Restoration of chiral symmetry in the large-Nclimit

Abstract

We study the large-Nc behavior of the critical temperature Tc for chiral symmetry restoration in the framework of the Nambu-Jona-Lasinio (NJL) model and the linear σ-model. While in the NJL case Tc scales as N 0 c and is, as expected, of the same order as ΛQCD (just as the deconfinement phase transition), in the σ-model the scaling behavior reads Tc ∝ N 1/2 c . We investigate the origin of the different scaling behavior and present two improvements of the σ-model: (i) a simple, phenomenologically motivated temperature dependence of the parameters and (ii) the coupling to the Polyakov loop. Both approaches lead to the scaling Tc ∝ N 0 c .PACS numbers: 11.30. Rd ,11.15.Pg, 11.10.Wx, 11.30.Qc Although quantum chromodynamics (QCD) is a theory for quarks and gluons, in the vacuum they are confined inside hadrons. It is, however, expected that at sufficiently high temperature and/or density a phase transition to a deconfined gas of interacting quarks and gluons takes place [1,2]. Moreover, it is also expected that this deconfinement phase transition is related to the so-called chiral phase transition: chiral symmetry is broken in the vacuum and restored in a hot and/or dense medium, see Ref.[3] and the lattice simulations of Refs. [4,5].The precise connection between the deconfinement and the chiral phase transition is not yet clear. This is also due to the fact that both transitions can only be precisely defined in limiting situations which are not realized in nature. Namely, in the context of pure Yang-Mills theory (QCD with infinitely heavy quarks) the Polyakov loop is the order parameter for the deconfinement phase transition [6]: the expectation value of the Polyakov loop vanishes for low T and µ (confined matter) and approaches unity in the deconfined phase. On the other hand, in the limit of zero quark masses the QCD Lagrangian is invariant under chiral transformations. The chiral condensate qq is the order parameter for the chiral phase transition: it is nonzero in the vacuum, decreases for increasing T and µ, and vanishes in the chirally restored phase. Nature is somewhere in between: the quark masses are neither infinite nor zero. The chiral condensate and the Polyakov loop are therefore only approximate order parameters.Since QCD cannot be directly solved, various methods are used to perform explicit calculations. Besides the already mentioned lattice simulations, effective models containing quark degrees of freedom only, such as the NJL model [7][8][9][10], and purely hadronic models, such as the linear σ-model [11,12], have been used to study the thermodynamics of QCD. Both approaches cannot describe the deconfinement phase transition: the degrees of freedom of NJL models are deconfined quarks at all temperatures and densities, while linear σ-models feature hadronic degrees of freedom in which quarks are always confined. However, in both approaches the critical temperature for the chiral phase transition is in agreement with recent lattice studies.In order to amend these problems, generalizations of th...

Inhomogeneous chiral condensates

“…Recently Heinz, Giacosa and Rischke (HGR) have therefore revisited the question of inhomogeneous chiral condensates in nuclear matter [101] using an extended version of the linear sigma model (eLSM), developed in Refs. [102] (see also [103] for an earlier version).…”

“…Treating the baryonic contribution in no-sea approximation, the total thermodynamic potential at T = 0 and baryon chemical potential µ is then given by [101] …”

Magnetic catalysis in nuclear matter

“…is a nucleon doublet where each of the components ψ 1 and ψ 2 (themselves being doublets in isospin space) transform oppositely under chiral transformations ("mirror assignment") [20][21][22][23][24][25][26]. The Dirac operator iγ µ D µ + γ 0 µ is diagonal in this "mirror space".…”

Mirror versus naive assignment in chiral models for the nucleon

“…The need of a sizable m 0 was also corroborated in the recent study of Ref. [34], in which the emergence of an inhomogenous chiral condensate was studied: it was shown that a value of m 0 500 MeV is necessary to have an homogeneous liquid nuclear matter ground state. In this paper we have also evaluated the pion-nucleon sigma term, which turns out to be about 15 MeV in the mirror case and about 18 MeV in the naive one.…”