Some variations of the reduction of one-loop Feynman tensor integrals

TL;DR

This paper introduces a new algorithm for reducing one-loop tensor Feynman integrals with up to four external legs to scalar integrals, avoiding inverse Gram determinants and enabling stable calculations at arbitrary phase space points.

Contribution

The paper presents a novel reduction algorithm that avoids inverse Gram determinants and extends numerical stability for one-loop tensor integrals with up to four external legs.

Findings

Reduces tensor integrals to scalar integrals without inverse Gram determinants.

Extends numerical stability to small, non-zero Gram determinants.

Allows stable reduction of n-point functions with n≤6 at arbitrary phase space points.

Abstract

We present a new algorithm for the reduction of one-loop \emph{tensor} Feynman integrals with $n\leq 4$ external legs to \emph{scalar} Feynman integrals $I_n^D$ with $n=3,4$ legs in $D$ dimensions, where $D=d+2l$ with integer $l \geq 0$ and generic dimension $d=4-2\epsilon$, thus avoiding the appearance of inverse Gram determinants $()_4$. As long as $()_4\neq 0$, the integrals $I_{3,4}^D$ with $D>d$ may be further expressed by the usual dimensionally regularized scalar functions $I_{2,3,4}^d$. The integrals $I_{4}^D$ are known at $()_4 \equiv 0$, so that we may extend the numerics to small, non-vanishing $()_4$ by applying a dimensional recurrence relation. A numerical example is worked out. Together with a recursive reduction of 6- and 5-point functions, derived earlier, the calculational scheme allows a stabilized reduction of $n$-point functions with $n\leq 6$ at arbitrary phase space points. The algorithm is worked out explicitely for tensors of rank $R\leq n$.