Algebraic evaluation of rational polynomials in one-loop amplitudes

TL;DR

The paper develops a systematic, algebraic method to extract the rational (non-cut-constructible) part of arbitrary one-loop amplitudes by projecting onto their ultraviolet-sensitive tensor form factors within a GOLEM-based reduction framework. By defining rational parts relative to IR-regulated amplitudes and using a tensor reduction scheme, the authors obtain compact, automatable expressions that reproduce known results and illuminate when rational terms vanish. They apply the method to diverse processes (Higgs production, light-by-light scattering, 4-gluon scattering, γγ ggg, and 6-photon amplitudes), demonstrating consistency with unitarity-based approaches and highlighting cases where rational terms are purely UV-driven or cancel. The approach complements unitarity methods and, with the GOLEM representation, is well-suited for automation and handling massive internal propagators, potentially enabling efficient evaluation of complex multi-leg one-loop amplitudes.

Abstract

One-loop amplitudes are to a large extent determined by their unitarity cuts in four dimensions. We show that the remaining rational terms can be obtained from the ultraviolet behaviour of the amplitude, and determine universal form factors for these rational parts by applying reduction techniques to the Feynman diagrammatic representation of the amplitude. The method is valid for massless and massive internal particles. We illustrate this method by evaluating the rational terms of the one-loop amplitudes for gg --> H, the 4-photon and 4-gluon amplitude, the amplitude for 2 photons plus 3 gluons and the 6-photon amplitude.