Universality (beyond leading log) of soft radiative corrections to $$ \hat{q} $$ in p⊥ broadening and energy loss
Peter Arnold
Abstract
A bstract It has been known for many years that soft radiation can give potentially large double-logarithm corrections to p ⊥ broadening of a high-energy particle traveling through QCD matter, but that this soft radiation correction can be absorbed into an effective value $$ \hat{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> eff for the medium p ⊥ -broadening parameter $$ \hat{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> . 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 → gg , where double logarithms can also be absorbed using the same effective $$ \hat{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> 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 c limit) that single logarithms from soft radiation in the case of p ⊥ broadening also determine the single logarithms from soft radiative corrections to hard g → gg splitting. Along the way, there is an interesting variation of the original BDMPS-Z calculation of splitting rates in the $$ \hat{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> approximation. I also discuss how, for soft-radiative corrections to hard splitting processes, there are two different types of “ $$ \hat{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> eff ” that come into play, which differ by “ iπ ” terms that multiply single logarithms.