On BCFW shifts of integrands and integrals

Abstract

In this article a first step is made towards the extension of Britto-CachazoFeng-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.

One-loop calculations in quantum field theory: From Feynman diagrams to unitarity cuts

“…It is an interesting and non-trivial question to what extent real needs of the physics program at the LHC can benefit from those developments. While it appears that many of the developments in that field rely on high degree of supersymmetry, a number of ideas such as the recursion relations for the loop integrand [145,146,147] may have more general nature and, perhaps, can be successfully used in non-supersymmetric quantum field theories, such as QCD and the Standard Model. Partially related to this is the barely explored question of whether it is possible to apply unitarity-based ideas to multi-loop computations.…”

The Amplituhedron

“…A program to find a reformulation along these lines was initiated in [14,15], and in the context of a planar N = 4 SYM was pursued in [16][17][18], leading to a new physical and mathematical understanding of scattering amplitudes [19]. This picture builds on BCFW recursion relations for tree [6,7] and loop [18,20] amplitudes, and represents the amplitude as a sum over basic building blocks, which can be physically described as arising from gluing together the elementary three-particle amplitudes to build more complicated on-shell processes. These "on-shell diagrams" (which are essentially the same as the "twistor diagrams" of [16,21,22]) are remarkably connected with "cells" of a beautiful new structure in algebraic geometry, that has been studied by mathematicians over the past number of years, known as the positive Grassmannian [19,23].…”

Note on recursion relations for the Q $$ \mathcal{Q} $$ -cut representation

“…The key for the progress lies in expressing planar loop integrands from forward limits of tree amplitudes [19][20][21], which has been very successful for cases without forward singularities, such as super-Yang-Mills at one loop and planar N = 4 SYM to all loops [20]. However, for general theories the afore-mentioned difficulties have only been resolved in the Q-cut construction.…”