On BCFW shifts of integrands and integrals

TL;DR

The work proposes extending BCFW on-shell recursion from tree-level amplitudes to loop-level integrands and their integrals in gauge theories, showing that integrands exhibit tree-like large-z scaling and can be reconstructed from single-cut residues using a special lightcone gauge. It analyzes the relation between integrand and integral shifts, identifying UV/IR divergences as potential sources of discrepancy and exploring one-loop scalar and gauge contributions with background-field methods. A string theory perspective and general loop-level considerations support the conjecture that integrand shifts follow a universal large-z structure, with non-adjacent shifts suppressed and potential loop-level recursion guided by single-cut residues. The paper outlines preliminary steps toward loop-level on-shell recursion, discusses forward-limit issues, and highlights future directions toward higher-loop computations and connections to gravity.

Abstract

In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their `single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choice employed earlier at tree level by Arkani-Hamed and Kaplan. The relation between shifts of integrands and shifts of its integrals is investigated explicitly at one loop. Two particular sources of discrepancy between the integral and integrand are identified related to UV and IR divergences. This is cross-checked with known results for helicity equal amplitudes at one loop. The nature of the on-shell residue at each of the single-cut singularities of the integrand is commented upon. Several natural conjectures and opportunities for further research present themselves.