The Bulk Viscosity of High-Temperature QCD

TL;DR

This work evaluates the bulk viscosity $\zeta$ of high-temperature QCD within kinetic theory, establishing it at leading order in $\alpha_s(T)$ and showing it is parametrically smaller than the shear viscosity $\eta$ for realistic couplings. The authors reveal that bulk viscosity vanishes in a conformal theory and grows with the square of conformal-symmetry breaking, arising from the beta function and quark masses, yielding $\zeta \sim \alpha_s^2 T^3/\log(1/\alpha_s)$ for massless QCD and $\zeta \sim m_0^4/(T\alpha_s^2\log(1/\alpha_s))$ when a heavy mass is present. They solve a linearized Boltzmann equation using a variational approach, deriving the source term $q^a(p)$ and expressing $\zeta$ as $\zeta=(\mathcal S,\chi)$ with $\mathcal C\chi=\mathcal S$, while addressing exact and approximate zero modes of the collision operator. The analysis shows that leading-log expansions are unreliable except at very small $\alpha_s$, and that number-changing processes in QCD do not bottleneck bulk relaxation; instead, soft gluon dynamics and $2\leftrightarrow 2$ scattering govern the parametric behavior. Overall, the results indicate that bulk viscosity is typically suppressed by several orders of magnitude relative to $\eta$ in perturbative QCD, supporting the common practice of neglecting $\zeta$ in many QGP and early-universe applications.

Abstract

We compute the bulk viscosity zeta of high-temperature QCD to leading order in powers of the running coupling alpha_s(T). We find that it is negligible compared to shear viscosity eta for any alpha_s that might reasonably be considered small. The physics of bulk viscosity in QCD is a bit different than in scalar phi^4 theory. In particular, unlike in scalar theory, we find that an old, crude estimate of zeta as 15 ((1/3)-v_s^2)^2 eta gives the correct order of magnitude, where v_s is the speed of sound. We also find that leading-log expansions of our result for zeta are not accurate except at very small coupling.

Figures (8)

• Figure 1: Bulk viscosity for massless QCD at several values of $N_{\rm f}$, as a function of the coupling $\alpha_{\rm s}$.
• Figure 2: Bulk viscosity when it is dominated by a single quark flavor's mass, as a function of $\alpha_{\rm s}$, for several values of $N_{\rm f}$.
• Figure 3: Shear versus bulk viscosity: $\eta/s$ and $\zeta/s$ ($s$ the entropy density) as a function of $\alpha_{\rm s}$, for $N_{\rm f}{=}3$ QCD, neglecting quark masses. Bulk viscosity $\zeta$ has been rescaled by a factor of 1000.
• Figure 4: The ratio $\zeta/\alpha_{\rm s}^4\eta$ for $N_{\rm f}{=}3$ QCD, neglecting quark masses. The dashed line shows the crude estimate of (\ref{['eq:crudezeta']}) with (\ref{['eq:deltavs']}). As $\alpha_{\rm s}\to0$ (and leading-log approximations to the leading-order result become applicable), the ratio approaches the limit $\zeta/\alpha_{\rm s}^4\eta \to 0.973$.
• Figure 5: Examples of (a) number conserving and (b) number changing processes in $\phi^4$ theory.