Gauge theory one-loop amplitudes and the BCFW recursion relations

TL;DR

This paper extends BCFW recursion relations to gauge theory one-loop amplitudes by demonstrating that the integrand's infinite shift contribution vanishes, allowing these amplitudes to be expressed in terms of on-shell tree and lower-loop amplitudes.

Contribution

It introduces a method to apply BCFW recursion to one-loop gauge theory amplitudes, avoiding singularities from forward amplitudes and enabling recursive calculations.

Findings

Calculated explicit 2, 3, 4-point one-loop amplitudes.

Showed the infinite shift contribution vanishes after loop integration.

Outlined the generalization to higher-point amplitudes.

Abstract

We calculate gauge theory one-loop amplitudes with the aid of the complex shift used in the Britto-Cachazo-Feng-Witten (BCFW) recursion relations of tree amplitudes. We apply the shift to the integrand and show that the contribution from the limit of infinite shift vanishes after integrating over the loop momentum, by a judicious choice of basis for polarization vectors. This enables us to write the one-loop amplitude in terms of on-shell tree and lower point one-loop amplitudes. Some of the tree amplitudes are forward amplitudes. We show that their potential singularities do not contribute and the BCFW recursion relations can be applied in such a way as to avoid these singularities altogether. We calculate in detail $n$-point one-loop amplitudes for $n=2,3,4$, and outline the generalization of our method to $n>4$.