Effective potential of the O(N) linear sigma-model at finite temperature

We study the O(N) symmetric linear sigma-model at finite temperature as the low-energy effective models of quantum chromodynamics (QCD) using the Cornwall-Jackiw-Tomboulis (CJT) effective action for composite operators. It has so far been claimed that the Nambu-Goldstone theorem is not satisfied at finite temperature in this framework unless the large-N limit in the O(N) symmetry is taken. We show that this is not the case. The pion is always massless below the critical temperature, if one determines the propagator within the form such that the symmetry of the system is conserved, and defines the pion mass as the curvature of the effective potential. We use a regularization for the CJT effective potential in the Hartree approximation, which is analogous to the renormalization of auxiliary fields. A numerical study of the Schwinger-Dyson equation and the gap equation is carried out including the thermal and quantum loops. We point out a problem in the derivation of the sigma meson mass without quantum correction at finite temperature. A problem about the order of the phase transition in this approach is also discussed. Received: 21 June 2000 / Accepted: 13 September 2000     

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.     

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.     

Meson Properties at Finite Temperature in the Linear Sigma Model

The meson masses are investigated at finite temperature in the framework of the linear sigma model with an explicit chiral symmetry breaking term. The imaginary-time thermo-field dynamics and effective potential have been used for the calculation of the meson masses. We found that the behavior of the sigma and pion masses at finite temperature is in agreement with previous works. The critical temperature, the order of the phase transition, and the dependence of the meson fields on the temperature are discussed.     

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.     

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.     

On the scaling behavior of the chiral phase transition in QCD in finite and infinite volume

We study the scaling behavior of the two-flavor chiral phase transition using an effective quark–meson model. We investigate the transition between infinite-volume and finite-volume scaling behavior when the system is placed in a finite box. We can estimate effects that the finite volume and the explicit symmetry breaking by the current quark masses have on the scaling behavior which is observed in full QCD lattice simulations. The model allows us to explore large quark masses as well as the chiral limit in a wide range of volumes, and extract information about the scaling regimes. In particular, we find large scaling deviations for physical pion masses and significant finite-volume effects for pion masses that are used in current lattice simulations.     

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.     

Behavior of the topological susceptibility at finite T and <Emphasis Type="Italic">μ</Emphasis> and signs of restoration of chiral symmetries

We investigate the possible restoration of chiral and axial symmetries across the phase transition at finite temperature and chemical potential, by analyzing the behavior of several physics quantities, such as the quark condensates and the topological susceptibility, the respective derivatives with respect to the chemical potential, and the masses of meson chiral partners. We discuss whether only chiral symmetry or both chiral and axial symmetries are restored and what the role of the strange quark is. The results are compared with recent lattice results.     

Massive Meson Fluctuation in NJL Model

Based on the self-consistent scheme beyond the mean-field approximation in the large N_c expansion, including current quark mass explicitly, a general scheme of SU(2) NJL model is developed. To ensure the quark self-energy expanded in the proper order of N_c,an approximate internal meson propagator is deduced, which is in order of O(l/N_c). In our scheme, adopting the method of external momentum expansion, all the Feynman diagrams are calculated in a unified way by only expanding the quark propagator. Our numerical results show that, different &om the mean-field approximation in which the explicitly chiral symmetry breaking is invisible, the effect of finite pion mass can be seen clearly when beyond the meanfield approximation.     

Quark potential in a quark–meson plasma

We investigate the effect of the restoration of chiral symmetry on the quark potential in a quark–meson plasma by considering meson exchanges in the two flavor Nambu–Jona-Lasinio model at finite temperature and density. There are two possible oscillations in the chiral restoration phase; one is the Friedel oscillation due to the sharp quark Fermi surface at high density, and the other is the Yukawa oscillation driven by the complex meson poles at high temperature. The quark–meson plasma is strongly coupled in the temperature region 1≤T/T _c≤3, with T _c being the critical temperature of the chiral phase transition. The maximum coupling in this region is located at the phase transition point.     

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.     

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.     

Zpif's law in the liquid gas phase transition of nuclei

Zpif's law in the field of linguistics is tested in the nuclear disassembly within the framework of isospin dependent lattice gas model. It is found that the average cluster charge (or mass) of rank n in the charge (or mass) list shows exactly inversely to its rank, i.e., there exists Zpif's law, at the phase transition temperature. This novel criterion shall be helpful to search the nuclear liquid gas phase transition experimentally and theoretically. In addition, the finite size scaling of the effective phase transition temperature at which the Zpif's law appears is studied for several systems with different mass and the critical exponents of ν and β are tentatively extracted. Received: 2 June 1999 / Revised version: 8 October 1999     

Thermodynamic properties of the SU(2)f chiral quark–loop soliton

We consider a chiral one-loop hedgehog soliton of the bosonized SU(2)_f Nambu & Jona-Lasinio model which is embedded in a hot medium of constituent quarks. Energy and radius of the soliton are determined in self-consistent mean-field approximation. Quasi-classical corrections to the soliton energy are derived by means of the pushing and cranking approaches. The corresponding inertial parameters are evaluated. It is shown that the inertial mass is equivalent to the total internal energy of the soliton. Corrected nucleon and δ isobar masses are calculated in dependence on temperature and density of the medium. As a result of the self-consistently determined internal structure of the soliton the scaling between constituent quark mass, soliton mass and radius is noticeably disturbed. Received: 26 September 1997     

Coexistence of color superconductivity and chiral symmetry breaking within the NJL model

The phase diagram for quark matter is investigated within a simple Nambu-Jona-Lasinio model without vector correlations. It is found that the phase structure in the temperature-density plane depends sensitively on the parametrization of the model. We present two schemes of parametrization of the model where, within the first one, a first-order phase transition from a phase with broken chiral symmetry to a color superconducting phase for temperatures below the triple point at T _t = 55 MeV occurs, whereas for the second one a second-order phase transition for temperatures below T _t = 7 MeV is found. In the latter case, there is also a coexistence phase of broken chiral symmetry with color superconductivity, which is a new finding within this class of models. Possible consequences for the phenomenology of the QCD phase transition at high baryon densities are discussed. Received: 3 January 2003 / Accepted: 21 February 2003 / Published online: 24 April 2003     

Pion and kaon electromagnetic form factors in a SUL(3) ⊗ SUR(3) effective Lagrangian

A SU(2) effective Lagrangian is extended to a SU _L(3) ⊗SU _R(3) by including the vector and axial vector meson. With this effective Lagrangian, electromagnetic form factors of charged pion and kaon are calculated in both time- and space-like regions. The pseudoscalar meson loops are taken into account. Good agreement with experimental data is obtained for those form factors and charged pseudoscalar meson radii. Decay widths of ρ→ππ and φ→K ⁺ K ^- are also calculated and shown to agree with experimental data very well. Received: 20 December 1999 / Accepted: 12 October 2000     

Electromagnetic interactions in a chiral effective lagrangian for nuclei

Electromagnetic (EM) interactions are incorporated in a recently proposed effective field theory of the nuclear many-body problem. Earlier work with this effective theory exhibited EM couplings that are correct only to lowest order in both the pion fields and the electric charge. The Lorentz-invariant effective field theory contains nucleons, pions, isoscalar scalar (σ) and vector (ω) fields, and isovector vector (ρ) fields. The theory exhibits a nonlinear realization of SU(2)_L × SU(2)_R chiral symmetry and has three desirable features: it uses the same degrees of freedom to describe the currents and the strong-interaction dynamics, it satisfies the symmetries of the underlying QCD, and its parameters can be calibrated using strong-interaction phenomena, like hadron scattering or the empirical properties of finite nuclei. It has been verified that for normal nuclear systems, the effective lagrangian can be expanded systematically in powers of the meson fields (and their derivatives) and can be truncated reliably after the first few orders. The complete EM lagrangian arising from minimal substitution is derived and shown to possess the residual chiral symmetry of massless, two-flavor QCD with EM interactions. The uniqueness of the minimal EM current is proved, and the properties of the isovector vector and axial-vector currents are discussed, generalizing earlier work. The residual chiral symmetry is maintained in additional (non-minimal) EM couplings expressed as a derivative expansion and in implementing vector meson dominance. The role of chiral anomalies in the EM lagrangian is briefly discussed.     

Effect of thermal ground state correlations on the statistical properties of the Lipkin model

The renormalized random phase approximation for hot finite Fermi systems is evaluated with the use of the thermo field dynamics formalism. This approximation treats vibrations of a hot finite Fermi system as harmonic ones but takes into account the Pauli principle in a more proper way than the usual thermal RPA, thus incorporating a new type of correlations in a thermal ground state. To demonstrate advantages of the approximation and to analyze a range of its validity, it is applied to the exactly solvable Lipkin model. A comparison is made with the exact grand canonical ensemble calculations, results of the thermal Hartree – Fock approximation and the thermal random phase approximation. The intrinsic energy of the system, the heat capacity, the average value of the quasispin operator z-projection and the particle number variance are calculated as functions of temperature. On the whole, the thermal renormalized RPA appears to be a better approximation than the other two. Its advantage is especially evident in the vicinity of the phase transition point. It is found that within TRRPA the phase transition occurs at lower temperature than in THFA and TRPA. Received: 4 January 1999 / Revised version: 10 March 1999     

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.