BCJ Identities and $d$-Dimensional Generalized Unitarity

TL;DR

This paper derives relations between one-loop integral coefficients in dimensionally regulated QCD amplitudes using BCJ identities and integrand reduction, simplifying calculations of scattering amplitudes.

Contribution

It introduces a novel method connecting BCJ identities with integrand reduction to relate different amplitude residues in dimensional regularization.

Findings

Relations hold for both cut-constructible and rational parts.

Explicit examples provided for multi-gluon one-loop amplitudes.

Simplifies the computation of one-loop amplitudes in QCD.

Abstract

We present a set of relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. Within dimensional regularization, the combined use of color-kinematics duality and integrand reduction yields the existence of relations between the integrand residues of partial amplitudes with different orderings of the external particles. These relations can be established for the cut-constructible contributions as well for the ones responsible for rational terms. Starting from the general parametrization of one-loop residues and applying Laurent expansion in order to extract the coefficients of the amplitude decomposition in terms of master integrals, we show that the full set of relations can be obtained by considering BCJ identities between d-dimensional tree-levels. We provide explicit examples for multi-gluon scattering amplitudes at one-loop.