Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts

TL;DR

This paper extends amplitude relation techniques to loop level in gauge theories using non-adjacent BCFW shifts, revealing new identities and generalizations of tree-level relations to one loop.

Contribution

It demonstrates that non-adjacent BCFW shifts lead to loop integrand relations in gauge theories, generalizing tree-level BCJ relations to all loop orders.

Findings

Loop integrand relations are independent of loop order for generic shifts.

New identities for bubble and triangle terms in one-loop amplitudes.

Generalization of tree-level amplitude relations to one-loop rational terms.

Abstract

This article studies methods to obtain relations for scattering amplitudes at the loop level, with concrete examples at one loop. These methods originate in the analysis of large so-called Britto-Cachazo-Feng-Witten shifts of tree level amplitudes and loop level integrands. In particular BCFW shifts for particles which are not color adjacent and some particular generalizations of this situation are analyzed in some detail in four and higher dimensions. For generic non-adjacent shifts our results are independent of loop order for integrands and hold for generic minimally coupled gauge theories with possible scalar potential and Yukawa terms. By a standard argument this result indicates a generalization of the Bern-Carrasco-Johansson relations for tree level amplitudes exists to the integrand at all loop levels. A concrete relation is presented at one loop. Furthermore, inspired by results in QED it is shown that the results on generalized BCFW shifts of tree level amplitudes imply relations for the so-called rational, bubble and triangle terms of one loop amplitudes in pure Yang-Mills theory. Bubble and triangle terms for instance are shown to obey a five photon decoupling identity, while a three photon decoupling identity is demonstrated for the rational terms. Along the same lines recently conjectured relations for helicity equal amplitudes at one loop are shown to generalize to helicity independent relations for the massive box coefficient of the rational terms.