The Dipole Formalism for Next-to-Leading Order QCD Calculations with Massive Partons

TL;DR

This work extends the dipole subtraction method to QCD processes with massive final-state partons, addressing the challenge of quasi-collinear logarithms $\ln(Q^2/M^2)$ that destabilize numerical calculations when $Q\gg M$. The authors develop dipole factorization formulae that interpolate soft, collinear, and quasi-collinear regions for massive partons, deriving universal integrated dipoles and insertion operators $\mathbf{I}$, $\mathbf{P}$, and $\mathbf{K}$ that consistently cancel infrared and collinear singularities across mass regimes. They provide explicit analytic expressions for dipole splitting functions, integrated dipoles, and phase-space factorization, covering final-state emitters with both final-state and initial-state spectators, as well as initial-state emitters. The formulation preserves the massless limit, enabling cross-checks against known massless results, and lays out the complete ingredients for implementing NLO QCD corrections to arbitrary observables in processes involving heavy quarks, squarks, and gluinos in a numerically stable way. The framework is designed to be integrated into general-purpose NLO Monte Carlo programs, facilitating precise predictions for SM and SUSY processes at current and future colliders such as the LHC. It also clarifies how to handle heavy-flavor PDFs and fragmentation in hadron collisions, ensuring smooth behavior across $Q/M$ variations.

Abstract

The dipole subtraction method for calculating next-to-leading order corrections in QCD was originally only formulated for massless partons. In this paper we extend its definition to include massive partons, namely quarks, squarks and gluinos. We pay particular attention to the quasi-collinear region, which gives rise to terms that are enhanced by logarithms of the parton masses, M. By ensuring that our subtraction cross section matches the exact real cross section in all quasi-collinear regions we achieve uniform convergence both for hard scales Q ~= M and Q >> M. Moreover, taking the masses to zero, we exactly reproduce the previously-calculated massless results. We give all the analytical formulae necessary to construct a numerical program to evaluate the next-to-leading order QCD corrections to arbitrary observables in an arbitrary process.

Figures (1)

• Figure 1: Effective diagrams for the different emitter--spectator cases.