Superconformal invariants for scattering amplitudes in N=4 SYM theory

TL;DR

The paper addresses how dual and conventional superconformal symmetries constrain scattering amplitudes in planar ${ m N}=4$ SYM to all loops by classifying the invariants. It presents a general dual-invariant construction as a Grassmannian-type integral $R_n^k(W)=ig[{ m D}tig]_{n,k}\prod_{a=1}^k\, hetaig( ext{sum of }t^i_aW_iig)$, with a measure ${[ ext{D}t]_{n,k}}$ not fixed by dual symmetry alone. A twistor transform then yields a conformal Ward identity that fixes the measure uniquely, yielding invariants that generalize tree- and one-loop results and align with leading-singularity coefficients expressed as Grassmannian contour integrals. The work establishes the precise interplay between dual and conventional conformal symmetries and highlights the geometric interpretation of invariants as configurations of intersecting twistor lines, with a unique, symmetry-driven measure tying the formalism to Grassmannian methods. These results provide a foundational step toward a complete, symmetry-determined description of ${ m N}=4$ amplitudes, with implications for all-loop structure and potential dual- Wilson loop connections.

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.