Quenching through the QCD chiral phase transition

TL;DR

This work investigates the out-of-equilibrium dynamics of Model G, the dynamic universality class associated with the QCD chiral transition in the chiral limit. Through 3D Langevin simulations of an $O(4)$ order parameter coupled to conserved charges, the authors reveal a regime where the chiral condensate grows and drives a long-lived enhancement of soft pions, with equal-time correlators $G_{\pi\pi}(t,k)$ peaking above their equilibrium values. A scaling analysis identifies three key timescales—relaxation, ballistic transport, and diffusion—and shows that the growth is governed by the non-linear, non-dissipative dynamics of an $SU(2)_L\times SU(2)_R$ superfluid, yielding a parametrically large, persistent enhancement roughly scaling as $\sim 1/(k\xi)$ for $k\xi\ll 1$. Lattice quenches corroborate the scaling predictions, while a mean-field treatment captures the qualitative growth but misses the correct $k$-dependence and the ballistic growth phase. The results imply a potentially observable soft-pion yield enhancement in heavy-ion collisions and provide a pathway to determine hydrodynamic parameters from lattice QCD within the chiral scaling window.

Abstract

We present a detailed numerical and analytical study of the out-of-equilibrium dynamics of Model G, the dynamical universality class relevant to the chiral phase transition. We perform numerical 3D stochastic (Langevin) simulations of the $O(4)$ critical point for large lattices in the chiral limit. We quench the system from the high-temperature unbroken phase to the broken phase and study the non-equilibrium dynamics of pion fields. Strikingly, the non-equilibrium evolution of the two-point functions exhibits a regime of growth, a parametrically large enhancement, and a subsequent slow relaxation to equilibrium. We analyze our numerical results using dynamic critical scaling and mean-field theory. The growth of the two point functions is determined by the non-linear dynamics of an ideal non-abelian superfluid, which is a limit of Model G that reflects the broken chiral symmetry. We also relate the non-equilibrium two-point functions to a long-lived parametric enhancement of soft pion yields relative to thermal equilibrium following a quench.

Figures (11)

• Figure 1: Cartoon of the quench protocol in terms of correlation length $\xi$. The quenches studied in \ref{['sec:quenches']} transition from the high temperature phase to the low temperature phase, ${\sf t}_{\sf r}^0 \rightarrow -{\sf t}_{\sf r}^0$ with $\xi({\sf t}_{\sf r}^0)=\xi(-{\sf t}_{\sf r}^0)$ up to subleading corrections, see \ref{['tab:subleading']}. The colors from top to bottom correspond to ${\sf t}_{\sf r}^0 =0.009, 0.015, 0.023, 0.04$ and are correlated with the lines in the upper panels of \ref{['fig:VeVquench', 'fig:quench', 'fig:sgquenchk']}. In the the lower panels of \ref{['fig:VeVquench', 'fig:quench', 'fig:sgquenchk']} and \ref{['fig:quench_fixed_k']} we vary the lattice size while ${\sf t}_{\sf r}^0 = 0.04$.
• Figure 2: (a) Schematic of a quench from the high-temperature phase to the broken phase in a spatial region of size $\ell$, which is several correlation lengths long (e.g., $\ell \sim 3\xi$). The red dashed line shows the initial potential at temperature ${\sf t}_{\sf r}^0$, and the blue line shows the post-quench potential. The ball represents a local order parameter $\bar{\phi}_a(t)$, averaged over the region. The orange Gaussian depicts the initial thermal distribution for $\bar{\phi}_a$ before the quench, with variance $\chi/\ell^d$. Over a time of order $t\sim \tau_R$, $\bar{\phi}_a$ evolves to the well’s bottom in a random direction, forming a local equilibrated domain of size $\ell$. (b) Field configuration for $t \gg \tau_R$ in a region of size $L \gg \ell \sim \xi$, showing randomly oriented domains of the chiral condensate. The domains merge over a time of order $t \sim L/v$ with dynamics governed by the ideal superfluid equations given in \ref{['eq:idealsuper']}.
• Figure 3: A sketch of different stages in the typical evolution of the (pion) two-point function: early evolution setting the initial conditions $t\sim \tau_R$, rapid growth until the maximum, and slow decay to its equilibrium value in an oscillatory manner. See \ref{['sec:scaling']} for detailed discussion.
• Figure 4: Time evolution of magnetization for symmetric quenches in \ref{['fig:sim_recap']}, with (top) different starting ${\sf t}_{\sf r}^0$, but same lattice size $L=192$ and (bottom) with ${\sf t}_{\sf r}^0=0.04$ and different lattice sizes. All curves are normalized by the initial equilibrium value, \ref{['eq:chi']}. Colors indicate the system size in units of correlation length $L/\xi$. The time is given in units of relaxation time $t/\tau_R\propto t/\xi^{d/2}$ and pion ballistic propagation time $L/v$. Dashed lines show the expected equilibrium value with the leading order finite-size correction \ref{['eq:Hasenfratz']}.
• Figure 5: (top left) Enhancement of equal time pion-pion field correlator over thermal equilibrium expectation during symmetric quench for different starting ${\sf t}_{\sf r}^0$ and wavenumbers $k=2\pi/L$, where $L=192$. The time is given in units of relaxation time $t/\tau_R\propto t/\xi^{d/2}$. (top right) The same plot multiplied with $k\xi$ on $y$-axis and time given in units of pion ballistic propagation time $L/v\sim \tau_R (L/\xi)$, where $v$ is the pion velocity at corresponding ${\sf t}_{\sf r} ={\sf t}_{\sf r}^0$. (bottom left and right) Enhancement of pion-pion field correlator over thermal equilibrium expectation during symmetric quench from ${\sf t}_{\sf r} =0.04$ to ${\sf t}_{\sf r} =-0.04$ for different wavenumbers $k= 2\pi/L$, where $L=192,128,96,64$.