Asymptotics of thermal spectral functions

TL;DR

The paper develops a Minkowski-space OPE framework to study thermal spectral functions in the deeply virtual regime $\omega \gg T$, establishing that leading thermal corrections scale as $\sim T^4$ and, at weak coupling, as $\sim g^2 T^4$ with an undetermined shear-channel coefficient. It demonstrates infrared safety of spectral functions up to order $g^8 T^4$, proves convergence of the shear and bulk viscosity sum rules in asymptotically free theories, and highlights significant ultraviolet tails in the bulk channel that influence sum-rule analyses away from the phase transition. The analysis provides explicit OPE coefficients for currents and stress tensors, clarifies contact-term ambiguities, and links RG running to the high-frequency behavior of spectral functions through anomalous dimensions and operator mixing. It also shows that in strongly coupled ${\cal N}=4$ SYM there are no medium-dependent power corrections, with high-energy spectra decaying exponentially, underscoring a qualitative difference between weakly coupled QCD-like theories and holographic models.

Abstract

We use operator product expansion (OPE) techniques to study the spectral functions of currents and stress tensors at finite temperature, in the high-energy time-like region $ω\gg T$. The leading corrections to these spectral functions are proportional to $\sim T^4$ expectation values in general, and the leading corrections $\sim g^2T^4$ are calculated at weak coupling, up to an undetermined coefficient in the shear viscosity channel. Spectral functions are shown to be infrared safe, in the deeply virtual regime, up to order $g^8T^4$. The convergence of (vacuum subtracted) sum rules in the shear and bulk viscosity channels is established in QCD to all orders in perturbation theory, though numerically significant tails $\sim T^4/(\logω)^3$ are shown to exist in the bulk viscosity channel. We argue that the spectral functions of currents and stress tensors in infinitely coupled $\mathcal{N}=4$ super Yang-Mills do not receive any medium-dependent power correction.

Figures (3)

• Figure 1: Feynman diagrams giving the leading order OPE coefficients for currents and stress tensors.
• Figure 2: Real-time diagrams of first topology contributing to $\Pi^>$ (with the complex conjugate diagrams omitted). The arrows show the time flow along retarded propagators, not the charge flow; the doubly-dashed propagator is the fluctuation function $G_{rr}$.
• Figure 3: Real-time diagrams of second topology contributing to $\Pi^>$, in the notation of Fig. \ref{['fig:diags1']}. Not shown, the complex conjugates to (b)-(c).