Second order hydrodynamic coefficients from kinetic theory

TL;DR

This work computes the leading-order second-order hydrodynamic coefficients in weakly coupled gauge theories using kinetic theory, extending the landscape beyond strongly coupled ${\cal N}=4$ SYM results. The analysis identifies a robust conformal relation $\lambda_2 = -2\eta\tau_{\Pi}$ and finds $\kappa=0$ at leading order, while determining nonzero $\eta\tau_{\Pi}$ and $\lambda_1$ through detailed treatment of $2P^{\mu}\partial_{\mu}f_1$, ${\cal C}_{11}[f_1]$, and ${\cal C}_{1;{\cal M}_1}[f_1]$ terms; the magnitude of these coefficients is presented as dimensionless ratios to $(\epsilon+P)$ and $\eta$, showing weak-coupling scaling with $m_D/T$. The methodology relies on a near-equilibrium Boltzmann framework with conformal, massless quasiparticles and includes both elastic and inelastic scatterings, with screening effects and infrared sensitivities carefully addressed. The results provide a bridge between weakly coupled kinetic theory and phenomenological viscous hydrodynamics for the quark-gluon plasma, offering quantitative benchmarks for second-order transport coefficients and their coupling dependence, and highlighting the physical interpretations and limitations of the kinetic approach.

Abstract

In a relativistic setting, hydrodynamic calculations which include shear viscosity (which is first order in an expansion in gradients of the flow velocity) are unstable and acausal unless they also include terms to second order in gradients. To date such terms have only been computed in supersymmetric N=4 Super-Yang-Mills theory at infinite coupling. Here we compute these second-order hydrodynamic coefficients in weakly coupled QCD, perturbatively to leading order in the QCD coupling, using kinetic theory. We also compute them in QED and scalar lambda phi^4 theory.

Figures (3)

• Figure 1: Coupling dependence of the ratio $(\epsilon{+}P) \tau_{\Pi}/\eta$. This ratio compares the relaxation time scale for $\Pi_{\mu\nu}$, $\tau_{\Pi}$, to the time scale implied by the viscosity $\eta$.
• Figure 2: Coupling dependence of the nonlinearity parameter $\lambda_1$, expressed as the dimensionless ratio $(\epsilon{+}P)\lambda_1 / \eta^2$. As explained in Section \ref{['sec:discussion']}, there is an unresolved uncertainty in these curves, but it is smaller than the line widths.
• Figure 3: Illustration of the physical origin of $\eta$ and of $\lambda_2$. Under Bjorken contraction (left), the momentum distribution becomes prolate long the $z$ axis. But under rotation with $\partial_z v_y > 0$ (right), the prolate axis gets rotated to have a $y$ component, so $T_{yz}>0$.