A Duality For The S Matrix

TL;DR

This paper introduces a dual formulation of the S Matrix for N=4 SYM, linking scattering amplitudes to Grassmannian geometry, and provides a framework for understanding leading singularities at all loop orders.

Contribution

It proposes a novel dual approach to the S Matrix that captures leading singularities through Grassmannian integrals, offering explicit contour prescriptions at tree and one-loop levels.

Findings

Explicit integral representations for leading singularities

Identification of contours for NMHV and NNMHV amplitudes

Relations among residues from the global residue theorem

Abstract

We propose a dual formulation for the S Matrix of N = 4 SYM. The dual provides a basis for the "leading singularities" of scattering amplitudes to all orders in perturbation theory, which are sharply defined, IR safe data that uniquely determine the full amplitudes at tree level and 1-loop, and are conjectured to do so at all loop orders. The scattering amplitude for n particles in the sector with k negative helicity gluons is associated with a simple integral over the space of k planes in n dimensions, with the action of parity and cyclic symmetries manifest. The residues of the integrand compute a basis for the leading singularities. A given leading singularity is associated with a particular choice of integration contour, which we explicitly identify at tree level and 1-loop for all NMHV amplitudes as well as the 8 particle NNMHV amplitude. We also identify a number of 2-loop leading singularities for up to 8 particles. There are a large number of relations among residues which follow from the multi-variable generalization of Cauchy's theorem known as the "global residue theorem". These relations imply highly non-trivial identities guaranteeing the equivalence of many different representations of the same amplitude. They also enforce the cancellation of non-local poles as well as consistent infrared structure at loop level. Our conjecture connects the physics of scattering amplitudes to a particular subvariety in a Grassmannian; space-time locality is reflected in the topological properties of this space.