Asymptotics of thermal spectral functions

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 $\omega\gg T$. 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 $\sim T^4/(\log\omega)^3$ 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.