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We perform a systematic study of integrated four-point functions of half-BPS operators in four-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SU(N)$. These observables, defined by a certain spacetime integral of $\langle\mathcal{O}_2\mathcal{O}_2\mathcal{O}_p\mathcal{O}_p\rangle$ where $\mathcal{O}_p$ is a superconformal primary of charge $p$, are known to be computable by supersymmetric localization, yet are non-trivial functions of the complexified gauge coupling $\tau$. We find explicit and remarkably simple results for several classes of these observables, exactly as a function of $N$ and $\tau$. Their physical and formal properties are greatly illuminated upon employing the $SL(2,\mathbb{Z})$ spectral decomposition: in this S-duality-invariant eigenbasis, the integrated correlators are fixed simply by polynomials in the spectral parameter. These polynomials are determined recursively by linear algebraic equations relating different $N$ and $p$, such that all integrated correlators are ultimately fixed in terms of the integrated stress tensor multiplets in the $SU(2)$ theory. Our computations include the full matrix of integrated correlators at low values of $p$, and a certain infinite class involving operators of arbitrary $p$. The latter satisfy an open lattice chain equation for all $N$, reminiscent of the Toda equation obeyed by extremal correlators in $\mathcal{N}=2$ superconformal theories. We compute ensemble averages of these observables and analyze our solutions at large $N$, confirming and predicting features of semiclassical AdS$_5\, \times$ S$^5$ supergravity amplitudes., Comment: 42+24 pages

A dark sector with non-abelian gauge symmetry provides a sound framework to justify stable dark matter (DM) candidates. We consider scalar fields charged under a $SU(N)$ gauge group, and show that the centre of $SU(N)$, the discrete subgroup $Z_N$ also known as $N$-ality, can ensure the stability of scalar DM particles. We analyse in some details two minimal DM models of this class, based on $SU(2)$ and $SU(3)$, respectively. These models have non-trivial patterns of spontaneous symmetry breaking, leading to distinctive phenomenological implications. For the $SU(2)$ model these include a specific interplay of two DM states, with the same interactions but different masses, and several complementary DM annihilation regimes, either within the dark sector or through the Higgs portal. The $SU(3)$ model predicts dark radiation made of a pair of dark photons with a unique gauge coupling, as well as regimes where DM semi-annihilations become dominant and testable., 27 pages, 5 figures

Three-dimensional gravity is a topological field theory, which can be quantized as the Ponzano-Regge state-sum model built from the $\{3nj\}$-symbols of the recoupling of the $\SU(2)$ representations, in which spins are interpreted as quantized edge lengths in Planck units. It describes the flat spacetime as gluing of three-dimensional cells with a fixed boundary metric encoding length scale. In this paper, we revisit the Ponzano-Regge model formulated in terms of spinors and rewrite the quantum geometry of 3D cells with holomorphic recoupling symbols. These symbols, known as Schwinger's generating function for the $\{6j\}$-symbols, are simply the squared inverse of the partition function of the 2D Ising model living on the boundary of the 3D cells. They can furthermore be interpreted, in their critical regime, as scale-invariant basic elements of geometry. We show how to glue them together into a discrete topological quantum field theory. This reformulation of the path integral for 3D quantum gravity, with a rich pole structure of the elementary building blocks, opens a new door toward the study of phase transitions and continuum limits in 3D quantum gravity, and offers a new twist on the construction of a duality between 3D quantum gravity and a 2d conformal theory., 42 pages + appendices, 18 figures

Spontaneous symmetry breaking (SSB) is a property of Hamiltonian equilibrium states which, in the thermodynamic limit, retain a finite average value of an order parameter even after a field coupled to it is adiabatically turned off. In the case of quantum spin models with continuous symmetry, we show that this adiabatic process is also accompanied by the suppression of the fluctuations of the symmetry generator -- namely, the collective spin component along an axis of symmetry. In systems of $S=1/2$ spins or qubits, the combination of the suppression of fluctuations along one direction and of the persistence of transverse magnetization leads to spin squeezing -- a much sought-after property of quantum states, both for the purpose of entanglement detection as well as for metrological uses. Focusing on the case of XXZ models spontaneously breaking a U(1) (or even SU(2)) symmetry, we show that the adiabatically prepared states have nearly minimal spin uncertainty; that the minimum phase uncertainty that one can achieve with these states scales as $N^{-3/4}$ with the number of spins $N$; and that this scaling is attained after an adiabatic preparation time scaling linearly with $N$. Our findings open the door to the adiabatic preparation of strongly spin-squeezed states in a large variety of quantum many-body devices including e.g. optical lattice clocks., 5+3 pages, 3+6 figures

In this article we consider a path integral formulation of the Hubbard model based on a SU(2)-symmetrical Hubbard-Stratonovich transformation that couples auxiliary field to the local electronic density. This decoupling is known to have a regular saddle-point structure: each saddle point is a set of elementary field configurations localized in space and imaginary time which we coin instantons. We formulate a classical partition function for the instanton gas that has predictive power. Namely, we can predict the distribution of instantons and show that the instanton number is sharply defined in the thermodynamic limit, thus defining a unique dominant saddle point. Despite the fact that the instanton approach does not capture the magnetic transition inherent to the Hubbard model on the honeycomb lattice, we were able to describe the local moment formation accompanied by short-ranged anti-ferromagnetic correlations. This aspect is also seen in the single particle spectral function that shows clear signs of the upper and lower Hubbard bands. Our instanton approach bears remarkable similarities to local dynamical approaches, such as dynamical mean field theory, in the sense that it has the unique property of allowing for local moment formation without breaking the SU(2) spin symmetry. In contrast to local approaches, it captures short-ranged magnetic fluctuations. Furthermore, it also offers possibilities for systematic improvements by taking into account fluctuations around the dominant saddle point. Finally, we show that the saddle point structure depends upon the choice of lattice geometry. For the square lattice at half-filling, the saddle point structure reflects the itinerant to localized nature of the magnetism as a function of the coupling strength. The implications of our results for Lefschetz thimbles approaches to alleviate the sign problem are also discussed., Comment: 32 pages, 29 figures

A local flavour symmetry acting on the quarks of the Standard Model can automatically give rise to an accidental global $U(1)$ which remains preserved from sources of explicit breaking up to a large operator dimension, while it gets spontaneously broken together with the flavour symmetry. Such non-fundamental symmetries are often endowed with a mixed QCD anomaly, so that the strong CP problem is automatically solved via the axion mechanism. We illustrate the general features required to realise this scenario, and we discuss a simple construction based on the flavour group $SU(3)\times SU(2) \times U(1)_F$ to illustrate how mass hierarchies can arise while ensuring at the same time a high quality Peccei-Quinn symmetry., Comment: 11 pages, no figures

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $\gamma$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models., Comment: 29 pages and 3 figures. v2:minor corrections and a subsection added to match published version

In a previous work [arXiv:2009.03428] we proposed a new model for Quantum GRavity(QGR) and cosmology, dubbed $SU(\infty)$-QGR. One of the axioms of this model is that Hilbert spaces of the Universe and its subsystems represent $SU(\infty)$ symmetry group. In this framework, the classical spacetime is interpreted as being the parameter space characterizing states of the $SU(\infty)$ representing Hilbert spaces. Using quantum uncertainty relations, it is shown that the parameter space - the spacetime - has a 3+1 dimensional Lorentzian geometry. Here after a review of $SU(\infty)$-QGR, including the demonstration that its classical limit is Einstein gravity, we compare it with several QGR proposals, including: string and M-theories, loop quantum gravity and related models, and QGR proposals inspired by holographic principle and quantum entanglement. The purpose is to find their common and analogous features, even if they apparently seem to have different roles and interpretations. The hope is that such exercise gives a better understanding of gravity as a universal quantum force and clarifies the physical nature of the spacetime. We identify several common features among the studied models: importance of 2D structures; algebraic decomposition to tensor products; special role of $SU(2)$ group in their formulation; necessity of a quantum time as a relational observable. We discuss how these features can be considered as analogous in different models. We also show that they arise in $SU(\infty)$-QGR without fine-tuning, additional assumptions, or restrictions., 39 pages, no figure

Journal of high energy physics 04(4), 023 (2023). doi:10.1007/JHEP04(2023)023, We study the eikonal scattering of two gravitationally interacting bodies, in the regime of large angular momentum and large center of mass energy. We show that eikonal exponentiation of the scattering phase matrix is a direct consequence of the group contraction $SU(2)\to ISO(2)$, from rotations to the isometries of the plane, in the large angular momentum limit. We extend it to all orders in the scattering angle, and for all masses and spins. The emergence of the classical limit is understood in terms of the continuous-spin representations admitted by $ISO(2)$. We further investigate the competing classical vs quantum corrections to the leading classical eikonal scattering, and find several interesting examples where quantum corrections are more important than Post-Minkowskian's. As a case of study, we analyse the scattering of a photon off a massless neutral scalar field, up to next-to-leading order in the Newton constant, and to leading order in the fine structure constant. We investigate the causal structure of the eikonal regime and establish an infinite set of non-linear positivity bounds, of which positivity of time delay is the simplest., Published by SISSA, [Trieste]

We show that the parity (evenness or oddness) of a nonclassical state of light has a dominant influence on the interference effects at a balanced beam splitter, irrespective of the state initially occupying the other input mode. Specifically, the parity of the nonclassical state gives rise to destructive interference effects that result in deep valleys in the output joint number distribution of which the Hong-Ou-Mandel (HOM) effect is a limiting case. The counter-intuitive influence of even a single photon to control the output of a beam splitter illuminated by any field, be it a coherent or even a noisy thermal field, demonstrates the extraordinary power of non-classicality. The canonical example of total destructive interference of quantum amplitudes leading to the absence of coincidence counts from a 50/50 beam splitter is the celebrated HOM effect, characterized by the vanishing of the joint probability of detecting singe photons in each of the output beams. We show that this is a limiting case of more general input states upon which a 50/50 BS can create total, or near total, destructive interference of quantum amplitudes. For odd photon number input Fock states of arbitrary value n>0 we show that the joint photon number probabilities vanish when detecting identical photon numbers in each output beams. We examine the mixing of photon number states of n = 1, 2, and 3 with a CV state, such as a coherent state of arbitrary amplitude, and a thermal state. These vanishing joint probabilities form what we call a central nodal line -- a contiguous set of zeros representing complete destructive interference of quantum amplitudes. For odd or even photon number Fock states with n>1 there will be additional off-diagonal pseudo-nodal curves along which the joint photon number probabilities are either zero, or near zero, which constitute a near, but not complete, destructive interference., 27 pages, 13 figures; v2: minor editorial fixes (minor notational corrections in Appendix D), additional clarification comments at end of Appendix A

We present a coarse perspective on relations of the $SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the $SU(2)$-TQFT, as developed by Blanchet, Habegger, Masbaum and Vogel, we define the quantum intersection number between pants decompositions of a closed surface. We show that the quantum intersection number admits two sided bounds in terms of the geometric intersection number and we use it to obtain a metric on the pants graph of surfaces. Using work of Brock we show that the pants graph equipped with this metric is quasi-isometric to the Teichm\"uller space with the Weil-Petersson metric and that the translation length of our metric provides two sided linear bounds on the volume of hyperbolic fibered manifolds. We also obtain a characterization of pseudo-Anosov mapping classes in terms of asymptotics of the quantum intersection number under iteration in the mapping class group and relate these asymptotics with stretch factors., Comment: 38 pages, 6 Figures. Minor changes in introduction, reference updated

The renormalization properties of two local gauge invariant composite operators $(O,R^a_{\mu})$ corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the $SU(2)$ Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean space-time. The present analysis generalizes earlier results presented in the case of the $U(1)$ Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators $R^a_{\mu}$ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization $Z$-factors of the corresponding external sources, needed to define the operators $R^a_{\mu}$ at the quantum level, do not receive any quantum corrections. Another consistency feature of our analysis is that the longitudinal component of the two-point correlation function $\langle R^a_\mu(p) R^b_\nu(-p) \rangle$ exhibits only a tree level non-vanishing contribution which, moreover, is momentum independent, being thus not associated to any physical propagating mode. Finally, we point out that the renowned non-renormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in presence of the Higgs field.

We discuss the prediction of critical behavior of lattice observables in SU(2) and SU(3) gauge theories. We show that feed-forward neural network, trained on the lattice configurations of gauge fields as input data, finds correlations with the target observable, which is also true in the critical region where the neural network has not been trained. We have verified that the neural network constructs a gauge-invariant function and this property does not change over the entire range of the parameter space., 7 pages, 2 figures, 1 table, contribution to the proceedings of XXXIII International (ONLINE) Workshop on High Energy Physics "Hard Problems of Hadron Physics: Non-Perturbative QCD & Related Quests" November 8-12, 2021, Submission to SciPost

We consider three-flavor QCD and perform a determination of the low-energy coupling $\hat{g}_\chi$ of SU(2) Heavy Meson Chiral Perturbation Theory. It is the $B^*B\pi$ coupling in the limit of static heavy and chiral light quarks and has not been determined with precision thus far. The calculation is performed on a large set of the $2+1$ flavor CLS ensembles with pion masses from 420 MeV down to 130 MeV. This allows us to significantly reduce the systematic uncertainty from the chiral extrapolation compared to previous works. Only a weak dependence on the lattice spacing is visible in our results., Comment: 9 pages, 4 figures, The 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

In 1949, Ramsey's method of separated oscillating fields was proposed boosting over many decades metrological performances of atomic clocks and becoming the standard technique for very high precision spectroscopic measurements. A generalization of this interferometric method is presented replacing the two single coherent excitations by arbitrary composite laser pulses. The rotation of the state vector of a two-level system under the effect of a single pulse is described using the Pauli matrices basis of the SU(2) group. It is then generalized to multiple excitation pulses by a recursive Euler-Rodrigues-Gibbs algorithm describing a composition of rotations with different rotation axes. A general analytical formula for the phase-shift associated with the clock's interferometric signal is derived. As illustrations, hyper-clocks based on three-pulse and five-pulse interrogation protocols are studied and shown to exhibit nonlinear cubic and quintic sensitivities to residual probe-induced light-shifts. The presented formalism is well suited to derive and optimize composite phase-shifts produced by tailored quantum algorithms in order to design a new generation of optical frequency standards, qubit rotations with reduced ac Stark-shift distortion for quantum computation and robust engineering control of atomic interferences in AMO physics targeting ultra-high precision laser spectroscopy on cold matter and anti-matter.

We consider the problem of the decomposition of the R\'enyi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider $SU(2)_k$ as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size $L$ the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on $L$ but only on the dimension of the representation. Moreover, a $\log\log L$ contribution to the R\'enyi entropies exhibits a universal form related to the underlying symmetry group of the model, i.e. the dimension of the Lie group., Comment: 31 pages, v2: minor changes

This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d $\mathcal{N}=2$ SCFTs in the planar limit. At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry algebra, defined by the $R$-matrix read off from the superpotential. This implies that when orbifolding the $\mathcal{N}=4$ symmetry algebra down to the $\mathcal{N}=2$ one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. Importantly, we demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. Concretely, for the $\mathbb{Z}_2$ quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach. We confirm our analytic results by numerical comparison with the explicit diagonalisation of the Hamiltonian for short closed chains.

We propose a novel approach to the confinement-deconfinement transition in Yang-Mills theories in the context of gauge-fixed calculations. The method is based on a background-field generalisation of the Landau gauge (to which it reduces at vanishing temperature) with a given, center-symmetric background. This is to be contrasted with most implementations of background field methods in gauge theories, where one uses a variable, self-consistent background. Our proposal is a bona fide gauge fixing that can easily be implemented on the lattice and in continuum approaches. The resulting gauge-fixed action explicitly exhibits the center symmetry of the nonzero temperature theory that controls the confinement-deconfinement transition. We show that, in that gauge, the electric susceptibility diverges at a second order transition [e.g., in the SU(2) theory], so that the gluon propagator is a clear probe of the transition. We implement our proposal in the perturbative Curci-Ferrari model, known for its successful description of various infrared aspects of Yang-Mills theories, including the confinement-deconfinement transition. Our one-loop calculation confirms our general expectation for the susceptibility while providing transition temperatures in excellent agreement with the SU(2) and SU(3) lattice values. Finally, the Polyakov loops above the transition show a more moderate rise, in contrast to previous implementations of the Curci-Ferrari model using a self-consistent background, and our SU(3) result agrees quite well with lattice results in the range $[0,2T_c]$., Comment: 13 pages, 5 figures

Journal of high energy physics 08(8), 127 (1-86) (2021). doi:10.1007/JHEP08(2021)127, This is the first in a series of papers devoted to the study of spin chains capturing the spectral problem of 4d $ \mathcal{N} $ = 2 SCFTs in the planar limit. At one loop and in the quantum plane limit, we discover a quasi-Hopf symmetry algebra, defined by the R-matrix read off from the superpotential. This implies that when orbifolding the $ \mathcal{N} $ = 4 symmetry algebra down to the $ \mathcal{N} $ = 2 one and then marginaly deforming, the broken generators are not lost, but get upgraded to quantum generators. Importantly, we demonstrate that these chains are dynamical, in the sense that their Hamiltonian depends on a parameter which is dynamically determined along the chain. At one loop we map the holomorphic SU(3) scalar sector to a dynamical 15-vertex model, which corresponds to an RSOS model, whose adjacency graph can be read off from the gauge theory quiver/brane tiling. One scalar SU(2) sub-sector is described by an alternating nearest-neighbour Hamiltonian, while another choice of SU(2) sub-sector leads to a dynamical dilute Temperley-Lieb model. These sectors have a common vacuum state, around which the magnon dispersion relations are naturally uniformised by elliptic functions. Concretely, for the ���$_{2}$ quiver theory we study these dynamical chains by solving the one- and two-magnon problems with the coordinate Bethe ansatz approach. We confirm our analytic results by numerical comparison with the explicit diagonalisation of the Hamiltonian for short closed chains., Published by SISSA, [Trieste]

International audience; In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3 + 1 space-time dimensions, the boundary theory lives on the 2 + 1-dimensional time-like boundary and is supposed to describe the time evolution of gravitational boundary modes—‘edge modes’—living on the two-dimensional boundary of space, i.e. the space-time corner. We do not analyse the boundary structures of general relativity and their quantization, and focus instead on investigating which boundary theories can be supported by the standard loop quantum gravity formalism and spin network states. Focussing on ‘electric’ excitations—quanta of area—living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background metric on the 2 + 1-dimensional time-like boundary. This leads to a conjecture of a deeper correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a connection, transporting the spinors on the boundary surface and whose SU(2) component would define ‘magnetic’ excitations (tangential Ashtekar–Barbero connection), thereby opening the door to writing the loop quantum gravity boundary dynamics as a 2 + 1-dimensional gauge theory.

A manifestly Lorentz-covariant formulation of Loop Quantum Gravity (LQG) is given in terms of finite-dimensional representations of the Lorentz group. The formulation accounts for discrete symmetries, such as parity and time-reversal, and it establishes a link with Wigner classification of particles. The resulting quantum model can be seen as LQG with the internal $SU(2)\otimes SU(2)$ group and it is free of the Immirzi parameter, while the scalar constraint is just the Euclidean part., 5 pages, to appear in Classical and Quantum Gravity

In the recent article Phys. Rev. D 100, no. 4, 043533 (2019) a compact phase space generalization of the flat de Sitter cosmology has been proposed. The main advantages of the compactification is that physical quantities are bounded, and the quantum theory is characterized by finite dimensional Hilbert space. Furthermore, by considering the $\mathbb{S}^2$ phase space, quantum description is constructed with the use $SU(2)$ representation theory. The purpose of this article is to apply effective methods to extract semi-classical regime of the quantum dynamics. The analysis is performed both without prior solving of the quantum constraint and by extracting physical Hamiltonian of the model. At the effective level, the results of the two procedures are shown to be equivalent. We find a nontrivial behavior of the fluctuations around the recollapse of the universe, which is distinct from what is found after quantization with the standard flat phase space. The behavior is reflected at the level of the modified Friedmann equation with quantum back-reaction effects, which is derived. Finally, an unexpected relation between the quantum fluctuations of the cosmological sector and the holographic Bousso bound is shown., 13 pages, 4 figures

Physical review / D 102(9), 094510 (1-21) (2020). doi:10.1103/PhysRevD.102.094510, We study the dependence of the electric conductivity on chemical potential in finite-density SU(2) gauge theory with Nf=2 flavors of rooted staggered sea quarks, in combination with Wilson-Dirac and domain-wall valence quarks. The pion mass is reasonably small with mπ/mρ≈0.4. We concentrate in particular on the vicinity of the chiral crossover, where we find the low-frequency electric conductivity to be most sensitive to small changes in fermion density. Working in the low-density QCD-like regime with spontaneously broken chiral symmetry, we obtain an estimate of the first nontrivial coefficient c(T) of the expansion of conductivity σ(T,μ)=σ(T,0)(1+c(T)(μ/T)2+O(μ4)) in powers of μ, which has rather weak temperature dependence and takes its maximal value c(T)≈0.10±0.07 around the critical temperature. At larger densities and lower temperatures, the conductivity quickly grows toward the diquark condensation phase and also becomes closer to the free-quark result. As a by-product of our study we confirm the conclusions of previous studies with heavier pion that for SU(2) gauge theory the ratio of crossover temperature to pion mass Tc/mπ≈0.4 at μ=0 is significantly smaller than in real QCD., Published by Inst., Melville, NY