Reducing full one-loop amplitudes to scalar integrals at the integrand level

TL;DR

The paper tackles the problem of reducing arbitrary one-loop amplitudes to scalar integrals by performing an integrand-level decomposition into 4-, 3-, 2-, and 1-point scalar functions plus a rational part, using minimal analytic input.It introduces a systematic handling of spurious, q-dependent terms that vanish upon integration and shows how to extract the scalar coefficients through evaluations of the integrand at special loop momenta, yielding a triangular solution order by order in the number of denominators.The rational part is reconstructed via a mass-shift approach that introduces extra integrals, with coefficients obtained from limiting procedures or multiple choices of the shifted parameter, ensuring complete amplitude reconstruction.The method is validated through practical tests on tensor reductions and rational-term extraction, and is positioned as especially valuable for numerically driven, recursion-based one-loop computations.

Abstract

We show how to extract the coefficients of the 4-, 3-, 2- and 1-point one-loop scalar integrals from the full one-loop amplitude of arbitrary scattering processes. In a similar fashion, also the rational terms can be derived. Basically no information on the analytical structure of the amplitude is required, making our method appealing for an efficient numerical implementation