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

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 ${\hat{q}}_{\mathrm{eff}}$ 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 ${\hat{q}}_{\mathrm{eff}}$. 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-$N_{\mathrm{c}}$ 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 "${\hat{q}}_{\mathrm{eff}}$" that come into play, which differ by "$i\pi$" terms that multiply single logarithms.

35 pages, 13 figures [only minor change from v2 (fixed a figure reference)]

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• Transport properties of the QCD medium
• Anomalous dimension of transverse momentum broadening in planar $\mathcal{N}=4$ SYM
• Universality aspects of quantum corrections to transverse momentum broadening in QCD media