The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM

TL;DR

This work provides an explicit all-loop integrand for scattering amplitudes in planar ${\cal N}=4$ SYM, making the full Yangian symmetry manifest and extending BCFW recursion to all loop orders. It develops a momentum-twistor formulation and Grassmannian framework that connects leading singularities to a dual Grassmannian, and introduces forward-limit based recursion together with GL(2) integration to assemble loop amplitudes from lower-loop data. The authors demonstrate concrete multi-loop results, including all 2-loop MHV amplitudes and 2-loop NMHV and 3-loop MHV amplitudes, expressed in a local basis of chiral tensor integrals with unit leading singularities. They argue that IR anomalies drive the simplicity of the resulting amplitudes and discuss extensions to other planar theories, offering a new paradigm for loop amplitudes beyond traditional local integral decompositions.

Abstract

We give an explicit recursive formula for the all L-loop integrand for scattering amplitudes in N=4 SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This generalizes the BCFW recursion relation for tree amplitudes to all loop orders, and extends the Grassmannian duality for leading singularities to the full amplitude. It also provides a new physical picture for the meaning of loops, associated with canonical operations for removing particles in a Yangian-invariant way. Loop amplitudes arise from the "entangled" removal of pairs of particles, and are naturally presented as an integral over lines in momentum-twistor space. As expected from manifest Yangian-invariance, the integrand is given as a sum over non-local terms, rather than the familiar decomposition in terms of local scalar integrals with rational coefficients. Knowing the integrands explicitly, it is straightforward to express them in local forms if desired; this turns out to be done most naturally using a novel basis of chiral, tensor integrals written in momentum-twistor space, each of which has unit leading singularities. As simple illustrative examples, we present a number of new multi-loop results written in local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point 3-loop MHV amplitude. The structure of the loop integrand strongly suggests that the integrals yielding the physical amplitudes are "simple", and determined by IR-anomalies. We briefly comment on extending these ideas to more general planar theories.