QCD splitting/joining functions at finite temperature in the deep Landau-Pomeranchuk-Migdal regime

Abstract

There exist full leading-order-in-${\ensuremath{\alpha}}_{\mathrm{s}}$ numerical calculations of the rates for massless quarks and gluons to split and join in the background of a quark-gluon plasma through hard, nearly collinear bremsstrahlung and inverse bremsstrahlung. In the limit of partons with very high energy $E$, where the physics is dominated by the Landau-Pomeranchuk-Migdal effect, there are also analytic leading-log calculations of these rates, where the logarithm is $\mathrm{ln}(E/T)$. We extend those analytic calculations to next-to-leading-log order. We find agreement with the full result to within roughly 20% for ${E}_{<}\ensuremath{\gtrsim}10T$, where ${E}_{<}$ is the energy of the least energetic parton in the splitting/joining process. We also discuss how to account for the running of the coupling constant in the case that $E/T$ is very large. Our results are also applicable to isotropic nonequilibrium plasmas if the plasma does not change significantly over the formation time associated with particle splitting.