Bulk viscosity and spectral functions in QCD

TL;DR

This paper investigates the bulk viscosity in QCD by analyzing the spectral function of the trace of the stress tensor, $\rho(\omega)$, in two analytically tractable regimes: weak coupling at high $T$ and near a second-order phase transition. Using the Kubo relation $\zeta=\tfrac{1}{2}\lim_{\omega\to 0}\frac{\rho(\omega)}{\omega}$ and universality arguments, the authors derive a narrow peak in $\rho(\omega)/\omega$ at low frequency in both regimes, with a perturbative height $\sim g^4 T^4$ and area $\sim g^7 T^5$ at weak coupling, and a divergent height controlled by $t^{-z\nu+\alpha}$ near the critical point (with $\nu\simeq0.630$, $\alpha\simeq0.110$, $z\simeq3$). They show that the peak complicates analytic continuation from Euclidean lattice data, and propose lattice-based strategies to estimate the peak area and correlation length to bound $\zeta$. The results highlight different mechanisms of slow relaxation—quasiparticle scattering at weak coupling and critical order-parameter fluctuations near the transition—and their implications for bulk viscosity in the QCD plasma and heavy-ion phenomenology.

Abstract

We examine the behavior of the spectral function for the trace of the stress tensor in QCD in the two regimes where it is possible to make analytical progress; weak coupling, and close to a second order QCD phase transition. We determine the behavior of the bulk viscosity in each regime. We discuss the problem of analytic continuation of the (lattice) Euclidean correlation function to determine the spectral function. In each case the spectral function has a narrow peak at small frequency; its shape would be challenging to extract accurately from lattice data with error bars.