Energy and momentum dependence of the soft-axion interaction rate

Abstract

Axions coupled to thermal non-Abelian gauge fields may have cosmological significance. As the heat bath defines a frame, its influence depends separately on energy and momentum. A light-like momentum ($k \approx \omega$) is relevant for the axion contribution to the effective number of light neutrinos, $\Delta N^{ }_\mathrm{eff}$, whereas a vanishing momentum ($k=0$) plays a role for warm natural inflation or ultralight dark matter, and has been employed in lattice estimates (both classical and quantum-statistical) of the strong sphaleron rate. Focussing on soft energies ($\alpha_\mathrm{s}^{ }T \ll \omega \ll \pi T$), we carry out an HTL computation to show how the domains $k=0$ and $k \approx \omega$ interpolate to each other. We then compare with lattice data at $k=0$, and connect our analysis to NLO computations at $k \approx \omega \ge \pi T$. Assembling the current best input, we re-investigate light QCD axion decoupling dynamics at $T \ge 200$ MeV, showing that efficient interactions in the ultrasoft domain increase $\Delta N^{ }_\mathrm{eff}$ from $\sim 0.03$ to $\sim 0.04$ at $f^{ }_a = 4\times 10^8_{ }$ GeV.