Chiral phase transition at finite temperature in the linear sigma model

We study the chiral phase transition at finite temperature in the linear sigma model by employing a self-consistent Hartree approximation. This approximation is introduced by imposing self-consistency conditions on the effective meson mass equations which are derived from the finite temperature one-loop effective potential. It is shown that in the limit of vanishing pion mass, namely when the chiral symmetry is exact, the phase transition becomes a weak first order accompanying a gap in the order parameter as a function of temperature. This is caused by the long range fluctuations of meson fields whose effective masses become small in the transition region. It is shown, however, that with an explicit chiral symmetry breaking term in the Lagrangian which generates the realistic finite pion mass the transition is smoothed out irrespective of the choice of coupling strength. Recieved: 19 September 1997 / Revised version: 30 October 1997     

Dilute neutron matter on the lattice at next-to-leading order in chiral effective field theory

We discuss lattice simulations of the ground state of dilute neutron matter at next-to-leading order in chiral effective field theory. In a previous paper the coefficients of the next-to-leading-order lattice action were determined by matching nucleon-nucleon scattering data for momenta up to the pion mass. Here the same lattice action is used to simulate the ground state of up to 12 neutrons in a periodic cube using Monte Carlo simulations. We explore the density range from 2% to 8% of normal nuclear density and analyze the ground-state energy as an expansion about the unitarity limit with corrections due to finite scattering length, effective range, and P -wave interactions.     

An effective lagrangian with broken scale and chiral symmetry III. Mesons at finite temperature

We investigate the finite temperature behavior of the meson sector of an effective Lagrangian which describes nuclear matter. A method is developed for evaluating the logarithmic terms in the effective potential which involves expansion and resummation; the result is written in terms of the exponential integral. In the absence of explicit chiral symmetry breaking, a phase transition restores the symmetry at a temperature of 190 MeV; when the pion has a mass the transition is smooth. At a much higher temperature a first order phase transition restores scale symmetry.     

Chiral Phase Transition at Finite Isospin Density in Linear Sigma Model

Using the linear sigma model, we have introduced the pion isospin chemical potential. The chiral phase transition is studied at finite temperatures and finite isospin densities. We have studied the μ-T phase diagram for the chiral phase transition and found the transition cannot happen below a certain low temperature because of the Bose-Einstein condensation in this system. Above that temperature, the chiral phase transition is studied by the isotherms of pressure versus density. We indicate that the transition, in the chiral limit, is a first-order transition from a low-density phase to a high-density phase like a gas-liquid phase transition.     

Chiral symmetry breaking and pion properties at finite temperatures

Spontaneous and explicit chiral symmetry breaking is analyzed in Coulomb gauge QCD at finite temperatures, using an instantaneous approximation for the quark interaction and incorporating confinement through a running coupling constant. The thermodynamics of the quarks is treated approximatively by assuming that the momentum-dependent constituent quark mass sets the scale for thermodynamic fluctuations of colour singlet excitations. We investigate the class of a temperature independent and a temperature dependent interaction between quarks. In the chiral limit both temperature independent and a smooth temperature dependent interaction yields a second order chiral phase transition with critical exponents close to the values for a BCS super-conductor. For explicit chiral symmetry breaking we find a nearly constant pion mass below the transition temperature, but a strongly overdamped mode above. For a first order deconfining transition in the gluonic sector also the quark sector shows a first order chiral phase transition. The relevance of our results for relativistic heavy ion collisions is briefly discussed.     

An effective lagrangian with broken scale and chiral symmetry IV: Nucleons and mesons at finite temperature

We study the finite temperature properties of an effective chiral Lagrangian which describes nuclear matter. Thermal fluctuations in both the nucleon and the meson fields are considered. The logarithmic and square root terms in the effective potential are evaluated by expansion and resummation with the result written in terms of the exponential integral and the error function, respectively. In the absence of explicit chiral symmetry breaking a phase transition restores the symmetry, but when the pion has a mass the transition is smooth. The nucleon and meson masses as a functions of density and temperature are discussed.     

On the phase structure of QCD in a finite volume

The chiral phase transition in QCD at finite chemical potential and temperature can be characterized for small chemical potential by its curvature and the transition temperature. The curvature is accessible to QCD lattice simulations, which are always performed at finite pion masses and in finite simulation volumes. We investigate the effect of a finite volume on the curvature of the chiral phase transition line. We use functional renormalization group methods with a two flavor quark-meson model to obtain the effective action in a finite volume, including both quark and meson fluctuation effects. Depending on the chosen boundary conditions and the pion mass, we find pronounced finite-volume effects. For periodic quark boundary conditions in spatial directions, we observe a decrease in the curvature in intermediate volume sizes, which we interpret in terms of finite-volume quark effects. Our results have implications for the phase structure of QCD in a finite volume, where the location of a possible critical endpoint might be shifted compared to the infinite-volume case.     

Phase structure of the linear sigma model with the standard symmetry breaking term

The linear sigma model at finite isospin chemical potential μ and temperature T is systematically studied by means of the Cornwal–Jackiw–Tomboulis (CJT) effective potential calculated in the improved Hartree–Fock (HF) approximation, where the Goldstone theorem and the thermodynamic consistency are respected. It results that in the chiral limit, for μ=0 the chiral phase transition is second order as expected from the general universality arguments, and for μ≠0 the phase diagram for the pion condensation in the (T,μ) plane exhibits a tricritical point which is crossover from first-order to second-order phase transitions. In the physical world, where the chiral symmetry is explicitly broken, the pion condensation occurs at μ=m _π , the pion mass in vacuum, and its phase diagram is basically in agreement with those found from the chiral perturbation theory. The chiral symmetry gets restored at high values of T for fixed μ and of μ for fixed T.     

Meson Effects on the Chiral Condensate at Finite Density

Meson corrections on the chiral condensate up to next-to-leading order in a 1/N_c expansion at finite density are investigated in the NJL model with explicit chiral symmetry breaking. Compared with mean-field results, the chiral phase transition is still of the first order while the properties near the critical density for chiral phase transition are found to change significantly.     

In-medium chiral perturbation theory

We report on how to tackle the problem of establishing a chiral effective field theory in nuclear matter with explicit pion fields and in the presence of external sources (Ann. Phys. 297, 27 (2002)). We have made use of the results of J.A. Oller (Phys. Rev. C 65, 025204 (2002)) where the generating functional for the in-medium chiral SU(2) x SU(2) Lagrangian has been derived. Within this approach we develop the so-called standard power counting rules for the calculation of in-medium pion properties if the residual nucleon energies are of the order of the pion mass. In addition, for the case of vanishing residual nucleon energies, a modified scheme (non-standard counting) is introduced. For both schemes the pertinent scales where the chiral expansions have to break down are established as well. We have performed a systematic analysis of n-point in-medium Green functions up to and including next-to-leading order when the standard rules apply. These include the in-medium contributions to quark condensates, pion propagators, pion masses and couplings of the axial-vector, vector and pseudoscalar currents to pions.Received: 30 September 2002, Published online: 22 October 2003PACS: 12.39.Fe Chiral Lagrangians - 11.30.Rd Chiral symmetries - 21.65. + f Nuclear matter     

Effects of Composite Pions on the Chiral Condensate within the PNJL Model at Finite Temperature

We investigate the effect of composite pions on the behaviour of the chiral condensate at finite temperature within the Polyakov-loop improved NJL model. To this end we treat quark-antiquark correlations in the pion channel (bound states and scattering continuum) within a Beth–Uhlenbeck approach that uses medium-dependent phase shifts. A striking medium effect is the Mott transition which occurs when the binding energy vanishes and the discrete pion bound state merges the continuum. This transition is triggered by the lowering of the continuum edge due to the chiral restoration transition. This in turn also entails a modification of the Polyakov-loop so that the SU(3) center symmetry gets broken at finite temperature and dynamical quarks (and gluons) appear in the system, taking over the role of the dominant degrees of freedom from the pions. At low temperatures our model reproduces the chiral perturbation theory result for the chiral condensate while at high temperatures the PNJL model result is recovered. The new aspect of the current work is a consistent treatment of the chiral restoration transition region within the Beth–Uhlenbeck approach on the basis of mesonic phase shifts for the treatment of the correlations.     

Meson Effects on the Chiral Condensate at Finite Density

Meson corrections on the chiral condensate up to next-to-leading order in a 1/Nc expansion at finite densityare investigated in the NJL model with explicit chiral symmetry breaking. Compared with mean-field results, the chiralphase transition is still of the first order while the properties near the critical density for chiral phase transition are foundto change significantly.     

In-medium chiral perturbation theory

This talk will report about a systematical implementation of a chiral effective field theory in nuclear matter with explicit pion fields and in the presence of external sources[1]. Within the generating functional approach of Ref.[2] the so-called standard power counting rules for the calculation of in-medium pion properties are developed that apply if the residual nucleon energies are of the order of the pion mass. In addition, for the case of vanishing residual nucleon energies, a modified scheme (non-standard counting) is introduced. For both schemes the pertinent scales where the chiral expansions have to break down are established as well. We report about a systematic analysis of n-point in-medium Green functions up to and including next-to-leading order when the standard rules apply. These include the in-medium contributions to quark condensates, pion propagators, pion masses and couplings of the axial-vector, vector and pseudoscalar currents to pions.     

Dispersion Relation of σ Meson and Pion at Finite Nuclear Density in Chiral σ Model

The propagators of pion and sigma meson at a finite nuclear density and zero temperature are studied in chiral σ model. Their dispersion relations are calculated numerically in one-loop approximation. In order to avoid the so-called tachyon pole appearing in the one-loop propagators of pion and sigma meson, we regard the mass of sigma meson m_σ as a free parameter and adjust it to fit the nuclear saturation properties. For m_σ equal to 3075 MeV, the tachyon pole does not appear at the normal nuclear density. Thus the dispersion relation can be calculated in chiral σ model in one-loop level for the first time.     

No-go theorem for critical phenomena in large-N(c) QCD

We derive some rigorous results on the chiral phase transition in QCD and QCD-like theories with a large number of colors, N(c), based on the QCD inequalities and the large-N(c) orbifold equivalence. We show that critical phenomena and associated soft modes are forbidden in flavor-symmetric QCD at finite temperature T and finite but not so large quark chemical potential μ for any nonzero quark mass. In particular, the critical point in QCD at a finite baryon chemical potential μ(B)=N(c)μ is ruled out, if the coordinate (T, μ) is outside the pion condensed phase in the corresponding phase diagram of QCD at a finite isospin chemical potential μ(I)=2μ.     

Strong isospin breaking of the nucleon and delta masses on the lattice

Strong isospin breaking in the spectrum of the nucleons and deltas can be studied in lattice QCD with the help of chiral perturbation theory. At leading order in the chiral expansion, the mass splittings between the proton and neutron and between the deltas are linear in the quark mass difference. The next-to-leading order contributions to these splittings vanish even away from the strong-isospin limit. Therefore, any non-linear quark mass dependence of these mass splittings is a signal of the next-to-next-to-leading order mass contributions, thus providing access to low energy constants at this order. We determine the mass splittings of the nucleons and deltas in two-flavor, heavy baryon chiral perturbation theory to next-to-next-to-leading order. We also derive expressions for the nucleon and delta masses in partially quenched chiral perturbation theory to the same order. The resulting mass expressions will be useful both for the extrapolation of lattice data on baryon masses, and for the study of strong isospin breaking.     

Baryon chiral perturbation theory A.D. 1994

In these lectures, the status of baryon chiral perturbation theory is reviewed. Particular emphasis is put on the two-flavor sector and the physics related to electromagnetic probes. I discuss in some detail the structure of the effective Lagrangian at next-to-leading order, the meaning of low-energy theorems in Compton scattering and pion photoproduction, and confront the chiral predictions with the existing data. Some remaining problems and challenges are outlined.Lectures given at the Indian-Summer School on Electron Scattering from Nucleons and Nuclei, Prague, Czech Republic, September 1994.First, I would like to thank the organizers for their kind invitation and hospitality. I am also grateful to Véronique Bernard and Norbert Kaiser for fruitful collaborations and allowing me to present some material before publication. I would like to thank Gerhard Ecker and Jürg Gasser for sharing with me their insight into the chiral dynamics.     

The equation of state of quarks and mesons in a renormalization group flow picture

Non-perturbative flow equations within an effective linear sigma model coupled to constituent quarks for two quark flavours are derived and solved. A heat kernel regularization is employed for a renormalization group improved effective potential. We determine the initial values of the coupling constants in the effective potential at zero temperature. Solving the evolution equations with the same initial values at finite temperature in the chiral limit, we find a second-order phase transition at T_c≈150 MeV. Due to the smooth decoupling of massive modes, we can directly link the low-temperature four-dimensional theory to the three-dimensional high-temperature theory. We calculate the equation of state in the chiral limit and for finite pion masses and determine universal critical exponents.     

Vector manifestation of chiral symmetry

We propose "vector manifestation" (VM) of the Wigner realization of chiral symmetry in which the symmetry is restored at the critical point by the massless degenerate pion (and its flavor partners) and the rho meson (and its flavor partners) as the chiral partner, in sharp contrast to the traditional manifestation á la the linear sigma model where the symmetry is restored by the degenerate pion and the scalar meson. The application to the chiral phase transition of large N(f) QCD is performed using the hidden local symmetry Lagrangian. Combined with the Wilsonian matching proposed recently, VM determines the critical number of massless flavors N(f) approximately equal to 5 without much ambiguity.