Effective theories for real-time correlations in hot plasmas

TL;DR

This work analyzes real-time current correlations in hot, weakly coupled gauge theories by building a tower of effective descriptions: perturbation theory for early times, kinetic theory for intermediate times, and hydrodynamics for very late times. It shows that the current–current correlator develops a long-time power-law tail due to coupling to hydrodynamic modes, with a $t^{-3/2}$ decay in the continuum and a $t^{-5/2}$ decay on the lattice, reflecting differences in conserved quantities and lattice artifacts. The authors also discuss finite-volume effects and caution against assuming classical lattice simulations automatically reproduce quantum real-time dynamics, highlighting where classical results can be informative and where they fail. Overall, the paper clarifies how to connect microscopic scattering physics to macroscopic transport and long-time behavior, and it delineates the limitations of classical simulations for real-time observables in hot gauge theories.

Abstract

We discuss the sequence of effective theories needed to understand the qualitative, and quantitative, behavior of real-time correlators < A(t) A(0) > in ultra-relativistic plasmas. We analyze in detail the case where A is a gauge-invariant conserved current. This case is of interest because it includes a correlation recently measured in lattice simulations of classical, hot, SU(2)-Higgs gauge theory. We find that simple perturbation theory, free kinetic theory, linearized kinetic theory, and hydrodynamics are all needed to understand the correlation for different ranges of time. We emphasize how correlations generically have power-law decays at very large times due to non-linear couplings to long-lived hydrodynamic modes.

Figures (8)

• Figure 1: A qualitative picture of the real part of the current-current correlation $V\langle {\bf j}(t) {\bf j}(0) \rangle$ in a hot plasma, for the case of bosonic charge carriers. The labels on the axes are only meant to denote orders of magnitude, except that the "$2 T^3$" mark is in fact twice the "$T^3$" mark in the small-coupling limit. The time axis should be thought of as linear before the break and logarithmic after; the vertical axis should be thought of as linear. $\ln$ is short for $\ln(g^{-2})$. The transient shown at $t \sim 1/T$ decays exponentially. If the current is carried by fermions instead of scalars, the resulting graph is identical except for the initial transient, which starts from minus the "$T^3$" plateau value instead of twice that value.
• Figure 3: The leading-order perturbative diagram for $\langle{\bf j}(t) {\bf j}(0)\rangle$.
• Figure 4: A pictorial example of a typical fluctuation with net current ${\bf j}$.
• Figure 5: t-channel gauge boson exchange. The solid lines represent any sort of particles that couple to the gauge force.
• Figure 6: (a) The randomization of a particle's velocity by successive scatterings in the plasma. (b) The same process in the background of a large column of fluid moving with net velocity. In this case, the particle's velocity is not completely randomized but is biased by the net velocity ${\bf v}_{\rm net}$.