Reduction of one-loop tensor 5-point integrals

TL;DR

The paper presents a novel reduction of one-loop tensor 5-point integrals to 4-point integrals that entirely avoids inverse Gram determinants, addressing numerical instabilities near phase-space boundaries. It derives a four-dimensional reduction framework based on linear dependence of the loop momentum, introduces finite and determinant-weighted terms, and provides explicit closed-form expressions for all 5-point tensor coefficients up to rank 4 in terms of standard 1–4 point integrals (including shifted 4-point functions). The authors validate the approach against the conventional Passarino–Veltman method for non-exceptional points, demonstrate dramatic improvements in numerical stability, and discuss generalizations to dimensional regularization and IR singularities. The method is applicable to common one-loop processes in electroweak and QCD corrections, facilitating stable evaluation of 2→3 and related amplitudes with up to four external gauge bosons.

Abstract

A new method for the reduction of one-loop tensor 5-point integrals to related 4-point integrals is proposed. In contrast to the usual Passarino-Veltman reduction and other methods used in the literature, this reduction avoids the occurrence of inverse Gram determinants, which potentially cause severe numerical instabilities in practical calculations. Explicit results for the 5-point tensor coefficients are presented up to rank 4. The expressions for the reduction of the relevant 1-, 2-, 3-, and 4-point tensor coefficients to scalar integrals are also included; apart from these standard integrals no other integrals are needed.