Universality (beyond leading log) of soft radiative corrections to $\hat q$ in $p_\perp$ broadening and energy loss

TL;DR

This paper investigates whether soft radiative corrections beyond leading log order can be universally absorbed into an effective $\

Contribution

It demonstrates that single logarithmic soft radiative corrections in $p_\perp$ broadening and hard splitting are related and can be incorporated into an effective $\

Findings

Single logarithms from soft radiation in $p_\perp$ broadening are linked to those in hard splitting.

Two types of $\hat q_{\rm eff}$ emerge, differing by $i\pi$ terms, affecting single logarithms.

The study extends the understanding of soft corrections beyond leading logs in QCD media.

Abstract

It has been known for many years that soft radiation can give potentially large double-logarithm corrections to $p_\perp$ broadening of a high-energy particle traveling through QCD matter, but that this soft radiation correction can be absorbed into an effective value $\hat q_{\rm eff}$ for the medium $p_\perp$-broadening parameter $\hat q$. Here "soft" means high energy compared to medium scales but soft compared to the original high-energy particle traveling through the medium. A similar situation arises in the case of soft corrections to hard splitting of a high-energy particle, such as hard $g{\to}gg$, where double logarithms can also be absorbed using the same effective $\hat q_{\rm eff}$. In this paper, I study whether the same holds true for potentially large, subleading, *single*-logarthim corrections. The correspondence is more indirect for single logarithms, but I show (in the large-$N_{\rm c}$ limit) that single logarithms from soft radiation in the case of $p_\perp$ broadening also determine the single logarithms from soft radiative corrections to hard $g{\to}gg$ splitting. Along the way, there is an interesting variation of the original BDMPS-Z calculation of splitting rates in the $\hat q$ approximation. I also discuss how, for soft-radiative corrections to hard splitting processes, there are two different types of "$\hat q_{\rm eff}$" that come into play, which differ by "$i\pi$" terms that multiply single logarithms.