Thermodynamical second-order hydrodynamic coefficients

TL;DR

This work distinguishes dynamical and thermodynamical properties in non-conformal hydrodynamics and derives Kubo relations for the thermodynamical second-order coefficients $\kappa$, $\lambda_3$, and $\lambda_4$, computing their leading weak-coupling values in a general massless theory. By evaluating free-field contributions from scalars, gauge fields, and fermions, it provides explicit formulas for these coefficients in terms of the matter content, and shows $\lambda_4$ vanishes at this order while $\kappa$ and $\lambda_3$ depend on $N_0$, $N_{1/2}$, and $N_1$. The paper also discusses practical lattice approaches to extract these coefficients from Euclidean correlators, highlighting operator renormalization, contact terms, and the need to rotate to access off-diagonal stress-tensor components. For ${\mathcal{N}}=4$ SYM, the leading weak-coupling results yield zero for both $\kappa$ and $\lambda_3$, illustrating sensitivity to the matter content and suggesting nonzero values may arise at subleading orders or nonperturbative regimes. Overall, the results provide a bridge between finite-temperature lattice QCD and non-conformal second-order hydrodynamics, offering a pathway to quantify coupling-dependent thermodynamical transports.

Abstract

Transport coefficients in non-conformal second-order hydrodynamics can be classified as either dynamical or thermodynamical. We derive Kubo formuale for the thermodynamical coefficients and compute them at leading perturbative order in a theory with general matter content. We also discuss how to approach their evaluation on the lattice.

Figures (3)

• Figure 1: Leading order scalar diagram contributing to $\langle T^{xy}(-k) T^{xy}(k)\rangle$, necessary for evaluation of $\kappa$. The crosses are $T$ insertions, the solid lines are scalar propagators, and the arrows indicate the momenta flowing on lines and entering or leaving $T$ insertions.
• Figure 2: Three-point correlation function $\langle T^{xt}(p) T^{yt}(q) T^{xy}(-p-q) \rangle$ that contributes to the Kubo formula of $\lambda_{3}$; the leftmost vertex is $T^{xy}$, the other vertices are $T^{xt}$ and $T^{yt}$.
• Figure 3: How to handle a rotation which mixes a time and a space direction, on the lattice.