Off-shell amplitudes as boundary integrals of analytically continued Wilson line slope

TL;DR

This paper introduces a novel method to compute off-shell gluon amplitudes using boundary integrals of analytically continued Wilson line operators, revealing a helicity-independent gauge invariant structure.

Contribution

It demonstrates that off-shell amplitudes can be derived from boundary integrals of Wilson line matrix elements with shifted slopes, extending BCFW-like recursion to off-shell cases.

Findings

Decomposition of off-shell currents into gauge invariant parts.

Application to five-point next-to-MHV helicity configuration.

Boundary integrals correspond to off-shell amplitudes.

Abstract

One of the methods to calculate tree-level multi-gluon scattering amplitudes is to use the Berends-Giele recursion relation involving off-shell currents or off-shell amplitudes, if working in the light cone gauge. As shown in recent works using the light-front perturbation theory, solutions to these recursions naturally collapse into gauge invariant and gauge-dependent components, at least for some helicity configurations. In this work, we show that such structure is helicity independent and emerges from analytic properties of matrix elements of Wilson line operators, where the slope of the straight gauge path is shifted in a certain complex direction. This is similar to the procedure leading to the Britto-Cachazo-Feng-Witten (BCFW) recursion, however we apply a complex shift to the Wilson line slope instead of the external momenta. While in the original BCFW procedure the boundary integrals over the complex shift vanish for certain deformations, here they are non-zero and are equal to the off-shell amplitudes. The main result can thus be summarized as follows: we derive a decomposition of a helicity-fixed off-shell current into gauge invariant component given by a matrix element of a straight Wilson line plus a reminder given by a sum of products of gauge invariant and gauge dependent quantities. We give several examples realizing this relation, including the five-point next-to-MHV helicity configuration.