Supercooled Goldstone Bosons at the QCD Chiral Phase Transition

Abstract

We discuss a universal non-equilibrium enhancement of long-wavelength Goldstone bosons induced by quenches to the broken phase in Model G -- the dynamical universality class of an $O(4)$-antiferromagnet and the chiral phase transition in QCD. Scaling arguments for the coarsening dynamics describing the formation of the chiral condensate predict a parametric enhancement in the infrared spectra of Goldstones, a prediction confirmed by stochastic simulations of the transition. The details of the enhancement are determined by the non-linear dynamics of a superfluid effective theory, which is a limit of Model G reflecting the broken $O(4)$ symmetry. Our results translate to a parametric enhancement of low-momentum pions in heavy-ion collisions at the LHC, which are underpredicted in current hydrodynamic models without critical dynamics.

Figures (4)

• Figure 1: Illustration of instantaneous quenches in Model G. (Left) At every point in space, the effective potential of the theory suddenly changes from a convex potential to a double-well potential. After a time of order $\tau_R$, the field locally reaches a new minimum in field space, leading to many small randomly oriented domains shown in the right panel. (Right) The domains merge in a time $t_B \sim L/v$ with coarsening dynamics given by the superfluid limit of Model G (see text).
• Figure 2: Schematic time evolution of the pion-pion correlator $G_{\pi\pi}(t,k)$ at small momenta $k\xi\ll 1$, following a symmetric quench at $t=0$ from the restored to the broken phase in three dimensions ($d=3$). The system first relaxes locally for $t\sim\tau_R$, seeding local condensate domains and setting the initial conditions for correlator growth. Pions of size $1/k\gg\xi$ then propagate ballistically over time $t\sim\tau_R/(k\xi) \sim 1/vk$, and the condensate domains grow to size $\sim 1/k$. This coarsening dynamics in $d=3$ enhances $G_{\pi\pi}$ by $\sim 1/k\xi$ over equilibrium, $G_{\pi\pi}^{\rm eq}\sim \xi^{2-\eta}/(k\xi)^2$. Finally, the pion correlator returns to equilibrium (the dashed line) via diffusion for $t\sim\tau_R/(k\xi)^2$.
• Figure 3: The chiral condensate (estimated by $\sqrt{\left\langle M_a M_a\right\rangle}$) versus time following a symmetric quench, ${\sf t}_{\sf r}^0 \rightarrow -{\sf t}_{\sf r}^0$, for several reduced temperatures, ${\sf t}_{\sf r}^0$. The upper panel shows the raw data, while the lower panel shows the data with the $x$ and $y$ axes rescaled by $L/v$ and $\bar{\sigma}_{\rm eq}$ respectively. The data collapse (up to $\xi({\sf t}_{\sf r} )/L$ corrections) confirms the non-equilibrium scaling predictions of \ref{['eq:scaling_G0']}.
• Figure 4: Pion-pion correlator versus time at $k=2\pi/L$ following a symmetric quench, ${\sf t}_{\sf r}^0\rightarrow -{\sf t}_{\sf r}^0$, for several initial reduced temperatures, ${\sf t}_{\sf r}^0$. The upper panel shows the raw data, while the lower panel shows the data with the $x$ and $y$ axes rescaled by $vk$ and $k\xi/G_{\pi\pi}^{\rm eq}$ respectively. The data collapse at intermediate times in the lower panel supports the non-equilibrium scaling in \ref{['eq:Gpipik']} and implies that $G_{\pi\pi}(t,k)$ is parametrically enhanced by $\sim 1/k\xi$ at these times. At late times, the scaled correlators thermalize and the scaled correlators do not collapse.