Chiral matrix model of the semi-QGP in QCD

Robert D. Pisarski and Vladimir V. Skokov

Phys. Rev. D 94, 034015 – Published 8 August 2016

Abstract

Previously, a matrix model of the region near the transition temperature, in the “semi”quark gluon plasma, was developed for the theory of $S U (3)$ gluons without quarks. In this paper we develop a chiral matrix model applicable to QCD by including dynamical quarks with $2 + 1$ flavors. This requires adding a nonet of scalar fields, with both parities, and coupling these to quarks through a Yukawa coupling, $y$ . Treating the scalar fields in mean field approximation, the effective Lagrangian is computed by integrating out quarks to one loop order. As is standard, the potential for the scalar fields is chosen to be symmetric under the flavor symmetry of $S U (3)_{L} \times S U (3)_{R} \times Z (3)_{A}$ , except for a term linear in the current quark mass, $m_{qk}$ . In addition, at a nonzero temperature $T$ it is necessary to add a new term, $\sim m_{qk} T^{2}$ . The parameters of the gluon part of the matrix model are identical to those for the pure glue theory without quarks. The parameters in the chiral matrix model are fixed by the values, at zero temperature, of the pion decay constant and the masses of the pions, kaons, $η$ , and $η^{'}$ . The temperature for the chiral crossover at $T_{χ} = 155 MeV$ is determined by adjusting the Yukawa coupling $y$ . We find reasonable agreement with the results of numerical simulations on the lattice for the pressure and related quantities. In the chiral limit, besides the divergence in the chiral susceptibility there is also a milder divergence in the susceptibility between the Polyakov loop and the chiral order parameter, with critical exponent $β - 1$ . We compute derivatives with respect to a quark chemical potential to determine the susceptibilities for baryon number, the $χ_{2 n}$ . Especially sensitive tests are provided by $χ_{4} - χ_{2}$ and by $χ_{6}$ , which changes in sign about $T_{χ}$ . The behavior of the susceptibilities in the chiral matrix model strongly suggests that as the temperature increases from $T_{χ}$ , that the transition to deconfinement is significantly quicker than indicated by the measurements of the (renormalized) Polyakov loop on the lattice.