Solution to the Ward Identities for Superamplitudes

TL;DR

This paper provides a new, explicit solution to supersymmetry and R-symmetry Ward identities for superamplitudes in N=4 and N=8 theories, revealing a simplified, invariant form applicable at tree and loop levels.

Contribution

It introduces a novel, manifestly supersymmetric and R-invariant representation of superamplitudes, with a basis linked to Young diagrams, and analyzes symmetry-based reductions in amplitude complexity.

Findings

Superamplitudes expressed as sums of simple invariant polynomials.

Number of basis amplitudes matches Young diagram dimensions.

Symmetry considerations reduce the functional basis of amplitudes.

Abstract

Supersymmetry and R-symmetry Ward identities relate on-shell amplitudes in a supersymmetric field theory. We solve these Ward identities for (Next-to)^K MHV amplitudes of the maximally supersymmetric N=4 and N=8 theories. The resulting superamplitude is written in a new, manifestly supersymmetric and R-invariant form: it is expressed as a sum of very simple SUSY and SU(N)_R-invariant Grassmann polynomials, each multiplied by a "basis amplitude". For (Next-to)^K MHV n-point superamplitudes the number of basis amplitudes is equal to the dimension of the irreducible representation of SU(n-4) corresponding to the rectangular Young diagram with N columns and K rows. The linearly independent amplitudes in this algebraic basis may still be functionally related by permutation of momenta. We show how cyclic and reflection symmetries can be used to obtain a smaller functional basis of color-ordered single-trace amplitudes in N=4 gauge theory. We also analyze the more significant reduction that occurs in N=8 supergravity because gravity amplitudes are not ordered. All results are valid at both tree and loop level.