Algebraic reduction of one-loop Feynman graph amplitudes

TL;DR

The paper presents an algorithm that simplifies one-loop Feynman integrals by transforming tensor integrals into scalar integrals and applying recurrence relations, especially effective for 5- and 6-point functions.

Contribution

It introduces a novel algebraic reduction method for one-loop tensor integrals using dimension-shifting and recurrence relations, including special cases where Gram determinants vanish.

Findings

Explicit recurrence relations for 5- and 6-point functions are derived.

The method simplifies calculations when Gram determinants vanish in four dimensions.

The approach reduces complex tensor integrals to basic scalar integrals efficiently.

Abstract

An algorithm for the reduction of one-loop n-point tensor integrals to basic integrals is proposed. We transform tensor integrals to scalar integrals with shifted dimension and reduce these by recurrence relations to integrals in generic dimension. Also the integration-by-parts method is used to reduce indices (powers of scalar propagators) of the scalar diagrams. The obtained recurrence relations for one-loop integrals are explicitly evaluated for 5- and 6-point functions. In the latter case the corresponding Gram determinant vanishes identically for d=4, which greatly simplifies the application of the recurrence relations.