v3 = 0 (x1x3) (1 + β3)v2 − 3β2v3 = 0 (x2x3) β1v0 + β0v1 = 0 (x0x1) β2v0 + (1− β0)v2 = 0 (x0x2) (1 − β2)v1 + β1v2 = 0 (x1x2) 2β1v1 − v2 = λ− j=0 βjvj − v3 (x21) −v0 + 2β2v2 = λ− j=0 βjvj − v3 (x22) (3 + 2β0)v0 = λ− j=0 βjvj − v3 (x20) 2β3v3 = λ− j=0 βjvj − v3 (x23) The points of SC are the solutions v of the previous linear system, as a linear system whose coefficients depend on β0, . . . , β3. It is easy to prove that it has only a solution equal to (v0, v1, v2, v3) = (0, 0, 0, λ) if β0 = β1 = β2 = β3 = 0 and it has not solutions otherwise, (see [5], page 98). By the previous lemma, we conclude that the point [0 : 0 : 0 : 1] ∈ P3 is the only point which belongs to six tangent lines to the canonical curve C ⊂ P3 which is intersection of the cubic surface of equation F3(x0, x1, x2, x3) = x 3 + (x 0 + x 1 − x22)x3 + x30 + x0x22 − x21x2 = 0 and the quadric of equation = 3x23 + x 0 + x 1 − x22 = 0. It follows that, on the normalization curve D of the plane curve Γ′ corresponding to the general point of Σ1 ⊂ Σ66,0 there exists only a finite number of linear series of dimension two with six ramification points. � Remark 2.9. By using the notation introduced in the proof of corollary 2.8, we observe that in this corollary we have proved that if C is a general canonical curve of genus four such that the set SC is not empty, then SC is finite. Actually, C. Ciliberto pointed out to our attention that it is possible to show, with a very simple argument, that for every canonical curve C of genus four such that SC is not empty, we have that SC is finite. Finally, we observe that, by remark 2.7, for every canonical curve C of genus four, the set SC coincides with the set of points of P3 which are contained in six tangent lines to C. Acknowledgment. The results of this paper are contained in my PhD-thesis. I would like to thank my advisor C. Ciliberto for introducing me into the subject and for providing me very useful suggestions. I have also enjoyed and benefited from conversation with G. Pareschi and my college M. Pacini. References [1] A. Degtyarev: On deformations of singular plane sextics, math.AG/0511379, appearing on Journal of Algebraic Geometry. [2] S. Diaz and J. Harris: Ideals associated to deformations of singular plane curves, Transactions of the American Mathematical Society, vol. 309, n. 2, 433–468 (1988). [3] J. Harris and I. Morrison: Moduli of curves, Graduate texts in mathematics, vol. 187, Springer, New York, 1988. [4] C. Galati: Number of moduli of irreducible families of plane curves with nodes and cusps, Collect. Math. 57,3 (2006), 319-346. [5] C. Galati: Number of moduli of plane curves with nodes and cusps, PhD the- sis, Università degli Studi di Tor Vergata, 2004-2005, available on the homepage http://dspace.uniroma2.it/dspace/items-by-author?author=Galati http://arxiv.org/abs/math/0511379 http://dspace.uniroma2.it/dspace/items-by-author?author=Galati ON THE NUMBER OF MODULI OF PLANE SEXTICS WITH SIX CUSPS 9 [6] E. Sernesi: On the existence of certain families of curves, curves. Invent. Math. vol. 75 (1984), no. 1, 25–57. [7] O. Zariski: Dimension theoretic characterization of maximal irreducible sistems of plane nodal curves, Amer. J. Math. vol. 104 (1982), no. 1, 209–226. [8] O. Zariski: Algebraic surfaces, Classics in mathematics, Springer. Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende (CS) E-mail address: [email protected] 1. Introduction 2. On the number of moduli of complete irreducible families of plane sextics with six cusps Acknowledgment References

0704.0623

Algorithm for anisotropic diffusion in hydrogen-bonded networks

Algorithm for anisotropic diffusion in hydrogen-bonded networks. Edoardo Milotti Dipartimento di Fisica, Università di Trieste, and INFN – Sezione di Trieste, Via Valerio, 2, I-34127 Trieste, Italy∗ (Dated: November 4, 2018) Abstract In this paper I describe a specialized algorithm for anisotropic diffusion determined by a field of transition rates. The algorithm can be used to describe some interesting forms of diffusion that occur in the study of proton motion in a network of hydrogen bonds. The algorithm produces data that require a nonstandard method of spectral analysis which is also developed here. Finally, I apply the algorithm to a simple specific example. PACS numbers: 05.40.-a, 02.70.-c, 83.10.Rs ∗Electronic address: [email protected] http://arxiv.org/abs/0704.0623v1 mailto:[email protected] I. INTRODUCTION Protons migrating in water have an anomalously high mobility [1] and their diffusion is actually limited by the continuous rearrangement of hydrogen bonds [1, 2]. Indeed protons migrating in ice move faster than protons in water, as the transition rate from one water molecule to the next is enhanced by the higher molecular order in ice. Proton mobility increases whenever water molecules are constrained, as in carbon nanotubes [3, 4]. Local electric fields also orientate water molecules, and thus should lead to a local increase of proton mobility, and indeed it is now known that there is a definite water dipole orientational order in the hydration water close to ionizable residues in hydrated proteins [5, 6, 7, 8]: this is a collective property, which is somewhat independent of the individual fluctuations of the water dipoles. Here I am not concerned with the detailed simulation of proton motion which is the sub- ject of several specialized papers like [9, 10, 11], but rather I wish to set up the framework for a simulation of the random walks performed by protons in some interesting context, like proton migration on the surface of hydrated proteins [13]. The basic idea is that protons move faster in the network of hydrogen bonds just where there is a higher molecular order, i.e., the transition rate is higher where there is higher spatial order, and because of the continuous rearrangement of the water molecules which make up a fluctuating bond struc- ture, the random walk performed by a single proton can actually be viewed as a walk in continuous space and continuous time, as long as the time resolution of the process is longer than the relaxation time of water dipole motion. Here I take for granted that there is some induced order in the hydrogen bond network, like the dipole field described in [5, 6], and I introduce a corresponding field of transition rates γ(r, t), such that the time-dependent probability density ρ(r, t) and the associated probability ∆p = ρ∆V of finding a random walker (a proton) in the small volume ∆V at position r and time t, yield the following equation for the decrease of ∆p, due to random walker escape from the region, ∆p(r, t) = −γ(r, t)∆p(r, t) (1) I also assume that γ(r, t) is a continuous, differentiable function. The situation is illustrated in figure 1, which shows a random subdivision of a plane region: a set of positions – marked by the large black dots – is associated to small surrounding regions; the arrows in the figure mark the flow of random walkers in the central region to and from the bordering regions. If the area (actually, the line length in this 2D representation) of the interface between the central region and the kj-th region is Ai,kj , and the total interface area of the kj-th region is Akj , then it is easy to see that the total derivative of ∆pi is = −γi∆pi + Ai,kj γkj∆pkj (2) and the global flow in this discretized system is described by a system of coupled linear differential equations. More importantly, we can define currents for the inflow and outflow of random walkers from a modified form of Fick’s law Ji→k = α γiρi − γkρk where ∆i,k denotes the distance between the centroids of the bordering i-th and k-th region, and the parameter α is akin to the diffusion coefficient, but is measured in different units (it has the dimensions of a surface) [14]. The current in the previous formula is actually a projection along the direction that connects the centroids of the bordering region and it is easy to generalize to the continuum case and find the outflowing current J = −α∇(γρ) (4) so that finally one finds the following Fokker-Planck equation from the conservation of the total number of random walkers = α∇2(γρ) (5) assuming that α does not depend on position. In the following sections I describe an algorithm to simulate this kind of diffusive motion: first I discuss the angular distribution, then confinement to motion on surfaces, and in section IV I show how to extend the algorithm for asynchronous updates. In section V I give a recipe to analyze asynchronous data. In section VI I discuss a simple example, and finally in section VII I give a short summary and outlook for the utilization of the algorithm. II. ANGULAR DISTRIBUTION From equation (4) we see that the current actually contains two contributions J = −α∇(γρ) = −α (ρ∇γ + γ∇ρ)) (6) however when we consider the problem at hand – namely, the diffusion of protons in the network of hydrogen bonds, and we remark that we wish to describe the individual proton motion, then we notice that we are only interested in situations where ∇ρ = 0. In fact, pro- tons repel other protons that are too close, and obey a sort of effective exclusion principle – which is actually independent of their fermionic nature – and the position of the individual proton corresponds to a peak of the instantaneous probability density: therefore the current defined in (4) has the same direction as ∇γ in all cases of practical interest. This direction corresponds to the average proton motion, but for a single transition to a nearby site it can only define the axis of an angular probability distribution. Here I make the simplest possi- ble choice, namely that the angular probability distribution is a simple dipole distribution defined by the normalized conditional probability density for the unit vector n P (n|n0) = (1 + ∆Pn · n0) (7) where I0 is the normalization factor (I0 = 2π in the 2D case and I0 = 4π in the 3D case), and n0 is the unit vector n0(r) = so that the decrease of the density ρ due to the flow in the angular range ∆Ω, during a given time interval ∆t, is (1 + ∆Pn · n0)∆t∆Ω (9) The constant inside the parenthesis corresponds to the isotropic loss term, while the other term is associated to the current (6). From a comparison of the elementary flows of random walkers in direction n we find J · n = γρ n · n0 (10) so that ∆P = I0α and the conditional angular probability density is P (n|n0) = 1 + I0α n · n0 The conditional angular probability density (12) can be used to generate random walks discarding the time information. Here I take the following time-independent expression for the transition rate γ(r, t) = γ(r) = |r|2 + Γ2 +B (13) which has an obvious symmetry center, located in the origin, which corresponds to the po- sition of the peak value as well. This transition rate is motivated by the considerations put forward in the introduction: if protons migrate in a hydrogen bonded network with polariza- tion centers that create partial ice-like order in their neighborhood, then the transition rate (13) is highest, and saturates, close to the polarization centers, and decays to a constant value with a 1/r2 behavior for r ≫ Γ (i.e., it has a radial dependence like the potential of electric dipole fields). Notice also that the anisotropy coefficient is |r|2 + Γ2 The techniques to generate random angles which are distributed according to the probability density (12) are reviewed in appendix A, and figures 2-5 show some examples: in these examples all length and distance units are in arbitrary units. Figure 2 shows random walks around a single center with transition rate (13): the random walker starts at the origin, with a fixed step length = 0.005 arbitrary units; the horizontal and vertical scales are also labeled with the same arbitrary length units; the parameters of the transition rate function are the same in these simulations, A = 1, B = 0.1, and Γ = 1, while α changes in the three cases displayed in the figure. Larger values of α correspond to higher anisotropy, and we see that as the anisotropy grows, the random walk becomes more and more compact. Figure 3 shows a random walk with two centers at positions r1 = (−1, 0), r2 = (1, 0) (arbitrary units): the random walker starts at the origin, with a fixed step length = 0.01 arbitrary units; the horizontal and vertical scales are also labeled with the same arbitrary length units. In this case the transition rate is similar to (13), but with two centers, γ(r) = |r− r1|2 + Γ21 |r− r2|2 + Γ22 +B (15) with A1 = A2 = 1, B = 0.1, and Γ1 = Γ2 = 1, and α = 1. Here the random walker explores the regions around both centers. Figure 4 shows a situation which is similar to figure 3, although it is more complex. The transition rate is once again similar to (13), but now it has ten centers, γ(r) = |r− rk|2 + Γ2k +B (16) with Ak = 1, B = 0.1, and Γk = 0.1, and α = 0.025; the step length is = 0.01 arbitrary units. The centers are scattered randomly, with a lower bound on the minimum distance between them; the figure shows three snapshots at different times in the simulation, as the random walker starts from the center of the figure, drifts to one of the centers and later migrates to other neighboring centers. Finally figure 5 shows a random walk in space about two centers at r1 = (−1, 0, 0), r2 = (1, 0, 0) (arbitrary units), which is very similar to the random walk in figure 3: the transition rate is still given by expression (15), with A1 = A2 = 1, B = 0.1, and Γ1 = Γ2 = 1, and α = 0.5, with a fixed step length = 0.1 arbitrary units. Once again the random walker explores the regions around both centers. III. DIFFUSION ON SURFACES In many cases it is important to confine the motion of the random walkers to some particular portion of space, for instance in the case of protons on hydrated proteins the motion is confined to the thin hydration layer. The simulation method outlined in the previous section can be adapted to provide such a confinement to a surface: in this case one can define at each step the tangent plane at the position of the random walker and proceed as in the 2D case. Obviously the gradient of the transition rate γ used in the formulas of the previous section must be projected on the tangent plane, and moreover the directions must be generated according to the 2D angular distribution (see appendix B). Figure 6 shows a random walk on a spherical surface with 10 centers as in the examples in the previous section: the sphere has radius 1 (arb. units), the transition rate is given by expression (16), with Ak = 10, B = 1, and Γk = 0.25, and α = 0.2; the step length is = 0.1 arbitrary units. IV. ASYNCHRONOUS TRANSITIONS In the previous sections we have discussed the space behavior of the random walks, but obviously we can use the transition rate function γ(r, t) to describe the time behavior as well. It is possible to choose a fixed time step ∆t and use the transition rates to compute the probability of generating a transition in the time interval (t, t+∆t) (synchronous update), however this is inconvenient if the function γ(r, t) spans a wide range of values, because it means that the choice ∆t ≪ min(1/γ(r, t)) which is required for an accurate simulation, produces very long waiting times where the transition rate is very small (and therefore, very large amounts of sampled data). It is actually much more practical to use the transition rate to generate directly the transition times of each step, which we assume to be indepen- dent (asynchronous update) from one another. With this – rather natural – assumption of independency, it is very simple to generate the transition times, as explained in appendix B, although this leads to uneven sampling, and requires a specialized form of spectral analysis. V. FOURIER ANALYSIS OF ASYNCHRONOUS DATA Using asynchronous sampling times it is not possible to use the standard Fourier or other similar spectral analysis techniques [15]. However the signal produced by the time-domain simulation is “exact”, at least in the sense that there are no algorithmic artifacts due to sampling and it is desirable to extract as much information as possible. To this end, I notice that any function of the position of the random walkers must be stationary between successive transitions, and that it is possible to make direct use of the definition of Fourier transform F (ω) = s(t)e−iωtdt (17) where s(t) is any signal produced in the simulation, which depends on the positions of the random walkers (e.g., a component of the electric dipole moment if the random walkers are charged particles). The signal s(t) has the fixed value sn in the time interval (tn, tn + 1), where tn is the time of the n-th transition, and we find: F (ω) = ∫ tn+1 e−iωtdt −iω(tn+1+tn)/2(tn+1 − tn) sin [ω(tn+1 − tn)/2] ω(tn+1 − tn)/2 sn sin ω(tn+1 − tn) ω(tn+1 + tn) sn sin ω(tn+1 − tn) ω(tn+1 + tn) Using equation (19) the Fourier transform can be evaluated exactly for all frequencies, and without aliasing: in practice this is possible, practical, and actually useful only for a small finite set of frequencies. If we had used a Discrete Fourier Transform (DFT) algorithm [15] to analyze N real samples, we would have found N/2 independent Fourier coefficients, and using a Fast Fourier Transform (FFT) algorithm the time complexity of the calculation would be O(N logN). If we use the algorithm defined by equation (19) to compute M values (M ≤ N/2) of the Fourier transform, the time complexity is clearly O(NM), so that in a practical calculation we can only compute a reduced number of Fourier coefficients. However, I remark that in addition to being exact, the algorithm has another major advantage over the standard DFT calculations: there is no limitation to the set of frequencies that can be computed, and in particular one can choose a set of frequencies that is not evenly spaced and that is denser close to the origin, which is particularly useful in this case since the random walk – when considered as a noise process – is expected to produce a spectrum with a large power-law peak at low frequencies. This kind of analysis is actually limited by the finite time span of the generated signal: we see from equation (18) that the Fourier transform of the generated signal is a sum of sinc functions, and therefore it is not useful to represent the transform for frequencies lower than ωmin = π/T where T is the signal duration and ωmin is the lowest positive zero of the corresponding sinc function. With this limitation, we can sample the Fourier transform at frequency values that are evenly spaced on a logarithmic scale and obtain a better representation of the transform close to the origin than is possible with conventional methods. In this approach we evaluate the Fourier transform of the simulated signal in the time interval (0, tN) and we implicitly assume that the signal vanishes outside this interval: this is different from the standard (implicit) assumption in standard DFT analysis, where the observed signal is assumed to repeat periodically outside the observation interval [15]. If we introduce a rectangular window with a width equal to the observation interval (tN), we see that the present method returns a Fourier transform that is the convolution of the transform of the signal with the transform of the rectangular window, which is exp(−iωt)dt = tN exp(−iωtN/2) sin(ωtN/2) (ωtN/2) As a consequence of the convolution associated to the rectangular window we see that a constant nonzero level produces a sharp peak centered at zero frequency, with a shape given by equation (20); this peak corresponds to the standard DC peak in DFT analysis, and has tails with a 1/ω2 spectral behavior that may mimic the low-frequency behavior of a standard Debye relaxation with a very small decay rate. The mean level of the simulated signal is ∫ tn+1 sn (tn+1 − tn) (21) and thus we can correct for the DC peak by subtracting its transform exp(−iωtN/2) sin(ωtN/2) (22) from the signal transform. VI. RANDOM WALK ABOUT A SINGLE CENTER As an example, I consider here a complete simulation (3D space and time data) for random walks about a single center (as defined by the transition rate (13) ) located in the origin. The transition rate function used in this example is given again by expression (13), with A = 100, B = 10−8, Γ = 0.1, α = 0.001, and with a step length = 0.01. The values of A and B have been chosen to maximize the range of γ sampled by the random walker, while still keeping a rather short simulation time. I have generated 500 random walks and 10000 transitions for each walk; the random walker always starts at the origin (where the center of (13) is located). The results of the simulation are shown in figures 7-11: figure 7 is the superposition of a few walks, and it is qualitatively clear that the density profile is very similar to that of the standard random walk in the plane. Figure 8 shows instead the x projection of the position signal vs. time, and this is not very different, e.g., from an electric dipole component if the random walkers are charged particles. The insets show parts of the signal with increasing magnification, and the last inset displays clearly the stationary parts of the signal between successive transitions. Figure 9 shows the (unnormalized) distribution of time intervals between successive transitions: the figure demonstrates clearly that although the times have been generated according to the interval distribution in appendix B, the distribution in figure 9 is not a simple exponential, but rather it contains two different power-law regions (marked by the dotted lines in this log-log plot), which reflects the way in which the transition probability function is sampled by the random walks. One well-known property of ordinary random walks is that their mean square radius is proportional to time, i.e., 〈r2〉 ∝ t: here we see (figure 10) that this linearity is recovered only asymptotically, as random walkers explore regions that are far away from the origin. Finally, I have used the power spectral estimation method of section V to analyze the x position signals (as those in figure 8): the result is shown in figure 11. Figure 11a is the power spectrum obtained in a single realization of the random walk, while figure 11b is the average of 400 spectra. In each part of figure 11, the thin gray line represents an ideal power-law spectral density with the same slope as the the average of 400 spectra, i.e., a 1/f 2 spectrum, which is usually expected in these types of processes. In fact the random walks simulated here effectively sample asymptotically only a rather limited range of transition rates – even though the A/B ratio is quite high – and this means that the usual superposition argument that leads to more general power-law spectra [16] does not apply here and it is quite natural to find a 1/f 2 spectrum. VII. DISCUSSION Before concluding this paper it is important to note that the correct continuum formu- lation of the diffusion equation in an inhomogeneous environment has been the subject of much discussion in the past and is still debated (see [17, 18], and references therein). The generalization is unclear because the microscopic details seem to matter [17]. Moreover, there is also some interest towards the diffusion equation in various forms of anomalous diffusion [19]. Here I wish to stress again that the results presented in this paper are spe- cialized and are meant to address diffusion in structures like those described in [5, 6], unlike other approaches described in the existing literature that deal with more general diffusion problems [20, 21, 22]; still, the diffusion equation (5) is similar to equation (5) in reference [17], and therefore it is interesting to give one further look at its structure. In section II we have seen that the current (4) has two components, and the gradient term that generates the random walks discussed here roughly corresponds to the so-called “spurious” drift term (using the terminology of reference [17]). The other term in the current, namely −αγ∇ρ has been neglected because the single random walkers considered here are charged fermions and obey a sort of effective exclusion principle. Indeed, in a context like that of [5, 6] pro- tons repel because of their charge, while their spin structure, and therefore also their true fermionic character – and the Pauli exclusion principle – do not matter much; this effective exclusion principle has been used in the past for an Ising-like modeling of proton motion, where the presence or absence of a proton at a given position is treated like a pseudo-spin variable (see the discussion in the review paper [12]). The situation would be very different if space could be filled by a cloud of random walkers: in a case such as this – which roughly corresponds to random walkers that effectively behave as bosons – the neglected term should be included. Thus the actual importance of the different terms of the diffusion equation (5) depends on the bosonic or fermionic character of the random walkers. One prominently missing term in the diffusion equation (5) is the usual drift term asso- ciated to external fields, however if we look at the structure of the current (3), we see that we can easily produce a flow unbalance with a space- (and possibly time-) dependent α, so that we obtain a modified diffusion equation = ∇ · [α∇(γρ)] = ∇α · ∇(γρ) + α∇2(γρ) (23) with an additional α-dependent drift term. A final, important comment, is that the algorithm presented here has a sort of backward approach with respect to other existing algorithms for random walks and diffusion in inho- mogeneous environments, as it starts directly from the transition rates, instead of deriving them from a given diffusion equation (as, e.g., [20]). To conclude, in this paper I have described a novel algorithm for anisotropic diffusion, which is continuous both in space and time, and I have discussed its application to a simple example, in anticipation of further work that shall be carried out in a realistic simulation of noise in proton conduction in weakly hydrated proteins [23]. APPENDIX A: ANGULAR DISTRIBUTIONS Here I consider first the planar angular distribution defined by the normalized probability density P (ϕ; g) = (1 + g cosϕ) (A1) Using the standard inversion method (described in many textbooks, see, e.g., [24]), one finds that the solution ϕ of the nonlinear equation (ϕ+ g sinϕ) + = y (A2) has the distribution described by (A1) if y is a uniform variate on the (0, 1) interval. The generation of a random direction in space from the normalized probability density P (n; g) = (1 + gn · n0) (A3) requires two angles, a zenithal angle θ and azimuthal angle ϕ, where n0 defines the zenithal axis, so that the probability of finding a unit vector n in the interval (θ, θ+dθ), and (ϕ, ϕ+dϕ) dP (θ, ϕ; g) = (1 + g cos θ) d cos θ (1 + gx) dx i.e., the probability density is the product of two independent densities, one uniform with respect to ϕ on the (0, 2π) interval, and the other linear with respect to x = cos θ on the (−1, 1) interval. Using again the inversion method, one finds that has the required linear distribution if y is a uniform variate on the (0, 1) interval. Since n0 is not usually parallel to the z direction, two rotations are also required: one first rotates the reference frame so that n0 is parallel to the z axis, this is followed by the angle generation step, and finally one must transform back to the original reference frame. APPENDIX B: TIME DISTRIBUTION Time transitions are generated according to the exponential distribution, which has the probability density = γ exp(−γt) (B1) where γ is the transition rate. The standard inversion method [24] can be used again to generate times t = − ln y (B2) that are exponentially distributed if y is a uniform variate on the (0, 1) interval. ACKNOWLEDGMENTS I wish to thank warmly Giorgio Careri for his insightful comments and suggestions: he was the first to pinpoint the problem that spurred the research described here, and this paper would not have been written without his encouragement. I also wish to thank Alessio Del Fabbro for his careful reading of the manuscript and for several interesting discussions. [1] N. Agmon, Chem. Phys. Lett. 244 (1995) 456. [2] N. Agmon, J. Chim. Phys. (Paris) 93 (1996) 1714. [3] D. J. Mann and M. D. Halls, Phys. Rev. Lett. 90 (2003) 195503. [4] R. Jay Mashi, Sony Joseph, N. R. Aluru, and Eric Jakobsson, Nano Lett. 3 (2003) 589. [5] J. Higo, M Sasai, H. Shirai, H. Nakamura, and T. Kugimiya, PNAS 98 (2001) 5961. [6] J. Higo and M. Nakasako, J. Comput. Chem. 23 (2002) 1323. [7] T. Yokomizo, M. Nakasako, T. Yamazaki, H. Shindo, and J. Higo, Chem. Phys. Lett. 401 (2005) 332. [8] P. Kumar, G. Franzese, S. V. Buldyrev, and H. E. Stanley, Phys. Rev. E 73 (2006) 041505. [9] J. Halding and P. S. Lomdahl, Phys. Rev. A 37 (1988) 2608. [10] I. Chochliouros and J. Pouget, J. Phys.: Condensed Matter 7 (1995) 8741. [11] U. W. Schmitt and G. A. Voth, J. Chem. Phys. 111 (1999) 9361. [12] G. Careri, Prog. Biophys. Mol. Biol. 70 (1998) 223. [13] G. Careri and E. Milotti, Phys. Rev. E 67 (2003) 051923. [14] The distance ∆i,k in equation (3) in the text is defined exactly only in the continuous limit, however this is not important because the discrete formulation in section I is used only to introduce the diffusion equation and is not relevant for the algorithm. [15] S. M. Kay and S. L. Marple, Proc. IEEE 69 (1981) 1380. [16] M. B. Weissman, Rev. Mod. Phys. 60 (1988) 537. [17] N. G. van Kampen, J. Phys. Chem. Solids 49 (1988) 673. [18] M. Christensen and J. Boiden Pedersen, J. Chem. Phys. 119 (2003) 5171. [19] S. Sellers and J. A. Barker, Phys. Rev. E 74 (2006) 061103. [20] M. Christensen, J. Comp. Phys. 201 (2004) 421. [21] K. Kikuchi, M. Yoshida, T. Maekawa, and H. Watanabe, Chem. Phys. Lett. 185 (1991) 335. [22] K. Kikuchi, M. Yoshida, T. Maekawa, and H. Watanabe, Chem. Phys. Lett. 196 (1992) 57. [23] G. Careri and E. Milotti, in preparation. [24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery: “Numerical recipes in C: the art of scientific computing, 2nd edition”, section 7.2 (Cambridge Univ. Press, Cambridge, 1992). FIG. 1: Here space has been subdivided in elementary regions (the dots represent the centroids of the elementary regions): each region is characterized by a particular transition rate γi and by an elementary volume Vi. In this figure the transitions between the central region i and the adjacent regions (denoted here by an index kj) are marked by the gray arrows. FIG. 2: Random walks around a single center as explained in section II. The random walker starts at the origin, with a fixed step length = 0.005 arbitrary units; the horizontal and vertical scales are also labeled with the same arbitrary length units. The width parameter is Γ = 1 in all simulations, and the other parameters have fixed values as well, A = 1, B = 0.1, while a. α = 1; b. α = 10; c. α = 10. Larger values of α correspond to higher anisotropy, and we see here that as the anisotropy grows, the random walk becomes more and more compact. FIG. 3: Random walk with two centers, as explained in the text. The walk is superposed on the contour plot of the transition rate γ(r). The random walker starts at the origin, with a fixed step length = 0.01 arbitrary units; the horizontal and vertical scales are also labeled with the same arbitrary length units. FIG. 4: Random walk with several (10) centers, a. after 10000 steps; b. after 40000 steps; c. after 70000 steps. The walk is superposed on the contour plot of the transition rate γ(r). FIG. 5: Random walk in space with two centers, located at the positions marked by the gray arrows. FIG. 6: Random walk on a spherical surface, with several (10) centers, some of which have not yet been visited by the random walker. FIG. 7: Superposition of several random walks about a single center in the origin. Isodensity contour lines are superposed on the density plot. FIG. 8: This figure shows the x projection of the position signal (this is not very different from an electric dipole component if the random walkers are charged particles) vs. time for one of the random walks in the example of section VI; both position and time are in arbitrary units. The insets show parts of the signal with increasing magnification, and the last inset displays clearly the stationary parts of the signal between successive transitions. FIG. 9: Unnormalized distribution of the time intervals ∆t between transitions in the set of 400 random walks of 10000 steps each described in the text: a. logarithm of the relative frequency vs. ∆t (both in arbitrary units), which shows that the distribution is not a simple exponential, but rather contains two different power-law regions (dotted lines); b. log-log plot of the same distribution, where the two power laws are identified by the nearly straight sections. FIG. 10: Mean squared distance 〈r2〉 vs. time. In an ordinary random walk the mean squared distance is a linear function of time: here we see that linearity is recovered only asymptotically, as the random walkers explore regions that are further away from the origin, where the γ is nearly constant. FIG. 11: Spectral density calculated with the method of section V, sampled at logarithmically spaced frequencies: a. spectrum obtained from the x position signal for a single random walk; b. average of 400 spectra. Spectral densities and frequencies are in arbitrary units. The thin gray line represents an ideal 1/f2 spectrum, which is expected for this kind of processes: the computed spectrum deviates from the ideal spectrum only at very low frequency, because of the limited observation time. Notice also that there is no upward bend at very high frequency – a hint of the absence of aliasing. In part a. it is clearly visible that the spectrum is sampled at (500) logarithmically spaced frequencies, because there is no crowding at the high end of the spectrum, and no rarefaction at low frequency, unlike spectra obtained with the Fast Fourier Transform or other similar algorithms. Introduction Angular distribution Diffusion on surfaces Asynchronous transitions Fourier analysis of asynchronous data Random walk about a single center Discussion Angular distributions Time distribution Acknowledgments References

0704.0624

Temporal Evolution of Step-Edge Fluctuations Under Electromigration Conditions

Temporal Evolution Of Step-Edge Fluctuations Under Electromigration Conditions P.J. Rous∗ and T.W. Bole Department of Physics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 (Dated: October 31, 2018) The temporal evolution of step-edge fluctuations under electromigration conditions is analysed using a continuum Langevin model. If the electromigration driving force acts in the step up/down direction, and step-edge diffusion is the dominant mass-transport mechanism, we find that significant deviations from the usual t1/4 scaling of the terrace-width correlation function occurs for a critical time τ which is dependent upon the three energy scales in the problem: kBT , the step stiffness, γ, and the bias associated with adatom hopping under the influence of an electromigration force, ±∆U . For (t < τ ), the correlation function evolves as a superposition of t1/4 and t3/4 power laws. For t ≥ τ a closed form expression can be derived. This behavior is confirmed by a Monte-Carlo simulation using a discrete model of the step dynamics. It is proposed that the magnitude of the electromigration force acting upon an atom at a step-edge can by estimated by a careful analysis of the statistical properties of step-edge fluctuations on the appropriate time-scale. PACS numbers: 05.40.-a, 66.30.Qa, 68.35.Ja I. INTRODUCTION During the past decade, continuum models and dis- crete lattice simulations have been applied to understand direct imaging observations of the thermal fluctuations of step edges in which the step position is monitored as a function of time [1, 2]. Of particular interest has been the study of the dynamics of the equilibration of terrace width distributions where the temporal evolution of step edge fluctuations are driven by the exchange of atoms between the step and the adjacent terrace and/or by mo- tion of adatoms along the step edge itself [3, 4, 5, 6, 7, 8]. It is well known that the position of the step edge, as de- scribed by its temporal correlation function, has a time dependence that scales as a power law with an expo- nent characteristic of specific atomic processes driving the step fluctuations; tβ . In cases where mass transport at the step is dominated by diffusion of atoms along the step edge; β = 1/4. When mass transport proceeds via exchange of atoms between the step edge and the terrace β = 1/2. Careful experiments allow a crossover from t1/4 to t1/2 scaling to be observed as a function of the sam- ple temperature [9]. Further, experimental measurement of the correlation functions have been used to determine thermodynamic properties of the steps, such as the step stiffness [1, 2]. In this paper we investigate how the scaling of the step edge fluctuations is changed by the presence of an electro- migration force [10] acting upon atoms diffusing along the step edges. The primary motivation for this study is the possibility of using measurements of these changes to ob- ∗Electronic address: [email protected] tain information about the electromigration force itself. In conducting materials, a surface electromigration force can be generated by passing an electrical current through the sample [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In terms of a simple discrete model, the presence of the electromigration force introduces a small bias in the dif- fusion of atoms at the step edge, parallel to the current (and field). By convention, this bias can be expressed as an energy difference between atoms diffusing parallel or anti-parallel to the current ∆U = Z∗eEa⊥ where E is the electric field applied to the sample, Z∗ is the effec- tive valence of adatoms and a⊥ is the lattice spacing per- pendicular to the step edge. A characteristic property of surface electromigration is that the electromigration bias, ∆U , is several orders of magnitude smaller than the other energy scales in the problem: γa⊥, where γ is the step stiffness and the energy associated with thermal fluctua- tions; kBT . This suggests that thermal fluctuations will completely dominate the short-time behavior of the step fluctuations with the effect of the electromigration bias emerging only on much longer time scales. Nevertheless, such time scales (of the order of seconds) are accessible to experiment offering the possibility that the observation of step fluctuations under electromigration conditions may allow us to obtain quantitative information about the magnitude of the force itself; a quantity that, to date, has been hidden from experimental study. This paper is organized as following. In section 2 we present a continuum theory of step edge fluctuations un- der electromigration conditions. In section 3 we test the fidelity of the theory by showing the results of a Monte- Carlo simulation of the temporal evolution step correla- tion function. Concluding remarks are contained in sec- tion 4. http://arxiv.org/abs/0704.0624v1 mailto:[email protected] II. THEORY In order to describe the dynamics of a step-edge evolv- ing under electromigration conditions we employ the usual Langevin formalism where each degree of freedom diffuses towards lower energy with a velocity that is pro- portional to the energy gradient, subject to random ther- mal fluctuations. The position of step edge is described by it’s edge profile y(x, t) where the x-axis is oriented parallel to the step edge. y(x, t) is the position of the step edge at x and at time t. In this paper we consider the limiting case where the step motion occurs most easily by adatom diffusion along the step edge itself. Adatom exchange between the step- edge and the adjacent terraces, via attachment and de- tachment, is neglected. It is well known that atomic diffu- sion along a step edge is driven by the step curvature [23], which generates a flux Dsδsγ ∇sκ (1) where Jc is the curvature-driven flux, Ds = a ||/τh is the surface diffusion constant, a|| is the atomic diameter, δs = a⊥ the width of an atom perpendicular to the step, τh is the average time between hopping events and γ is the step stiffness. ∇sκ is the gradient of the curvature along the step edge. Mass conservation determines the normal velocity of the step edge, n̂ = −Ω∇s · Jc (2) where Ω is the area of a single atom and n̂ is normal to the step edge. The inclusion of a thermal noise term, η and lineariza- tion of the above equation leads to the well-known equa- tion of motion for a step edge [3] − Γhγ y(x, t) = η(x, t). (3) where we have defined a3||a . (4) We model the effects of electromigration by adding to this equation of motion a term generated by a constant force, F = Z∗eE, felt by atoms at the step edge, which arises from the application of a electric field, E, to the material oriented parallel to the y-axis. Z∗ is the effective valence of an atom at the step edge. The electromigration force generates an additional flux, JEM = DsδsZ ∗eE|| DsδsZ 1 + y2x where E|| is component of the electric field along the step edge. yx indicates an x derivative. The energy of any step configuration, relative to the energy of the straight step, is determined by the step stiff- ness γ and the electromigration force, F , felt by atoms at the step edge. If the force acts perpendicular to the step edge (i.e. when F > 0, the force acts in the +y, or step down, direction) then, for small fluctuations, one can linearize the stochastic equation of motion for the step edge to obtain the following Langevin equation for the step dynamics, − Γhγ − ΓhF kTa||a⊥ y(x, t) = η(x, t) (6) where a|| and a⊥ are the lattice parameter parallel and perpendicular to the step edge. The noise term, η, de- scribes the thermal fluctuations and is correlated be- cause, in our model, of the random hopping of atoms occurs only between between adjacent step edge sites. 〈η(x, t)η(x′, t′)〉 = 2Γhδ(t− t′) δ(x − x′) (7) To first order, the electromigration force does not al- ter the correlation properties of the noise so that in the single-jump regime equation 4 remians valid. Before proceeding, for notational simplicity we rewrite eqn. 6 as − α ∂ ∓ αq2c y(x, t) = η(x, t) (8) where α ≡ Γhγ , qc ≡ a||a⊥ The parameter qc depends only on the magnitude of the electromigration force and in eqn. 6 the ∓ denotes the force acting in the step down(-) or step up (+) direction. In the case where there is no random thermal noise (i.e. T → 0), eqn. 6 predicts that the amplitude of a small sinusoidal fluctuation of the step-edge profile, with wavevector q, evolves according to the following disper- sion relation, iω = αq2 q2 ± q2c This is the same dispersion relation as the one that de- scribes step flow under growth conditions [24, 25, 26]. If the electromigration force acts in the step up (+) direc- tion the step fluctuations are linearly stable. For a force acting in the step-down direction there exists a linear in- stability which initiates the well-known phenomenon of electromigration-induced step-wandering for fluctuations with wavevector smaller than qc [27, 28, 29]. The critical wavevector is an important parameter in our Langevin model that, from eqn. 9, is determined by the ratio of the electromigration force and the step stiffness. In order to determine the statistical properties of the solutions of eqn. 6 we take the usual approach and first derive the Green’s function for the problem using stan- dard Fourier transform methods, G(x, t|x′, t′) = i e−α(k k2)(t−t′)eik(x−x In terms of the Green’s function, the displacement of the step at time t is: y(x, t′′) = y(x, t′) + ∫ t′′ G(x, t′′|x′, t)η(x′, t′)dx′dt We can now compute the time correlation function of the step edge, G(t), after time t = t′′ − t′ has elapsed, g(t) ≡< (y(x, t′′)− y(x, t′))2 > (13) Substituting eqn. 11 into 12 and using the correlation properties of the noise (eqn. 7) we obtain we find, after some calculation, g±(t) = 21/4πα (k2 ± q2c ) 1− e−2α(k or, using substitution, g±(t) = t (kT ) × (15) (u2 ± αtq4c ) 1− e−2u When there is no electromigration force (i.e. |F | = 0, qc = 0), the integral in eqn. 16 is clearly time- independent and we recover the result derived by Bartelt et al. [3] where the g(t) the step edge fluctuations evolves with the well-known t1/4 scaling law characteristic of step-edge limited diffusion. g±(t) ≡ g0(t) = t1/4 (kT ) It is helpful to re-express eqn. 16 in terms of the average time, τ0, that it takes for the mean-squared width of an initially straight step to reach a value equal to g20(τ0) = a2⊥ (i.e. one square lattice spacing): g0(t) = a where τ0 ≡ τh 2kTa|| 4Γ (3/4) When the electromigration force is finite ( i.e. |F | > 0), it is apparent from eqn. 16 that this scaling behavior is modified by the appearance of an explicit time depen- dence in the integral. This can be seen more clearly by defining a critical time, τ , and a rescaled time, ζ = t/τ , where, = 2τh 2kTa|| , (19) Then, eqn. 16 can be rewritten as g±(t) = a where I± (ζ) ≡ 21/4Γ(3/4) × (21) (u2 ± 1− e−2u I± is a universal function of the rescaled time, ζ and is normalized such that I±(t → 0) = 1. The integral appearing in eqn. 22 is easily evaluated numerically and is shown in fig. 1 in which we display I± plotted as a function of the rescaled time ζ (i.e. time is expressed in units of τ). The solid curves show I± obtained for F > 0 (step up) , F < 0 (step down) electromigration forces (For F = 0, I±(ζ) = 1). From fig. 1. it is apparent that g±(t) (eqn. 20) deviates very significantly from the t1/4 scaling behavior of g0(t) (eqn. 17) as t approaches τ (ζ → 1). These deviations be- gin to appear at earlier times, t ∼ 0.1τ , when the effect of the force on the evolution of the step fluctuations begins to be felt. This can be seen more clearly in fig. 2 which displays the correlation function of a step plotted as a function of time for τ0 = 5s and τ = 10 +4s. The values for these parameters were chosen so that τ/τ0 ∼ 104, a ratio typical of accelerated electromigration experiments. As noted above, deviations from t1/4 scaling start to ap- pear when t ∼ 0.1τ = 100s. The results shown in figs. 1 and 2 have a simple qualitative interpretation; the short time behavior of the step fluctuations (t ≪ τ) is com- pletely dominated by the thermal fluctuations and the effect of the electromigration bias emerges only at later FIG. 1: The integral function I(ζ) (eqn. 22) plotted as a function of the rescaled time ζ = t/τ for no electromigration force (F = 0), the electromigration force in the step down direction (F > 0) and in the step-up direction (F < 0). FIG. 2: The time corrleation function, g(t), of the step- edge distribution predicted by the continuum Langevin model (eqns. 20 and 22), plotted as a function of time for τ0 = 5s and τ = 104s. times. Such behavior is typical of the dynamics of diffu- sion driven by weak external forces. It is instructive to perform a power series expansion of the integral about ζ = 0 such that eqn. 20 becomes, g±(t) = a 1∓ a 1 . . . The expansion coefficents can be obtained analytically, = 0.3487, a1 = = 0.1500, Shown as the dashed lines in fig. 1 are the results of this series approximation for I±(ζ) (eqn. 22), evaluated up to, and including the terms linear in time. Clearly, this truncated expression is a reasonable approximation for t ≤ 0.4τ . III. SIMULATION In order to test the predictions of the continuum Langevin model described above, we developed a Monte Carlo simulation of step edge fluctuations in the pres- ence of an external force. In this model, atomic diffu- sion was restricted to the step edges with atoms jump- ing between adjacent step sites only. Only nearest- neighbor interactions on a square lattice were permit- ted (a⊥ = a|| = a = 1) and were modelled by a single bond energy ǫ. In order to describe the electromigration force, the atomic jump probabilities for motion parallel and anti-parallel to the force were biased by a small en- ergy differential ∆U . In terms of the electromigration force, F , and the lattice spacing perpendicular to the step edge, a; ∆U = ±Fa. Simulations were performed for steps of length ℓ = 10000a|| fluctuation on a square lattice. Periodic bound- ary conditions were employed. The bond energy was set to ǫ = 2.0kBT and the magnitude of the electromigra- tion bias was |∆U | = 0.01kBT , a factor of 103 smaller than the typical binding energy of an atom to the step edge. This value was chosen to generate statistically sig- nificant deviations from the (no-force) t1/4 scaling within reasonable simulation times. If ǫ = 0.1eV and it is as- sumed that a ∼ 1.5Å (typical of metals) then this bias corresponds to an electric field with a magnitude of or- der 1000V cm−1 acting on an atom with effective valence of Z∗ ∼ ±10e. In actual accelerated electromigration experiments, a field of 0.1− 1V cm−1 is typical. Figure 3 shows the results of the simulation where the correlation function of an isolated step is plotted as a function of the time measured in Monte-Carlo steps per step-edge site (MCS). Shown is g(t) obtained when the electromigration force acts in the step-up and step-down directions and when ∆U = 0 (i.e. no electromigration force is acting at the step edge).We define one Monte FIG. 3: Results of the a kinetic Monte Carlo simulation where the correlation function of an isolated step is plotted as a function of the time (measured in Monte-Carlo steps per step- edge site, the lattice spacing is a = 1). Shown is g(t) obtained when the electromigration force acts in the step-up and step- down directions and when ∆U = 0 (i.e. no electromigration force is acting at the step edge). The curves shown were obtained by averaging the data from 200 randomly generated replicas after each was equilibrated for 105 Monte Carlo steps. Carlo step to be equal to the average time needed for ev- ery atom at the step edge to attempt a hop. The results shown in fig. 3 were obtained by averaging the data from 100 randomly generated replicas after each was equili- brated for 105 Monte Carlo iterations per site. Compar- ing the simulation results (fig. 3.) to the predictions of the Langevin theory (fig 4.) it is apparent that the qual- itative features of the continuum theory are reproduced by the Monte Carlo simulation. These same data are pre- sented in the form of a log-log plot in fig. 4. In the ab- sence of the force (F = 0 , ∆U = 0), a least-squares fit of the no-force simulation data shows that g0(t) is very well fit by a tβ power-law where β = 0.25± 0.01. Therefore, when there is no electromigration force present in the simulation, the correlation function of the step evolves according well-known t1/4 scaling, as predicted by the Langevin analysis (eqn. 17). By least squares fitting the simulation results to eqn. 17 we obtain a value of τ0 = 5 We now consider the results of the simulation obtained for a finite electromigration force, also shown in figs. 3 and 4. Equation 22 suggests that the value of τ can be FIG. 4: A log-log plot of the simulation data shown in fig. 3. The dashed line shows the best fit of a power law, (t/τ0) β, to the no-force (F = 0) data; β = 0.25, τ0 = 5 MCS. extracted from the simulation results by considering the scaling of the difference between the correlation functions for the force in the up and down step directions: ∆(t) = g+ − g− = 2a 1 . . . (24) For the simulation results, this normalized difference is plotted in fig. 5. The behavior of this quantity is well fit by the leading order term in eqn. 24 from which we obtain a value of τ = 62000± 10000 MCS. For comparison with the fits to the continuum Langevin model , we can estimate τ from the microscopic parameters employed in the discrete Monte Carlo model used to generate the simulation data. Combining eqn. 19 with the step stiffness appropriate for our model 2kTa|| 2kBTa|| we obtain an estimate of τ in units of MCS: τ = 2 2kBTa|| The ratio of the hopping time in the Langevin model , τh to the time between hopping attempts in the Monte Carlo simulation τa can be obtained by monitoring the FIG. 5: For the simulation results shown in figs. 3 and 4; a log-log plot of the normalized difference (eqn. 24) plotted as a function of time (MCS). The behavior of this quantity is well fit by the leading order term in eqn. 24 (t1/2, dashed curve) from which we obtain a value of τ = 62000 ± 10000 MCS. success rate of hops between adjacent lattice sites in the simulation. We find that τa/τh = 3.6 ± 0.1. Thus our estimate for the value of τ in the Monte Carlo simu- lations, used to generate the data shown in figs 3-5, is τ = 62000±10000 MCS. Clearly, the agreement between the continuum Langevin theory (τ = 71000 MCS ) and the microscopic model (τ = 62000± 10000 MCS) is rea- sonable. Finally we note that in the high temperature limit (kBT ≫ ǫ) the ratio of the electromigration bias to the binding energy at a step edge, ǫ is related directly to τ that would be obtained from experiment: where τh can be determined from eqn. 18, if the step stiffness is known. IV. CONCLUSIONS The temporal evolution of step-edge fluctuations un- der electromigration conditions has been analyzed using a continuum Langevin model for the case where diffu- sion is limited by mass transport along the step edge. We find that the presence of the electromigration force, felt by atoms at the step edge, causes deviations from the usual t1/4 scaling of the terrace-width distribution driven by thermal fluctuations alone. We have identified a critical time τ that is a function of the three energy scales in the problem: kBT , the step stiffness, γ, and the bias associated with adatom hopping under the influence of an electromigration force, ±∆U . For (t < τ), the cor- relation function evolves, to a good approximation, as a superposition of t1/4 and t3/4 power laws. For all τ a closed form expression was derived. This behavior was confirmed by a Monte-Carlo simulation using a discrete model of the step dynamics. Finally we propose that the magnitude of the electromigration force acting upon an atom at a step-edge could be determined directly by care- ful measurement and analysis of the statistical properties of step-edge fluctuations on the appropriate time-scale. Acknowledgments We acknowledge helpful discussions with E.D. Williams. This work has been supported by the US Department of Energy grant DE-FG02-01ER45939 and by the NSF- Materials Research Science and Engineering Center un- der grant DMR-00-80008. [1] H.-C. Jeong and E. D. Williams, Surface Science Reports 34, 171 1999. [2] M. Giesen, Progress in Surface Science 68, 1 2001. [3] N. C. Bartelt, J. L. Goldberg, T. L. Einstein, and E. D. Williams, Surf. Sci. 273, 252 1992. [4] S. V. Khare and T. L. Einstein, Phys. Rev. B 57, 4782 1998. [5] T. Ihle, C. Misbah, and O. Pierre-Louis, Phys. Rev. B 58, 2289 1998. [6] H.-C. Jeong and J. D. Weeks, Surf. Sci. 432 , 101 1999. [7] C. P. Flynn, Phys. Rev. B 66, 155405 2002. [8] N. C. Bartelt, T. L. Einstein, and E. D. Williams, Surf. Sci. 521 , L669 2002. [9] M. Giesen, Surf. Sci. 442 , 543 1999. [10] R. S. Sorbello, Solid State Physics eds. H. Ehrenreich and F. Spaepen 51, 159 1999. [11] D. Schumacher, Surface Scattering Experiments With Conduction Electrons Springer Tracts in Modern Physics (Springer, Berlin) 128, 1993. [12] T. W. Duryea and H. B. Huntington, Surf. Sci. 199, 261 1988. [13] H. Ishida, Phys. Rev. B 49, 14610 1994. [14] H. Ishida, Phys. Rev. B 52, 10819 1995. [15] H. Ishida, Phys. Rev. B 57, 4140 1998. [16] H. Ishida, Phys. Rev. B 60, 4532 1999. [17] H. Ishida, Phys. Rev. B 54, 10905 1996. [18] P. J. Rous, T. L. Einstein, and E. D. Williams, Surf. Sci. Lett. 315, 995 1994. [19] P. J. Rous, Phys. Rev. B 61 , 8475 2000. [20] P. J. Rous, Phys. Rev. B 61 , 8475 2000. [21] P. J. Rous, J. Appl. Phys. 87, 2780 2000. [22] H. Yasunaga and A. Natori, Surf. Sci. Rep. 15, 205 1992. [23] W. W. Mullins, J. Appl. Phys. 28 , 333 1957. [24] G. Bales and A. Zangwill, Phys. Rev. B 41 , 5500 1990. [25] O. Pierre-Louis, M. D’Orsogna, and T. Einstein, Phys. Rev. Lett. 82, 3661 1999. [26] M. R. Murty and B. Cooper, Phys. Rev. Lett. 83, 352 1999. [27] J. Krug, Multiscale Modeling of Epitaxial Growthed. ed. A Voight (Birkhauser) ), 2004. [28] M. Degawa et al., Surf. Sci. 487 , 171 2001. [29] T. Zhao, J. D. Weeks, and D. Kandel, Phys. Rev. B 70 , 161303 2004.

0704.0625

Renormgroup origin and analysis of Split Higgsino scenario

Renormgroup origin and analysis of Split Higgsino scenario V. A. Beylin,∗ V. I. Kuksa,† and G. M. Vereshkov‡ Institute of Physics, Southern Federal University (former Rostov State University), Rostov-on-Don 344090, Russia R. S. Pasechnik§ Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia (Dated: November 4, 2018) Abstract We present a renormalization group motivation of scale hierarchies in SUSY SU(5) model. The Split Higgsino scenario with a high scale of the SUSY breaking is considered in detail. Its mani- festations in experiments are discussed. PACS numbers: 12.60.Jv, 95.35.+d, 95.30.Cq ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] §Electronic address: [email protected] http://arxiv.org/abs/0704.0625v1 mailto:[email protected] mailto:[email protected] mailto:[email protected] mailto:[email protected] I. INTRODUCTION Supersymmetric generalization of the SM permits one to solve some inner problems of the theory. However, the MSSM is an ambiguous model: despite the defined set of physical fields, only one characteristic MSSM parameter – electroweak scale MEW – has been fixed in experiment. The scales of scalar quarks and leptons M0, gaugino M1/2, and Higgsino µ, can be picked out as arbitrary ones. Expectations of new dynamical effects of the supersymmetry at the LHC are induced in the MSSM by some specific choice of the SUSY breaking scale that is not very far from the EW scale, MSUSY ∼ M1/2 ∼ M0 ∼ O (1 TeV). This scale hierarchy seems “natural”, providing regularization of quadratic divergencies long before MGUT . Thus, the MSSM as it is motivated by some theoretical and phenomenological arguments – successful RG evolution, “natural” choice of the renormalization scale and the reasonable dark matter (DM) description – works well for the specific choice of the SUSY breaking scale only. However, the “naturalness” as one of the MSSM principles does not seem to be the obligatory requirement from the QFT point of view. So establishing of the genuine SUSY scales hierarchy is a problem which can be unambiguously solved beyond the MSSM. The arbitrariness in the choice of the scale hierarchy can be sufficiently diminished when the gauge coupling unification is considered as the most fundamental theoretical basis. With this consideration as a starting point, we have found that the one-loop RG study, involving SUSY SU(5) degrees of freedom placed near MGUT , allows one to select some specific classes of the scale hierarchies. The proton stability is provided simultaneously. It was shown that states nearMGUT should be taken into account as threshold corrections; they are crucially important for the selection of scales, giving sufficiently high unification point. Then the RG consideration at the one-loop level results in two classes of scenarios with an opposite arrangement of µ and M1/2 scales. At the same time, the analysis does not fix the characteristic scalar scale M0 due to a specific form of the RG equations which contain the squark and slepton scales as a ratio Mq̃/Ml̃ only. For the first class of scenarios the hierarchy |µ| ≫ M1/2 takes place. The second class is defined by the hierarchy |µ| ≪ M1/2. Due to arbitrariness of M0 it is possible to select some subscenarios with various M0 arrangements. Among them there can be found some hierarchies corresponding to Split Symmetry and some ideologically close scenarios which do not reject fine-tuning and are motivated, on the one hand, by the anthropic principle and, on the other hand, multi-vacua string landscape arguments [1, 2, 3, 4, 5, 6] (see, however, comments in [7]). In particular, the RG analysis results in the hierarchy M0 ∼ M1/2 ≫ |µ| > MEW , (1.1) whose spectrum contains two lightest neutralinos degenerated in mass (almost pure Higgsino) and one of charginos as the states that are the nearest to the electroweak scale. In this Split Higgsino scenario both M0 and M1/2 are shifted to scales ∼ (107 − 1010) TeV. In this paper, we consider some features and manifestations of the last scenario only (a preliminary version of this work was presented in [8], see also [9]). As it will be shown, direct observation of neutralino (Higgsino) in χ − N scattering is impossible nowadays due to a very small interaction cross section. As to collider experiments, the signature of neutralino and chargino production and decays at the LHC crucially depends on the mass splitting values for these degenerate states [9, 10, 11, 12, 13]. On the one hand, observation of these hardly detectable effects means that it is possible to get important information on the high superstate structure, in particular, t̃1,2 and their mixing, from one- loop calculations of the mass splitting [14, 15, 16]. On the other hand, if only these low-lying SUSY degrees of freedom are detected in experiments near a TeV scale, the conventional MSSM spectrum should necessarily be splitted in some manner to produce a high scale for other superstates. The value predicted for the diffuse gamma flux from the Galactic halo can be measured by GLAST or ACT in the near future but it is not peculiar to this scenario. Signals from direct photons do not contain any specific effects too [9]. Thus, the Split Higgsino scenario is an example of SUSY models, in which the lightest degrees of freedom manifest themselves in some subtle effects as specific decay channels, correlating with the value of diffuse (or direct) gamma flux. Note, even if collider experiments will demonstrate the absence of the lowest neutralino signals (due to a high degeneracy of states, for example, it will certainly testify to some fine-tuning in the SUSY states spectrum), the SUSY, as a theoretical principle, cannot be rejected. At low O (1 TeV) energies the SUSY, in a sense, is barely hidden, so it exists at a high scale. The structure of the paper is as follows. In Section 2, we discuss RG analysis of the SUSY SU(5) in detail, indicating the most interesting scenarios. In Section 3, the Lagrangian of the Split Higgsino model and its analysis are presented. There is also an estimation of the lowest Higgsino mass from the relic abundance data. We discuss expected experimental manifestations of the scenario in Section 4 and some conclusions are in Section 5. II. RENORMGROUP ANALYSIS We consider one-loop RG equations for running couplings and start from their values at the MZ scale α−1(MZ) = 127.922± 0.027, αs(MZ) = 0.1200± 0.0028, sin2 θW (MZ) = 0.23113± 0.00015. (2.1) In renormgroup equations the following values are used as initial: (MZ) = α−1(MZ) cos 2 θW (MZ) = 59.0132∓ (0.0384)sin2 θW ± (0.0124)α, (MZ) = α −1(MZ) sin 2 θW (MZ) = 29.5666± (0.0192)sin2 θW ± 0.0062)α, (2.2) (MZ) = α s (MZ) = 8.3333± 0.1944. At the one-loop level the equations for running couplings are well-known α−1i (Q2) = α i (Q1) + , bi = bij . (2.3) In the sum all states with masses Mj < Q2/2 at Q2 > Q1 are taken into account. Namely, we include in the RG analysis the following degrees of freedom: singlet quarks and their superpartners (DL, D̃L), (DR, D̃R) contained in super-Higgs quintets of SUSY SU(5); chiral superfields (ΦL, Φ̃L) and (ΨL, Ψ̃L) in adjoint representations of SU(2) and SU(3), respectively, which survive from super Higgs 24-plet. In the minimal SUSY SU(5) masses of the states M5 = (MD, MD̃), M24 = (MΨ, MΨ̃, MΦ, MΦ̃) are generated by an interaction with Higgs condensate at the GUT scale, but the interaction couplings are not fixed. For the analysis we assume masses M5, M24 to be slightly lower thanMGUT . Note that heavy states near MGUT are of especial importance for the RG equations. The necessity of taking into account of such heavy states in the RG analysis as so-called threshold corrections is well-known [17, 18, 19, 20, 21, 22]. For one-loop running couplings at the scale q2 = (2MGUT ) 2 we have (2MGUT ) = α (MZ)− ln 2 + −7 lnMGUT + lnMD + lnMD̃ lnMq̃ + lnMl̃ + lnMH + lnMt + (2MGUT ) = α (MZ)− ln 2 + −3 lnMGUT + lnMq̃ + lnMl̃ + lnMW̃ + lnMH + lnMt − (2MGUT ) = α (MZ) + ln 2 + (− lnMGUT + 2 lnMΨ̃ + lnMΨ lnMD + lnMD̃ + 2 lnMq̃ + 2 lnMg̃ + lnMt − (2.4) Here M0 = (Mq̃, Ml̃) are masses of scalar quarks and leptons averaged over chiralities and generations; Mt is t-quark mass; other parameters were introduced above. In (2.4) it is supposed that the lightest Higgs boson mass mh is near to MEW and other Higgs bosons H, A, H± are placed at some high MH scale. For extra heavy states the only condition is M24, M5 < MGUT and we suppose that this inequality is fulfilled with accuracy within 1 - 2 orders of magnitude. As to SUSY degrees of freedom M0, M1/2, µ, MH , equations (2.4) do not depend on any specific arrangement of these scales. So when MGUT is the highest scale in the system, there is no necessity to establish an initial scale hierarchy. The only demand is that the RG equations should lead to coupling unification at sufficiently high MGUT scale for the proton stability. Then the set of possible hierarchies of energy scales arises as the final result of the study. Note also that if masses of singlet superstates and residual Higgs superfields are equal to MGUT , equations (2.4) return to the standard form automatically. The above considerations together with experimental restrictions on the SUSY mass spectrum and general expectation of the lightest Higgs boson scale (it is not very far from the EW scale) are sufficient for the RG analysis. As the first step, all couplings were recalculated at the 2MZ scale, all the SM states contribute to running of couplings despite W±, Z0, Higgs bosons, and t-quark. At the same time, terms with ln 2 occur which are quantitatively important for calculations. Between the (2MZ , 2Mt) scales the following states emerge: W ±, Z0 and one Higgs doublet containing light h-boson and longitudinal degrees of freedom ofW±, Z0. At these stages, Zq̄q vertex was used for calculations of α−1 (2Mt), starting from α (2MZ) value. Above 2Mt calculations were carried out in a standard manner. Now, equating couplings at MGUT , from (2.4) we get the following expressions: MGUT = Ak1MZ , µ = Bk2MZ , (2.5) where k1 = K −1/12 GUT1 ≡ )1/12( M ′GUT k2 = K K−1GUT2 ≡ )1/4( )1/4( )5/2( M ′′GUT 1/2 ≡ (MW̃Mg̃) 1/2, M ′GUT ≡ (MΨ̃MΦ̃) 1/3(MΨMΦ) 1/6 ≤ MGUT , M ′′GUT ≡ 7/6(M2DMD̃) MΦ)4/3 ≤ MGUT , A = exp (5α−1 (MZ)− 3α−12 (MZ)− 2α−13 (MZ))− = (1.57×1.09 0.92) · 1014, B = exp (5α−1 (MZ)− 12α−12 (MZ) + 7α−13 (MZ)) + = (2.0×0.15 6.56) · 103. Here all dimensionless parameters K with various indices are defined as quantities having values larger than unity (see also [23]). Note that KGUT1, KGUT2 are not under theoretical control either in the MSSM or in the SUSY SU(5). So we assume that 1 ≤ KGUT1 ≃ KGUT2 ≤ 10. For the lightest Higgs boson there is an experimental restriction [24]: Mh > 114.4 GeV, as to other (heavy) Higgs bosons the following interval 2 ≤ KHt ≤ 10 is supposed for the numerical analysis. Values of Kq̃l̃, Kg̃W̃ can be determined from the renormgroup evolution from MGUT to M0, M1/2. Here we suppose that 1.5 ≤ Kq̃l̃ ≃ Kg̃W̃ ≤ 2.5. Now all dimensionless parameters are fixed in some intervals and we analyze M ′ 1/2 and µ as functions of MGUT : 1/2(MGUT ) = (Ak1) GUT , µ(MGUT ) = (Ak1)3/2M ×M3/2GUT . (2.6) As it is seen, the initial system of equations for three running couplings can be rewritten as a system of two equations for the effective gauginoM ′ and µ scales, depending onMGUT . It is very essential that all other characteristic scales arise in the equations as dimensionless ratios only. (This method of RG analysis was used in [23] for investigation of SUSY SU(5) and E6 energy scale hierarchies; see also [20, 22].) As an important feature, we have to note that the renormgroup study remains a common scalar scale M0 as an arbitrary one: the scale M0 turns into (2.6) through the ratio Mq̃/Ml̃ only. For numerical predictions the ratio value was estimated as O(1). Certainly, the splitting of the Mq̃ and Ml̃ scales will affect the hierarchies of other scales. However, to change these hierarchies crucially, the ratio Mq̃/Ml̃ must be ∼ O(100) or larger, and here we do not consider this possibility. Also, we suppose that radiative corrections to superscalar masses do not change the ratio substantially (numerically, these corrections can be as large as (1 − 5)%, see [14]). Conclude, the scale M0 should be fixed by some extra arguments independently. Contributions of heavy states M24, M5 do not affect the RG hierarchy of SUSY scales radically because their scales are combined into a ratio too. Nevertheless, the ratio value is important to fix the MGUT scale: the ratio M24/M5 depends on MZ logarithmically, so numerical coefficients of the ratio can change the MGUT value. For the analysis we used the known restrictions for the proton lifetime (τp ≥ 1032 yr at MGUT ≥ 1015 GeV) and for MSUSY that is ∼ M ′1/2 > 100 GeV when MGUT < 3 · 1016 GeV). We hope also that loop corrections to all masses in the model do not change results of the RG analysis drastically, it is supposed that for the scenario they contribute no more than ∼ (10− 15)% [14, 15]. 1015 1016 Higgsino 1: K =1.5 , K =3 , K 2: K =1.5 , K =5 , K 3: K =2.5 , K =2 , K 4: K =2.0 , K =8 , K , GeV FIG. 1: SUSY SU(5) hierarchies of scales permitted by the one-loop RG analysis. Numerical results following from (2.6) are shown in Fig. 1. It is clearly seen that the proposed method of the one-loop RG analysis results in the selection of two proper sets of SUSY scales only (further we will omit differences between M ′ and M1/2). For this conclusion we assume that the characteristic µ scale is in the O(1 TeV) region, providing the neutralino mass value that is sufficient for explanation of the DM properties. The first class of scenarios is defined by the hierarchy |µ| ≫ M1/2 & MEW . As it was noted, M0 value does not affect this scale splitting dictated by the one-loop RG evolution, and can be placed at an arbitrary but physically reasonable scale. So the class could be divided into some subscenarios with hierarchies (a) M0 ≫ |µ| ≫ M1/2 & MEW , (b) M0 ∼ |µ| ≫ M1/2 & MEW , (c) |µ| ≫ M0 ≫ M1/2 & MEW , (d) |µ| ≫ M0 ∼ M1/2 & MEW . The subscenarios (a), (b) and (c) are close to the known Split SUSY (Gaugino) models [1, 2, 3], but the variant (d), keeping most part of the SUSY degrees of freedom (except Higgsino) near the EW scale, is similar to the MSSM spectrum with “naturalness”. For these variants we get MGUT = (1− 5) · 1016GeV and gaugino masses can be as small as (0.1− 1.0) TeV. The second set of scenarios is generated by the hierarchy M1/2 ≫ |µ| > MEW . In this set we have the following subscenarios: (e) M0 & M1/2 ≫ |µ| > MEW , (f) M1/2 ≫ M0 ∼ |µ| > MEW , (g) M1/2 ≫ M0 ≫ |µ| > MEW , (h) M1/2 ≫ |µ| > M0 & MEW . Except the last variant (h) with too low M0 scale as contradicting known experimental data, the other scenarios of this class seem reasonable. All of them introduce Higgsino as the DM carrier. Note that the hierarchy (f) is close to the MSSM spectrum again, despite gaugino states at a high scale. Scenarios (e) and (g) shift M0 and M1/2 scales into a multi- TeV region, splitting the spectrum. In these cases, the µ scale is the nearest to the EW scale and should be near 1TeV, as it follows from the RG analysis. As to the unification scale, we have the value MGUT = (1− 2) · 1015GeV which makes the proton stable. Obviously, Split SUSY models with highM0 andM1/2 scales lead to the damping of heavy states (squarks, sleptons) contributions near the EW scale. Another important consequence of this type of scenarios is (nearly) degeneration in mass of the lowest neutralino and chargino states [9, 11, 13, 16, 25, 26] (experimental searches of this degenerated states were carried out at LEP energies [27] without any evident signals). Further, we will consider the Split Higgsino scenario with hierarchy M0 & M1/2 ≫ |µ| > MEW . Taking MGUT value as & 10 15 GeV, for µ = 1.0 TeV the SUSY breaking scale is MSUSY ∼ M1/2 ∼ (0.1− 2.5) · 108 GeV; when µ = 1.4 TeV we get MSUSY ∼ (0.5− 2.8) · 108 GeV. For these cases MGUT ∼ (1.0− 1.7) · 1015 GeV. The used values of µ, as it will be seen further, agree with modern data on relic DM abundance. Intervals of MSUSY values follow mainly from uncertainties in the parameters KGUT1 and KGUT2. Most evidently, the scenario is selected when KGUT1, KGUT2 ∼ 5 − 8; it means that the effect of threshold corrections is essentially important for the discovery of hierarchy. Namely, if masses M24, M5 are equal to MGUT , it is questionable to select several possible scenarios that are compatible with the DM data. These one-loop results should be refined with taking into account of two-loop β-functions and loop corrections to masses. By now we expect that two-loop RG analysis does not change the above hierarchies radically as well as mass corrections. Certainly, various numerical coefficients will change, arrangements of the scales will move somehow, but the relative spacing between µ and M1/2 scales will not change, as we hope. Nevertheless, the M0 scale can be splitted by the RG at the two-loop level, so positions of the Mq̃ and Ml̃ scales can be determined independently. More carefully this question will be analyzed in a forthcoming paper. Now we will consider the Lagrangian of the Split Higgsino scenario, its structure, and features. III. THE SPLIT HIGGSINO SCENARIO A. The model Lagrangian and its features When the group of symmetry is fixed, the Lagrangian as a local gauge group invariant can be written in a standard manner. Nevertheless, its specific form depends on the chosen field representation. In this paper, we use the formalism of Majorana spinors and define all physical fields as having positive masses only. More precisely, in the scenario considered the neutralino mixing parameter is proportional to the ratio MZ/MSUSY , so it is negligibly small due to a high scale of SUSY breaking. However, the mixing can be important when the model restrictions are analyzed in the framework of a specific mass spectrum with a small splitting of the lightest neutralino masses. As to neutralino masses, i.e., the scale of µ parameter, diagonalization of the neutralino mass matrix by an orthogonal matrix usually leads to emerging of neutralino state with negative mass. This should be taken into account in calculations by the corresponding definition of the propagator (if negative mass neutralino is in the intermediate state) or in the neutralino polarization matrix (if the neutralino is in the initial or in the final state). To evade this inconvenience, we redefine the neutralino field with the negative mass in the following way: χ → iγ5χ. This operation provides positive neutralino mass keeping standard calculation rules simultaneously. Moreover, it does not change Majorana type of the field. It also has been shown that this procedure is equivalent to the diagonalization by the unitary matrix instead of the orthogonal one, for details see [28]. Hence, in the scenario two lowest neutralino states are Majorana Higgsino-like ones. The set of scales leads to the strong splitting in the neutralino and chargino spectra, so masses of these lightest neutralinos χ0 1, 2 and the light chargino χ emerge near the µ scale. Light neutralino states are built from the initial fields h1,2 in the limit of pure Higgsino when mZ/µ → 0 and mZ/MSUSY → 0; here MSUSY ∼ M1 ∼ M2. Masses of the lightest states are ≃ µ− M Z(1 + sin 2β) 2M1M2 (M1 cos 2 θW +M2 sin 2 θW ), ≃ µ+ M Z(1− sin 2β) 2M1M2 (M1 cos 2 θW +M2 sin 2 θW ), (3.1) ≃ µ− µ sin 2β. It is seen that the spectrum of the lowest states, χ0 1, 2 and χ , is nearly degenerated. For the Higgsino DM this fact is known (see, for example, [16, 25, 26, 29]). Two other neutralino states, χ0 3, 4, and heavy chargino χ are placed far from the lightest ones, near the high scale MSUSY . The initial Lagrangian of Z −h1,2 interactions, including the µ-term, is the following (for more details see [28]) gZZµ(h̄1Lγ µh1L + h̄2Rγ µh2R)− µ(h̄1Rh2L + h̄1Lh2R). (3.2) Taking into account that h̄aLγ µhaL = −h̄aRγµhaR, h̄1Rh2L = h̄2Rh1L, from (3.2) in the limit of pure neutral Higgsino it follows: L = − gZZµh̄Dγ µhD − µh̄DhD, (3.3) where hD = h1R + h2L is the Dirac neutral field and gZ = g2/ cos θW . When the field hD is expressed through Majorana fields χ1 and χ2 as hD ≡ h1R + h2L = (χ1 − iχ2)/ 2, (3.4) for the transition (h1, h2) → (χ1, χ2) we get h1,2 = (χ1 ∓ iγ5χ2); χ1 = (h1 + h2), χ2 = (h1 − h2). (3.5) All processes near the EW scale are described by the SM Lagrangian together with extra Lagrangian of the Higgsino interactions with photons and vector bosons: ∆L ≃ − eAµ + 2 cos θ cos 2θZµ µχc + 2 cos θ W+µ (iχ̄ + χ̄0 )γµχc + W−µ χ̄cγ µ(iχ0 ), (3.6) where χc denotes the lowest chargino state, χ . Remember that χ0 1, 2 states are the Majorana spinors and chargino χc is the Dirac spinor. In the above Lagrangian the vertices are considered in the limit of zero mixing, because contributions to the vertices induced by a mixing are of an order of MZ/MSUSY . So correc- tions induced by mixing are negligibly small and can be omitted. However, the same small mixing contributions to the particle mass spectrum should be involved into analysis. The reason is that the decay widths depend on the mass splitting crucially. B. Split Higgsino as the DM carrier Supposing that the lightest Higgsino-like neutralino is a carrier of the DM in the Universe, we evaluate its mass from the DM relic abundance. In accordance with the known method for the relic abundance calculation (see, for ex- ample, [30, 31, 32, 33] and [34] with references therein) before freeze-out neutralinos are in the thermodynamical equilibrium with other components of the cosmological plasma. Relic is formed by the irreversible annihilation process – from the moment, when the freeze- out is reached, up to the present day when temperature is approximately equal to abso- lute zero (this approximation is sufficient for the case). With the standard estimation of xf = Mχ/Tf ≈ 20−25 the value of Tf can be well above TEW ∼ 100 GeV only ifmχ > 2 TeV. Now, if the lowest Higgsino masses were well above TEW , the irreversible annihilation pro- cess could start before electro-weak phase transition (it is considered as the first order phase transition) in the high-symmetric phase of the cosmological plasma. Then the plasma does not contain any Higgs condensate, and all standard particles, except Higgsino, are massless (more exactly, their masses ≪ T ). In the ordinary low-symmetric case, the neutralino annihilation cross section is calculated with the Lagrangian (3.6). In relic calculations all possible coannihilation channels were taken into account (there is no coannihilation with squarks and/or sleptons), namely χαχ̄β → ZZ, W+W− χcχ̄c → ZZ, W+W−, f f̄ , γγ, Zh χαχ̄c → ZW, lνl, qiq̄j , γW, Wh, (3.7) where α, β = 1, 2 denote the lowest neutralino states. In the scenario the effective neutralino annihilation cross section is: < (σv)ann >= 128 πM2χ 27 + 4(3 + 5t4W )(c W + s W )− kc2W + (c4W + s 2 c4W (c4W + s W ) + (c W − s2W )4 2 c2W 10− 1 + 2kc2W (3.8) where tW = tanΘW , sW = sinΘW , cW = cosΘW and k = M W . To compare the calculated value of Ωh2 with an experimental corridor of the relic data [35], we use known values of the above parameters and extract the following LSP (Higgsino) mass: Mχ = 1.0 − 1.4 TeV for xf = 25 and Mχ = 1.4 − 1.6 TeV for xf = 20. These values do not spoil the gauge coupling convergence and are in agreement with the results from [2, 3, 9, 29]. Thus, in the model where two lightest neutralinos and one chargino are the closest to the EW scale having masses O(1 TeV). Further, we will use Mχ = 1.4 TeV for all numerical estimations as an average value. To answer whether the neutralino annihilation process can start in the high-symmetric phase, we calculate the annihilation cross section with the Lagrangian which contains phys- ical states presented by chiral fermions and gauge fields B,Wa (a = 1, 2, 3). Due to the absence of the Higgs condensate neutralino and chargino degrees of freedom join into the fundamental SU(2) representation, i.e., Dirac field χD. All states of this field are dynam- ically equivalent and correspond to quantum numbers of the restored (unbroken) SU(2) symmetry. In t- and s-channels all cross sections of Higgsino annihilation into gauge bosons and massless fermions were calculated analogously to the QCD calculation technology. The only difference is that it is necessary to consider all channels with initial and final states which have an arbitrary color in two dimensions, corresponding to the restored SU(2) symmetry. Calculating the total neutralino annihilation cross section, from the known corridor of the relic abundance values we get the lowest Higgsino mass ∼ 1 TeV again. It contradicts the initial supposition that Mχ > 2 TeV to provide freeze-out temperature higher than the EW phase transition temperature. So in the framework of orthodox notions (mechanisms of extra entropy production or superheavy states decays, etc. are not considered) this scenario does not allow the DM to be created in the high symmetric phase. IV. SUSY SCALES AND EXPERIMENTAL POSSIBILITIES A. χ−N scattering and collider signals It is important to understand how an experimental study of the Split Higgsino scenario can be realized. Here we discuss some possibilities that can be given by neutralino-nucleon scattering processes, collider experiments with SUSY particles creation and the data on photon spectrum from neutralino annihilation. The last one will be considered in the next subsection. An experimental observation of the scenario manifestations depends on neutralino mass splitting parameters ∆Mχ0 = Mχ0 and ∆Mχc = Mχc −Mχ0 crucially. These splittings are determined by the sum of tree splittings and radiative corrections to masses. Depending on the M0 spacing and structure of high energy states, loop corrections can be comparable with tree values of ∆Mχ (∆Mχ0 or ∆Mχc) or even exceed them [12, 14, 15, 16, 26]. As it is known [16], the hierarchy of t̃1 and t̃2 states and their mixing angle Θt drive the value of the mass difference when squarks dominate in loops ∆Mχc ∼ ∆Mχ0 ∼ m3t sin(2Θt) · ln(m2t̃1/m Due to these (large) corrections the mass splittings can be increased up to ∼ 10GeV [16, 26]. Then let us consider two possible variants. Firstly, let there be large mass splittings ∆Mχ ∼ (1−10) GeV or even larger. When this is the case, the lowest Higgsino and chargino states can be detected at the LHC, in particular, due to specific decay channels of χ0 and χc – the corresponding analysis was considered in some detail in [9, 10, 11, 12], so we do not repeat it here. In the Split Higgsino hierarchy M0 & M1/2 ≫ µ, squarks (sleptons) slip out of the LHC experiment (together with the high energy gaugino), keeping specific chargino (neutralino) decays as the observable only. If, however, M1/2 ≫ M0, there are two additional possibilities. Namely, if M1/2 ≫ M0 ∼ µ, i.e., squarks (sleptons) have masses ∼ (1 − 2) TeV, an occasion arises to observe their signals at the LHC. (If mq̄ and/or ml̄ are sufficiently close to the lowest neutralino mass, some coannihilation contribution to the effective neutralino annihilation cross section is produced.) When M1/2 ≫ M0 ≫ µ there are no observable effects of squarks (sleptons) at the LHC scale. Both of these last subscenarios are very peculiar due to a large splitting between M1/2 and M0 – it can produce relatively long-lived superscalars at TeV (or higher) scale. In this case, specific manifestations of these states both at colliders (long-lived squarks and /or sleptons, changes in their decay modes hierarchy etc.), and in neutralino-nucleon scattering (large contribution to SI interaction) should be. In this paper, however, we concentrate only on the scenario with M1/2 ∼ M0 where superscalars are very heavy. Neutralino-nucleon interaction in the scenario behaves as a threshold process due to the formalism accepted – the corresponding term in the Lagrangian is nondiagonal in neutralino fields: i(g2/2 cos θ)Zµχ̄ . In the case of pure neutralino states, zero order contribu- tions to χ − N interaction correspond to the spin-independent (SI) inelastic process. This conclusion results from the interaction Lagrangian – nondiagonal Higgsino current χ0 interacts with Zµ as a vector, rather than an axial vector, it is a consequence of the real 4-dimensional Majorana formalism used (see [28]). At the tree level the zero order SI cross section for χ0 − N reaction takes the following threshold form σSIχ0N = g4M2N 64π cos4 θWM EN −∆Mχ0 . (4.1) In the non-relativistic case EN = Wk(mN/Mχ), where Wk is an average kinetic energy of the neutralino in the Sun neighborhood, Wk = Mχv r/2. For Mχ0 ∼ 1 TeV this energy Wk ∼ 1 MeV. So for ∆Mχ0 as large as (1 − 10)GeV the threshold value for the reaction energy is unattainable. Then the process is forbidden, and cosmic neutralinos cannot be detected in the direct terrestrial experiments. The cross section for the neutralino-nucleon scattering with chargino production (recharge process) is similar to (4.1) and for large ∆Mχc ≈ 0.5∆Mχ0 this channel is also closed. Corrections induced by a nonzero mixing and/or loop diagrams cannot make these non- diagonal reactions visible because part of correction amplitudes is damped in the limit of pure Higgsino; contributions from squark exchanges are small due to large squark masses (see also [36, 37]). Furthermore, due to Majorana nature of neutralino, nonzero loop contri- butions are proportional to the small parameter q2/M2Z ≪ 1. As concerns elastic (diagonal) channels, they contribute to the χ−N cross section via loops or due to a nonzero mixing, so their yield is small as well. Returning to some RG arguments, it seems that when M0 & M1/2 we can expect ∆Mχ somewhat lower than for the case M1/2 ≫ M0 when the splitting between t̃1 and t̃2 can be larger. Certainly, the relative splitting and mixing of these states are really unknown. So let us consider the second case with ∆Mχ < 1GeV and it can be as low as ∼ (100− 300)MeV if mass splittings are mainly determined by tree contributions. As to collider signature, in this case only photon, neutrino pair or low energy e−e+, µ−µ+, π−π+ pairs can be created in the final states, but it is hard to select these events from the background (see also [9, 10, 11]). So in the case of low ∆Mχ, degenerated Higgsino and chargino are indeed invisible at the LHC. The Higgsino-nucleon nondiagonal interaction takes place again in this case, contributing to the SI cross section. Comparing the model predictions for χ − N reaction with experi- mental restrictions on the SI cross section [38, 39] it is possible to estimate the MSUSY value. From the inequality ∆Mχ0 = (M Z/(M1M2)(M1 cos 2(θW ) +M2 sin 2(θW ) < Wk(MN/Mχ), it follows that the process χ0 N → χ0 N ′ is closed when MSUSY ≤ 8.3 ·109 GeV and MSUSY ≤ 1.2 · 1010 GeV for Mχ0 = 1.0 TeV and 1.4 TeV, respectively. These estimations depend on tan β weakly and are in agreement with the ones given by the RG analysis. So we conclude that in this scenario the SI inelastic Higgsino-nucleon scattering cannot be observed experimentally today. It seems that the inverse inelastic reaction χ0 N → χ0 N ′ is possible, but χ0 states are unstable, so they decayed a long time ago. In other words, the only inequality τχ ≤ T0 takes place, where T0 is the age of the Universe. An upper estimation for τχ follows from the width Γ(χ2 → χ1νν̄) (when ∆Mχ0 is reasonably small we consider only the most ”soft” channel) and we have ∆M5χ. (4.2) Then the following restriction emerges MSUSY ≤ M2Z G2FT0 . (4.3) With the value T0 = 3.15 · 1017 s we get MSUSY ≤ 4.25 · 109GeV. It is again in accordance with the RG results and slightly more stringent than the restrictions following from the threshold inequality ∆Mχ ≥ Wk(MN/Mχ). Electroweak corrections to the splitting ∆Mχ± can be as small as ∼ 100MeV due to loops with γ, Z and W [14]. When ∆Mχ± ∼ 100MeV (squark loop contributions are too small), recharge process χ0 n → χ± p accompanied by a track of χ± seems as possible due to very energetic neutralinos, but their cross section is strongly damped and the reaction is entirely exotic. So the recharge process cannot be detected experimentally too. Depending on mass splitting the chargino lifetime can be estimated in the following manner: from the channel χ± → χ0eν̄e we get τχ± = (30π 3/G2F )∆M but in the intermediate interval of ∆Mχ± = (0.1 − 1.0) GeV there are also chargino decay channels with the final µ and π-meson. For these decays the corresponding formulae are more cumbersome, and we do not write them here. Gathering all contributions, we get approximately τχ± ∼ (10−7 − 10−12) s. Thus, for both the cases – large or small ∆Mχ – the subscenarios of this class do not produce practically any visible signal in various channels of neutralino-nucleon scattering at modern measuring tools. If ∆Mχ values are sufficiently large, there is a chance to discover at the LHC some decay modes of the lowest neutralino and chargino states. Moreover, if some specific squark (slepton) effects occur at low TeV scale too, but gaugino manifestations are absent, it may correspond to the subscenario M1/2 ≫ M0 ∼ µ. The existence of specific neutralino-chargino decays together with the absence of other SUSY states at the TeV scale can be understood in the framework of the subscenarios M0 & M1/2 ≫ µ or M1/2 ≫ M0 ≫ µ. Absence in an experiment of various decay modes manifestations can indicate that the subscenarios with M0 & M1/2 ≫ µ or M1/2 ≫ M0 ≫ µ are realized, providing the low- est neutralino and chargino with a well degenerate spectrum that cannot be observed in experiment. Note, for all scenarios with a large splitting between M1/2 and M0 a large contribution to the SI neutralino-nucleon cross section is possible due to the squark exchange, especially if the superscalar scale is closed to µ. Naturally, a study of collider and χ− N data corre- lations is necessary, making details of the state spectrum more precise. Data on neutralino annihilation photon spectrum are also needed to complete an analysis of capabilities of the scenarios. B. Diffuse gamma spectrum from the Galactic halo As it follows from above, in the Split Higgsino scenario obvious manifestations of SUSY can be in some latent form: direct interaction of neutralino with nuclei is too small and it is very questionable whether effects of degenerate neutralino and chargino can be detected at the LHC if ∆Mχ < 10GeV. It seems that a chance to verify this scenario is to study the photon spectra from neutralino annihilation in the Galactic halo. The process can produce gamma quants in two ways: direct photon creation through loop diagrams or formation of a diffuse (continuous) gamma spectrum due to secondary photons created in radiative decays of mesons. In this scenario the dominant mode of continuous gamma spectrum creation is Hig- gsino annihilation into WW and ZZ bosons followed by creation and radiative decays of light mesons, mainly through the channel π0 → 2γ. For the lowest Higgsino mass Mχ = 1.4 TeV we calculate the cross section of Higgsino annihilation into WW and ZZ and get (σv)WW+ZZ ≈ 0.7 · 10−26 cm3s−1. Further, the same spectrum e−7.76x where x = Eγ/Mχ, is used for both (W and Z) channels approximately ( [40, 41]). Then the total diffuse photon flux from halo can be calculated as Φγ(E0, Em) ≈ 9.3 · 10−13 cm−2 s−1 · (4.4) < σv >WW+ZZ 10−26 cm3 s−1 · J̄(∆Ω) ·∆Ω, where E0 is the threshold photon energy for an apparatus, Em is the maximal registered photon energy, J̄(∆Ω) is averaged over angle value of the integral J(Ψ) which contains information on the DM distribution in halo [41, 42]. Fixing some value J̄(10−3) ≈ 1.2 · 103 that is typical of the Navarro-Frenk-White profile, where ∆Ω = 10−3 sr is used (see [42, 43]), we evaluate the total continuous gamma flux that can be measured at space-based telescopes (EGRET or GLAST) or at ground based Atmospheric Cherenkov Telescopes (ACT) (HESS, in particular). Then from (4.5) for the total flux we get EGRET ≈ 0.17 · 10 −10 cm−2 s−1, E0 = 1GeV, Em = 20GeV, GLAST ≈ 0.19 · 10 −10 cm−2 s−1, E0 = 1GeV, Em = 300GeV, HESS ≈ 0.82 · 10 −12 cm−2 s−1, E0 = 60GeV, Em = 1TeV. (4.5) As it is seen, the calculated values are beyond experimental possibilities of these telescopes – at present, only GLAST has some chance to measure the total continuous gamma flux, because it can detect a gamma flux as small as Φ GLAST (exp) ≈ 10−10 cm−2 s−1. In the near future, however, ACT like HESS, for example, will be able to fix the flux due to high sensitivity level [41] Φ HESS(exp) ≈ 10−14 cm−2 s−1. Note also that similar results were derived in [8] for direct photon signals. A characteristic value of the flux is not characteristic feature of the scenario, nearly the same values arise in all schemes with highly degenerate Higgsino as the lowest state. Nevertheless, from comparison with other model predictions (MSSM, mSUGRA, etc., see, for example [41, 42, 43, 44, 45, 46, 47]) we note that channel of neutralino annihilation into quarks increases the total flux up to one order of magnitude, so the flux could be well over the GLAST sensitivity threshold. Then the Split Higgsino scenario prediction for the diffuse gamma flux can be discriminated from predictions of models where there is a large contribution of quark and/or Higgs annihilation modes. Nevertheless, some conclusion on the Split Higgsino scenario realizability can be made only from the whole data analysis, using χ−N scattering, collider experiments, and photon spectrum data together. V. CONCLUSIONS We have pointed out that from the one-loop RG analysis of the SUSY SU(5) theory there can be extracted a few sets of energy scales which are compatible with conventional ideas on the DM structure and manifestations. Threshold corrections, induced by heavy states near MGUT – M24, M5, – are especially important for establishment of the hierarchies. Due to a specific form of the RG equations the scalar scale M0 remains arbitrary, and it occurs the variety of scenarios divided in the two classes: M1/2 ≫ µ or M1/2 ≪ µ. Note also that the refined RG analysis with two-loop β-functions and mass spectrum improved by radiative corrections can make the energy scale set more precise. Namely, the M0 scale can be split to establish squark and slepton scales separately, while the energy scale hierarchies as two global classes should remain. In this paper, the hierarchy M0 & M1/2 ≫ µ (the Split Higgsino model) was considered in detail. In this case, the renormalization group approach determines the SUSY breaking scale as MSUSY ∼ 108 − 109GeV. Due to the degenerate mass spectrum of the lightest states in the model the coannihilation channels are essential for the effective annihilation cross section. With the calculated value of < σeffv > the relic abundance value is provided by the lowest Higgsino mass in the interval 1.2 − 1.6TeV. Experimental observation of this scenario effects crucially depends on ∆Mχ0 and ∆Mχc . If these differences are ∼ 10GeV, products of χ0 and χ± decays can be, in principle, detected at the LHC, for small splittings these decay modes are invisible. At the same time, if superscalars are close to the lowest Higgsino scale, their specific manifestations are possible too. In particular, such squarks at a TeV scale should increase significantly the SI neutralino- nucleon cross section. Heavy scalars from the hierarchy M0 ≫ µ do not change the tree level χ−N cross section significantly. Even if there are signals from neutralino and /or chargino decays, they are hardly de- tected. Despite these (possible) effects the Split Higgsino model is characterized by χ− N scattering with the small and therefore yet unregistered SI cross section and some typical value of a continuous annihilation gamma flux. For small ∆Mχ values the MSUSY scale is evaluated as . 1010GeV in agreement with the RG results leading to unobserved SI χ−N scattering. 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[48] A. Provenza, M. Quiros, P. Ullio, ArXiv: hep-ph/0609059; Y. Mambrini, C. Munoz, J. Cosm. Astropart. Phys. 10, 003 (2004). http://arxiv.org/abs/hep-ph/0702148 http://arxiv.org/abs/hep-ph/0601041 http://arxiv.org/abs/hep-ph/0512090 http://arxiv.org/abs/hep-ph/0509209 http://arxiv.org/abs/hep-ph/0407158 http://arxiv.org/abs/astro-ph/0305075 http://arxiv.org/abs/hep-ph/0407342 http://arxiv.org/abs/hep-ph/0309029 http://arxiv.org/abs/astro-ph/0212509 http://arxiv.org/abs/hep-ph/0609059 Introduction Renormgroup analysis The Split Higgsino scenario The model Lagrangian and its features Split Higgsino as the DM carrier SUSY scales and experimental possibilities -N scattering and collider signals Diffuse gamma spectrum from the Galactic halo Conclusions References

0704.0626

Probing dark energy with steerable wavelets through correlation of WMAP and NVSS local morphological measures

Mon. Not. R. Astron. Soc. 000, 1–12 (2007) Printed 25 October 2018 (MN LaTEX style file v2.2) Probing dark energy with steerable wavelets through correlation of WMAP and NVSS local morphological measures J. D. McEwen1⋆, Y. Wiaux2, M. P. Hobson1, P. Vandergheynst2, A. N. Lasenby1 1Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, UK 2Signal Processing Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Accepted 27 November 2007. Received 27 November 2007; in original form 12 April 2007 ABSTRACT Using local morphological measures on the sphere defined through a steerable wavelet analy- sis, we examine the three-year WMAP and the NVSS data for correlation induced by the inte- grated Sachs-Wolfe (ISW) effect. The steerable wavelet constructed from the second deriva- tive of a Gaussian allows one to define three local morphological measures, namely the signed- intensity, orientation and elongation of local features. Detections of correlation between the WMAP and NVSS data are made with each of these morphological measures. The most sig- nificant detection is obtained in the correlation of the signed-intensity of local features at a significance of 99.9%. By inspecting signed-intensity sky maps, it is possible for the first time to see the correlation between the WMAP and NVSS data by eye. Foreground contamination and instrumental systematics in the WMAP data are ruled out as the source of all significant detections of correlation. Our results provide new insight on the ISW effect by probing the morphological nature of the correlation induced between the cosmic microwave background and large scale structure of the Universe. Given the current constraints on the flatness of the Universe, our detection of the ISW effect again provides direct and independent evidence for dark energy. Moreover, this new morphological analysis may be used in future to help us to better understand the nature of dark energy. Key words: cosmic microwave background – cosmology: observations – methods: data anal- ysis – methods: numerical. 1 INTRODUCTION A cosmological concordance model has emerged recently, explain- ing many observations of our Universe to very good approxi- mation. In this model our Universe is dominated by an exotic dark energy component that may be described by a cosmologi- cal fluid with negative pressure, interacting only gravitationally to counteract the attractive gravitational nature of matter. Dark energy may be strongly inferred from observations of the cos- mic microwave background (CMB), such as the Wilkinson Mi- crowave Anisotropy Probe (WMAP) data (Bennett et al. 2003b; Hinshaw et al. 2007), together with either observations of Type Ia Supernovae (Riess et al. 1998; Perlmutter et al. 1999) or of the large scale structure (LSS) of the Universe (e.g. Allen et al. 2002). Although dark energy dominates the energy density of our Uni- verse, we know very little about its origin and nature. Indeed, a consistent model to describe dark energy in the framework of par- ticle physics is lacking. The integrated Sachs-Wolfe (ISW) effect (Sachs & Wolfe 1967) provides an independent physical phenomenon that may be ⋆ E-mail: [email protected] used to detect and probe dark energy. As CMB photons travel to- wards us from the surface of last scattering they pass through gravi- tational potential wells due to the LSS. If the gravitational potential evolves during the photon propagation, then the photon undergoes an energy shift. The ISW effect is the integrated sum of this en- ergy shift along the photon path. In a matter-dominated Einstein-de Sitter universe the gravitational potential remains constant with re- spect to conformal time, thus there is no ISW effect. However, if the universe deviates from matter domination due to curvature or dark energy then an ISW effect is induced. Strong constraints have been placed on the flatness of the Universe by WMAP (Spergel et al. 2007), hence a detection of the ISW effect may be inferred as di- rect and independent confirmation of dark energy. It is difficult to isolate the ISW contribution to CMB anisotropies directly, therefore Crittenden & Turok (1996) sug- gested that the ISW effect be detected by cross-correlating CMB anisotropies with tracers of the local matter distribution, such as the nearby galaxy density distribution. A positive large-scale correlation will be induced by the ISW effect as a consequence of decaying gravitational potentials due to dark energy. First attempts to detect the ISW effect using Cosmic Background Explorer-Differential Microwave Radiometer (COBE-DMR) data c© 2007 RAS http://arxiv.org/abs/0704.0626v2 2 McEwen et al. failed, concluding that greater sensitivity and resolution than that provided by COBE were required (Boughn & Crittenden 2002). Fortunately, the WMAP mission soon provided suitable CMB data. Correlations indicative of the ISW effect have now been detected between both the first- and three-year WMAP data and a large number of tracers of the LSS. Detections have been made using NRAO VLA Sky Survey (NVSS; Condon et al. 1998) data (Boughn & Crittenden 2004; Nolta et al. 2004; Vielva et al. 2006; McEwen et al. 2007b; Pietrobon et al. 2006), hard X-ray data provided by the High Energy Astronomy Observatory- 1 satellite (HEAO-1; Boldt 1987) (Boughn & Crittenden 2005, 2004), APM galaxy survey (Maddox et al. 1990) data (Fosalba & Gaztañaga 2004), Sloan Digital Sky Survey (SDSS; York D. G. et al. (SDSS collaboration) 2000) data (Scranton et al. 2003; Fosalba et al. 2003; Padmanabhan et al. 2005; Cabré et al. 2006; Giannantonio et al. 2006), the Two Micron All Sky Survey Extended Source Catalogue (2MASS XSC; Jarrett et al. 2000) (Afshordi et al. 2004; Rassat et al. 2006) and combinations of the aforementioned data sets (Gaztañaga et al. 2006). Furthermore, a number of other works have focused on theoretical detectability, future experiments and/or error analyses related to ISW detections (Afshordi 2004; Hu & Scranton 2004; Pogosian 2005, 2006; Pogosian et al. 2005; Corasaniti et al. 2005; LoVerde et al. 2007; Cabré et al. 2007). Many different analysis techniques have been employed in these previous works to detect correlations between the data sets, each of which have their own merits and limitations. In this work we focus on wavelet-based techniques. Wavelets provide an ideal analysis tool to search for the ISW effect. This is due to the localised nature of the effect, in both scale and position on the sky, and the simultaneous scale and position localisation afforded by a wavelet analysis. Since the ISW effect is cosmic variance limited, it is desirable to examine as great a sky cover- age as possible. In this near full-sky setting the geometry of the sphere should be taken into account, thus wavelet analyses on the sphere are required. The first analysis using wavelets on the sphere to search for correlations between the CMB and LSS was per- formed by Vielva et al. (2006) using the axisymmetric spherical Mexican hat wavelet. McEwen et al. (2007b) extended this anal- ysis to directional wavelets, as correlated features induced by the ISW effect may not necessarily be rotationally invariant. Indeed, it is known that statistically isotropic Gaussian random fields are characterised by local features that are not rotationally invariant (Barreiro et al. 1997, 2001). Recently, Pietrobon et al. (2006) ap- plied a new wavelet construction on the sphere called needlets to search for correlations. All of these works examined the WMAP and NVSS data (due to the large sky coverage of the NVSS data) and made significant detections of the ISW effect. Moreover, in each analysis the detection of the ISW effect was used to constrain dark energy parameters. In this paper we present a new search for correlation between the three-year WMAP and the NVSS data using wavelets on the sphere. However, the approach taken here differs fundamentally to previous wavelet analyses. We use steerable wavelets on the sphere to extract, from each of the two data sets, measures of the morphol- ogy of local features, such as their signed-intensity, their orienta- tion, or their elongation. We then correlate the WMAP and NVSS data through these quantities. This represents the first direct anal- ysis of local morphological measures on the sphere resulting from the steerability of wavelets. A positive detection of correlation us- ing any of these morphological probes would give a direct indi- cation of the ISW effect, provided it were not due to unremoved foregrounds or measurement systematics, which could again be in- ferred as direct and independent evidence for dark energy. More- over, such a detection would give an indication of the morphologi- cal nature of the correlation. This further insight might help to bet- ter understand the nature of dark energy. However, the derivation of the theoretical correlation of the newly defined morphological measures as a function of the dark energy content of the Universe is not easily tractable. Consequently, corresponding constraints on dark energy parameters are left to a future work. The remainder of this paper is organised as follows. In sec- tion 2 we present the physical and signal processing background behind the analyses performed. The ISW effect is described and the use of local morphological measures defined through a steer- able wavelet analysis is presented as a probe to search for the ISW effect. In section 3 the data and analysis procedures employed are described in detail. Two distinct analysis techniques are proposed to compute correlations from morphological measures: a local mor- phological analysis and a matched intensity analysis. The results of these analyses are presented in section 4 and section 5 respectively. Concluding remarks are made in section 6. 2 ISW EFFECT AND STEERABLE WAVELETS The existence of an ISW effect induces a cross-correlation between the CMB and NVSS signals on the celestial sphere. From the spec- tral point of view, the corresponding two-point cross-correlation function may be expressed in terms of the cross-correlation angu- lar power spectrum. In this paper, we go beyond this spectral anal- ysis thanks to a decomposition of the CMB and NVSS signals with steerable wavelets on the sphere. The wavelet analysis enables one to probe local features at each analysis scale. In this section we first review the ISW effect and the correlation between the CMB and NVSS data that it induces. Secondly, we discuss wavelets on the sphere and describe how the steerability of wavelets is used to define local morphological measures on the sphere. 2.1 ISW effect The secondary temperature anisotropy induced in the CMB by the ISW effect is related to the evolution of the gravitational poten- tial. Any recent acceleration of the scale factor of the Universe due to dark energy will cause local gravitational potentials to decay. CMB photons passing through over dense regions of decaying po- tential suffer blue shifts, resulting in a positive correlation between the induced anisotropy and the local matter distribution. It can be shown that the relative temperature fluctuation induced in the CMB is given by T(ω) = ∆T (ω) dη Φ̇(η, ω) (1) (Sachs & Wolfe 1967; Nolta et al. 2004), where ∆T is the induced temperature perturbation, T0 is the mean temperature of the CMB, η is conformal time, Φ is the gravitational potential and the dot represents a derivative with respect to conformal time. A point ω on the sky is represented in spherical coordinates as ω = (θ, ϕ), with co-latitude θ and longitude ϕ. The integral is computed over the photon path from emission to observation, i.e. from today back to the last scattering surface. In a matter-dominated Einstein-de Sitter universe (with zero cosmological constant) the potential evolves as Φ ∼ δ/R, where δ is the matter perturbation and R is the scale factor of the universe. In this setting the matter perturbation evolves with c© 2007 RAS, MNRAS 000, 1–12 Probing dark energy with steerable wavelets 3 the scale factor, δ ∝ R. Consequently, Φ̇ = 0 and there is no ISW effect, as discussed previously. We use the NVSS galaxy count distribution projected onto the sky as a tracer of the local matter distribution. It is as- sumed that the two corresponding relative fluctuations, respec- tively δN(z, ω) and δ(z, ω), are related by the linear bias factor b(z): δN(z, ω) = b(z) δ(z, ω), where z is redshift. Hereafter we take the bias to be redshift independent since the redshift epoch over which the ISW effect is produced is small. The galaxy source count fluc- tuation observed on the sky is therefore given by N(ω) = b δ(z, ω) , (2) where dN/ dz is the mean number of sources per steradian at red- shift z and the integral is performed from today to the epoch of recombination, i.e. last scattering. We are now in a position to consider the correlation between the galaxy count and CMB temperature fluctuations. We consider the cross-power spectrum CNT defined by the ensemble average of the product of the spherical harmonic coefficients of the two signals observed on the sky: 〈∆Nℓn ∆ ℓ′n′ 〉 = δℓℓ′δnn′ C ℓ , (3) where ∆ℓn = 〈Yℓn|∆〉 are the spherical harmonic coefficients of ∆(ω), 〈·|·〉 denotes the inner product on the sphere, Yℓn are the spher- ical harmonic functions for multipole ℓ ∈ N, n ∈ Z, |n| 6 ℓ and δi j is the Kronecker delta symbol. In writing the cross-correlation in this manner we implicitly assume that the galaxy density and CMB random fields on the sphere are homogeneous and isotropic, which holds under the basic assumption of the cosmological principle. Representing the gravitational potential and the matter density per- turbation in Fourier space and substituting (1) and (2) into (3), it is possible to show (e.g. Nolta et al. 2004) that CNTℓ = 12π Ωm H0 δ(k) F ℓ (k) F ℓ (k) , (4) where Ωm is the matter density, H0 is the Hubble parameter, (k) = k3Pδ(k)/2π 2 is the logarithmic matter power spectrum, Pδ(k) = 〈|δ(k)|2〉 is the matter power spectrum and the filter func- tions for the galaxy density and CMB are given by FNℓ (k) = b D(z) jℓ[kη(z)] (5) FTℓ (k) = jℓ[kη(z)] (6) respectively. The integration required to compute FT (k) is per- formed over z from zero to the epoch of recombination, whereas, in practice, the integration range for FN (k) is defined by the source redshift distribution dN/ dz. D(z) is the linear growth factor for the matter distribution: δ(z, k) = D(z)δ(k), with δ(k) = δ(0, k). The function g(z) ≡ (1 + z)D(z) is the linear growth suppression factor and jℓ(·) is the spherical Bessel function. We have represented in harmonic space the expected correlation between the galaxy source count fluctuations and the CMB temperature fluctuations induced by the ISW effect. We next turn our attention to steerable wavelets on the sphere as a potential tool for detecting this correlation. 2.2 Wavelets on the sphere The WMAP and NVSS data may be understood as the sampling of continuous signals on the sphere, which can be analysed in the framework of a continuous wavelet formalism. The analysis of a signal on the sphere with a wavelet, which is a local analysis func- tion, yields a set of wavelet coefficients. These coefficients result from the scalar products between the signal and the wavelet di- lated at any scale, rotated around itself by any angle, and trans- lated at any point on the sphere. The so-called steerable wavelets allow, from these wavelet coefficients, the definition of local mor- phological measures of the signal. In particular, the second Gaus- sian derivative wavelet used in the present analysis gives access to the measures of orientation, signed-intensity, and elongation of the signal’s local features. The remainder of this section is devoted to the explicit definition of these local morphological measures. The reader not directly interested in these formal and technical details may proceed directly to section 3. The continuous wavelet formalism on the sphere originally proposed by Antoine & Vandergheynst (1998) was recently further developed in a practical approach by Wiaux et al. (2005). It de- fines the wavelet decomposition of a signal on the sphere in the following way. We consider an orthonormal Cartesian coordinate system (o, ox̂, oŷ, oẑ) centred on the (unit) sphere, with the direc- tion oẑ defining the North Pole. To relate this coordinate system to the spherical coordinates ω = (θ, ϕ) defined previously, we let the polar angle, or co-latitude, θ ∈ [0, π] represent the angle between the vector identifying ω and the axis oẑ, and the azimuthal angle, or longitude, ϕ ∈ [0, 2π) represent the angle between the orthogonal projection of this vector in the plane (o, ox̂, oŷ) and the axis ox̂. The signal to be analysed is represented by a square-integrable function F(ω) on the sphere, i.e. F(ω) ∈ L2(S 2, dΩ), for the invari- ant measure dΩ = d(cos θ) dϕ. A square-integrable analysis func- tion Ψ(ω) is defined, the so-called mother wavelet, initially centred at the North Pole. This wavelet may be dilated at any scale a > 0. The dilation of a wavelet on the sphere in this formalism may be uniquely related to the usual dilation in the tangent plane at the north pole (as discussed in more detail below). The corresponding Ψa can cover arbitrarily small or large regions of the sphere, re- spectively corresponding to high or low frequencies. Through the appropriate three-dimensional rotation, the wavelet may also be ro- tated on itself by any angle χ ∈ [0, 2π) and translated to any point ω0 = (θ0, ϕ0) on the sphere. The final dilated, rotated and translated wavelet is denoted Ψω0 ,χ,a. At each scale, the wavelet coefficients of the signal are defined by the directional correlation with Ψa, i.e. by the simple scalar product (ω0, χ, a) = 〈Ψω0 ,χ,a|F〉 ≡ dΩΨ∗ω0 ,χ,a (ω) F (ω) , (7) where the superscript ∗ denotes complex conjugation. These wavelet coefficients hence characterise the signal locally around each point ω0, at scale a and orientation χ. The choice of possi- ble mother wavelets is submitted to a condition ensuring that the original signal can be reconstructed exactly from its wavelet co- efficients. In practice, this wavelet admissibility condition on the sphere is not easy to check for a given candidate function. It is therefore difficult to build wavelets directly on the sphere. A correspondence principle has been established (Wiaux et al. 2005), which states that the inverse stereographic projection of a wavelet on the plane provides a wavelet on the sphere. The stereo- graphic projection of a point ω = (θ, ϕ) on the sphere gives by defi- nition a point ~x = (r, ϕ) on its tangent plane at the North pole, which is co-linear with ω and the South Pole. It relates the radial variables on the two manifolds through the bijection r(θ) = 2 tan(θ/2), and simply identifies the angular variables ϕ (see Fig. 1). If the corre- sponding unitary operator on functions is denoted Π, the correspon- c© 2007 RAS, MNRAS 000, 1–12 4 McEwen et al. ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ����������� ϕ(r, ) θ ϕ( , ) Figure 1. Wavelets on the sphere can be defined through simple inverse stereographic projection Π−1 of wavelets on the tangent plane at the North pole. The corresponding functions are illustrated here by the shadow on the sphere and the localised region on the plane. (Illustration reproduced from Wiaux et al. 2005.) dence principle states that, if ψ(~x) is a wavelet on the plane, then Ψ(ω) = [Π−1ψ](ω) is a wavelet on the sphere. The dilation of a wavelet on the sphere actually corresponds to the conjugate, through the stereographic projection, of the usual dilation in the tangent plane at the north pole. This conjugation relation also trivially holds for the rotation of wavelets by χ, which only affects the longitude, while the stereographic projection leaves it invariant. In other words, the dilation and rotation of the wavelet on itself may be performed through the natural dilation and rotation on the plane before projection: Ψχ,a(ω) = [Π −1ψχ,a](ω) . (8) Only the translation by ω0 must be performed explicitly on the sphere through the appropriate rotation in three dimensions. As wavelets on the plane are well-known and may be easily built as simple zero-mean filters, the correspondence principle enables one to define a large variety of wavelets on the sphere by simple projec- tion. This principle also allows one to transfer wavelet properties from the plane onto the sphere, such as the notion of steerability discussed in the next section. The wavelet analysis on the sphere described above is compu- tationally demanding. The directional correlation described by (7) requires the evaluation of a two-dimensional integral over a three- dimensional domain. Direct computation of this integral by sim- ple quadrature is not computationally feasible for practical CMB data, such as the currently available 3 megapixel WMAP maps and the forthcoming 50 megapixel Planck (Planck collaboration 2005) maps. Fast algorithms to compute the wavelet decomposition of signals on the sphere have been developed and implemented by Wiaux et al. (2006) and McEwen et al. (2007a) (for a review see Wiaux et al. invited contribution, 2007). In this work we utilise the implementation described by McEwen et al. (2007a). 2.3 Steerability and morphological measures Firstly, we review the notion of steerable wavelets on the sphere. Just as on the plane, a function on the sphere is said to be ax- isymmetric if it is invariant under rotations around itself. Any non- axisymmetric function is called directional. The directionality of a wavelet is essential in allowing one to probe the direction of local features in a signal. A wavelet is said to be steerable if its rota- tion around itself by any angle χ may be written as a simple linear combination of non-rotated basis filters. On the sphere, for M basis filters Ψm, a steerable wavelet may be written, by definition, as Ψχ(ω) = km(χ)Ψm(ω) (9) (Wiaux et al. 2005). The weights km(χ), with 1 6 m 6 M, are called interpolation functions. In particular cases, the basis filters may be specific rotations by angles χm of the original wavelet: Ψm = Ψχm . Steerable wavelets have a non-zero angular width in the azimuthal angle ϕ. This property makes them sensitive to a range of directions and enables them to satisfy their relation of definition. In spherical harmonic space, this non-zero angular width corresponds to an az- imuthal angular band limit N in the integer index n associated with the azimuthal variable ϕ:Ψℓn = 0 for |n| > N. Typically, the number M of interpolating functions is of the same order as the azimuthal band limit N. Notice that the steerability is a property related to the azimuthal variable ϕ only. Again as the stereographic projection only affects the radial variables, steerable wavelets on the sphere can easily be built as simple inverse stereographic projection of steerable wavelets on the plane. The derivatives of order Nd in direction x̂ of radial functions φ(r) on the plane are steerable wavelets. The corresponding steer- able wavelets on the sphere produced by inverse stereographic pro- jection may be rotated in terms of M = Nd + 1 basis filters, and are band-limited in ϕ at N = Nd + 1. The second derivative Ψ x̂ of any radial function has a band limit N = 3, and contains the frequen- cies n = {0,±2} only. Ψ∂ x̂ may be rotated in terms of three basis filters: the second derivatives in directions x̂ and ŷ, Ψ1 = Ψ x̂ and Ψ2 = Ψ ∂2ŷ , and the cross derivativeΨ3 = Ψ ∂x̂∂ŷ , with the correspond- ing weights k1(χ) = cos 2 χ, k2(χ) = sin 2 χ, and k3(χ) = sin 2χ. Fig. 2 illustrates the explicit example of the second derivative of a Gaussian, which is the wavelet employed in the subsequent analy- sis. In this case, the original radial function on the plane is given as φ(Gauss)(r) = − 4/3πe−r 2/2, with proper normalisation ensuring that the mother wavelet Ψ∂ (Gauss) on the sphere is normalised to unity. The explicit basis filters obtained by differentiation are ana- lytically defined by Wiaux et al. (2006). Recall that the natural dila- tion on the tangent plane at the North Pole sends the radial variable r(θ) onto r(θ)/a. The value of the scale a therefore identifies with the dispersion of the Gaussian. Consequently, we define the angu- lar size of our second Gaussian derivative wavelet on the sphere as twice the half-width of the wavelet, where the half-width is defined by θhw = 2 arctan(a/2), which is closely approximated by a at small scales. We use wavelet steerability to define local morphological measures of real signals F on the sphere. Local geometrical quanti- ties have previously been defined in real space, and have been used to analyse CMB data and test cosmological models (Barreiro et al. 1997, 2001; Monteserı́n et al. 2005; Gurzadyan et al. 2005). Our approach in wavelet space is completely novel. By linearity of the wavelet decomposition, the steerability relation also holds on the wavelet coefficients of a signal F, (ω0, χ, a) = km (χ) W (ω0, a) , (10) where the coefficients WF (ω0, a) result from the correlation of F with the non-rotated (χ = 0) filter Ψm at scale a. Technically these M coefficients thus gather all the local information retained by the steerable wavelet analysis at the point ω0 and at the scale a. One c© 2007 RAS, MNRAS 000, 1–12 Probing dark energy with steerable wavelets 5 (a) Ψ∂ x̂(Gauss) (b) Ψ (Gauss) (c) Ψ∂x̂∂ŷ(Gauss) (d) Ψ∂ (Gauss) rotated by χ = π/4 Figure 2. Mollweide projection of second Gaussian derivative wavelet on the sphere for wavelet half-width θhw ≃ 0.4 rad. The rotated wavelet illus- trated in panel (d) can be constructed from a sum of weighted versions of the basis wavelets illustrated in panels (a) through (c). may be willing to reorganise this information in terms of M quan- tities with an explicit local morphological meaning. For M = 1, the steerable wavelet is actually axisymmetric, and only a measure of signed-intensity IF(ω0, a) of local features in the signal is ac- cessible (given by the wavelet coefficient WF (ω0, a) directly). For M > 2, the steerable wavelet is directional and the orientation, or direction, DF (ω0, a) of local features may be accessed through the value which maximises the wavelet coefficient. The signed- intensity IF (ω0, a) of the local feature is given by that coefficient of maximum absolute value. Additional local morphological mea- sures may be defined for M > 3, notably a measure of the elonga- tion of local features, probing their non-axisymmetry, in addition to their signed-intensity and orientation. As the wavelet coefficient in each orientation naturally probes the extension of the local feature in the corresponding direction, this elongation EF (ω0, a) can sim- ply be measured from the absolute value of the ratio of the wavelet coefficient of maximum absolute value, to the wavelet coefficient in the perpendicular direction. The higher the number M of basis filters, the wider the accessible range of local morphological mea- sures. Any second derivative of a radial function, such as the sec- ond Gaussian derivative, is defined from M = 3 basis filters. The signed-intensity, orientation, and elongation are therefore accessi- ble in this case. Because the mother wavelet Ψ∂ x̂ oscillates in the tangent direction x̂, it actually detects features oriented along the tangent direction ŷ. The orientation of local features is therefore given as 6 DF (ω0, a) ≡ χ0 + , (11) where the orientation χ0 ≡ χ0(ω0, a) that maximises the wavelet co- efficient can be analytically defined from (10). Notice that the sec- ond derivative of a radial function is invariant under a rotation by π around itself. Consequently, the local orientations define headless vectors in the tangent plane atω0. By convention the corresponding vectors are chosen to point towards the Northern hemisphere only, and the local orientations are therefore in the range [π/2, 3π/2). For a second derivative of a radial function, as for any real steer- able wavelet, the signed-intensity of local features is given by the wavelet coefficient in the direction χ0: IF (ω0, a) ≡ WF (ω0, χ0, a) . (12) The elongation of local features analysed by the second derivative of a radial function is explicitly defined by 0 6 EF (ω0, a) ≡ 1 − ω0, χ0 + (ω0, χ0, a) 6 1 . (13) Numerical tests performed on elliptical Gaussian-profile features show that this elongation measure for the second Gaussian deriva- tive increases monotonously in the range [0, 1] with the intrinsic eccentricity e ∈ [0, 1] of the features. While it is possible to de- fine alternative elongation measures, these numerical tests indicate that the chosen definition is not an arbitrary measure of the non- axisymmetry of local features, but represents a rough estimate of the eccentricity of a Gaussian-profile local feature. 3 ANALYSIS PROCEDURES The data and analysis procedures that we use in an attempt to detect the ISW effect are described in detail in this section. We propose two distinct analysis techniques based on the correlation of the local morphological measures on the sphere defined in section 2.3. After describing the data, we discuss the generic procedure to search for any correlation between the NVSS and WMAP data. We then de- scribe in detail the two approaches proposed, motivating the analy- ses and highlighting the differences between them. 3.1 Data and simulations In this work we examine the three-year release of the WMAP data and the NVSS data for correlations induced by the ISW effect. Here we briefly describe the data, preprocessing of the data and the sim- ulations performed to quantify any correlations detected in the data. We examine the co-added three-year WMAP sky map. This map is constructed from a noise weighted sum of the maps ob- served by the Q-, V- and W- channels of WMAP, in order to enhance the signal-to-noise ratio of the resultant map. The co- added map was first introduced by the WMAP team in their non- Gaussianity analysis (Komatsu et al. 2003) and has since been used in numerous analyses. We use the template based foreground cleaned WMAP band maps (Bennett et al. 2003a) to construct the co-added map. The conservative Kp0 mask provided by the WMAP team is used to remove remaining Galactic emission and bright point sources. Since the ISW effect is expected to induce correla- tions on scales > 2◦ (Afshordi 2004), we downsample the co-added map to a pixel size of ∼55′ (a HEALPix1 (Górski et al. 2005) reso- lution of Nside = 64). This reduces the computation requirements of the subsequent analysis considerably, while ensuring the analysis remains sensitive to ISW induced correlations. The large sky coverage and source distribution of the NVSS data make it an ideal probe of the local matter distribution to use when searching for the ISW effect. Sources in the catalogue are thought to be distributed in the range 0 < z < 2, with a peak dis- tribution at z ∼ 0.8 (Boughn & Crittenden 2002). This corresponds closely to redshift regions where the ISW signal that we hope to detect is produced (Afshordi 2004). The NVSS source distribution is projected onto the sky in a HEALPix representation at the same resolution as the WMAP co-added map considered (i.e. Nside = 64) and an important systematic in the data is corrected. The correc- tion of this systematic and the resulting preprocessed NVSS data examined here are both identical to that analysed in McEwen et al. (2007b), hence we refer the interested reader to this work for more 1 http://healpix.jpl.nasa.gov/ c© 2007 RAS, MNRAS 000, 1–12 http://healpix.jpl.nasa.gov/ 6 McEwen et al. (a) WMAP (b) NVSS Figure 3. WMAP co-added three-year and NVSS maps (Mollweide projec- tion) after application of the joint mask. The maps are downsampled to a pixel size of ∼55′. The WMAP temperature data are reported in mK, while the NVSS data are reported in number-of-counts per pixel. details of the data preprocessing. Not all of the sky is sufficiently observed in the NVSS catalogue. We construct a joint mask to ex- clude from our subsequent analysis those regions of the sky not observed in the catalogue, in addition to the regions excluded by the WMAP Kp0 mask. The WMAP and NVSS data analysed sub- sequently, with the joint mask applied, are illustrated in Fig. 3. In order to constrain the statistical significance of any detec- tion of correlation between the WMAP and NVSS data we perform Monte Carlo simulations. 1000 simulations of the WMAP co-added map are constructed, modelling carefully the beam and anisotropic noise properties of each of the WMAP channels and mimicking the co-added map construction procedure (including the downsam- pling stage). The analyses subsequently performed on the data are repeated on the simulations and compared. 3.2 Generic procedure Two different analysis procedures are applied to test the data for correlation. The first procedure correlates the local morphological measures of signed-intensity, orientation and elongation, extracted independently from the two data sets. This procedure does not rely on any assumption about the correlation in the data. The sec- ond procedure correlates local features in the WMAP data that are matched in orientation to local features extracted from the NVSS data. This approach explicitly assumes that local features of the LSS are somehow included in the CMB. In the absence of any ISW effect one would not expect to detect any significant correlation in either of these analyses. The analysis procedures are described in more detail in the following subsections. Firstly, we describe the generic part of the procedure common to both techniques. We consider only those scales where the ISW effect is expected to be significant (Afshordi 2004), i.e. scales approximately corresponding to wavelet half-widths θhw of {100′, 150′, 200′, 250′ , 300′, 400′ , 500′ , 600′}. The local morpho- logical measures defined in section 2.3 are computed for the WMAP and NVSS data and are used to compute various correlation statistics (see the following subsections for more detail). Identical statistics are computed for the Monte Carlo simulations in order to measure the statistical significance of any correlation detected in the data. A statistically significant correlation in the data appar- ent from any of these statistics is an indication of the ISW effect, provided it is not due to foreground contributions or systematics. The procedure described previously would be sufficient for data with full-sky coverage. Unfortunately this is not the case and the application of the joint mask must be taken into account. The application of the mask distorts those morphological quantities constructed from wavelets with support that overlaps the mask ex- clusion regions. The associated local morphological measures must be excluded from the analysis. An extended mask is computed for each scale to remove all contaminated values. The extended masks are constructed by extending the central masked region by the wavelet half-width, whilst maintaining the size of point source regions in the mask. We use identical masks to those applied by McEwen et al. (2007b) and refer the interested reader to this work for more details. Before proceeding with our morphological analyses it is im- portant to check that the WMAP and NVSS data do not contain pre- dominantly axisymmetric features, corresponding to local elonga- tion trivially equal to zero and for which no local orientation might be defined. To do this we examine the distribution of our morpho- logical measure of elongation defined in section 2.3. In Fig. 4 we plot the histograms of the elongation measures computed from the data (outside of the extended exclusion masks). These histograms are built from all scales but the distributions obtained for each indi- vidual scale also exhibit similar structure. Many elongation values computed from both data sets lie far from zero, thereby justifying the morphological analyses that we propose. In order to give some intuition on the magnitude of various statistics computed in the remainder of this paper, we also quote the orders of magnitude for the means and standard deviations of the different local morphological measures, as estimated from the two data sets. Firstly, the mean and standard deviation of the elonga- tion in the WMAP data are respectively µ̂TE = 0.61 and σ̂ E = 0.27, when data from all scales are gathered. The corresponding values for the NVSS data are µ̂NE = 0.62 and σ̂ E = 0.26. Secondly, re- call that the galaxy density and CMB random fields on the sphere are assumed to be homogeneous and isotropic. The local orienta- tions computed from the data sets should therefore reflect a uni- form distribution in the range [π/2, 3π/2). Consistently, the mean and standard deviation of the orientation in the WMAP data are re- spectively µ̂TD = 3.10 and σ̂ D = 0.93, again gathering data from all scales. The corresponding values for the NVSS data are µ̂ND = 3.10 and σ̂ND = 0.90. Finally, notice that the local signed-intensity of a signal is expressed, a priori on the whole real line, in the same units as the signal itself. The signed-intensity of the WMAP temperature fluctuation data is given in mK, while the signed-intensity of the NVSS galaxy source count fluctuation data is given in number- of-counts per pixel (see Fig. 3). The mean and standard devia- tion of the signed-intensity in the WMAP data are respectively µ̂TI = −2×10 −4 and σ̂TI = 6×10 −3, still accounting for data from all scales. The corresponding values for the NVSS data are µ̂NI = 0.001 and σ̂NI = 0.01. c© 2007 RAS, MNRAS 000, 1–12 Probing dark energy with steerable wavelets 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag replacements (a) WMAP 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag replacements (b) NVSS Figure 4. Histograms of the elongation values computed from the WMAP and NVSS data for all scales. Many elongation values lie far from zero, thereby justifying the morphological analyses that we propose. 3.3 Local morphological analysis In the local morphological analysis that follows we correlate the WMAP and NVSS data through each of the morphological mea- sures provided by the second derivative of a Gaussian wavelet sep- arately, i.e. the signed-intensity, orientation and elongation of lo- cal features. Local features are extracted independently from the WMAP and NVSS data. We then compute the correlation by XNTS i (a) = S Ni (ω0, a) S i (ω0, a) S Ni (ω0, a) S Ti (ω0, a) , (14) where S i = {I, D, E} is the morphological measure examined, with superscript N or T representing respectively the NVSS and WMAP data (note that the morphological measures are computed at Np dis- crete points on the sky only). This analysis provides the most gen- eral approach to computing correlations and does not make any assumption about possible correlation in the data. In the absence of an ISW effect the WMAP and NVSS data should be indepen- dent and none of the correlation estimators defined by (14) should exhibit a significant deviation from zero. 3.4 Matched intensity analysis In the matched intensity analysis that follows we first compute the orientation and signed-intensity of local features for the NVSS data. Using the local orientations extracted from the NVSS data, we compute the signed-intensity of local features in the WMAP data that are matched in orientation to the local features in NVSS data. For this matched intensity analysis it no longer makes sense to examine the correlation of orientations any further since they are identical in both data sets, nor to examine the elongation measures any further since the elongation does not measure the eccentricity of local features in the WMAP data when the orientation is ex- tracted from the NVSS data. We therefore only correlate, through (14) again, the signed-intensity of features extracted from the data. This analysis is based on the assumption that local features in the NVSS data are somehow included in the WMAP data. This is a conceptually different analysis to the local morphological analysis described in the preceding subsection. Before proceeding with this analysis it is important to check that it differs in practice to the local morphological analysis. To do this we examine the difference in the orientation of local features, extracted independently in each data set, at each position on the sky. From relation (11) it can be seen that this difference techni- cally lies in the range [−π, π). It is however defined modulo π, and the range is explicitly restricted to ∆D ∈ [−π/2, π/2). If orientation differences are predominantly zero, then we know a priori that the two analysis procedures are essentially the same and would give the same result for the correlation of signed-intensities. In Fig. 5 we plot the histogram of the differences in orientation between the two data sets (outside of the extended exclusion masks), built from all scales (the distributions obtained for each individual scale ex- hibit similar structure). The orientation differences are broadly dis- tributed about zero, thus the two analysis techniques proposed are indeed different practically, as well as conceptually. Moreover, the relative flatness of the distribution suggests no obvious correlation of orientations between the WMAP and NVSS data. This is exam- ined in detail through the local morphological analysis when ap- plied to orientations. Notice that the local morphological measure of the orienta- tion defined by (11) depends on the coordinate system consid- ered through the definition of the origin χ = 0 at each point on the sphere. The estimator for the correlation of orientations be- tween the two data sets given by (14) also depends on the coordi- nate system. This dependence does not affect the statistical signifi- cance of any correlation detected, which simply measures a possi- ble anomaly between the data and the simulations. One could sub- stitute the estimator (14) by a measure of flatness for the distribu- tion of the difference in the orientation ∆D between the two data sets. This difference is indeed independent of the coordinate sys- tem and has a direct physical meaning. However, this would simply provide an alternative measure of the statistical significance of any correlation detected in the orientation of local features. Either es- timator is therefore suitable; for consistence we choose the former case. 4 LOCAL MORPHOLOGICAL CORRELATIONS In this section we present the results obtained from the local mor- phological analysis described in section 3.3 to examine the WMAP and NVSS data for possible correlation. Correlations are detected and are examined to determine whether they are due to foreground contamination or instrumental systematics, or whether they are in- deed induced by the ISW effect. c© 2007 RAS, MNRAS 000, 1–12 8 McEwen et al. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 PSfrag replacements ∆D (rad) Figure 5. Histogram of the difference in orientation of local features ex- tracted independently in the WMAP and NVSS data for all scales. This difference ∆D is defined modulo π, in the range [−π/2, π/2). The distribu- tion is not concentrated about zero, indicating that the local morphological and matched intensity analyses are indeed different practically, as well as conceptually. 4.1 Detections The correlation statistics computed from the WMAP and NVSS data for each morphological measure are displayed in Fig. 6. Sig- nificance levels computed from the 1000 Monte Carlo simulations described in section 3.1 are also shown on each plot. A non-zero correlation signal is clearly present in the signed-intensity of local features. A strong detection at 99.3% significance is made in the correlation of signed-intensities for wavelet half-width θhw ≃ 400′ . This correlation deviates from the mean of the Monte Carlo sim- ulations by 2.8 standard deviations. Moderate detections are also made in the correlation of the orientations and elongations of local features for θhw ≃ 400′ at 93.3% significance and for θhw ≃ 600′ at 97.2% significance respectively. The statistical significance of these findings is based on an a posteriori selection of the scale corresponding to the most sig- nificant correlation. Nevertheless, such statistics have been widely used to quantify detections of the ISW effect, and are satisfactory provided that the a posteriori nature of the analysis is acknowl- edged. However, in this work we also consider a χ2 analysis to examine the significance of correlation between the data when the correlation statistics for all scales are considered in aggregate. This a priori statistic is more conservative in nature; we consider it as a simple alternative detection measure which takes into account the correlation between scales. For the correlation computed for each local morphological measure we compute the χ2 defined by χ2S i = X C−1S i X , (15) where XNTS i is a concatenated vector of X (a) for all scales, the matrix CS i is a similarly ordered covariance matrix of the mor- phological correlation statistics computed from the Monte Carlo simulations and † represents the matrix transpose operation. Com- paring the χ2 values computed from the data to those computed from the Monte Carlo simulations, it is possible to quantify the sta- tistical significance of correlation in the data. Moderate detections are made in the correlation of signed-intensities and orientations of local features at 94.5% and 93.2% significance respectively. The detection in the correlation of elongations drops considerably to 84.5% significance. The steerable wavelet analysis performed allows one to lo- calise on the sky those regions that contribute most significantly to the correlation detected in the data. This type of localisation was performed by McEwen et al. (2007b). However, it was found in this work that, although the localised regions were a significant source of correlation between the data, they by no means provided the sole contribution of correlation. Similar findings were also ob- tained by Boughn & Crittenden (2005) using a different analysis technique. These results are consistent with intuitive predictions based on the ISW effect, namely that one would expect to observe correlations weakly distributed over the entire sky, rather than only a few highly correlated localised regions. For these reasons we do not repeat the localisation performed by McEwen et al. (2007b) for the current analysis (especially since we shown in section 4.2 that the detection is likely due to the ISW effect and not contam- ination or systematics). Instead, we display in Fig. 7 the signed- intensity values on the sky for the scale (θhw ≃ 400′) correspond- ing to the most significance detection of correlation. Due to the strength of the correlation, it is possible to see it by eye between the WMAP (Fig. 7 (a)) and NVSS data (Fig. 7 (c)). We consider only the signed-intensity of local features here since this morpho- logical measure corresponds to the most significant detection of correlation made in these local morphological analyses. 4.2 Foregrounds and systematics To test whether the correlation detected in the preceding subsection is perhaps due to foreground contamination or instrumental system- atics in the WMAP data, we examine the correlation signals com- puted using individual WMAP bands and difference maps in place of the WMAP co-added map. Again, we consider only the signed- intensity of local features here since this morphological measure corresponds to the most significant detection of correlation made in these local morphological analyses. In Fig. 8 (a) we plot the mor- phological signed-intensity correlation of the NVSS data with each of the individual WMAP band maps (constructed from the mean signal observed by all receivers in that band), while in Fig. 8 (b) we plot the correlation for maps constructed from band differences (constructed from differences of signals observed by various re- ceivers). Any correlation induced by unremoved foreground contami- nation in the WMAP data is expected to exhibit a frequency de- pendence, reflecting the emission law of the foreground compo- nent. However, the same correlation signal is observed in each of the WMAP bands and no frequency dependence is apparent (see Fig. 8 (a)). Furthermore, the difference map W−V−Q is clearly contaminated by foreground contributions, while having a minimal CMB contribution. No correlation is detected in this difference map (see Fig. 8 (b)). Consequently, we may conclude that foreground contamination is not the source of the correlation detected. The same correlation signal is observed in each WMAP band (see Fig. 8 (a)). Furthermore, this signal is not present in any of the difference maps constructed for each given band (see Fig. 8 (b)), which contain no CMB or foreground contributions. These findings suggest that the correlation detected is not due to systematics in the WMAP data. The preliminary tests performed in this subsection indicate that it is unlikely that either foregrounds or systematics are respon- sible for the correlation detected in the signed-intensity of local features. This strongly suggests that the ISW effect is responsible for the statistically significant correlation detected in this section between the WMAP and NVSS data. c© 2007 RAS, MNRAS 000, 1–12 Probing dark energy with steerable wavelets 9 100 150 200 250 300 350 400 450 500 550 600 PSfrag replacements Wavelet half-width θhw (arcmin) (a) Signed-intensity 100 150 200 250 300 350 400 450 500 550 600 −0.15 −0.05 PSfrag replacements Wavelet half-width θhw (arcmin) (b) Orientation 100 150 200 250 300 350 400 450 500 550 600 −0.015 −0.01 −0.005 0.005 0.015 PSfrag replacements Wavelet half-width θhw (arcmin) (c) Elongation Figure 6. Correlation statistics computed for each morphological measure in the local morphological analysis, from the WMAP co-added map and the NVSS map. Significance levels obtained from the 1000 Monte Carlo simulations are shown by the shaded regions for 68% (yellow/light-grey), 95% (magenta/grey) and 99% (red/dark-grey) levels. The different ranges of the correlations for the signed-intensity, orientation, and elongation are consistent with our rough estimations of the standard deviations for each local morphological measure in each of the two data sets (see section 3.2). 5 MATCHED INTENSITY CORRELATION In this section we present the results obtained from the matched in- tensity analysis described in section 3.4 to examine the WMAP and NVSS data for possible correlation. In this setting the orientation of local features extracted from the WMAP data at each position on the sky are matched to orientations extracted from the NVSS data. It therefore no longer makes sense to correlate the orientation or elongation of features between the two data sets. Correlations are detected between the signed-intensity of matched features and are examined to determine whether they are due to foreground con- tamination or instrumental systematics, or whether they are indeed induced by the ISW effect. (a) WMAP for independent features (b) WMAP for features matched to NVSS orientations (c) NVSS Figure 7. Morphological signed-intensity maps (Mollweide projection) cor- responding to the scale (θhw ≃ 400′) on which the maximum detections of correlation are made. In panel (a) signed-intensities are shown for local fea- tures extracted independently from the WMAP co-added data, whereas in panel (b) signed-intensities are shown for local features in the WMAP co- added data that are matched in orientation to local features in the NVSS data. Due to the strength of the correlation in the data, it is possible to ob- serve the correlation both between maps (a) and (c) and between maps (b) and (c) by eye. 5.1 Detections The correlation statistics computed from the WMAP and NVSS data for the signed-intensity of matched features are displayed in Fig. 9. Significance levels computed from the Monte Carlo simula- tions are again shown on each plot. A non-zero correlation signal is clearly present. A strong detection at 99.2% significance is made in the correlation of signed-intensities of local features for wavelet half-width θhw ≃ 400′. This correlation deviates from the mean of the Monte Carlo simulation by 2.6 standard deviations. Note that these significance measures are again based on an a posteriori scale selection. We also compute the a priori significance of the detection of correlation when considering all scales in aggregate, taking into account the correlation between scales. This is based on the χ2 test described in section 4.1. For this case we detect a highly significant correlation at a level of 99.9%. Although it is possible to localise regions on the sky that con- c© 2007 RAS, MNRAS 000, 1–12 10 McEwen et al. 100 150 200 250 300 350 400 450 500 550 600 PSfrag replacements Wavelet half-width θhw (arcmin) (a) Individual band maps 100 150 200 250 300 350 400 450 500 550 600 W−V−Q Q1−Q2 V1−V2 W1−W2+W3−W4 PSfrag replacements Wavelet half-width θhw (arcmin) (b) Difference maps Figure 8. Correlation statistics computed for the signed-intensity of features in the local morphological analysis, from the WMAP individual band and difference maps and the NVSS map. Significance levels obtained from the 1000 Monte Carlo simulations are shown by the shaded regions for 68% (yellow/light-grey), 95% (magenta/grey) and 99% (red/dark-grey) levels. 100 150 200 250 300 350 400 450 500 550 600 PSfrag replacements Wavelet half-width θhw (arcmin) Figure 9. Correlation statistics computed for signed-intensity in the matched intensity analysis, from the WMAP co-added map and the NVSS map. Significance levels obtained from the 1000 Monte Carlo simulations are shown by the shaded regions for 68% (yellow/light-grey), 95% (ma- genta/grey) and 99% (red/dark-grey) levels. tribute most significantly to the correlation detected, we again do not do this since the ISW signal is expected to be weakly distributed over the entire sky (as described in section 4.1). Instead, we display in Fig. 7 the signed-intensity of matched features on the sky for the scale (θhw ≃ 400′) corresponding to the most significance detection of correlation using this analysis. Due to the strength of the cor- relation, it is again possible to see it by eye between the WMAP (Fig. 7 (b)) and NVSS data (Fig. 7 (c)). 100 150 200 250 300 350 400 450 500 550 600 PSfrag replacements Wavelet half-width θhw (arcmin) (a) Individual band maps 100 150 200 250 300 350 400 450 500 550 600 W−V−Q Q1−Q2 V1−V2 W1−W2+W3−W4 PSfrag replacements Wavelet half-width θhw (arcmin) (b) Difference maps Figure 10. Correlation statistics computed for the signed-intensity in the matched intensity analysis, from the WMAP individual band and difference maps and the NVSS map. Significance levels obtained from the 1000 Monte Carlo simulations are shown by the shaded regions for 68% (yellow/light- grey), 95% (magenta/grey) and 99% (red/dark-grey) levels. 5.2 Foregrounds and systematics For this analysis procedure we also check whether the correlation detected is perhaps due to foreground contamination or system- atics. We follow the same procedure outlined in section 4.2. The correlation signals computed for the matched intensity analysis us- ing individual WMAP bands and differences maps in place of the WMAP co-added data are plotted in Fig. 10. No frequency depen- dence is observed between the signals computed for the individual band maps (see Fig. 10 (a)), neither is any correlation observed in the foreground contaminated W−V−Q map (see Fig. 10 (b)). These findings suggest that foreground contributions are not re- sponsible for the correlation detected in this analysis. Furthermore, the same correlation signal is observed in each individual band (see Fig. 10 (a)), but is not present in any of the difference maps for each given band (see Fig. 10 (b)). This indicates that systematics are not responsible for the correlation detected in this analysis. These re- sults again strongly suggest that the ISW effect is responsible for the correlation detected between the WMAP and NVSS data in this second analysis based on the signed-intensity of matched features. 6 CONCLUSIONS We have presented the first direct analysis of local morphological measures on the sphere defined through a steerable wavelet analy- sis. Using the wavelet on the sphere constructed from the second derivative of a Gaussian, we are able to define three quantities to characterise the morphology of local features: the signed-intensity, c© 2007 RAS, MNRAS 000, 1–12 Probing dark energy with steerable wavelets 11 orientation and elongation. These local morphological measures provide additional probes to search for correlation between the CMB and LSS of the Universe that may have been induced by the ISW effect. Based on the morphological measures defined, we perform two distinct analyses to search for correlation between the WMAP and NVSS data. The first procedure, the local morphological anal- ysis, correlates the morphological measures of local features ex- tracted independently in the two data sets. This provides the most general analysis and does not make any assumption regarding pos- sible correlation in the data. The second procedure, the matched in- tensity analysis, is based on the assumption that local features of the LSS are somehow included in the local features of the CMB. Lo- cal features are extracted from the WMAP data that are matched in orientation to features in the NVSS data and the signed-intensities of these features are correlated between the two data sets. Although the two analysis procedures performed are obviously different con- ceptually, we have shown that they differ practically also. Regard- less of the procedure adopted, in the absence of an ISW effect the WMAP and NVSS data should be independent and no correlation should be present in any of the local morphological measures con- sidered. However, strong detections are obtained in the correlation of the signed-intensities of local features using both analysis proce- dures. The most significant detection is made in the matched inten- sity analysis at 99.9% significance, when all scales are considered in aggregate. Moreover, in the local morphological analysis, mod- erate detections are also made in the correlation of the orientation and elongation of local features. In both analyses foreground con- tamination and instrumental systematics have been ruled out as the source of the correlation observed. This strongly suggests that the correlation detected is indeed due to the ISW effect. Since the ISW effect only exists in a universe with dark energy or a non-flat uni- verse, and strong constraints have been placed on the flatness of the Universe (Spergel et al. 2007), the detection of the ISW effect made here can be inferred as direct and independent evidence for dark energy. In previous wavelet analyses to detect the ISW effect, theo- retical predictions are derived for the wavelet correlation in or- der to constrain dark energy parameters of cosmological models (Vielva et al. 2006; McEwen et al. 2007b; Pietrobon et al. 2006). As already discussed, our detections give a new insight on the na- ture of the correlation between the CMB and the LSS, in terms of the signed-intensity, orientation and elongation of local features. This might in turn help to better understand the nature of dark en- ergy through new constraints on dark energy parameters. However, our steerable wavelet analysis is complicated by allowing the ori- entation of wavelet coefficients to vary as a function of the data. The derivation of the theoretical correlation for each local morpho- logical measure is not easily tractable. We are currently exploring this issue in more detail and leave any attempts to constrain stan- dard dark energy parameters to a future work. We also intend to explore the use of this, and other wavelet-based, ISW detections to investigate perturbations in the dark energy fluid and to constrain the sound speed of dark energy (see e.g. Weller & Lewis 2003; Bean & Doré 2004). 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J., 120, 1579, astro-ph/0006396 c© 2007 RAS, MNRAS 000, 1–12 http://arXiv.org/abs/astro-ph/0408456 http://arXiv.org/abs/astro-ph/0004318 http://arXiv.org/abs/astro-ph/0302223 http://arXiv.org/abs/astro-ph/0611539 http://arXiv.org/abs/astro-ph/0506308 http://arXiv.org/abs/astro-ph/0602398 http://arXiv.org/abs/astro-ph/0511308 http://arXiv.org/abs/astro-ph/0305097 http://arXiv.org/abs/astro-ph/0410360 http://arXiv.org/abs/astro-ph/9812133 http://arXiv.org/abs/astro-ph/0606475 http://arXiv.org/abs/astro-ph/0604069 http://arXiv.org/abs/astro-ph/0409059 http://arXiv.org/abs/astro-ph/0606626 http://arXiv.org/abs/astro-ph/0506396 http://arXiv.org/abs/astro-ph/0610911 http://arXiv.org/abs/astro-ph/9805201 http://arXiv.org/abs/astro-ph/0307335 http://arXiv.org/abs/astro-ph/0603449 http://arXiv.org/abs/astro-ph/0408252 http://arXiv.org/abs/astro-ph/0307104 http://arXiv.org/abs/astro-ph/0502486 http://arXiv.org/abs/astro-ph/0508516 http://arXiv.org/abs/arXiv:0704.3144 http://arXiv.org/abs/astro-ph/0006396 Introduction ISW effect and steerable wavelets ISW effect Wavelets on the sphere Steerability and morphological measures Analysis procedures Data and simulations Generic procedure Local morphological analysis Matched intensity analysis Local morphological correlations Detections Foregrounds and systematics Matched intensity correlation Detections Foregrounds and systematics Conclusions

0704.0627

Electromagnetic structure and weak decay of meson K in a light-front QCD-inspired

Electromagnetic structure and weak decay of meson K in a light-front QCD-inspired model∗ Fabiano P. Pereira a, J. P. B. C. de Melo b †, T. Frederico c, and Lauro Tomio d aInstituto de F́ısica, Universidade Federal Fluminense, 24210-900, Niterói, RJ, Brazil bUniversidade Cruzeiro do Sul, CETEC, 08060-070, São Paulo, SP, Brazil cInstituto Tecnológico de Aeronáutica, 12228-900, São José dos Campos, SP, Brazil dInstituto de F́ısica Teórica, UNESP, 01405-900, São Paulo, SP, Brazil The kaon electromagnetic (e.m.) form factor is reviewed considering a light-front con- stituent quark model. In this approach, it is discussed the relevance of the quark-antiquark pair terms for the full covariance of the e.m. current. It is also verified, by considering a QCD dynamical model, that a good agreement with experimental data can be obtained for the kaon weak decay constant once a probability of about 80% of the valence component is taken into account. 1. INTRODUCTION The kaon, as quark-antiquark bound states, is one appropriate system to study aspects of QCD at low and intermediate energy regions. By using quantum field theory at the light-front the subnuclear structure can be more easily studied [ 1, 2, 3]. Within the light- front framework and an appropriate choice of the frame, it is possible to obtain the pion electromagnetic form factor at both space- and time-like regimes[ 4]. Using the light-cone components J+K = J 0+J3 and J−K = J 0−J3 of the kaon electromagnetic current, one can obtain the corresponding form factors in the light-front formalism, with a pseudoscalar coupling for the quarks and considering the Breit frame (q+ = 0, ~q⊥ = (qx, 0) 6= 0) [ 5]. In the case of J+K there is no pair term contribution in the Breit frame. However, for the J−K component of the electromagnetic current, the pair term contribution is different from zero and necessary to preserve the rotational symmetry of the current. In the next section, we outline the main equations of the model for the kaon electromag- netic current, detailed in [ 5], with the corresponding results obtained for the kaon elastic form factor. In section 3, we briefly review a QCD inspired model, presenting results for the weak decay pseudoscalar constants compared to data. In section 4 we present our conclusions. ∗Work partially supported by the Brazilian funding agencies FAPESP and CNPq †JPBC de Melo thanks Instituto de F́ısica Teórica, UNESP, for supporting facilities http://arxiv.org/abs/0704.0627v1 2. ELECTROMAGNETIC FORM FACTOR The initial light-front wave function considered in the present model is given by: ΦiQ(x, k⊥) = (1− x)2 −M20 )(m −M2(mQ, mR)) , (1) where N is a normalization constant, Q ≡ {q̄, q} is the quark or antiquark index with mQ is the corresponding quark mass, m2 is the kaon mass, x = k+/P+ is the momentum fraction, and M2(mQ, mR) ≡ k2⊥ +m (P − k)2⊥ +m − P 2⊥, (2) with the free quark-mass operator given by M20 = M 2(mq̄, mq). mR is a mass constant chosen to regularize the triangle diagram. For the corresponding final wave-functions, q̄ and Φ q , we just need to exchange P ↔ P ′ in (1) and (2). The relation between the electromagnetic current Jµ and the space-like kaon electromagnetic form factor FK+(q is given by 〈P ′|Jµ|P ′〉 = (P ′ + P )µFK+(q 2) . In terms of the initial (Φiq̄) and final (Φ light-front wave functions, we have F+q̄ (q 2) = −eq̄ N2g2Nc 4π3P+ d2k⊥dx N+q̄ θ(x)θ(1− x) Φ q̄ (x, k⊥)Φ q̄(x, k⊥) , (3) F+q (q 2) = [ q ↔ q̄ in F+q̄ (q 2) ] , (4) where Nc is the color number, g is the coupling constant, eQ is the charge of quark Q, and N+q̄ = (−1/4)Tr[(/k +mq̄)γ 5(/k − /P ′ +mq)γ +(/k − /P +mq)γ . In the light-front approach, beside the valence contribution, we have also the non-valence contributions to the currents. In the case of the J+ component, the non-valence component does not contribute to the corresponding matrix elements [ 5]. The kaon electromagnetic form factor obtained with J+ is the sum of two contributions from quark and antiquark currents: (q2) = F+q (q 2) + F+q̄ (q 2) normalized such that F+ (0) = 1. (5) In the case that we consider the J− component, to obtain the kaon electromagnetic form factor, after considering the contribution from the interval 0 < k+ < P+ (interval I), we need to add a second contribution, which is originated from the pair terms, and non-zero in the interval P+ < k+ < P ′+ (interval II). The contribution is obtained after a Cauchy integral in k− is performed in the limit P ′+ → P+ [ 5]. So, instead of (5), we will have: (q2) = F−q (q 2) + F−q̄ (q F−(q2) , (6) normalized by the charge conservation to F− (0) = 1. The parameters of the model are the constituent quark masses, mq = mu = md = 220 MeV, ms = 419 MeV and the regulator mass mR =946 MeV, adjusted to fit the electro- magnetic radius of the kaon. The electromagnetic radius is related to the corresponding form factor, with the mean-square-radius given by 〈r2K+〉 = 6 dFK+(q . (7) With the parameters adjusted as given above, we have 〈r2 〉 = 0.354 fm2, which is very close to the experimental value 〈r2 〉|exp = 0.340 fm 2 [ 6]. Our results for the kaon electromagnetic form factor are presented in Fig. 1, in compar- ison with available experimental data [ 6]. We observe that the full kaon electromagnetic form factor is covariant only after the inclusion of the pair terms or non-valence contri- bution to the J− component of the electromagnetic current. 0.01 0.10 1.00 [(GeV/c) Figure 1. The kaon electromagnetic form factor is obtained with the plus and minus components of the e.m. current (both cases are shown by the solid-line results) and compared with experimental data [ 6]. The dashed-line curve shows the form factor without the pair terms contribution in J− 3. WEAK DECAY CONSTANTS IN A QCD INSPIRED MODEL Next, we briefly review the calculation of the pseudoscalar constants, in a light-front QCD-inspired dynamical model. In this case, the constituent quark masses need to be readjusted in view of the fact that, differently from the approach outlined in section 2, the wave-function is obtained from an eigenvalue equation, as follows. The valence wave function is obtained by solving an eigenvalue equation for the effective square mass operator M2ps [ 7]: ps ψ(x, ~k⊥) = M 0 (x, k⊥) ψ(x, ~k⊥)− dx′d~k′⊥θ(x ′)θ(1− x′) x(1− x)x′(1− x′) 4m1m2 − λpsg(M 0 (x, k⊥))g(M ′, k′⊥)) ψ(x′, ~k′⊥) , (8) where M20 (x, k⊥) ≡ ( ~k2⊥ + m 1)/x + ( ~k2⊥ + m 2)/(1 − x) is the free square mass operator in the meson rest frame, m1,2 are the constituent quark masses, α gives the strength of the Coulomb-like interaction. g(K) is the model form factor, with λps the strength of the separable interaction. We consider two expressions for the form factors: g(a)(K2) = β(a) +K2 and g(b)(K2) = , (9) where the parameters β(a,b) and λps are adjusted to reproduce the experimental val- ues of the pion electromagnetic radius and mass, mπ. For α = 0.5, we have β (a) = −(634.5 MeV)2 and β(b) = (1171 MeV)2. mu = 384 MeV, ms = 508 MeV. In Table 1, we have the results compared with experimental data [ 8]. Table 1 Results for the kaon and pion weak decay constants, compared with experimental data. The model is adjusted to reproduce pion radius and mass. qq f (a)ps (MeV) f ps (MeV) f ps (MeV) M ps (MeV) M ps (MeV) [ 8] π+(ud) 110 110 92.4± .07± 0.25 [ 8] 140 140 K+(us) 126 121 113.0± 1.0± 0.31[ 8] 490 494 4. CONCLUSIONS Considering a light-front model wave-function we have observed a good agreement of the results for the kaon electromagnetic form factor with experimental data. The electro- magnetic form factor was obtained using the plus and minus components of the electro- magnetic current. The inclusion of the non-valence component of the current was shown to be essential in this approach to obtain covariant results for the calculated matrix ele- ments. We also show that a good agreement with experimental data is obtained for the kaon weak decay constants once a probability of the valence component of about 80% is taken into account. REFERENCES 1. F. Cardarelli, I. L. Grach, I. M. Narodetsky, E. Pace, G. Salme, S. Simula, Phys. Rev. D 53 (1996) 6682. 2. J. P. B. C. de Melo, H. W. Naus and T. Frederico, Phy. Rev. C 59 (1999) 2278. 3. B. L. G. Bakker, H.-M. Choi and C.-R. Ji, Phys. Rev. D 63 (2001) 074014. 4. J. P. B. C. de Melo, T. Frederico, E. Pace and G. Salmè, Phy. Rev. D 73 (2006) 074013; J. P. B. C. de Melo, T. Frederico, E. Pace and G. Salmè, Phy. Lett. B 581 (2004) 75. 5. F.P. Pereira, J.P.B.C. de Melo, T. Frederico and L. Tomio, Phys. of Part. and Nucl. 36 (2005) 5217; F.P. Pereira, Fatores de Forma Eletromagnéticos do Ṕıon e do Kaon na Frente de Luz, Msc Dissertation, IFT, São Paulo, 2005. 6. S. R. Amendolia et al., Phys. Lett. B 178 (1986) 435. 7. T. Frederico and H.-C. Pauli, Phy. Rev. D 64 (2001) 054004; L. A. M. Salcedo, J. P. B. C. de Melo, D. Hadjmichef and T. Frederico, Eur. Phys. J. A 27 (2006) 213. 8. W.-M. Yao et al., Journal of Physics G 33 (2006) 1. INTRODUCTION ELECTROMAGNETIC FORM FACTOR WEAK DECAY CONSTANTS IN A QCD INSPIRED MODEL CONCLUSIONS

0704.0628

Black hole puncture initial data with realistic gravitational wave content

Black hole puncture initial data with realistic gravitational wave content B. J. Kelly,1, 2 W. Tichy,3 M. Campanelli,4, 2 and B. F. Whiting5, 2 1Gravitational Astrophysics Laboratory, NASA Goddard Space Flight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771, USA 2Center for Gravitational Wave Astronomy, Department of Physics and Astronomy, The University of Texas at Brownsville, Brownsville, Texas 78520 3Department of Physics, Florida Atlantic University, Boca Raton Florida 33431-0991 4Center for Computational Relativity and Gravitation, School of Mathematical Sciences, Rochester Institute of Technology, 78 Lomb Memorial Drive, Rochester, New York 14623 5Department of Physics, University of Florida, Gainsville Florida 32611-8440 (Dated: October 26, 2018) We present improved post-Newtonian-inspired initial data for non-spinning black-hole binaries, suitable for numerical evolution with punctures. We revisit the work of Tichy et al. [W. Tichy, B. Brügmann, M. Campanelli, and P. Diener, Phys. Rev. D 67, 064008 (2003)], explicitly calculating the remaining integral terms. These terms improve accuracy in the far zone and, for the first time, include realistic gravitational waves in the initial data. We investigate the behavior of these data both at the center of mass and in the far zone, demonstrating agreement of the transverse- traceless parts of the new metric with quadrupole-approximation waveforms. These data can be used for numerical evolutions, enabling a direct connection between the merger waveforms and the post-Newtonian inspiral waveforms. PACS numbers: 04.25.Dm, 04.25.Nx, 04.30.Db, 04.70.Bw I. INTRODUCTION Post-Newtonian (PN) methods have played a funda- mental role in our understanding of the astrophysical im- plications of Einstein’s theory of general relativity. Most importantly, they have been used to confirm that the ra- diation of gravitational waves accounts for energy loss in known binary pulsar configurations. They have also been used to create templates for the gravitational waves emit- ted from compact binaries which might be detected by ground-based gravitational wave observatories, such as LIGO [1, 2], and the NASA/ESA planned space-based mission, LISA [3, 4]. However, PN methods have not been extensively used to provide initial data for binary evolution in numerical relativity, nor, until recently (see [5, 6]), have they been extensively studied so that their limitations could be well identified and the results of nu- merical relativity independently confirmed. Until the end of 2004, the field of numerical relativ- ity had been struggling to compute even a single or- bit for a black-hole binary (BHB). Although debate oc- curred on the advantages of one type of initial data over another, the primary focus within the numerical rela- tivity community was on code refinement which would lead to more stable evolution. Astrophysical realism was very much a secondary issue. However, this situation has radically changed in the last few years with the in- troduction of two essentially independent, but equally successful techniques: the generalized harmonic gauge (GHG) method developed by Pretorius [7] and the “mov- ing puncture” approach, independently developed by the UTB and NASA Goddard groups [8, 9]. Originally in- troduced by Brandt & Brügmann [10] in the context of initial data, the puncture method explicitly factored out the singular part of the metric. When used in numerical evolution in which the punctures remained fixed on the numerical grid, it resulted in distortions of the coordi- nate system and instabilities in the Baumgarte-Shapiro- Shibata-Nakamura (BSSN) [11, 12] evolution scheme. The revolutionary idea behind the moving puncture ap- proach was precisely, not to factor out the singular part of the metric, but rather evolve it together with the reg- ular part, allowing the punctures to move freely across the grid with a suitable choice of the gauge. A golden age for numerical relativity is now emerging, in which multiple groups are using different computer codes to evolve BHBs for several orbits before plunge and merger [13, 14, 15, 16, 17, 18, 19, 20, 21]. Comparison of the numerical results obtained from these various codes has taken place [22, 23, 24], and comparison with PN inspiral waveforms has also been carried out with encour- aging success [5, 6, 25, 26]. The application of successful numerical relativity tools to study some important as- trophysical properties (e.g. precession, recoil, spin-orbit coupling, elliptical orbits, etc) of spinning and/or un- equal mass-black hole systems is currently producing ex- tremely interesting new results [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. It now seems that the primary obstacle to further progress is simply one of computing power. In this new situation, it is perhaps time to return to the question of what initial data will best describe an astrophysical BHB. To date, the best-motivated description of pre-merger BHBs has been supplied by PN methods. We might ex- pect, then, that a PN-based approach would give us the most astrophysically correct initial data from which to run full numerical simulations. In practice, PN results are frequently obtained in a form ill-adapted to numeri- http://arxiv.org/abs/0704.0628v2 cal evolution. PN analysis often deals with the full four- metric, in harmonic coordinates; numerical evolutions frequently use ADM-type coordinates, with a canonical decomposition of the four-metric into a spatial metric and extrinsic curvature. Fortunately, many PN results have been translated into the language of ADM by Ohta, Damour, Schäfer and collaborators. Explicit results for 2.5PN BHB data in the near zone were given by Schäfer [43] and Jaranowski & Schäfer (JS) [44], and these were implemented numer- ically by Tichy et al. [45]. Their insight was that the ADM-transverse-traceless (TT) gauge used by Schäfer was well-adapted to a puncture approach. To facilitate comparison with this earlier work [45], we continue to use the results of Schäfer and co-workers, anticipating that higher-order PN results should eventually become available in a useful form. The initial data provided previously by Tichy et al. already include PN information. They are accurate up to order (v/c)5 in the near zone (r ≪ λ), but the accuracy drops to order (v/c)3 in the far zone (r ≫ λ) [here λ ∼ π r312/G(m1 +m2) is the gravitational wave- length]. These data were incomplete in the sense that they did not include the correct TT radiative piece in the metric, and thus did not contain realistic gravita- tional waves. In this paper, we revisit the PN data problem in ADM-TT coordinates, with the aim of supplying Numer- ical Relativity with initial BHB data that extend as far as necessary, and contain realistic gravitational waves. To do this, we have evaluated the “missing pieces” of Schäfer’s TT metric for the case of two non-spinning par- ticles. We have analyzed the near- and far-zone behavior of these data, and incorporated them numerically in the Cactus [46] framework. In principle, the most accurate PN metric available could be used at this step, but it is not currently available in ADM-TT form. The remainder of this paper is laid out as follows. In Section II, we summarize the results of Schäfer (1985) [43], and Jaranowski & Schäfer (1997) [44] and their ap- plication by Tichy et al. (2003) [45], to the production of puncture data for numerical evolution. In Section III, we describe briefly the additional terms necessary to complete hTT to order (v/c)4, deferring details to the Appendix. In Section IV, we study the full data both analytically and numerically. Section V summarizes our results, and lays the groundwork for numerical evolution of these data, to be presented in a subsequent article. II. ADM-TT GAUGE IN POST-NEWTONIAN The “ADM-TT” gauge [43, 47] is a 3+1 split of data where the three-metric differs from conformal flatness precisely by a TT radiative part: gij = ηij + h ij , (1) πii = 0. (2) The fields φ, πij and hTTij can all be expanded in a post- Newtonian series. Solving the constraint equations of 3+1 general relativity in this gauge, [43, 44] obtained ex- plicit expressions valid up to O(v/c)5 in the near zone, incorporating an arbitrary number of spinless point par- ticles, with arbitrary masses mA. For N particles, the lowest-order contribution to the conformal factor is1: φ(2) = 4G , (3) where rA = ~x− ~xA is the distance from the field point to the location of particle A. In principle hTTij is computed from hTTij = −δTT klij ✷−1retskl, (4) where ✷−1ret is the (flat space) inverse d’Alembertian (with a “no-incoming-radiation” condition [48]), skl is a non- local source term and δTT klij is the TT-projection oper- ator. In order to compute hTTij we first rewrite Eq. (4) hTTij = −δTT klij ∆−1 + (✷−1ret −∆−1) TT (NZ) ij − δ TT kl ij (✷ ret −∆−1)skl. (5) Note that the near-zone approximation h TT (NZ) ij of h has already been computed in [43] up to order O(v/c)4 (see also Eq. 12 below). The last term in Eq. (5) is diffi- cult to compute because skl = 16πG pAk pAl δ(x− xA) + ,l (6) is a non-local source. However, we can approximate skl s̄kl = pAk pAl B 6=A nABk nABl ×16πGδ(x− xA). (7) and show that hTTij,(div) = −δ TT kl ij (✷ ret−∆−1)(skl− s̄kl) ∼ O(v/c)5 (8) 1 We explicitly include the gravitational constant G in all expres- sions here, as the standard convention G = 1 used in Numerical Relativity differs from the convention 16πG = 1 employed by [43, 44]. in the near zone. Furthermore, outside the near zone ij,(div) ∼ 1/r2, so that hTT ij,(div) falls off much faster than rest of hTTij , which falls off like 1/r. Hence hTTij = h TT (NZ) ij − δ TT kl ij (✷ ret −∆−1)s̄kl + hTTij,(div), (9) where hTT ij,(div) can be neglected if we only keep terms up to O(v/c)4 generally, and O(1/r) at infinity. The full expression for hTTij for N interacting point particles from Eq. (4.3) of [43] is: hTTij = h TT (NZ) ij + h ij,(div) + 16πG d3~k dω dτ (2 π)4 pAi pAj B 6=A nABi nABj × (ω/k) ~k·(~x−~xA)−i ω (t−τ) k2 − (ω + i ǫ)2 . (10) The first term in (10), h TT (NZ) ij can be expanded in v/c TT (NZ) ij = h TT (4) ij + h TT (5) ij +O(v/c) 6. (11) The leading order term at O(v/c)4, is given explicitly by Eq. (A20) of [44]: hTT (4)ij = mA rA ‖ ~pA ‖2 −5 (n̂A · ~pA)2 δij + 2 piA p 3(n̂A · ~pA)2 − 5 ‖ ~pA ‖2 niA n A + 12(n̂A · ~pA)n B 6=A niABn AB + 2 rA + rB niA n rABrA + 3rA rABrA , (12) where sAB ≡ rA + rB + rAB. The other two terms in Eq. (10) can be shown to be small in the near zone (r ≪ λ, where the characteristic wavelength λ ∼ 100M for rAB ∼ 10M). However, hTT (NZ)ij is only a valid ap- proximation to hTTij in the near zone, and becomes highly inaccurate when used further afield. Setting aside these far-field issues, Tichy et al. [45] ap- plied Schäfer’s formulation, in the context of a black-hole binary system, to construct initial data that are accurate up to O(v/c)5 in the near zone. They noted that the ADM-TT decomposition was well-adapted to the use of a puncture approach to handle black-hole singularities. This approach is essentially an extension of the method introduced in [10]. It allows a simple numerical treatment of the black holes without the need for excision. The PN-based puncture data of Tichy et al. have not been used for numerical evolutions. This is in part because these data, just like standard puncture data [10, 49, 50, 51], do not contain realistic gravitational waves in the far zone: h TT (NZ) ij does not even vaguely agree with the 2PN approximation to the waveform amplitude nor with the quadrupole approximation to the waveform phase for realistic inspiral. To illustrate this, we restrict to the case of two point sources, and compute the “plus” and “cross” polariza- tions of the near-zone approximation for hTTij : + = h TT (NZ) θ, (13) × = h TT (NZ) φ. (14) For comparison, the corresponding polarizations of the quadrupole approximation for the gravitational-wave strain are given by (paraphrasing Eq. (3.4) of [52]): (1+cos2 θ)(πGMfGW)2/3cos(ΦGW), (15) cos θ(πGMfGW)2/3sin(ΦGW), (16) where M ≡ ν3/5M is the “chirp mass” of the binary, given in terms of the total PN mass of the system M = m1 +m2, and the symmetric mass ratio ν = m1m2/M The angle θ is the “inclination angle of orbital angular momentum to the line of sight toward the detector”; that is, just the polar angle to the field point, when the binary moves in the x-y plane. ΦGW and fGW are the phase and frequency of the radiation at time t, exactly twice the orbital phase Φ(t− r) and orbital frequency Ω(t− r)/2π. The lowest-order PN prediction for radiation-reaction effects yields a simple inspiral of the binary over time, with orbital phasing given by Φ(τ) = Φ(tc)− Θ5/8, (17) Ω(τ) = Θ−3/8, (18) where Θ ≡ ν (tc − τ)/5GM , M and ν are given below (16), and tc is a nominal “coalescence time”. To evaluate (13-14), we need the transverse momentum p correspond- ing to the desired separation r12. The simplest expression for this is the classical Keplerian relation, which we give parameterized by Ω(τ): r12 = G 1/3M(MΩ)−2/3, (19) p = Mν(GMΩ)1/3. (20) In Fig. 1 we compare the plus polarization of the two waveforms (13) and (15) at a field point r = 100M , θ = π/4, φ = 0, for a binary in the x-y plane, with ini- tial separation r12 = 10M . The orbital frequency of the binary is related to the separation r12 and momenta p en- tering (13) by (19-20). To this level of approximation, the binary has a nominal PN coalescence time tc ≈ 780M . As might have been anticipated, both phase and ampli- tude of h TT (4) ij are wrong outside the near zone. This means that the data constructed from h TT (4) ij have the wrong wave content, but nevertheless these data are still accurate up to order (v/c)3 in the far zone. It is evident from the present-time dependence of (12) that it cannot actually contain any of the past history of an inspiralling binary. We would expect that a cor- rect “wave-like” contribution should depend rather on the retarded time of each contributing point source. It seems evident that the correct behavior must, in fact, be contained in the as-yet unevaluated parts of (10). The requisite evaluation is what we undertake in the next sec- tion. III. COMPLETING THE EVALUATION OF hTTij To move forward, we will simplify (10) and (12) to the case of only two particles. Then (10) reduces to: hTTij = h TT (NZ) ij + 16πG p1 i p1 j ~k ·(~x−~x1) + p2 i p2 j ~k ·(~x−~x2) − G n12i n12j ~k ·(~x−~x1) n21i n21j ~k ·(~x−~x2) · (ω/k) 2 e−i ω (t−τ) k2 − (ω + i ǫ)2 d3~k dω dτ (2 π)4 + hTTij,(div) (21) TT (NZ) ij +H +HTT2ij −HTT1ij Gm1m2 2 r12 −HTT2ij Gm1m2 2 r12 +hTTij,(div), (22) where HTTAij [~u] := 16πG d3~k dω (2 π)4 [ui uj ] (ω/k)2 k2 − (ω + i ǫ)2 ~k·(~x−~xA(τ)) e−i ω (t−τ). (23) Here, the “TT projection” is effected using the operator i := δ i − ki kj/k2. For an arbitrary spatial vector ~u, [ui uj ] TT = uc ud (P Pij P = ui uj + ki kj u(i kj) . (24) Details on the evaluation of these terms are presented in Appendix A. After calculation, we write the result as a sum of terms evaluated at the present field-point time t, the retarded time trA defined by t− trA − rA(trA) = 0, (25) and integrals between trA and t, TTA[~u] = H TTA[~u; t] +H TTA[~u; t TTA[~u; t A → t], (26) where the three parts are given by: 0 100 200 300 400 500 600 700 800 -0.0015 -0.001 -0.0005 0.0005 0.001 (quadrupole) (NZ) FIG. 1: Plus polarization of the quadrupole (black/solid) and near-zone (red/dashed) strains observed at field point r = 100M , θ = π/4, φ = 0. The binary orbits in the x-y plane, with initial separation r12 = 10M , and a nominal coalescence time tc ≈ 780M . Both phase and amplitude of h TT (4) very wrong outside the near zone. TTA[~u; t] = − rA(t) u2 − 5 (~u · n̂A)2 δi j + 2 ui uj + 3 (~u · n̂A)2 − 5 u2 niA n +12 (~u · n̂A)u(i nj)A , (27) TTA[~u; t −2 u2 + 2 (~u · n̂A)2 δi j + 4 ui uj + 2 u2 + 2 (~u · n̂A)2 niA n −8 (~u · n̂A)u(i nj)A , (28) TTA[~u; t A → t] = −G (t− τ) rA(τ)3 −5 u2 + 9 (~u · n̂A)2 δi j + 6 ui uj − 12 (~u · n̂A)u(i nj)A 9 u2 − 15 (~u · n̂A)2 niA n (t− τ)3 rA(τ)5 u2 − 5 (~u · n̂A)2 δi j + 2 ui uj − 20 (~u · n̂A)u(i nj)A −5 u2 + 35 (~u · n̂A)2 niA n . (29) In Fig. 2, we show the retarded times calculated for each particle, as measured at points along the x axis, for the same orbit as in Fig. 1. We also show the corre- sponding retarded times for a binary in an exactly cir- cular orbit. Since the small-scale oscillatory effect of the finite orbital radius would be lost by the overall linear trend, we have multiplied by the orbital radius. A. Reconciling with Jaranowski & Schäfer’s h TT (4) From the derivation above it is clear that hTTij includes retardation effects, so it will not depend solely on the present time. We might even expect that all “present- time” contributions should vanish individually, or should cancel out. It can be seen easily from (27) that the “t” part of the second and third terms of Eq. (22) exactly cancel out the “kinetic” part (first line) of Eq. (12). Thus, we can simply remove that line in Eq. (12), and use the 500 1000 1500 particle 1 (circular) particle 1 (inspiral) particle 2 (circular) particle 2 (inspiral) FIG. 2: Retarded times for particles 1 and 2, as measured by observers along the x axis at the initial time t = 0, for the binary of Fig. 1. To highlight the oscillatory effect of the finite-radius orbit on tr, we first divide by the average field distance r. “tr” part instead. One may similarly inquire whether the “t” parts of the fourth and fifth terms of Eq. (22) above, TT (pot,now) ij ≡ −H Gm1m2 2 r12 n̂12; t −HTT2ij Gm1m2 2 r12 n̂12; t , (30) also cancel the remaining, “potential” parts of Eq. (12). The answer is “not completely”; expanding in powers of 1/r, we find: TT (pot,4) ij + h TT (pot,now) G2m1m2 r12 16 r3 (3 + 14W 2 − 25W 4) δi j − 4 (1 + 5W 2)n12i n12j −5 (1 + 6W 2 − 7W 4)n1i n1j + 2W (7 + 9W 2) (n12i n1j + n12j n1i) +O(1/r4),(31) where W ≡ sin θ cos(φ − Φ(t)), and Φ(t) is the orbital phase of particle 1 at the present time t. That is, the “new” contribution cancels the 1/r and 1/r2 pieces of TT (4) ij entirely. In the far zone the result is thus smaller than the hTT ij,(div) term which we are ignoring everywhere, since it is small both in the near and the far zone [43]. We note here two general properties of the contribu- tions to the full hTTij . 1. In the near zone h TT (4) ij is the dominant term since all other terms arise from (✷−1ret − ∆−1)skl. Thus all other terms must cancel within the accuracy of the near-zone approximation. TT (4) ij is wrong far from the sources; thus, the new corrections should “cancel” h TT (4) ij entirely, far from sources. Note, however, that while hij = −✷−1retskl depends only on retarded time, its TT- projection hTTij = δ TT kl ij hkl has a more complicated causal structure; E.g. the finite time integral comes from applying the TT-projection. [Proof: Even if we had a source given exactly by s̄kl, h TT (4) ij would depend only the present time, hij would depend only on retarded time, and hTTij would (as we have computed) contain a finite time integral term.] Additionally, the full hTTij agrees well with quadrupole predictions, which we demonstrate in Section IV. IV. NUMERICAL RESULTS AND INVARIANTS A. Phasing and Post-Keplerian Relations It has been known for some time (see for example [53]) that gravitational wave phase plays an even more impor- tant part in source identification than does wave ampli- tude. In PN work, phase and amplitude are estimated somewhat separately; the amplitude requires knowledge of the time-dependent multipoles, used in developing the the full metric, while the phase can be relatively simply approximated from the orbital equations of motion, tak- ing into account the gravitational wave flux at infinity to evolve the orbital parameters [54]. The quadrupole waveform introduced for the compar- ison in Fig. 1 had an amplitude accurate to O(v/c)4 and the simplest available time evolution for the phase. Waveform phase is a direct consequence of orbital phase. To lowest order, we could have assumed a binary mov- ing in a circular orbit (of zero eccentricity) since, up to 2PN order, we can have circular orbits, where the linear momentum, p, of each particle is related to the separa- tion r12 by, say, Eq. (24) of [45]. Nevertheless, circular orbits are physically unrealistic – since radiation reac- tion will lead to inspiral and merger of the particles – and Eqs. (17-18) already include leading-order radiation- reaction effects. Moreover, the phase errors that would accrue from using purely circular orbits would be larger, the further from the sources we tried to compute them. The calculations of section III lead to waveform am- plitudes that are accurate at O(v/c)4 everywhere. How- ever, we desire that our initial-data wave content already encode the phase as accurately as possible. Highly ac- curate phase for our initial data (via hTT), and hence in the leading edge of the waveforms we would extract from numerical evolution, is critical for parameter estimation following a detection. For demonstrative purposes, in this section, we will restrict ourselves to the simplest phasing relations con- sistent with radiation-reaction inspiral as given by Eqs. (17-18), while using higher-order PN expressions than Eqs. (19 -20) for relating the orbit to the phase. For ex- ample, from [55], we have found to second PN (beyond leading) order: r12(Ω) = (GMΩ)−2/3 − (3− ν) − (18− 81ν − 8ν (GMΩ)2/3, (32) = (GMΩ)1/3 + (15− ν) (GMΩ) (441− 324ν − ν2) (GMΩ)5/3, (33) and we note that higher-order equivalents of these can be computed from [56]. In the numerical construction of initial data, the pri- mary input is the coordinate separation of the holes. In placing the punctures on the numerical grid, the separa- tion must be maintained exactly. To ensure this, we in- vert Eq. (32) to obtain the exact Ωr corresponding to our desired r12. Then we use Eq. (18) with t = 0 to find the coalescence time tc that yields this Ωr. Once we have ob- tained tc, we then find the orbital phase Φ and frequency Ω at any source time τ directly from Eqs. (17-18), and the corresponding separation r12 and momentum p from Eqs. (32-33), or their higher-order equivalents. In Fig. 3, we show a representative component of the retarded-time part of hTTij for both circular and leading- order inspiral orbits. For both orbits, we use the ex- tended Keplerian relations (32) and (33); otherwise the orbital configuration is that of Fig. 1. The coalescence time is now tc ∼ 1100M . We can see that the cumu- lative wavelength error of the circular-orbit assumption becomes very large at large distances from the sources. This demonstrates that using inspiral orbits instead of circular orbits will significantly enhance the phase accu- racy of the initial data, even though circular orbits are in principle sufficient when we include terms only up to O(v/c)4 as done in this work. From now on we use only inspiral orbits. 0 500 1000 1500 circular inspiral FIG. 3: The xx component of the full hTTij for a binary with initial separation r12 = 10M in a circular (black/solid) or inspiralling (red/dashed) orbit. Both fields have been rescaled by the observer radius r = z to compensate for the leading 1/r fall-off. The orbital configuration is the same as for Fig. 1, apart from the Keplerian relations, where we have used the higher-order relations (32-33), yielding tc ∼ 1100M . Note the frequency broadening at more distant field points. 0 100 200 300 400 500 600 700 800 -0.0015 -0.001 -0.0005 0.0005 0.001 (quadrupole) (full) (quadrupole) (full) FIG. 4: Plus and cross polarizations of the strain observed at field point r = 100M , θ = π/4, φ = 0. Both the quadrupole- approximation waveform (black/solid and green/dot-dashed) and the full (red/dashed and blue/dotted) waveforms coming from hTTij are shown. The orbital configuration is the same as for Fig. 1. Next, we compare our full waveform hTTij (expressed as the combinations h+ and h×) at an intermediate-field position (r = 100M , θ = π/4, φ = 0) to the lowest-order quadrupole result. In Fig. 4, the orbital configuration is the same as for Fig. 1. As one can see, both the + and × polarizations of our hTTij agree very well with quadrupole results, as they should. We demonstrate the near- and intermediate-zone behavior of the new data on 50 100 150 200 250 300 350 400 450 500 -0.05 -0.05 quadrupole FIG. 5: Plus and cross polarizations of the strain observed at t = 0 along the z axis. We show the near-zone (solid/black), the quadrupole (dashed/red) and full (dot-dashed/green) waveforms. All waveforms have been rescaled by the observer radius r = z to compensate for the leading 1/r fall-off. The orbital configuration is the same as for Fig. 1. the initial spatial slice in Fig. 5. The quadrupole and full solutions agree very well outside ∼ 100M . However, the full solution’s phase and amplitude approach the NZ solution closer to the sources. B. Numerical Implementation After having confirmed that we have a PN three-metric gij that is accurate up to errors of order O(v/c) 5, and that correctly approaches the quadrupole limit outside the near zone, we are now ready to construct initial data for numerical evolutions. In order to do so, we need the intrinsic curvature Kij , which can be computed as in Tichy et al. [45] from the conjugate momentum. The difference is that here we use the full ḣTTij instead of the near-zone approximation ḣ TT (4) ij to obtain the conjugate momentum [43]. The result is Kij = −ψ−10PN ḣTTij + (φ(2)π̃ +O(v/c)6, (34) where the error term comes from neglecting terms like ij,(div) at O(v/c)5 in hTTij , and where ψPN , π̃ φ(2) can be found in Tichy et al. [45]. An additional difference is that the time derivative of hTTij is evaluated numerically in this work. Note that the results for gij are accurate up to O(v/c)4, while the results for Kij are 0 0.5 1 1.5 = 10M = 20M = 50M = 100M 0 0.5 1 1.5 FIG. 6: Upper panel: Hamiltonian constraint violation along the y axis of our new data in the near zone, as a function of binary separation r12. Lower panel: Momentum constraint (y-component) violation of the same data along the x axis. The orbital configuration is that of Fig. 3. Distances have been scaled relative to r12, so that the punctures are initially at y/r12 = ±0.5. accurate up O(v/c)5, because Kij contains an additional time derivative [45, 57, 58]. Next we show the violations of the Hamiltonian and momentum constraints computed from gij and Kij , as functions of the binary separation r12. As we can see in both panels of Fig. 6, the constraints become smaller for larger separations, because the post-Newtonian approxi- mation gets better. Note that, as in [45], the constraint violation remains finite everywhere, and is largest near each black hole. C. Curvature Invariants and Asymptotic Flatness In analysis of both initial and evolved data, it is often instructive to investigate the behavior of scalar curva- ture invariants, as these give some idea of the far-field properties of our solution. We expect, for an asymptoti- cally flat space-time, that in the far field, the speciality index S ≡ 27J 2/I3 will be close to unity. This can be seen from the following arguments. Let us choose a tetrad such that the Weyl tensor components ψ1 and ψ3 are both zero. Further, we assume that in the far field ψ0 and ψ4 are both perturbations of order ǫ off a Kerr background. Then S ≈ 1− 3ψ0ψ4 +O(ǫ3), (35) which is indeed close to one. Note however, that this argument only works if the components of the Weyl ten- sor obey the peeling theorem, such that ψ2 ∼ O(r−3), ψ0 ∼ O(r−5) and ψ4 ∼ O(r−1). In particular, if ψ0 falls off more slowly than O(r−5), S will grow for large r. Now observe that ψ0 ∼ O(r−5) ∼ M3/r5 is formally of O(v/c)6. Thus, in order to see the expected behavior of S ≈ 1 in the far-field we need to go to O(v/c)6. If we only go to O(v/c)4 (as done in this work) ψ0 consists of un- controlled remainders only, which should in principle be dropped. When we numerically compute S we find that for our data, S deviates further and further from unity for large distances from the binary. This reflects the fact that the so-called “incoming” Weyl scalar ψ0 only falls off as 1/r3, due to uncontrolled remainders at O(v/c)6, which arise from a mixing of the background with the TT waveform. V. DISCUSSION AND FUTURE WORK Exploring and validating PN inspiral waveforms is cru- cially important for gravitational-wave detection and for our theoretical understanding of black-hole binaries. Our goal has been to provide a step forward in this under- standing by building a direct interface between the PN approach and numerical evolution, along the lines ini- tially outlined in Ref. [45]. In this paper we have essentially completed the calculation of the transverse- traceless part of the ADM-TT metric to O(v/c)4 pro- vided in [45], yielding data that, on the initial Cauchy slice, will describe the space-time into the far-field. We have incorporated this formulation into a numerical initial-data routine adapted to the “puncture” topology that has been so successful recently, and have explored these data’s numerical properties on the initial slice. Our next step is to evolve these data with moving punctures, and investigate how the explicit incorporation of post-Newtonian waveforms in the initial data affects both the ensuing slow binary inspiral of the sources and the release of radiation from the system. We note es- pecially that our data are non-conformally flat beyond O(v/c)3. We expect our data to incorporate smaller unphysical initial distortions in the black holes than is possible with conformal flatness, and hence less spurious gravitational radiation during the numerical evolution. We see this as a very positive step toward providing fur- ther validation of numerical relativity results for multiple orbit simulations, since it permits comparison with PN results where they are expected to be reliable. Our initial data will also allow us to fully evaluate the validity of PN results for merging binaries by enabling comparison with the most accurate numerical relativity results. We expect that further development of these data will certainly involve the use of more accurate orbital phas- ing information than the leading order given by Eqs. (17- 18). This information is available in radiative coordinates (see, e.g. Eq. (6.29) of [59]) appropriate for far-field eval- uation of the gravitational radiative modes; it may be possible to produce them in ADM-TT coordinates via a contact transformation, or by direct calculation (see, e.g. [60]). For initial separations similar to the fiducial test case of this paper, r12 = 10M , the order necessary for clean matching of the initial wave content with the new radiation generated in evolution should not be par- ticularly high [26]. As noted, the Keplerian relations Eqs. (32-33) can easily be extended to higher PN order. The data presented already allow for arbitrary initial mass ratios ν; this introduces the possibility of significant gravitational radiation in odd-l multipoles, together with associated phenomena, such as in-plane recoil “kicks”. An interesting future development of these data will be the inclusion of spin angular momenta on the pre-merger holes. This will open our initial-data prescription to de- scribing an even richer spectrum of binary radiation. Acknowledgments We would like to thank L. Blanchet and G. Schäfer for generous assistance and helpful discussion. M.C., B.K. and B.W. gratefully acknowledge the sup- port of the NASA Center for Gravitational Wave Astron- omy (NAG5-13396). M.C. and B.K. also acknowledge the NSF for financial support under grants PHY-0354867 and PHY-0722315. B.K. also acknowledges support from the NASA Postdoctoral Program at the Oak Ridge Associ- ated Universities. The work of W.T. was supported by NSF grant PHY-0555644. W.T. also acknowledges par- tial support from the NCSA under Grant PHY-060040T. The work of B.W. was also supported by NSF grants PHY-0245024 and PHY-0555484. APPENDIX A: DETAILS OF INTEGRAL CALCULATION Here we present some more details of the calculations that lead to the three contributions to Eq. (23): Eqs. (27-29). Inserting Eq. (24) in the general integral (23), we can write H TTA[~u] as a combination of scalar and tensor terms: HTTAij [~u] = 16πG ui uj − Iij A + [uc ud IcdA δij 2 uc u(i Icj)A + [uc ud I cdij A ,(A1) where the “I” integrals are defined as: d3~k dω (2 π)4 (ω/k)2 ei k rA cos θ−i ω T k2 − (ω + i ǫ)2 , (A2) d3~k dω (2 π)4 ki kj × (ω/k) 2 ei k rA cos θ−i ω T k2 − (ω + i ǫ)2 , (A3) i j c d d3~k dω (2 π)4 ki kj kc kd × (ω/k) 2 ei k rA cos θ−i ω T k2 − (ω + i ǫ)2 . (A4) Here T ≡ t − τ , and ~rA ≡ ~x − ~xA. We have also taken our integration coordinates such that ~rA lies in the z di- rection, so that the dummy momentum vector ~k satisfies ~k · ~rA = k rA cos θ, (A5) d3~k = k2 dk sin θ dθ dφ. (A6) Define the unit orthogonal vectors n̂A ≡ (0, 0, 1) , ℓ̂ ≡ (cosφ, sinφ, 0). Then we can write ~k = k cos θ n̂A + k sin θ ℓ̂ ⇒ ~k · ~rA = rA ~k · n̂A. We can also define a projector tensor onto ℓ̂: Qa b ≡ δa b − na nb ⇒ Qab = δab − na nb ⇒ QacQcb = Qab , Qab nb = 0 , Qab ℓb = ℓa. 1. Angular integration We will neglect the A subscript for now, until it be- comes relevant again. To calculate the integrals (A2-A4), we begin with the φ integration. The only φ dependence comes from the ~ℓ parts of the ~k terms. It can be seen from elementary trigonometric integrals that: dφ ℓa = dφ ℓa ℓb ℓc = 0, dφ ℓa ℓb = π Qa b, dφ ℓa ℓb ℓc ℓd = Qa bQc d +Qa cQb d +Qa dQb c We use these to calculate the φ integrals for Ia bA and Ia b c dA . Define w ≡ cos θ. Then dφ 1 = 2 π, ka kb = 2 π w2 na nb + π (1− w2)Qa b, ka kb kc kd = 2 π w4 na nb nc nd +6 πw2 (1− w2)Q(a b nc nd) (1− w2)2Q(a bQc d). So the next integrals will differ in their θ dependence, contained in the powers of w above. The θ integrals will contain the following basic types: g0(a) ≡ dw eaw = 2 sinh a , (A7) g2(a) ≡ dw w2 eaw = 2 sinh a − 4 cosha sinh a g4(a) ≡ dw w4 eaw = 2 sinh a − 8 cosha sinh a − 48 cosha sinh a . (A9) Now Ia b and Ia b c d can be written as the linear combi- nations: Ia b = Qa b I (na nb − 1 Qa b)K , (A10) Ia b c d = na nb nc nd − 3Q(a b nc nd) + 3 Q(a bQc d) 3Q(a b nc nd) − Q(a bQc d) Q(a bQc d) I . (A11) I here can be expressed in terms of g0(a) above: (2 π)3 (ω/k)2 k2 − (ω + i ǫ)2 e i k r cos θ−i ω T (2 π)3 ω2 e−i ω T k2 − (ω + i ǫ)2 g0(i k r) (2 π)3 ω2 e−i ω T J0. (A12) The 1/2 factor is because we moved to integrating k over the whole real line instead of the positive half-line (this is permissible as gn(a) is an even function of a). K and L are defined analogously to I, but with extra even powers of cos θ = w: (2 π)3 (ω/k)2 k2 − (ω + i ǫ)2 ei k r cos θ−i ω T cos2 θ (2 π)3 ω2 e−i ω T k2 − (ω + i ǫ)2 g2(i k r) (2 π)3 ω2 e−i ω T J2, (A13) (2 π)3 (ω/k)2 k2 − (ω + i ǫ)2 ei k r cos θ−i ω T cos4 θ (2 π)3 ω2 e−i ω T k2 − (ω + i ǫ)2 g4(i k r) (2 π)3 ω2 e−i ω T J4. (A14) 2. Momentum integration Now we address the k integrals, defined as: dk fn(k) = dk f+n (k) + dk f−n (k), where we collect the positive exponents in the gn in the integrand of f+n (k), and the negative exponents in f n (k): f+n (k) ≡ g+n (i k r)/2 k2 − (ω + i ǫ)2 , f−n (k) ≡ g−n (i k r)/2 k2 − (ω + i ǫ)2 We calculate this as the sum of contour integrals of the “plus” and “minus” integrands (necessary, as the oppo- site signs require different contours). Each of these has poles at k = 0, k = k+ ≡ ω + i ǫ, and k = k− ≡ −ω − i ǫ (the first of these is from the gn). We integrate the “plus” integrands anticlockwise around the contour C1, and the “minus” integrands anticlockwise around the contour C2 (see Fig. 7); taking the limit |k| → ∞, the contribu- tion from the curved segments vanishes, and the residue theorem gives us: Jn = 2 π iRes[f n , k+]− 2 π iRes[f−n , k−] +π iRes[f+n , 0]− π iRes[f−n , 0]. (A15) Calculating the residues, we find the values of each of the Jn: π ei r (ω+i ǫ) r (ω + i ǫ)2 r (ω + i ǫ)2 , (A16) π ei r (ω+i ǫ) r (ω + i ǫ)2 π ei r(ω+i ǫ) [−2 + 2 i r (ω + i ǫ)] r3 (ω + i ǫ)4 r3 (ω + i ǫ)4 , (A17) π ei r (ω+i ǫ) r (ω + i ǫ)2 4π ei r(ω+i ǫ) r5 (ω + i ǫ)6 [6− 6 i r (ω + i ǫ) −3 r2 (ω + i ǫ)2 + i r3 (ω + i ǫ)3 − 24 π r5 (ω + i ǫ)6 . (A18) 3. Frequency integration Now we perform the ω integration. Inserting the re- sults (A16-A18) into (A12-A14) respectively, we see that each of I, K and L contains a delta function, which we can extract: 4 π r [δ(T − r) − δ(T )], 4 π r δ(T − r) + e−r ǫ (2 π)3 e−iω (T−r) F2a(ω) (2 π)3 e−i ω T F2b(ω), 4 π r δ(T − r) + e−r ǫ (2 π)3 e−iω (T−r) F4a(ω) (2 π)3 e−i ω T F4b(ω), where the new terms on the right-hand side come from the Jn above, grouped by exponential, as that is what determines the contours chosen during integration (see Fig. 7): F2a(ω) = π ω2 [−2 + 2 i r (ω + i ǫ)] r3 (ω + i ǫ)4 F2b(ω) = 2 π ω2 r3 (ω + i ǫ)4 F4a(ω) = r5 (ω + i ǫ)6 [24− 24 i r (ω + i ǫ) −12 r2 (ω + i ǫ)2 + 4 i r3 (ω + i ǫ)3 F4b(ω) = − 24 π ω2 r5 (ω + i ǫ)6 Now the residues are as follows (taking the ǫ→ 0 limit): e−iω (T−r) F2a(ω),−i ǫ 2 π i T e−iω T F2b(ω),−i ǫ = −2 π i T e−iω (T−r) F4a(ω),−i ǫ 4 π i T 3 e−iω T F4b(ω),−i ǫ = −4 π i T The only pole is at ω = −i ǫ, so if we can close the contour in the upper half-plane, we will get zero. • For T < 0, both the “a” and “b” integrals can be closed in C1. Result: zero contribution. • For 0 < T < r, the “a” integrals can be closed in C1, but the “b” integrals must be closed in C2. Result: “b” contribution. • For T > r, both the “a” and “b” integrals must be closed in C2. But then the “a” and “b” residues cancel out. Result: zero contribution. Thus the only interesting contribution happens in the interval 0 < T < r ⇔ t− r(τ) < τ < t. In this case, the final integrals yield (2 π)3 e−iω (T−r) F2b(ω) = − 2 π r3 (2 π)3 e−iω (T−r) F4b(ω) = − −ω − iǫ ω + iǫ FIG. 7: Contours needed to complete integration over k (left) and ω (right). leading to the final result for K and L: 4 π r δ(T − r)− 1 4 π r δ(T ), 4 π r δ(T − r)−Θ(T )Θ(r − T ) T 2 π r3 4 π r δ(T − r)−Θ(T )Θ(r − T ) T We use these to calculate the Ii j and Ii j k l: Ii j = ni nj 4 π r δ(T − r)−Θ(T )Θ(r − T ) T 2 π r3 4 π r δ(T ) + Θ(T )Θ(r − T ) T 2 π r3 ,(A19) Ii j k l = ni nj nk nl 4 π r δ(T − r) −Θ(T )Θ(r − T ) T − 3Q(i j nk nl) Θ(T )Θ(r − T ) 2 π r3 Q(i j Qk l) 4 π r δ(T ) + Θ(T )Θ(r − T ) . (A20) 4. Time integration The final integrations will be over the source time τ . The “crossing times” for the two Θ functions are τ = t and τ = tr, where t is the present field time, and tr the corresponding retarded time defined by (25). 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0704.0629

Measurement of the Decay Constant $f_D{_S^+}$ using $D_S^+ --> ell^+ nu

CLNS 07/1989 CLEO 07-01 Measurement of the Decay Constant f using D+ → ℓ+ν M. Artuso,1 S. Blusk,1 J. Butt,1 S. Khalil,1 J. Li,1 N. Menaa,1 R. Mountain,1 S. Nisar,1 K. Randrianarivony,1 R. Sia,1 T. Skwarnicki,1 S. Stone,1 J. C. Wang,1 G. Bonvicini,2 D. Cinabro,2 M. Dubrovin,2 A. Lincoln,2 D. M. Asner,3 K. W. Edwards,3 P. Naik,3 R. A. Briere,4 T. Ferguson,4 G. Tatishvili,4 H. Vogel,4 M. E. Watkins,4 J. L. Rosner,5 N. E. Adam,6 J. P. Alexander,6 D. G. Cassel,6 J. E. Duboscq,6 R. Ehrlich,6 L. Fields,6 L. Gibbons,6 R. Gray,6 S. W. Gray,6 D. L. Hartill,6 B. K. Heltsley,6 D. Hertz,6 C. D. Jones,6 J. Kandaswamy,6 D. L. Kreinick,6 V. E. Kuznetsov,6 H. Mahlke-Krüger,6 D. Mohapatra,6 P. U. E. Onyisi,6 J. R. Patterson,6 D. Peterson,6 J. Pivarski,6 D. Riley,6 A. Ryd,6 A. J. Sadoff,6 H. Schwarthoff,6 X. Shi,6 S. Stroiney,6 W. M. Sun,6 T. Wilksen,6 S. B. Athar,7 R. Patel,7 J. Yelton,7 P. Rubin,8 C. Cawlfield,9 B. I. Eisenstein,9 I. Karliner,9 D. Kim,9 N. Lowrey,9 M. Selen,9 E. J. White,9 J. Wiss,9 R. E. Mitchell,10 M. R. Shepherd,10 D. Besson,11 T. K. Pedlar,12 D. Cronin-Hennessy,13 K. Y. Gao,13 J. Hietala,13 Y. Kubota,13 T. Klein,13 B. W. Lang,13 R. Poling,13 A. W. Scott,13 A. Smith,13 P. Zweber,13 S. Dobbs,14 Z. Metreveli,14 K. K. Seth,14 A. Tomaradze,14 J. Ernst,15 K. M. Ecklund,16 H. Severini,17 W. Love,18 V. Savinov,18 O. Aquines,19 A. Lopez,19 S. Mehrabyan,19 H. Mendez,19 J. Ramirez,19 G. S. Huang,20 D. H. Miller,20 V. Pavlunin,20 B. Sanghi,20 I. P. J. Shipsey,20 B. Xin,20 G. S. Adams,21 M. Anderson,21 J. P. Cummings,21 I. Danko,21 D. Hu,21 B. Moziak,21 J. Napolitano,21 Q. He,22 J. Insler,22 H. Muramatsu,22 C. S. Park,22 E. H. Thorndike,22 and F. Yang22 (CLEO Collaboration) Syracuse University, Syracuse, New York 13244 Wayne State University, Detroit, Michigan 48202 Carleton University, Ottawa, Ontario, Canada K1S 5B6 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 Cornell University, Ithaca, New York 14853 University of Florida, Gainesville, Florida 32611 George Mason University, Fairfax, Virginia 22030 University of Illinois, Urbana-Champaign, Illinois 61801 Indiana University, Bloomington, Indiana 47405 University of Kansas, Lawrence, Kansas 66045 Luther College, Decorah, Iowa 52101 University of Minnesota, Minneapolis, Minnesota 55455 Northwestern University, Evanston, Illinois 60208 State University of New York at Albany, Albany, New York 12222 State University of New York at Buffalo, Buffalo, New York 14260 University of Oklahoma, Norman, Oklahoma 73019 University of Pittsburgh, Pittsburgh, Pennsylvania 15260 University of Puerto Rico, Mayaguez, Puerto Rico 00681 Purdue University, West Lafayette, Indiana 47907 Rensselaer Polytechnic Institute, Troy, New York 12180 University of Rochester, Rochester, New York 14627 (Dated: November 1, 2018) We measure the decay constant f using the D+s → ℓ +ν channel, where the ℓ+ designates either a µ+ or a τ+, when the τ+ → π+ν. Using both measurements we find f = 274 ± 13 ± 7 MeV. Combining with our previous determination of fD+ , we compute the ratio fD+ /fD+ = 1.23±0.11± 0.04. We compare with theoretical estimates. PACS numbers: 13.20.Fc, 13.66.Bc To extract precise information on the size of CKM matrix elements from Bd and Bs mixing measure- ments the ratio of “decay constants,” that are re- lated to the heavy and light quark wave-function overlap at zero separation, must be well known [1]. Recent measurement of B0s mixing by CDF [2] has shown the urgent need for precise numbers. De- cay constants have been calculated for both B and http://arxiv.org/abs/0704.0629v3 D mesons using several methods, including lattice QCD [3]. Here we present the most precise measure- ment to date of f , and combined with our previ- ous determination of fD+ [4, 5], we find fD+ /fD+ . In the Standard Model (SM) purely leptonic Ds decay proceeds via annihilation through a virtual W+. The decay rate is given by [6] s → ℓ |Vcs| where M is the D+s mass, mℓ is the lepton mass, GF is the Fermi constant, and |Vcs| is a CKM matrix element with a value of 0.9738 [7]. In this Letter we report measurements of both B(D+s → µ +ν) and B(D+s → τ +ν), when τ+ → π+ν (D+s → π +νν). More details are given in a compan- ion paper [8]. The ratio Γ(D+s → τ +ν)/Γ(D+s → µ+ν) predicted in the SM via Eq. 1 depends only on well-known masses, and equals 9.72; any devia- tion would be a manifestation of new physics as it would violate lepton universality [9]. New physics can also affect the expected widths; any undiscov- ered charged bosons would interfere with the SM W+ [10]. The CLEO-c detector [11] is equipped to mea- sure the momenta of charged particles, identify them using dE/dx and Cherenkov imaging (RICH) [12], detect photons and determine their directions and energies. We use 314 pb−1 of data produced in e+e− collisions using CESR near 4.170 GeV. Here the cross-section for our analyzed sample, D∗+s D s , is ∼1 nb. Other charm produc- tion totals ∼7 nb [13], and the underlying light- quark “continuum” is ∼12 nb. We fully reconstruct oneD−s as a “tag,” and examine the properties of the D+s . (Charge conjugate decays are used.) Track se- lection, particle identification, π0, η, and K0S criteria are the same as those described in Ref. [4], except that RICH identification now requires a minimum momentum of 700 MeV/c. Tag modes are listed in Table I. For resonance de- cays we select intervals in invariant mass within ±10 MeV of the known mass for η′ → π+π−η, ±10 MeV for φ → K+K−, ±100 MeV for K∗0 → K−π+, and ±150 MeV for ρ− → π−π0. We require tags to have momentum consistent with coming from DsD s pro- duction. The distribution for the K+K−π− mode (44% of all the tags) is shown in Fig. 1. To select tags, we first fit the invariant mass dis- tributions to the sum of two Gaussians centered at MDs . The r.m.s. resolution (σ) is defined as σ ≡ f1σ1 + (1 − f1)σ2, where σ1 and σ2 are the in- dividual widths and f1 is the fractional area of the FIG. 1: Invariant mass of K+K−π− candidates after requiring the total energy to be consistent with the beam energy. The curve shows a fit to a two-Gaussian signal function plus a polynomial background. TABLE I: Tagging modes and numbers of signal and background events, within cuts, from two-Gaussian fits to the invariant mass plots, and the number of γ tags in each mode, within ±2.5σ from a fit to the signal Crys- tal Ball function (see text) and a 5th order Chebychev background polynomial and the associated background. Mode Invariant Mass MM∗2 Signal Bkgrnd Signal Bkgrnd K+K−π− 13871±262 10850 8053± 211 13538 − 3122±79 1609 1933±88 2224 ηπ− 1609± 112 4666 1024±97 3967 η′π− 1196±46 409 792±69 1052 φρ− 1678±74 1898 1050±113 3991 π+π−π− 3654±199 25208 2300±187 15723 K∗−K∗0 2030±98 4878 1298±130 5672 ηρ− 4142±281 20784 2195±225 17353 Sum 31302 ± 472 70302 18645±426 63520 first Gaussian. We require the invariant masses to be within ± 2.5σ (±2σ for the ηρ− mode) of MDs . We have a total of 31302±472 tag candidates. Then we add a γ candidate that satisfies our shower shape re- quirement. Regardless of whether or not the γ forms a D∗s with the tag, for real D sDs events, the missing mass squared, MM∗2, recoiling against the γ and the D−s tag should peak at M . We calculate MM∗2 = (ECM − EDs − Eγ) −→pCM − −→pDs − where ECM ( −→pCM) is the center-of-mass energy (mo- mentum), EDs ( −→pDs) is the energy (momentum) of the fully reconstructed D−s tag, Eγ ( −→pγ) is the en- ergy (momentum) of the additional γ. We use a kinematic fit that constrains the decay products of the D−s to MDs and conserves overall momentum and energy. All γ’s in the event are used, except for those that are decay products of the D−s tag. The MM∗2 distribution from K+K−π− tags is shown in Fig. 2. We fit all the modes individually to determine the number of tag events. This proce- dure is enhanced by having information on the shape of the signal function. We use fully reconstructed D−s D s events, and examine the signal shape when one Ds is ignored. The signal is fit to a Crystal Ball function [14], which determines σ and the shape of the tail. Though σ varies somewhat between modes, the tail parameters don’t change, since they depend on beam radiation and γ energy resolution. FIG. 2: The MM∗2 distribution from events with a γ in addition to the K+K−π− tag. The curve is a fit to the Crystal Ball function and a 5th order Chebychev background function. Fits of MM∗2 in each mode when summed show 18645±426 events within a ±2.5σ interval (see Ta- ble I). There is a small enhancement of (4.8± 1.0)% in our ability to find tags in µ+ν (or π+νν) events (tag bias) as compared with generic events. Addi- tional systematic errors are evaluated by changing the fitting range, using 4th and 6th order Chebychev background polynomials, and allowing the parame- ters of the tail of the fitting function to float, leading to an overall systematic uncertainty of 5%. Candidate µ+ν events are required to have only a single additional track oppositely charged to the tag with an angle >35.9◦ with respect to the beam line. We also require that there not be any neutral en- ergy cluster detected of more than 300 MeV, which is especially useful to reject D+s → π +π0 and ηπ+ decays. Since here we are searching for events in which there is a single missing ν, the missing mass squared, MM2, should peak at zero: MM2 = (ECM − EDs − Eγ − Eµ) −→pCM − −→pDs − −→pγ − where Eµ ( −→pµ) are the energy (momentum) of the candidate µ+ track. We also make use of a set of kinematical con- straints and fit each event to two hypotheses: (1) the D−s tag is the daughter of a D s and (2) the D∗+s decays into γD s . The kinemati- cal constraints, in the center-of-mass frame, are −→pDs + −→pD∗ = 0, ECM = EDs + ED∗ , ED∗ ECM/2+ −M2Ds /2ECM or EDs = ECM/2− −M2Ds /2ECM, MD∗ − MDs = 143.6 MeV. In addition, we constrain the invariant mass of the D−s tag to MDs . This gives a total of 7 constraints. The missing ν four-vector needs to be determined, so we are left with a three-constraint fit. We perform an iterative fit minimizing χ2. To eliminate system- atic uncertainties that depend on understanding the absolute scale of the errors, we do not make a χ2 cut but simply choose the γ and the decay sequence in each event with the minimum χ2. We consider three separate cases: (i) the track de- posits < 300 MeV in the calorimeter, characteristic of a non-interacting pion or a µ+; (ii) the track de- posits > 300 MeV in the calorimeter, characteristic of an interacting pion; or (iii) the track satisfies our electron selection criteria. The separation between muons and pions is not complete. Case (i) contains 99% of the muons but also 60% of the pions, while case (ii) includes 1% of the muons and 40% of the pions [5]. Case (iii) does not include any signal but is used for background estimation. For cases (i) and (ii) we insist that the track not be identified as an electron or a kaon. Electron candidates have a match between the momentum measured in the tracking system and the energy deposited in the CsI calorime- ter, and dE/dx and RICH measurements consistent with this hypothesis. For the µ+ν final state the MM2 distribution is modeled as the sum of two Gaussians centered at zero. A Monte Carlo (MC) simulation of the MM2 shows σ=0.025 GeV2 after the fit. We check the resolution using the D+s → K K+ mode. We search for events with at least one additional track identi- fied as a kaon using the RICH detector, in addition to a D−s tag. The MM 2 resolution is 0.025 GeV2 in agreement with the simulation. In the π+νν final state, the extra missing ν re- sults in a smeared MM2 distribution that is almost triangular in shape starting near -0.05 GeV2, peak- ing near 0.10 GeV2, and ending at 0.75 GeV2. FIG. 3: The MM2 distributions from data usingD−s tags, and one additional opposite-sign charged track and no extra energetic showers, for cases (i), (ii), and (iii). The MM2 distributions from data are shown in Fig. 3. The overall signal region is -0.05 < MM2 < 0.20 GeV2. The upper limit is chosen to prevent background from ηπ+ and K0π+ final states. The peak in Fig. 3(i) is due to D+s → µ +ν. Below 0.20 GeV2 in both (i) and (ii) we have π+νν events. The specific signal regions are: for µ+ν, −0.05 < MM2 < 0.05 GeV2, corresponding to ±2σ; for π+νν, in case (i) 0.05 < MM2 < 0.20 GeV2 and in case (ii) −0.05 < MM2 < 0.20 GeV2. In these regions we find 92, 31, and 25 events, respectively. We consider backgrounds from two sources: one from real D+s decays and the other from the back- ground under the single-tag signal peaks. For the latter, we estimate the background from data using side-bands of the invariant mass, shown in Fig. 1. For case (i) we find 3.5 (properly normalized) back- ground events in the µ+ν region and 2.5 back- grounds in the τ+ν region; for case (ii) we find 3 events. Our total background estimate summing over all of these cases is 9.0±2.3 events. The background from real D+s decays is evaluated by identifying specific sources. For µ+ν the only possible background is D+s → π +π0. Using a 195 pb−1 subsample of our data, we limit the branching fraction as < 1.1 × 10−3 at 90% C.L. [8]. This low rate coupled with the extra γ veto yields a negligible contribution. The real D+s backgrounds for π are listed in Table II. Using the SM expected ratio of decay rates we calculate a contribution of 7.4 π+νν events. TABLE II: Event backgrounds in the π+νν sample from real D+s decays. Source B(%) case (i) case (ii) Sum D+s → Xµ +ν 8.2 0+1.8 0 0+1.8 D+s → π +π0π0 1.0 0.03±0.04 0.08±0.03 0.11±0.04 D+s → τ +ν 6.4 τ+ → π+π0ν 1.5 0.55±0.22 0.64±0.24 1.20±0.33 τ+ → µ+νν 1.0 0.37±0.15 0 0.37±0.15 Sum 1.0+1.8 0.7±0.2 1.7+1.8 The event yield in the signal region, Ndet (92), is related to the number of tags, Ntag, the branching fractions, and the background Nbkgrd (3.5) as Ndet −Nbkgrd = Ntag · ǫ[ǫ ′B(D+s → µ +ν) (3) +ǫ′′B(D+s → π +νν)], where ǫ (80.1%) includes the efficiencies (77.8%) for reconstructing the single charged track including fi- nal state radiation, (98.3)% for not having another unmatched cluster in the event with energy greater than 300 MeV, and the correction for the tag bias (4.8%); ǫ′ (91.4%) is the product of the 99.0% µ+ calorimeter efficiency and the 92.3% acceptance of the MM2 cut of |MM2| < 0.05 GeV2; ǫ′′ (7.6%) is the fraction of π+νν events contained in the µ+ν sig- nal window (13.2%) times the 60% acceptance for a pion to deposit less than 300 MeV in the calorime- ter. Using B(τ+ → π+ν) of (10.90±0.07)% [7], the ratio of the π+νν to µ+ν widths is 1.059; we find: B(D+s → µ +ν) = (0.594± 0.066± 0.031)%. (4) We can also sum the µ+ν and τ+ν contributions for −0.05 < MM2 < 0.02 GeV2. Equation 3 still ap- plies. The number of signal and background events changes to 148 and 10.7, respectively. ǫ′ becomes 96.2%, and ǫ′′ increases to 45.2%. The effective branching fraction, assuming lepton universality, is Beff(D+s → µ +ν) = (0.638± 0.059± 0.033)%. (5) The systematic errors on these branching fractions are dominated by the error on the number of tags (5%). Other errors include: (a) track finding (0.7%), determined from a detailed comparison of the sim- ulation with double tag events where one track is ignored; (b) the error due to the requirement that the charged track deposit no more than 300 MeV in the calorimeter (1%), determined using two-body D0 → K−π+ decays [5]; (c) the γ veto efficiency (1%), determined by extrapolating measurements on fully reconstructed events. Systematic errors arising from the background estimates are negligible. The total systematic error for Eq. 4 is 5.2%, and is 5.1% for Eq. 5 as (b) doesn’t apply here. We also analyze the τ+ν final state independently. For case (i) we define the signal region to be the in- terval 0.05<MM2 <0.20 GeV2, while for case (ii) -0.05<MM2 <0.20 GeV2. The upper limit on MM2 is chosen to avoid background from the tail of the K0π+ peak. The fractions of the MM2 range ac- cepted are 32% and 45% for case (i) and (ii), respec- tively. We find 31 [25] events in the signal region with a background of 3.5 [5.1] events for case (i) [(ii)]. The branching fraction, averaging the two cases is B(D+s → τ +ν) = (8.0± 1.3± 0.4)%, (6) where the systematic error includes a contribution of 0.06% from the uncertainty on B(τ+ → π+ν). We measure 13.4 ± 2.6 ± 0.2 for the ratio of τ+ν to µ+ν rates using Eq. 4. Here the systematic er- ror is dominated by the uncertainty on the mini- mum ionization cut. We also set an upper limit of B(D+s → e +ν) < 1.3 × 10−4 at 90% C.L. Both of these results are consistent with SM predictions and lepton universality. We perform an overall check of our procedures by measuring B(D+s → K K+). We compute the MM2 (Eq. 2) using events with an additional charged track identified as a kaon. These track candidates have momenta of approximately 1 GeV/c; here the RICH has a pion to kaon fake rate of 1.1% with a kaon detection efficiency of 88.5% [12]. For this study, we do not veto events with extra charged tracks, or γ’s, because of the presence of the K0. We deter- mine B(D+s → K K+) = (2.90 ± 0.19 ± 0.18)%. This method gives a result in good agreement with preliminary CLEO-c results using double tags of (3.00 ± 0.19 ± 0.10)% [15]; these results are not in- dependent. We also performed the entire analysis on a MC sample that is 4 times larger than the data sam- ple. The input branching fraction is 0.5% for µ+ν and 6.57% for τ+ν, while our analysis measured (0.514±0.027)% for the case (i) µ+ν signal and (0.521±0.024)% for µ+ν and τ+ν combined. Using B(D+s → µ +ν) from Eq. 5, and Eq. 1 with a Ds lifetime of (500±7)×10 −15 s [7], we extract = 274± 13± 7 MeV. (7) We combine with our previous result fD+ = 222.6± 16.7+2.8 −3.4 MeV [4], and find /fD+ = 1.23± 0.11± 0.04. (8) Lattice QCD predictions for f and the ratio /fD+ have been summarized by Onogi [16]. Our measurements are consistent with most calculations; examples are unquenched Lattice that predicts 249± 3± 16 MeV and 1.24± 0.01± 0.07 for the ratio [17], while a recent quenched prediction gives 266± 10± 18 MeV and 1.13 ± 0.03 ± 0.05 [18]. There is no evidence yet for any suppression in the ratio due to the presence of a virtual charged Higgs [10]. The CLEO-c determination of f is the most accurate to date and consistent with other measure- ments [7, 8]. It also does not rely on the indepen- dent determination of any normalization mode (e.g. φπ+). (We note that a preliminary CLEO-c result using D+s → τ +ν, τ+ → e+νν [19] is consistent with these results.) We gratefully acknowledge the effort of the CESR staff in providing us with excellent luminosity and running conditions. This work was supported by the A.P. Sloan Foundation, the National Science Foun- dation, the U.S. Department of Energy, and the Nat- ural Sciences and Engineering Research Council of Canada. [1] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). [2] A. Abulencia et al. (CDF), Phys. Rev. Lett. 97, 242003 (2006). See also V. Abazov et al. (D0), Phys. Rev. Lett. 97, 021802 (2006). [3] C. Davies et al., Phys. Rev. Lett. 92, 022001 (2004). [4] M. Artuso et al. (CLEO), Phys. Rev. Lett. 95, 251801 (2005). [5] G. Bonvicini et al. (CLEO) Phys. Rev. D70, 112004 (2004). [6] D. Silverman and H. Yao, Phys. Rev. D38, 214 (1988). [7] W.-M. Yao et al., J. Phys. G33, 1 (2006). [8] T. K. Pedlar et al. (CLEO), arXiv:0704.0437[hep- ex], submitted to Phys. Rev. D. [9] J. Hewett, [hep-ph/9505246]; W.-S. Hou, Phys. Rev. D48, 2342 (1993). [10] A. G. Akeroyd, Prog. Theor. Phys. 111, 295 (2004). [11] D. Peterson et al., Nucl. Instrum. and Meth. A478, 142 (2002); Y. Kubota et al., Nucl. Instrum. and Meth. A320, 66 (1992). [12] M. Artuso et al., Nucl. Instrum. Meth. A554, 147 (2005). [13] R. Poling, [hep-ex/0606016]. [14] P. Rubin et al. (CLEO), Phys. Rev. D73, 112005 (2006). http://arxiv.org/abs/0704.0437 http://arxiv.org/abs/hep-ph/9505246 http://arxiv.org/abs/hep-ex/0606016 [15] N. E. Adam et al. (CLEO), [hep-ex/0607079]. [16] T. Onogi [hep-lat/0610115]. [17] C. Aubin et al., Phys. Rev. Lett. 95, 122002 (2005). [18] T. W. Chiu et al., Phys. Lett. B624, 31 (2005). [19] S. Stone [hep-ex/0610026]. http://arxiv.org/abs/hep-ex/0607079 http://arxiv.org/abs/hep-lat/0610115 http://arxiv.org/abs/hep-ex/0610026

0704.0630

The $e^+ e^-\to K^+ K^- \pi^+\pi^-$, $K^+ K^- \pi^0\pi^0$ and $K^+ K^- K^+ K^-$ Cross Sections Measured with Initial-State Radiation

BABAR-PUB-07/021 SLAC-PUB-12435 Phys. Rev. D76, 012008 (2007) The e+e− → K+K−π+π−, K+K−π0π0 and K+K−K+K− Cross Sections Measured with Initial-State Radiation B. Aubert, M. Bona, D. Boutigny, Y. Karyotakis, J. P. Lees, V. Poireau, X. Prudent, V. Tisserand, and A. Zghiche Laboratoire de Physique des Particules, IN2P3/CNRS et Université de Savoie, F-74941 Annecy-Le-Vieux, France J. Garra Tico and E. Grauges Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain L. Lopez and A. Palano Università di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy G. Eigen, B. Stugu, and L. Sun University of Bergen, Institute of Physics, N-5007 Bergen, Norway G. S. Abrams, M. Battaglia, D. N. Brown, J. Button-Shafer, R. N. Cahn, Y. Groysman, R. G. Jacobsen, J. A. Kadyk, L. T. Kerth, Yu. G. Kolomensky, G. Kukartsev, D. Lopes Pegna, G. Lynch, L. M. Mir, T. J. Orimoto, M. T. Ronan,∗ K. Tackmann, and W. A. Wenzel Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA P. del Amo Sanchez, C. M. Hawkes, and A. T. Watson University of Birmingham, Birmingham, B15 2TT, United Kingdom T. Held, H. Koch, B. Lewandowski, M. Pelizaeus, T. Schroeder, and M. Steinke Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany D. Walker University of Bristol, Bristol BS8 1TL, United Kingdom D. J. Asgeirsson, T. Cuhadar-Donszelmann, B. G. Fulsom, C. Hearty, N. S. Knecht, T. S. Mattison, and J. A. McKenna University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 M. Barrett, A. Khan, M. Saleem, and L. Teodorescu Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom V. E. Blinov, A. D. Bukin, V. P. Druzhinin, V. B. Golubev, A. P. Onuchin, S. I. Serednyakov, Yu. I. Skovpen, E. P. Solodov, and K. Yu Todyshev Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia M. Bondioli, S. Curry, I. Eschrich, D. Kirkby, A. J. Lankford, P. Lund, M. Mandelkern, E. C. Martin, and D. P. Stoker University of California at Irvine, Irvine, California 92697, USA S. Abachi and C. Buchanan University of California at Los Angeles, Los Angeles, California 90024, USA S. D. Foulkes, J. W. Gary, F. Liu, O. Long, B. C. Shen, and L. Zhang University of California at Riverside, Riverside, California 92521, USA H. P. Paar, S. Rahatlou, and V. Sharma University of California at San Diego, La Jolla, California 92093, USA J. W. Berryhill, C. Campagnari, A. Cunha, B. Dahmes, T. M. Hong, D. Kovalskyi, and J. D. Richman http://arxiv.org/abs/0704.0630v3 University of California at Santa Barbara, Santa Barbara, California 93106, USA T. W. Beck, A. M. Eisner, C. J. Flacco, C. A. Heusch, J. Kroseberg, W. S. Lockman, T. Schalk, B. A. Schumm, A. Seiden, D. C. Williams, M. G. Wilson, and L. O. Winstrom University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA E. Chen, C. H. Cheng, F. Fang, D. G. Hitlin, I. Narsky, T. Piatenko, and F. C. Porter California Institute of Technology, Pasadena, California 91125, USA G. Mancinelli, B. T. Meadows, K. Mishra, and M. D. Sokoloff University of Cincinnati, Cincinnati, Ohio 45221, USA F. Blanc, P. C. Bloom, S. Chen, W. T. Ford, J. F. Hirschauer, A. Kreisel, M. Nagel, U. Nauenberg, A. Olivas, J. G. Smith, K. A. Ulmer, S. R. Wagner, and J. Zhang University of Colorado, Boulder, Colorado 80309, USA A. M. Gabareen, A. Soffer, W. H. Toki, R. J. Wilson, F. Winklmeier, and Q. Zeng Colorado State University, Fort Collins, Colorado 80523, USA D. D. Altenburg, E. Feltresi, A. Hauke, H. Jasper, J. Merkel, A. Petzold, B. Spaan, and K. Wacker Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany T. Brandt, V. Klose, M. J. Kobel, H. M. Lacker, W. F. Mader, R. Nogowski, J. Schubert, K. R. Schubert, R. Schwierz, J. E. Sundermann, and A. Volk Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany D. Bernard, G. R. Bonneaud, E. Latour, V. Lombardo, Ch. Thiebaux, and M. Verderi Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France P. J. Clark, W. Gradl, F. Muheim, S. Playfer, A. I. Robertson, and Y. Xie University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom M. Andreotti, D. Bettoni, C. Bozzi, R. Calabrese, A. Cecchi, G. Cibinetto, P. Franchini, E. Luppi, M. Negrini, A. Petrella, L. Piemontese, E. Prencipe, and V. Santoro Università di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy F. Anulli, R. Baldini-Ferroli, A. Calcaterra, R. de Sangro, G. Finocchiaro, S. Pacetti, P. Patteri, I. M. Peruzzi,† M. Piccolo, M. Rama, and A. Zallo Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy A. Buzzo, R. Contri, M. Lo Vetere, M. M. Macri, M. R. Monge, S. Passaggio, C. Patrignani, E. Robutti, A. Santroni, and S. Tosi Università di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy K. S. Chaisanguanthum, M. Morii, and J. Wu Harvard University, Cambridge, Massachusetts 02138, USA R. S. Dubitzky, J. Marks, S. Schenk, and U. Uwer Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany D. J. Bard, P. D. Dauncey, R. L. Flack, J. A. Nash, M. B. Nikolich, and W. Panduro Vazquez Imperial College London, London, SW7 2AZ, United Kingdom P. K. Behera, X. Chai, M. J. Charles, U. Mallik, N. T. Meyer, and V. Ziegler University of Iowa, Iowa City, Iowa 52242, USA J. Cochran, H. B. Crawley, L. Dong, V. Eyges, W. T. Meyer, S. Prell, E. I. Rosenberg, and A. E. Rubin Iowa State University, Ames, Iowa 50011-3160, USA A. V. Gritsan, Z. J. Guo, and C. K. Lae Johns Hopkins University, Baltimore, Maryland 21218, USA A. G. Denig, M. Fritsch, and G. Schott Universität Karlsruhe, Institut für Experimentelle Kernphysik, D-76021 Karlsruhe, Germany N. Arnaud, J. Béquilleux, M. Davier, G. Grosdidier, A. Höcker, V. Lepeltier, F. Le Diberder, A. M. Lutz, S. Pruvot, S. Rodier, P. Roudeau, M. H. Schune, J. Serrano, V. Sordini, A. Stocchi, W. F. Wang, and G. Wormser Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France D. J. Lange and D. M. Wright Lawrence Livermore National Laboratory, Livermore, California 94550, USA C. A. Chavez, I. J. Forster, J. R. Fry, E. Gabathuler, R. Gamet, D. E. Hutchcroft, D. J. Payne, K. C. Schofield, and C. Touramanis University of Liverpool, Liverpool L69 7ZE, United Kingdom A. J. Bevan, K. A. George, F. Di Lodovico, W. Menges, and R. Sacco Queen Mary, University of London, E1 4NS, United Kingdom G. Cowan, H. U. Flaecher, D. A. Hopkins, P. S. Jackson, T. R. McMahon, F. Salvatore, and A. C. Wren University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom D. N. Brown and C. L. Davis University of Louisville, Louisville, Kentucky 40292, USA J. Allison, N. R. Barlow, R. J. Barlow, Y. M. Chia, C. L. Edgar, G. D. Lafferty, T. J. West, and J. I. Yi University of Manchester, Manchester M13 9PL, United Kingdom J. Anderson, C. Chen, A. Jawahery, D. A. Roberts, G. Simi, and J. M. Tuggle University of Maryland, College Park, Maryland 20742, USA G. Blaylock, C. Dallapiccola, S. S. Hertzbach, X. Li, T. B. Moore, E. Salvati, and S. Saremi University of Massachusetts, Amherst, Massachusetts 01003, USA R. Cowan, P. H. Fisher, G. Sciolla, S. J. Sekula, M. Spitznagel, F. Taylor, and R. K. Yamamoto Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA S. E. Mclachlin, P. M. Patel, and S. H. Robertson McGill University, Montréal, Québec, Canada H3A 2T8 A. Lazzaro and F. Palombo Università di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy J. M. Bauer, L. Cremaldi, V. Eschenburg, R. Godang, R. Kroeger, D. A. Sanders, D. J. Summers, and H. W. Zhao University of Mississippi, University, Mississippi 38677, USA S. Brunet, D. Côté, M. Simard, P. Taras, and F. B. Viaud Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 H. Nicholson Mount Holyoke College, South Hadley, Massachusetts 01075, USA G. De Nardo, F. Fabozzi,‡ L. Lista, D. Monorchio, and C. Sciacca Università di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy M. A. Baak, G. Raven, and H. L. Snoek NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands C. P. Jessop and J. M. LoSecco University of Notre Dame, Notre Dame, Indiana 46556, USA G. Benelli, L. A. Corwin, K. K. Gan, K. Honscheid, D. Hufnagel, H. Kagan, R. Kass, J. P. Morris, A. M. Rahimi, J. J. Regensburger, R. Ter-Antonyan, and Q. K. Wong Ohio State University, Columbus, Ohio 43210, USA N. L. Blount, J. Brau, R. Frey, O. Igonkina, J. A. Kolb, M. Lu, R. Rahmat, N. B. Sinev, D. Strom, J. Strube, and E. Torrence University of Oregon, Eugene, Oregon 97403, USA N. Gagliardi, A. Gaz, M. Margoni, M. Morandin, A. Pompili, M. Posocco, M. Rotondo, F. Simonetto, R. Stroili, and C. Voci Università di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy E. Ben-Haim, H. Briand, G. Calderini, J. Chauveau, P. David, L. Del Buono, Ch. de la Vaissière, O. Hamon, Ph. Leruste, J. Malclès, J. Ocariz, and A. Perez Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France L. Gladney University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA M. Biasini, R. Covarelli, and E. Manoni Università di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy C. Angelini, G. Batignani, S. Bettarini, M. Carpinelli, R. Cenci, A. Cervelli, F. Forti, M. A. Giorgi, A. Lusiani, G. Marchiori, M. A. Mazur, M. Morganti, N. Neri, E. Paoloni, G. Rizzo, and J. J. Walsh Università di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy M. Haire Prairie View A&M University, Prairie View, Texas 77446, USA J. Biesiada, P. Elmer, Y. P. Lau, C. Lu, J. Olsen, A. J. S. Smith, and A. V. Telnov Princeton University, Princeton, New Jersey 08544, USA E. Baracchini, F. Bellini, G. Cavoto, A. D’Orazio, D. del Re, E. Di Marco, R. Faccini, F. Ferrarotto, F. Ferroni, M. Gaspero, P. D. Jackson, L. Li Gioi, M. A. Mazzoni, S. Morganti, G. Piredda, F. Polci, F. Renga, and C. Voena Università di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy M. Ebert, H. Schröder, and R. Waldi Universität Rostock, D-18051 Rostock, Germany T. Adye, G. Castelli, B. Franek, E. O. Olaiya, S. Ricciardi, W. Roethel, and F. F. Wilson Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom R. Aleksan, S. Emery, M. Escalier, A. Gaidot, S. F. Ganzhur, G. Hamel de Monchenault, W. Kozanecki, M. Legendre, G. Vasseur, Ch. Yèche, and M. Zito DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France X. R. Chen, H. Liu, W. Park, M. V. Purohit, and J. R. Wilson University of South Carolina, Columbia, South Carolina 29208, USA M. T. Allen, D. Aston, R. Bartoldus, P. Bechtle, N. Berger, R. Claus, J. P. Coleman, M. R. Convery, J. C. Dingfelder, J. Dorfan, G. P. Dubois-Felsmann, D. Dujmic, W. Dunwoodie, R. C. Field, T. Glanzman, S. J. Gowdy, M. T. Graham, P. Grenier, C. Hast, T. Hryn’ova, W. R. Innes, J. Kaminski, M. H. Kelsey, H. Kim, P. Kim, M. L. Kocian, D. W. G. S. Leith, S. Li, S. Luitz, V. Luth, H. L. Lynch, D. B. MacFarlane, H. Marsiske, R. Messner, D. R. Muller, C. P. O’Grady, I. Ofte, A. Perazzo, M. Perl, T. Pulliam, B. N. Ratcliff, A. Roodman, A. A. Salnikov, R. H. Schindler, J. Schwiening, A. Snyder, J. Stelzer, D. Su, M. K. Sullivan, K. Suzuki, S. K. Swain, J. M. Thompson, J. Va’vra, N. van Bakel, A. P. Wagner, M. Weaver, W. J. Wisniewski, M. Wittgen, D. H. Wright, A. K. Yarritu, K. Yi, and C. C. Young Stanford Linear Accelerator Center, Stanford, California 94309, USA P. R. Burchat, A. J. Edwards, S. A. Majewski, B. A. Petersen, and L. Wilden Stanford University, Stanford, California 94305-4060, USA S. Ahmed, M. S. Alam, R. Bula, J. A. Ernst, V. Jain, B. Pan, M. A. Saeed, F. R. Wappler, and S. B. Zain State University of New York, Albany, New York 12222, USA W. Bugg, M. Krishnamurthy, and S. M. Spanier University of Tennessee, Knoxville, Tennessee 37996, USA R. Eckmann, J. L. Ritchie, A. M. Ruland, C. J. Schilling, and R. F. Schwitters University of Texas at Austin, Austin, Texas 78712, USA J. M. Izen, X. C. Lou, and S. Ye University of Texas at Dallas, Richardson, Texas 75083, USA F. Bianchi, F. Gallo, D. Gamba, and M. Pelliccioni Università di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy M. Bomben, L. Bosisio, C. Cartaro, F. Cossutti, G. Della Ricca, L. Lanceri, and L. Vitale Università di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy V. Azzolini, N. Lopez-March, F. Martinez-Vidal, D. A. Milanes, and A. Oyanguren IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain J. Albert, Sw. Banerjee, B. Bhuyan, K. Hamano, R. Kowalewski, I. M. Nugent, J. M. Roney, and R. J. Sobie University of Victoria, Victoria, British Columbia, Canada V8W 3P6 J. J. Back, P. F. Harrison, T. E. Latham, G. B. Mohanty, and M. Pappagallo§ Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom H. R. Band, X. Chen, S. Dasu, K. T. Flood, J. J. Hollar, P. E. Kutter, Y. Pan, M. Pierini, R. Prepost, S. L. Wu, and Z. Yu University of Wisconsin, Madison, Wisconsin 53706, USA H. Neal Yale University, New Haven, Connecticut 06511, USA (Dated: November 4, 2018) We study the processes e+e− → K+K−π+π−γ, K+K−π0π0γ and K+K−K+K−γ, where the photon is radiated from the initial state. About 34600, 4400 and 2300 fully reconstructed events, respectively, are selected from 232 fb−1 of BABAR data. The invariant mass of the hadronic final state defines the effective e+e− center-of-mass energy, so that the K+K−π+π−γ data can be compared with direct measurements of the e+e− → K+K−π+π− reaction; no direct measurements exist for the e+e− → K+K−π0π0 or e+e− → K+K−K+K− reactions. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from e+e− → φ(1020)f0(980) and study its structure near threshold. In the charmonium region, we observe the J/ψ in all three final states and several intermediate states, as well as the ψ(2S) in some modes, and measure the corresponding branching fractions. We see no signal for the Y (4260) and obtain an upper limit of BY (4260)→φπ+π− · Γ ee < 0.4 eV at 90% C.L. PACS numbers: 13.66.Bc, 14.40.Cs, 13.25.Gv, 13.25.Jx, 13.20.Jf I. INTRODUCTION Electron-positron annihilation at fixed center-of-mass (c.m.) energies has long been a mainstay of research in elementary particle physics. The idea of utilizing initial- state radiation (ISR) to explore e+e− reactions below the nominal c.m. energies was outlined in Ref. [1], and dis- cussed in the context of high-luminosity φ and B facto- ries in Refs. [2, 3, 4]. At high energies, e+e− annihilation is dominated by quark-level processes producing two or more hadronic jets. However, low-multiplicity exclusive processes dominate at energies below about 2 GeV, and the region near charm threshold, 3.0–4.5 GeV, features a number of resonances [5]. These allow us to probe a wealth of physics parameters, including cross sections, spectroscopy and form factors. Of particular current interest are the recently observed states in the charmonium region, such as the Y (4260) [6], and a possible discrepancy between the measured value of the anomalous magnetic moment of the muon, gµ− 2, and that predicted by the Standard Model [7]. Charmo- nium and other states with JPC = 1−− can be observed as resonances in the cross section, and intermediate states may be present in the hadronic system. Measurements of the decay modes and their branching fractions are im- portant in understanding the nature of these states. For example, the glue-ball model [8] predicts a large branch- ing fraction for Y (4260) into φππ. The prediction for gµ − 2 is based on hadronic-loop corrections measured from low-energy e+e− → hadrons data, and these dom- inate the uncertainty on the prediction. Improving this prediction requires not only more precise measurements, but also measurements over the entire energy range and inclusion of all the important subprocesses in order to un- derstand possible acceptance effects. ISR events atB fac- tories provide independent and contiguous measurements of hadronic cross sections from the production threshold to about 5 GeV. The cross section for the radiation of a photon of energy Eγ followed by the production of a particular hadronic final state f is related to the corresponding di- rect e+e− → f cross section σf (s) by dσγf (s, x) =W (s, x) · σf (s(1− x)) , (1) where s is the initial e+e− c.m. energy, x = 2Eγ/ is the fractional energy of the ISR photon and Ec.m. ≡ ∗Deceased †Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy ‡Also with Università della Basilicata, Potenza, Italy §Also with IPPP, Physics Department, Durham University, Durham DH1 3LE, United Kingdom s(1− x) is the effective c.m. energy at which the fi- nal state f is produced. The probability density function W (s, x) for ISR photon emission has been calculated with better than 1% precision (see e.g. Ref. [4]). It falls rapidly as Eγ increases from zero, but has a long tail, which com- bines with the increasing σf (s(1−x)) to produce a sizable cross section at very low Ec.m.. The angular distribution of the ISR photon peaks along the beam directions, but 10–15% [4] of the photons are within a typical detector acceptance. Experimentally, the measured invariant mass of the hadronic final state defines Ec.m.. An important feature of ISR data is that a wide range of energies is scanned simultaneously in one experiment, so that no structure is missed and the relative normalization uncertainties in data from different experiments or accelerator parame- ters are avoided. Furthermore, for large values of x the hadronic system is collimated, reducing acceptance issues and allowing measurements at energies down to produc- tion threshold. The mass resolution is not as good as a typical beam energy spread used in direct measurements, but the resolution and absolute energy scale can be mon- itored by the width and mass of well known resonances, such as the J/ψ produced in the reaction e+e− → J/ψγ. Backgrounds from e+e− → hadrons events at the nomi- s and from other ISR processes can be suppressed by a combination of particle identification and kinematic fitting techniques. Studies of e+e− → µ+µ−γ and sev- eral multi-hadron ISR processes using BABAR data have been reported [9, 10, 11, 12], demonstrating the viability of such measurements. The K+K−π+π− final state has been measured di- rectly by the DM1 collaboration [13] for s < 2.2 GeV, and we have previously published ISR measurements of the K+K−π+π− and K+K−K+K− final states [11] for Ec.m.< 4.5 GeV. We recently reported [14] an updated measurement of theK+K−π+π− final state with a larger data sample, along with the first measurement of the K+K−π0π0 final state, in which we observed a struc- ture near threshold in the φf0 intermediate state. In this paper we present a more detailed study of these two final states along with an updated measurement of the K+K−K+K− final state. In all cases we require detec- tion of the ISR photon and perform a set of kinematic fits. We are able to suppress backgrounds sufficiently to study these final states from their respective production thresholds up to 5 GeV. In addition to measuring the overall cross sections, we study the internal structure of the events and measure cross sections for a number of intermediate states. We study the charmonium region, measure several J/ψ and ψ(2S) branching fractions, and set limits on other states. II. THE BABAR DETECTOR AND DATASET The data used in this analysis were collected with the BABAR detector at the PEP-II asymmetric energy e+e− storage rings. The total integrated luminosity used is 232 fb−1, which includes 211 fb−1 collected at the Υ (4S) peak, s = 10.58 GeV, and 21 fb−1 collected below the resonance, at s = 10.54 GeV. The BABAR detector is described elsewhere [15]. Here we use charged particles reconstructed in the track- ing system, which comprises the five-layer silicon vertex tracker (SVT) and the 40-layer drift chamber (DCH) in a 1.5 T axial magnetic field. Separation of charged pi- ons, kaons and protons uses a combination of Cherenkov angles measured in the detector of internally reflected Cherenkov light (DIRC) and specific ionization measured in the SVT and DCH. For the present study we use a kaon identification algorithm that provides 90–95% efficiency, depending on momentum, and pion and proton rejection factors in the 20–100 range. Photon and electron energies are measured in the CsI(Tl) electromagnetic calorimeter (EMC). We use muon identification provided by the in- strumented flux return (IFR) to select the µ+µ−γ final state. To study the detector acceptance and efficiency, we use a simulation package developed for radiative pro- cesses. The simulation of hadronic final states, includ- ing K+K−π+π−γ, K+K−π0π0γ and K+K−K+K−γ, is based on the approach suggested by Czyż and Kühn[16]. Multiple soft-photon emission from the initial- state charged particles is implemented with a structure- function technique [17, 18], and photon radiation from the final-state particles is simulated by the PHOTOS package [19]. The accuracy of the radiative corrections is about 1%. We simulate the K+K−ππ final states both accord- ing to phase space and with models that include the φ(1020) → K+K− and/or f0(980) → ππ channels, and the K+K−K+K− final state both according to phase space and including the φ → K+K− channel. The generated events go through a detailed detector simula- tion [20], and we reconstruct them with the same software chain as the experimental data. Variations in detector and background conditions are taken into account. We also generate a large number of background pro- cesses, including the ISR channels e+e−→ π+π−π+π−γ and π+π−π0π0γ, which can contribute due to particle misidentification, and φηγ, φπ0γ, π+π−π0γ, which have larger cross sections and can contribute via missing or spurious tracks or photons. In addition, we study the non-ISR backgrounds e+e−→qq (q = u, d, s, c) generated by JETSET [21] and e+e−→ τ+τ− by KORALB [22]. The contribution from the Υ (4S) decays is found to be negligible. The cross sections for these processes are known with about 10% accuracy or better, which is suf- ficient for these measurements. III. EVENT SELECTION AND KINEMATIC FIT In the initial selection of candidate events, we con- sider photon candidates in the EMC with energy above 0.03 GeV and charged tracks reconstructed in the DCH or SVT or both that extrapolate within 0.25 cm of the beam axis in the transverse plane and within 3 cm of the nominal collision point along the axis. These criteria are looser than in our previous analysis [11], and have been chosen to maximize efficiency. We require a high- energy photon in the event with an energy in the initial e+e− c.m. frame of Eγ > 3 GeV, and either exactly four charged tracks with zero net charge and total momen- tum roughly opposite to the photon direction, or exactly two oppositely charged tracks that combine with a set of other photons to roughly balance the highest-energy photon momentum. We fit a vertex to the set of charged tracks and use it as the point of origin to calculate the photon direction. Most events contain additional soft photons due to machine background or interactions in the detector material. We subject each of these candidate events to a set of constrained kinematic fits, and use the fit results, along with charged-particle identification, both to select the fi- nal states of interest and to measure backgrounds from other processes. We assume the photon with the highest Eγ in the c.m. frame is the ISR photon, and the kine- matic fits use its direction along with the four-momenta and covariance matrices of the initial e+e− and the set of selected tracks and photons. Because of excellent res- olution for the momenta in the DCH and good angular resolution for the photons in the EMC, the ISR photon energy is determined with better resolution through four- momentum conservation than through measurement in the EMC. Therefore we do not use its measured energy in the fits, eliminating the systematic uncertainty due to the EMC calibration for high energy photons. The fitted three-momenta for each charged track and photon are used in further kinematical calculations. For the four-track candidates, the fits have three con- straints (3C). We first fit to the π+π−π+π− hypothesis, obtaining a χ24π. If the four tracks include one identified K+ and one K−, we fit to the K+K−π+π− hypothesis and retain the event as a K+K−π+π− candidate. For events with one identified kaon, we perform fits with each of the two oppositely charged tracks given the kaon hy- pothesis, and the combination with the lower χ2 KKπ+π− is retained if it is lower than χ24π. If the event contains three or four identified K±, we fit to the K+K−K+K− hypothesis and retain the event as a K+K−K+K− can- didate. For the events with two charged tracks and five or more photon candidates, we require both tracks to be identified as kaons to suppress background from ISR π+π−π0π0 and K±K0 π∓ events. We then pair all non-ISR pho- ton candidates and consider combinations with invariant mass within ±30 MeV/c2 of the π0 mass as π0 candi- dates. We perform a six-constraint (6C) fit to each set of 0 20 40 60 χ2(2K2π) FIG. 1: Distribution of χ2 from the three-constraint fit for K+K−π+π− candidates in the data (points). The open his- togram is the distribution for simulated signal events, normal- ized as described in the text. The cross-hatched (hatched) his- togram represents the background from non-ISR events (plus that from ISR 4π events), estimated as described in the text. two non-overlapping π0 candidates plus the ISR photon direction, the two tracks and the beam particles. Both π0 candidates are constrained to the π0 mass, and we retain the combination with the lowest χ2 KKπ0π0 IV. THE K+K−π+π− FINAL STATE A. Final Selection and Backgrounds The experimental χ2 KKπ+π− distribution for the K+K−π+π− candidates is shown in Fig. 1 as points, and the open histogram is the distribution for the simulated K+K−π+π− events. The simulated distribution is nor- malized to the data in the region χ2 KKπ+π− < 10 where the backgrounds and radiative corrections are insignif- icant. The experimental distribution has contributions from background processes, but the simulated distribu- tion is also broader than the expected 3C χ2 distribution. This is due to multiple soft-photon emission from the ini- tial state and radiation from the final-state charged par- ticles, which are not taken into account by the fit, but are present in both data and simulation. The shape of the χ2 distribution at high values was studied in detail [11, 12] using specific ISR processes for which a very clean sample can be obtained without any limit on the χ2 value. The cross-hatched histogram in Fig. 1 represents the background from e+e−→ qq events, which is based on the JETSET simulation. It is dominated by events with a hard π0 producing a fake ISR photon, and the similar kinematics cause it to peak at low values of χ2 KKπ+π− We evaluate this background in a number of Ec.m. ranges by combining the ISR photon candidate with another photon candidate in both data and simulated events, and comparing the π0 signals in the resulting γγ invari- ant mass distributions. The simulation gives an Ec.m.- dependence consistent with the data, so we normalize it by an overall factor. The hatched histogram repre- sents the sum of this background and that from ISR e+e−→π+π−π+π− events with one or two misidentified π±, which also contributes at low χ2 values. We estimate the contribution as a function of Ec.m. from a simulation using the known cross section [11]. All remaining background sources are either negligible or give a χ2 KKπ+π− distribution that is nearly uniform over the range shown in Fig. 1. We therefore define a signal region χ2 KKπ+π− <30, and estimate the sum of the remaining backgrounds from the difference between the number of data and simulated entries in a control region, 30<χ2 KKπ+π− < 60. This difference is normalized to the corresponding difference in the signal region, as described in detail in Refs. [11, 12]. The signal region contains 34635 data and 14077 simulated events, and the control region contains 4634 data and 723 simulated events. 1 2 3 4 5 m(K+K-π+π-) (GeV/c2) FIG. 2: The invariant mass distribution for K+K−π+π− candidates in the data (points): the cross-hatched, hatched and open histograms represent, cumulatively, the non-ISR background, the contribution from ISR π+π−π+π− events, and the ISR background from the control region of Fig. 1. Figure 2 shows the K+K−π+π− invariant mass dis- tribution from threshold up to 5.0 GeV/c2 for events in the signal region. Narrow peaks are apparent at the J/ψ and ψ(2S) masses. The cross-hatched histogram repre- sents the qq background, which is negligible at low mass but becomes large at higher masses. The hatched region represents the ISR π+π−π+π− contribution, which we es- timate to be 2.4% of the selected events on average. The open histogram represents the sum of all backgrounds, including those estimated from the control region. They total 6–8% at low mass but account for 20-25% of the ob- served data near 4 GeV/c2 and become the largest con- tribution near 5 GeV/c2. We subtract the sum of backgrounds in each mass bin to obtain a number of signal events. Considering un- certainties in the cross sections for the background pro- cesses, the normalization of events in the control region and the simulation statistics, we estimate a systematic uncertainty on the signal yield that is less than 3% in the 1.6–3 GeV/c2 mass region, but increases to 3–5% in the region above 3 GeV/c2. B. Selection Efficiency The selection procedures applied to the data are also applied to the simulated signal samples. The resulting K+K−π+π− invariant-mass distributions in the signal and control regions are shown in Fig. 3(a) for the phase space simulation. The broad, smooth mass distribution is chosen to facilitate the estimation of the efficiency as a function of mass, and this model reproduces the observed distributions of kaon and pion momenta and polar angles. We divide the number of reconstructed simulated events in each mass interval by the number generated in that interval to obtain the efficiency shown as the points in Fig. 3(b). The 3rd order polynomial fit to the points is used for further calculations. We simulate events with the ISR photon confined to the angular range 20–160◦ with respect to the electron beam in the e+e− c.m. frame, which is about 30% wider than the EMC acceptance. This efficiency is for this fiducial region, but includes the acceptance for the final-state hadrons, the inefficiencies of the detector subsystems, and event loss due to additional soft-photon emission. The simulations including the φ(1020)π+π− and/or K+K−f0(980) channels have very different mass and an- gular distributions in the K+K−π+π− rest frame. How- ever, the angular acceptance is quite uniform for ISR events, and the efficiencies are consistent with those from the phase space simulation within 3%. To study possible mis-modeling of the acceptance, we repeat the analysis with the tighter requirements that all charged tracks be within the DIRC acceptance, 0.45<θch<2.4 radians, and the ISR photon be well away from the edges of the EMC, 0.35< θISR < 2.4 radians. The fraction of selected data events satisfying the tighter requirements differs from the simulated ratio by 3.7%. We conservatively take the sum in quadrature of this variation and the 3% model varia- tion (5% total) as a systematic uncertainty due to accep- tance and model dependence. We correct for mis-modeling of the shape of the KKπ+π− distribution by (3.0±2.0)% and the track find- ing efficiency following the procedures described in de- tail in Ref. [11]. We use a comparison of data and sim- ulated χ24π distributions in the much larger samples of ISR π+π−π+π− events. We consider data and simulated 1 2 3 4 m(K+K-π+π-) (GeV/c2) 1 2 3 4 m(K+K-π+π-) (GeV/c2) FIG. 3: (a) The invariant mass distributions for simulated K+K−π+π− events in the phase space model, reconstructed in the signal (open) and control (hatched) regions of Fig. 1; (b) net reconstruction and selection efficiency as a function of mass obtained from this simulation (the curve represents a 3rd order polynomial fit). 1 1.5 2 2.5 3 3.5 4 4.5 5 - DM1 - BaBar ISR Ec.m. (GeV) FIG. 4: The e+e−→K+K−π+π− cross section as a function of the effective e+e− c.m. energy measured with ISR data at BABAR (dots). The direct measurements from DM1 [13] are shown as the open circles. Only statistical errors are shown. events that contain a high-energy photon plus exactly three charged tracks and satisfy a set of kinematical cri- teria, including a good χ2 from a kinematic fit under the hypothesis that there is exactly one missing track in the event. We find that the simulated track-finding efficiency is overestimated by (0.8 ± 0.5)% per track, so we apply a correction of +(3± 2)% to the signal yield. TABLE I: Measurements of the e+e−→ K+K−π+π− cross section (errors are statistical only). Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 1.4125 0.00± 0.02 2.1375 2.83± 0.13 2.8625 0.50± 0.05 3.5875 0.12± 0.03 4.3125 0.04± 0.02 1.4375 0.01± 0.02 2.1625 2.71± 0.12 2.8875 0.51± 0.05 3.6125 0.13± 0.03 4.3375 0.04± 0.02 1.4625 0.00± 0.02 2.1875 2.46± 0.12 2.9125 0.54± 0.05 3.6375 0.12± 0.03 4.3625 0.03± 0.02 1.4875 0.04± 0.02 2.2125 1.84± 0.10 2.9375 0.46± 0.05 3.6625 0.11± 0.03 4.3875 0.06± 0.02 1.5125 0.03± 0.02 2.2375 1.66± 0.10 2.9625 0.45± 0.05 3.6875 0.25± 0.03 4.4125 0.01± 0.02 1.5375 0.11± 0.03 2.2625 1.59± 0.09 2.9875 0.46± 0.05 3.7125 0.07± 0.03 4.4375 0.03± 0.02 1.5625 0.15± 0.04 2.2875 1.66± 0.09 3.0125 0.36± 0.04 3.7375 0.08± 0.02 4.4625 0.06± 0.02 1.5875 0.32± 0.05 2.3125 1.50± 0.09 3.0375 0.39± 0.04 3.7625 0.11± 0.03 4.4875 0.03± 0.02 1.6125 0.48± 0.06 2.3375 1.65± 0.09 3.0625 0.31± 0.04 3.7875 0.11± 0.03 4.5125 0.04± 0.02 1.6375 0.85± 0.08 2.3625 1.56± 0.09 3.0875 2.95± 0.10 3.8125 0.10± 0.03 4.5375 0.01± 0.02 1.6625 1.42± 0.10 2.3875 1.49± 0.09 3.1125 1.51± 0.08 3.8375 0.08± 0.02 4.5625 0.02± 0.02 1.6875 1.86± 0.11 2.4125 1.46± 0.09 3.1375 0.37± 0.04 3.8625 0.12± 0.03 4.5875 0.05± 0.02 1.7125 2.36± 0.13 2.4375 1.48± 0.09 3.1625 0.35± 0.04 3.8875 0.09± 0.02 4.6125 0.02± 0.02 1.7375 2.67± 0.13 2.4625 1.17± 0.08 3.1875 0.28± 0.04 3.9125 0.09± 0.02 4.6375 0.01± 0.02 1.7625 3.51± 0.15 2.4875 1.16± 0.08 3.2125 0.35± 0.04 3.9375 0.08± 0.02 4.6625 0.04± 0.02 1.7875 3.98± 0.16 2.5125 1.21± 0.08 3.2375 0.31± 0.04 3.9625 0.10± 0.02 4.6875 0.02± 0.02 1.8125 4.10± 0.16 2.5375 0.94± 0.07 3.2625 0.30± 0.04 3.9875 0.04± 0.02 4.7125 0.03± 0.02 1.8375 4.68± 0.17 2.5625 0.95± 0.07 3.2875 0.24± 0.04 4.0125 0.06± 0.02 4.7375 0.01± 0.02 1.8625 4.49± 0.17 2.5875 0.84± 0.07 3.3125 0.22± 0.04 4.0375 0.07± 0.02 4.7625 0.02± 0.02 1.8875 4.26± 0.17 2.6125 0.85± 0.07 3.3375 0.25± 0.04 4.0625 0.05± 0.02 4.7875 0.01± 0.02 1.9125 4.30± 0.16 2.6375 0.90± 0.07 3.3625 0.16± 0.03 4.0875 0.06± 0.02 4.8125 0.00± 0.02 1.9375 4.20± 0.16 2.6625 0.82± 0.06 3.3875 0.17± 0.03 4.1125 0.06± 0.02 4.8375 0.02± 0.02 1.9625 4.13± 0.16 2.6875 0.70± 0.06 3.4125 0.18± 0.03 4.1375 0.05± 0.02 4.8625 0.00± 0.02 1.9875 3.74± 0.15 2.7125 0.86± 0.06 3.4375 0.12± 0.03 4.1625 0.06± 0.02 4.8875 0.04± 0.02 2.0125 3.45± 0.15 2.7375 0.81± 0.06 3.4625 0.17± 0.03 4.1875 0.05± 0.02 4.9125 0.05± 0.02 2.0375 3.38± 0.14 2.7625 0.76± 0.06 3.4875 0.17± 0.03 4.2125 0.05± 0.02 4.9375 0.02± 0.02 2.0625 3.17± 0.14 2.7875 0.73± 0.06 3.5125 0.21± 0.03 4.2375 0.08± 0.02 4.9625 0.00± 0.02 2.0875 3.23± 0.14 2.8125 0.64± 0.05 3.5375 0.14± 0.03 4.2625 0.04± 0.02 4.9875 0.04± 0.02 2.1125 3.15± 0.14 2.8375 0.56± 0.05 3.5625 0.16± 0.03 4.2875 0.08± 0.02 We correct the simulated kaon identification efficiency using e+e−→φ(1020)γ→K+K−γ events. Events with a hard ISR photon and two charged tracks, one of which is identified as a kaon, with a K+K− invariant mass near the φ mass provide a very clean sample, and we com- pare the fractions of data and simulated events with the other track also identified as a kaon, as a function of mo- mentum. The data-simulation efficiency ratio averages 0.990 ± 0.001 in the 1–5 GeV/c momentum range with variations at the 0.01 level. We conservatively apply a correction of +(1.0 ± 1.0)% per kaon, or +(2.0 ± 2.0)% to the signal yield. C. Cross Section for e+e− → K+K−π+π− We calculate the e+e−→K+K−π+π− cross section as a function of the effective c.m. energy from σKKπ+π−(Ec.m.) = dNKKπ+π−γ(Ec.m.) dL(Ec.m.) · ǫKKπ+π−(Ec.m.) , (2) where Ec.m. ≡ mKKπ+π−c2, mKKπ+π− is the measured invariant mass of the K+K−π+π− system, dNKKπ+π−γ is the number of selected events after background sub- traction in the interval dEc.m., and ǫKKπ+π−(Ec.m.) is the corrected detection efficiency. We calculate the differ- ential luminosity, dL(Ec.m.), in each interval dEc.m. from ISR µ+µ−γ events with the photon in the same fiducial range used for the simulation; the procedure is described in Refs. [11, 12]. From data-simulation comparison we conservatively estimate a systematic uncertainty on dL of 3%. This dL has been corrected for vacuum polar- TABLE II: Summary of corrections and systematic uncer- tainties on the e+e−→K+K−π+π− cross section. The total correction is the linear sum of the components and the total uncertainty is the sum in quadrature. Source Correction Uncertainty Rad. Corrections – 1% Backgrounds – 3%, mKKπ+π−< 3 GeV/c 5%, mKKπ+π−> 3 GeV/c Model Dependence – 5% KKπ+π− Distn. +3% 2% Tracking Efficiency +3% 2% Kaon ID Efficiency +2% 2% ISR Luminosity – 3% Total +8% 7%, mKKπ+π−< 3 GeV/c 9%, mKKπ+π−> 3 GeV/c ization and final-state soft-photon emission; the former should be excluded when using these data in calculations of gµ−2. For the cross section measurement we use the tighter angular criteria on the charged tracks and the ISR pho- ton, discussed in Sec. IVB, to exclude possible errors from incorrect simulation of the EMC and DCH edge ef- fects. We show the cross section as a function of Ec.m. in Fig. 4, with statistical errors only, and provide a list of our results in Table I. The result is consistent with the direct measurement by DM1 [13], and with our previous measurement of this channel [11] but has much better statistical precision. The systematic uncertainties, sum- marized in Table II, affect the normalization, but have little effect on the energy dependence. The cross section rises from threshold to a peak value of about 4.7 nb near 1.85 GeV, then generally decreases with increasing energy. In addition to narrow peaks at the J/ψ and ψ(2S) masses, there are several possible wider structures in the 1.8–2.8 GeV region. Such struc- tures might be due to thresholds for intermediate reso- nant states, such as φf0(980) near 2 GeV. Gaussian fits to the simulated line shapes give a resolution on the mea- sured K+K−π+π− mass that varies between 4.2 MeV/c2 in the 1.5–2.5 GeV/c2 region and 5.5 MeV/c2 in the 2.5– 3.5 GeV/c2 region. The resolution function is not purely Gaussian due to soft-photon radiation, but less than 10% of the signal is outside the 25 MeV/c2 mass bin. Since the cross section has no sharp structure other than the J/ψ and ψ(2S) peaks discussed in Sec. VIII below, we apply no correction for resolution. D. Substructure in the K+K−π+π− Final State Our previous study [11] showed many intermediate res- onances in the K+K−π+π− final state. With the larger data sample used here, they can be seen more clearly and, in some cases, studied in detail. Figure 5(a) shows a scatter plot of the invariant mass of the K−π+ pair versus that of the K+π− pair, and Fig. 5(b) shows the sum of the two projections. Here we have suppressed the contributions from φπ+π− and K+K−ρ(770) by requir- ing |m(K+K−)−m(φ)| > 10 MeV/c2 and |m(π+π−)− m(ρ)| > 100 MeV/c2, where m(φ) and m(ρ) values are taken from the Particle Data Group (PDG) tables [5]. Bands and peaks corresponding to the K∗0(892) and K∗02 (1430) are visible. In Fig. 5(c) we show the sum of projections of the K∗0(892) bands, defined by lines in Fig. 5(a), with events in the overlap region plotted only once. No K∗0(892) signal is seen, confirming that the e+e−→K∗0(892)K∗0(892) cross section is small. We ob- serve associated K∗0(892)K∗02 (1430) production, but it is mostly from J/ψ decays (see Sec. VIII). We combine K∗0/K∗0 candidates within the lines in Fig. 5(a) with the remaining pion and kaon to ob- tain the K∗0π+− invariant mass distribution shown in Fig. 6(a), and the K∗0π+− vs.K∗0K−+ mass scatter plot 1 1.5 2 m(K+π-) (GeV/c2) 1 1.5 2 m(K+π-, K-π+) (GeV/c2) 2000 (c) m(K+π-,K-π+) (GeV/c2) FIG. 5: (a) Invariant mass of the K−π+ pair versus that of the K+π− pair; (b) sum of projections of (a); (c) sum of projections of the K∗0 bands of (a), with events in the overlap region taken only once. The φπ+π− and KKρ are vetoed. in Fig. 6(b). The bulk of Fig. 6(b) shows a strong posi- tive correlation, characteristic ofK∗0Kπ final states with no higher resonances. The horizontal band in Fig. 6(b) corresponds to the peak region in Fig. 6(a), and is consis- tent with contributions from the K1(1270) and K1(1400) resonances. There is also an indication of a vertical band in Fig. 6(b), perhaps corresponding to a K∗0K resonance at ∼1.5 GeV/c2. 1 1.5 2 2.5 3 m(K*π+-) (GeV/c2) 1 1.5 2 2.5 3 m(K*K+-) (GeV/c2) FIG. 6: (a) The K∗0π invariant mass distribution; (b) the K∗0π mass versus K∗0K mass. We now suppress K∗0Kπ by considering only events outside the lines in Fig. 5(a). In Fig. 7 we show the K±π+π− invariant mass (two entries per event) vs. that of the π+π− pair, along with its two projections. There is a strong ρ(770)→π+π− signal, and theK±π+π− mass projection shows further indications of the K1(1270) and K1(1400) resonances, both of which decay into Kρ(770). There are suggestions of additional structure in the π+π− mass distribution, including possible f0(980) shoulder and a possible enhancement near the f2(1270), however the current statistics do not allow us to make definitive statements. The separation of all these, and any other, intermedi- 0.5 1 m(π+π− ) (GeV/c2) m(Kππ) (GeV/c2) (b)(a) 0.2 1 1.8 m(π+π− ) (GeV/c2) FIG. 7: (a) Invariant mass of the K±π+π− combinations versus that of the π+π− pair; (b) the K±π+π− and (c) π+π− mass projections of (a). ate states involving relatively wide resonances requires a partial wave analysis. This is beyond the scope of this paper. Here we present the cross section for the sum of all states including a K∗0(892), and study intermediate states that include a narrow φ or f0 resonance. E. The e+e−→K∗0Kπ Cross Section Signals for theK∗0(892) andK∗02 (1430) are clearly vis- ible in the K±π∓ mass distributions in Fig. 5(b) and, with a different bin size, in Fig. 8(a). We perform a fit to this distribution using P-wave Breit-Wigner (BW) functions for the K∗0 and K∗02 signals and a third-order polynomial function for the remainder of the distribu- tion taking into account the Kπ threshold. The result is shown in Fig. 8(a). The fit yields a K∗0 signal of 19738 ± 266 events with m(Kπ) = 896.2 ± 0.3 MeV/c2 and Γ(Kπ) = 50.6 ± 0.9 MeV, and a K∗02 signal of 1786±127 events withm(Kπ) = 1428.5±3.9 MeV/c2 and Γ(Kπ) = 113.7± 9.2 MeV. These values are consistent with current world averages [5], and the fit describes the data well, indicating that contributions from any other resonances decaying into K±π∓ are small. We perform a similar fit to the data in bins of the K+K−π+π− invariant mass, with the resonance masses and widths fixed to the values obtained by the overall fit. Since there is at most one K∗0 per event, we convert the resultingK∗0 yield in each bin into an “inclusive” e+e−→ K∗0Kπ cross section, following the procedure described in Sec. IVC. This cross section is shown in Fig. 8(b) and listed in Table III for the effective c.m. energies from threshold up to 3.5 GeV. At higher energies the signals are small and contain an unknown, but possibly large, contribution from e+e−→qq events. There is a rapid rise from threshold to a peak value of about 4 nb at 1.84 GeV, 0.75 1 1.25 1.5 1.75 m(K+-π-+) (GeV/c2) 1.5 2 2.5 3 3.5 Ec.m. (GeV) FIG. 8: (a) The K±π∓ mass distribution (two entries per event) for all selected K+K−π+π− events: the solid line rep- resents a fit including two resonances and a polynomial func- tion (see text), shown separately as the dashed line; (b) the e+e−→K∗0Kπ cross section obtained from the K∗0(892) sig- nal by a similar fit in each 25 MeV/c2 mass bin. followed by a very rapid decrease with increasing energy. There are suggestions of narrow structure in the peak region, but the only statistically significant structure is the J/ψ peak, which is discussed below. The e+e−→K∗0Kπ contribution is a large fraction of the total K+K−π+π− cross section at all energies above its threshold, and dominates in the 1.8–2.0 GeV region. We are unable to extract a meaningful measurement of the K∗02 Kπ cross section from this data sample because it is more than ten times smaller. The K+K−ρ0(770) in- termediate state makes up the majority of the remainder of the cross section and it can be estimated as a difference of the values in Table I and Table III for theK+K−π+π− and K∗0Kπ final states. F. The φ(1020)π+π− Intermediate State Intermediate states containing relatively narrow reso- nances can be studied more easily. Figure 9(a) shows a scatter plot of the invariant mass of the π+π− pair versus that of the K+K− pair. Horizontal and vertical bands corresponding to the ρ0(770) and φ, respectively, are visible, and there is a concentration of entries on the φ band corresponding to the correlated production of φ and f0(980). The φ signal is also visible in the K mass projection, Fig. 9(c). The large contribution from the ρ(770), coming from the K1 decay, is nearly uniform in the K+K− mass, and the cross-hatched histogram shows the non-K+K−π+π− background estimated from the control region in χ2 KKπ+π− . The cross-hatched his- TABLE III: Measurements of the e+e− → K∗0(892)Kπ cross section (errors are statistical only). Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 1.5875 0.16 ± 0.11 2.0875 2.36 ± 0.16 2.5875 0.54 ± 0.07 3.0875 1.73 ± 0.10 1.6125 0.31 ± 0.08 2.1125 1.92 ± 0.16 2.6125 0.63 ± 0.06 3.1125 0.92 ± 0.07 1.6375 0.81 ± 0.13 2.1375 1.99 ± 0.14 2.6375 0.57 ± 0.06 3.1375 0.21 ± 0.04 1.6625 0.79 ± 0.12 2.1625 1.19 ± 0.15 2.6625 0.46 ± 0.06 3.1625 0.24 ± 0.04 1.6875 1.33 ± 0.15 2.1875 1.24 ± 0.14 2.6875 0.46 ± 0.06 3.1875 0.08 ± 0.03 1.7125 1.63 ± 0.15 2.2125 1.25 ± 0.11 2.7125 0.64 ± 0.06 3.2125 0.15 ± 0.03 1.7375 1.87 ± 0.14 2.2375 0.90 ± 0.10 2.7375 0.56 ± 0.06 3.2375 0.14 ± 0.04 1.7625 2.12 ± 0.17 2.2625 0.79 ± 0.11 2.7625 0.46 ± 0.06 3.2625 0.16 ± 0.03 1.7875 2.51 ± 0.20 2.2875 1.15 ± 0.10 2.7875 0.36 ± 0.06 3.2875 0.13 ± 0.03 1.8125 2.96 ± 0.21 2.3125 0.99 ± 0.09 2.8125 0.31 ± 0.05 3.3125 0.12 ± 0.03 1.8375 4.35 ± 0.20 2.3375 0.91 ± 0.11 2.8375 0.35 ± 0.05 3.3375 0.14 ± 0.03 1.8625 4.11 ± 0.20 2.3625 1.11 ± 0.09 2.8625 0.27 ± 0.04 3.3625 0.12 ± 0.06 1.8875 3.26 ± 0.23 2.3875 0.83 ± 0.09 2.8875 0.27 ± 0.05 3.3875 0.09 ± 0.03 1.9125 3.90 ± 0.20 2.4125 0.87 ± 0.09 2.9125 0.34 ± 0.05 3.4125 0.10 ± 0.03 1.9375 3.53 ± 0.20 2.4375 1.00 ± 0.09 2.9375 0.29 ± 0.04 3.4375 0.11 ± 0.03 1.9625 3.42 ± 0.21 2.4625 0.86 ± 0.08 2.9625 0.25 ± 0.04 3.4625 0.10 ± 0.05 1.9875 2.81 ± 0.18 2.4875 0.88 ± 0.09 2.9875 0.38 ± 0.05 3.4875 0.08 ± 0.03 2.0125 2.47 ± 0.17 2.5125 0.69 ± 0.07 3.0125 0.21 ± 0.04 2.0375 2.26 ± 0.16 2.5375 0.62 ± 0.07 3.0375 0.24 ± 0.04 2.0625 2.00 ± 0.16 2.5625 0.55 ± 0.07 3.0625 0.22 ± 0.04 togram also shows a φ peak, but this is a small fraction of the events. Subtracting this background and fitting the remaining data gives 1706±56 events produced via the φπ+π− intermediate state. To study the φπ+π− channel, we select candidate events with a K+K− invariant mass within 10 MeV/c2 of the φ mass, indicated by the inner vertical lines in Figs. 9(a,c), and estimate the non-φ contribution from the mass sidebands between the inner and outer verti- cal lines. In Fig. 9(b) we show the π+π− invariant mass distributions for φ candidate events, sideband events and χ2 control region events as the open, hatched and cross- hatched histograms, respectively, and in Fig. 9(d) we show the numbers of entries from the candidate events minus those from the sideband and control region. There is a clear f0 peak over a broad mass distribution, with no indication of associated ρ0 production. A coherent sum of two Breit-Wigner functions is suf- ficient to describe the invariant mass distribution of the π+π− pair recoiling against a φ in Fig. 9(d). We fit with the function: F (m) = 1− 4m2π/m2 · |A1(m) + eiψA2(m)|2 , (3) Ai(m) = miΓi Ni/(m i −m2 + imiΓi) , where m is the π+π− invariant mass, mi and Γi are the parameters of the ith resonance, ψ is their relative phase and Ni are normalization parameters, corresponding to the number of events under each BW. One BW corre- sponds to the f0(980), but a wide range of values of the other parameters can describe the data. Fixing the rela- tive phase to ψ = π and the parameters of the first BW to m1 = 0.6 GeV/c 2 and Γ1 = 0.45 GeV (which could be interpreted as the f0(600) [5]), we obtain the fit shown in Fig. 9(d). It describes the data well and gives an f0(980) 0.98 1 1.02 1.04 1.06 m(K-K+) (GeV/c2) 0.4 0.6 0.8 1 1.2 m(π+π-) (GeV/c2) 0.98 1 1.02 1.04 1.06 m(K+K-) (GeV/c2) 0.4 0.6 0.8 1 1.2 m(π+π-) (GeV/c2) FIG. 9: (a) The π+π− vs. the K+K− invariant masses for all selected K+K−π+π− events; (b) the π+π− invari- ant mass projections for events in the φ peak (open his- togram), sidebands (hatched) and background control region (cross-hatched); (c) theK+K− mass projections for all events (open) and control region (cross-hatched); (d) the difference between the open and the sum of the other histograms in (b). signal of 262±30 events, withm2 = 0.973±0.003 GeV/c2 and Γ2 = 0.065 ± 0.013 GeV, consistent with the PDG values [5]. There is a suggestion of an f2(1270) peak in the data, but it is much smaller than the f0 peak and we do not consider it further. We obtain the number of e+e− → φπ+π− events in bins of φπ+π− invariant mass by fitting the K+K− in- variant mass projection in that bin after subtracting non- K+K−π+π− background. Each projection is a subset of Fig. 9(c), where the curve represent a fit to the full sam- ple. In each mass bin, all parameters are fixed to the values obtained from the overall fit except the numbers of events in the φ peak and the non-φ component. The efficiency may depend on the details of the pro- duction mechanism. Using the two-pion mass distribu- tion in Fig. 9(d) as input, we simulate the π+π− system as an S-wave comprising two scalar resonances, with pa- rameters set to the values given above. To describe the φπ+π− mass distribution we use a simple model with one resonance, the φ(1680), of mass 1.68 GeV/c2 and width 0.2 GeV, decaying to φf0. The simulated reconstructed spectrum is shown in Fig. 10(a). There is a sharp in- crease at about 2 GeV/c2 due to the φf0(980) threshold. All other structure is determined by phase space and a m−2 falloff with increasing mass. 1 2 3 4 m(K+K-π+π-) (GeV/c2) 1 2 3 4 m(K+K-π+π-) (GeV/c2) FIG. 10: (a) The K+K−π+π− invariant mass distributions from the φπ+π− simulation described in the text, recon- structed in the signal (open) and control (hatched) regions; (b) net reconstruction and selection efficiency as a function of mass: the solid line represents a cubic fit, and the dashed line the corresponding fit for the space phase model shown in Fig. 3. Dividing the number of reconstructed events in each bin by the number of generated ones, we obtain the effi- ciency as a function of φπ+π− mass shown in Fig. 10(b). The solid line represents a fit to a third order polyno- mial, and the dashed line the corresponding fit to the phase space model from Fig. 3. The model dependence is weak, giving confidence in the efficiency calculation. We calculate the e+e− → φπ+π− cross section as de- scribed in Sec. IVC but using the efficiency from the fit to Fig. 10(b) and dividing by the φ→K+K− branching fraction of 0.491 [5]. We show our results as a func- tion of energy in Fig. 11 and list them in Table IV. The cross section has a peak value of about 0.6 nb at about 1.7 GeV, then decreases with increasing energy until φ(1020)f0(980) threshold, around 2.0 GeV. From this point it rises, falls sharply at about 2.2 GeV, and then decreases slowly. Except in the charmonium region, the results at energies above 2.9 GeV are not meaningful due to small signals and potentially large backgrounds, and are omitted from Table IV. Figure 11 displays the cross-section up to 4.5 GeV to show the signals from the J/ψ and ψ(2S) decays. They are discussed in Sec. VIII. There are no previous measurements of this cross section, and our results are consistent with the upper limits given in Ref. [13]. 1.5 2 2.5 3 3.5 4 Ec.m. (GeV) FIG. 11: The e+e−→ φπ+π− cross section as a function of the effective e+e− c.m. energy. We perform a study of the angular distributions in the φ(1020)π+π− final state by considering all K+K−π+π− candidate events with mass below 3 GeV/c2, binning them in terms of the cosine of the angles defined below, and fitting the background-subtracted K+K− mass pro- jections. The efficiency is nearly uniform in these angles, so we study the number of events in each bin. We define the φ production angle, Θφ as the angle between the φ momentum and the e− beam direction in the rest frame of the φπ+π− system. The distribution of cosΘφ, shown in Fig. 12(a), is consistent with the uniform distribution expected if S-wave two-body channels φX , X → π+π− dominate the φπ+π− system. We define the pion and kaon helicity angles, Θπ+ and ΘK+ as those between the π+ and the π+π−-system momenta in the π+π− rest frame and between the K+ and ISR photon momenta in the φ rest frame, respectively. The distributions of cosΘπ+ and cosΘK+ , shown in Figs. 12(b) and 12(c), re- spectively, are consistent with those expected from scalar and vector meson decays. TABLE IV: Measurements of the e+e− → φ(1020)π+π− cross section (errors are statistical only). Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 1.4875 0.01 ± 0.02 1.8375 0.39 ± 0.10 2.1875 0.32 ± 0.06 2.5375 0.09 ± 0.03 1.5125 0.03 ± 0.03 1.8625 0.44 ± 0.10 2.2125 0.22 ± 0.05 2.5625 0.03 ± 0.02 1.5375 0.09 ± 0.04 1.8875 0.23 ± 0.08 2.2375 0.15 ± 0.04 2.5875 0.06 ± 0.02 1.5625 0.13 ± 0.04 1.9125 0.34 ± 0.09 2.2625 0.10 ± 0.03 2.6125 0.07 ± 0.02 1.5875 0.21 ± 0.06 1.9375 0.37 ± 0.08 2.2875 0.11 ± 0.04 2.6375 0.08 ± 0.03 1.6125 0.23 ± 0.06 1.9625 0.31 ± 0.08 2.3125 0.08 ± 0.03 2.6625 0.06 ± 0.02 1.6375 0.54 ± 0.08 1.9875 0.36 ± 0.07 2.3375 0.13 ± 0.03 2.6875 0.04 ± 0.02 1.6625 0.61 ± 0.09 2.0125 0.38 ± 0.07 2.3625 0.10 ± 0.04 2.7125 0.08 ± 0.03 1.6875 0.64 ± 0.10 2.0375 0.29 ± 0.07 2.3875 0.13 ± 0.04 2.7375 0.06 ± 0.02 1.7125 0.38 ± 0.09 2.0625 0.42 ± 0.07 2.4125 0.12 ± 0.04 2.7625 0.07 ± 0.02 1.7375 0.64 ± 0.10 2.0875 0.30 ± 0.06 2.4375 0.15 ± 0.04 2.7875 0.02 ± 0.02 1.7625 0.55 ± 0.11 2.1125 0.49 ± 0.07 2.4625 0.06 ± 0.03 2.8125 0.06 ± 0.02 1.7875 0.55 ± 0.11 2.1375 0.30 ± 0.06 2.4875 0.09 ± 0.03 2.8375 0.04 ± 0.02 1.8125 0.31 ± 0.09 2.1625 0.49 ± 0.07 2.5125 0.09 ± 0.03 2.8625 0.03 ± 0.01 -1 -0.5 0 0.5 1 cos(Θφ) -1 -0.5 0 0.5 1 cos(Θπ+) -1 -0.5 0 0.5 1 cos(ΘK+) FIG. 12: Distributions of the cosines of (a) the φ production angle, (b) the pion helicity angle, and (c) the kaon helicity angle (see text) for e+e−→φπ+π− events: the lines represent the distributions expected if the π+π− system recoiling against a vector φ meson is produced in an S-wave, normalized to the number of events in the data. G. The φ(1020)f0(980) Intermediate State The narrow f0(980) peak seen in Fig. 9(d) allows the selection of a fairly clean sample of φf0 events. We re- peat the analysis just described with the additional re- quirement that the π+π− invariant mass be in the range 0.85–1.10 GeV/c2. The fit to the full sample yields about 700 events; all of these contain a true φ, but about 10% are from e+e− → φπ+π− events where the pion pair is not produced through the f0(980). We convert the numbers of fitted events in each mass bin into a measurement of the e+e−→ φ(1020)f0(980) cross section as described above and dividing by the f0→ π+π− branching fraction of two-thirds. The cross section is shown in Fig. 13 as a function of the effective c.m. energy and is listed in Table V. Its behavior near threshold does not appear to be smooth, but is more consistent with a steep rise to a value of about 0.3 nb at 1.95 GeV followed by a slow decrease that is interrupted by a structure around 2.175 GeV. Possible interpreta- tions of this structure are discussed in Sec. VII. Again, the values are not meaningful for the effective c.m. above about 2.9 GeV/c2, except for the J/ψ and ψ(2S) signals, discussed in Sec. VIII. V. THE K+K−π0π0 FINAL STATE A. Final Selection and Backgrounds The K+K−π0π0 sample contains background from the ISR processes e+e− → K+K−π0γ and K+K−ηγ, in which two soft photon candidates from machine- or detector-related background combine with the relatively energetic photons from the π0 or η to form two fake π0 candidates. We reduce this background using the helic- ity angle between each reconstructed π0 direction and the direction of its higher-energy photon daughter calculated in its rest frame. If the cosines of both helicity angles are higher than 0.85, we remove the event. Figure 14 shows the distribution of χ2KKπ0π0 for the remaining candidates together with the simulated TABLE V: Summary of the e+e− → φ(1020)f0(980) cross section measurement. Errors are statistical only. Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 1.8875 0.02 ± 0.02 2.1625 0.56 ± 0.08 2.4375 0.13 ± 0.04 2.7125 0.05 ± 0.02 1.9125 0.07 ± 0.04 2.1875 0.38 ± 0.07 2.4625 0.09 ± 0.04 2.7375 0.02 ± 0.01 1.9375 0.30 ± 0.07 2.2125 0.21 ± 0.06 2.4875 0.08 ± 0.03 2.7625 0.06 ± 0.02 1.9625 0.27 ± 0.07 2.2375 0.17 ± 0.05 2.5125 0.09 ± 0.03 2.7875 0.03 ± 0.02 1.9875 0.25 ± 0.07 2.2625 0.12 ± 0.04 2.5375 0.07 ± 0.03 2.8125 0.04 ± 0.02 2.0125 0.25 ± 0.07 2.2875 0.13 ± 0.05 2.5625 0.02 ± 0.02 2.8375 0.05 ± 0.02 2.0375 0.28 ± 0.07 2.3125 0.12 ± 0.04 2.5875 0.06 ± 0.03 2.8625 0.02 ± 0.02 2.0625 0.43 ± 0.08 2.3375 0.14 ± 0.04 2.6125 0.07 ± 0.03 2.8875 0.01 ± 0.01 2.0875 0.28 ± 0.07 2.3625 0.13 ± 0.05 2.6375 0.06 ± 0.03 2.9125 0.02 ± 0.01 2.1125 0.54 ± 0.09 2.3875 0.13 ± 0.04 2.6625 0.05 ± 0.03 2.9375 0.00 ± 0.00 2.1375 0.46 ± 0.08 2.4125 0.14 ± 0.05 2.6875 0.03 ± 0.02 2.9625 0.02 ± 0.01 1.5 2 2.5 3 3.5 4 Ec.m. (GeV) FIG. 13: The e+e− → φ(1020)f0(980) cross section as a function of the effective e+e− c.m. energy obtained from the K+K−π+π− final state. K+K−π0π0 events. Again, the distributions are broader than those for a typical 6C χ2 due to higher order ISR, and we normalize the histogram to the data in the region χ2KKπ0π0 < 10. The cross-hatched histogram in Fig. 14 represents background from e+e−→qq events, evaluated in the same way as for the K+K−π+π− final state. The hatched histogram represents the sum of this background and that from ISR π+π−π0π0 events with both charged pions misidentified as kaons, evaluated using the simula- tion. The dominant background in this case is from resid- ual ISR K+K−π0 and K+K−η events, as well as ISR K+K−π0π0π0 events. Their simulated contribution, shown as the dashed histogram in Fig. 14, is consistent with the data in the high χ2 KKπ0π0 region. All other back- grounds are either negligible or distributed uniformly in KKπ0π0 . We define a signal region, χ2 KKπ0π0 <50, con- taining 4425 data and 6948 simulated events, and a con- trol region, 50 < χ2 KKπ0π0 < 100, containing 1751 data 0 20 40 60 80 100 χ2(2K2π0) FIG. 14: Distribution of χ2 from the six-constraint fit for K+K−π0π0 candidates in the data (points). The open his- togram is the distribution for simulated signal events, nor- malized as described in the text. The cross-hatched, hatched and dashed histograms represent, cumulatively, the back- grounds from non-ISR events, ISR π+π−π0π0 events, and ISR K+K−π0, K+K−η and K+K−π0π0π0 events. and 848 simulated events. Figure 15 shows the K+K−π0π0 invariant mass dis- tribution from threshold up to 5 GeV/c2 for events in the signal region. The qq background (cross-hatched his- togram) is negligible at low masses but forms a large frac- tion of the selected events above about 4 GeV/c2. The ISR π+π−π0π0 contribution (hatched region) is negligi- ble except in the 1.5–2.5 GeV/c2 region. The sum of all other backgrounds, estimated from the control region, is the dominant contribution below 1.6 GeV/c2 and non negligible everywhere. The total background in the 1.6– 2.5 GeV/c2 region is 15–20% (open histogram). We subtract the sum of backgrounds from the number of selected events in each mass bin to obtain a number of signal events. Considering uncertainties in the cross sections for the background processes, the normalization 1 2 3 4 5 m(K+K-π0π0) (GeV/c2) FIG. 15: Invariant mass distribution for K+K−π0π0 candi- dates in the data (points). The cross-hatched, hatched and open histograms represent, cumulatively, the non-ISR back- ground, the contribution from ISR π+π−π0π0 events, and the ISR background from the control region of Fig. 14. of events in the control region and the simulation statis- tics, we estimate a systematic uncertainty on the signal yield after background subtraction as less than 5% in the 1.6–3.0 GeV/c2 region, but increases to 10% in the region above 3 GeV/c2. B. Selection Efficiency The detection efficiency is determined in the same manner as in Sec. IVB. Figure 16(a) shows the sim- ulated K+K−π0π0 invariant mass distributions in the signal and control regions from the phase space model. We divide the number of reconstructed events in each 40 MeV/c2 mass interval by the number generated ones in that interval to obtain the efficiency shown as the points in Fig. 16(b); a third order polynomial fit to the efficiency is used to calculate the cross section. Again, the simulation of the ISR photon covers a limited angu- lar range, about 30% wider than EMC acceptance, and shown efficiency is factor 0.7 lower than for the hadronic system alone. Simulations assuming dominance of the φ→K+K− and/or f0→π0π0 channels give consistent re- sults, and we apply the same 5% systematic uncertainty for possible model dependence as in Sec. IVB. We correct for mis-modeling of the track finding and kaon identification efficiencies as in Sec. IVB, and for the shape of the χ2 KKπ0π0 distribution analogously, using the result in Ref. [12], (0 ± 6)%. We correct the π0- finding efficiency using the procedure described in detail in Ref. [12]. From ISR e+e−→ωπ0γ→π+π−π0π0γ events selected with and without the π0 from the ω decay, we 1 2 3 4 m(K+K-π0π0) (GeV/c2) 1 2 3 4 m(K+K-π0π0) (GeV/c2) FIG. 16: (a) Invariant mass distribution for simulated K+K−π0π0 events in the signal (open) and control (hatched) regions (see Fig. 14); (b) net reconstruction and selection ef- ficiency as a function of mass obtained from this simulation (the curve represents a third order polynomial fit). find that the simulated efficiency for one π0 is too high by (2.8±1.4)%. Conservatively we apply a correction of +(5.6± 2.8)% for two π0 in the event. C. Cross Section for e+e− → K+K−π0π0 We calculate the cross section for e+e− → K+K−π0π0 in 40 MeV Ec.m. intervals from the analog of Eq. 2, using the invariant mass of the K+K−π0π0 system to deter- mine the effective c.m. energy. We show the first mea- surement of this cross section in Fig. 17 and list the re- sults obtained in Table VI. The cross section rises to a peak value near 1 nb at 2 GeV, falls sharply at 2.2 GeV, then decreases slowly. The only statistically significant structure is the J/ψ peak. The drop at 2.2 GeV is sim- ilar to that seen in the K+K−π+π− mode. Again, dL includes corrections for vacuum polarization that should be omitted from calculations of gµ−2. The simulated K+K−π0π0 invariant mass resolution is 8.8 MeV/c2 in the 1.5–2.5 GeV/c2 mass range, and in- creases with mass to 11.2 MeV/c2 in the 2.5–3.5 GeV/c2 range. Since less than 20% of the events in a 40 MeV/c2 bin are reconstructed outside that bin and the cross sec- tion has no sharp structure other than the J/ψ peak, we again make no correction for resolution. The point-to- point systematic errors are much smaller than statistical ones, and the errors on the normalization are summa- rized in Table VII, along with the corrections that were applied to the measurements. The total correction is +9.2%, and the total systematic uncertainty is 10% at low mass, increasing to 14% above 3 GeV/c2. TABLE VI: Measurements of the e+e− → K+K−π0π0 cross section (errors are statistical only). Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 1.4200 0.00 ± 0.05 2.3400 0.35 ± 0.06 3.2600 0.13 ± 0.03 4.1800 0.02 ± 0.01 1.4600 0.12 ± 0.07 2.3800 0.29 ± 0.06 3.3000 0.09 ± 0.03 4.2200 0.03 ± 0.01 1.5000 0.00 ± 0.07 2.4200 0.38 ± 0.06 3.3400 0.09 ± 0.03 4.2600 0.03 ± 0.01 1.5400 0.01 ± 0.08 2.4600 0.38 ± 0.06 3.3800 0.08 ± 0.02 4.3000 0.03 ± 0.01 1.5800 0.03 ± 0.09 2.5000 0.22 ± 0.05 3.4200 0.11 ± 0.03 4.3400 0.03 ± 0.01 1.6200 0.09 ± 0.09 2.5400 0.25 ± 0.05 3.4600 0.06 ± 0.02 4.3800 0.01 ± 0.01 1.6600 0.31 ± 0.11 2.5800 0.25 ± 0.05 3.5000 0.04 ± 0.02 4.4200 0.05 ± 0.01 1.7000 0.35 ± 0.11 2.6200 0.25 ± 0.05 3.5400 0.06 ± 0.02 4.4600 0.03 ± 0.01 1.7400 0.49 ± 0.11 2.6600 0.28 ± 0.05 3.5800 0.07 ± 0.02 4.5000 0.04 ± 0.01 1.7800 0.51 ± 0.12 2.7000 0.16 ± 0.04 3.6200 0.04 ± 0.02 4.5400 0.00 ± 0.01 1.8200 0.84 ± 0.12 2.7400 0.22 ± 0.04 3.6600 0.06 ± 0.02 4.5800 0.02 ± 0.01 1.8600 0.94 ± 0.11 2.7800 0.21 ± 0.04 3.7000 0.08 ± 0.02 4.6200 0.02 ± 0.01 1.9000 0.95 ± 0.12 2.8200 0.13 ± 0.04 3.7400 0.09 ± 0.02 4.6600 0.02 ± 0.01 1.9400 0.80 ± 0.11 2.8600 0.21 ± 0.04 3.7800 0.02 ± 0.02 4.7000 0.04 ± 0.01 1.9800 0.87 ± 0.11 2.9000 0.11 ± 0.03 3.8200 0.05 ± 0.01 4.7400 0.02 ± 0.01 2.0200 1.00 ± 0.10 2.9400 0.12 ± 0.04 3.8600 0.04 ± 0.01 4.7800 0.01 ± 0.01 2.0600 0.96 ± 0.10 2.9800 0.12 ± 0.04 3.9000 0.03 ± 0.02 4.8200 0.01 ± 0.01 2.1000 0.90 ± 0.10 3.0200 0.21 ± 0.04 3.9400 0.02 ± 0.01 4.8600 0.01 ± 0.01 2.1400 0.82 ± 0.10 3.0600 0.16 ± 0.04 3.9800 0.03 ± 0.01 4.9000 0.03 ± 0.01 2.1800 0.58 ± 0.08 3.1000 0.92 ± 0.07 4.0200 0.05 ± 0.01 4.9400 0.04 ± 0.02 2.2200 0.56 ± 0.08 3.1400 0.19 ± 0.04 4.0600 0.04 ± 0.01 4.9800 0.04 ± 0.02 2.2600 0.37 ± 0.07 3.1800 0.12 ± 0.03 4.1000 0.03 ± 0.01 2.3000 0.43 ± 0.07 3.2200 0.14 ± 0.03 4.1400 0.03 ± 0.01 TABLE VII: Summary of corrections and systematic uncer- tainties on the e+e−→K+K−π0π0 cross section. The total correction is the linear sum of the components and the total uncertainty is the sum in quadrature. Source Correction Uncertainty Rad. Corrections – 1% Backgrounds – 5%, mKKπ0π0< 3 GeV/c 10%, mKKπ0π0> 3 GeV/c Model Dependence – 5% χ2KKπ0π0 Distn. 0% 6% Tracking Efficiency +1.6% 0.8% Kaon ID Efficiency +2% 2% π0 Efficiency +5.6% 2.8% ISR Luminosity – 3% Total +9.2% 10%, mKKπ0π0< 3 GeV/c 14%, mKKπ0π0> 3 GeV/c D. Substructure in the K+K−π0π0 Final State A scatter plot of the invariant mass of the K−π0 versus that of the K+π0 pair is shown in Fig. 18(a) with two en- tries per event selected in the χ2 signal region. Horizontal and vertical bands corresponding to the K∗+(892) and K∗−(892), respectively, are visible. Figure 18(b) shows as points the sum of the two projections of Fig. 18(a); a large K∗±(892) signal is evident. Fitting this distri- bution with the function discussed in Sec. IVE gives a good χ2 and the curve shown on Fig. 18(b). The K∗±(1430):K∗±(892) ratio is consistent with that for 1 2 3 4 5 Ec.m. (GeV) FIG. 17: The e+e−→K+K−π0π0 cross section as a function of the effective e+e− c.m. energy measured with ISR data at BABAR. The errors are statistical only. neutral K∗ seen in the K+K−π+π− channel, and the number of K∗±(892) in the peak is consistent with one per selected event. The hatched histogram in Fig. 18(b) represents theK±π0 mass in events with the otherK∓π0 mass within the lines in Fig. 18(a), but with events in the overlap region used only once, and shows no K∗±(892) signal. These results indicate that the e+e−→K∗±K∗∓ cross section is small and that the K∗±(892)K∓π0 chan- 1 1.5 2 m(K+π0) (GeV/c2) 1 1.5 2 m(K+π0, K-π0) (GeV/c2) FIG. 18: (a) Invariant mass of the K−π0 pair versus that of the K+π0 pair in selected K+K−π0π0 events (two entries per event); (b) sum of projections of (a) (dots, four entries per event). The curve represents the result of the fit described in the text. The hatched histogram is the K±π0 distribution for events in which the other K∓π0 combination is within the K∗±(892) bands indicated in (a), with events in the overlap region taken only once. nels dominate the overall cross section. We find no signals for resonances in the K+K−π0 or K±π0π0 decay modes. Since the K∗±(892)K∓π0 chan- nels dominate and the statistics are low in any mass bin, we do not attempt to extract a separate K∗±(892)K∓π0 cross section. The total K+K−π0π0 cross section is roughly a factor of four lower than the K∗0(892)K±π∓ cross section observed in the K+K−π+π− final state. This is consistent with what one might expect from isospin and the charged vs. neutral K∗ branching frac- tions into charged kaons. E. The φ(1020)π0π0 Intermediate State The selection of events containing a φ(1020)→K+K− decay follows that in Section IVF. Figure 19(a) shows a scatter plot of the invariant mass of the π0π0 pair versus that of the K+K− pair. A vertical band corresponding to the φ is visible, whose intensity decreases with increas- ing π0π0 mass except for an enhancement in the f0(980) region. The φ signal is also visible in the K+K− invari- ant mass projection shown in Fig. 19(c). The relative non-φ background is smaller than in the K+K−π+π− mode, but there is a large background from ISR φπ0, φη and/or φπ0π0π0 events, as indicated by the control region histogram (hatched) in Fig. 19(c). The contribu- tions from non-ISR and ISR π+π−π0π0 events are negli- gible. Selecting φ candidate and side band events as for the K+K−π+π− mode (vertical lines in Figs. 19(a,c)), we obtain the π0π0 mass projections shown as the open and cross-hatched histograms, respectively, in Fig. 19(b). Control region events (hatched histogram) are concen- trated at low masses. A peak corresponding to the f0(980) is visible over a relatively low background. In Fig. 19(d) we show the numbers of entries from the candidate events minus those from the sideband and control regions. A sum of two Breit-Wigner functions is again sufficient to describe the data. Fitting Eq. 3 0.98 1 1.021.041.06 m(K-K+) (GeV/c2) 0.4 0.6 0.8 1 1.2 1.4 m(π0π0) (GeV/c2) 0.98 1 1.021.041.06 m(K+K-) (GeV/c2) 0.4 0.6 0.8 1 1.2 1.4 m(π0π0) (GeV/c2) FIG. 19: (a) Scatter plot of the π0π0 invariant mass vs. the K+K− invariant mass for all selected K+K−π0π0 events; (b) the π0π0 invariant mass projections for events in the φ peak (open histogram), sidebands (cross-hatched) and control region (hatched); (c) the K+K− mass projection for events in the signal (open) and control (hatched) regions; (d) difference between the open and other histograms in (b). with the parameters of one BW fixed to the values given in Sec. IVF, corresponding to the f0(600), we obtain a good fit, shown as the curve in Fig. 19(d). This fit yields a f0(980) signal of 54±9 events with a massm = 0.970± 0.007 GeV/c2 and width Γ = 0.081± 0.021 GeV consis- tent with PDG values [5]. Due to low statistics and high backgrounds, we do not extract an e+e− → φ(1020)π0π0 cross section. F. The φ(1020)f0(980) Intermediate State Since the background under the f0(980) peak in Figs. 19(b,d) is relatively low we are able to extract the φ(1020)f0(980) contribution. As in Sec. IVG, we require the dipion mass to be in the range 0.85–1.10 GeV/c2 and fit the background-subtractedK+K− mass projection in each bin of K+K−π0π0 mass to obtain a number of φf0 events. Again, about 10% of these are φπ0π0 events in which the π0π0 pair is not produced through the f0, but this does not affect the conclusions. We convert the number of fitted events in each mass bin into a measurement of the e+e−→ φ(1020)f0(980) cross section as described above and dividing by the f0(980)→ π0π0 branching fraction of one-third. The cross section is shown in Fig. 20 as a function of Ec.m. and is listed in Table VIII. Due to smaller number of events, we have used larger bins at higher energies. The overall shape is consistent with that obtained in the K+K−π+π− mode (see Fig. 13), and there is a sharp TABLE VIII: Measurements of the e+e− → φ(1020)f0(980) cross section (f0 → π 0π0, errors are statistical only). Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 1.88-1.92 0.078+0.082−0.053 2.12-2.16 0.397 +0.164 −0.137 2.44-2.52 0.120 +0.053 −0.042 1.92-1.96 0.220+0.114−0.085 2.16-2.20 0.408 +0.143 −0.118 2.52-2.68 0.050 +0.024 −0.019 1.96-2.00 0.136+0.104−0.075 2.20-2.24 0.070 +0.064 −0.042 2.68-2.84 0.026 +0.017 −0.012 2.00-2.04 0.446+0.160−0.131 2.24-2.28 0.174 +0.095 −0.071 2.84-3.00 0.026 +0.015 −0.011 2.04-2.08 0.315+0.142−0.113 2.28-2.36 0.069 +0.042 −0.030 3.00-3.16 0.032 +0.017 −0.013 2.08-2.12 0.519+0.169−0.141 2.36-2.44 0.112 +0.051 −0.040 3.16-3.32 0.012 +0.012 −0.008 1.5 2 2.5 3 3.5 4 Ec.m. (GeV) FIG. 20: Cross section for the reaction e+e− → φ(1020)f0(980) as a function of effective e +e− c.m. energy obtained from the K+K−π0π0 final state. drop near 2.2 GeV/c2, but the statistical errors are large and no conclusion can be drawn from this mode alone. Possible interpretations are discussed in Section VII. VI. THE K+K−K+K− FINAL STATE A. Final Selection and Background Figure 21 shows the distribution of χ24K for the K+K−K+K− candidates as points, and the open his- togram is the distribution for simulated K+K−K+K− events, normalized to the data in the region χ24K < 5 where the backgrounds and radiative corrections are small. The hatched histogram represents the back- ground from e+e− → qq events, evaluated as for the other modes. The cross-hatched histogram represents the background from simulated ISR K+K−π+π− events with both charged pions misidentified as kaons. We define signal and control regions of χ24K < 20 and 20< χ24K < 40, respectively. The signal region contains 2,305 data and 20,616 simulated events, and the con- trol region contains 463 data and 1,601 simulated events. 0 20 40 60 χ2(4K) FIG. 21: Distribution of χ2 from the three-constraint fit for K+K−K+K− candidates in the data (points). The open histogram is the distribution for simulated signal events, nor- malized as described in the text. The hatched histogram rep- resents the background from non-ISR events, estimated as described in the text. The cross-hatched histograms is for simulated ISR K+K−π+π− events. Figure 22 shows the K+K−K+K− invariant mass distri- bution from threshold up to 5 GeV/c2 for events in the signal region as points with errors. The qq background (hatched histogram) is small at low masses, but dominant above about 4.5 GeV/c2. Since the ISR K+K−π+π− background does not peak at low χ24K values, we in- clude it in the background evaluated from the control region, according to the method explained in Sec. IVA. It dominates this background, which is 10% or lower at all masses. The total background is shown as the open histogram in Fig. 22. We subtract the sum of backgrounds from the number of selected events in each mass bin to obtain a number of signal events. Considering uncertainties in the cross sections for the background processes, the normalization of events in the control region, and the simulation statis- tics, we estimate a systematic uncertainty on the signal yield of less than 5% in the 2–3 GeV/c2 region, increasing to 10% in the region above 3 GeV/c2. 2 3 4 5 m(K+K-K+K-) (GeV/c2) FIG. 22: Invariant mass distribution for K+K−K+K− can- didates in the data (points). The hatched and open his- tograms represent, cumulatively, the non-ISR background and the ISR background from the control region, which is domi- nated by the contribution from ISR K+K−π+π− events. 2 3 4 m(K+K-K+K-) (GeV/c2) 2 3 4 m(K+K-K+K-) (GeV/c2) FIG. 23: (a) Invariant mass distributions for simulated K+K−K+K− events in the signal (open) and control (hatched) regions (see Fig. 21); (b) net reconstruction and selection efficiency as a function of mass obtained from this simulation (the curve represents a 3rd order polynomial fit). B. Selection Efficiency The detection efficiency is determined as for the other two final states. Figure 23(a) shows the simulated K+K−K+K− invariant-mass distributions in the signal and control regions from the phase space model. We di- vide the number of reconstructed events in each mass interval by the number of generated ones in that interval to obtain the efficiency shown as the points in Fig. 23(b). It is quite uniform, and we fit a third order polynomial, which we use to extract the cross section. A factor of 0.7 is again applicable for only hadronic system efficiency due to the limited angular coverage of the ISR photon simula- tion. A simulation assuming dominance of the φK+K− channel, with the K+K− pair in an S-wave, gives con- sistent results, and we apply the same 5% systematic uncertainty as for the other final states. We correct for mis-modeling of the track finding and kaon identifica- tion efficiencies as in Sec. IVB, and for the shape of the χ24K distribution analogously, using the result in Ref. [11], (3.0± 2.0)%. C. Cross Section for e+e− → K+K−K+K− We calculate the e+e−→K+K−K+K− cross section in 40 MeV Ec.m. intervals from the analog of Eq. 2, us- ing the invariant mass of the K+K−K+K− system to determine the effective c.m. energy. We show this cross section in Fig. 24 and list it in Table IX. It rises to a peak value near 0.1 nb in the 2.3–2.7 GeV region, then decreases slowly with increasing energy. The only sta- tistically significant narrow structure is the J/ψ peak. Again, dL includes corrections for vacuum polarization that should be omitted from calculations of gµ−2. This supersedes our previous result [11]. 2 3 4 5 Ec.m. (GeV) FIG. 24: The e+e−→K+K−K+K− cross section as a func- tion of the effective e+e− c.m. energy measured with ISR data at BABAR. The errors are statistical only. The simulated K+K−K+K− invariant mass resolu- tion is 3.0 MeV/c2 in the 2.0–2.5 GeV/c2 range, increas- ing with mass to 4.7 MeV/c2 in the 2.5–3.5 GeV/c2 range. Since the cross section has no sharp structure except for the J/ψ peak, we again make no correction for resolution. TABLE IX: Measurements of the e+e− → K+K−K+K− cross section (errors are statistical only). Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) Ec.m. (GeV) σ (nb) 2.02 0.006 ± 0.004 2.66 0.075 ± 0.015 3.30 0.025 ± 0.009 3.94 0.012 ± 0.006 2.06 0.011 ± 0.006 2.70 0.102 ± 0.015 3.34 0.025 ± 0.009 3.98 0.012 ± 0.006 2.10 0.019 ± 0.007 2.74 0.069 ± 0.014 3.38 0.024 ± 0.010 4.02 0.010 ± 0.006 2.14 0.034 ± 0.011 2.78 0.063 ± 0.014 3.42 0.034 ± 0.010 4.06 0.009 ± 0.005 2.18 0.040 ± 0.013 2.82 0.051 ± 0.012 3.46 0.036 ± 0.009 4.10 0.006 ± 0.005 2.22 0.087 ± 0.016 2.86 0.024 ± 0.011 3.50 0.032 ± 0.009 4.14 0.008 ± 0.006 2.26 0.064 ± 0.018 2.90 0.054 ± 0.012 3.54 0.025 ± 0.009 4.18 0.011 ± 0.005 2.30 0.082 ± 0.017 2.94 0.045 ± 0.013 3.58 0.031 ± 0.009 4.22 0.011 ± 0.005 2.34 0.079 ± 0.018 2.98 0.045 ± 0.011 3.62 0.014 ± 0.007 4.26 0.012 ± 0.005 2.38 0.084 ± 0.017 3.02 0.063 ± 0.013 3.66 0.019 ± 0.008 4.30 0.004 ± 0.005 2.42 0.070 ± 0.016 3.06 0.049 ± 0.012 3.70 0.028 ± 0.008 4.34 0.003 ± 0.006 2.46 0.092 ± 0.018 3.10 0.287 ± 0.024 3.74 0.008 ± 0.005 4.38 0.013 ± 0.005 2.50 0.082 ± 0.015 3.14 0.050 ± 0.012 3.78 0.026 ± 0.007 4.42 0.012 ± 0.005 2.54 0.091 ± 0.015 3.18 0.054 ± 0.011 3.82 0.020 ± 0.007 4.46 0.006 ±-0.004 2.58 0.077 ± 0.017 3.22 0.049 ± 0.011 3.86 0.007 ± 0.006 4.50 -0.001 ±-0.004 2.62 0.095 ± 0.015 3.26 0.045 ± 0.010 3.90 0.012 ± 0.006 4.54 0.000 ± 0.004 The errors shown in Fig. 24 and Table IX are statisti- cal only. The point-to-point systematic errors are much smaller than this, and the errors on the normalization are summarized in Table X, along with the corrections ap- plied to the measurement. The total correction is +10%, and the total systematic uncertainty is 9% at low mass, increasing to 13% above 3 GeV/c2. TABLE X: Summary of corrections and systematic uncer- tainties on the e+e−→K+K−K+K− cross section. The total correction is the linear sum of the components and the total uncertainty is the sum in quadrature. Source Correction Uncertainty Rad. Corrections – 1% Backgrounds – 5% m4K < 3 GeV/c 10% m4K > 3 GeV/c Model Dependence – 5% χ24K Distribution +3% 2% Tracking Efficiency +3% 2% Kaon ID Efficiency +4% 4% ISR Luminosity – 3% Total +10% 9% m4K < 3 GeV/c 13% m4K > 3 GeV/c D. The φ(1020)K+K− Intermediate State Figure 25 shows the invariant mass distribution for all K+K− pairs in the selected K+K−K+K− events (4 en- tries per event) as the open histogram. A prominent φ peak is visible, along with possible peaks at 1.5, 1.7 and 2.0 GeV/c2. The hatched histogram is for the pair in each event with mass closest to the nominal φ mass, and indicates that the φK+K− channel dominates the K+K−K+K− final state. Our previous finding of very 1 1.5 2 m(K+K-) (GeV/c2) FIG. 25: Invariant mass distributions for all K+K− pairs in selected e+e−→K+K−K+K− events (open histogram) and for the combination in each event closest to the φ-meson mass (hatched). little φ signal [11] was incorrect due to an error in the analysis algorithm. If the pair mass closest to the φ mass is within 10 MeV/c2 of the φ mass, then we include the invariant mass of the other K+K− combination in Fig. 26. The contribution from events in the J/ψ peak is shown as the hatched histogram which is in agreement with the BES experiment [24] which studied the structures around 1.5, 1.7 and 2.0 GeV/c2 in detail. There is no evidence for the φf0 channel, but there is an enhancement at thresh- old that can be interpreted as the tail of the f0(980). This is expected in light of the φf0 cross sections mea- sured above in the K+K−π+π− and K+K−π0π0 final 1 1.2 1.4 1.6 1.8 2 2.2 2.4 m(K+K-) (GeV/c2) FIG. 26: Invariant mass distribution for K+K− pairs in events in which the other K+K− pair has mass closest to and within 10 MeV/c2 of the nominal φ mass (open histogram). The hatched histogram is for the subset with a K+K−K+K− mass in the J/ψ peak. states. However the statistics and uncertainties in the f0(980) → K+K− lineshape do not allow a meaningful extraction of the cross section in this final state. We observe no significant structure in the K+K−K± mass distribution. We use these events to study the pos- sibility that part of our φπ+π− signal is due to φK+K− events with the two kaons not from the φ taken as pions. No structure is present in the reconstructedK+K−π+π− invariant mass distribution from these events. VII. e+e− → φf0 NEAR THRESHOLD The behavior of the e+e− → φf0 cross section near threshold shows a structure near 2150 MeV/c2, and we have published this result in Ref. [14]. Here we pro- vide a more detailed study of this cross section in the 1.8–3 GeV region. In Fig. 27 we superimpose the cross sections measured in the K+K−π+π− and K+K−π0π0 final states (shown in Figs. 13 and 20); they are con- sistent with each other. The K+K−K+K− cross section (Sec. VID) is also consistent with the presence of a struc- ture near 2150 MeV/c2 and shows a contribution from the φf0 channel, but since we cannot extract a meaning- ful φf0 cross section, we do not discuss this final state further. First, we attempt to reproduce this spectrum with a smooth threshold function. In the absence of resonances, the only theoretical constraint on the cross section well above threshold is that it should decrease smoothly with increasing Ec.m.. However the form of the cutoff at threshold is determined by the properties of the interme- 1.8 2 2.2 2.4 2.6 2.8 3 Ec.m. (GeV) FIG. 27: The e+e−→φ(1020)f0(980) cross section measured in the K+K−π+π− (circles) and K+K−π0π0 (squares) final states. The hatched histogram shows the simulated cross sec- tion, assuming no resonant structure. The solid (dashed) line represents the result of the one-resonance (no-resonance) fit described in the text. diate resonances and the final state particle spins, phase space and detector resolution. The model discussed in Sec. IVF takes the φ and f0(980) lineshapes, the spins of all particles and their phase space into account, and postulates a simple E−4c.m. dependence of the cross section. For the e+e−→φf0 reaction, it predicts the cross section shown as the hatched histogram in Fig. 27, normalized to the same total area as the data. It shows a sharp rise from the threshold with a peak near 2070 MeV and is inconsistent with the data. To account for uncertainties in the f0 width and the shape of the cross section well above threshold, we seek a functional form that describes the simulation and whose parameters can be varied to cover a reasonable range of possibilities. This can be achieved by the product of a phase space term, an exponential rise and a second order polynomial: σnr(µ) = P (µ) ·Anr(µ), (4) Anr(µ) = σ0 · (1 − e−(µ/a1) ) · (1 + a2µ+ a3µ2), P (µ) = 1−m20/(m0 + µ)2, µ = Ec.m.−m0, where the ai are free parameters, σ0 is a normalization factor, and P (µ) is a good approximation of the two-body phase space for particles with similar masses. Both the φ(1020) and f0(980) have small but finite widths, and our selection criterion of m(ππ) > 0.85 GeV/c2 defines an effective minimum mass, m0 = 1.8 GeV/c 2. Fitting Eq. 4 to the simulated cross section yields the ai values listed in the first column of Table XI. Fitting to the data with all ai fixed to these values and σ0 floating yields χ2/n.d.f.= 86/(56 − 2). Floating a2 and a3 in addition, we obtain χ2/n.d.f.=85/(56− 4) with a confidence level (C.L.) of 0.0025. If we float all parameters in Eq. 4, the fit yields χ2/n.d.f.= 80/(56− 5) with C.L. of 0.0053. The results of these fits are listed in Table XI, and the latter is shown as the dashed curve on Fig. 27; all fits are inconsistent with the data. TABLE XI: Parameter values, χ2 values and confidence levels from the fits of Eq. 4 to the data described in the text. An asterisk denotes a value that was fixed in that fit. Fit All ai fixed Only a1 fixed All ai free σ0 1.19±0.03 1.23±0.03 1.09±0.01 a1 0.218* 0.218* 0.174±0.012 a2 −1.68* −1.51±0.15 −1.49±0.12 a3 0.81* 0.66±0.14 0.63±0.11 χ2/n.d.f. 86.4/54 85.3/52 80.5/51 P(χ2) 0.0035 0.0025 0.0053 We now add a resonance and fit the data with the function σ1r(µ) = P (µ) P (m1) ∣Anr(µ)e iψ1 +Ar1(µ) , (5) Ar1(µ) = σ1m1Γ1 m21 − E2c.m.− iEc.m.Γ1 where m1 and Γ1 are the mass and width of the reso- nance, σ1 is its peak cross section, and ψ1 is its phase rela- tive to the non-resonant component. We obtain good fits both assuming no interference between the two compo- nents, ψ1=π, and with ψ1 floating. The result of the lat- ter fit is shown as the solid curve on Fig. 27. The data are somewhat above this curve near 2.4 GeV/c2 and a fit with two resonances can also describe the data. Due to the sharp drop near 2.2 GeV/c2, the single-resonance fit with interference gives a resonance mass about 30 MeV/c2 higher than the other two fits. All these fits, with or with- out resonances, give a peak non-resonant cross section in the range 0.3–0.4 nb, which is of independent theoretical interest, because it can be related to the φ → f0(980)γ decay studied at the φ-factory [25]. Under the hypothesis of one resonance interfering with the non-resonant component, the fit gives the resonance parameters σ1 = 0.13± 0.04 nb, m1 = 2.175± 0.010 GeV/c2, Γ1 = 0.058± 0.016 GeV, and ψ1 = −0.57± 0.30 radians, along with χ2/n.d.f.=37.6/(56− 9) (C.L. 0.84). We can estimate the product of its electronic width and branch- ing fraction to φf0 as Bφf0 · Γee = Γ1σ1m = (2.5± 0.8± 0.4) eV , where we fit the product Γ1σ1 to reduce correlations, and the conversion constant C = 0.389 mb(GeV/c2)2. The second error is systematic and corresponds to the nor- malization errors on the cross section. The significance of the structure calculated from the change in χ2 between the best fit and the null hypothe- sis is 6.2 standard deviations. Since this calculation can be unreliable in the case of low statistics and functions that vary rapidly on the scale of the bin size, we per- form a set of simulations in which we generate a number of events according to a Poisson distribution about the number observed in the data and with a mass distribu- tion given by either the simulation or fitted function in Fig. 27 without resonant structure. On each sample, we perform fits to Eqs. 4 and 5 and calculate the difference in χ2. The fraction of trials giving a χ2 difference larger than that seen in the data corresponds to a significance of approximately 5 standard deviations. We search for this structure in other submodes with different and/or fewer intermediate resonances. The to- tal cross sections are dominated by K∗Kπ channels, and the K∗0K+π− cross section is shown in Fig. 8. There is no significant structure in the 2.1–2.5 GeV region, but the point-to-point statistical uncertainties are large. If we remove events within the bands in Figs. 5 and 18, then most of the events containing a K∗ are eliminated and we obtain the raw mass distributions shown as the points with errors in Figs. 28 and 29, respectively. Both distri- bution show evidence of a structure around 2.15 GeV/c2 and the K+K−π+π− distribution also shows a structure near 2.4 GeV/c2. We cannot exclude the presence of these structures in events with a K∗, but we can conclude that they do not dominate those events, whereas they com- prise a substantial fraction of the remaining events in that mass region. Applying the further requirement that the dipion mass be in the range 0.85–1.10 GeV/c2, we remove most of the events without an f0, and obtain the mass distribu- tions shown as the hatched histograms in Figs. 28 and 29. Peaks are visible at both 2.15 GeV/c2 and 2.4 GeV/c2 in both distributions, and they contain enough events to account for the corresponding structures in the distribu- tions for all non-K∗ events. These peaks contain at least as many events as are present in the φf0 samples, but the non-resonant components are higher and there is a substantial kinematic overlap between K+K−f0 events and K∗Kπ events in this mass range. Since this f0(980) band appears to contain a large frac- tion of the events within the structure, we now consider all selected events with a dipion mass inside or outside this range. Figure 31 shows the mass distribution for all selected K+K−π0π0 events as the open histogram, and the subsets of events with π0π0 mass inside and outside the range 0.85–1.10 GeV/c2 as the hatched and cross- hatched histograms, respectively. It is evident that the K+K−f0 channel contains the majority of the structure in the 2.0–2.6 GeV/c2 range. We show the corresponding distributions for the 1 1.5 2 2.5 3 3.5 4 4.5 5 m(K+K-π+π-) (GeV/c2) FIG. 28: Invariant mass distribution for all selected K+K−π+π− events lying outside the K∗0(892) bands of Fig. 5 (points), and the subset of these events with 0.85 < m(π+π−) < 1.10 GeV/c2 (hatched). 1 1.5 2 2.5 3 3.5 4 4.5 5 m(K+K-π0π0) (GeV/c2) FIG. 29: Invariant mass distribution for all selected K+K−π0π0 events lying outside the K∗0(892) bands of Fig. 18 (points), and the subset of these events with 0.85 < m(π0π0) < 1.10 GeV/c2 (hatched). K+K−π+π− events in Fig. 30. Due to the presence of the ρ0, the relative f0 contribution is much smaller in this final state, but the events in the f0 band show clear in- dications of structure in the 2.0–2.4 GeV/c2 region. The remaining events may also have structure in this region, but the statistical significance is marginal and it could be due to other sources, such as the φf2(1270) threshold at 2.3 GeV/c2. Figures 32 and 33 show enlarged views of the mass distributions within the f0 bands from Figs. 30 and 31, respectively. The two-peak structure is more evident 1 1.5 2 2.5 3 3.5 4 4.5 5 m(K+K-π+π-) (GeV/c2) FIG. 30: The K+K−π+π− invariant mass distribution for all selected events (open histogram), and for those with a π+π− mass inside (cross-hatched) or outside (hatched) the f0 band as defined in the text. 1 1.5 2 2.5 3 3.5 4 4.5 5 m(K+K-π0π0) (GeV/c2) FIG. 31: The K+K−π0π0 invariant mass distribution for all selected events (open histogram), and for those with a π0π0 mass inside (cross-hatched) or outside (hatched) the f0 band as defined in the text. here than in the φf0 events. The 0.85 < m(ππ) < 1.10 GeV/c2 requirement gives enough phase space for K+K− invariant mass to cover the region from threshold to ∼1.3 GeV/c2 for m(K+K−ππ) ≈ 2.15 GeV/c2. From the measured kaon form factor we expect to find only about two-thirds of K+K− P-wave in our fitted φ peak. Since the non-ISR and ISR ππππ backgrounds have not been subtracted and the samples contain an unknown mixture of intermediate states, we fit them with a modi- TABLE XII: Summary of parameters obtained from the fits described in the text to the K+K−π+π− and K+K−π0π0 events with dipion mass in the f0(980) band. An asterisk denotes a value that was fixed in that fit. No Resonance One Resonance Two Resonances Fit K+K−π+π− K+K−π0π0 K+K−π+π− K+K−π0π0 K+K−π+π− K+K−π0π0 Nnr 7204±775 991±202 8466±334 722±112 6502±476 117±89 a1 0.181±0.012 0.134±0.017 0.224±0.024 0.197±0.048 0.201±0.035 0.143±0.053 a2 −0.75±0.21 −1.47±0.38 −0.89±0.17 −0.36±0.10 −0.44±0.15 5.80±2.36 a3 0.09±0.17 0.75±0.35 0.17±0.08 −0.28±0.14 −0.15±0.12 −5.26±1.75 a4 0.75* 0.50* 0.75* 0.50* 0.75* 0.50* N1 0* 0* 116±95 149±36 163±70 192±44 m1 (GeV/c 2) – – 2.192±0.014 2.169±0.020 2.187±0.013 2.154±0.029 Γ1 (GeV) – – 0.071±0.021 0.102±0.027 0.066±0.018 0.110±0.022 ψ1 (rad) – – −0.60±0.41 −1.02±0.19 −1.10±0.14 −1.04±0.23 N2 0* 0* 0* 0* 16±16 6±5 m2 (GeV/c 2) – – – – 2.47±0.07 2.45±0.04 Γ2 (GeV) – – – – 0.077±0.065 0.062±0.102 ψ2 (rad) – – – – 0.28±1.06 1.41±1.29 χ2/n.d.f. 62.8/41 38.1/21 35.6/37 13.0/17 31.4/34 9.7/13 P(χ2) 0.016 0.012 0.54 0.74 0.60 0.72 fied version of Eq. 5 that allows up to two resonances, F (µ) = (a4 · Anr)2 (6) + |(1 − a4)Anr +Ar1eiψ1 +Ar2eiψ2 |2. Here, the normalization is in terms of events rather than cross section (σi → Ni) and a fraction a4 of the non- resonant component does not interfere with the reso- nances. We first fit the distribution with no resonances (and a4 =1). The results are shown as the dashed lines in Figs. 32 and 33 and listed in Table XII; both are in- consistent with the data. We next include one resonance in the fit. The param- eter a4 is not well constrained by the data and its value has a small influence on all other fit parameters except for the number of events assigned to the resonance, so we present results with a4 fixed to the reasonable values of 0.75 and 0.50 for theK+K−π+π− andK+K−π0π0 data, respectively. The results are shown as the solid lines in Figs. 32 and 33 and listed in Table XII. The fit quality is good in both cases, the fitted resonance parameters are consistent with those from the φf0 study, and the cal- culated significance of the structure for the K+K−π+π− data is similar, 5.2 standard deviations. TheK+K−π0π0 data show much more pronounced structure than in the φf0 study, allowing a full fit to this sample with a signif- icance of 5.0 standard deviations. We then add a second resonance to the fit, keeping a4 fixed and floating all other parameters. The results are shown as the dotted lines in Figs. 32 and 33, and listed in Table XII. These fits are also of good quality, but do not change the χ2 CL or the parameters of the first resonance significantly. We also perform fits with no in- terference between the non-resonant component and any resonance (a4 = 1), obtaining good quality fits for both one resonance and two resonances with relative phase π/2. The fitted resonance parameters are consistent in all cases, except that the mass of the first resonance is lower by about 50 MeV/c2, similar to the 30 MeV/c2 shift seen in the φf0 study. From these studies we conclude that we have observed a new vector structure at a mass of about 2150 MeV/c2 with a significance of over six standard deviations. It decays into K+K−f0(980), with the K +K− pair pro- duced predominantly via the φ(1020). There is an ad- ditional structure at about 2400 MeV/c2, and the two structures can be described by either two resonances or a single resonance that interferes with the non-resonant K+K−f0(980) process. More data and searches in other final states are needed to understand the nature of these structures. If the main structure is due to a resonance, then it is relatively narrow and might be interpreted as the strange analog of the recently observed charmed Y(4260) state [6], which decays to J/ψπ+π−. The value of Bφf0 ·Γee = (2.5± 0.8± 0.4) eV measured here is similar to the value of BY→J/ψπ+π− · ΓYee = (5.5± 1.0± 0.8) eV reported in Ref. [6]. VIII. THE CHARMONIUM REGION The data at masses above 3 GeV/c2 can be used to measure or set limits for the branching fractions of nar- row resonances, such as charmonia, and the narrow J/ψ and ψ(2S) peaks allow measurements of our mass scale and resolution. Figures 34, 35 and 36 show the in- variant mass distributions for the selected K+K−π+π−, K+K−π0π0 and K+K−K+K− events, respectively, in this region, with finer binning than in the corresponding Figs. 2, 15 and 22. We do not subtract any background FIG. 32: The K+K−π+π− invariant mass distribution in the K+K−f0(980) threshold region for events with a π mass inside the f0 band. The lines represent the results of the fits including no (dashed), one (solid) and two (dotted) resonances described in the text. FIG. 33: The K+K−π0π0 invariant mass distribution in the K+K−f0(980) threshold region for events with a π 0π0 mass inside the f0 band. The lines represent the results of the fits including no (dashed), one (solid) and two (dotted) reso- nances described in the text. from the K+K−π+π− or K+K−K+K− data, since it is small and nearly uniformly distributed, but we use the KKπ0π0 control region to subtract part of the ISR back- ground from the K+K−π0π0 data. Signals from the J/ψ are visible in all three distributions, and the ψ(2S) is vis- ible in the K+K−π+π− mode. We fit each of these distributions using a sum of two Gaussian functions to describe the J/ψ and ψ(2S) signals plus a polynomial to describe the remainder of the dis- 3 3.25 3.5 3.75 m(K+K-π+π-) (GeV/c2) FIG. 34: Raw invariant mass distribution for all selected e+e−→K+K−π+π− events in the charmonium region. The line represents the result of the fit described in the text. tribution. We take the signal function parameters from the simulation, but let the overall mean and width float in the fit, along with the amplitude and the coefficients of the polynomial. The fits are of good quality and are shown as the curves on Figs. 34, 35 and 36. In all cases, the fitted mean value is within 1 MeV/c2 of the PDG [5] J/ψ or ψ(2S) mass, and the width is consistent within 10% with the simulated resolution discussed in Sec. IVC, VC or VIC. The fits yield 1586± 58 events in the J/ψ peak for the K+K−π+π− final state, 203± 16 events for K+K−π0π0 and 156±15 events for K+K−K+K−. From these num- bers of observed events in each final state f , NJ/ψ→f , we calculate the product of the J/ψ branching fraction to f and the J/ψ electronic width: BJ/ψ→f · ΓJ/ψee = NJ/ψ→f ·m2J/ψ 6π2 · dL/dE · ǫf(mJ/ψ) · C , (7) where dL/dE = 89.8 nb−1/MeV and ǫf (mJ/ψ) are the ISR luminosity and corrected selection efficiency, respec- tively, at the J/ψ mass and C is the conversion constant. We estimate ǫK+K−π+π− =0.202, ǫK+K−π0π0 =0.069 and ǫK+K−K+K− =0.176. Using Γ ee = 5.40 ± 0.18 keV [5], we obtain the branching fractions listed in Table XIII, along with the measured products and the current PDG val- ues. The systematic errors include a 3% uncertainty ee . The branching fractions to K +K−π+π− and K+K−K+K− are more precise than the current PDG values, which were dominated by our previous results of (6.25±0.80)×10−3 and (7.4±1.8)×10−4, re- spectively [11]. This is the first measurement of the K+K−π0π0 branching fraction. 3 3.25 3.5 3.75 m(K+K-π0π0) (GeV/c2) FIG. 35: Invariant mass distribution for e+e−→K+K−π0π0 events in the charmonium region, after partial background subtraction. The line represents the result of the fit described in the text. 3 3.25 3.5 3.75 m(K+K-K+K-) (GeV/c2) FIG. 36: Raw invariant mass distribution for all selected e+e−→K+K−K+K− events in the charmonium region. The line represents the result of the fit described in the text. These fits also yield 91±15 K+K−π+π− events in the ψ(2S) peak, but no other significant signals. We expect 6.3 events from ψ(2S)→J/ψπ+π−→K+K−π+π− from the relevant branching fractions [5], which is less than the statistical error. Subtracting this contribution and using a calculation analogous to Eq. 7, with dL/dE = 115.3 nb−1/MeV, we obtain the product of the ψ(2S)→ K+K−π+π− branching fraction and its electronic width. Dividing by the world average value of Γ ψ(2S) ee [5], we obtain the branching fraction listed in Table XIII; it is consistent with the current PDG value [5]. As noted in Sec. IVD and shown in Fig. 5, the K+K−π+π− final state is dominated by the K∗0(892)Kπ channels, with a small fraction seen in the K∗0(892)K∗02 (1430) + c.c. channels. Figure 37 shows a scatter plot of the invariant mass of a K±π∓ pair ver- sus that of the K+K−π+π− system in events with the other K∓π± pair near the K∗0(892) mass, i.e. within the bands in Fig. 5(a) with overlapped region taken only once. There is a large concentration of entries in the J/ψ band with K±π∓ masses near 1430 MeV/c2, but no solid evidence for a horizontal band corresponding to K∗02 (1430) production other than in J/ψ decays. We show the K±π∓ mass projection for the subset of events with a K+K−π+π− mass within 50 MeV/c2 of the J/ψ mass in Fig. 38 as the open histogram. The hatched his- togram is the projection for events with a K+K−π+π− mass between 50 and 100 MeV/c2 below the J/ψ mass. 2 3 4 m(K+K-π+π-) (GeV/c2) FIG. 37: The K±π∓ invariant mass versus K+K−π+π− invariant mass for events with the other K∓π± combination in the K∗0(892) bands of Fig. 5(a). The overlapped region is taken only once. The J/ψ component appears to be dominated by the K∗02 (1430). Also seen is a small signal from K ∗0(892) indicating the K∗0(892)K̄∗0(892) decay of J/ψ: this is also seen as an enhancement in the vertical J/ψ band in Fig. 37. The enhancement at 1.8 GeV/c2 of Fig. 38 can be explained by the J/ψ decay into K∗0(892)K2(1770)+c.c. (or K∗0(892)K2(1820)+ c.c.), a mode which has not pre- viously been reported. Subtracting the number of side- band events from the number in the J/ψ mass window, we obtain 317±23 events with a K±π∓ mass in the range 1200–1700 MeV/c2, which we take as a measure of J/ψ decays into K∗0(892)K∗02 (1430), 25 ± 8 events in the 0.8–1.0 GeV/c2 window for the K∗0(892)K̄∗0(892) de- cay and 110 ± 14 events for the K∗0(892)K2(1770) or K∗0(892)K2(1830) final state in the 1.7–2.0 GeV/c 2 re- gion. We convert these to branching fractions using Eq. 7 and dividing by the known branching fractions of excited 1 2 3 m(K+π-,K-π+) (GeV/c2) FIG. 38: The K±π∓ mass projection from Fig. 37 for events with a K+K−π+π− mass within 50 MeV/c2 of the J/ψ mass (open histogram) and 50–100 MeV/c2 below (hatched). kaons [5]. The results are listed in Table XIII: they are considerably more precise than the PDG values. We cannot calculate BJ/ψ→K∗0K2(1770) because no branching fractions of K2(1770) or K2(1830) to Kπ are reported. We study decays into φπ+π− and φπ0π0 using the mass distributions shown in Figs. 39 and 40, respectively. The open histograms are for the events with a K+K− mass within the φ bands of Figs. 9(c) and 19(c). The cross-hatched histogram in Fig. 39 is from the φ side- bands of Fig. 9(c) and represents the dominant back- ground in the φπ+π− mode. The hatched histogram in Fig. 40 is from the χ2KKπ0π0 control region and represents the dominant background in the φπ0π0 mode. Subtract- ing these backgrounds, we find 103±12 J/ψ → φπ+π− events, 23±6 J/ψ → φπ0π0 events, and 10±4 ψ(2S) → φπ+π− events. We convert these to branching fractions and list them in Table XIII. This is the first measure- ment of the J/ψ → φπ0π0 branching fraction, and the other two are consistent with current PDG values. If the Y (4260) has a substantial branching fraction into φπ+π−, then we would expect to see a signal in Fig. 39. In the mass range |m(φπ+π−) − m(Y )| < 0.1 GeV/c2, we find 10 events, and assuming a uniform distribu- tion we estimate 9.2 background events from the 3.8– 5.0 GeV/c2 region. This corresponds to a signal of 0.8± 3.3 events or a limit of < 5 events at the 90% C.L. Using dL/dE = 147.7 nb−1/MeV at the Y (4260) mass, we calculate BY→φπ+π− ·ΓYee<0.4 eV which is well below the value of BY→J/ψπ+π− ·ΓYee = (5.5± 1.0± 0.8) eV [6]. No Y (4260) signal is seen in any other mode studied here. Figures 41(a) and 42 show the corresponding mass dis- tributions for φf0 events, i.e. the subsets of the events in Figs. 39 and 40 with a dipion mass in the range 0.85– 1.10 GeV/c2. Signals at the J/ψ mass are visible in 3 3.5 4 4.5 5 m(K+K-π+π-) (GeV/c2) FIG. 39: Raw invariant mass distributions for candidate e+e− → φπ+π− events (open histogram) and events in the φ side bands of Fig. 9(c) (cross-hatched) in the charmonium region. The vertical lines indicate the region used for the Y (4260) search. 3 3.5 4 4.5 5 m(K+K-π0π0) (GeV/c2) FIG. 40: Raw invariant mass distributions for candidate e+e− → φπ0π0 events (open histogram) and events in the χ2KKπ0π0 control region (hatched) in the charmonium region. both cases, but φf0 is not the dominant mode of the J/ψ→ φπ+π− decay. Figure 41(b) shows the π+π− in- variant mass distribution for events in the J/ψ peak of Fig. 39, 3.05<m(K+K−π+π−)< 3.15 GeV/c2. A two- peak structure is visible that can interpreted as due to the f0(980) and f2(1270) resonances. Fitting the dis- tribution in Fig. 41(b) with a sum of two Breit-Wigner functions with parameters fixed to PDG values [5], we find 19.5 ± 4.5 J/ψ→φf0 events and 44 ± 7 J/ψ→φf2 events. From Fig. 42 we estimate 7.0± 2.8 φf0 events in the π0π0 mode. TABLE XIII: Summary of the J/ψ and ψ(2S) branching fractions measured in this article. Measured Measured J/ψ or ψ(2S) Branching Fraction (10−3) Quantity Value ( eV) Calculated, this work PDG2006 ee ·BJ/ψ→K+K−π+π− 36.3 ±1.3 ±2.1 6.72±0.24±0.40 6.2 ±0.7 ee ·BJ/ψ→K+K−π0π0 13.6 ±1.1 ±1.3 2.52±0.20±0.25 no entry ee ·BJ/ψ→K+K−K+K− 4.11±0.39±0.30 0.76±0.07±0.06 0.78 ±0.14 ee ·BJ/ψ→K∗0K∗0 · BK∗0→Kπ · BK∗0 →Kπ 7.3 ±0.5 ±0.6 2.7 ±0.2 ±0.2 6.7 ±2.6 ee ·BJ/ψ→K∗0K̄∗0 · BK∗0→Kπ · BK̄∗0→Kπ 0.57±0.18±0.05 0.11±0.04±0.01 <0. 5 at 90% C.L. ee ·BJ/ψ→K∗0K2(1770) · BK∗0→Kπ · BK2→Kπ 2.5 ±0.3 ±0.2 – no entry ee ·BJ/ψ→φπ+π− · Bφ→K+K− 2.61±0.30±0.18 0.98±0.11±0.07 0.94 ±0.15 ee ·BJ/ψ→φπ0π0 · Bφ→K+K− 1.54±0.40±0.16 0.58±0.15±0.06 no entry ee ·BJ/ψ→φf0 · Bφ→K+K− · Bf0→π+π− 0.50±0.11±0.04 0.28±0.07±0.02 0.32 ±0.09 (s=1.9) ee ·BJ/ψ→φf0 · Bφ→K+K− · Bf0→π0π0 0.47±0.19±0.05 0.54±0.21±0.05 0.32 ±0.09 (s=1.9) ee ·BJ/ψ→φf2 · Bφ→K+K− · Bf2→π+π− 1.12±0.18±0.09 0.50±0.08±0.04 <0. 37 at 90% C.L. ψ(2S) ee ·Bψ(2S)→K+K−π+π− 2.56±0.42±0.16 1.2 ±0.2 ±0.08 0.72 ±0.05 ψ(2S) ee ·Bψ(2S)→φπ+π− · Bφ→K+K− 0.28±0.11±0.02 0.27±0.11±0.02 0.113 ±0.029 ψ(2S) ee ·Bψ(2S)→φf0 · Bφ→K+K− · Bf0→π+π− 0.17±0.08±0.02 0.26±0.12±0.03 0.090 ±0.033 3 3.5 4 4.5 5 m(φ f0) (GeV/c 0.5 1 1.5 m(π+π-) (GeV/c2) FIG. 41: (a) Raw invariant mass distribution for candidate φf0, f0→π +π− events (open histogram) and events in the φ side bands (cross-hatched) in the charmonium region; (b) the π+π− invariant mass distribution for φπ+π− events from the J/ψ peak of Fig. 39. The line represents the result of the fit described in the text. Using Eq. 7 and dividing by the appropriate branching fractions, we obtain the J/ψ branching fractions listed in Table XIII. The measurements of BJ/ψ→φf0 in the π+π− and π0π0 decay modes of the f0 are consistent with each other and with the PDG value, and combined they have roughly the same precision as listed in the PDG [5]. This is the first measurement of BJ/ψ→φf2 , and the value is consistent with the previous upper limit [5]. We also 3 3.5 4 4.5 5 m(φ f0) (GeV/c FIG. 42: Raw invariant mass distribution for candidate φf0, 0π0 events (open histogram) and events in the χ2KKπ0π0 control region (hatched) in the charmonium region. observe 6 ± 3 ψ(2S)→φf0, f0→π+π− events, which we convert to the branching fraction listed in Table XIII; it is consistent with the PDG value [5], assuming Bf0→π+π− = In the Y (4260) region we have 4 events with an esti- mated background of about 1 event. This corresponds to 3±2 events, or a 90% CL upper limit of 5.6 events. We convert this to the limits BY→φf0 · ΓYee · Bφ→K+K− · Bf0→π+π− < 0.14 eV , BY→φf0 · ΓYee < 0.43 eV, 90%CL, which is again much lower than the corresponding quan- tity for the Y (4260)→J/ψπ+π− decay. IX. SUMMARY We use the excellent charged particle tracking and identification, and photon detection of the BABAR detector to fully reconstruct events of the type e+e− → γe+e− → γK+K−π+π−, γK+K−π0π0 and γK+K−K+K−, where the γ is radiated from the ini- tial state e+ or e−. Such events are equivalent to direct e+e− annihilation at an effective c.m. energy correspond- ing to the mass of the hadronic system, and we study the annihilation into these three final states at low Ec.m., from their respective production thresholds up to 5 GeV. The K+K−π+π− and K+K−K+K− measurements are consistent with, and supersede, our previous results [11]. This is the first measurement of the K+K−π0π0 final state, although some of the results were also presented in Ref. [14]. The systematic uncertainties on the normaliza- tion of the e+e− → K+K−π+π−, K+K−π0π0 and K+K−K+K− cross sections are 8%, 10% and 9%, re- spectively, for Ec.m.<3 GeV, and 10%, 14% and 13% in the 3–5 GeV range. The obtained cross sections are con- siderably more precise than previous measurements and cover this low energy range completely, so they provide important input to calculations of the hadronic correc- tions to the anomalous magnetic moment of the muon and the fine structure constant at the Z0 mass. These final states exhibit complex resonant substruc- ture. In the K+K−π+π− mode we measure the cross sections for the first time for the specific channels e+e−→ K∗0(890)K−π+, φπ+π− and φf0. We also observe sig- nals for the ρ0(770), K1(1270), K1(1400), K 2 (1430) and f∗02 (1270) resonances. It is difficult to disentangle these contributions to the final state, and we make no attempt to do so in this paper. We note that the ρ0 signal is consistent with being due entirely to K1 de- cays, and the total cross section is dominated by the K∗0(892)K−π+ + c.c. channels, but there is no signifi- cant signal for e+e−→K∗0(892)K∗0(892). In the K+K−π0π0 mode we measure cross sections for e+e− → φf0 and observe signals for the K∗±(892) and K∗±2 (1430) resonances. Again, the total cross sec- tion is dominated by the K∗±(892)K∓π0 channels, and there is no signal for e+e−→K∗+(892)K∗−(892). The K+K−π0π0 final state is not accessible to intermediate states containing K1 resonances, and we note that the cross section is roughly a factor of four smaller over most of the range than the K+K−π+π− cross section, consis- tent with K∗Kπ dominance with a factor of two isospin suppression of the π0π0 final state and another factor of two for the relative branching fractions of the neutral and charged K∗ to charged kaons. In the K+K−K+K− mode we find e+e−→φK+K− to be the dominant channel. With the current data sample we can say little about the other K+K− combination, except that there is an enhancement near threshold, con- sistent with the φf0 channel, and that in J/ψ decays there is structure in the 1.5–2.0 GeV region, consistent with that observed by BES [24]. The φf0 cross section measured in the K +K−π+π− final state shows structure around 2.15 GeV and pos- sibly 2.4 GeV, and the corresponding measurement in the K+K−π0π0 final state is consistent, as reported in Ref. [14]. Further investigation here reveals consistent re- sults in the K+K−K+K− final state and clear signals in the K+K−f0 channels, with f0→π+π− and π0π0. The signals are predominantly from φf0, but the relaxation of the K+K− mass requirement reveals a strong signal in the K+K−π0π0 final state. This structure can be in- terpreted as a strange partner (with c-quarks replaced by s-quarks) of the recently observed Y (4260), which has the analogous decay mode J/ψπ+π−, or as an ssss state that decays predominantly to φf0. We also study charmonium decays into these final states and their intermediate channels. All nine of the J/ψ branching fractions and one of the three ψ(2S) branching fractions listed in Table XIII are as precise or more precise than the current world averages. We do not observe the Y (4260) in any decay mode. In particular, we find that the branching fraction for the Y (4260)→φπ+π− decay, that a glueball model [8] predicts, is less than one- tenth of that to J/ψπ+π−. X. ACKNOWLEDGMENTS We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminos- ity and machine conditions that have made this work pos- sible. The success of this project also relies critically on the expertise and dedication of the computing organiza- tions that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospi- tality extended to them. This work is supported by the US Department of Energy and National Science Foun- dation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atom- ique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsge- meinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Re- search on Matter (The Netherlands), the Research Coun- cil of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Educación y Cien- cia (Spain), and the Science and Technology Facilities Council (United Kingdom). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation. [1] V. N. Baier and V. S. Fadin, Phys. Lett. B27, 223 (1968). [2] A. B. Arbuzov et al., J. High Energy Phys. 9812, 009 (1998). [3] S. Binner, J. H. Kühn and K. Melnikov, Phys. Lett. B459, 279 (1999). [4] M. Benayoun et al., Mod. Phys. Lett. A14, 2605 (1999). [5] Review of Particle Physics, W.-M. Yao et al., J. Phys. G:Nucl. Part. Phys. 33, 1 (2006). [6] BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 95, 142001 (2005). [7] M. Davier, S. Eidelman, A. Hocker, Z. Zhang, Eur. Phys. J. C31, 503 (2003). [8] Shi-Lin Zhu, Phys. Lett. B625, 212 (2005). [9] BABAR Collaboration, B. Aubert et al., Phys. Rev. D69, 011103 (2004). [10] BABAR Collaboration, B. Aubert et al., Phys. Rev. D70, 072004 (2004). [11] BABAR Collaboration, B. Aubert et al., Phys. Rev. D71, 052001 (2005). [12] BABAR Collaboration, B. Aubert et al., Phys. Rev. D73, 052003 (2006). [13] DM-1 Collaboration, Cordier et al., Phys. Lett. B110, 335 (1982). [14] BABAR Collaboration, B. Aubert et al., Phys. Rev. D74, 091103(R) (2006). [15] BABAR Collaboration, B. Aubert et al., Nucl. Instrum. Methods Phys. Research A479, 1 (2002). [16] H. Czyż and J. H. Kühn, Eur. Phys. J. C18, 497 (2001). [17] A. B. Arbuzov et al., J. High Energy Phys. 9710, 001 (1997). [18] M. Caffo, H. Czyz, E. Remiddi, Nuovo Cim. A110, 515 (1997); Phys. Lett. B327, 369 (1994). [19] E. Barberio, B. van Eijk and Z. Was, Comput. Phys. Commun. 66, 115 (1991). [20] GEANT4 Collaboration, S. Agostinelli et al., Nucl. In- strum. Methods Phys. Res., Sect. A 506, 250 (2003). [21] T. Sjostrand, Comput. Phys. Commun. 82, 74 (1994). [22] S. Jadach and Z. Was, Comput. Phys. Commun. 85, 453 (1995). [23] S. Eidelman and F. Jegerlehner, Z. Phys. C67, 585 (1995). [24] BES Collaboration, M. Ablikim et al., Phys. Lett. B607, 243 (2005). [25] KLOE Collaboration, A. Aloisio et al., Phys. Lett.B537, 21 (2002). Introduction The BABAR detector and dataset Event Selection and Kinematic Fit The K+ K- +- final state Final Selection and Backgrounds Selection Efficiency Cross Section for e+e- K+ K- +- Substructure in the K+ K- +- Final State The e+e- K*0 K Cross Section The (1020)+- Intermediate State The (1020) f0(980) Intermediate State The K+ K-00 Final State Final Selection and Backgrounds Selection Efficiency Cross Section for e+e- K+ K- 00 Substructure in the K+ K- 00 Final State The (1020)00 Intermediate State The (1020) f0(980) Intermediate State The K+ K- K+ K- Final State Final Selection and Background Selection Efficiency Cross Section for e+e- K+ K- K+ K- The (1020) K+ K- Intermediate State e+e- f0 Near Threshold The Charmonium Region Summary Acknowledgments References

0704.0633

d-wave superconductivity from electron-phonon interactions

d-wave superconductivity from electron-phonon interactions J.P.Hague Dept. of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH and Dept. of Physics, Loughborough University, Loughborough, LE11 3TU (Dated: 4th May 2005) I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon frequency regime of the quasi-2D Holstein model. I use an extended Migdal–Eliashberg theory which includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is unique close to half-filling. The order parameter undergoes a transition to s-wave superconductivity on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulomb pseudopotential on the superconductivity (such as is found in contemporary superconducting ma- terials like the cuprates), which results in the destruction of the s-wave states, while leaving the d-wave states unmodified. Published as: Phys. Rev. B 73, 060503(R) (2006) PACS numbers: 71.10.-w, 71.38.-k, 74.20.-z The discovery of high transition temperatures and a d-wave order parameter in the cuprate superconductors are remarkable results and have serious implications for the theory of superconductivity. The presence of large Coulomb interactions in the cuprates which have the po- tential to destroy conventional s-wave BCS states has prompted the search for new mechanisms that can give rise to superconductivity. However, electron-phonon me- diated superconductivity is still not well understood, es- pecialy in lower dimensional systems. In particular, the electron-phonon problem is particularly difficult at inter- mediate couplings with large phonon frequency (such as found in the cuprates) and the electron-phonon mech- anism cannot be fully ruled out. It is therefore of paramount importance to develop new theories to under- stand electron-phonon mediated superconductivity away from the BCS limit. The assumption that electron-phonon interactions can- not lead to high transition temperatures and unusual or- der parameters was made on the basis of calculations from BCS theory, which is a very-weak-coupling mean- field theory (although of course highly successful for pre-1980s superconductors)1. In the presence of strong Coulomb interaction, the BCS s-wave transition temper- ature is vastly reduced. However, the recent measure- ment of large couplings between electrons and the lat- tice in the cuprate superconductors means that exten- sions to the conventional theories of superconductivity are required2,3,4. In particular, low dimensionality, in- termediate dimensionless coupling constants of ∼ 1 and large and active phonon frequencies of ∼ 75meV mean that BCS or the more advanced Migdal–Eliashberg (ME) theory cannot be applied. In fact, the large coupling con- stant and a propensity for strong renormalization in 2D systems, indicate that the bare unrenormalized phonon frequency could be several times greater than the mea- sured 75 meV5. Here I apply the dynamical cluster approxima- tion (DCA) to introduce a fully self-consistent momentum-dependent self-energy to the electron-phonon problem5,6,7,8. Short ranged spatial fluctuations and low- est order vertex corrections are included, allowing the sequence of phonon absorption and emission to be re- ordered once. In particular, the theory used here is second order in the effective electron-electron coupling U = −g2/Mω20, which provides the correct weak coupling limit from small to large phonon frequencies18. In this paper, I include symmetry broken states in the anoma- lous self energy to investigate unconventional order pa- rameters such as d-wave. No assumptions are made in advance about the form of the order parameter. DCA6,8,9 is an extension to the dynamical mean-field theory for the study of low dimensional systems. To ap- ply the DCA, the Brillouin zone is divided into NC sub- zones within which the self-energy is assumed to be mo- mentum independent, and cluster Green functions are determined by averaging over the momentum states in each subzone. This leads to spatial fluctuations with characteristic range, N c . In this paper, Nc = 4 is used throughout. This puts an upper bound on the strength of the superconductivity, which is expected to be reduced in larger cluster sizes10. To examine superconducting states, DCA is extended within the Nambu formalism7,8. Green functions and self-energies are described by 2 × 2 matrices, with off diagonal terms relating to the super- conducting states. The self-consistent condition is: G(K, iωn) = Di(ǫ)(ζ(Ki, iωn)− ǫ) |ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2 F (K, iωn) = − Di(ǫ)φ(Ki, iωn) |ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2 where ζ(Ki, iωn) = iωn + µ−Σ(Ki, iωn), µ is the chem- ical potential, ωn are the Fermionic Matsubara frequen- cies, φ(K, iω) is the anomalous self energy and Σ(K, iω) is the normal self energy. G(K, iωn) must obey the lat- tice symmetry. In contrast, it is only |F (K, iωn)| which is constrained by this condition, since φ is squared in the denominator of Eqn. 1. Therefore the sign of φ http://arxiv.org/abs/0704.0633v1 FIG. 1: Diagrammatic representation of the current approx- imation. Series (a) represents the vertex-neglected theory which corresponds to the Migdal–Eliashberg approach, valid when the phonon energy ω0 and electron-phonon coupling U are small compared to the Fermi energy. Series (b) repre- sents additional diagrams for the vertex corrected theory. The phonon self energies are labeled with Π, and Σ denotes the electron self-energies. Lines represent the full electron Green function and wavy lines the full phonon Green function. can change. For instance, if the anomalous self energy has the rotational symmetry φ(π, 0) = −φ(0, π), the on-diagonal Green function, which represents the elec- tron propagation retains the correct lattice symmetry G(π, 0) = G(0, π). Therefore, only inversion symmetry is required of the anomalous Green function representing superconducting pairs and the anomalous self energy. Here I examine the Holstein model11 of electron- phonon interactions. It treats phonons as nuclei vibrat- ing in a time-averaged harmonic potential (representing the interactions between all nuclei), i.e. only one fre- quency ω0 is considered. The phonons couple to the local electron density via a momentum-independent coupling constant g11. H = − <ij>σ tc iσcjσ + iσ niσ(gri − µ) The first term in this Hamiltonian represents hopping of electrons between neighboring sites and has a dispersion ǫk = −2t i=1 cos(ki). The second term couples the lo- cal ion displacement, ri to the local electron density. The last term is the bare phonon Hamiltonian, i.e. a sim- ple harmonic oscillator. The creation and annihilation of electrons is represented by c i (ci), pi is the ion momen- tum and M the ion mass. The effective electron-electron interaction is, U(iωs) = ω2s + ω where, ωs = 2πsT , s is an integer and U = −g 2/Mω20 represents the magnitude of the effective electron- electron coupling. D = 2 with t = 0.25, resulting in a non-interacting band width W = 2. A small interpla- nar hopping t⊥ = 0.01 is included. This is necessary to stabilise superconductivity, which is not permitted in a pure 2D system12. Perturbation theory in the effective electron-electron interaction (Fig. 1) is applied to second order in U , us- ing a skeleton expansion. The electron self-energy has two terms, ΣME(ω,K) neglects vertex corrections (Fig. 1(a)), and ΣVC(ω,K) corresponds to the vertex corrected case (Fig. 1(b)). ΠME(ω,K) and ΠVC(ω,K) correspond to the equivalent phonon self energies. At large phonon frequencies, all second order diagrams including ΣV C are essential for the correct description of the weak coupling limit. The phonon propagator D(z,K) is calculated from, D(iωs,K) = ω2s + ω 0 −Π(iωs,K) and the Green function from equations 1 and 2. Σ = ΣME+ΣVC and Π = ΠME+ΠVC. Details of the transla- tion of the diagrams in Fig. 1 and the iteration procedure can be found in Ref. 7. Calculations are carried out along the Matsubara axis, with sufficient Matsubara points for an accurate calculation. The equations were iterated un- til the normal and anomalous self-energies converged to an accuracy of approximately 1 part in 103. Since the anomalous Green function is proportional to the anomalous self energy, initializing the problem with the non-interacting Green function leads to a non- superconducting (normal) state. A constant supercon- ducting field with d-wave symmetry was applied to the system to induce superconductivity. The external field was then completely removed. Iteration continued with- out the field until convergence. This solution was then used to initialize self-consistency for other similar val- ues of the parameters. The symmetry conditions used in Refs 5 and 7 have been relaxed to reflect the additional breaking of the anomalous lattice symmetry in the d-wave state. This does not affect the normal state Green func- tion, but does affect the anomalous state Green function. In Fig. 2, the anomalous self energy is examined for n = 1.0 (half-filling). The striking feature is that sta- ble d-wave superconductivity is found. This is mani- fested through a change in sign of the anomalous self energy, which is negative at the (π, 0) point and positive at the (0, π) point. The electron Green function (equa- tion 1) depends on φ2, so causality and lattice symmetry are maintained. Since the gap function φ(iωn)/Z(iωn) is directly proportional to φ(iωn), and Z(iωn,K(π,0)) = Z(iωn,K(0,π)), then the sign of the order parameter i.e. the sign of the superconducting gap changes under 90o rotation. Z(iωn) = 1− Σ(iωn)/iωn. Figure 3 shows the variation of superconducting pair- ing across the Brillouin zone. ns(k) = T n F (iωn,k). U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. The d-wave order can be seen very clearly. The largest anomalous densities are at the (π, 0) and (0, π) points, with a node situated at the (π/2, π/2) point and a sign change on 90o rotation. Pairing clearly occurs between electrons close to the Fermi surface. So far, the model has been analyzed at half filling. Figure 4 demonstrates the evolution of the order param- -0.025 -0.02 -0.015 -0.01 -0.005 0.005 0.01 0.015 0.02 0.025 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (0,0) (π,0) (0,π) (π,π) FIG. 2: Anomalous self-energy at half-filling. The anomalous self energy is real. It is clear that φ(π, 0) = −φ(0, π). This is characteristic of d-wave order. Similarly, the electron self energy has the correct lattice symmetry Σ(π, 0) = Σ(0, π), which was not imposed from the outset. The gap function is related to the anomalous self energy via φ(iωn)/Z(iωn). -0.25 0.25 -0.25 0.25 ns(k) FIG. 3: Variation of superconducting (anomalous) pairing density across the Brillouin zone. ns(k) = T F (iωn,k). U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. The d-wave order can be seen very clearly, with a change in sign on 90o rota- tion and a node situated at the (π/2, π/2) point. The largest anomalous (superconducting) densities are at the (π, 0) and (0, π) points. eter as the number of holes is first increased, and then decreased. The total magnitude of the anomalous den- sity, ns = |ns(Ki)| is examined. When the number of holes is increased, stable d-wave order persists to a filling of n = 1.18, while decreasing monotonically. At the critical point, there is a spontaneous transition to s- wave order. Starting from a high filling, and reducing the number of holes, there is a spontaneous transition from s to d-wave order at n = 1.04. There is therefore hys- teresis associated with the self-consistent solution. It is reassuring that the d-wave state can be induced without the need for the external field. As previously established, s-wave order does not exist at half-filling as a mainfesta- 0.02 0.04 0.06 0.08 1 1.05 1.1 1.15 1.2 1.25 1.3 External field FIG. 4: Hysteresis of the superconducting order parameters. |ns(Ki)|. Starting from a d-wave state at half- filling, increasing the chemical potential increases the filling and decreases the d-wave order. Eventually, at n = 1.18 the system changes to an s-wave state. On return from large filling, the s-wave superconductivity is persistent to a low filling of n = 1.04, before spontaneously reverting to a d- wave state. The system is highly susceptible to d-wave order, and application of a very small external superconducting field to an s-wave state results in a d-wave state. Note that d- and s-wave channels are coupled in the higher order theory, so the transition can take place spontaneously, unlike in the standard gap equations. tion of Hohenberg’s theorem7, so the computed d-wave order at half-filling is the ground state of the model. It is interesting that the d- and s-channels are able to co- exist, considering that the BCS channels are separate on a square lattice. This is due to the vertex corrections, since the self consistent equations are no longer linear in the gap function (the 1st order gap equation vanishes in the d-wave case, leaving 2nd order terms as the leading contribution). I finish with a brief discussion of Coulomb effects. In the Eliashberg equations, a Coulomb pseudopotential may be added to the theory as, φC = UCT F (iωn,K) (6) It is easy to see the effect of d-wave order on this term. Since the sign of the anomalous Green function is mod- ulated, the average effect of d-wave order is to nullify the Coulomb contribution to the anomalous self-energy (i.e. φCd = 0). This demonstrates that the d-wave state is stable to Coulomb perturbations, presumably be- cause the pairs are distance separated. In contrast, the s-wave state is not stable to Coulomb interaction, with a corresponding reduction of the transition temperature (TC = 0 for λ < µC). Thus, such a Coulomb filter selects the d-wave state (see e.g. Ref. 13). Since large local Coulomb repulsions are present in the cuprates (and in- deed most transition metal oxides), then this mechanism seems the most likely to remove the hysteresis. Without the Coulomb interactions, it is expected that the s-wave state will dominate for n > 1.04, since the anomalous order is larger. I note that a further consequence of strong Coulomb repulsion is antiferromagnetism close to half-filling. Typ- ically magnetic fluctuations act to suppress phonon medi- ated superconducting order. As such, one might expect a suppression of superconducting order close to half-filling, with a maximum away from half filling. The current the- ory could be extended to include additional anomalous Green functions related to antiferromagnetic order. This would lead to a 4x4 Green function matrix. A full anal- ysis of antiferromagnetism and the free energy will be carried out at a later date. a. Summary In this paper I have carried out simu- lations of the 2D Holstein model in the superconducting state. Vertex corrections and spatial fluctuations were included in the approximation for the self-energy. The anomalous self energy and superconducting order param- eter were calculated. Remarkably, stable superconduct- ing states with d-wave order were found at half-filling. d-wave states persist to n = 1.18, where the symmetry of the parameter changes to s-wave. Starting in the s- wave phase and reducing the filling, d-wave states spon- taneously appear at n = 1.04. The spontaneous appear- ance of d-wave states in a model of electron-phonon in- teractions is of particular interest, since it may negate the need for novel pairing mechanisms in the cuprates19. The inclusion of vertex corrections and spatial fluctua- tions was essential to the emergence of the d-wave states in the Holstein model, which indicates why BCS and ME calculations do not predict this phenomenon. For very weak coupling, the off diagonal Eliashberg self-energy has the form −UT Q,n F (iωn,Q)D0(iωs − iωn), so it is clear (for the same reasons as the Coulomb pseudopo- tential) that this diagram has no contribution in the d- wave phase (the weak coupling phonon propagator is mo- mentum independent for the Holstein model). Therefore, vertex corrections are the leading term in the weak cou- pling limit. Furthermore, I have discussed the inclusion of Coulomb states to lowest order, which act to desta- bilize the s-wave states, while leaving the d-wave states unchanged. Since the Coulomb pseudopotential has no effect then it is possible that electron-phonon interactions are the mechanism inducing d-wave states in real mate- rials such as the cuprates. The Coulomb filtering mech- anism works for p-wave symmetry and higher, so it is possible that electron-phonon interactions could explain many novel superconductors. Certainly, such a mecha- nism cannot be ruled out. The doping dependence of the order qualitatively matches that of La2−xSrxCuO4 (here order extends to x = 0.18, in the Cuprate to x = 0.3). Antiferromagnetism is only present in the cuprate very close to half filling (up to approx x = 0.02), and on a mean-field level does not interfere with the d-wave su- perconductivity at larger dopings. It has been determined experimentally that strong electron-phonon interactions and high phonon frequen- cies are clearly visible in the electron and phonon band structures of the cuprates, and are therefore an essential part of the physics3,4. Similar effects to those observed in the cuprates are seen in the electron and phonon band structures of the 2D Holstein model in the normal phase5. It is clearly of interest to determine whether other fea- tures and effects in the cuprate superconductors could be explained with electron-phonon interactions alone. b. Acknowledgments I thank the University of Le- icester for hospitality while carrying out this work. I thank E.M.L.Chung for useful discussions. I am currently supported under EPSRC grant no. EP/C518365/1. 1 J.Bardeen, L.N.Cooper, and J.R.Schrieffer, Phys. Rev. B 108, 1175 (1957). 2 G.M.Zhao, M.B.Hunt, H.Keller, and K.A.Müller, Nature 385, 236 (1997). 3 A.Lanzara, P.V.Bogdanov, X.J.Zhou, S.A.Kellar, D.L.Feng, E.D.Lu, T.Yoshida, H.Eisaki, A.Fujimori, K.Kishio, et al., Nature 412, 6846 (2001). 4 R.J.McQueeney, Y.Petrov, T.Egami, M.Yethiraj, G.Shirane, and Y.Endoh, Phys. Rev. Lett. 82, 628 (1999). 5 J.P.Hague, J. Phys. Condens. Matt 15, 2535 (2003). 6 M.H.Hettler, A.N.Tahvildar-Zadeh, M.Jarrell, T.Pruschke, and H.R.Krishnamurthy, Phys. Rev. B 58, R7475 (1998). 7 J.P.Hague, J. Phys.: Condens. Matter 17, 5663 (2005). 8 T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys 77, 1027 (2005). 9 M.H.Hettler, M.Mukherjee, M.Jarrell, and H.R.Krishnamurthy, Phys. Rev. B 61, 12739 (2000). 10 M.Jarrell, Th.Maier, C.Huscroft, and S.Moukouri, Phys. Rev. B 64, 195130 (2001). 11 T.Holstein, Ann. Phys. 8, 325 (1959). 12 P.C.Hohenberg, Phys. Rev. 158, 383 (1967). 13 J.F.Annett, Superconductivity, Superfluidity and Conden- sates (Oxford University Press, 2004). 14 C.Grimaldi, L.Pietronero, and S.Strässler, Phys. Rev. Lett. 75, 1158 (1995). 15 A.A.Abrikosov, Physica C 244, 243 (1995). 16 A.A.Abrikosov, Phys. Rev. B 52, R15738 (1995). 17 R. J. Birgeneau,and G. Shirane, in Physical Properties of High Temperature Superconductors I, edited by D. M.Ginsberg (World Scientific, Singapore, 1989). 18 I also note the extensions to Eliashberg theory carried out by Grimaldi et al.14. 19 On the basis of a screened electron-phonon interaction, Abrikosov claims to have found stable d-wave states in a BCS like theory15,16. However with an unscreened Holstein potential, the transition temperature it the d-wave chan- nel given by the standard theory is zero. Also, the assumed order parameter in his work does not clearly have d-wave symmetry.

0704.0634

A Finite Element framework for computation of protein normal modes and mechanical response

Microsoft Word - Bathe 2007.doc A Finite Element framework for computation of protein normal modes and mechanical response Mark Bathe Arnold Sommerfeld Zentrum für Theoretische Physik and Center for NanoScience Ludwig–Maximilians–Universität München Theresienstrasse 37, 80333 Munich, Germany Abstract A coarse-grained computational procedure based on the Finite Element Method is proposed to calculate the normal modes and mechanical response of proteins and their supramolecular assemblies. Motivated by the elastic network model, proteins are modeled as homogeneous isotropic elastic solids with volume defined by their solvent- excluded surface. The discretized Finite Element representation is obtained using a surface simplification algorithm that facilitates the generation of models of arbitrary prescribed spatial resolution. The procedure is applied to compute the normal modes of a mutant of T4 phage lysozyme and of filamentous actin, as well as the critical Euler buckling load of the latter when subject to axial compression. Results compare favorably with all-atom normal mode analysis, the Rotation Translation Blocks procedure, and experiment. The proposed methodology establishes a computational framework for the calculation of protein mechanical response that facilitates the incorporation of specific atomic-level interactions into the model, including aqueous-electrolyte-mediated electrostatic effects. The procedure is equally applicable to proteins with known atomic coordinates as it is to electron density maps of proteins, protein complexes, and supramolecular assemblies of unknown atomic structure. Introduction Equilibrium conformational fluctuations of proteins about their folded, native structure play an important role in their biological function.1-4 Three prominent approaches used to compute conformational fluctuations of proteins are, in order of increasing computational efficiency and decreasing modeling resolution, molecular dynamics (MD), all-atom normal mode analysis (NMA), and coarse-grained elastic NMA (eNMA). MD attempts to sample the equilibrium distribution of states in the vicinity of the native structure via time-integration of Newton’s equations of motion, typically modeling solvent explicitly.5 All-atom NMA assumes harmonic fluctuations about the native state in solving the free vibration problem for the protein while treating the solvent implicitly.2,3,6 Finally, eNMA employs a coarse-grained elastic description of the protein in which specific atomic interactions are replaced by a simple network of linear elastic springs, typically connecting Cα atoms within an arbitrary cut-off radius.7,8 Successively coarser and thus computationally more efficient eNMA descriptions are obtained by reducing the total number of interaction sites in the system.9-12 The idea of treating proteins as effective elastic media in calculating their normal modes dates back at least to Suezaki and Go.13 Despite their relative simplicity, elastic coarse-grained models have proven remarkably successful in calculating the slow, large length-scale vibrational modes of proteins and their supramolecular assemblies. As shown recently by Lu and Ma,14 their success may partially be attributed to the fact that biomolecular shape plays a dominant role in determining the lowest normal modes of proteins. Indeed, large length-scale modes naturally average over heterogeneous interactions present at atomic length-scales, thereby rendering elastic descriptions valid in this regime. Global structural averages such as backbone fluctuations and inter-residue correlations are in turn also successfully predicted because they are dominated by these low frequency modes. The success of eNMA motivates the current work, in which the elastic network model for proteins is cast in the framework of the well established Finite Element Method (FEM).15,16 In formulating the model, the protein is defined by its mass density, ρ, isotropic elastic modulus, E, and solvent-excluded surface (SES), which is obtained by rolling a water molecule-size probe-sphere over its van der Waals surface.17-20 As an initial exploration of the utility of the FEM in analyzing protein mechanical response, the normal modes of a mutant of T4 lysozyme and of F-actin are computed, as well as the critical Euler buckling load of F-actin when subject to axial compression. NMA results for T4 lysozyme are compared with all-atom NMA, the Rotation Translation Blocks (RTB) procedure,21,22 which treats residues as rigid but retains atomic-level interactions as modeled by the implicit solvent force-field EEF1,23 and experiment. Similar to eNMA, the proposed FE-based procedure offers several advantages over all-atom NMA, including the elimination of costly energy minimization that may distort the initial protein structure, direct applicability to x-ray data of proteins with unknown atomic structure,24-26 and a significant speed-up of the NMA itself due to a drastic reduction in the number of degrees of freedom simulated. Additionally, the FEM offers several distinct advantages over existing elastic network models that provide the primary motivation for the current work. Principal among these is the suitability of the FEM to calculate the mechanical response of proteins and their supramolecular assemblies to applied bending, buckling, and other generalized loading scenarios, which is needed to probe the structure-function relation of supramolecular assemblies such as viral capsids,27,28 microtubules,29,30 F-actin bundles,31,32 and molecular motors.33,34 Moreover, casting the coarse-grained elastic model in the framework of the FEM opens two important avenues of model refinement that are currently being pursued. First, the atomic Hessian can be projected onto the FE- space in order to incorporate atomic-level interactions into the model, thereby eliminating the a priori assumption of homogeneous isotropic elastic response. This idea is similar to the initial version of the Rotation Translation Blocks (RTB) procedure proposed in Durand et al.,35, as well as related works in modeling crystals.36,37 The incorporation of atomic-level interactions may be particularly important in modeling binding interfaces present between constituent monomers in supramolecular assemblies such as F-actin, MTs, and viral coat protein subunits, particularly near the onset of mechanical failure. Second, the FE-based protein model may be coupled directly to field calculations including the Poisson–Boltzmann Equation to model solvent-mediated electrostatic interactions38-40 and the Stokes Equations to model solvent-damping in dynamic response calculations.41,42 Methods The FEM is a mature field that is discussed in detail in references such as Bathe15 and Zienkiewicz and Taylor.16 Accordingly, the focus here is on its application to proteins and readers are kindly referred to the above-referenced books for details on its theoretical foundations. Generation of the FE model requires three steps: (1) definition and discretization of the protein volume; (2) definition of the local effective mass density and constitutive behavior of the protein; and (3) application of boundary conditions such as displacement- or force-based loading. The protein volume is defined by its bounding SES, which is also called the Richards Molecular Surface or simply the Molecular Surface. This surface is defined by the closest point of contact of a solvent-sized probe-sphere that is rolled over the van der Waals surface of the protein, which defines the molecular volume that is never penetrated by any part of the solvent probe-sphere.17-19 The SES is computed using MSMS ver. 2.6.1, which generates a high density triangulated approximation (one triangular vertex per Å2) to the exact SES.20 The MSMS-discretized SES is subsequently decimated to arbitrary prescribed spatial resolution using the surface simplification algorithm QSLIM.43-45 The QSLIM algorithm employs iterative vertex-pair contraction together with a quadric error metric to retain a near-optimal representation of the original surface while reducing the total number of faces by an arbitrary, user-specified amount.43 The protein volume that is bounded by the closed SES is subsequently discretized with 3D tetrahedral finite elements via automatic mesh generation using the commercial Finite Element program ADINA ver. 8.4 (Watertown, MA, USA). Application of the proposed FE-based procedure directly to x-ray data would require definition of the molecular volume from the electron density map using Voronoi tessellation or a similar procedure, as proposed by Wriggers et al.,46, and performed by Ming et al.24 The protein constitutive response is modeled using the standard Hooke’s law, which treats the protein as a homogeneous, isotropic, elastic continuum with Young’s modulus E and Poisson ratio ν.47 While this is conceptually similar to elastic-network based models, it is rigorously distinct: Elastic network models typically connect Cα atoms by springs of equal stiffness, which results in general in a locally anisotropic and inhomogeneous elastic material with length-scale dependent mechanical properties. In contrast, the FE-model defined here treats the protein as strictly homogeneous, with an isotropic elastic material response that is length-scale invariant. The mass density of the protein is taken to be homogeneous, although it could equally be defined as a spatially- varying function from the underlying atomic constitution or from electron density data. Finally, arbitrary boundary conditions consisting of displacement- or force-based loading may be applied to the molecule, modeling the effects of the protein environment. In the current application, the free vibration problem is solved for T4 lysozyme and F- actin in the absence of any boundary condition and the linearized buckling problem for F- actin is solved by applying co-axial compressive point loads to the ends of the molecule. Given the protein volume, constitutive behavior, and boundary conditions, the FEM uses numerical volume-integration to derive a set of algebraic equations that is linear in the finite element nodal displacement degrees of freedom, u , + =Mu Ku R (1) where M is the diagonal mass-matrix, K is the elastic stiffness matrix, and R is a forcing vector that results from natural (force-based) boundary conditions.15 In the case of the free vibration problem relevant to the NMA of proteins, 0=R . Substitution of the oscillatory solution, cos( )tω γ= +u y , into the free-vibration form of Eq. (1) results in the generalized eigenvalue problem, 2 0ω− =Ky My (2) which after definition of the eigenvalues, 2:λ ω= , may be written in secular form, det 0λ− =K M . (3) Various efficient FE procedures exist to obtain the solution to the generalized eigenvalue problem, yielding the eigenvalues and eigenvectors, ( , )i iλ y . In the present application an accelerated subspace-iteration method15,48 is used for T4 lysozyme and F-actin. The substructure synthesis procedure49 commonly available in structural mechanics FE programs could also be applied to calculate the normal modes of F-actin, as recently proposed by Ming et al.50 The eigenvectors corresponding to the FE nodal degrees of freedom are linearly interpolated to the Cα positions given by the atomic coordinates that were used to define the FE model. Standard equilibrium thermal averages may then be computed in the standard way, including the fluctuation of Cα atom i due to mode k, 2 2 /ik B ik k ir k Ta mλΔ = , the total fluctuation of Cα atom i due to all modes, i ikk r rΔ = Δ∑ , correlations in positional fluctuations of Cα atoms i and j, /ij i j i i j jC r r r r r r= Δ ⋅Δ Δ ⋅Δ Δ ⋅Δ , where ( )i j B ik jk k i jkr r k T a a m mλΔ ⋅Δ = ∑ , and the overlap, ijR , between normal modes i and j, defined by the inner product of the modes, ij i j i jR = ⋅a a a a , where (1 1)ijR≤ ≤ − . 6 As with elastic network models, the protein stiffness-scale (E) is unknown. Accordingly, the acoustic wave speed, E ρ , which is the relevant physical unit in the free-vibration problem, is adjusted to best-fit the pertinent Cα fluctuation data, which is either experimental or that from the all-atom NMA. In the case of F-actin, the average mass density, ρ, is set explicitly and the Young’s modulus is determined by matching its stretching stiffness to experiment,51 as also performed by ben-Avraham and Tirion52 and described in more detail below. The Poisson ratio is taken to be 0.3 for T4 lysozyme and F-actin, which is typical of crystalline solids. While the choice of 0.3ν = has, to the best of the author’s knowledge, no rigorous justification, it is noted that its precise value does not affect the computed results within the range of (0.3 0.5)ν≤ ≤ . This is typical of response calculations such as those performed here, in which material compressibility does not play an important role. Two important considerations in generating the FE model are the choice of the probe-sphere radius used to define the protein volume and the degree of surface simplification performed. Regarding the choice of probe-sphere radius, two approaches were deliberated here. In the first, the probe-sphere radius is treated as an adjustable parameter, akin to the cut-off radius used in elastic network models. In this case, as the radius of the probe-sphere is increased, protein cavities in which solvent would normally be present become part of the effective elastic medium constituting the protein. Accordingly, the shape of the protein becomes a function of the probe-sphere radius, which will affect its mechanical response. In the second approach, the probe-sphere radius is treated as a fixed, physically-based parameter that is approximately equal to the size of a water molecule, as in electrostatic field calculations.53-55 The homogeneous elastic medium of the protein is then strictly applied to those volumetric regions in which dense intramolecular packing involving close-ranged van der Waals, hydrogen-bond, and bonded interactions are present, and the molecular surface is a well-defined physical feature of the protein. The latter approach was taken here in order to retain the physical connection to atomic packing in solids. An important theoretical property of the FEM is that it guarantees convergence to the exact solution of the underlying mathematical model as the FE mesh is refined, where the mathematical model is defined by the protein’s analytical SES, constitutive behavior, and boundary conditions.15 Thus, any normal mode or mechanical response calculation performed using the proposed FE-based procedure should in principle systematically refine the discretized representation in order to ensure convergence of the computed model property to its exact result. In practice, however, the permissible degree of surface simplification using QSLIM or similar algorithm will depend on the sensitivity of the computed observable to details of molecular shape, which must be evaluated on a case- by-case basis, as addressed below for T4 lysozyme and F-actin. T4 lysozyme The initial structure of the 164 residue (18.7 kDa) mutant T4 phage lysozyme is taken from Matsumura et al.,56 (Protein Data Bank ID 3LZM).57 CHARMM ver. 33a158 is used with the implicit solvation model EEF123 to build in coordinates missing in the crystal structure and to perform energy minimization and NMA. Steepest descent minimization followed by adopted-basis Newton–Raphson minimization is performed in the presence of successively reduced harmonic constraints on backbone atoms to achieve a final root-mean-square (RMS) energy gradient of 5×10–4 kcal/(mol Å) with corresponding RMS deviation between the x-ray and energy-minimized structures of 1.3 Å (Fig. 1a). All-atom (ATM) and RTB NMA21 are used as implemented in CHARMM,22 using one-block per residue for the RTB calculations. Fig. 1 T4 lysozyme (a) crystal structure (Protein Data Bank ID 3lmz), (b) MSMS- triangulated SES, and (c) QSLIM-decimated SES used for the FE computation. Atomic structure rendered with VMD ver. 1.8.559 and triangulated models rendered with ADINA ver. 8.4. To define the FE model, MSMS is used to compute the SES of the energy- minimized structure of T4 lysozyme using the MSMS-default 1.5 Å radius probe ignoring hydrogens. As noted previously, the FE model may be defined directly from the atomic structure without initial energy minimization, however, the energy-minimized structure is used here to be consistent with the ATM model, which requires minimization. MSMS generates a triangulated approximation to the analytical SES that consists of 17,300 triangular faces (Fig. 1b). This model is decimated using QSLIM to a reduced model consisting of 2,000 faces (Fig. 1c). The decimated surface-mesh is read into ADINA ver. 8.4 and used as a template to generate 6,843 4-node tetrahedral finite elements consisting of 1,627 nodes. Calculation of the 100 lowest non-rigid-body modes using an accelerated subspace-iteration method15,48 required 27 MB of RAM and about 10 seconds on a 2.1 GHz Intel Core2Duo processor. Refining drastically the surface representation from 2,000 faces to 17,300 faces (and associated volume discretization) or computing more than 100 normal modes did not alter the Cα fluctuations significantly. F-actin The atomic structure of F-actin (52 protomers, 2.2 MDa molecular weight) is generated using FilaSitus ver. 1.460 based on the Holmes fiber model61 and the structure of G-actin:ADP:Ca2+ from the actin-gelsolin segment-1 complex.62,63 This structure of F- actin-ADP models the filament in its “young” state when the DNase I binding region of subdomain 2 of G-actin (residues 40–48) is in its disordered loop conformation as opposed to its ordered α-helix conformation.63-65 Importantly, in its disordered loop conformation this region forms intramolecular contacts in F-actin that stabilize the filament and have direct consequences on its mechanical properties.66-68 Calculation of the SES using MSMS and a 3 Å radius probea results in a model with 1,248,038 triangular faces, which is subsequently decimated in several seconds using QSLIM to a reduced model with 40,000 triangular faces. The decimated surface- mesh is read into ADINA ver. 8.4 and used as a template to generate 134,883 4-node tetrahedral finite elements consisting of 31,881 nodes (Fig. 2b). Planar axial stretching is used to determine the effective Young’s modulus of F-actin, E = 2.69 GPa, by fitting its computed value to its experimentally-measured value in the absence of tropomyosin, 43.7 nN.51 The homogeneous mass density, ρ = 1,170 kg/m3, is based on the 42 kDa molecular weight of G-actin and the calculated molecular volume of F-actin, which is equal to 3.1×106 Å3 for the 52-mer considered. Normal mode analysis using the accelerated subspace-iteration procedure in ADINA requires 22 MB and less than 10 seconds to calculate the lowest 10 modes on a 2.1 GHz Intel Core2Duo processor. To test a Use of the MSMS-default 1.5 Å radius probe resulted in QSLIM-decimated surface models that were poorly formed with multiple intersecting and degenerate triangles due to re-entrant surfaces of F-actin. Use of a 3 Å radius probe resolved this problem of SES-representation and is not expected to affect significantly the large length-scale normal modes of F-actin, which has relatively large minor and major diameters of ~40 and 80 Å, respectively. convergence of the FE solution to the exact solution, the FE mesh was coarsened considerably to a model consisting of only 7,558 4-node tetrahedral volume elements (4,000 surface triangles), for which the lowest four eigen-frequencies increased by at most 15% with respect to the more detailed model. Further mesh refinement beyond 40,000 surface elements was precluded by the problematic surface mesh generated by the proposed procedure, in which substantial element intersections were present. Fig. 2 (a) Atomic structure of the 52-monomer F-actin filament analyzed and (b) the triangulated SES used to define the FE model. Atomic structure is rendered with VMD ver. 1.8.5 59 and the FEM model rendered using ADINA ver. 8.4. Results T4 lysozyme Equilibrium thermal fluctuations of Cα atoms aid in understanding protein function as mediated by local conformational flexibility and provide a first quantitative test for the proposed coarse-grained procedure. Experimental fluctuations are related to the experimental temperature- or B-factor by, 2 28 / 3i iB rπ= Δ , where irΔ is the mean-squared fluctuation of atom i. While both coarse-grained models capture well the overall experimental variation in flexibility of T4 lysozyme (Fig. 3a and Table 1),56 local differences are evident in disordered loop regions where conformational flexibility is overestimated significantly by both the RTB and FEM procedures (e.g., residue numbers 35–40). Comparison with the all-atom model indicates that these discrepancies are inherent to the protein structure, however, and not artifacts of the RTB and FEM procedures (Fig. 3b). Indeed, Cα fluctuations calculated with the RTB and FEM models correlate notably better with fluctuations calculated with the all-atom model than with experiment (Table 1). residue index RMSF (Å) 0 20 40 60 80 100 120 140 160 residue index RMSF (Å) 0 20 40 60 80 100 120 140 160 (a) (b)residue index residue index Fig. 3. Coarse-grained RMSF of Cα atoms in T4 lysozyme compared with (a) experiment and (b) all-atom NMA. 100 modes are used to compute the all-atom, RTB, and FEM fluctuations. Correlation coefficients provided in Table 1. Table 1 Correlation coefficients corresponding to Cα atom RMSF in Figure 3. Experiment ATM RTB 0.73 0.95 FEM 0.68 0.89 Inter-residue spatial correlations measured at Cα atoms provide additional insight into protein function,69,70 as well as a further test of the proposed coarse-grained procedure. Interestingly, the RTB and FEM procedures provide similar information with respect to the all-atom model, as measured over either the lowest 10 or 100 modes (Fig. 4 top and bottom, respectively). The fact that the correlation maps are largely determined with as few as ten modes reconfirms numerous previous findings that the lowest modes of proteins dominate their free vibration response.14,71,72 The similarity in the FEM and ATM correlation maps provides additional evidence that T4 lysozyme behaves remarkably similar to a homogeneous isotropic elastic solid in free vibration. Fig. 4 T4 lysozyme inter-Cα correlations computed using (top) 10 modes and (bottom) 100 modes for the (a) ATM, (b) RTB, and (c) FEM models. The lowest four mode shapes computed using the FEM may be projected onto the ground-state (energy-minimized) structure of T4 lysozyme to visualize their nature (Fig. 5). Similar to the native hen egg lysozyme, the lowest mode is a hinge-bending mode,1,73 whereas the three higher modes are a combination of hinge- and twist-deformations. Quantitative comparison between the coarse-grained and all-atom models is made in Table 2 for the lowest four mode shapes, and the lowest 200 frequencies in Figure 6. Fig. 5 Lowest four eigenmodes computed by the FEM superimposed on the minimized structure of T4 lysozyme. Overlap with the all-atom model is given in Table 2. Images rendered using VMD ver. 1.8.5 59. Table 2 Overlap of coarse-grained model and all-atom normal modes as measured at Cα positions. Mode 1 Mode 2 Mode 3 Mode 4 RTB 0.97 0.93 0.82 0.28 FEM 0.91 0.86 0.76 0.71 The modal frequency distributions provide a final quantitative evaluation of the FEM and RTB approach for T4 lysozyme (Fig. 6). While the overall correlation between the FEM and all-atom frequencies is reasonable, particularly for low mode-numbers, the FEM tends to underestimate the “exact” frequency computed using the all-atom model at high mode-numbers. This suggests that the FEM models the protein as overly compliant in this regime, which is to be expected because higher modes excite shorter wavelength, stiffer degrees of freedom in the all-atom protein resulting from chain connectivity, whereas the elastic solid approximation assumes a compliance that is length-scale invariant. Backbone Cα fluctuations as well as Cα correlations are apparently unaffected by this approximation because the low modes dominate these observables. Interestingly, the opposite tendency was observed by Tama et al.,21 for the RTB-approach with successively larger blocks. This is also to be expected because the assumption of rigid blocks in the protein renders the structure overly stiff on short length scales (high frequency modes), and the length-scale at which this deviation from the all-atom model becomes significant increases with increasing block-size. ωCG [cm−1] ωATM [cm−1] 0 5 10 15 20 25 30 35 40 ωATM [cm–1] ωCG [cm–1] Fig. 6 Correlation between coarse-grained (CG) and all-atom (ATM) model frequencies for the lowest 200 modes. F-actin F-actin is a highly dynamic biopolymer with a considerable degree of internal plasticity in the state of tilt and twist of its constituent protomers, which depends on the bound nucleotide-state (ATP/ADP), bound actin-binding protein, and solvent conditions.74-78 Additionally, the bending stiffness of F-actin has been shown to increase by a factor of two in the presence of phalloidin, by 50% in the F-ADP-P versus F-ADP state, and to be regulated by tropomyosin in a Ca2+-dependent fashion.66 Thus, any modeling attempt to predict the mechanical properties of F-actin and investigate their relation to its detailed internal structure and composition must consider such variations. Modeling attempts to investigate the structure-function relation of F-actin include an early study by ben-Avraham and Tirion,68 who treated G-actin monomers as internally rigid and connected to their nearest neighbor monomers by compliant springs, a more recent study by Ming et al.,79 in which conventional eNMA is used together with substructure-synthesis to calculate the large wavelength normal modes of a micron-long F-actin molecule, and most recently an all-atom MD study by Chu and Voth,67 who found that the loop-helix transition of the DNase I binding region of subdomain 2 of G-actin plays a central role in respectively stabilizing-destabilizing F-actin by disrupting inter- monomer interactions. Chu and Voth67 also calculated the apparent persistence length of F-actin and found that the loop-to-helix transition between the ATP- and ADP-bound states accounted for the approximately 50% decrease in associated bending stiffness observed experimentally.66 The normal modes of F-actin (52-mer, 0.14 μm length) computed here in free planar-vibration yield four bending modes as the lowest modes (Fig. 7). Association of F- actin with a homogeneous elastic rod in free vibration80 results in an apparent bending stiffness, κ = 6.8×10–26 Nm2, for the lowest mode, which is near the upper limit of bending stiffness typically reported experimentally.66,81,82 Subjecting F-actin to an axially compressive load and performing a linearized buckling analysis yields the lowest critical Euler buckling load, Pcrit = 33 pN. Association of the filament again with a homogeneous elastic Euler–Bernoulli beam yields the effective bending stiffness, 2 2 26 2/ 6.9 10 NmcritP Lκ π −= = × , which is similar to the bending stiffness calculated from the lowest bending mode because that mode of deformation is the same as the lowest Euler buckling mode. Mode 1 Mode 4 Mode 3 Mode 2 Fig. 7 Four lowest free vibration modes of F-actin (52-mer, 0.144 μm length) in planar deformation. The corresponding angular frequencies are, 0.18×10–2, 0.48×10–2, 0.92×10– 2, and 0.16×10–1 rad/psec. The bending stiffness calculated here for F-actin is consistent with experimental measurements of the ATP-bound-state in which the DNase binding region (residues 40– 48) in subdomain 2 of G-actin is in its disordered loop conformation, thereby stabilizing inter-monomer interactions.66,67 While this is to be expected given the structure of G- actin:ADP:Ca2+ employed, in which the DNase binding region is also in its disordered loop conformation, a similar coarse-grained analysis of F-actin must be performed in which the DNase binding region of G-actin is in its ordered α-helical structure. Only then may it be stated definitively whether the observed mechanical behavior is due solely to this detailed structural difference or to some other source, such as a lack of modeling resolution. While a more detailed investigation of this type is of direct interest in evaluating the full utility of the proposed procedure, it is also of interest fundamentally to investigate the respective roles of molecular shape versus molecular interactions on determining the mechanical properties of supramolecular assemblies such as F-actin, MTs, and viral capsids. In particular, an intriguing hypothesis is that mechanical response is determined solely by molecular shape, in which case the mechanical properties of supramolecular assemblies would be robust to amino acid mutations that do not alter molecular shape. A competing hypothesis is that mechanical response is sensitive to both molecular shape and detailed molecular interactions, in which case amino acid mutations would be more tightly constrained. In either case, investigation of the respective roles of molecular shape versus specific interactions on protein mechanics clearly requires that all-atom models be considered, either directly or via incorporation into coarse-grained models. Such investigations are currently underway and are expected to provide fundamental insight into the origin and robustness of the mechanical properties of supramolecular assemblies. Concluding discussion A coarse-grained FE-based procedure is proposed to compute the normal modes and mechanical response of proteins and their supramolecular assemblies. The procedure takes as input the atomic structure to define uniquely the volume associated with the SES, mass density, and elastic stiffness of the protein. The initial, high resolution SES discretized at atomic resolution is simplified using a quadric simplification algorithm to obtain a molecular surface representation of arbitrary prescribed spatial resolution. While the proposed procedure is applied to proteins with known atomic structure, the molecular volume could equally be defined from electron density data, rendering the procedure applicable to a broad class of biomolecules and biomolecular complexes for which only a rough approximation to the molecular volume is known. As with existing coarse-grained elastic network models, energy minimization is not required prior to the NMA because the initial structure is assumed to be the ground-state structure. Ongoing development of the proposed procedure is directed towards three areas of improvement. First, the atomic-based Hessian from all-atom force-fields such as CHARMM58 will be projected onto the FE-space such that the model optimally converges to the “exact” all-atom solution as the FE mesh is refined to atomic length- scales. Such a procedure will enable the systematic coarsening of protein structure and interactions without the a priori assumption of elastic response. Indeed, an intriguing and as of yet unresolved question regards the relative effects of molecular shape versus specific molecular interactions on the mechanical response of supramolecular assemblies such as F-actin, MTs, and viral capsids. Second, the Poisson–Boltzmann equation used to model aqueous electrolyte-mediated electrostatic interactions in proteins may be coupled directly to the elastic-based FE model, so that it may be included in computations of normal modes and mechanical response. Langevin dynamics may also be incorporated into the model by coupling the protein-domain to the Stokes equations to model solvent damping.41,83 Finally, the proposed surface discretization and simplification procedure requires improvement because it often results in surface meshes with intersecting or degenerate triangles, as encountered here for F-actin. The utility of the proposed FE-based procedure is explored here for one specific globular protein and supramolecular assembly, namely T4 lysozyme and F-actin. Clearly, in order to evaluate the utility of the proposed procedure thoroughly, a set of proteins of drastically varying structure must be analyzed, as well as additional supramolecular assemblies. Additional response variables and the effects of internal structural variations of the molecules examined should also be investigated. Notwithstanding these additional analyses and the foregoing model improvements, the current communication establishes an effective theoretical framework for the computation of the normal modes and generalized mechanical response of proteins and their supramolecular assemblies based on the elastic medium theory of proteins. Acknowledgements Discussions with Marco Cecchini, Martin Karplus, Klaus–Jürgen Bathe, and Michael Garland are gratefully acknowledged, as is funding from the Alexander von Humboldt Foundation in the form of a post-doctoral fellowship. The author additionally thanks Michael Sanner for bringing QSLIM to his attention. References 1. Levitt M, Sander C, Stern PS. Protein Normal-mode Dynamics: Trypsin Inhibitor, Crambin, Ribonuclease and Lysozyme. Journal of Molecular Biology 1985;181(3):423-447. 2. Brooks B, Karplus M. Harmonic dynamics of proteins - Normal Modes and fluctuations in bovine pancreatic trypsin inhibitor. Proceedings of the National Academy of Sciences of the United States of America 1983;80(21):6571-6575. 3. Go N, Noguti T, Nishikawa T. Dynamics of a small globular protein in terms of low frequency vibrational modes. Proceedings of the National Academy of Sciences of the United States of America 1983;80(12):3696-3700. 4. Bruccoleri RE, Karplus M, McCammon JA. 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0704.0635

Reply to 'Comment on 'Heavy element production in inhomogeneous big bang nucleosynthesis''

Reply to ’Comment on ’Heavy element production in inhomogeneous big bang nucleosynthesis” Shunji Matsuura Department of Physics, School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Shin-ichirou Fujimoto Department of Electronic Control, Kumamoto National College of Technology, Kumamoto 861-1102, Japan Masa-aki Hashimoto Department of Physics, School of Sciences, Kyushu University, Fukuoka 810-8560, Japan Katsuhiko Sato Department of Physics, School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan and Research Center for the Early Universe, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan This is a reply report to [1]. We studied heavy element production in the high baryon density region in the early universe[3]. However it is claimed by [1] that a small scale but high baryon density region contradicts observations for the light element abundance or, in order not to contradict to the observations the high density region must be so small that it cannot affect the present heavy element abundance. In this paper we study big bang nucleosynthesis in the high baryon density region and show that in certain parameter spaces it is possible to produce enough amount of the heavy element without contradiction to cosmic microwave background and light element observations. PACS numbers: 26.35.+c, 98.80.Ft, 13.60.Rj I. INTRODUCTION In a standard scenario, big bang nucleosynthesis (BBN) can produce only light elements, up to 7Li, and all heavy elements have been synthesized in stars. However, many phase transitions in the early universe could have printed their trace in a non-standard way. For example, some baryogenesis models[2] predict very high baryon density islands in ordinary low density backgrounds. In the previous paper[3], we studied heavy element production in inhomogeneous BBN from this point of view. 1 However we limited ourselves to the heavy element abundance and did not discuss about the light element abundance and consistency with observations. This is because we assumed that high baryon density region is very local and do not affect the global light element abundance. In [1], Rauscher pointed out that in order not to contradict to observations, the high baryon density region must be very small and cannot affect the present heavy element abundance. In this paper, we show that there is a parameter region in which the heavy element can be produced enough to affect observation while keeping the light element abundance consistent with observations. We consider that the disagreement between Rauscher’s opinion and our opinion comes from two points. One is that we are looking at some parameter regions in which neutrons in high baryon density do not diffuse so much as to cause disaster in standard BBN. We would like to emphasize this point. The other is that the relevant quantity is not the spatial size of the high baryon density region but the amount of baryon in high density regions. We will discuss the following issues: In section II, we discuss the light element abundance in the homogeneous high baryon density region and after mixing the high and the low baryon density region. In section III, we study the heavy element(Ru,Mo) abundance in high and averaged baryon density and show that heavy elements can be produced enough without contradicting the light element observation. In section IV, we briefly comment on the diffusion scale of the high baryon density region. 1 For previous works on the inhomogeneous big bang nucleosynthesis, see [4, 5]. Heavy elements production is also mentioned in [6]. http://arxiv.org/abs/0704.0635v3 II. LIGHT ELEMENT ABUNDANCE A. Homogeneous BBN We calculate homogeneous BBN with various values of η(baryon photon ratio). In Table.I and II, we show the numerical result of the mass fraction and the number fraction of each light element for η = 10−3 and 3.162× 10−10. η = 10−3 name mass fraction number fraction H 5.814 × 10−1 8.475 × 10−1 4He 4.185 × 10−1 1.525 × 10−1 3He 4.842 × 10−13 1.614 × 10−13 7Li+7Be 1.559 × 10−12 2.227 × 10−13 D 1.577 × 10−22 7.883 × 10−23 Table. I: The mass and the number fractions of light elements for the homogeneous BBN with η = 10−3 η = 3.162 × 10−10 name mass fraction number fraction H 7.58 × 10−1 9.26× 10−1 4He 2.419 × 10−1 7.39× 10−2 3He 4.299 × 10−5 1.433 × 10−5 7Li + 7Be 8.239 × 10−10 1.177 × 10−10 D 1.345 × 10−4 6.723 × 10−5 Table. II: The mass and the number fractions of light elements for the homogeneous BBN with η = 3.162 × 10−10 As baryon density becomes higher, more protons and neutrons are bounded to form 4He. At η = 10−3, most of the final product of 7Li comes from 7Be which decays to 7Li after BBN. Details on light element production for various η can also be found in [7]. In this paper we almost concentrate on a case in which the high baryon density region has η = 10−3. We expect that compared to η ≥ 10−3, the profile of the abundance for η = 10−3 is more different from standard BBN because most of the light element abundances change monotonically with respect to η and if this case does not contradict to observations, other cases would also be consistent. Briefly, the amount of H decreases and 4He increases monotonically as η become larger. The number fraction of D is less than 10−20 for η greater than 10−7. For 3He, the number fraction drastically decreases around η = 10−4 down to O(10−13), and for 7Li, the number fraction increases until η = 10−6 and drastically decreases for a larger value of η. In the following sections, we will see that this non-standard setup does not strongly contradict to the observations. For simplicity we ignore the diffusion effect before and during BBN, and after BBN both high and low baryon density regions are completely mixed. Detailed analysis such as the case in which the high baryon density region doesn’t completely mixed, or taking into account diffusion effects are left for future work. B. Parameters and Basic equations In this section, we summarize the relations among parameters. Notations : n, nH , nL are averaged, high, and low baryon number density. fH , fL are the volume fractions of the high and the low baryon density region. yi, y i , y i are the mass fractions of each element (i) in averaged-, high- and low-density regions. The basic relations are fH + fL = 1 (1) fHnH + fLnL = n (2) yHi f HnH + yLi f LnL = yin. (3) Under the assumption that the temperature of the universe is homogeneous, the above equation can be written as fHηH + fLηL = η (4) yHi f HηH + yLi f LηL = yiη (5) where η = n ,ηH,L = n Conventional parameters for inhomogeneous BBN are η, f and density ratio R = n . Here we use a different combination of parameters. Relevant values for the abundance analysis are products fH,L × ηH,L and ηH,L. fH,Lv × η H,L determines the amount of baryon from high- and low- density regions. ηH,L determines the mass fraction of each species of nuclei. For convenience, we write the ratio of baryon number contribution from high density region as a, i.e., fHηH : fLηL = a : (1 − a). There are 5 parameters(nH,L, n and fH,L) and 2 constraints (Eq.(1) and Eq.(2)). We calculate the light element abundance for various values of ηH,L. η can also take any value, but in order not to contradict observational constraints, we choose η from 3.162 × 10−10 to 10−9. a is determined by Eq(4). The aim of the analysis in this section is not to find parameter regions which precisely agree with the observational light element abundance and η from CMB. Our model is too simple to determine the constraints to parameters. For example, we completely ignore the diffusion effect before and during BBN. Instead we see that at least our analysis in previous paper is physically reasonable. C. Theoretical predictions and observations of light elements We consider the cases of ηH = 10−3 and ηL = 3.162 × 10−10. The mass fractions of and H and 3He in the high density region are 0.5814 and 4.842 × 10−13, respectively, while those in the low density region are 0.758 and 4.299× 10−5. From Eq.(5), we have fHηHyH3He + f LηLyL3He = ηy3He (6) 4.842× 10−13 × a+ 4.299× 10−5 × (1− a) = y3He (7) fHηHyH + fLηLyL = ηyH (8) 0.5814× a+ 0.758× (1− a) = yH. (9) We can calculate an averaged value of the abundance ratio of 3He to H as 4.842× 10−13 × a+ 4.299× 10−5 × (1− a) 0.5814× a+ 0.758× (1 − a) . (10) where a is related to η as η − ηL ηH − ηL η − 3.162× 10−10 10−3 − 3.162× 10−10 η − 3.162× 10−10 . (13) Here a varies from 0 to 0.9 for reasonable values of η, or 3.162× 10−10 − 10−9. Similarly, for ηH = 10−3 the number fractions are 1.577× 10−22 × a+ 1.345× 10−4 × (1− a) 0.5814× a+ 0.758× (1 − a) 0.0e+00 2.0e-05 4.0e-05 6.0e-05 8.0e-05 1.0e-04 1e-97e-105e-10 Fig. 1: Averaged ratio of D to H,(D/H) vs η 0.0e+00 5.0e-06 1.0e-05 1.5e-05 2.0e-05 1e-97e-105e-10 Fig. 2: Same as Fig.1 but for (3He/H) 0.0e+00 5.0e-11 1.0e-10 1.5e-10 2.0e-10 1e-97e-105e-10 Fig. 3: Same as Fig.1 but for (7Li/H) 1.559× 10−12 × a+ 8.239× 10−10 × (1− a) 0.5814× a+ 0.758× (1− a) . (15) Fig.1,2 and 3 represent the averaged abundance ratio, (D/H), (3He/H) and (7Li/H) respectively. We can see that the light element abundance is the same order around η ∼ 5 × 10−10 − 10−9 as observations [8, 9, 10, 11, 12, 13, 14, 15]. )obs = (1.5− 6.7)× 10 −5 (16) )obs = (0.59− 4.1)× 10 −10. (17) We do not discuss detail about diffusion here. But at least above result suggest that our analysis is not beside the point. III. THEORETICAL PREDICTIONS AND OBSERVATIONS OF HEAVY ELEMENTS (92,94MO, 96,98RU) The same analysis can be applied for heavy elements such as 92Mo, 94Mo, 96Ru and 98Ru. We are interested in these elements because in many models of supernovae nucleosynthesis, these p-nuclei are less produced. We will see that some amount of these heavy elements can be synthesized in BBN. η = 10−3 name mass fraction H 5.814 × 10−1 4He 4.185 × 10−1 92Mo 1.835 × 10−5 94Mo 4.1145 × 10−6 96Ru 1.0789 × 10−5 98Ru 1.0362 × 10−5 Table. III: The mass fractions of nuclei for homogeneous BBN with η = 10−3 From Table.III, we can derive the expected value of these elements. 1.835× 10−5 × a 0.5814× a+ 0.758× (1 − a) 4.1145× 10−6 × a 0.5814× a+ 0.758× (1 − a) 1.0789× 10−5 × a 0.5814× a+ 0.758× (1− a) 1.0362× 10−5 × a 0.5814× a+ 0.758× (1− a) . (21) We plot expected value of these quantities in Fig.4. These values should be compared with the solar abundance(Table.IV)[17]. 0.0e+00 5.0e-08 1.0e-07 1.5e-07 2.0e-07 2.5e-07 1e-97e-105e-10 Mo/H) Mo/H) Ru/H) Ru/H) Fig. 4: (92Mo/H),(94Mo/H),(96Ru/H) and (98Ru/H) vs η. Red, green, blue and pink lines represent the ratio (92Mo/H),(94Mo/H),(96Ru/H),(98Ru/H) respectively. name number fraction ratio to H H 7.057280 × 10−1 1 92Mo 8.796560 × 10−10 1.2465 × 10−9 94Mo 5.611420 × 10−10 7.9512 × 10−10 96Ru 2.501160 × 10−10 3.5441 × 10−10 98Ru 8.676150 × 10−11 1.2294 × 10−10 Table. IV: The abundances of 92,94Mo and 96,98Ru in the solar system[17] Compared those observational values with Fig.4, it is clear that the heavy element produced in BBN can affect the solar abundance heavy element. Some of them are produced too much. But this is not a problem of the previous work [3], because we assumed that high density regions are very small and do not disturb standard BBN. The analysis here suggest that even if we assume the density fluctuations are completely mixed, heavy element can have enough affect to the solar abundance. IV. DIFFUSION DURING BBN In the previous analysis, we assumed that the diffusion effect can be ignored during BBN and both high density regions and low density regions are completely mixed after BBN. In this section, we determine the scale of high baryon density island in which the diffusion effect during BBN is very small enough and our assumption is valid. We do not discuss the diffusion after BBN here. A detail analysis of the comoving diffusion distance of the baryon, the neutron and the proton is in [16]. From Fig.1 in [16], in order to safely ignore the diffusion effect, it is necessary for the high baryon density island to be much larger than 105cm at T=0.1MeV(1.1× 109K). Notice that T ∝ 1 , where A is a scale factor. For scale d now corresponds to d/(4.0× 108) at BBN epoch. Present galaxy scale is O(1020)cm, which corresponds to O(1012)cm >> 105cm at BBN epoch. temperature and scale temperature scale 1.1× 109K (BBN) d 3000K (decouple) 3.7× 106 × d 2.725K (now) 4.0× 108 × d Table. V: Relation between temperature and scale The maximum angular resolution of CMB is lmax ∼2000. The size of universe is ∼ 5000Mpc. In order not to contradict to CMB observation, the fluctuation of baryon density must be less than ∼ 16Mpc now. This corresponds to 1017cm at BBN. Since the density fluctuation size in Dolgov and Silk’s model[2] is a free parameter, the above brief estimation sug- gests that we can take the island size large enough to ignore the diffusion effect without contradicting to observations, i.e., the reasonable size of 105cm −1017cm at the BBN epoch. We can choose distances between high density islands so that we obtain a suitable value of f . V. SUMMARY In this paper, we studied the relation between the heavy element production in high baryon density regions during BBN and the light element observation. By averaging the light element abundances in the high and the low density regions we showed that it is possible to produce a relevant amount of heavy element without contradicting to observa- tions. However we should stress that in this paper we restricted ourselves to some parameter regions where neutrons in high baryon density regions do not destroy the standard BBN. 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O’Meara, “QSO 0130-4021: A third QSO showing a low Deuterium to Hydrogen Abundance arXiv:astro-ph/9907128. http://arxiv.org/abs/astro-ph/0604264 http://arxiv.org/abs/astro-ph/0507439 http://arxiv.org/abs/astro-ph/0101292 http://arxiv.org/abs/astro-ph/0405459 http://arxiv.org/abs/astro-ph/9911242 http://arxiv.org/abs/astro-ph/0406663 http://arxiv.org/abs/astro-ph/0302006 http://arxiv.org/abs/astro-ph/0011179 http://arxiv.org/abs/astro-ph/9907128 [12] J. L. Linsky, arXiv:astro-ph/0309099. [13] S. G. Ryan, J. E. Norris and T. C. Beers, Astrophys. J. 523, 654 (1999) [arXiv:astro-ph/9903059]. [14] P. Bonifacio et al., arXiv:astro-ph/0204332. [15] M. H. Pinsonneault, G. Steigman, T. P. Walker and V. K. Narayanans, Astrophys. J. 574, 398 (2002) [arXiv:astro-ph/0105439]. [16] J. H. Applegate, C. J. Hogan and R. J. Scherrer, Phys. Rev. D 35, 1151 (1987). [17] E. Anders and N. Grevesse, Geochim. Cosmochim. Acta 53, 197 (1989). http://arxiv.org/abs/astro-ph/0309099 http://arxiv.org/abs/astro-ph/9903059 http://arxiv.org/abs/astro-ph/0204332 http://arxiv.org/abs/astro-ph/0105439 Introduction Light element abundance Homogeneous BBN Parameters and Basic equations Theoretical predictions and observations of light elements Theoretical predictions and observations of heavy elements (92,94Mo, 96,98Ru) Diffusion during BBN Summary Acknowledgements References

0704.0636

Sum-over-states vs quasiparticle pictures of coherent correlation spectroscopy of excitons in semiconductors; femtosecond analogues of multidimensional NMR

Sum-over-states vs quasiparticle pictures of coherent correlation spectroscopy of excitons in semiconductors; femtosecond analogues of multidimensional NMR Shaul Mukamel, Rafal Oszwaldowski, Darius Abramavicius Chemistry Department, University of California, Irvine, CA 92697-2025, United States (Dated: August 28, 2021) Abstract Two-dimensional correlation spectroscopy (2DCS) based on the nonlinear optical response of excitons to sequences of ultrafast pulses, has the potential to provide some unique insights into carrier dynamics in semiconductors. The most prominent feature of 2DCS, cross peaks, can best be understood using a sum-over-states picture involving the many-body eigenstates. However, the optical response of semiconductors is usually calculated by solving truncated equations of motion for dynamical variables, which result in a quasiparticle picture. In this work we derive Green’s function expressions for the four wave mixing signals generated in various phase-matching directions and use them to establish the connection between the two pictures. The formal connection with Frenkel excitons (hard-core bosons) and vibrational excitons (soft-core bosons) is pointed out. PACS numbers: 78.47.+p,71.35.-y http://arxiv.org/abs/0704.0636v1 I. INTRODUCTION Exciton models are widely used to describe the linear and nonlinear optical properties of many types of systems, including semiconductor nanostructures (quantum wells, dots and wires), molecular aggregates and crystals,1,2,3 as well as vibrations in proteins.4,5 In semi- conductors, nonlinear optical experiments reveal a wealth of interesting phenomena.6,7,8,9,10 For instance, such experiments provide information about many-exciton states such as biex- citons, their interactions, relaxation and dissociation.11,12,13,14 The introduction of multidimensional techniques had revolutionized NMR in the seventies15 and established it as a powerful tool for studying complex systems and iden- tifying specific structural and dynamical correlations.16 In such experiments the system is subjected to a sequence of well separated pulses. Correlation plots of the signals vs. two (or more) time delay periods then provide multidimensional spectroscopic windows into the system. The correlated dynamics of spins carefully prepared by the pulse sequence is very sensitive to their interactions. Analysis of these correlation plots then provides a powerful probe for molecular geometries and dynamical correlations. These techniques were recently extended to the infrared and the visible regime and were shown to be very useful for Frenkel excitons in molecular systems.5,17,18,19 There are some differences between the optical and the NMR techniques. NMR uses strong saturating fields whereas optical pulses are most effective in the weak field regime. NMR signals are essentially isotropic in space whereas coherent optical signals are generated in well defined (phase-matching) directions. These differences were explored in detail in Refs. 20,21,22. Nevertheless the NMR and optical techniques are conceptually similar and many ideas of pulse sequences developed in NMR may be adopted in the optical regime, where the millisecond NMR time-scale is pushed to the femtosecond regime. The same ideas may be extended to study interband and inter- suband excitations in semiconductors.23,24,25,26,27 Multidimensional analysis of the nonlinear optical response of semiconductors to sequences of femtosecond pulses could provide a novel probe for many-body interactions. In a recent work23 on semiconductor Quantum Wells, 2D correlation spectra from three 3rd order optical techniques have been calculated. The unique character of 2D spectroscopy allowed to easily recognize and classify features due to different types of biexcitons. Such features are sometimes difficult to separate in the usual one-dimensional mode of displaying non-linear spectra, due to the strong line broadening and the highly congested exciton spectra. Two types of approaches have been traditionally used towards modeling the nonlinear op- tical response of excitonic systems. The first is based on the many-body eigenstates obtained by exact diagonalization of the Hamiltonian.28 Sum-over-states (SOS) expressions can then be derived for the nonlinear response functions and optical signals. This method is practical in many applications to electronic and vibrational Frenkel excitons in molecules29,30,31 and allows clear identification and classification of possible single- and multi-photon resonances. Calculating the eigenstates is a serious computational bottleneck in extended structures. For an N site tight-binding Frenkel-exciton model the number of single and two-exciton states scales as ∼ N and ∼ N2 respectively. For Wannier excitons in semiconductors these scalings are ∼ N2 and ∼ N4, making the simulations prohibitively expensive. This is why the approach is not widely used for electron-hole excitations in semiconductors. Instead, one adopts a second strategy, which describes the response in terms of quasiparticles (QP), and the many-particle eigenstates are never calculated.2,28,32,33,34,35 Calculations are per- formed by solving equations of motion for microscopic coherences, which are coupled to other dynamical variables. Even for a simple system such as a single semiconductor quan- tum well, solving the equations numerically to create a 2D map of a nonlinear response function is computationally expensive,23 since these equations must be solved repeatedly for different pulse delays. Only after obtaining the optical signal on a 2D time grid, a Fourier transform can be performed to get the 2DCS. Apart from direct, numerical solutions of equations of motion36,37 there exist other theoretical approaches to exciton correlation ef- fects, such as memory kernel representation38,39 or Coupled Cluster Expansion for doped semiconductors.40,41 In this paper we derive closed expressions for 2DCS of semiconductors by solving the Nonlinear Exciton Equations (NEE)3,42 for the third order response. Both time-ordered and non-ordered forms of the response function which represent time and frequency domain techniques, respectively, are derived. Our QP expressions for the response are given in terms of the single exciton Green’s function and the exciton scattering matrix. The SOS response functions, in contrast, are expressed in terms of many-exciton eigenstates. Even though the response functions calculated using both techniques must be identical, the relation between the two pictures is not obvious. The expressions look very different and it is not possible to see their equivalence by a simple inspection. The SOS expressions contain large terms, which grow with system size and have opposite signs, thus they almost cancel. This complicates their numerical implementation. In contrast these cancellations are built-in from the outset in the QP approach, which uses a harmonic reference system. The nonlinearities are then attributed to exciton-exciton scattering which is absent in the harmonic reference system. The second goal of this paper is to show precisely how the two pictures of many-body correlations are connected. We write down the SOS expressions using the Keldysh loop and then derive the QP expressions directly from the SOS ones. This provides a time- domain interpretation for the interference effects. The SOS and the QP expressions provide complementary views into the origin of features seen in 2D spectrograms. In Sec. II we present the SOS expressions for the third order response obtained from time-dependent perturbation theory. Their QP counterparts are derived in Sec. III. We use the method developed in Refs. 3,34 to transform the Hamiltonian to a form typical for interacting oscillators. The starting many-electron Hamiltonian can be written in an ab-initio,43 tight-binding44 or a k · p basis. One of the key results of this paper, i.e., the equivalence of the SOS and QP pictures is proven in Sec. IV. In Sec. V we derive closed expressions for 2D correlation signals. The QP approach provides a unified description for electron-hole excitations in semiconductors as well as to Frenkel excitons in molecular aggregates (Paulions) and anharmonic vibrations (bosons), which are described by the same general Hamiltonian. QP formulae for nonlinear response have been derived previously along similar lines for Frenkel excitons. This connection is shown in Appendix F. In the last Section (VI) we discuss the results. II. SUM-OVER-STATES EXPRESSIONS FOR THE TIME-ORDERED NONLIN- EAR RESPONSE We consider a 4 wave-mixing experiment performed with three femtosecond laser pulses (Fig. 1). The optical electric field is: E (r, t) = Ej (r, t) = E +(r, t) + E−(r, t), (1) E+(r, t) = E+j (t− τj)e −iωjteikjr, (2) E−(r, t) = E−j (t− τj)e iωjte−ikjr. (3) The j-th pulse is centered at τj, has an envelope Ej(t − τj), carrier frequency ωj , and wavevector kj . E + (E−) denotes the positive (negative) frequency part of the field, and E−j = . The induced polarization in the system is recorded as a function of time- delays between pulses. Assuming the dipole interaction with the optical field ĤI = µ̂ · E(r, τ), where µ̂ is the dipole operator, the third-order contribution to the system’s polarization can be written as P (r, τ4) = ∫∫∫ ∞ dτ3dτ2dτ1S (SOS)(τ4, τ3, τ2, τ1)E(r, τ3)E(r, τ2)E(r, τ1), (4) where the response function S(SOS), which connects the induced polarization with the laser field envelopes, is given by (throughout this paper we set ~ = 1): (SOS)(τ4, τ3, τ2, τ1) = i 3 [θ(τ43)θ(τ32)θ(τ21) 〈µ̂(τ4)µ̂(τ3)µ̂(τ2)µ̂(τ1)〉 (5) − θ(τ43)θ(τ42)θ(τ21) 〈µ̂(τ3)µ̂(τ4)µ̂(τ2)µ̂(τ1)〉 + θ(τ42)θ(τ23)θ(τ41) 〈µ̂(τ3)µ̂(τ2)µ̂(τ4)µ̂(τ1)〉 −θ(τ41)θ(τ12)θ(τ23) 〈µ̂(τ3)µ̂(τ2)µ̂(τ1)µ̂(τ4)〉] . We shall use double-sided Feynman diagrams to represent the time ordering of various interactions.45 The four terms in Eq. (5) are represented by diagrams a, b, c, d shown on Figure 2. These diagrams should be read starting at the bottom left and proceeding along the loop, clockwise, as indicated by the arrows. The τi variables are ordered on the Keldysh- Schwinger loop, but not necessarily in real (physical) time. τi in diagrams (a) and (d) are also ordered in real time. This is not the case for diagrams (b) and (c): in (b) τ3 can come either before or after τ1 and τ2, whereas in (c) τ1 can come either before or after τ3 and τ2. If the eigenstates |a〉 and eigenvalues εa of the system are known, Eq. (5) may be expanded in terms of the corresponding matrix elements: 〈µ(τ4)µ(τ3)µ(τ2)µ(τ1)〉 (6) a1,a2,a3 µga3µa3a2µa2a1µa1ge (εa3−εg)τ4+(εa2−εa3)τ3+(εa1−εa2)τ2+(εg−εa1)τ1 So far we considered a general multilevel system. We next turn to the response of excitons, where the energy levels form manifolds, classified by the number of excitons: the ground state (g), single exciton (e), two-exciton (f) (or biexciton), etc. (Fig. 3). We shall assume that the dipole operator can only create and annihilate a single exciton at a time. Only the single and the two-exciton states then contribute to the third order signals. We further partition the dipole operator as µ̂ = µ̂+ + µ̂−, where µ̂+ is the positive frequency part which induces upward g to e and e to f transitions, while its Hermitian conjugate µ̂− (the negative frequency part) induces the opposite transitions. We thus write εν>εν′ µνν′ |ν〉 〈ν εν<εν′ µνν′ |ν〉 〈ν Invoking the rotating-wave approximation (RWA), we neglect all terms where at least one of the transitions is not in resonance with one of the incident carrier frequencies. The system-field interaction term then becomes HI (t) = −µ̂ +(r, τ)− µ̂−E−(r, τ) Each correlation function in Eq. (5) will split into 24 = 16 terms upon substituting µ̂ = µ̂+ + µ̂−. Assuming that the system is initially in the ground state, only two of these contributions are non-zero 〈µ̂µ̂µ̂µ̂〉 = . (7) Substitution of Eq. (7) into Eq. (5) gives (SOS)(τ4, τ3, τ2, τ1) = i θ(τ43)θ(τ32)θ(τ21) −(τ4)µ̂ −(τ3)µ̂ +(τ2)µ̂ +(τ1) (a1) (8) + θ(τ43)θ(τ32)θ(τ21) −(τ4)µ̂ +(τ3)µ̂ −(τ2)µ̂ +(τ1) − θ(τ43)θ(τ42)θ(τ21) −(τ3)µ̂ −(τ4)µ̂ +(τ2)µ̂ +(τ1) +θ(τ42)θ(τ23)θ(τ41) −(τ3)µ̂ +(τ2)µ̂ −(τ4)µ̂ +(τ1) + c.c. The four terms represented by the diagrams in Fig. 4 were obtained by taking µ̂(τ4) = µ̂−(τ4) for the last interaction, µ̂(τ4) = µ̂ +(τ4) gives the complex conjugates. Hereafter left/right direction of the arrows corresponds to µ̂−/µ̂+ in Eq.(8). Note that time-reversal symmetry implies 〈µ̂−(τ4)µ̂ −(τ3)µ̂ −(τ2)µ̂ −(τ1)〉 = 〈µ̂+(τ1)µ̂ +(τ2)µ̂ +(τ3)µ̂ +(τ4)〉. If the pulse envelopes are much shorter than their delays, the system is forced to interact se- quentially first with pulse k1, then k2 and finally k3. This means that in the integral of Eq. (4) one must replace E(r, τj) with one of the Ej, depending on the time-ordering of the integration variables in real (physical) time. We note that the first and the second terms in Eq. (8) impose a full time ordering of the integration variables while the third and the fourth terms do not. Term (b) is only partially time ordered. Depending on the position of τ3 relative to the τ1 < τ2 < τ4 sequence, the diagram can be separated into three fully time ordered terms: τ3 < τ1, τ1 < τ3 < τ2 or τ2 < τ3 < τ4. Formally we do that by separating the product of step functions as follows: θ(τ43)θ(τ42)θ(τ21) = θ(τ42)θ(τ21)θ(τ13) + θ(τ42)θ(τ23)θ(τ31) + θ(τ43)θ(τ32)θ(τ21). Using this relation, diagram (b) of Fig. 4 is split into (b3), (b2) and (b1) as shown in the first line of Fig. 5. The interactions on the l.h.s. of this diagrammatic equation are ordered on the loop. On the other hand, the arrows in the open, double-sided diagrams on the r.h.s. are ordered in real (physical) time. All diagrams on the r.h.s. are obtained from (b) by moving the arrows while preserving their order along the loop (but not in physical time!). Similarly we write for term (c) θ(τ23)θ(τ42)θ(τ41) = θ(τ42)θ(τ21)θ(τ13) + θ(τ42)θ(τ23)θ(τ31) + θ(τ41)θ(τ12)θ(τ23) and the diagram is split into (c2), (c3), (c1). S(3) can now be recast in the fully time-ordered (SOS)(τ4, τ3, τ2, τ1) = i θ(τ43)θ(τ32)θ(τ21) −(τ4)µ̂ −(τ3)µ̂ +(τ2)µ̂ +(τ1) (a1) (9) + θ(τ43)θ(τ32)θ(τ21) −(τ4)µ̂ +(τ3)µ̂ −(τ2)µ̂ +(τ1) − θ(τ43)θ(τ32)θ(τ21) −(τ3)µ̂ −(τ4)µ̂ +(τ2)µ̂ +(τ1) − θ(τ42)θ(τ23)θ(τ31) −(τ3)µ̂ −(τ4)µ̂ +(τ2)µ̂ +(τ1) − θ(τ42)θ(τ21)θ(τ13) −(τ3)µ̂ −(τ4)µ̂ +(τ2)µ̂ +(τ1) + θ(τ41)θ(τ12)θ(τ23) −(τ3)µ̂ +(τ2)µ̂ −(τ4)µ̂ +(τ1) + θ(τ42)θ(τ23)θ(τ31) −(τ3)µ̂ +(τ2)µ̂ −(τ4)µ̂ +(τ1) +θ(τ42)θ(τ21)θ(τ13) −(τ3)µ̂ +(τ2)µ̂ −(τ4)µ̂ +(τ1) + c.c. The labels on the right correspond to the various diagrams shown in Figs. (4) and (5). Once split into fully time-ordered contributions, it is convenient to change the integra- tion variables in Eq. (4) from τ4, τ3, τ2, τ1 that label the actual interaction times with the fields, to the three delays t3, t2, t1 between successive interactions. Note that the correla- tion functions are invariant to time translation 〈µ̂(τ4 − τ)µ̂(τ3 − τ)µ̂(τ2 − τ)µ̂(τ1 − τ)〉 = 〈µ̂(τ4)µ̂(τ3)µ̂(τ2)µ̂(τ1)〉. Eq. (4) thus assumes the form P (r, τ4) = ∫∫∫ +∞ dt3dt2dt1S (3)(t3, t2, t1)E3(r, τ4−t3)E2(r, τ4−t2−t3)E1(r, τ4−t1−t2−t3), where t1 = τ2 − τ1, t2 = τ3 − τ2, t3 = τ4 − τ3. In the impulsive limit, where all pulses are shorter than all system’s response time scales, we can substitute Eqs. (1)-(3) in Eq. (10) and eliminate the time integrations. This gives P (r, τ4) = S (SOS)(t3, t2, t1)E 3 (τ4 − t3)E 2 (τ4 − t3 − t2)E 1 (τ4 − t3 − t2 − t1)× (11) ei(λ1k1+λ2k2+λ3k3)re−i(λ1ω1+λ2ω2+λ3ω3)τ4ei(λ1ω1+λ2ω2+λ3ω3)t3ei(λ1ω1+λ2ω2)t2eiλ1ω1t1 The polarization is created along 8 possible directions ks = λ1k1+λ2k2+λ3k3 with λi = ±1 P (r, τ4) = P (ks, ωs) e iksr−iωsτ4 + c.c. (12) where P (ks, ωs) = S (SOS) s (t3, t2, t1) E k1 + k2 + k3 vanishes for the assumed dipole selection rules in our model. Since P (−ks,−ωs) = P ∗ (ks, ωs), we are left with three independent combinations kI ≡ −k1 + k2 + k3, kII ≡ +k1 − k2 + k3, kIII ≡ +k1 + k2 − k3: (SOS)(t3, t2, t1) = S (SOS) I (t3, t2, t1) + S (SOS) II (t3, t2, t1) + S (SOS) III (t3, t2, t1). We can classify the diagrams in Fig. 5 according to the directions of the arrows: arrow pointing to the right (left) represents +k (−k), arrows are read from the bottom up on either side. We obtain for kI (Fig. 6) (SOS) I = i 3θ(t1)θ(t2)θ(t3) −(0)µ̂+(t1 + t2)µ̂ −(t1 + t2 + t3)µ̂ +(t1) (c3) (13) −(0)µ̂+(t1)µ̂ −(t1 + t2 + t3)µ̂ +(t1 + t2) −(0)µ̂−(t1 + t2 + t3)µ̂ +(t1 + t2)µ̂ +(t1) . (b3) For the kII technique we similarly have (Fig. 7): (SOS) II = i 3θ(t1)θ(t2)θ(t3) −(t1 + t2 + t3)µ̂ +(t1 + t2)µ̂ −(t1)µ̂ (a2) (14) −(t1)µ̂ +(t1 + t2)µ̂ −(t1 + t2 + t3)µ̂ −(t1)µ̂ −(t1 + t2 + t3)µ̂ +(t1 + t2)µ̂ Finally kIII is given by (Fig. 8): (SOS) III = i 3θ(t1)θ(t2)θ(t3) −(t1 + t2 + t3)µ̂ −(t1 + t2)µ̂ +(t1)µ̂ (a1) (15) −(t1 + t2)µ̂ −(t1 + t2 + t3)µ̂ +(t1)µ̂ Each term is labelled according to Eq. (9). Eqs. (13-15) can be used to express the third order SOS response in terms of transition dipoles, system frequencies and dephasing rates (see App. E and Sec. V). In the next section we employ the EOM approach to derive the alternative QP expres- sions for these signals. These will then be connected with the current SOS expressions in Section IV. III. QUASIPARTICLE EXPRESSIONS FOR WANNIER EXCITONS IN SEMI- CONDUCTORS Interband transitions in semiconductors may be described by the two-band many-electron Hamiltonian:42,46 ĤT = Ĥ0 + ĤC + ĤI , (16) with the single-particle part Ĥ0 = m1,n1 t(1)m1,n1c cn1 + m2,n2 t(2)m2,n2d dn2 , where c† create electrons and d† create holes. The Coulomb interaction is: ĤC = m1,n1,k1,l1 m1n1k1l1 c†m1c ck1cl1 + m2,n2,k2,l2 m2n2k2l2 d†m2d dk2dl2 m1,n2,k2,l1 Wm1n2l1k2c d†n2dk2cl1 , while ĤI = − m1,m2 + (t)µ∗m1m2c d†m2 +E − (t)µm1m2dm2cm1 is the dipole interaction with light, and the optical electric field E will be treated as a scalar for simplicity. ĤT can describe both bulk and low-dimensional semiconductor systems. All the steps in this Section are independent of the single-electron basis used. Ĥ0 would be diagonal in the basis of the system’s single-particle eigenstates, i.e., t m1,n1 = ε m1δm1n1. In this paper we focus on the coherent response and we neglect coupling with phonons, which would result in additional, relevant dynamical variables and new contributions to the response function.2,3 The SOS and QP pictures should be equivalent also when dephasing is included. In that case, however, the theory becomes more complicated. For the sake of simplicity and transparency we restrict the following analysis to the coherent response, where we do not include phonons explicitly. Dephasing effects, necessary for a realistic description, will be simply introduced by adding imaginary parts to excitonic frequencies. To introduce the exciton representation we define electron-hole operators:34 B̂m ≡ dm2cm1 , B̂ m ≡ c d†m2 , where we have employed shorthand notation for pairs of indices: m ≡ (m1, m2). Using these operators we construct an effective Hamiltonian Ĥ (see App. A): hmnB̂ mB̂n + UmnklB̂ nB̂kB̂l − + (t)µ∗mB̂ − (t)µmB̂m . (17) The Hamiltonians Ĥ and ĤT are equivalent in the single and double excitations subspace, which is relevant for the response to third order in E.28 This transformation from fermion to exciton variables is crucial for our approach, since it allows us to view the electronic degrees of freedom as a system of coupled oscillators. The parameters of the transformed Hamiltonian Ĥ are given by: hmn = t m1,n1 δm2n2 + t m2,n2 δm1n1 −Wm1m2n1n2 , (18) Umnkl = − m1,k1 δm2k2δn1l1δn2l2 + t m2,k2 δm1k1δn1l1δn2l2+ n1,l1 δm1k1δm2k2δn2l2 + t n2,l2 δm1k1δm2k2δn1l1 m1n1k1l1 δm2k2δn2l2 + V m2n2k2l2 δm1k1δn1l1 The commutation relations for the B̂ operators can be obtained using the elementary fermion anticommutators: c†m1 , ck1 = δm1,k1. Within the subspace of |0〉 and B̂ i |0〉 states (i.e., the ground state and single excitations), we get34 B̂m, B̂ = δmn − 2 PmnpqB̂ pB̂q , (19) where δmn = δm1n1δm2n2 and Pmnpq = δm1q1δp1n1δm2p2δn2q2 + δm2q2δp2n2δm1p1δn1q1. (20) Eqs. (19) and (20) are obtained in a similar way to (17) and (18). Terms with additional i B̂j pairs (e.g. B̂ †B̂†B̂B̂) are neglected in (19), because they would introduce corrections higher than O (E3) to the nonlinear response. Note the symmetry Pmnpq = Pmnqp. Using Eqs. (17) and (19) we obtain the nonlinear exciton equations (see Appendix B) for single-exciton variables :2,17,35,47 − µ∗mE + (t) + Vmnkl B̂kB̂l + 2E+ (t) Pmnpq where V is given by Vnmpq = 2Unmpq − 2 Pnmlphlq − 2 PnmklUklpq. (22) Here Ymn ≡ B̂mB̂n are two-exciton variables. The Heisenberg equations give: mn,kl Ykl −E + (t) + 2E+ (t) Pmnkl Calculating the optical response by numerical integration of these equations23,48 is straightforward but numerically expensive. An alternative, more tractable approach, which further provides a better insight into the nature of the response, is to integrate the equations formally using one-exciton Green’s functions G (t) and exciton scattering matrix Γ (t). The scattering matrix depends on quasiparticle statistics through the P matrix (Eqs. B4, B5) as well as on exciton-exciton coupling. This results in closed quasiparticle expressions for the 3rd order contributions SI , SII and SIII to the response function (for details see Appendix C and Ref. 49) (QP ) I (τ4, τ3, τ2, τ1) = (24) − 2θ (τ43) θ (τ32) θ (τ21) n4...n1 dτ ′′s τ ′′s dτ ′s (τ ′′s − τ s)Gn4n′4 (τ s)Gn′3n3 (τ43 − τ s )Gn′2n2 (τ42 − τ (τ41 − τ where τ43 = τ4 − τ3, etc. and Gmn (t) = −iθ (t) [exp (−iht)]mn. The response functions for the other phase-matching directions can be derived along the same lines. We get (QP ) II (τ4, τ3, τ2, τ1) = (25) − 2θ (τ43) θ (τ32) θ (τ21) n4...n1 dτ ′′s τ ′′s dτ ′s ...n′ (τ ′′s − τ s)Gn4n′4(τ s)Gn′3n3(τ43 − τ (τ42 − τ s)Gn′1n1(τ41 − τ (QP ) III (τ4, τ3, τ2, τ1) = (26) − 2θ (τ43) θ (τ32) θ (τ21) n4...n1 µn4µn3µ dτ ′′s τ ′′s dτ ′s ...n′ (τ ′′s − τ s)Gn4n′4(τ (τ43 − τ s)Gn′2n2(τ42 − τ s )Gn′1n1(τ41 − τ Just as in the SOS case, time translation symmetry implies that these response functions only depend on the three pulse delays t3, t2, t1. Eqs. (24-26) will be used next to connect the QP and the SOS pictures. IV. CONNECTING THE SUM-OVER-STATES AND THE QUASIPARTICLE PICTURES We first recast Eqs. (13-15) using Green’s functions (in all expressions t1 > 0, t2 > 0, t3 > 0): (SOS) −Ĝ†(t1 + t2 + t3)µ̂ −Ĝ(t3)µ̂ +Ĝ(t2)µ̂ (b3) (27) −Ĝ†(t1 + t2)µ̂ †(t3)µ̂ −Ĝ(t2 + t3)µ̂ −Ĝ† (t1) µ̂ †(t2 + t3)µ̂ −Ĝ(t3)µ̂ , (c1) (SOS) II =− −Ĝ(t3)µ̂ Ĝ(t2)µ̂ −Ĝ(t1)µ̂ (a2) (28) −Ĝ†(t2 + t3)µ̂ −Ĝ(t3)µ̂ +Ĝ(t1 + t2)µ̂ −Ĝ†(t2)µ̂ †(t3)µ̂ −Ĝ(t1 + t2 + t3)µ̂ , (c2) (SOS) III =− −Ĝ(t3)µ̂ −Ĝ(t2)µ̂ +Ĝ(t1)µ̂ (a1) (29) −Ĝ†(t3)µ̂ −Ĝ(t2 + t3)µ̂ +Ĝ(t1)µ̂ , (b1) Here Ĝ(t) ≡ −iθ(t) exp(−iĤt) and Ĝ†(t) ≡ +iθ(t) exp(iĤt) represent the retarded and the advanced Green’s function respectively; Ĝ, Ĝ and Ĝ describe the evolution within the ground-state, single-exciton and double-exciton blocks of the Hamiltonian (Eq. 17) respec- tively. We also set the ground state energy εg to zero. Our goal is to show the equivalence of the QP and SOS pictures by deriving Eqs. (24-26) from Eqs. (27-29). To that end we adopt a harmonic reference system of noninteracting quasiparticle and expand the SOS response in anharmonicities. Harmonic oscillators are linear, and their nonlinear response vanishes identically2,28,32,50, as can be easily seen from the Heisenberg equations of motion. This means that the various Liouville space pathways for all nonlinear response function interfere destructively. Exploiting this property in the following derivation, we show that the quasiparticle physical picture has built-in cancellations in the reference harmonic system. We shall use the Dyson equation for the two particle Green’s function, also known as the Bethe-Salpeter equation33 Ĝ (ω) = Ĝ0 (ω) + Ĝ0 (ω) Γ (ω) Ĝ0 (ω) , (30) or in the time domain: Ĝ(τ) = Ĝ0(τ) + ∫ τ ′ dτ ′′Ĝ0 (τ − τ ′) Γ (τ ′ − τ ′′) Ĝ0(τ ′′). (31) Ĝ0 is taken to be the Green’s function of a doubly excited, harmonic system. It can be factorized into the product of a single-exciton Green’s functions Ĝ0(τ)n1n2n3n4 = iĜn1n3(τ)Ĝn2n4(τ). The exciton scattering matrix Γ is defined by Eq. (31). Let us start with the SI technique and show the equivalence of Eq. (27) to (24). The second and third terms of Eq. (27) (diagrams (c1) and (c3) in Fig. 6) are purely harmonic, independent on the quasiparticle interactions. This is a direct result of the ordering of µ±, whereby the system only evolves in the ground and first excited state. Exciton-exciton interactions influence the evolution only in the second excited manifold. The first term in Eq. (31), i.e. Ĝ0, represents harmonic evolution in the two-exciton manifold. Thus the first term of Eq. (27) with Ĝ replaced by Ĝ0 must cancel the other two terms, because the nonlinear response of a harmonic system vanishes. Substituting the second term from Eq. (31) in Eq. (27) we obtain a single term for SI : (SOS) I = −θ (t3) θ (t2) θ (t1) ∫ τ ′ dτ ′′× (32) −Ĝ†(t1 + t2 + t3)µ̂ −Ĝ0(t3 − τ ′)Γ(τ ′ − τ ′′)Ĝ0(τ ′′)µ̂+Ĝ(t2)µ̂ The equivalence of the Eqs. (32) and (24) can be directly seen using the diagrams shown in Fig. (9). In these diagrams the scattering matrix Γ is represented by dashed regions. Note that G (t)G† (t) = θ (t) exp(−iht) exp(iht) = θ (t). The QP diagram in Fig. (9) is obtained from the SOS one by changing the integration variables τ ′ = t3 − τ s and τ ′′ = t3 − τ This completes the derivation of the QP expression for SI (Eq. 24) starting from the SOS expression (Eq. 27). SII can be calculated similarly. By combining Eqs. (28) and (31) the same type of cancellation of harmonic terms yields (SOS) II = −θ (t3) θ (t2) θ (t1) ∫ τ ′ dτ ′′× (33) −Ĝ† (t2 + t3) µ̂ −Ĝ0 (t3 − τ ′) Γ (τ ′ − τ ′′) Ĝ0 (τ ′′) µ̂+Ĝ(t1 + t2)µ̂ Eq. (33) is identical to Eq. (25) as illustrated in Fig. (10). We finally turn to SIII , (Eq. 29). Using again the Bethe Salpeter equation (31) and the fact that terms that only depend on Ĝ0 must cancel (harmonic reference), we get (SOS) III = −θ (t3) θ (t2) θ (t1)× (34) ∫ τ ′ dτ ′′ −Ĝ(t3)µ̂ −Ĝ0(t2 − τ ′)Γ(τ ′ − τ ′′)Ĝ0(τ ′′)µ̂+Ĝ(t1)µ̂ ∫ t2+t3 ∫ τ ′ dτ ′′ −Ĝ†(t3)µ̂ −Ĝ0(t2 + t3 − τ ′)Γ(τ ′ − τ ′′)Ĝ0(τ ′′)µ̂+Ĝ(t1)µ̂ The equivalence of QP (Eq. 26) and SOS (Eq. 29) expressions can be shown as follows: the two terms in Eq. (34) are labeled (SOSa) and (SOSb). The term (SOSb) can further be split into two terms (SOSb1) and (SOSb2), the first corresponding to τ ′ < t2, the second to τ ′ > t2 (Fig. 11). (SOSb1) is identical to (SOSa), but with opposite sign coming from Ĝ†(t3). Only the second term (SOSb2) remains, and it is equivalent to the (QP) diagram. We thus obtained Eq. (26) from Eq. (34). V. 2D CORRELATION SIGNALS 2D signals are displayed as correlation plots obtained by the double Fourier transforms of the various signals.17 We shall denote the frequencies conjugate to the pulse delay times t1, t2 and t3 by Ω1, Ω2 and Ω3. Starting with Eq. (11), and deleting some inessential factors, we obtain the induced polarization, which depends parametrically on the delay times t1, t2 and t3: Ps(t3, t2, t1) = Ss(t3, t2, t1)e i(λ1ω1+λ2ω2+λ3ω3)t3ei(λ1ω1+λ2ω2)t2eiλ1ω1t1 . (35) Specifying the three possible signals by a proper choice of λ factors we obtain: PI(t3, t2, t1) = SI(t3, t2, t1)e i(−ω1+ω2+ω3)t3ei(−ω1+ω2)t2e−iω1t1 , PII(t3, t2, t1) = SII(t3, t2, t1)e i(ω1−ω2+ω3)t3ei(ω1−ω2)t2eiω1t1 , PIII(t3, t2, t1) = SIII(t3, t2, t1)e i(ω1+ω2−ω3)t3ei(ω1+ω2)t2eiω1t1 . The 2DCS for PI and PII is defined as Pα(Ω3, t2,Ω1) ≡ dt1Pα(t3, t2, t1) exp {iΩ3t3 + iΩ1t1} , α = I, II (36) For the SOS picture we use the expansions in eigenstates given by Eqs. (E1), (E2) and (E3). The QP expressions for P (QP ) I , P (QP ) II and P (QP ) III are obtained along the lines presented in App. C. Dephasing is introduced phenomenologically by adding a decay rate γ to the Green’s functions. We thus obtain PI (SOS) I (Ω3, t2,Ω1) = (37) e2,e1 µge1µ µge2I (−Ω1 + ω1)I g (t2)Ig(t2)Ie1(Ω3 − ω1 + ω2 + ω3) e2,e1 µge1µ µge2I (−Ω1 + ω1)I (t2)Ie1(t2)Ie1(Ω3 − ω1 + ω2 + ω3) e2,e1f µe2fµ µge2I (−Ω1 + ω1)I (t2)Ie1(t2)Ffe2(Ω3 − ω1 + ω2 + ω3), (QP ) I (Ω3, t2,Ω1) = (38) e1,e2,e3,e4 (t2)Ie2(t2)I (−Ω1 − ω1)Ie4(Ω3 − ω1 + ω2 + ω3) × Γe4e1e3e2(Ω3 − ω1 + ω2 + ω3 + εe1 + iγe1)G0 e3e2(Ω3 − ω1 + ω2 + ω3 + εe1 + iγe1). The Green’s function Fourier transform is defined as G(ω) = dt exp(iωt)G(t) [and G(t) = exp(−iωt)G(ω)]. We have Ie(ω) ≡ Ĝ (ω) = (ω − εe + iγe) −1, (39) G0 e2e1(ω) ≡ Ĝ0(ω) ω − εe2 − εe1 + i (γe2 + γe1) . (40) Eq. (40) is obtained by transforming G0 kljr (t) to the single-exciton basis and performing the Fourier transform. We also define Fab(t) ≡ −iθ(t) exp (i (εb − εa) t− (γa + γb) t) , Fab(ω) = (ω − εa + εb + iγa + iγb) Similarly we obtain for PII : (SOS) II (Ω3, t2,Ω1) = (41) e2,e1 µge2µge1µ Ie1(Ω1 + ω1)I g (t2)Ig(t2)Ie2(Ω3 + ω1 − ω2 + ω3) e2,e1 µge1µ µge2µ Ie1(Ω1 + ω1)I (t2)Ie1(t2)Ie1(Ω3 + ω1 − ω2 + ω3) e2,e1,f fµge2µ Ie1(Ω1 + ω1)I (t2)Ie1(t2)Ffe2(Ω3 + ω1 − ω2 + ω3), (QP ) II (Ω3, t2,Ω1) = (42) e4..e1 I∗e2(t2)Ie1(t2)Ie1(Ω1 + ω1)Ie4(Ω3 + ω1 − ω2 + ω3) × Γe4e2e3e1(Ω3 + ω1 − ω2 + ω3 + εe2 + iγe2)G0 e3e1(Ω3 + ω1 − ω2 + ω3 + εe2 + iγe2). The PIII 2DCS signal is defined as PIII(Ω3,Ω2, t1) ≡ dt2PIII(t3, t2, t1) exp {iΩ3t3 + iΩ2t2} . (43) This yields: (SOS) III (Ω3,Ω2, t1) = (44) e2,e1,f µge1µe1fµ Ie2(t1)If(Ω2 + ω1 + ω2)Ie1(Ω3 + ω1 + ω2 − ω3) e2,e1,f µge1µe1fµ Ie2(t1)If(Ω2 + ω1 + ω2)Ffe1(Ω3 + ω1 + ω2 − ω3), (QP ) III (Ω3,Ω2, t1) = (45) e4...e1 µe4µe3µ Ie1(t1)Ie4(Ω3 + ω1 + ω2 − ω3)I (Ω2 − Ω3 + ω3)× [Γe4e3e2e1(Ω2 + ω1 + ω2)G0 e2e1(Ω2 + ω1 + ω2) −Γe4e3e2e1(Ω3 + ω1 + ω2 − ω3 + εe3 + iγe3)G0 e2e1(Ω3 + ω1 + ω2 − ω3 + εe3 + iγe3)] . Both P (SOS) and P (QP ) depend on the single-exciton energies. However, the SOS expres- sions contain two-exciton eigenenergies (εf) explicitly, while the QP counterparts contain the scattering matrix Γ instead. The equivalence of the two representations has been established in Sec. IV. Eqs. (37-45) constitute our final expressions for the various 2DCS signals. In this form they may be readily used in numerical simulations. The SOS expressions (Eqs. 37, 41 and 44) were recently used to survey the various possible resonances and cross-peaks in 2DCS of semiconductors.23 VI. DISCUSSION The quasiparticle representation is obtained using the Heisenberg equations for the ex- citon oscillator variables. These equations form an infinite hierarchy involving successively higher numbers of excitons.3,51 The hierarchy may be truncated, depending on the observable of interest. For instance, the absorption originates from single-exciton creation/annihilation. Only single-exciton variables should then be considered, and exciton-exciton interaction terms may be neglected. The nonlinear response depends on the exciton interactions, thus single- and double-exciton variables need to be treated explicitly. The two coupled NEE equations (21, 23) describe the third order response. These equations are exact in the ab- sence of dephasing. When dephasing is included by adding linear coupling to a phonon bath, two additional variables B̂†B̂ B̂†B̂B̂ must be included in the NEE to describe the third order response.3 Without dephasing these may be factorized as B̂†B̂ B̂†B̂B̂ . We then recover the coherent limit considered in this arti- cle. For some techniques the present equations provide a good approximation even in the presence of dephasing. B̂†B̂ describes incoherent exciton transport and is only relevant during t2, while B̂†B̂B̂ is generated during t3. It describes the optical coherence between one-exciton and two-exciton manifolds, which are represented by The quasiparticle approach avoids the explicit calculation of multiple exciton states: their influence is represented by the scattering matrix, which can be calculated provided the exci- ton interactions are known. We have shown how the quasiparticle expressions for the various third order techniques, ordinarily derived by solving equations of motion, can be obtained directly from the sum-over-states expressions by employing the Bethe-Salpether equation. These expressions explicitly contain the two-exciton Green’s functions and have many inter- fering terms with large cancellations,32 which complicate their numerical implementation. In the QP picture, on the other hand, these interference effects are built-in, considerably simplifying the expressions for the nonlinear response.35 The interpretation of 2DCS signals using the SOS expressions is straightforward.23 In the kI technique one-exciton coherences are observed during t1, and the coherences between excitons and biexcitons are observed during t3. Thus the 2DCS shows peaks along Ω1 and Ω3 corresponding to these resonances. kIII shows biexciton resonances along the Ω2 axis, this technique is known in NMR as double-quantum coherence. We have established the connection between the SOS and the QP pictures by using time-ordering on the Keldysh- Schwinger loop, which only maintains partial time ordering in real (physical) time. The Hamiltonian Ĥ (Eq. 17) can describe several microscopic models other than the Wan- nier excitons considered here (ĤT ). Vibrational excitations (soft-core bosons) and Frenkel excitons (hard-core bosons, Paulions) in molecules can be mapped into the same model.19,50,51 The equations of motion for these other systems are similar, but not identical, because of the different commutation relations (QP statistics). Eq. (19) provides a unified description for all of these systems, by specifying the proper commutation rules:3,34 for bosons Pmnpq = 0 and for Paulions Pmnpq = δmnδmqδnp. These expressions for P may be substituted into our final expressions for the response functions, where they only affect the exciton scattering matrix, which in the frequency domain reads (App. D) Γ(ω) = (I− V G0(ω)) V G0(ω) (I− P)G 0 (ω)− PG 0 (ω), where V is given in Eq. (22), G0 is the free two–exciton Green’s function (App. B) and I is the tetradic identity matrix. The nonlinearity of the system depends on QP interactions as well as non-boson statistics; both enter through Γ. For noninteracting bosons, where U ≡ P ≡ 0, Γ vanishes and so does the nonlinear response. In Appendix F we present Γ for bosons and Paulions. We have used the symmetry Pmnpq = Pmnqp in our derivation. Since the boson commuta- tion relations are simpler than for Fermi or Pauli operators, a considerable effort has been de- voted to mapping the original problem with complicated commutation relations into a boson picture.52,53 The resulting boson Hamiltonian contains additional interactions which com- pensate for the statistics. For instance, the Frenkel exciton Hamiltonian for Paulions may be mapped into an anharmonic Hamiltonian of bosons with quartic couplings. Bosonization34 is very convenient for describing exciton scattering: the response functions derived for bosons can be applied for arbitrary operators, provided we modify the Hamiltonian and express it in terms of boson operators. Acknowledgments We wish to thank Dr Igor V. Schweigert for valuable discussions. This research was supported by the National Science Foundation Grant no. CHE-0446555 and the National Institutes of Health 2RO1-GM59230-05. APPENDIX A: EXCITON REPRESENTATION OF THE TWO-BAND HAMIL- TONIAN FOR FERMIONS By construction, the Hamiltonians H (Eq. (17)) and ĤT (Eq. (16)) are equivalent only in the physically relevant space of single and double excitations. This is sufficient to calculate the response to third order in the field E (t). Ĥ may be constructed using the following rules: • since the Hamiltonian (16) conserves the number of excitons, it should only contain products with equal number of B̂† and B̂ operators (except for the HI term, which does change the number of excitons) • a term B̂†a1B̂ . . . B̂†apB̂b1B̂b2 . . . B̂bp gives zero when acting on states with less than p excitations and only affects manifolds with p excitations and higher. The parameters of Ĥ can be obtained as follows. First we note that no con- stant term k should be added to (17), since it would yield: 〈0 |k| 0〉 6= 0, while 〈0 |HT | 0〉 = 0. The m1,n2,k2,l1Wm1n2l1k2c d†n2dk2cl1 term of HT can be written directly m1,m2,n1,n2 Wm1m2n1n2c d†m2dn2cn1 = WmnB̂ mB̂n. Also the term describing the interaction with light can be obtained directly. Using the second rule given above we imme- diately see that no terms higher than B̂†a1B̂ B̂b1B̂b2 are necessary in the sub-space defined by functions |0〉, B̂ i |0〉 and B̂ j |0〉. We thus obtain the form given in (17). We next calculate, in this sub-space, matrix elements of Ĥ , and compare to matrix elements of ĤT . In this way a one-to-one correspondence of the parameters of Ĥ and ĤT can be established. Additional terms must be included in Ĥ in order to describe higher order response func- tions. This can be done using the same rules. APPENDIX B: THE NONLINEAR EXCITON EQUATIONS The Heisenberg equation of motion (NEE) for the Hamiltonian (17) reads: − µ∗nE Vnmpq B̂†mB̂pB̂q Pnmpq B̂†mB̂p B̂†mB̂q Here we invoked RWA and used the notation of Eq. (1). Employing (20) we see that Pnmpq = Pnmqp, so the last two terms in Eq. (B1) can be recast as: 2E Pnmpq B̂†mB̂q We now make the following factorization: B̂†mB̂p B̂†mB̂pB̂q B̂pB̂q , (B2) which is exact for pure states when dephasing is neglected33 and is a good approximation in the absence of incoherent exciton transport. Eq. (B1) then yields the Eqs. (21) and (23), where mn,kl = δmkhnl + δnlhmk + Vmnkl ≡ h̄ + V. (B3) We next expand the EOMs in orders of E. Using B m for we obtain: n − µ + (t) , mn,kl −E+ (t) B(1)n µ + 2E+ (t) PmnklB VmnklB + 2E+ (t) PmnpqB The Green’s function (tetradic matrix) for Y (2) is G (t) = −iθ (t) −ih(Y )t Y (2)mn (t) = Gmnkl (t− τ)E + (τ) (τ)µ∗k +B (τ)µ∗l PklpqB p (τ)µ Gmnkl (t− τ)× Gla (τ − τ ′)µ∗k +Gka (τ − τ ′)µ∗l − 2 PklpqGpa (τ − τ ′)µ∗q + (τ)E+ (τ ′) . We also define the zero-order tetradic Green’s function G0 for Y (2) for the case V = 0, i.e. G0mnkl (t) = −iθ (t) −ih̄t , it will be used later. B m is given as B(3)m (t Gmn (t ′ − t)× VnpklB p (t) Y kl (t) + 2E + (t) PnkpqB k (t)B q (t)µ This expression can be simplified using the symmetry Gklfg = Gklgf . At this point we introduce the tetradic exciton scattering matrix Γ defined as: Γ (ω)G0 (ω) = V G (ω) (I−P)−P, (B4) which in time domain can be written as (see App. D): V G (t− τ) (I− P) = Pδ (t− τ) + dτ1Γ (t− τ1)G0 (τ1 − τ) , (B5) where the tetradic identity matrix Ifgjr = δfjδgr. Since G (t− τ) ∼ θ (t− τ) is retarded, Γ must be retarded as well, i.e., Γ (t− τ1) ∼ θ (t− τ1). To proceed further we take advantage of the factorization: G0 kljr (t) = iGkj (t)Glr (t) , (B6) which can be easily shown in the single-exciton eigenbasis. After a rearrangement of terms we obtain: n4 (τ4) = −2 ∫ ∫ ∫ ∫ ∫ dτ ′dτ2dτ1dτ ′′dτ3 n1,n2,n3 Gn4n′4 (τ4 − τ ′′) Γn′ (τ ′′ − τ ′)G∗n′ (τ ′′ − τ3)× n2 (τ ′ − τ2)Gn′ n1 (τ ′ − τ1) θ (τ2 − τ1)µ + (τ2)E + (τ1)E − (τ3) . The 3rd order polarization is P (3) (τ4) = + µ∗n4B ∫ ∫ ∫ dτ3dτ2dτ1 n4,n3,n2,n1 µn4µn3µ θ (τ2 − τ1) dτ ′′ (τ ′′ − τ ′)× Gn4n′4 (τ4 − τ ′′)G∗n′ (τ ′′ − τ3)Gn′ n2 (τ ′ − τ2)Gn′ n1 (τ ′ − τ1)× − (τ3)E + (τ2)E + (τ1) + complex conjugate, where we used Γn′ (t) = Γn′ The above expression is finite only for τ2 > τ1. Hence there are 3 possible intervals for τ3, that define three contributions to the third order response function S(QP ) P (τ4) = dτ1θ (τ2 − τ1)× (B7) (QP ) I dτ3 + (QP ) II dτ3 + (QP ) III dτ3 − (τ3)E + (τ2)E + (τ1) + c.c. This definition of S (QP ) I , S (QP ) II and S (QP ) III is used in Appendix (C) to obtain the Eqs. (24-26). APPENDIX C: RESPONSE FUNCTIONS OF QUASIPARTICLES For calculating each of the contributions to Eq. (B7) we need to switch to a different set of time-ordered variables. For S (QP ) I we set: τ1 → τ2, τ2 → τ3, τ3 → τ1: (QP ) I (τ4, τ3, τ2, τ1) = −2 n4,n3,n2,n1 µn4µn3µ dτ ′′ (τ ′′ − τ ′)Gn4n′4 (τ4 − τ ′′)G∗n′ (τ ′′ − τ1)Gn′ n2 (τ ′ − τ3)Gn′ n1 (τ ′ − τ2) , where τ4 > τ3 > τ2 > τ1. Substituting τ ′s = τ4 − τ ′′ and τ ′′s = τ4 − τ ′ and exchanging the dummy indices n1 → n2, n2 → n3, n3 → n1 (same for primed indices) we obtain Eq. (24). Integration limits for τ have been limited by Gn4n′4 (τ s) and Γ (τ s − τ s), while for τ s by Gn′3n3 (τ43 − τ s ). Eqs. (25) and (26) are obtained in a similar way. Eq. (24) can be simplified considerably by performing the double time-integrations ana- lytically. We first express the exciton Green’s function G (τ) and Γ (τ) in the one-exciton basis ψe, defined by: hmnψen = εeψem, (C1) where hmn is given by Eq. (18). The energies εe define the lowest optically-excited manifold of the system, i.e., single excitons. In this basis we express the time and frequency-domain one-exciton Green’s functions: Gmn (τ) = ψemIe (τ)ψ en ⇒ Ie (τ) = −iθ (τ) exp (−iεet) , (C2) where we introduce dephasing via ε → ε − iγe. The tetradic exciton scattering matrix is given by Γm4m3m2m1 (τ) = e1...e4 ψe4m4ψe3m3Γe4e3e2e1 (τ)ψ ψ∗e1m1 . and the transformed dipole matrix elements µeg are µe = µmψem. Using these quantities we express the S (QP ) I in the single-exciton basis: (QP ) I (τ4, τ3, τ2, τ1) = −2θ (τ43) θ (τ32) θ (τ21) e1...e4 µe3 (C3) ∫ τ43 dτ ′′s ∫ τ ′′s dτ ′sΓe4e3e2e1(τ s − τ × Ie4(τ s)Ie2(τ43 − τ s )Ie1(τ42 − τ (τ41 − τ We next introduce Γ (t) = e−iωtΓ (ω). Since the response function depends only on the pulse delays and not on the absolute times, we denote SI(τ4, τ3, τ2, τ1) = SI(t3, t2, t1), where t3 = τ43, t2 = τ32 and t1 = τ21. We perform a Fourier transform with respect to the first and last arguments. We thus obtain (QP ) I (Ω3, t2,Ω1) = −2iθ (t2) e1...e4 (−Ω1) I (t2) Ie (t2) Ie (Ω3)× (C4) (ω)G0 e (ω) I∗e1 (ω − Ω3) , The ω integration can be performed by noting that Γ(ω)G0(ω) ∼ ω − 2ε+ 2iγ which is obtained from Eq. (B4) by noting that G (ω) has poles only at two-exciton energies, and that 2ε−2iγ is a good approximation for two-exciton energy and dephasing rate. Hence, if we close the Cauchy integration path in the positive half-plane, there will be only a single pole at ω = Ω3 + εe1 + iγe1 as seen from (39). This finally gives (QP ) I (Ω3, t2,Ω1) = −2 e4...e1 (t2)Ie2(t2)I (−Ω1)Ie4(Ω3) (C5) × Γe4e1e3e2(Ω3 + εe1 + iγ1)G0 e3e2(Ω3 + εe1 + iγ1), To account for carrier frequencies ω1, ω2 and ω3 appearing in the polarization (Eq. 35) at this stage, we can perform the substitution Ω1 → Ω1 + ω1 and Ω3 → −Ω1 + ω1 + ω2 + ω3. In this way we obtain Eq. (38). Eqs. (42) and (45) are derived similarly. APPENDIX D: THE EXCITON SCATTERING-MATRIX In order to use equations (24-26) for calculating the quasiparticle response function, we should calculate the scattering matrix Γ. We first write G (ω) and G0 (ω) in an operator G0 (ω) = ω − h̄ G (ω) = ω − h(Y ) ω − h̄− V where h̄ is defined in (B3). The Dyson equation then reads G = G0 + G0V G (ω) = G0 + G0V G0 + G0V G0V G0 + . . . , which can be recast in the form V G = (1− V G0) V G0. Using Eq. (B4), we obtain: ΓG0 = (I− V G0) V G0 (I− P)− P, which results in the final expression for Γ Γ = (I− V G0) V G0 (I− P)G 0 −PG 0 (D1) The l.h.s. of Eq. (B4) can be expressed as a convolution: dt1Γ (t1)G0 (τ ′ − t1) exp (iωτ The r.h.s. can be written as dτ ′G (τ ′) (I−P) exp (iωτ ′)− dτ ′Pδ (τ ′) exp (iωτ ′) , note that P is independent on τ ′ or ω. Since l.h.s.=r.h.s. for any ω, we must have: dt1Γ (t1)G0 (τ ′ − t1) = V G (τ ′) (I−P)− Pδ (τ ′) . Substituting τ ′ → t− τ and t1 → t− τ1 we obtain Eq. (B5). APPENDIX E: SOS EXPRESSIONS FOR THIRD ORDER TECHNIQUES. Upon expansion in the eigenstates for the exciton level scheme shown in Fig. 3 we get Expanding Eqs. (13 - 15) in the eigenstates, we obtain the sum-over-states expressions for the third-order response functions: (SOS) I (t3, t2, t1) = i 3θ(t3)θ(t2)θ(t1) µge′µe′gµgeµegI e′ (t1) Ie (t3) (E1) + i3θ(t3)θ(t2)θ(t1) µge′µe′gµgeµegI e′ (t2 + t1) Ie (t2 + t3) − i3θ(t3)θ(t2)θ(t1) µge′µe′fµfeµegI e′ (t1 + t2 + t3) If (t3) Ie (t2) , (SOS) II (t3, t2, t1) = i 3θ(t3)θ(t2)θ(t1) µge′µe′gµgeµegIe′ (t3) Ie (t1) (E2) + i3θ(t3)θ(t2)θ(t1) µge′µe′gµgeµegI e′ (t2) Ie (t1 + t2 + t3) − i3θ(t3)θ(t2)θ(t1) µge′µe′fµfeµegI e′ (t2 + t3) If (t3) Ie (t1 + t2) , (SOS) III (t3, t2, t1) = i 3θ(t3)θ(t2)θ(t1) µge′µe′fµfeµegIe′ (t3) If (t2) Ie (t1) (E3) − i3θ(t3)θ(t2)θ(t1) µge′µe′fµfeµegI e′ (t3) If (t2 + t3) Ie (t1) , where Ie(t), defined in Eq. (C2), is the Green’s function in the single-exciton eigenstate basis. Eqs. (37,41,44) immediately follow by substituting Eqs. (E1,E2,E3) in Eq. (36) and (43). APPENDIX F: QUASIPARTICLE PICTURE FOR SOFT-CORE AND HARD- CORE BOSONS In this Appendix we apply our QP expressions to two other types of quasiparticles with different statistics. These two examples demonstrate the generality of our approach discussed briefly in Sec. VI. We first consider the Hamiltonian of a system of coupled anharmonic oscillators (soft-core bosons): hmnB̂ mB̂n + UmnklB̂ nB̂kB̂l, where B̂+m and B̂n are boson creation and annihilation operators with commutation [Bm, B n ] = δmn hmm is the fundamental transition energy of the mth oscillator, while hmn is the coupling between the mth and the nth oscillators. Umnkl is the anharmonic coupling. This Hamiltonian has been used to describe infrared nonlinear spectra of proteins.49,51 For this model the scattering matrix can be obtained from (D1) by putting P = 0. In the site representation it reads: Γ = (I− V G0) here Γ is a tetradic matrix, V = 2U and G0 (ω) is defined in Eq. (30). We next turn to electronic excitations in molecular aggregates or crystals with weakly interacting molecules. These are described using the Frenkel Exciton Hamiltonian. If the excited-state absorption frequency of each molecule is well separated from the ground state absorption, the excitations can be modelled as coupled two-level systems.33,54 The Hamilto- nian is hmnB̂ mB̂n. The nonlinearities are now hidden in the statistics of exciton creation (B̂+m) and annihilation (B̂n) operators. These are bosonic for different oscillators (units) and fermionic for the same oscillator. Their Pauli commutation relation is [B̂m, B̂ n ] = δmn 1− 2B̂†nB̂n . The commutation relation ensures that two excitations are not allowed to reside on the same site (hard-core bosons). The scattering matrix in this case is given by: Γmnkl = δmnδklΓ̄mn , Γ̄ = −Ḡ (ω) Ḡmn (ω) = δmm1δnn1G0mm1nn1 (ω) . 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FIG. 1: The sequence of light pulses in a time-domain Four Wave Mixing Experiment: the pulses are centered at times τ1, τ2, τ3, while the delays are t1, t2 and t3. (The latter are sometimes denoted as τ , T and t.) The signal is generated in the kS direction. (a) (b) (c) (d) FIG. 2: Diagrams representing the four partially time-ordered terms contributing to the third-order polarization (Eq. 5). FIG. 3: Energy levels of the exciton model. (a1) (a2) (b) (c) FIG. 4: Loop diagrams representing the partially time-ordered terms contributing to the third- order polarization [Eq. (8)] within the rotating wave approximation. Arrows pointing to the right represent µ+ and arrows pointing to the left µ−. The diagrams are obtained by adding arrows to the interactions in Fig. (2). (b2)(b1) (b3) (c2) (c3)(c1) FIG. 5: Diagrams representing the fully time-ordered terms contributing to the third-order polar- ization within the rotating wave approximation (Eq. (9)). τj repesent the interaction times with the various fields. Arrows pointing to the right (left) represent µ+ (µ−). Time variables in loop diagrams (b) and (c) on the left are ordered in the loop. The other open diagrams are fully ordered in physical time. 'e'egg FIG. 6: Feynman diagrams for the kI technique (Eq. 13). g gg g g g g g g g FIG. 7: Feynman diagrams for the kII technique (Eq. 14). (a1) g gg g g g 'e'e FIG. 8: Feynman diagrams for the kIII technique (Eq. 15). xx Ws” (SOS) FIG. 9: Loop diagrams showing the equivalence of the SI expressions in the QP (Eq. 24) and SOS (Eq. 32) pictures. Dotted, single and double lines show ground, single exciton and double exciton states’ evolution respectively. Dashed region represents scattering matrix. xx Ws” (SOS) FIG. 10: Loop diagrams showing the order of time variables in the QP (Eq. 25) and SOS (Eq. 33) expressions for SII . FIG. 11: Loop diagrams showing the order of time variables in the QP (Eq. 26) and SOS (Eq. 34) expressions for SIII . (SOSb1) cancels with the (SOSa) term in (Eq. 34) (not shown). The remaining diagram (SOSb2) is identical to the (QP) diagram with a simple change of time variables. Introduction Sum-over-states expressions for the time-ordered nonlinear response Quasiparticle expressions for Wannier excitons in semiconductors Connecting the sum-over-states and the quasiparticle pictures 2D correlation signals Discussion Acknowledgments Exciton representation of the two-band Hamiltonian for fermions The Nonlinear Exciton Equations Response functions of quasiparticles The exciton scattering-matrix SOS expressions for third order techniques. Quasiparticle picture for soft-core and hard-core bosons References

0704.0637

Are constant loop widths an artifact of the background and the spatial resolution?

Are constant loop widths an artifact of the background and the spatial resolution? M. C. López Fuentes1,2,3, P. Démoulin4, J. A. Klimchuk2 ABSTRACT We study the effect of the coronal background in the determination of the di- ameter of EUV loops, and we analyze the suitability of the procedure followed in a previous paper (López Fuentes, Klimchuk & Démoulin 2006) for characterizing their expansion properties. For the analysis we create different synthetic loops and we place them on real backgrounds from data obtained with the Transition Region and Coronal Explorer (TRACE ). We apply to these loops the same pro- cedure followed in our previous works, and we compare the results with real loop observations. We demonstrate that the procedure allows us to distinguish con- stant width loops from loops that expand appreciably with height, as predicted by simple force-free field models. This holds even for loops near the resolution limit. The procedure can easily determine when loops are below resolution limit and therefore not reliably measured. We find that small-scale variations in the measured loop width are likely due to imperfections in the background subtrac- tion. The greatest errors occur in especially narrow loops and in places where the background is especially bright relative to the loop. We stress, however, that these effects do not impact the ability to measure large-scale variations. The result that observed loops do not expand systematically with height is robust. Subject headings: Sun: corona – Sun: magnetic fields – Sun: UV radiation 1Instituto de Astronomı́a y F́ısica del Espacio, CONICET-UBA, CC. 67, Suc. 28, 1428 Buenos Aires, Argentina 2Naval Research Laboratory, Code 7675, Washington, DC 20375 3Member of the Carrera del Investigador Cient́ıfico, Consejo Nacional de Investigaciones Cient́ıficas y Técnicas, Argentina 4Observatoire de Paris, LESIA, UMR 8109 (CNRS), F-92195, Meudon Principal Cedex, France http://arxiv.org/abs/0704.0637v2 – 2 – 1. Introduction Since the solar corona is optically thin, studies based on coronal loop observations must include some form of subtraction of the background contribution (see e.g., Klimchuk 2000, Schmelz & Martens 2006, López Fuentes, Klimchuk & Démoulin 2006). In their recent statistical study based on observations from the Transition Region and Coronal Explorer (TRACE, see Handy et al. 1999), Aschwanden, Nightingale & Boerner (2007) showed that the background can be several times brighter than the loops themselves. It is likely that the background corona is formed by a number of loops that are too faint to produce a large enough contrast to make them detectable. However, these unobserved structures constitute a spatially fluctuating background for actual observed loops. Therefore, even for loops with constant intensity along their length, fluctuations due to the structuring of the background are expected. The determination of morphological properties of a loop, such as its diameter, can be affected by the characteristics of the background, and therefore it is important that the background be taken into account during such analyses. In a recent paper (López Fuentes, Klimchuk & Démoulin 2006, henceforth LKD06) we explored the problem of the apparent constant width of coronal loops. Since loops are the trace of magnetic flux tubes rooted in the photosphere, we might expect on the basis of simple force-free magnetic field models that most loops would expand with height. However, observations show that this is not the case; both X-ray loops (Klimchuk 2000) observed with the Soft X-ray Telescope (SXT, see Tsuneta et al. 1991) aboard Yohkoh, and EUV loops (Watko & Klimchuk 2000, and LKD06) observed with TRACE, seem to correspond to constant cross-sections. In LKD06, we compared a number of observed TRACE loops with corresponding model flux tubes obtained from force-free extrapolations of magnetogram data from the Michelson Doppler Imager (MDI, see Scherrer et al. 1995) aboard the Solar and Heliospheric Ob- servatory (SOHO). To quantify the expansion of the loops and flux tubes, we defined the expansion factor Γ as the ratio between the widths averaged over the middle and footpoint sections. We found that the mean expansion factor of the model flux tubes is about twice that of the corresponding observed loops. Another important result is that the cross sec- tion is much more asymmetric (from footpoint to footpoint) for the model flux tubes than for observed loops. We suggest that the origin of this asymmetry lies is the complexity of the magnetic connectivity of the solar atmosphere. In LKD06 we proposed a mechanism to explain the observed symmetry of real loops. Although the measured widths of observed loops have very little global variation, there are short distance fluctuations as large as 25% of the average width. In LKD06 the loop background was subtracted by linearly interpolating between the intensities on either side – 3 – of the loop. Since the background intensity can be as much as three to five times the intrinsic intensity of the loop, we might expect the width determination to be less reliable at positions where the ratio of loop to background intensities is smaller. If so, then measured width variations might be partly or largely an artifact of imperfect background subtraction. It is also possible that the width fluctuations are indicative of real structural properties of the loops. For instance, loops may be bundles of thinner unresolved magnetic strands that wrap around each other. If there are only a few such strands, then we might expect the width of the bundle to fluctuate on top of a global trend. Furthermore, the width should be anti-correlated with the intensity, since the bundle will be thinner and brighter in places where the strands are lined up along the line of sight, and it will be fatter and fainter in places where the stands are side-by-side across the plane of the sky. The filamentary nature of coronal loops, and the solar corona in general, has been progressively evident from the combination of models and observations (for a review, see e.g., Klimchuk 2006). Our ability to discern the internal structure of loops is limited by the instrument resolution. It can be seen from TRACE images that structures many times wider than the instrument Point Spread Function (PSF) are clearly made of thinner strands. It is not surprising then that recognizable “individual” loops are no thicker than a few times the PSF width. Since identifiable individual loops are close to the resolution limit, it has been suggested that the apparent constant width may just be an artifact of the resolution (see the recent paper by DeForest 2007). If loops are everywhere much smaller than the PSF, then they will appear to have a constant width equal to that of the PSF, even if the true width is varying greatly. We have carefully accounted for the PSF in our earlier studies and concluded that this is not a viable explanation for the observed constant widths. What we have not addressed in as much detail is the possible role of imperfect background subtraction. This paper describes a study that addresses both the background subtraction and finite resolution and the extent to which they influence the measured widths of loops. Our approach is to produce synthetic loops with constant and variable cross-sections, and place them on real TRACE backgrounds to simulate loop observations. We then process the synthetic data following the same procedure used in LKD06, so we can compare them with actual TRACE loops. This allows us to determine whether the procedure followed in LKD06 is able to distinguish expanding loops from constant cross-section loops. We will answer the question of whether the lack of global expansion in observed loops is real or simply an observational artifact, as suggest by DeForest (2007). We will also investigate the reliability of the shorter length scale fluctuations that are often observed. In Section 2 we describe the main properties of the set of loops studied in LKD06. We explain the synthetic loop construction in Section 3. In Section 4 we compare synthetic and – 4 – observed loops, and we discuss and conclude in Section 5. 2. Observed TRACE loops In LKD06 we studied a set of 20 loops from TRACE images in the 171 Å passband. To determine the width of the loops we followed a procedure based on the measurement of the second moment (the standard deviation) of the cross-axis intensity profile at each position along the loop. The measurement is done on a straightened version of the loop, as described in LKD06. Assuming circular cross-sections and uniform emissivity, the cross- section diameter (that we refer to as the width) will be 4 times the standard deviation of the profile. The same procedure had been used in previous studies (see Klimchuk et al. 1992, Klimchuk 2000, Watko & Klimchuk 2000). The actual background is estimated by linear interpolation of the background pixels at both sides of the loop. The obtained width is corrected for instrumental resolution (i.e. the combined PSF due to telescope smearing and detector pixelation). The typical length of the studied loops is around 150 TRACE pixels or 54 Mm, though loops as long as 300 pixels (108 Mm) are included in the set. The average width for all loops in the set is 4.2 pixels or 1.5 Mm. Figure 1 shows a typical case having the average width and length. The upper panel shows the loop as observed in the TRACE image, and the lower panel is the “straightened” version. For the resolution correction we use a conversion curve (see Figure 4 in LKD06) to transform each measured standard deviation value to width. The curve has been obtained assuming a Gaussian PSF with a full width at half maximum of 2.25 pixels and loop cross- sections that are circular and uniformly filled. The chosen PSF width is an upper limit for values obtained in different studies (Golub et al. 1999, Gburek, Sylwester & Martens 2006). The resolution correction curve plays a two-fold role. First, it allows us to obtain a more realistic width value from a measured quantity like the standard deviation. Second, it is a filter for measurements that are clearly unreliable. Standard deviation measurements smaller than a minimum value equal to the standard deviation of the PSF itself (where the conversion curve crosses the abscissa axis, see LKD06 Figure 4) are considered untrustworthy, and the corresponding width is set to zero (i.e., rejected). Problems resulting from significant errors in the background subtraction can also be identified in this way. It is worth remarking that our approach is quite cautious in that the PSF we have assumed is wider than the most recent estimates (see Gburek, Sylwester & Martens 2006). Some of the measurements we reject as being unresolved may in fact be valid. – 5 – Figure 2 is a plot of width (asterisks) versus position along the loop shown in Figure 1. The horizontal line corresponds to the average width. It is nearly identical to the mean width of all the loops in the set. The three “zero width” values that lie on the abscissa axis correspond to standard deviation measurements that were below the resolution limit as explained above. To quantify the expansion of loops from footpoint to top we defined expansion factors as follows: Γm/se = Ws +We , Γm/s = and Γm/e = , (1) where Wm, Ws and We are the average width of portions that cover 15% of the loop length at the middle, start footpoint, and end footpoint, respectively. Start and end refer to the magnetic field line traces used to define the magnetic flux tubes in the extrapolation models. The model flux tubes expand much more than the corresponding observed loops (LKD06). Their expansion factors are 1.5 to 2 times larger. As explained in Section 1, the loop width fluctuates as much as 25% over short dis- tances (see e.g., Figure 2). We tried alternate measures of the loop width (full width at half maximum and equivalent width of the intensity profile), and the same fluctuations are present. Our conclusion is that the fluctuations are most likely due to the influence of the background (see below). Since these fluctuations have a short length scale and vary quasi randomly around a global trend, they do not significantly affect the measured expansion factors. 3. Synthetic loops In this study we create a set of synthetic loops with similar characteristics to the TRACE loops studied in LKD06, and we overlay them on real TRACE backgrounds. The axis of the loop is linear and its cross-section is circular. To analyze the possibility that the apparent constant width is due to a resolution effect we create loops of two kinds: loops with constant diameter along their length, and loops that are wider in the middle than at the footpoints. For the second class of synthetic loops we use an expansion factor that is typical of the model flux tubes obtained in LKD06. We set the diameter of the loop at the mid point to be twice the diameter at the ends, and we assume that the diameter varies quadratically with position. Since the expansion factor defined in Equation (1) involves averages along 15% sections of the loop, Γm/se = 1.57 rather than 2.0. – 6 – We have chosen two kinds of background for the synthetic loops. Background I, shown in the top-left panel in Figure 3, corresponds to a typical TRACE loop background: it has similar intensity magnitude and fluctuations, and it contains moss (see Berger et al. 1999, Martens et al. 2000) and other intense features. Background II, shown in the top-right panel, is fainter and fluctuates less than Background I. Although it does not correspond very well to real loop backgrounds, we consider it interesting to study how this kind of background affects the width determination. The average intensities of backgrounds I and II are approximately 70 and 30 DN (Data Numbers), respectively. For comparison, the typical intrinsic (background subtracted) intensity of observed loops is between 20 and 40 DN. Both background areas have been extracted from a TRACE image in the 171 Å band obtained at 01:45 UT on July 30, 2002. To create a simulated TRACE image containing the synthetic loop we proceed as follows. We create an image of the loop without background. The maximum intensity of the loop (at the axis) is set proportional to the average intensity of the background image on which it will be later superposed. The constant of proportionality is referred to as the intensity factor Φ. Since the background intensity tends to be higher than the intrinsic loop intensity, Φ is generally smaller than 1. To simulate the finite resolution, we smooth the image of the loop using a gaussian profile with a full width at half maximum of 2.25 pixels corresponding to the instrument PSF. The resulting loop is then placed on the previously selected background (I or II) from the TRACE image. The images in Figure 3 have been created as described above. Both panels in each row use the same synthetic loop placed in one case on background I (left) and the other case on Background II (right). The four loops differ in the following ways. The loop in the second row (panels a and b) has a constant diameter of 4 pixels, corresponding to Γm/se = 1. The loop in the third row (panels c and d) has a diameter that expands from 2.5 pixels at the ends to 5 pixels at the center, corresponding to Γm/se = 1.57. The loop in fourth row (panels e and f) has a constant diameter of 3 pixels. Finally, the loop in the bottom row (panels g and h) expands from 2 pixels at the ends to 4 pixels in the middle. Notice that the ends of this last loop are narrower than the PSF. The intensity factor Φ has been adjusted so that the resulting loops look similar, by eye, to typical TRACE loops. We used Φ = 0.5 for Background I and Φ = 0.7 for Background II. Considering the average intensities of Backgrounds I and II, this gives intrinsic loop intensities of around 35 DN and 25 DN, respectively. These values are consistent with the intrinsic intensities of observed loops. The photon statistical noise associated with TRACE data is given by N , where N is the number of photon counts per pixel (Handy et al. 1999). Since 1 DN corresponds to 12 photon counts, the photon noise as a percentage of the signal is: – 7 – PN% = 100 , (2) where I is the intensity of the signal in DN/pix. The synthetic loop data constructed here includes the photon noise present in the TRACE image used for the background. As we now demonstrate, this contribution dominates the noise from the loop itself, so we can safely ignore the loop contribution. Let us first consider the extreme case of low background and loop intensities, namely: Ib = 30 DN and Il = 10 DN. According to Equation (2) the photon noise of the total signal is PN% = 30/ 40 or 4.7%. On the other hand, for our synthetic images (photon noise from the background only) it is PN% = 30/ 30 or 4.1%, meaning a difference of 0.6%. For a more typical case of Ib = 70 DN and Il = 25 DN, the same percentages are 3.1% and 2.6% respectively, implying a difference of 0.5% of the total signal. These differences are minor and will have a negligible effect on the results of the following sections. 4. Results 4.1. Can we detect expanding loops? From the set of loop images, we measured the width following the same procedure used in LKD06 for real loops and described in Section 2. We used the conversion curve (Figure 4 in LKD06) to correct for the instrument PSF. The non-linearity of the curve increases the dispersion of the resulting widths at smaller values approaching the width of the PSF. In Figure 4 we plot the “measured” width (asterisks) versus position along the loop for the eight cases in Figure 3. The format of the figures is the same. For comparison, we also plot as continuous lines the actual diameters used to construct the images. It can be seen that, despite the fluctuations, expanding and constant width loops are clearly distinguishable. This is true for loops that are relatively wide (top two rows) and loops that are relatively narrow (bottom two rows). This demonstrates convincingly that, if loops expanded as expected from standard force-free extrapolation models, then it would be noticeable from observations even when they are very close to the resolution limit (last row). Since that it not the case, this may imply that actual magnetic fields have more complexity than is present in the standard models. We know, for example, that the field is comprised of many thin flux strands (elemental kilogauss tubes) that are tangled by photospheric convection. We believe this can explain the symmetry of observed loops with respect to their summit (see discussion in LKD06 and Klimchuk 2006), but whether it can also explain the lack of a general expansion with height is unclear. – 8 – It is interesting to note from the plots in Figure 4 that the measured width is systemat- ically smaller than the width set in the construction of the loops. This tendency appears in all loops and is very likely due to an underestimation of the real width in the measurement procedure. The procedure requires a subjective selection of the loop edges for the purpose of defining the background and computing the standard deviation. During this step one can miss the faint tail of the cross-axis intensity profile that blends in with the background. We have verified that there is a tendency to define the loop edges to be slightly inside the actual edges. This causes the measured width to be artificially small, both because the tail of the profile is missing from the standard deviation computation and because too strong a background is subtracted from the loop. We expect the effect to be greatest for loops that are especially faint or especially narrow, as discussed below. If this explanation is correct, we can conclude that the TRACE loops studied in LKD06 are actually slightly wider than our measurements seem to indicate. The fact that the measured width is a lower bound for the real width gives further support to our assertion that the analyzed loops are instrumentally resolved. In LDK06, we estimated the width uncertainties associated with background subtraction by repeating each measurement using different choices for the loop edges. We concluded that rule-of-thumb error bars range from 10% below to 20% above the measured best value. It now appears that the actual error bars may be somewhat larger. However, we stress that this does not impact our ability to distinguish expanding loops from non-expanding loops, as is readily apparent from Figure 4. To quantify this claim, we computed the expansion factors (Γm/se in Equation 1) of all the synthetic loops shown in Figure 4. These are listed in Table 1. The upper and lower limits that define the error bars are the expansion factors Γm/s and Γm/e. For comparison, in the case of the observed loop of Figures 1 and 2, the expansion factor computed in the same way is 1.03±0.04. The values given in Table 1 clearly confirm our conclusion that loops with constant and expanding cross section can be easily distinguished. It is interesting to note that the expansion factors for the same loops placed in different backgrounds can be notably different. The same is true for loops of the same kind (expanding or not) but with different characteristic size (wide or narrow). Compare row 1 with row 3, and row 2 with row 4 in Figure 4 and the table. The error bars are also different in all cases. Part of these differences may be due to the subjective part of the analysis procedure (the selection of the loop edges). However, repeating the width measurements we obtain approximately the same expansion factors. Therefore, the distribution of the background emission and the characteristic size of the loop both play a role in determining the precise value of the expansion factors and the error bars. In particular, Backgrounds I and II tend to give an under and over estimation of the expansion factors, respectively (compared to the values set during the loop construction). – 9 – Nevertheless, we want to stress that the measured expansion factors of the expanding and non-expanding synthetic loops are clearly clustered around the actual values, implying that loops with constant and expanding cross section are readily distinguishable. Next, we study how the observed loop expansion is affected by the relative intensity of the loop compared to the background. To test this, we created synthetic data in the way described in Section 3, for different values of the loop-to-background intensity ratio Φ. In Figure 5 we plot the expansion factor Γm/se (Equation 1) versus Φ for 4 narrow synthetic loops with similar characteristics to those shown in panels e) to h) of Figures 3 and 4. The difference is that the loops of Figure 5 are 300 pixels long, instead of 200 pixels. The definition of Γ is not affected by the change of length. On the other hand, longer loops provide more measurements and better statistics for studying how loop expansion depends on the loop-to-background intensity ratio. We chose thin loops for Figure 5 because their expansion is more challenging to measure and they are more affected by the background. If the intensity ratio Φ is too small, it is difficult to detect a loop above the background, much less measure its width. Our previous studies of observed loops have therefore avoided such cases. We subjectively define a lower limit for loop visibility of around Φ = 0.3. Below that, the width determination is unreliable. The upper value Φ = 1.5 is extreme for most TRACE loops, but it is interesting for analysis and may be appropriate to other datasets. The intermediate Φ values are 0.5, 0.7 and 1.0. Figure 5 provides strong additional support for our claim that the expansion factors of expanding and non-expanding loops can be clearly distinguished, even for the most critical cases of very low intensities and narrow widths. In no case do the error bars of expanding and non-expanding loops overlap. It is interesting to note that the synthetic loops used for Figure 5 overlap with more of the background image than do the shorter synthetic loops used for Figures 3 and 4. The footpoint and middle sections therefore combine with different portions of the background. Since the expansion factors are qualitatively similar in the corresponding cases, we can be confident that our results are not an artifact of the particular loop-background combinations. So far we have not considered loops that are completely below the resolution limit. In Figure 6 we show two cases of unresolved synthetic loops. Both have a constant diameter of 0.5 pix, and both use Background I (Figure 3). The loops differ only in the intensity ratio Φ, which is set to 1 for case (a) and 3 for case (b). Note, however, that because the loops occupy only a fraction of a pixel, the “observed” intensity ratios are much smaller: around 0.25 for the Φ=1 loop and 0.5 for the Φ=3 loop. Figure 7 shows the widths of the loops as measured in the usual way, including correction for instrument resolution. A majority of the measurements are equal to zero, meaning that – 10 – the computed standard deviation is below that of the PSF. This is especially true for the fainter loop of case (a). We can understand this behavior as follows. Due to the influence of the variable background, we expect some measurements to be too large and others to be too small. However, because of the systematic effects associated with loop edge selection, discussed above, we expect more of the measurements to be too small. The conversion from standard deviation to width is very sensitive at small values, where the conversion curve is nonlinear, and it only takes small errors in the standard deviation to produce a zero width value. The solid line in Figure 7 indicates the actual loop width of 0.5 pix, while the dashed line indicates the full width at half maximum of the PSF. The most important conclusion to draw from the figure is that our measurement technique can easily detect when loops are unresolved, i.e., when they are thinner than the PSF. As we stated before, the loops analyzed in LKD06 and previous works are all wider than the PSF (see also Section 5 below). Finally, in Figure 8 we plot width versus position along a synthetic loop with a footpoint width of 2.5 pixels and a model expansion factor of Γm/se = 2.2. Our measurement procedure tracks the loop expansion very well. The expansion factor computed from the observed width as in Table 1 gives Γm/se = 2.1± 0.2. Therefore, the loop can be readily distinguished from the Γm/se = 1.57 loop having the same footpoint size in Figure 4, panel (d). 4.2. Short length scale width fluctuations As discussed in Section 1 the measured widths of observed loops fluctuate as much as 25% over short length scales. It is important to know whether these variations are real or an artifact of the background. Comparison of panels a) and b) in Figure 4 with Figure 2 shows that synthetic and observed loops with similar characteristic width exhibit similar width fluctuations. For the observed loop of Figure 2, the amplitude of the fluctuations computed as the ratio of the standard deviation of the measured width to its average is 18%. The corresponding ratios for the synthetic loops of panels (a) and (b) in Figure 4, are 17% and 25%, respectively. This suggests that the fluctuations are not real and argues against loops being comprised of a small number of braided strands (the possibility that they are bundles of many tangled strands is not affected). This is not a firm conclusion, however, since the fluctuations are somewhat more coherent for the observed loop than for the synthetic loop. We return to this issue below. Narrower loops (e.g., panels (e) and (f) in Figure 4) show larger amplitude fluctuations (21% and 38%, respectively) mostly because of the non-linearity of the resolution correction curve (LKD06 Figure 4), which exaggerates differences at smaller widths. – 11 – To study how the background fluctuations affect the width determination, we analyze the relationships between the width and the loop and background intensities. In Figure 9 we plot as a function of position along the loop: the intensity of the background pixels at either side of the loop (from which the loop background is linearly interpolated; continuous lines), the loop width (dotted), the loop intensity (maximum intensity of the background-subtracted profile; dashed), and the absolute value of the difference between the two background pixel intensities (dot-dashed). The loop width is given in pixels and multiplied by 10 for easier comparison with the intensities. The upper panel corresponds to the observed loop example of Figures 1 and 2, and the lower panel correspond to the synthetic loop of Figures 3 and 4, panels (a). Figure 9 shows that our synthetic loop data share the main qualitative characteristics of real loops. The fluctuations of the background intensity and its difference at the sides of the loop, and the loop intrinsic intensity and its fluctuations, are similar in both cases. There are obvious differences due to the spatial structure unique to each background that can easily be identified in the images. For example, the bumps between 30 and 60 for the observed loop, and between positions 0 and 40 and between 90 and 130 for the synthetic loop, can be traced to patches of enhanced emission in Figures 1 and 3. Another difference is the global variation in the intensity of the observed and synthetic loops. The measured intensity of the synthetic loop is nearly constant because the loop was constructed that way (small fluctuations come entirely from imperfect background subtraction). The measured intensity of the observed loop, on the other hand, tends to diminish systematically toward the right end. This is likely to be real and not an artifact of the background subtraction. Despite of these expected differences, the comparison shows that the synthetic loops reproduce the main properties of the observed cases. We have suggested that small-scale fluctuations of the measured intensities and widths of loops are due to imperfect background subtraction. To further assess this, we look for statistical correlations between these quantities. In the upper panels of Figure 10 we plot width versus intensity for all positions along the observed and synthetic loops, respectively. We find that there is a small direct correlation between the width and intensity in both cases: wider sections of the loops tend to be brighter. The lines in the scatter plots are least- squares fits, which have the indicated slopes and intercepts. The correlation between width and intensity can be explained by the tendency, during the interactive analysis procedure, to miss the wings of the intensity profile and define the loop edges to be inside the true edges. As described earlier, this causes an over estimation of the background intensity and produces artificially narrow loop widths and artificially faint loop intensities. We expect the magnitude of this effect to vary depending on the brightness of the background relative to the loop. It will be stronger (i.e., the underestimates of width and intensity will be greater) – 12 – when the background is relatively bright. This is confirmed in the second row of Figure 10. It shows an inverse correlation between the measured width and the background-to-loop intensity ratio for both the observed and synthetic loops. The background intensity used here is the average of the sloping background subtracted during the analysis (i.e., the average of the values on either side of the loop). Notice also that for the synthetic loop, the measured width tends to be smaller than the model width (4 pixels) when the relative intensity of the background is larger. This effect is almost certainly responsible for the width-intensity correlation of the synthetic loop and seems a likely explanation for the observed loop, as well. Whether it is strong enough to allow the possibility that loops are bundles of a few (3-5) intertwined strands is unclear. Recall that such loops would exhibit an inverse correlation between width and intensity if the measurements were perfect. Are measurement errors large enough to negate this inverse correlation and produce a small direct correlation, as observed? Only more involved modeling can answer this question. It seems plausible that cross-loop gradients in the background could also have an effect on the measured width. Certainly small scale inhomogeneities are more difficult to subtract than a flat background. In the upper panels of Figure 11, we plot width versus the absolute value of the background intensity difference on the two sides of the loop. No correlation is apparent for either the observed or synthetic loops. We confirmed a lack of correlation using a non-parametric statistical analysis. We also find no correlation between the intrinsic loop intensity and the background intensity difference. The right bottom panel of Figure 11 indicates how the known error in width measure- ment for the synthetic loop correlates with the background intensity gradient. The ordinate is the absolute value of the difference between the measured width and the width used dur- ing the loop construction. The abscissa is the absolute value of the background intensity difference on the two sides, normalized by the loop intensity. The normalization is meant to compensate for the fact that background gradients should have a lesser impact on bright loops. The left bottom panel of Figure 11 is a corresponding scatter plot for the observed loop. Since the actual width is not known, the ordinate is replaced by the absolute value of the deviation of the measured width from its mean. In neither case is there a correlation, as confirmed by statistical analysis. We conclude that the magnitude of the background has a bigger effect on the width measurements than does the difference in the background on the two sides of the loop (the cross loop gradient). – 13 – 5. Discussion and conclusion In this paper we study the effect of the background and the instrument PSF in the determination of the apparent width of EUV coronal loops observed by TRACE. Our main motivation is to extend the results obtained in our previous work: López Fuentes, Klimchuk & Démoulin (2006; LKD06). There, we compared a set of observed TRACE loops with corresponding force-free model flux-tubes, and we found that observed loops do not expand with height as expected from the extrapolation model. Here, we construct artificial loops with expansion factors similar to those of the studied loops and the model flux-tubes, and we overlay them on real TRACE backgrounds. We repeat on these synthetic loops the same procedure followed in LKD06, and compare the results back with real loops. We find that even for loops close to the resolution limit the procedure followed in LKD06 discerns expanding and non-expanding cross-sections. The method includes a resolution correction that identifies measurements that are below the resolution limit and therefore unreliable (see explanation in Section 2). We used a gaussian Point Spread Function (PSF) for the instrument with a FWHM of 2.25 pixels, which is an upper bound for values found by different authors (Golub et al. 1999, Gburek et al. 2006). In a recent paper, DeForest (2007) has proposed the interesting idea that most thin individual loops observed by TRACE are actually extremely bright structures well under the resolution limit. In this scenario, the loop apparent width would be given by the instrument PSF. In this way, loops may actually expand, but their size both at the top and the footpoints would be unresolved and would appear the same. The motivations for this conjecture are the apparent constant width of loops, and the observation that TRACE loops have an intensity scale height that is considerably larger than expected for static equilibrium (Winebarger et al. 2003, Aschwanden et al. 2001) or steady flow (Patsourakos & Klimchuk 2004). More precisely, for expanding loops that are everywhere unresolved, the density gradient present in the corona is larger than inferred from the observations under the assumption of constant cross section. According to the above explanation, we should expect all individual TRACE loops to have a true width less than that of the PSF and an apparent width roughly equal to that of the PSF. However, observations do not support this. The mean width of the loops studied in LKD06 is 4.2 pix after correction for the instrument resolution (see also Watko & Klimchuk 2000). As shown in Figure 7 and discussed in Section 4.1, our method can easily identify loops that are intrinsically more narrow than the PSF. The loops selected for our studies are clearly not of this type. As we discussed in Section 1, coronal structures that are many times wider than the TRACE PSF are observed to be formed by thinner individual loops. Therefore, there is an – 14 – intermediate range of widths – let us say between one and three PSF widths – for which the profiles produced by unresolved threads could overlap to form apparently wider loops. This, together with the effect of a fluctuating and intense background, are the arguments provided by DeForest (2007) to explain loops wider than the PSF. A key point in this discussion is that unresolved neighbor threads might be expected to separate from each other with height for the same reasons that individual strands might be expected to expand with height (e.g., if the field behaves like simple force-free extrapolation models predict). In this respect, a structure formed by diverging threads does not differ from the expanding loops studied in Section 3. As we discussed there, the plots in Figures 4 and 5 and the Γ factors given in Table 1 show clearly that our procedure for the width determination would be able to detect the expansion if it existed, even for loops near the resolution limit. It is interesting to compare the synthetic images in Figure 2 in DeForest’s article with our Figure 3. There, he claims that synthetic loops made from a single unresolved thread of constant width and from two diverging threads are indistinguishable from each other and from actual TRACE loops. In our Figure 3 it is also very difficult, by eyeball, to determine which loops have expanding widths or constant widths. However, the plots in Figure 4 show that a careful examination through a quantitative measurement provides the answer. One of DeForest’s main arguments is that it is difficult to measure the width of features that are at or near the instrument resolution due to effects such as the smearing from the telescope, pixilation from the detector, and the presence of background emission. We agree, but these claims need to be quantified. It is not sufficient to make eye-ball comparisons of features. Quantitative measures must be used. We have adopted the standard deviation of the loops cross-axis intensity profile as one such measure. We have been very careful in our work to indicate when the measurements are reliable and when they are not. Measured widths that are very close to the instrument resolution have very large error bars that we show (see LKD06) and that we take into account. We have paid particularly careful attention to the effects of the combined PSF, which accounts for both smearing and pixilation. DeForest is correct that measurements of very thin features depend critically on the PSF. We have therefore adopted a conservative value for the PSF width that is greater than the estimates determined by the instrument teams and others. Furthermore, features as narrow as our assumed PSF are routinely observed, which would not be possible if the actual PSF were wider. DeForest is also correct that background emission can be important and may lead to spurious results. It is therefore vital to subtract the background before making measure- ments, as we have done. In LKD06, we have avoided loops where the background is especially bright or complicated. We attempted in our earlier studies to estimate the uncertainties as- – 15 – sociated with imperfect background subtraction, but this was not as careful as our treatment of resolution effects. The main purpose of this paper was to rigorously evaluate the effects of background on the measurement of loop widths. Regarding the importance of quantitative measurements versus visual inspection, we concur with DeForest that the visual determination of the edge of a feature is subjective and largely based on the intensity gradient across the feature. This can lead to erroneous conclusions about width variations if there is a systematic variation of intensity along the feature, such as decreasing intensity with height. Our quantitative measure of width based on the standard deviation of the intensity profile is by construction moderating such bias. The positive correlation found between the loop width and the maximum intensity (top panels in Figure 10) could be a remnant of this effect or an intrinsic property of the loops. DeForest correctly points out that, with optically-thin coronal emission, the observed intensity scale height of a hydrostatic structure is larger for an expanding loop than for a constant cross section loop, especially if the loop is unresolved. In fact, for the 1-2 MK model examples he shows (Figures 5 and 6), the intensity actually increases with height by a factors of 2-3 to a maximum brightness at altitudes near 7 × 109 cm). Whether actual TRACE loops have this property is unknown and should be investigated. The variation of temperature with height combined with the transmission properties of the filter used will complicate the interpretation. We note that the observation of super-hydrostatic scale heights is different from the observation of excess densities in TRACE loops. For most TRACE loops, the density inferred from the observed emission measure and diameter is much larger than that expected from static equilibrium theory, given the observed temperature and loop length (Aschwanden et al. 2001, Winebarger et al. 2003). DeForest’s idea of unresolved loops would make this discrepancy even worse, since a higher density is required to produce the same emission measure from a smaller volume. The loops identified and measured by DeForest are qualitatively much different from the loops identified and measured in our studies. We chose cases that are not obviously composed of a few resolved or quasi-resolved strands (although we believe that our loops may be composed of large numbers of elemental strands that are far below the resolution limit). The only one of his loops with no apparent internal structure (Loop 6 in his Figure 8) would have not been selected by us, because it is barely discernable above the background. On the other hand, some of the thinner structures within DeForest’s loop bundles (e.g., at the bottom edge of his Loop 3) are not unlike the loops we have investigated. In this regard, we must clarify a comment attributed to one of us at the end of Section 5 in his paper. We suggest that researchers seeking to study monolithic-looking loops will tend to select cases – 16 – that are only a few resolution elements across. Significantly wider loops (e.g, all except Loop 6 in DeForest’s sample) usually show evidence of internal structure and will be rejected. We agree fully with DeForest that collections of loops (loop bundles) expand appreciably with height. However, we stand by our claim that individual loops that are clearly discernable within a bundle have a much more uniform width. This is not an artifact of the resolution. A hare and hounds exercise, as currently planned, is one useful way to clarify any remaining differences of opinion. An important topic of the present study has been the analysis of how the properties of the background affect the loop width determination. We searched for correlations between the width and: the loop intrinsic intensity, the background intensity to loop intensity ratios, and the absolute value of the background difference. The background intensity is computed as the average between the pixels at both sides of the loop, which is used for the estimation of the actual loop background, while the background difference is the difference between those pixels. We found a direct correlation between the width and the maximum intensity of the loop profile (see Figure 10, upper panels). This is probably due to the fact that we tend to miss the “tails” of the loop profile at positions where the loop is less intense with respect to the background, and therefore, the measured profile tends to be narrower. This is confirmed by the inverse correlation found between the width and the ratio of background intensity to loop intensity (see Figure 10, bottom panels). It can be seen from the plots that the width tends to be abnormally narrower, and the points more disperse, for larger background to loop intensity ratios. We found no evidence of correlation between the width and the the background differ- ence. This shows that the background gradients are less important in the determination of the width than the background relative intensity. We stress, however, that this does not affect our ability to determine the global expansion properties of loops, and that despite of the background contribution we are readily able to distinguish constant width loops from loops that expand as predicted from simple force-free magnetic models. The results presented here are extremely intriguing and provide clues and new questions to guide future investigations. However, it is expected that definitive answers will come from improved observations using new generations of solar instruments with higher resolution. We acknowledge the Transition Region and Coronal Explorer (TRACE ) team. We wish to thank Craig DeForest for fruitful discussions about the nature of observed loops. We also thank our anonymous referee for his/her valuable suggestions and comments. The authors acknowledge financial support from CNRS (France) and CONICET (Argentina) through their cooperative science program (N0 20326). MLF thanks the Secretary of Science and – 17 – Technology of Argentina, through its RAICES program, for travel support. This work was partially funded by NASA and the Office of Naval Research. REFERENCES Aschwanden,M.J., Nightingale,R.W., & Boerner,P. 2007, ApJ, 656, 577 Aschwanden, M. J., Schrijver, C. J., & Alexander, D. 2001, ApJ, 550, 1036 Berger, T. E., De Pontieu, B., Fletcher, L., Schrijver, C. J., Tarbell, T. D., & Title, A. M. 1999, Sol. Phys., 190, 409 DeForest, C. E. 2007, ApJ, 661, 532 Gburek, S., Sylwester, J., & Martens, P. 2006, Sol. Phys., 239, 531 Golub, L., et al. 1999, Physics of Plasmas, 6, 2205 Handy, B. N., et al. 1999, Sol. Phys., 187, 229 Klimchuk, J. A. 2000, Sol. Phys., 193, 53 Klimchuk, J. A., Antiochos, S. K., & Norton, D. 2000, ApJ, 542, 504 Klimchuk, J. A. 2006, Sol. Phys., 234, 41 Klimchuk, J. A., Lemen, J. R., Feldman, U., Tsuneta, S., & Uchida, Y. 1992, PASJ, 44, López Fuentes, M. C., Klimchuk, J. A., & Démoulin, P. 2006, ApJ, 639, 459 (LKD06) Martens, P. C. H., Kankelborg, C. C., & Berger, T. E. 2000, ApJ, 537, 471 Patsourakos, S. & Klimchuk, J. A. 2004, ApJ, 603, 322 Scherrer, P. H. et al. 1995, Sol. Phys., 162, 129 Schmelz, J. T., & Martens, P. C. H. 2006, ApJ, 636, L49 Watko, J. A. & Klimchuk, J. A. 2000, Sol. Phys., 193, 77 Winebarger, A. R., Warren, H. P., & Mariska, J. T. 2003, ApJ, 587, 439 This preprint was prepared with the AAS LATEX macros v5.2. – 18 – Table 1: Expansion factors Γm/se (Equation 1) for the synthetic loops shown in Figures 3 and 4 (see detailed explanation in Section 4.1). Synthetic loop Imposed Background I Background II Const. width (4 pix) 1 0.85± .02 0.95± .04 Variable width (2.5-5 pix) 1.57 1.38± .25 1.76± .01 Const. width (3 pix) 1 0.82± .05 1.03± .03 Variable width (2-4 pix) 1.57 1.59± .28 2.11± .20 – 19 – Fig. 1.— The loop shown is an example of the TRACE loops studied in LKD06. The lower panel shows the straightened version of the loop that is used for the width determination. – 20 – Fig. 2.— Measured width versus position along the loop of Figure 1. Background subtraction and PSF correction have been applied, as described in Section 2. The horizontal line shows the mean value. – 21 – Fig. 3.— Synthetic data created by superposing loops with different specified properties on real TRACE backgrounds. The top panels show the background used in each column. The ends (footpoints) and middle of the loops have an imposed diameter (in pixels) of: (a,b) 4-4, (c,d) 2.5-5, (e,f) 3-3, (g,h) 2-4. This provides both constant and expanding (by a factor 2) synthetic loops close to the spatial resolution. For a detailed description of the panels see Section 3. Fig. 4.— Measured width corrected for instrument resolution (asterisks) versus position along the synthetic loops shown in panels a) to h) of Figure 3. Continuous lines indicate the model width. – 23 – Fig. 5.— Expansion factor Γm/se versus loop-to-background intensity ratio Φ, for expanding and non-expanding synthetic narrow loops on backgrounds I and II (similar to loops in panels e-h of Figure 4). The error bars are defined by the expansion factors Γm/s and Γm/e (see Section 4.1). The horizontal line indicates the expansion factor of the model. – 24 – Fig. 6.— Two examples of unresolved synthetic loops constructed with a constant width of 0.5 pixels. The loops differ only in the loop-to-background intensity ratio Φ, which is set to 1 for case (a) and 3 for case (b) (see Section 4.1). – 25 – Fig. 7.— Width measurements for the two synthetic loops in Figure 6, corrected for the instrument resolution. The solid line indicates the actual model loop width, and the dashed line indicates the PSF full width at half maximum. – 26 – Fig. 8.— Width versus position along a synthetic loop with a model expansion factor of 2.2. The width has been corrected for instrument resolution. – 27 – Fig. 9.— Different loop and background properties versus position along the loop. Top panel: example loop from Figure 1; botton panel: synthetic loop from Figure 3, panel a). For a detailed description see Section 4.2. Fig. 10.— Scatter plots of measured quantities for the observed loop in Figure 1 (left column) and for the synthetic loop of Figures 3 and 4, panels (a) (right column). Top: width versus on-axis loop intensity. Bottom: width versus background intensity to loop intensity ratio (see Section 4.2). Continuous lines correspond to least-squares fits of the data. Slopes and intercepts are given in the respective panels. Fig. 11.— Scatter plots of measured quantities for the observed loop in Figure 1 (left column) and for the synthetic loop of Figures 3 and 4, panels (a) (right column). Top: width versus absolute value of the background intensity difference across the loop. Bottom left: width deviation from the mean versus the background intensity difference normalized by the loop intensity (see Section 4.2). Bottom right: same kind of plot, but the deviation is relative to the model width. Introduction Observed TRACE loops Synthetic loops Results Can we detect expanding loops? Short length scale width fluctuations Discussion and conclusion

0704.0638

Polarizations of J/psi and psi(2S) Mesons Produced in ppbar Collisions at 1.96 TeV

Polarizations of J/ψ and ψ(2S) Mesons Produced in pp Collisions at s = 1.96 TeV A. Abulencia,24 J. Adelman,13 T. Affolder,10 T. Akimoto,55 M.G. Albrow,17 S. Amerio,43 D. Amidei,35 A. Anastassov,52 K. Anikeev,17 A. Annovi,19 J. Antos,14 M. Aoki,55 G. Apollinari,17 T. Arisawa,57 A. Artikov,15 W. Ashmanskas,17 A. Attal,3 A. Aurisano,42 F. Azfar,42 P. Azzi-Bacchetta,43 P. Azzurri,46 N. Bacchetta,43 W. Badgett,17 A. Barbaro-Galtieri,29 V.E. Barnes,48 B.A. Barnett,25 S. Baroiant,7 V. Bartsch,31 G. Bauer,33 P.-H. Beauchemin,34 F. Bedeschi,46 S. Behari,25 G. Bellettini,46 J. Bellinger,59 A. Belloni,33 D. Benjamin,16 A. Beretvas,17 J. Beringer,29 T. Berry,30 A. Bhatti,50 M. Binkley,17 D. Bisello,43 I. Bizjak,31 R.E. Blair,2 C. Blocker,6 B. Blumenfeld,25 A. Bocci,16 A. Bodek,49 V. Boisvert,49 G. Bolla,48 A. Bolshov,33 D. Bortoletto,48 J. Boudreau,47 A. Boveia,10 B. Brau,10 L. Brigliadori,5 C. Bromberg,36 E. Brubaker,13 J. Budagov,15 H.S. Budd,49 S. Budd,24 K. Burkett,17 G. Busetto,43 P. Bussey,21 A. Buzatu,34 K. L. Byrum,2 S. Cabreraq,16 M. Campanelli,20 M. Campbell,35 F. Canelli,17 A. Canepa,45 S. Carilloi,18 D. Carlsmith,59 R. Carosi,46 S. Carron,34 B. Casal,11 M. Casarsa,54 A. Castro,5 P. Catastini,46 D. Cauz,54 M. Cavalli-Sforza,3 A. Cerri,29 L. Cerritom,31 S.H. Chang,28 Y.C. Chen,1 M. Chertok,7 G. Chiarelli,46 G. Chlachidze,17 F. Chlebana,17 I. Cho,28 K. Cho,28 D. Chokheli,15 J.P. Chou,22 G. Choudalakis,33 S.H. Chuang,52 K. Chung,12 W.H. Chung,59 Y.S. Chung,49 M. Cilijak,46 C.I. Ciobanu,24 M.A. Ciocci,46 A. Clark,20 D. Clark,6 M. Coca,16 G. Compostella,43 M.E. Convery,50 J. Conway,7 B. Cooper,31 K. Copic,35 M. Cordelli,19 G. Cortiana,43 F. Crescioli,46 C. Cuenca Almenarq,7 J. Cuevasl,11 R. Culbertson,17 J.C. Cully,35 S. DaRonco,43 M. Datta,17 S. D’Auria,21 T. Davies,21 D. Dagenhart,17 P. de Barbaro,49 S. De Cecco,51 A. Deisher,29 G. De Lentdeckerc,49 G. De Lorenzo,3 M. Dell’Orso,46 F. Delli Paoli,43 L. Demortier,50 J. Deng,16 M. Deninno,5 D. De Pedis,51 P.F. Derwent,17 G.P. Di Giovanni,44 C. Dionisi,51 B. Di Ruzza,54 J.R. Dittmann,4 M. D’Onofrio,3 C. Dörr,26 S. Donati,46 P. Dong,8 J. Donini,43 T. Dorigo,43 S. Dube,52 J. Efron,39 R. Erbacher,7 D. Errede,24 S. Errede,24 R. Eusebi,17 H.C. Fang,29 S. Farrington,30 I. Fedorko,46 W.T. Fedorko,13 R.G. Feild,60 M. Feindt,26 J.P. Fernandez,32 R. Field,18 G. Flanagan,48 R. Forrest,7 S. Forrester,7 M. Franklin,22 J.C. Freeman,29 I. Furic,13 M. Gallinaro,50 J. Galyardt,12 J.E. Garcia,46 F. Garberson,10 A.F. Garfinkel,48 C. Gay,60 H. Gerberich,24 D. Gerdes,35 S. Giagu,51 P. Giannetti,46 K. Gibson,47 J.L. Gimmell,49 C. Ginsburg,17 N. Giokarisa,15 M. Giordani,54 P. Giromini,19 M. Giunta,46 G. Giurgiu,25 V. Glagolev,15 D. Glenzinski,17 M. Gold,37 N. Goldschmidt,18 J. Goldsteinb,42 A. Golossanov,17 G. Gomez,11 G. Gomez-Ceballos,33 M. Goncharov,53 O. González,32 I. Gorelov,37 A.T. Goshaw,16 K. Goulianos,50 A. Gresele,43 S. Grinstein,22 C. Grosso-Pilcher,13 R.C. Group,17 U. Grundler,24 J. Guimaraes da Costa,22 Z. Gunay-Unalan,36 C. Haber,29 K. Hahn,33 S.R. Hahn,17 E. Halkiadakis,52 A. Hamilton,20 B.-Y. Han,49 J.Y. Han,49 R. Handler,59 F. Happacher,19 K. Hara,55 D. Hare,52 M. Hare,56 S. Harper,42 R.F. Harr,58 R.M. Harris,17 M. Hartz,47 K. Hatakeyama,50 J. Hauser,8 C. Hays,42 M. Heck,26 A. Heijboer,45 B. Heinemann,29 J. Heinrich,45 C. Henderson,33 M. Herndon,59 J. Heuser,26 D. Hidas,16 C.S. Hillb,10 D. Hirschbuehl,26 A. Hocker,17 A. Holloway,22 S. Hou,1 M. Houlden,30 S.-C. Hsu,9 B.T. Huffman,42 R.E. Hughes,39 U. Husemann,60 J. Huston,36 J. Incandela,10 G. Introzzi,46 M. Iori,51 A. Ivanov,7 B. Iyutin,33 E. James,17 D. Jang,52 B. Jayatilaka,16 D. Jeans,51 E.J. Jeon,28 S. Jindariani,18 W. Johnson,7 M. Jones,48 K.K. Joo,28 S.Y. Jun,12 J.E. Jung,28 T.R. Junk,24 T. Kamon,53 P.E. Karchin,58 Y. Kato,41 Y. Kemp,26 R. Kephart,17 U. Kerzel,26 V. Khotilovich,53 B. Kilminster,39 D.H. Kim,28 H.S. Kim,28 J.E. Kim,28 M.J. Kim,17 S.B. Kim,28 S.H. Kim,55 Y.K. Kim,13 N. Kimura,55 L. Kirsch,6 S. Klimenko,18 M. Klute,33 B. Knuteson,33 B.R. Ko,16 K. Kondo,57 D.J. Kong,28 J. Konigsberg,18 A. Korytov,18 A.V. Kotwal,16 A.C. Kraan,45 J. Kraus,24 M. Kreps,26 J. Kroll,45 N. Krumnack,4 M. Kruse,16 V. Krutelyov,10 T. Kubo,55 S. E. Kuhlmann,2 T. Kuhr,26 N.P. Kulkarni,58 Y. Kusakabe,57 S. Kwang,13 A.T. Laasanen,48 S. Lai,34 S. Lami,46 S. Lammel,17 M. Lancaster,31 R.L. Lander,7 K. Lannon,39 A. Lath,52 G. Latino,46 I. Lazzizzera,43 T. LeCompte,2 J. Lee,49 J. Lee,28 Y.J. Lee,28 S.W. Leeo,53 R. Lefèvre,20 N. Leonardo,33 S. Leone,46 S. Levy,13 J.D. Lewis,17 C. Lin,60 C.S. Lin,17 M. Lindgren,17 E. Lipeles,9 A. Lister,7 D.O. Litvintsev,17 T. Liu,17 N.S. Lockyer,45 A. Loginov,60 M. Loreti,43 R.-S. Lu,1 D. Lucchesi,43 P. Lujan,29 P. Lukens,17 G. Lungu,18 L. Lyons,42 J. Lys,29 R. Lysak,14 E. Lytken,48 P. Mack,26 D. MacQueen,34 R. Madrak,17 K. Maeshima,17 K. Makhoul,33 T. Maki,23 P. Maksimovic,25 S. Malde,42 S. Malik,31 G. Manca,30 F. Margaroli,5 R. Marginean,17 C. Marino,26 C.P. Marino,24 A. Martin,60 M. Martin,25 V. Marting,21 M. Mart́ınez,3 R. Mart́ınez-Ballaŕın,32 T. Maruyama,55 P. Mastrandrea,51 T. Masubuchi,55 H. Matsunaga,55 M.E. Mattson,58 R. Mazini,34 P. Mazzanti,5 K.S. McFarland,49 P. McIntyre,53 R. McNultyf ,30 A. Mehta,30 P. Mehtala,23 S. Menzemerh,11 A. Menzione,46 P. Merkel,48 C. Mesropian,50 A. Messina,36 T. Miao,17 N. Miladinovic,6 J. Miles,33 R. Miller,36 C. Mills,10 M. Milnik,26 A. Mitra,1 G. Mitselmakher,18 A. Miyamoto,27 S. Moed,20 N. Moggi,5 B. Mohr,8 C.S. Moon,28 R. Moore,17 http://arxiv.org/abs/0704.0638v2 M. Morello,46 P. Movilla Fernandez,29 J. Mülmenstädt,29 A. Mukherjee,17 Th. Muller,26 R. Mumford,25 P. Murat,17 M. Mussini,5 J. Nachtman,17 A. Nagano,55 J. Naganoma,57 K. Nakamura,55 I. Nakano,40 A. Napier,56 V. Necula,16 C. Neu,45 M.S. Neubauer,9 J. Nielsenn,29 L. Nodulman,2 O. Norniella,3 E. Nurse,31 S.H. Oh,16 Y.D. Oh,28 I. Oksuzian,18 T. Okusawa,41 R. Oldeman,30 R. Orava,23 K. Osterberg,23 C. Pagliarone,46 E. Palencia,11 V. Papadimitriou,17 A. Papaikonomou,26 A.A. Paramonov,13 B. Parks,39 S. Pashapour,34 J. Patrick,17 G. Pauletta,54 M. Paulini,12 C. Paus,33 D.E. Pellett,7 A. Penzo,54 T.J. Phillips,16 G. Piacentino,46 J. Piedra,44 L. Pinera,18 K. Pitts,24 C. Plager,8 L. Pondrom,59 X. Portell,3 O. Poukhov,15 N. Pounder,42 F. Prakoshyn,15 A. Pronko,17 J. Proudfoot,2 F. Ptohose,19 G. Punzi,46 J. Pursley,25 J. Rademackerb,42 A. Rahaman,47 V. Ramakrishnan,59 N. Ranjan,48 I. Redondo,32 B. Reisert,17 V. Rekovic,37 P. Renton,42 M. Rescigno,51 S. Richter,26 F. Rimondi,5 L. Ristori,46 A. Robson,21 T. Rodrigo,11 E. Rogers,24 S. Rolli,56 R. Roser,17 M. Rossi,54 R. Rossin,10 P. Roy,34 A. Ruiz,11 J. Russ,12 V. Rusu,13 H. Saarikko,23 A. Safonov,53 W.K. Sakumoto,49 G. Salamanna,51 O. Saltó,3 L. Santi,54 S. Sarkar,51 L. Sartori,46 K. Sato,17 P. Savard,34 A. Savoy-Navarro,44 T. Scheidle,26 P. Schlabach,17 E.E. Schmidt,17 M.P. Schmidt,60 M. Schmitt,38 T. Schwarz,7 L. Scodellaro,11 A.L. Scott,10 A. Scribano,46 F. Scuri,46 A. Sedov,48 S. Seidel,37 Y. Seiya,41 A. Semenov,15 L. Sexton-Kennedy,17 A. Sfyrla,20 S.Z. Shalhout,58 M.D. Shapiro,29 T. Shears,30 P.F. Shepard,47 D. Sherman,22 M. Shimojimak,55 M. Shochet,13 Y. Shon,59 I. Shreyber,20 A. Sidoti,46 P. Sinervo,34 A. Sisakyan,15 A.J. Slaughter,17 J. Slaunwhite,39 K. Sliwa,56 J.R. Smith,7 F.D. Snider,17 R. Snihur,34 M. Soderberg,35 A. Soha,7 S. Somalwar,52 V. Sorin,36 J. Spalding,17 F. Spinella,46 T. Spreitzer,34 P. Squillacioti,46 M. Stanitzki,60 A. Staveris-Polykalas,46 R. St. Denis,21 B. Stelzer,8 O. Stelzer-Chilton,42 D. Stentz,38 J. Strologas,37 D. Stuart,10 J.S. Suh,28 A. Sukhanov,18 H. Sun,56 I. Suslov,15 T. Suzuki,55 A. Taffardp,24 R. Takashima,40 Y. Takeuchi,55 R. Tanaka,40 M. Tecchio,35 P.K. Teng,1 K. Terashi,50 J. Thomd,17 A.S. Thompson,21 E. Thomson,45 P. Tipton,60 V. Tiwari,12 S. Tkaczyk,17 D. Toback,53 S. Tokar,14 K. Tollefson,36 T. Tomura,55 D. Tonelli,46 S. Torre,19 D. Torretta,17 S. Tourneur,44 W. Trischuk,34 R. Tsuchiya,57 S. Tsuno,40 Y. Tu,45 N. Turini,46 F. Ukegawa,55 S. Uozumi,55 S. Vallecorsa,20 N. van Remortel,23 A. Varganov,35 E. Vataga,37 F. Vazquezi,18 G. Velev,17 G. Veramendi,24 V. Veszpremi,48 M. Vidal,32 R. Vidal,17 I. Vila,11 R. Vilar,11 T. Vine,31 I. Vollrath,34 I. Volobouevo,29 G. Volpi,46 F. Würthwein,9 P. Wagner,53 R.G. Wagner,2 R.L. Wagner,17 J. Wagner,26 W. Wagner,26 R. Wallny,8 S.M. Wang,1 A. Warburton,34 D. Waters,31 M. Weinberger,53 W.C. Wester III,17 B. Whitehouse,56 D. Whiteson,45 A.B. Wicklund,2 E. Wicklund,17 G. Williams,34 H.H. Williams,45 P. Wilson,17 B.L. Winer,39 P. Wittichd,17 S. Wolbers,17 C. Wolfe,13 T. Wright,35 X. Wu,20 S.M. Wynne,30 A. Yagil,9 K. Yamamoto,41 J. Yamaoka,52 T. Yamashita,40 C. Yang,60 U.K. Yangj,13 Y.C. Yang,28 W.M. Yao,29 G.P. Yeh,17 J. Yoh,17 K. Yorita,13 T. Yoshida,41 G.B. Yu,49 I. Yu,28 S.S. Yu,17 J.C. Yun,17 L. Zanello,51 A. Zanetti,54 I. Zaw,22 X. Zhang,24 J. Zhou,52 and S. Zucchelli5 (CDF Collaboration∗) 1Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China 2Argonne National Laboratory, Argonne, Illinois 60439 3Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain 4Baylor University, Waco, Texas 76798 5Istituto Nazionale di Fisica Nucleare, University of Bologna, I-40127 Bologna, Italy 6Brandeis University, Waltham, Massachusetts 02254 7University of California, Davis, Davis, California 95616 8University of California, Los Angeles, Los Angeles, California 90024 9University of California, San Diego, La Jolla, California 92093 10University of California, Santa Barbara, Santa Barbara, California 93106 11Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain 12Carnegie Mellon University, Pittsburgh, PA 15213 13Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637 14Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia 15Joint Institute for Nuclear Research, RU-141980 Dubna, Russia 16Duke University, Durham, North Carolina 27708 ∗ With visitors from aUniversity of Athens, bUniversity of Bristol, cUniversity Libre de Bruxelles, dCornell University, eUniversity of Cyprus, fUniversity of Dublin, gUniversity of Edinburgh, hUniversity of Heidelberg, iUniversidad Iberoamericana, jUniversity of Manchester, kNagasaki Institute of Applied Science, lUniversity de Oviedo, mUniversity of London, Queen Mary College, nUniversity of California Santa Cruz, oTexas Tech University, pUniversity of California Irvine, and qIFIC(CSIC-Universitat de Valencia). 17Fermi National Accelerator Laboratory, Batavia, Illinois 60510 18University of Florida, Gainesville, Florida 32611 19Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy 20University of Geneva, CH-1211 Geneva 4, Switzerland 21Glasgow University, Glasgow G12 8QQ, United Kingdom 22Harvard University, Cambridge, Massachusetts 02138 23Division of High Energy Physics, Department of Physics, University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland 24University of Illinois, Urbana, Illinois 61801 25The Johns Hopkins University, Baltimore, Maryland 21218 26Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany 27High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305, Japan 28Center for High Energy Physics: Kyungpook National University, Taegu 702-701, Korea; Seoul National University, Seoul 151-742, Korea; SungKyunKwan University, Suwon 440-746, Korea 29Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720 30University of Liverpool, Liverpool L69 7ZE, United Kingdom 31University College London, London WC1E 6BT, United Kingdom 32Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain 33Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 34Institute of Particle Physics: McGill University, Montréal, Canada H3A 2T8; and University of Toronto, Toronto, Canada M5S 1A7 35University of Michigan, Ann Arbor, Michigan 48109 36Michigan State University, East Lansing, Michigan 48824 37University of New Mexico, Albuquerque, New Mexico 87131 38Northwestern University, Evanston, Illinois 60208 39The Ohio State University, Columbus, Ohio 43210 40Okayama University, Okayama 700-8530, Japan 41Osaka City University, Osaka 588, Japan 42University of Oxford, Oxford OX1 3RH, United Kingdom 43University of Padova, Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, I-35131 Padova, Italy 44LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France 45University of Pennsylvania, Philadelphia, Pennsylvania 19104 46Istituto Nazionale di Fisica Nucleare Pisa, Universities of Pisa, Siena and Scuola Normale Superiore, I-56127 Pisa, Italy 47University of Pittsburgh, Pittsburgh, Pennsylvania 15260 48Purdue University, West Lafayette, Indiana 47907 49University of Rochester, Rochester, New York 14627 50The Rockefeller University, New York, New York 10021 51Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, University of Rome “La Sapienza,” I-00185 Roma, Italy 52Rutgers University, Piscataway, New Jersey 08855 53Texas A&M University, College Station, Texas 77843 54Istituto Nazionale di Fisica Nucleare, University of Trieste/ Udine, Italy 55University of Tsukuba, Tsukuba, Ibaraki 305, Japan 56Tufts University, Medford, Massachusetts 02155 57Waseda University, Tokyo 169, Japan 58Wayne State University, Detroit, Michigan 48201 59University of Wisconsin, Madison, Wisconsin 53706 60Yale University, New Haven, Connecticut 06520 We have measured the polarizations of J/ψ and ψ(2S) mesons as functions of their transverse momentum pT when they are produced promptly in the rapidity range |y| < 0.6 with pT ≥ 5 GeV/c. The analysis is performed using a data sample with an integrated luminosity of about 800 pb−1 collected by the CDF II detector. For both vector mesons, we find that the polarizations become increasingly longitudinal as pT increases from 5 to 30 GeV/c. These results are compared to the pre- dictions of nonrelativistic quantum chromodynamics and other contemporary models. The effective polarizations of J/ψ and ψ(2S) mesons from B-hadron decays are also reported. PACS numbers: 13.88.+e, 13.20.Gd, 14.40.Lb An effective field theory, nonrelativistic quantum chromodynamics (NRQCD) [1], provides a rigorous formalism for calculating the production rates of charmonium (cc) states. NRQCD explains the direct production cross sections for J/ψ and ψ(2S) mesons observed at the Tevatron [2, 3] and predicts their increasingly transverse polarizations as pT increases, where pT is the meson’s momentum component perpendicular to the colliding beam direction [4]. The first polarization measurements at the Tevatron [5] did not show such a trend. This Letter reports on J/ψ and ψ(2S) polarization measurements with a larger data sample than previously available. This allows the extension of the measurement to a higher pT region and makes a more stringent test of the NRQCD prediction. The NRQCD cross section calculation for cc production separates the long-distance nonperturbative contributions from the short-distance perturbative behavior. The former is treated as an expansion of the matrix elements in powers of the nonrelativistic charm-quark velocity. This expansion can be computed by lattice simulations, but currently the expansion coefficients are treated as universal parameters, which are adjusted to match the cross section measurements at the Tevatron [2, 3]. The calculation also applies to cc production in ep collisions, but HERA measurements of J/ψ polarization tend to disagree with the NRQCD prediction [6]. These difficulties have led some authors to explore alternative power expansions of the long-distance interactions for the cc system [7]. There are also new QCD-inspired models, the gluon tower model [8] and the kT -factorization model [9], that accomodate vector-meson cross sections at both HERA and the Tevatron and predict the vector-meson polarizations as functions of pT . These authors emphasize that measuring the vector-meson polarizations as functions of pT is a crucial test of NRQCD. The CDF II detector is described in detail elsewhere [3, 10]. In this analysis, the essential features are a muon system covering the central region of pseudorapidity, |η| < 0.6, and the tracking system, immersed in the 1.4 T solenoidal magnetic field and composed of a silicon microstrip detector and a cylindrical drift chamber called the central outer tracker (COT). The data used here correspond to an integrated luminosity of about 800 pb−1 and were recorded between June 2004 and February 2006 by a dimuon trigger, which requires two opposite-charge muon candidates, each having pT > 1.5 GeV/c. Decays of vector mesons V (either J/ψ or ψ(2S)) → µ+µ− are selected from dimuon events for which each track has segments reconstructed in both the COT and the silicon microstrip detector. The pT of each muon is required to exceed 1.75 GeV/c in order to guarantee a well-measured trigger efficiency. The muon track pair is required to be consistent with originating from a common vertex and to have an invariant mass M within the range 2.8 (3.4) < M < 3.4 (3.9) GeV/c2 to be considered as a J/ψ (ψ(2S)) candidate. To have a reasonable polarization sensitivity, the vector-meson candidates are required to have pT ≥ 5 GeV/c in the rapidity range |y (≡ 12 ln E+p|| E−p|| )| < 0.6, where E is the energy and p|| is the momentum parallel to the beam direction of the dimuon system. Events are separated into a signal region and sideband regions, as indicated in Fig. 1. The fit to the data uses a double (single) Gaussian for the J/ψ (ψ(2S)) signal and a linear background shape. The fits are used only to define signal and background regions. The signal regions are within 3σV of the fitted mass peaks MV , where σV is the width obtained in the fit to the invariant mass distribution. Both the background distribution and the quantity of background events under the signal peak are estimated by events from the lower and upper mass sidebands. The sideband regions are 7σJ/ψ (4σψ(2S)) away from the signal region for J/ψ (ψ(2S)). For each candidate, we compute ct =MLxy/pT , where t is the proper decay time and Lxy is the transverse distance between the beam line and the decay vertex in the plane normal to the beam direction. The ct distributions of the selected dimuon events are shown in Fig. 2. The ct distribution of prompt events is a Gaussian distribution centered at zero due to finite tracking resolution. For J/ψ, the prompt events are due to direct production or the decays of heavier charmonium states such as χc and ψ(2S); for ψ(2S), the prompt events are almost entirely due to direct production since heavier charmonium states rarely decay to ψ(2S) [11]. Both the J/ψ and the ψ(2S) samples contain significant numbers of events originating from long-lived B-hadron decays, as can be seen from the event excess at positive ct. We have measured the fraction of B → J/ψ + X events in the J/ψ sample and found agreement with other results [3]. We select prompt events by requiring the sum of the squared impact parameter significances of the positively and negatively charged muon tracks S ≡ ( d )2 + ( )2 ≤ 8. The impact parameter d0 is the distance of closest approach of the track to the beam line in the transverse plane. Vector-meson candidates from B-hadron decays are selected by requiring S > 16 and ct > 0.03 cm. This requirement retains a negligible fraction of prompt events in the B sample. To measure the polarizations of prompt J/ψ and ψ(2S) mesons as functions of pT , the J/ψ events are analyzed in six pT bins and the ψ(2S) events in three bins, shown in Table I. We determine the fraction of B-decay background remaining in prompt samples fbkd by subtracting the number of negative ct events from the number of positive ct events. Only a negligible fraction (< 0.2%) of B decays produce vector-meson events with negative ct. For both vector mesons, fbkd increases with pT , as listed in Table I. The prompt polarization from the fitting algorithm is corrected for this contamination. M (GeV/c 2.8 2.9 3 3.1 3.2 3.3 3.4 M (GeV/c 3.4 3.5 3.6 3.7 3.8 3.9 FIG. 1: Invariant mass distributions for (a) J/ψ and (b) ψ(2S) candidates. The curves are fits to the data. The solid (dashed) lines indicate the signal (sideband) regions. ct (cm) -0.3 -0.2 -0.1 0 0.1 0.2 0.3 510 (a) ct (cm) -0.2 -0.1 0 0.1 0.2 410 (b) FIG. 2: Sideband-subtracted ct distributions for (a) J/ψ and (b) ψ(2S) events. The prompt Gaussian peak, positive excess from B-hadron decays, and negative tail from mismeasured events are shown. The dotted line is the reflection of the negative ct histogram about zero. The polarization information is contained in the distribution of the muon decay angle θ∗, the angle of the µ+ in the rest frame of vector meson with respect to the vector-meson boost direction in the laboratory system. The decay angle distribution depends on the polarization parameter α: d cos θ∗ ∝ 1 + α cos2θ∗ (−1 ≤ α ≤ 1). For fully transverse (longitudinal) polarization, α = +1 (−1). Intermediate values of α indicate a mixture of transverse and longitudinal polarization. A template method is used to account for acceptance and efficiency. Two sets of cos θ∗ distributions for fully polarized decays of J/ψ and ψ(2S) events, one longitudinal (L) and the other transverse (T ), are produced with the CDF simulation program using the efficiency-corrected pT spectra measured from data [3, 12]. We use the muon trigger efficiency measured using data as a function of track parameters (pT , η, φ) to account for detector non- uniformities. The parametrized efficiency is used as a filter on all simulated muons. Events that pass reconstruction represent the behavior of fully polarized vector-meson decays in the detector. The fitting algorithm [5] uses two binned cos θ∗ distributions for each pT bin, one made by NS events from the signal region (signal plus background) and the other made by NB events from the sideband regions (background). The χ2 minimization is done simultaneously for both cos θ∗ distributions. The fitting algorithm includes an individual background term for each cos θ∗ bin, normalized to NB. Simulation shows that the cos θ ∗ resolution at all decay angles over the entire pT range is much smaller than the bin width of 0.05 (0.10 for ψ(2S)) used here. The data, fit, and template distributions for the worst fit (9% probability) in the J/ψ data are shown in Fig. 3. *θcos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (GeV/c) < 9T p≤ 7 FIG. 3: cos θ∗ distribution of data (points) and polarization fit for the worst χ2 probability bin in the J/ψ data. The dotted (dashed) line is the template for fully L (T) polarization. The fit describes the overall trend of the data well. All systematic uncertainties are much smaller than the statistical uncertainties. Varying the pT spectrum used in the simulation by 1σ changed the polarization parameter for J/ψ at most by 0.002. A systematic uncertainty of 0.007 was estimated by the change in the polarization parameter when a modification was made on all trigger efficiencies by ±1σ. For ψ(2S), the dominant systematic uncertainty came from the yield estimate because of the radiative tail and the large background. The total systematic uncertainties shown in Table I were taken to be the quadrature sum of these individual uncertainties. Other possible sources of systematic uncertainties - signal definition and cos θ∗ binning - were determined to be negligible. Corrections to prompt polarization from B-decay contamination were small, so that uncertainties on B-decay polarization measurements also had negligible effect. No φ-dependence of the polarizations was observed. The polarization of J/ψ mesons from inclusive Bu and Bd decays was measured by the BABAR collaboration [13]. In this analysis, the B-hadron direction is unknown, so we define θ∗ with respect to the J/ψ direction in the laboratory system. The resulting polarization is somewhat diluted. As discussed in Ref. [3], CDF uses a Monte Carlo procedure to adapt the BABAR measurement to predict the effective J/ψ polarization parameter. For the J/ψ events with 5 ≤ pT < 30 GeV/c, the CDF model for Bu and Bd decays gives αeff = −0.145± 0.009, independent of pT . We have measured the polarization of vector mesons from B-hadron decays. For J/ψ, we find αeff = −0.106±0.033 (stat)±0.007 (syst). At this level of accuracy, a polarization contribution by J/ψ mesons from Bs and b-baryon decays cannot be separated from the effective polarization due to those from Bu and Bd decays. We also report the first measurement of the ψ(2S) polarization from B-hadron decays: αeff = 0.36± 0.25 (stat)± 0.03 (syst). The polarization parameters for both prompt vector mesons corrected for fbkd using our experimental results on αeff are listed as functions of pT in Table I and are plotted in Fig. 4. The polarization parameters for J/ψ are negative over the entire pT range of measurement and become increasingly negative (favoring longitudinal polarization) as pT increases. For ψ(2S), the central value of the polarization parameter is positive at small pT , but, given the uncertainties, its behavior is consistent with the trend shown in the measurement of the J/ψ polarization. The polarization behavior measured previously with 110 pb−1 [5] is not consistent with the results presented here. This is a differential measurement, and the muon efficiencies in this analysis are true dimuon efficiencies. In Ref. [5], they are the product of independent single muon efficiencies. The efficiency for muons with pT < 4 GeV/c is crucial for good polarization sensitivity. In this analysis, the muon efficiency varies smoothly from 99% to 97% over this range. In the analysis of Ref. [5], it varied from 93% to 40% with significant jumps between individual data points. Data from periods of drift chamber aging were omitted from this analysis because the polarization results were inconsistent with the remainder of the data. Studies such as this were not done in the analysis of Ref. [5]. The systematics of the polarization measurement are much better understood in this analysis. These polarization measurements for the charmed vector mesons extend to a pT regime where perturbative QCD should be applicable. The results are compared to the predictions of NRQCD and the kT -factorization model in Fig. 4. The prediction of the kT -factorization model is presented for pT < 20 GeV/c and does not include the contribution pT (GeV/c) <pT > (GeV/c) fbkd(%) α χ 2/d.o.f J/ψ 5−6 5.5 2.8± 0.2 −0.004± 0.029± 0.009 15.5/21 6−7 6.5 3.4± 0.2 −0.015± 0.028± 0.010 24.1/23 7−9 7.8 4.1± 0.2 −0.077± 0.023± 0.013 35.1/25 9−12 10.1 5.7± 0.3 −0.094± 0.028± 0.007 34.0/29 12−17 13.7 6.7± 0.6 −0.140± 0.043± 0.007 35.0/31 17−30 20.0 13.6± 1.4 −0.187± 0.090± 0.007 33.9/35 ψ(2S) 5−7 5.9 1.6± 0.9 +0.314± 0.242± 0.028 13.1/11 7−10 8.2 4.9± 1.2 −0.013± 0.201± 0.035 18.5/13 10−30 12.6 8.6± 1.8 −0.374± 0.222± 0.062 26.9/17 TABLE I: Polarization parameter α for prompt production in each pT bin. The first (second) uncertainty is statistical (sys- tematic). <pT > is the average transverse momentum. (GeV/c)Tp 5 10 15 20 25 30 CDF Data NRQCD -factorizationTk (GeV/c)Tp 5 10 15 20 25 30 CDF Data NRQCD -factorizationTk FIG. 4: Prompt polarizations as functions of pT : (a) J/ψ and (b) ψ(2S). The band (line) is the prediction from NRQCD [4] (the kT -factorization model [9]). from the decays of heavier charmonium states for J/ψ production. The polarizations for prompt production of both vector mesons become increasingly longitudinal as pT increases beyond 10 GeV/c. This behavior is in strong disagreement with the NRQCD prediction of large transverse polarization at high pT . It is striking that the NRQCD calculation and the other models reproduce the measured J/ψ and ψ(2S) cross sections at the Tevatron, but fail to describe the polarization at high pT . This indicates that there is some important aspect of the production mechanism that is not yet understood. We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the Korean Science and Engineering Foundation and the Korean Research Foundation; the Particle Physics and Astronomy Research Council and the Royal Society, UK; the Institut National de Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation for Basic Research; the Comisión Interministerial de Ciencia y Tecnoloǵıa, Spain; the European Community’s Human Potential Programme; the Slovak R&D Agency; and the Academy of Finland. [1] G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D 51, 1125 (1995); Erratum, ibid. Phys. Rev. D 55, 5853 (1997); E. Braaten and S. Fleming, Phys. Rev. Lett. 74, 3327 (1995). [2] F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 79, 572 (1997). [3] D. Acosta et al. (CDF Collaboration), Phys. Rev. D 71, 032001 (2005). [4] P. Cho and M. Wise, Phys. Lett. B 346, 129 (1995); M. Beneke and I. Z. Rothstein, Phys. Lett. B 372, 157 (1996); Erratum, ibid. Phys. Lett. B 389, 769 (1996); E. Braaten, B. A. Kniehl, and J. Lee, Phys. Rev. D 62, 094005 (2000). [5] T. Affolder et al. (CDF Collaboration), Phys. Rev. Lett. 85, 2886 (2000). [6] C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 25, 41 (2002); S. Chekanov et al. (ZEUS Collaboration), Eur. Phys. J. C 44, 13 (2005). [7] S. Fleming, I. Z. Rothstein, and A. K. Leibovich, Phys. Rev. D 64, 036002 (2001). [8] V. A. Khoze, A. D. Martin, M. G. Ryskin, and W. J. Stirling, Eur. Phys. J. C 39, 163 (2005). [9] S. P. Baranov, Phys. Rev. D 66, 114003 (2002). [10] The CDF coordinate system has ẑ along the proton direction, x̂ horizontal pointing outward from the Tevatron ring, and ŷ vertical. θ (φ) is the polar (azimuthal) angle measured with respect to ẑ, and η is the pseudorapidity defined as −ln (tan (θ/2)). The transverse momentum of a particle is denoted as pT = p sin θ. [11] W.-M. Yao et al. (Particle Data Group), J. Phys. G 33, 1 (2006). [12] A Letter on ψ(2S) cross section measurement is in preparation. [13] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 67, 032002 (2003). References

0704.0639

A measure of the non-Gaussian character of a quantum state

A measure of the non-Gaussian character of a quantum state Marco G. Genoni,1 Matteo G. A. Paris,1, 2, ∗ and Konrad Banaszek3 1Dipartimento di Fisica dell’Università di Milano, I-20133, Milano, Italia. 2Institute for Scientific Interchange, I-10133 Torino, Italia 3Institute of Physics, Nicolaus Copernicus University, PL-87-100 Toruń, Poland (Dated: November 3, 2018) We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert-Schmidt distance between the state under examination and a reference Gaussian state. We analyze in details the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single- and multi-mode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows to define a corresponding measure for the non-Gaussian character of a quantum operation. PACS numbers: 03.67.-a, 03.65.Bz, 42.50.Dv I. INTRODUCTION Gaussian states play a crucial role in quantum information processing with continuous variables. This is especially true for quantum optical implementations since radiation at ther- mal equilibrium, including the vacuum state, is itself a Gaus- sian state and most of the Hamiltonians achievable within the current technology are at most bilinear in the field operators, i.e. preserve the Gaussian character [1, 2, 3]. As a matter of fact, using single-mode and entangled Gaussian states, lin- ear optical circuits and Gaussian operations, like homodyne detection, several quantum information protocols have been implemented, including teleportation, dense coding and quan- tum cloning [4]. On the other hand quantum information protocols required for long distance communication, as for example entangle- ment distillation and entanglement swapping, rely on non- Gaussian operations. In addition, it has been demonstrated that teleportation [5, 6, 7] and cloning [8] of quantum states may be improved by using non-Gaussian states and non- Gaussian operations. Indeed, de-Gaussification protocols for single-mode and two-mode states have been proposed [5, 6, 7] and realized [9]. It should be also noticed that any strongly superadditive function is minimized, at fixed covariance ma- trix, by Gaussian states. This is crucial to prove extremality of Gaussian states and Gaussian operations [10, 11] for what concerns various quantities as channel capacities [12], multi- partite entanglement measures [13] and distillable secret key in quantum key distribution protocols. Since in most cases these quantities can be computed only for Gaussian states, a non-Gaussianity measure may serve as a guideline to quan- tify them for the class of non-Gaussian states. Overall, non- Gaussianity is revealing itself as a resource for continuous variable quantum information, and thus we urge a measure able to quantify the non-Gaussian character of a quantum state. In this paper we introduce a novel quantity, the non- ∗Electronic address: [email protected] Gaussianity δ[̺] of a quantum state, which quantifies how much a state fails to be Gaussian. Our measure, which is based on the Hilbert-Schmidt distance between the state itself and a reference Gaussian state, can be easily computed for any state, either single-mode or multi-mode. The paper is structured as follows. In the next Section we introduce notation and review the basic properties of Gaussian states. Then, in Section III we introduce the formal definition of δ[̺] and study its properties in details. In Section IV we evaluate non-Gaussianity of relevant quantum states whereas in Section V we analyze the evolution of non-Gaussianity for known Gaussification and de-Gaussification maps. Section VI closes the paper with some concluding remarks. II. GAUSSIAN STATES For concreteness, we will use here the quantum optical ter- minology of modes carrying photons, but our theory applies to general bosonic systems. Let us consider a system of n modes described by mode operators ak, k = 1 . . . n, satis- fying the commutation relations [ak, a j ] = δkj . A quantum state ̺ of the n modes is fully described by its characteristic function [14] χ[̺](λ) = Tr[̺D(λ)] where D(λ) = k=1Dk(λk) is the n-mode displacement operator, with λ = (λ1, . . . , λn) T , λk ∈ C, and where Dk(λk) = exp{λka†k − λ is the single-mode displacement operator. The canonical op- erators are given by: (ak + a (ak − a†k) with commutation relations given by [qj , pk] = iδjk. Upon in- troducing the real vector R = (q1, p1, . . . , qn, pn) T , the com- http://arxiv.org/abs/0704.0639v4 mailto:[email protected] mutation relations rewrite as [Rk, Rj ] = iΩkj where Ωkj are the elements of the symplectic matrix Ω = k=1 σ2, σ2 being the y-Pauli matrix. The covariance ma- trix σ ≡ σ[̺] and the vector of mean values X ≡ X[̺] of a quantum state ̺ are defined as Xj = 〈Rj〉 σkj = 〈{Rk, Rj}〉 − 〈Rj〉〈Rk〉 where {A,B} = AB+BA denotes the anti-commutator, and 〈O〉 = Tr[̺ O] is the expectation value of the operator O. A quantum state ̺G is referred to as a Gaussian state if its characteristic function has the Gaussian form χ[̺G](Λ) = exp σΛ+XTΩΛ where Λ is the real vector Λ = (Reλ1, Imλ1, . . . ,Reλn, Imλn) T . Of course, once the covariance matrix and the vector of mean values are given, a Gaussian state is fully determined. For a single-mode system the most general Gaussian state can be written as ̺G = D(α)S(ζ)ν(nt)S †(ζ)D†(α), D(α) being the displacement operator, S(ζ) = exp[ 1 ζ(a†)2 − 1 ζ∗a2] the squeezing operator, α, ζ ∈ C, and ν(nt) = (1 + nt) −1[nt/(1 + nt)] a†a a thermal state with nt average number of photons. III. A MEASURE OF THE NON-GAUSSIAN CHARACTER OF A QUANTUM STATE In order to quantify the non-Gaussian character of a quan- tum state ̺ we use a quantity based on the distance between ̺ and a reference Gaussian state τ , which itself depends on ̺. Specifically, we define the non-Gaussianity δ[̺] of the state ̺ δ[̺] = D2HS [̺, τ ] where DHS [̺, τ ] denotes the Hilbert-Schmidt distance be- tween ̺ and τ D2HS [̺, τ ] = Tr[(̺− τ)2] = µ[̺] + µ[τ ] − 2κ[̺, τ ] , (3) with µ[̺] = Tr[̺2] and κ[̺, τ ] = Tr[̺τ ] denoting the purity of ̺ and the overlap between ̺ and τ respectively. The Gaussian reference τ is the Gaussian state such that X[̺] = X[τ ] σ[̺] = σ[τ ] i.e. τ is the Gaussian state with the same covariance matrix σ and the same vector X of the state ̺. The relevant properties of δ[̺], which confirm that it repre- sents a good measure of the non-Gaussian character of ̺, are summarized by the following Lemmas: Lemma 1: δ[̺] = 0 iff ̺ is a Gaussian state. Proof: If δ[̺] = 0 then ̺ = τ and thus it is a Gaussian state. If ̺ is a Gaussian state, then it is uniquely identified by its first and second moments and thus the reference Gaussian state τ is given by τ = ̺, which, in turn, leads to DHS [̺, τ ] = 0 and thus to δ[̺] = 0. Lemma 2: If U is a unitary map corresponding to a symplec- tic transformation in the phase space, i.e. if U = exp{−iH} with hermitianH that is at most bilinear in the field operators, then δ[U̺U †] = δ[̺]. This property ensures that displace- ment and squeezing operations do not change the Gaussian character of a quantum state. Proof: Let us consider ̺′ = U̺U †. Then the covariance ma- trix transforms as σ[̺′] = Σσ[̺]ΣT , Σ being the symplectic transformation associated to U . At the same time the vector of mean values simply translates to X ′ = X +X0, where X0 is the displacement generated by U . Since any Gaussian state is fully characterized by its first and second moments, then the reference state must necessarily transform as τ ′ = UτU †, i.e. with the same unitary transformation U . Since the Hilbert- Schmidt distance and the purity of a quantum state are invari- ant under unitary transformations the lemma is proved. Lemma 3: δ[̺] is proportional to the squared L2(Cn) dis- tance between the characteristic functions of ̺ and of the ref- erence Gaussian state τ . In formula: δ[̺] ∝ d2nλ [χ[̺](λ)− χ[τ ](λ)]2 . (4) Since the notion of Gaussianity of a quantum state is de- fined through the shape of its characteristic function, and since the characteristic function of a quantum state belongs to the L2(Cn) space [14], we address L2(C) distance to as a good indicator for the non Gaussian character of ̺. Proof: Since characteristic functions of self-adjoint operators are even functions of λ and by means of the identity Tr[O1O2] = χ[O1](λ)χ[O2](−λ) , we obtain D2HS [̺, τ ] = [χ[̺](λ)− χ[τ ](λ)]2 . Lemma 4: Consider a bipartite state ̺ = ̺A ⊗ ̺G. If ̺G is a Gaussian state then δ[̺] = δ[̺A]. Proof: we have µ[̺] = µ[̺A]µ[̺G] µ[τ ] = µ[τA]µ[τG] κ[̺, τ ] = κ[̺A, τA]κ[̺G, ̺G] . Therefore, since κ[̺G, ̺G] = µ[̺G] we arrive at δ[̺] = µ[̺A]µ[̺G] + µ[τA]µ[̺G]− 2κ[̺A, τA]κ[̺G, ̺G] 2µ[̺A]µ[̺G] = δ[̺A] (5) The four properties illustrated by the above lemmas are the natural properties required for a good measure of the non- Gaussian character of a quantum state. Notice that by using the trace distanceDT [̺, τ ] = Tr|̺−τ | instead of the Hilbert- Schmidt distance we would lose Lemmas 3 and 4, and that the invariance expressed by Lemma 4 holds thanks to the renor- malization of the Hilbert-Schmidt distance through the purity µ[̺]. We stress the fact that our measure of non-Gaussianity is a computable one: It may be evaluated for any quantum state of n modes by the calculation of the first two moments of the state, followed by the evaluation of the overlap with the corresponding Gaussian state. Notice that δ[̺] is not additive (nor multiplicative) with re- spect to the tensor product. If we consider a (separable) multi- partite quantum state in the product form ̺ = ⊗nk=1̺k, the non-Gaussianity is given by δ[̺] = k=1 µ[̺k] + k=1 µ[τk]− 2 k=1 κ[̺k, τk] k=1 µ[̺k] where τk is the Gaussian state with the same moments of ̺k. In fact, since the state ̺ is factorisable, we have that the cor- responding Gaussian τ is a factorisable state too. IV. NON-GAUSSIANITY OF RELEVANT QUANTUM STATES Let us now exploit the definition (2) to evaluate the non- Gaussianity of some relevant quantum states. At first we con- sider Fock number states |p〉 of a single mode as well as mul- timode factorisable states |p〉⊗n made of n copies of a num- ber state. The reference Gaussian states are a thermal state τp = ν(p) with average photon number p and a factorisable thermal state τN = [ν(p)] ⊗n with average photon number p in each mode [15]. Non-Gaussianity may be analytically eval- uated, leading to δ[|p〉〈p|] = 1 2p+ 1 δ[(|p〉〈p|)⊗n] = 1 2p+ 1 In the multimode case of |p〉⊗n, we seek for the number of copies that maximizes the non-Gaussianity. In Fig. 1 we show both δp ≡ δ[|p〉〈p|] and δ̄p = maxn δ[(|p〉〈p|)⊗n] as a function of p. As it is apparent from the plot non-Gaussianity of Fock states |p〉 increases monotonically with the number of photon p with the limiting value δp = 1/2 obtained for p → ∞. Upon considering multi-mode copies of Fock states we obtain larger value of non-Gaussianity: δ̄p is a decreasing function of p, approaching δ̄ = 1/2 from above. The value 1 5 10 15 20 25 30 FIG. 1: (Top): Non-Gaussianity of single mode Fock states (gray) |p〉 and of multi-mode Fock states |p〉⊗n (black) as a function of p. Non-Gaussianity for multi-mode states has been maximized over the number of copies n. (Bottom): Non-Gaussianity, as a function of the parameter φ, for the two-mode superpositions |Φ〉〉 (dashed gray), |Ψ〉〉 (solid gray), and for the single-mode superposition of coherent states |ψS〉 for α = 0.5 (solid black) and α = 5 (dashed black). of δ̄p corresponds to n = 3 for p < 26 and to n = 2 for 27 ≤ p . 250. Another example is the superposition of coherent states |ψS〉 = N−1/2 (cosφ|α〉 + sinφ| − α〉) (7) with normalization N = 1 + sin(2φ) exp{−2α2} which for φ = ±π/4 reduces to the so-called Schrödinger cat states, and whose reference Gaussian state is a displaced squeezed thermal state τS = D(C)S(r)ν(N)S †(r)D†(C), where the real parameters C, r, and N are analytical functions of φ and α. Finally we evaluate the non-Gaussianity of the two-mode Bell-like superpositions of Fock states |Φ〉〉 = cosφ|0, 0〉+ sinφ|1, 1〉 |Ψ〉〉 = cosφ|0, 1〉+ sinφ|1, 0〉, which for φ = ±π/4 reduces to the Bell states |Φ±〉 and |Ψ±〉. The corresponding reference Gaussian states are respectively a two mode squeezed thermal state τΦ = S2(ξ)[ν(N) ⊗ ν(N)]S†2(ξ), where S2(ξ) = exp(ξa ξ∗ab) denotes the two-mode squeezing operator, and τΨ = R(θ)[ν(N1)⊗ν(N2)]R†(θ), namely the correlated two-mode state obtained by mixing a single-mode thermal state with the vacuum at a beam splitter of transmissivity cos2 θ, i.e. R(θ) = exp[iθ(a 1a2+a 2a1)]. All the parameters involved in these reference Gaussian states are analytical functions of the superposition parameter φ. Non-Gaussianities are thus evalu- ated by means of (2) and are reported in Fig. 1 as a function of the parameter φ. As it is apparent from the plot, the non- Gaussianity of single-mode states does not surpass the value δ = 1/2, and this fact is confirmed by other examples not reported here. As concern the cat-like states, we notice that for small val- ues of α the non-Gaussianity of the superposition |ψS〉 shows a different behavior for positive and negative values of the pa- rameter φ: for φ > 0 and α = 0.5 we have almost zero δ, while higher values are achieved for φ < 0. For higher val- ues of α (α = 5 in Fig. 1), non-Gaussianity becomes an even function of φ. This different behavior can be understood by looking at the Wigner functions of even and odd Schrödinger cat states for different values of α: for small values of α the even cat’s Wigner function is similar to a Gaussian function, while the odd cat’s Wigner function shows a non-Gaussian hole in the origin of the phase space; increasing the value of α the Wigner functions of the two kind of states become similar and deviate from a Gaussian function. We have also done a numerical analysis of non-Gaussianity of single-mode quantum states represented by finite superpo- sition of Fock states n,k=0 ̺nk|n〉〈k| . (8) To this aim we generate randomly quantum states in a finite dimensional subspaces, dim(H) ≡ d+ 1 ≤ 21, following the algorithm proposed by Zyczkowski et al [16, 17], i.e. by gen- erating a random diagonal state (i.e. a point on the simplex) and a random unitary matrix according to the Haar measure. In Fig. 2 we report the distribution of non-Gaussianity δ[̺d], as evaluated for 105 random quantum states, for three different value of the maximum number of photons d. As it is apparent from the plots the distribution of δ[̺d] becomes Gaussian-like for increasing d. In the fourth panel of Fig. 2 we thus re- port the mean values and variances of the the distributions as a function of the maximum number of photons d. The mean value increases with the dimension whereas the variance is a monotonically decreasing function of d. Also for finite superpositions simulations we did not ob- serve non-Gaussianity higher than 1/2. Therefore, although we have no proof, we conjecture that δ = 1/2 is a limiting value for the non-Gaussianity of a single-mode state. Higher values are achievable for two-mode or multi-mode quantum states (e.g. δ = 2/3 for the Bell states |Ψ±〉〉). V. GAUSSIFICATION AND DE-GAUSSIFICATION PROCESSES We have also studied the evolution of non-Gaussianity of quantum states undergoing either Gaussification or de- Gaussification processes. First we have considered the Gaus- sification of Fock states due do the interaction of the system FIG. 2: Distribution of non-Gaussianity δ[̺d] as evaluated for 10 random quantum states, for three different value of the maximum number of photons d. Top: d = 2 (left), d = 10 (right); Bottom: d = 20 (left). (Bottom-right): Mean values and variances of the non- Gaussianities evaluated for 105 random quantum states, as a function of the maximum number of photons d. with a bath of oscillators at zero temperature. This is per- haps the simplest example of a Gaussification protocol. In fact the interaction drives asymptotically any quantum state to the vacuum state of the harmonic system, which, in turn, is a Gaussian state. The evolution of the system is governed by the Lindblad Master equation ˙̺ = γ L[a]̺, where ˙̺ denotes time derivative, γ is the damping factor and the Lindblad superop- erator acts as follows L[a]̺ = 2a†̺a − a†a̺ − ̺a†a. Upon writing η = e−γt the solution of the Master equation can be written as ̺(η) = Vm ̺ V m (9) Vm = [(1− η)m/m!] 2 am η (a†a−m) , where ̺ is the initial state. In particular for the system ini- tially prepared in a Fock state ̺p = |p〉〈p|, we obtain, after evolution, the mixed state ̺p(η) = Vm̺pV αl,p(η)|l〉〈l| (10) with αl,p(η) = (1−η)p−lηl. The reference Gaussian state corresponding to ̺p(η) is a thermal state τp(η) = ν(pη) with average photon number pη. Non-Gaussianity of ̺p(η) can be evaluated analytically, we have δpη ≡ δ[̺p(η)] 2(1− η)2m 2F1 −m,−m, 1; η2 (η−1)2 (1− η)2m 2F1 −m,−m, 1; η (η − 1)2 + (1 + 2mη)−1 − 2(1 + (m− 1)η) (1 +mη)m+1 2F1(a, b, c;x) being a hypergeometric function. We show the behavior of δpη in Fig. 3 as a function of 1 − η for different values of p. As it is apparent from the plot δpη is a monotoni- cally decreasing function of 1 − η as well as a monotonically increasing function of p. That is, at fixed time t the higher is the initial photon number p, the larger is the resulting non- Gaussianity. 0.2 0.4 0.6 0.8 1 1 - Η 0.2 0.4 0.6 0.8 1 FIG. 3: (Left): Non-Gaussianity of Fock states |p〉 undergoing Gaus- sification by loss mechanism due to the interaction with a bath of os- cillators at zero temperature. We show δηp as a function of 1 − η for different values of p: from bottom to top p = 1, 10, 100, 1000. (Right): Non-Gaussianity of ̺IPS as a function of T for r = 0.5 and for different values of ǫ = 0.2, 0.4, 0.6, 0.8 (from bottom to top). δIPS results to be a monotonous increasing function of T , while ǫ only slightly changes the non-Gaussian character of the state. Let us now consider the de-Gaussification protocol ob- tained by the process of photon subtraction. Inconclusive Pho- ton Subtraction (IPS) has been introduced for single-mode and two-mode states in [6, 7, 18] and experimentally realized in [9]. In the IPS protocol an input state ̺(in) is mixed with the vacuum at a beam splitter (BS) with transmissivity T and then, on/off photodetection with quantum efficiency ǫ is per- formed on the reflected beam. The process can be thus charac- terized by two parameters: the transmissivity T and the detec- tor efficiency ǫ. Since the detector can only discriminate the presence from the absence of light, this measurement is in- conclusive, namely it does not resolve the number of detected photons. When the detector clicks, an unknown number of photons is subtracted from the initial state and we obtain the conditional IPS state ̺IPS . The conditional map induced by the measurement is non-Gaussian [7], and the output state is de-Gaussified. Upon applying the IPS protocol to the (Gaus- sian) single-mode squeezed vacuum S(r)|0〉 (r ∈ R), where S(r) is the real squeezing operation we obtain [18] the con- ditional state ̺IPS , whose characteristic function χ[̺IPS ](λ) is a sum of two Gaussian functions and therefore is no longer Gaussian. The corresponding Gaussian reference state is a squeezed thermal state τIPS = S(ξIPS)ν(NIPS)S †(ξIPS) where the parameters ξIPS andNIPS are analytic functions of r, T and ǫ. Non-Gaussianity δIPS = δIPS(T, ǫ, r) has been evaluated, and in Fig. 3 (right) we report δIPS for r = 0.5 as a function of the transmittivity T for different values of the quantum efficiency ǫ. As it is apparent from the plot the IPS protocol indeed de-Gaussifies the input state, i.e. nonzero values of the non-Gaussianity are obtained. We found that δIPS is an increasing function of the transmissivity T which is the relevant parameter, while the quantum efficiency ǫ only slightly affects the non-Gaussian character of the output state. The highest value of non-Gaussianity is achieved in the limit of unit transmissivity and unit quantum efficiency T,η→1 δIPS = δ[|1〉〈1|] = δ[S(r)|1〉〈1|S†(r)], where the last equality is derived from Lemma 2. This result is in agreement with the fact that a squeezed vacuum state undergoing the IPS protocol is driven towards the target state S(r)|1〉 in the limit of T, ǫ → 1 [18]. Finally, we notice that for T, ǫ 6= 1 and for r → ∞ the non-Gaussianity vanishes. In turn, this corresponds to the fact that one of the coefficients of the two Gaussians of χ[̺IPS ](λ) vanishes, i.e. the output state is again a Gaussian one. VI. CONCLUSION AND OUTLOOKS Having at disposal a good measure of non-Gaussianity for quantum state allows us to define a measure of the non- Gaussian character of a quantum operation. Let us denote by G the whole set of Gaussian states. A convenient defi- nition for the non-Gaussianity of a map E reads as follows δ[E ] = max̺∈G δ[E(̺)], where E(̺) denotes the quantum state obtained after the evolution imposed by the map. Indeed, for a Gaussian map Eg , which transforms any input Gaussian state into a Gaussian state, we have δ[Eg] = 0. Work along this line is in progress and results will be reported elsewhere. In conclusion, we have proposed a measure of the non- Gaussian character of a CV quantum state. We have shown that our measure satisfies the natural properties expected from a good measure of non-Gaussianity, and have evaluated the non-Gaussianity of some relevant states, in particular of states undergoing Gaussification and de-Gaussification protocols. Using our measure an analogue non-Gaussianity measure for quantum operations may be introduced. Acknowledgments This work has been supported by MIUR project PRIN2005024254-002, the EC Integrated Project QAP (Con- tract No. 015848) and Polish MNiSW grant 1 P03B 011 29. APPENDIX A: GAUSSIAN REFERENCE WITH UNCONSTRAINED MEAN VALUE As we have seen from the above examples δ[̺] of Eq. (2) represents a good measure of the non-Gaussian character of a quantum state. A question arises on whether different choices for the reference Gaussian state τ may lead to alternative, valid, definitions. As for example (for single-mode states) we may define δ′[̺] = min D2HS [̺, τ ]/µ[̺], (A1) where τ = D(C)S(ξ)ν(N)S†(ξ)D†(C) is a Gaussian state with the same covariance matrix of ̺ and unconstrained vec- tor of mean values X = (ReC, ImC) used to minimize the Hilbert-Schmidt distance. Here we report few examples of the comparison between the results already obtained using (2) with that coming from (A1). As we will see either the two definitions coincide or δ′ and δ are monotone functions of each other. Since the definition (2) corresponds to an easily computable measure we conclude that it represents the most convenient choice. Let us first consider the Fock state ̺ = |p〉〈p|. According to (A1), the reference Gaussian state is given by a displaced thermal states τ ′ = D(C)νpD †(C). The overlap between ̺ and τ ′ is given by κ[|p〉〈p|, τ ′] = 1 1 + p 1 + p 1 + p p(1 + p) The maximum of (A2) is achieved forC = 0, which coincides with the assumptions C = Tr[a|p〉〈p|]. Let us consider the quantum state (10) obtained as the so- lution of the loss Master Equation for an initial Fock state |p〉〈p|. The unconstrained Gaussian reference is again a dis- placed thermal state τ ′ = D(C)νpηD †(C), and the overlap is given by κ[̺p(η), τ ′] = Tr[τ̺p(η)] = (1 + η(p− 1))p (1 + pη)p+1 η|C|2 (1 + pη)(η(1 − p)− 1) Again, since the overlap is maximum for C = Tr[a̺p(η)] = 0, both definitions give the same results for the non- Gaussianity. Let us now consider the Schrödinger cat-like states of (7). The reference Gaussian state is a displaced squeezed thermal state, with squeezing and thermal photons as calculated be- fore. The optimization over the free parameterC may be done numerically. In Fig. 4 we show the non-Gaussianitiy, both as resulting from (A1) and by choosing C = Tr[a̺S ] as in (2), as a function of ǫ. The two curves are almost the same, with no qualitative differences. [1] A. Ferraro, S. Olivares and M. G. A. 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