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

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 \emph{static} limit of critical phenomena is shown to be described by a Ginzburg-Landau effective action including \emph{spatial} gradients. Hence \emph{static} critical phenomena is described by the universality class of the O(4) Heisenberg ferromagnet. \emph{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 $\omega_{\pi}(k)=k$ described by isolated pion poles. The residue of the pion pole vanishes near $T_c$ as $Z \propto 1/|\ln(1-T/T_c)|$ and long-wavelength fluctuations are damped out by Landau damping on a time scale $t_\mathrm{rel}(k)\propto 1/k$, reflecting \emph{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 \emph{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 \emph{local} low energy effective action in terms of a derivative expansion in real time.