Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
Simon Caron-Huot
TL;DR
This work proposes a generalized, supersymmetric extension of planar N=4 SYM scattering amplitudes, dependent on additional Grassmann variables, to restore dual superconformal symmetry via a Wilson loop dual in full (x, θ, θ̄) superspace. It develops a concrete, gauge-invariant method (Lagrangian insertion with AdS-geodesic regularization) to derive a finite integral representation for the total differential of all n-point two-loop MHV remainder functions and computes its symbol, with extensive checks including hexagon consistency, integrability, and correct collinear and 2D kinematic limits. The results suggest strong all-loop constraints and a structured, potentially universal pattern for leading singularities, pointing towards deep symmetry-driven organization of amplitudes beyond conventional approaches. The framework also opens avenues for applying similar principles to other superconformal theories and higher-loop computations, potentially illuminating the connections between amplitudes, Wilson loops, and integrability.
Abstract
Scattering amplitudes in superconformal field theories do not enjoy this symmetry, because the definition of asymptotic states involve a notion of infinity. Concentrating on planar $\mathcal{N}=4$ Yang-Mills, we consider a generalization of scattering amplitudes which depends on twice as many Grassmann variables. We conjecture that it restores at least half of the superconformal symmetries, and all of the dual superconformal symmetries. The object arises naturally as the dual of a null polygonal Wilson loop in an $(x,θ,\barθ)$ superspace. We support the conjecture by using it to obtain the total differential of all $n$-point two-loop MHV amplitudes, and showing that the result passes consistency checks. Potential all-loop constraints are also discussed.