A simple method for multi-leg loop calculations

TL;DR

The paper tackles the problem of tensor reduction for multi-leg one-loop integrals with arbitrary internal masses and at least two massless external legs, where Gram determinant instabilities complicate standard reductions. It introduces a simple method based on $\gamma$-matrix algebra and spinor techniques to move all $/q$ factors together and replace $/q\,/q$ by $q^2$, yielding a decomposition with at most rank-1 $m$-point tensor functions plus lower-point, lower-rank tensors. The approach is formulated in four dimensions with a discussion of extending to $n=4+\epsilon$, and it provides a pre-processing simplification that avoids introducing new master functions while improving numerical stability. This method complements existing strategies (e.g., Campbell–Glover–Miller) by simplifying the diagram before applying broader reduction schemes, enabling more stable and efficient multi-leg loop calculations.

Abstract

In this paper, I present a technique to simplify the tensorial reduction of one-loop integrals with arbitrary internal masses, but at least two massless external legs. By applying the method to rank l tensor integrals, one ends up with at most rank 1 tensor functions with the initial number of denominators, plus tensor integrals with less denominators and rank < l. To illustrate the algorithm, I explicitly compute diagrams contributing to processes of physical interest and show how the usual numerical instabilities due to the appearance of Gram determinants can be controlled.