A recursive approach to the reduction of tensor Feynman integrals

TL;DR

This paper introduces a recursive tensor integral reduction scheme for one-loop n-point Feynman integrals, simplifying tensor calculations into scalar functions using algebraic formalism and recurrence relations.

Contribution

It presents a new recursive reduction method for tensor Feynman integrals, explicitly working out cases up to six external legs and tensor ranks up to n, with practical Fortran code implementation.

Findings

Effective reduction of tensor integrals to scalar functions demonstrated.

Explicit formulas provided for up to six external legs and tensor ranks.

Numerical results validate the reduction scheme.

Abstract

We describe a new, convenient, recursive tensor integral reduction scheme for one-loop $n$-point Feynman integrals. The reduction is based on the algebraic Davydychev-Tarasov formalism where the tensors are represented by scalars with shifted dimensions and indices, and then expressed by conventional scalars with generalized recurrence relations. The scheme is worked out explicitly for up to $n=6$ external legs and for tensor ranks $R\leq n$. The tensors are represented by scalar one- to four-point functions in $d$ dimensions. For the evaluation of them, the Fortran code for the tensor reductions has to be linked with a package like QCDloop or LoopTools/FF. Typical numerical results are presented.