Semi-Numerical Evaluation of One-Loop Corrections

TL;DR

This work develops a semi-numerical framework for evaluating one-loop virtual corrections using dimensional regularization, enabling calculations for processes with up to five external legs and massless internal lines. It combines tensor integral decomposition with integration-by-parts recursion to reduce tensor integrals to a basis of master integrals, and introduces robust modifications to maintain numerical stability in exceptional kinematic regions where Gram determinants vanish. The authors implement and validate the approach by comparing Higgs decays to four partons against analytic results, demonstrating stability near problematic phase-space points. The methodology paves the way for automated NLO one-loop evaluations of more complex final states and potential integration with parton shower programs.

Abstract

We present a semi-numerical algorithm to calculate one-loop virtual corrections to scattering amplitudes. The divergences of the loop amplitudes are regulated using dimensional regularization. We treat in detail the case of amplitudes with up to five external legs and massless internal lines, although the method is more generally applicable. Tensor integrals are reduced to generalized scalar integrals, which in turn are reduced to a set of known basis integrals using recursion relations. The reduction algorithm is modified near exceptional configurations to ensure numerical stability. To test the procedure we apply these techniques to one-loop corrections to the Higgs to four quark process for which analytic results have recently become available.

Figures (10)

• Figure 1: The generic $N$-point loop graph.
• Figure 2: Reduction scheme for triangle diagrams with three off-shell legs.
• Figure 3: Reduction scheme for box diagrams.
• Figure 4: Reduction scheme for pentagon diagrams.
• Figure 5: Reduction scheme for triangles with two off-shell legs.