Transport coefficients, spectral functions and the lattice

TL;DR

The paper analyzes transport coefficients through the finite-temperature spectral function of the energy-momentum tensor in weakly coupled scalar and nonabelian gauge theories. By computing the shear-related spectral function across all frequencies and linking it to Euclidean correlators, the study reveals a distinct two-region structure: a low-frequency, ladder-dominated sector and a high-frequency decay/creation sector, with a $\tau$-independent constant contribution to the Euclidean correlator that complicates lattice extraction of $\eta$. It argues that simple Breit-Wigner fits are inadequate for high-temperature spectral functions, and it proposes a two-term ansatz that separately models low- and high-frequency contributions to facilitate nonperturbative lattice determinations of transport coefficients. The results highlight the practical challenges of obtaining precise $\eta$ from lattice data and underscore the need for improved spectral parametrizations and careful treatment of ladder effects.

Abstract

Transport coefficients are determined by the slope of spectral functions of composite operators at zero frequency. We study the spectral function relevant for the shear viscosity for arbitrary frequencies in weakly-coupled scalar and nonabelian gauge theories at high temperature and compute the corresponding correlator in euclidean time. We discuss whether nonperturbative values of transport coefficients can be extracted from euclidean lattice simulations.

Figures (7)

• Figure 1: Typical configuration of poles in the complex $k^0$-plane for the evaluation of the one-loop spectral function $\rho_{\pi\pi}(\omega)$, $E_{\mathbf k}$ ($2\gamma_{\mathbf k}$) denotes the quasiparticle energy (width).
• Figure 2: Contribution to the spectral function $\rho_{\pi\pi}(\omega)/\omega^4$ (full line) from decay/creation processes [see Eq. (\ref{['eqhigh']})] as a function of $\omega/T$, with $m/T=0.1$. The contribution from the nearly pinching poles in the low-frequency region is discussed below. The dashed line shows the contribution proportional to the Bose distribution only. The dotted line indicates the asymptotic value.
• Figure 3: Ladder diagrams that contribute to $\rho_{\pi\pi}(\omega)$ in the scalar theory. In the low-frequency region, $\omega \lesssim \gamma$, the pinching-pole contributions from the ladder diagrams are equally important as the one-loop contribution. When $\omega \gg \gamma$, pinching-pole contributions from ladder diagrams are suppressed.
• Figure 4: Contribution to the spectral function $\rho_{\pi\pi}(\omega)/T^4$ (full line) from the nearly pinching poles in the low-frequency region as a function of $\omega/\gamma$, obtained by numerical integration of Eq. (\ref{['eqrhobelow']}). The dashed line is the analytical result (\ref{['eqapprox']}) when $\gamma\ll\omega\lesssim m$, neglecting nontrivial momentum dependence and finite mass corrections. The inset shows a blowup. The viscosity is determined by the slope for $\omega\to 0$.
• Figure 5: Complete one-loop spectral function $\rho_{\pi\pi}(\omega)/\omega^4$ (full line) as a function of $\omega/T$, with $m/T=0.1$. The dashed line is the contribution from decay/creation processes and vanishes below $\omega=2m$, the dot-dashed line is the contribution due to the nearly pinching poles at lower frequencies. The dotted line indicates the asymptotic value.