Soft gluons and gauge-invariant subtractions in NLO parton-shower Monte Carlo event generators

TL;DR

This work develops a gauge-invariant subtractive framework to decompose perturbative QCD graphs into region-specific contributions (soft, initial-state collinear, final-state collinear, and hard) for NLO corrections in parton-shower Monte Carlo event generators. Using Wilson-line operators and a graph-by-graph R-operation, it derives gauge-invariant counterterms that yield an integrable hard-scattering coefficient and MC-friendly collinear terms, with a formalization that ties rapidity cutoffs to homogeneous evolution equations. A detailed DIS example (γ* q → g q) provides explicit soft and collinear subtractions (M_S, M_I, M_F, M_H) and shows how η, η′ dependencies cancel, while selecting MC-compatible forms (M_I^{MC}, M_F^{MC}) that connect to standard splitting kernels and linked cutoffs. The results lay groundwork for incorporating NLO corrections into MC generators, while noting that integrating these subtractions into a complete shower algorithm requires further development.

Abstract

We address the problem of decomposing graphs in perturbative QCD into terms associated with particular regions. Motivated by asking how to incorporate next-to-leading order (NLO) QCD corrections in parton-shower algorithms, we require that: (a) The integrand for the hard part is to be integrable even if the corrections are applied to a process that is not infrared and collinear safe. (b) The splitting between the terms should be defined gauge-invariantly. (c) The dependence on cut-offs should obey homogeneous evolution equations. In the context of one-gluon-emission graphs for deep inelastic scattering, we explain a subtractive technique that is based on gauge-invariant Wilson-line operators. Appropriate organization of subtractions involving the soft region allows a connection to previous work where evolution equations with respect to the directions of the Wilson lines have been derived.