Algebraic reduction of one-loop Feynman graph amplitudes

TL;DR

The paper tackles the challenge of calculating one-loop tensor n-point amplitudes, especially for five- and six-point functions, where conventional methods become impractical. It introduces a framework that reduces tensor integrals to scalar integrals in shifted space-time dimensions and employs recurrence relations derived from integration-by-parts to lower indices and dimensions. By leveraging Melrose's modified Cayley determinant and signed minors, the authors obtain compact, implementable formulas and explicitly treat 5- and 6-point cases, including handling of special kinematic situations where Gram determinants vanish. The approach yields expressions in terms of lower-point master integrals, avoids solving large systems of equations, and is suitable for computer algebra implementation, providing a practical tool for precision calculations in multi-leg processes.

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.