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

TL;DR

This paper presents a recursive formula for the all-loop integrand of scattering amplitudes in planar N=4 SYM, revealing full Yangian symmetry and providing new insights into the structure of loop amplitudes.

Contribution

It generalizes the BCFW recursion to all loop orders and extends Grassmannian duality, offering a new physical picture of loops in terms of Yangian-invariant particle removal.

Findings

Explicit all-loop integrand formula in momentum-twistor space

New multi-loop amplitude results including 6- and 7-point 2-loop NMHV amplitudes

Concise expressions for 2-loop MHV and 3-loop MHV amplitudes

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.