Recursive numerical calculus of one-loop tensor integrals

TL;DR

This work presents a numerically stable recursive framework to reduce one-loop tensor integrals to a minimal scalar basis, enabling efficient radiative-correction calculations for multi-leg processes. It introduces a master 4D recursion that expresses high-rank tensors in terms of lower-rank objects and extends to $n=4+epsilon$ via a 4D+$b epsilon$ split, avoiding explicit subtraction of ultraviolet and infrared/collinear divergences. The method emphasizes Gram determinant stability by ensuring only inverse square roots of Gram determinants appear and provides special handling for exceptional kinematics and three-point tensors. It also offers a clear approach for including epsilon-dimension contributions and demonstrates applicability to QCD and EW one-loop calculations, with plans for a practical implementation.

Abstract

A numerical approach to compute tensor integrals in one-loop calculations is presented. The algorithm is based on a recursion relation which allows to express high rank tensor integrals as a function of lower rank ones. At each level of iteration only inverse square roots of Gram determinants appear. For the phase-space regions where Gram determinants are so small that numerical problems are expected, we give general prescriptions on how to construct reliable approximations to the exact result without performing Taylor expansions. Working in 4+epsilon dimensions does not require an analytic separation of ultraviolet and infrared/collinear divergences, and, apart from trivial integrals that we compute explicitly, no additional ones besides the standard set of scalar one-loop integrals are needed.