Contribution of quark loops to the effective action in a model with four-quark interaction

We treat a nonlinear model which includes a four-quark interaction and which accounts for the effect of spontaneous chiral symmetry breaking in the pseudoscalar particle sector. We obtain the Wess-Zumino-Witten action as a contribution to an effective chiral action from the quark loops with five meson ends. The coupling constant between the quarks and the pseudoscalar mesons is related by the Goldberger-Treiman relation to the mass of the constituent quarks and the meson decay constant. Accounting for the vector-meson sector and the corresponding contributions of the quark loops to the effective chiral action (ECA) leads to new Wess-Zumino terms in the ECA.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 19–22, June, 1990.In conclusion, we express deep gratitude to V. Ya. Fainberg for his interest in this work and discussion of some problems touched upon in this paper.     

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.     

Phase structure of two-flavor QCD at finite chemical potential

We study the phase diagram of two-flavor QCD at imaginary chemical potentials in the chiral limit. To this end we compute order parameters for chiral symmetry breaking and quark confinement. The interrelation of quark confinement and chiral symmetry breaking is analyzed with a new order parameter for the confinement phase transition. We show that it is directly related to both the quark density as well as the Polyakov loop expectation value. Our analytical and numerical results suggest a close relation between the chiral and the confinement phase transition.     

A Variational Approach for Hadronic Properties in NJL Model

We develop a variational approach for the Nambu-Jona-Lasinio (NJL) model with explicit chiral symmetry breaking and compute the constituent quark mass, the pion decay constant and the vacuum condensate. For a simple ansatz for the hadronic wavefunction, we also analyze the effects of the current quark mass and the ultraviolet cutoff on the static properties of hadrons.     

Phase transition and the nucleon as a soliton in the Nambu-Jona-Lasinio model at finite temperature and density

We study some bulk thermodynamical characteristics, meson properties and the nucleon as a baryon-number-one soliton in hot quark matter in the NJL model as well as in hot nucleon matter in a hybrid NJL model in which the Dirac sea of quarks is combined with a Fermi sea of nucleons. In both cases, working in the mean-field approximation, we find a chiral phase transition from the Goldstone to the Wigner phase. At finite density the chiral order parameter and the constituent quark mass have a non-monotonic temperature dependence — at finite temperatures not close to the critical one they are less affected than in cold matter. Whereas quark matter is rather soft against thermal fluctuations and the corresponding chiral phase transition is smooth, nucleon matter is much stiffer and the chiral phase transition is very sharp. The thermodynamical variables show large discontinuities which is an indication for a first-order phase transition. We solve the B = 1 solitonic sector of the NJL model in the presence of external hot quark and nucleon media. In the hot medium at intermediate temperature the soliton is more bound and less swelled than in the case of cold matter. At some critical temperature, which for nucleon matter coincides with the critical temperature for the chiral phase transition, we find no more a localized solution. According to this model scenario one should expect a sharp phase transition from nucleon to quark matter.     

Gluon condensates,chiral symmetry breaking and pion wave-function

We consider here chiral symmetry breaking in quantum chromodynamics arising from gluon condensates in vacuum. Through coherent states of gluons simulating a mean field type of approximation, we show that the off-shell gluon condensates of vacuum generate a mass-like contribution for the quarks, giving rise to chiral symmetry breaking. We next note that spontaneous breaking of global chiral symmetry links the four component quark field operator to the pion wave function. This in turn yields many hadronic properties in the light quark sector in agreement with experiments, leading to the conclusion that low energy hadron properties are primarily driven by the vacuum structure of quantum chromodynamics.     

Pion properties at finite isospin chemical potential with isospin symmetry breaking

Pion properties at finite temperature, finite isospin and baryon chemical potentials are investigated within the SU(2) NJL model. In the mean field approximation for quarks and random phase approximation fpr mesons, we calculate the pion mass, the decay constant and the phase diagram with different quark masses for the u quark and d quark, related to QCD corrections, for the first time. Our results show an asymmetry between μI 0 and μI 0 in the phase diagram, and different values for the charged pion mass(or decay constant) and neutral pion mass(or decay constant) at finite temperature and finite isospin chemical potential. This is caused by the effect of isospin symmetry breaking, which is from different quark masses.     

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     

The chiral phase transition at high baryon density from nonperturbative flow equations

We investigate the chiral phase transition at high baryon number density within the linear quark meson model for two flavors. The method we employ is based on an exact renormalization group equation for the free energy. Truncated nonperturbative flow equations are derived at nonzero chemical potential and temperature. Whereas the renormalization group flow leads to spontaneous chiral symmetry breaking in vacuum, we find a chiral symmetry restoring first order transition at high density. Combined with previous investigations at nonzero temperature, the result implies the presence of a tricritical point with long–range correlations in the phase diagram. Received: 24 August 1999 / Published online: 17 February 2000     

Possible pseudogap phase in QCD

Thermal pion fluctuations, in principle, can completely disorder the phase of quark condensate and thus restore chiral symmetry. If this happens before the quark condensate melts, strongly interacting matter will be in the pseudogap state just above the chiral phase transition. The quark condensate does not vanish locally, and quarks acquire constituent masses in the pseudogap phase, despite the fact that chiral symmetry is restored.     

Chiral symmetry breaking in Hamiltonian QCD in Coulomb gauge

Spontaneous breaking of chiral symmetry is investigated in the Hamiltonian approach to QCD in Coulomb gauge. The quark wave functional is determined by the variational principle using an ansatz which goes beyond the commonly used BCS-type of wave functionals and includes the coupling of the quarks to the transversal spatial gluons. Using the lattice gluon propagator as input it is shown that the low energy chiral properties of the quarks, like the quark condensate and the constituent quark mass, are substantially increased by the coupling of the quarks to the spatial gluons. Our results compare favorably with the phenomenological values.     

Transition magnetic moments of J~P=3/2~+ decuplet to J~P=1/2~+ octet baryons in the chiral constituent quark model

In light of the developments of the chiral constituent quark model(χ~(CQM)) in studying low energy hadronic matrix elements of the ground-state baryons, we extend this model to investigate their transition properties.The magnetic moments of transitions from the J~P=3/2~+ decuplet to J~P=1/2~+ octet baryons are calculated with explicit valence quark spin, sea quark spin and sea quark orbital angular momentum contributions. Since the experimental data is available for only a few transitions, we compare our results with the results of other available models. The implications of other complicated effects such as chiral symmetry breaking and SU(3) symmetry breaking arising due to confinement of quarks are also discussed.     

The quark-gluon vertex in Landau gauge QCD: Its role in dynamical chiral symmetry breaking and quark confinement

The infrared behavior of the quark-gluon vertex of quenched Landau gauge QCD is studied by analyzing its Dyson-Schwinger equation. Building on previously obtained results for Green functions in the Yang-Mills sector, we analytically derive the existence of power-law infrared singularities for this vertex. We establish that dynamical chiral symmetry breaking leads to the self-consistent generation of components of the quark-gluon vertex forbidden when chiral symmetry is forced to stay in the Wigner-Weyl mode. In the latter case the running strong coupling assumes an infrared fixed point. If chiral symmetry is broken, either dynamically or explicitly, the running coupling is infrared divergent. Based on a truncation for the quark-gluon vertex Dyson-Schwinger equation which respects the analytically determined infrared behavior, numerical results for the coupled system of the quark propagator and vertex Dyson-Schwinger equation are presented. The resulting quark mass function as well as the vertex function show only a very weak dependence on the current quark mass in the deep infrared. From this we infer by an analysis of the quark-quark scattering kernel a linearly rising quark potential with an almost mass independent string tension in the case of broken chiral symmetry. Enforcing chiral symmetry does lead to a Coulomb type potential. Therefore, we conclude that chiral symmetry breaking and confinement are closely related. Furthermore, we discuss aspects of confinement as the absence of long-range van der Waals forces and Casimir scaling. An examination of experimental data for quarkonia provides further evidence for the viability of the presented mechanism for quark confinement in the Landau gauge.     

Confining but chirally symmetric dense and cold matter

The possibility for existence of cold, dense chirally symmetric matter with confinement is reviewed. The answer to this question crucially depends on the mechanism of mass generation in QCD and interconnection of confinement and chiral symmetry breaking. This question can be clarified from spectroscopy of hadrons and their axial properties. Almost systematical parity doubling of highly excited hadrons suggests that their mass is not related to chiral symmetry breaking in the vacuum and is approximately chirally symmetric. Then there is a possibility for existence of confining but chirally symmetric matter. We clarify a possible mechanism underlying such a phase at low temperatures and large density. Namely, at large density the Pauli blocking prevents the gap equation to generate a solution with broken chiral symmetry. However, the chirally symmetric part of the quark Green function as well as all color non-singlet quantities are still infrared divergent, meaning that the system is with confinement. A possible phase transition to such a matter is most probably of the first order. This is because there are no chiral partners to the lowest lying hadrons.     

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.     

Nuclear Matter in a Chiral SU(3) Quark Mean-Field Model

A quark meson coupling model based on SU(3)_L×SU(3)_R symmetry and scale invariance is proposed. The quarks and mesons get masses through symmetry broken. We apply this SU(3) chiral constituent quark model to investigating the nuclear matter at finite temperature and density. The effective baryon masses, compression modulus and hyperon potentials are all reasonable. The critical temperature of liquid-gas phase transition is also calculated in this model.     

The Effect of Bare Quark Mass on the Pion Properties in BS Formalism Containing a Confining Term

Using the SD equation and the covariant BS equation, we investigate dynamical chiral symmetry breaking in the presence of a bare quark mass in a QCD motivated model containing a confining interaction. The bare quark mass m₀ corresponding to the physical mass of the pion and corrections to the dynamical quark mass and the pion decay constant due to mo are calculated. The results are consistent with the notion of chiral perturbatiyn and predictions of the current algebra.     

Generalized form factors and spin structures of the kaon

We investigate the spin structure of the kaon, based on the nonlocal chiral quark model from the instanton vacuum. We first revisit the electromagnetic form factors of the pion and kaon, improving the results for the kaon. We evaluate the generalized tensor form factors of the kaon in order to determine the probability density of transversely polarized quarks inside the kaon. We consider the effects of flavor SU(3) symmetry breaking, so that the probability density of the up and strange quarks are examined in detail. It is found that the strange quark behaves differently inside the kaon in comparison with the up quark.     

Electrical Neutrality and Symmetry Restoring Phase Transitions at High Density in a Two-Flavor Nambu-Jona-Lasinio Model

A general research on chiral symmetry restoring phase transitions at zero temperature and finite chemical potentials under electrical neutrality condition has been conducted in a Nambu-Jona-Lasinio model to describe twoflavor normal quark matter. Depending on whether mo/A, the ratio of dynamical quark mass in vacuum and the 3D momentum cutoff in the loop integrals, is less or greater than 0.413, the phase transition will be of the second or first order. A complete phase diagram of u quark chemical potential versus mo is given. With the electrical neutrality constraint, the region where the second order phase transition happens will be wider than the one without electrical neutrality limitation. The results also show that, for the value of m0/∧ from QCD phenomenology, the phase transition must be of the first order.     

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.