The Soft Quark Sudakov

TL;DR

This paper investigates the all-loop structure of soft fermion emissions, demonstrating Sudakov exponentiation in certain theories and identifying violations in QCD, leading to a conjecture of a universal soft quark Sudakov function at subleading power.

Contribution

It introduces the first all-order conjecture for the soft quark Sudakov in QCD, extending understanding of subleading power infrared logarithms and their exponentiation.

Findings

Soft quark Sudakov exponentiation in $ ext{N}=1$ QCD proven and verified to $ ext{O}( ext{α}_s^3)$.

Endpoint contributions in QCD violate simple Sudakov exponentiation at leading logarithmic order.

Conjecture of a universal soft quark Sudakov function in QCD, expressed via cusp anomalous dimensions.

Abstract

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 $\mathcal{N}=1$ QCD we perform an operator based factorization and resummation of the associated infrared logarithms, and prove that they exponentiate into a Sudakov due to their relation to soft gluon emission. We verify this result through explicit calculation to $\mathcal{O}(\alpha_s^3)$. 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 $\mathcal{N}=1$ result and our calculation of the endpoint contributions to $\mathcal{O}(\alpha_s^3)$, 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.