Energy and momentum dependence of the soft-axion interaction rate

TL;DR

This work computes the energy- and momentum-dependent soft-axion interaction rate in a thermal QCD plasma using a Hard Thermal Loop framework, and it explicitly interpolates between the $k=0$ (condensate) and $k\approx\omega$ (lightlike) regimes. It combines HTL calculations, lattice data for ultrasoft modes, and NLO hard-graph results to construct a full rate function $\gamma_{\rm full}(k)$ that feeds a kinetic equation for axions in the early universe. The analysis shows that ultrasoft, nonperturbative dynamics substantially enhance the axion interaction rate relative to purely perturbative HTL estimates, increasing the predicted contribution to $\Delta N_{\rm eff}$ for $f_a\sim 4\times 10^8$ GeV from about $0.03$ to $\sim0.04$. The assembled results offer a more robust estimate of light QCD axion decoupling dynamics at $T\gtrsim200$ MeV and emphasize the importance of nonperturbative soft physics in cosmological axion phenomenology. The paper also outlines a roadmap for future lattice studies at nonzero $k$ to further test the perturbative/nonperturbative interpolation and its cosmological implications.

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 ($k \approx ω$) is relevant for the axion contribution to the effective number of light neutrinos, $ΔN^{ }_\mathrm{eff}$, 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 ($α_\mathrm{s}^{ }T \ll ω\ll πT$), we carry out an HTL computation to show how the domains $k=0$ and $k \approx ω$ interpolate to each other. We then compare with lattice data at $k=0$, and connect our analysis to NLO computations at $k \approx ω\ge πT$. 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 $ΔN^{ }_\mathrm{eff}$ from $\sim 0.03$ to $\sim 0.04$ at $f^{ }_a = 4\times 10^8_{ }$ GeV.

Figures (12)

• Figure 1: Illustration of the kinematic domains discussed in this work. On-shell axions have a fixed mass, $m_a^2 = \omega^2_{ } - k^2_{ }$, shown with a dotted crimson line. If their mass is small, $m^{ }_a \ll \pi T$, they are almost lightlike for typical thermal momenta, $k\sim \pi T$. In contrast, when we consider how an axion condensate interacts with a thermal plasma, we restrict ourselves to the axis $k=0$. All existing lattice simulations also focus on $k=0$. The dotted indigo line shows the curve probed by would-be imaginary-time lattice measurements at $k > 0$. The dashed blue rays span the domain studied in the present work, and will be used for illustrating our results in fig. \ref{['fig:chi_htl']}.
• Figure 2: Illustration of the integration domain pertinent to eqs. \ref{['rho_K']} and \ref{['new_domain']}. The red dashed line corresponds to $q^{ }_+ = \omega/2$, separating different types of physical processes (cf. table \ref{['table:domains']} on p. \ref{['table:domains']}).
• Figure 3: The integration domain for eqs. \ref{['oneloopexpmd2p']} and \ref{['oneloopexpmdperp']}. The meanings of the domains and of the dotted line, indicating a contour of constant $p^{ }_\perp$, are explained between eqs. \ref{['oneloopexpmd2p']} and \ref{['oneloopexpmdperp']}.
• Figure 4: The integration contours corresponding to eq. \ref{['residue']}, leading to eq. \ref{['oneloopexpmdperpeucl']}.
• Figure 5: Comparison of $\gamma_{\hbox{\scriptsize asy}}$ from eq. \ref{['gammaasyresumhack']}, denoted by "general", with the first two approximations from eq. \ref{['gamma_asy_cases']}, denoted by $k^{ }_- \gg m_{\hbox{\tiny\rm{E}}}^{ }$ and $k^{ }_- \sim m_{\hbox{\tiny\rm{E}}}^{ }$, respectively. The line $k=0$ gives $\gamma^{ }_{\hbox{\scriptsize asy}}(\omega,0) = 1$. The variable $k^{ }_-\equiv(\omega-k)/2$ runs from 0 (at $k=\omega$) to $\omega/2$ (at $k=0$).