Subleading Effects in Soft-Gluon Emission at One-Loop in Massive QCD

TL;DR

This paper derives a soft operator for one-loop QCD amplitudes with massive quarks, enabling better approximations of gluon emissions in the soft limit, and provides subleading collinear expansions involving quark-antiquark pairs.

Contribution

It introduces a soft operator acting on color and spin space for massive QCD, and offers a complete subleading collinear expansion for tree-level amplitudes involving quark-antiquark pairs.

Findings

Derived a soft operator for massive QCD amplitudes.

Provided a subleading collinear expansion for quark-antiquark pairs.

Enhanced the accuracy of soft-gluon emission approximations.

Abstract

We provide the last missing ingredient necessary to approximate one-loop amplitudes in QCD with massive quarks in the limit of vanishing energy of a single gluon up to terms suppressed by this energy. Our main result is a soft operator acting in color and spin space that manipulates the momenta of the hard partons while keeping them on-shell and respecting momentum conservation. Additionally, we provide a complete expression for the subleading term of the expansion of an arbitrary tree-level amplitude in the limit where the momenta of a massless quark and a massless anti-quark of the same flavor become collinear. This limit is necessary to obtain the one-loop soft approximation whenever the process involves such a quark-anti-quark pair. Interestingly, the result involves a high-energy limit.