Superconformal invariants for scattering amplitudes in N=4 SYM theory

TL;DR

This paper explores the structure of superconformal invariants in N=4 SYM theory, revealing how combined symmetries uniquely determine scattering amplitude forms and generalize known invariants.

Contribution

It constructs a general integral representation of dual superconformal invariants and shows their form is uniquely fixed by symmetry considerations.

Findings

Integral representation with flexible measure

Unique solution for conformal Ward identity

Invariants match Grassmannian contour integrals

Abstract

Recent studies of scattering amplitudes in planar N=4 SYM theory revealed the existence of a hidden dual superconformal symmetry. Together with the conventional superconformal symmetry it gives rise to powerful restrictions on the planar scattering amplitudes to all loops. We study the general form of the invariants of both symmetries. We first construct an integral representation for the most general dual superconformal invariants and show that it allows a considerable freedom in the choice of the integration measure. We then perform a half-Fourier transform to twistor space, where conventional conformal symmetry is realized locally, derive the resulting conformal Ward identity for the integration measure and show that it admits a unique solution. Thus, the combination of dual and conventional superconformal symmetries, together with invariance under helicity rescalings, completely fixes the form of the invariants. The expressions obtained generalize the known tree and one-loop superconformal invariants and coincide with the recently proposed coefficients of the leading singularities of the scattering amplitudes as contour integrals over Grassmannians.