Nonequilibrium pion dynamics near the critical point in a constituent quark model

Abstract

We study static and dynamical critical phenomena of chiral symmetry breaking in a two-flavor Nambu–Jona-Lasinio constituent quark model. We obtain the low-energy effective action for scalar and pseudoscalar degrees of freedom to lowest order in quark loops and to quadratic order in the meson fluctuations around the mean field. The static limit of critical phenomena is shown to be described by a Ginzburg–Landau effective action including spatial gradients. Hence static critical phenomena is described by the universality class of the O(4) Heisenberg ferromagnet. Dynamical critical phenomena is studied by obtaining the equations of motion for pion fluctuations. We find that for T< T c the are stable long-wavelength pion excitations with dispersion relation ω π ( k)= k described by isolated pion poles. The residue of the pion pole vanishes near T c as Z∝1/|ln(1− T/ T c )| and long-wavelength fluctuations are damped out by Landau damping on a time scale t rel( k)∝1/ k, reflecting critical slowing down of pion fluctuations near the critical point. At the critical point, the pion propagator features mass shell logarithmic divergences which we conjecture to be the harbinger of a (large) dynamical anomalous dimension. We find that while the classical spinodal line coincides with that of the Ginzburg–Landau theory, the growth rate of long-wavelength spinodal fluctuations has a richer wavelength dependence as a consequence of Landau damping. We argue that Landau damping prevents a local low-energy effective action in terms of a derivative expansion in real time at least at the order studied.