Deformed Yangian invariants in N = 4 super-Yang–Mills theory

Abstract

Tree-level scattering amplitudes in maximally supersymmetric Yang–Mills theory in four dimensions are invariant under a large symmetry algebra known as the Yangian of the superconformal algebra Y[psu(2, 2|4)]. While the regularisation process spoils conformal invariance at loop level, the integrand of loop amplitudes displays remnants of such symmetry. Yangians arise as symmetry algebras in integrable field theories and integrable models, therefore it is very interesting to study in what sense the integrability of N = 4 superYang–Mills constrains the S-matrix. We construct more general invariants by relaxing the condition of physical external states and study their properties. We present two frameworks: the first is an extension of the Grassmannian formulation of scattering amplitudes, while the second is a purely algebraic approach to the construction of Yangian invariants. We show that these approaches are equivalent, and the features of both are explored. We subsequently analyse how such invariants can contribute to describe deformed scattering amplitudes and the role of the deformation at loop level. We find that the deformation is not compatible with the ordinary Britto–Cachazo–Feng–Witten recursion relations, and that the naive computation of deformed loop amplitudes yields surprising results. These facts suggest that the definition of deformed scattering amplitudes cannot come from a simple generalisation of the ordinary BCFW construction.