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

TL;DR

This work extends the subleading soft-gluon one-loop framework to massive QCD by deriving a color-spin soft operator ${\mathbf S}^{(1)}$ that preserves on-shell kinematics and momentum conservation, and by providing a complete subleading collinear term for $g\to q\bar{q}$ splittings. The main result decomposes ${\mathbf S}^{(1)}$ into non-abelian and abelian parts, expressed through a six-master-integral basis and detailed kinematic variables, and is supplemented by a thorough pole-cancellation analysis and finite contributions. The authors validate the formulation with numerical studies against full one-loop amplitudes, demonstrating reliable NLP accuracy for processes with massive quarks. The work lays groundwork for practical one-loop QCD amplitude approximations with masses and points toward future extensions to multi-soft emissions and higher orders, with potential impact on precision collider phenomenology.

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.

Figures (3)

• Figure 1: Diagrams included in the calculation of the soft operator. A double line is eikonal, while a dashed line corresponds to a (anti-)quark or a gluon. The shaded rectangle is a process-dependent sum of tree-level diagrams. The eikonal approximation is identical for (anti-)quarks and gluons if expressed through color operators. In this approximation, the quartic gluon vertex does not contribute. It does contribute, however, at subleading order of the soft expansion.
• Figure 2: Diagrams for $\gamma^* \to q\bar{q}$ (a)) and $\gamma^* \to q\bar{q}g$ (b)-h)). The cross in diagram f) represents a mass-renormalization counterterm.
• Figure 3: Relative error $\Delta_{\text{LP/NLP}}$ of the one-loop soft approximation to leading power (LP) and subleading power (NLP). The energy, $q_0$, of the soft gluon is normalised to the centre-of-mass energy, $\sqrt{s}$, of the process. The apparent breakdown of the approximation at low soft-gluon energies is due to the limited numerical precision of the one-loop integrals in OneLOop which impacts the result for the $(n+1)$-particle amplitudes.