The soft quark Sudakov
Ian Moult, Iain W. Stewart, Gherardo Vita, Hua Xing Zhu
Abstract
A bstract There has been recent interest in understanding the all loop structure of the subleading power soft and collinear limits, with the goal of achieving a systematic resummation of subleading power infrared logarithms. Most of this work has focused on subleading power corrections to soft gluon emission, whose form is strongly constrained by symmetries. In this paper we initiate a study of the all loop structure of soft fermion emission. In $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 QCD we perform an operator based factorization and resummation of the associated infrared logarithms using the formalism introduced in [1], and prove that they exponentiate into a Sudakov due to their relation to soft gluon emission. We verify this result through explicit calculation to $$ \mathcal{O}\left({\alpha}_s^3\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mfenced> </mml:math> . We show that in QCD, this simple Sudakov exponentiation is violated by endpoint contributions proportional to ( C A − C F ) n which contribute at leading logarithmic order. Combining our $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 1 result and our calculation of the endpoint contributions to $$ \mathcal{O}\left({\alpha}_s^3\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mfenced> </mml:math> , we conjecture a result for the soft quark Sudakov in QCD, a new all orders function first appearing at subleading power, and give evidence for its universality. Our result, which is expressed in terms of combinations of cusp anomalous dimensions in different color representations, takes an intriguingly simple form and also exhibits interesting similarities to results for large-x logarithms in the off diagonal splitting functions.