Formulae for a numerical computation of one-loop tensor integrals

TL;DR

This work addresses the numerical computation of one-loop tensor integrals by refining an existing $n$-dimensional reduction method. The authors introduce more symmetric recursion with $p_0^\mu \neq 0$ and a stable momentum basis to handle problematic kinematics, along with explicit treatments for three-point tensors up to rank 3 and rank-one reductions for general $m$, including guidance on extra integrals. The main contributions are a generalized recursion formula, explicit rank-2/3 three-point tensor reductions, and concrete expressions for rank-1, -2, and -3 cases that improve numerical stability and efficiency. The results enable robust, fast numerical implementations of one-loop tensor integrals across general kinematics, with direct relevance for precision calculations in particle physics.

Abstract

In a previous paper a new approach has been introduced for computing, recursively and numerically, one-loop tensor integrals. Here we describe a few modifications of the original method that allow a more efficient numerical implementation of the algorithm.