Solution to the Ward identities for superamplitudes

Supersymmetry and R-symmetry Ward identities relate on-shell amplitudes in a supersymmetric field theory. We solve these Ward identities for N [superscript K] MHV amplitudes of the maximally supersymmetric =4 and =8 theories. The resulting superamplitude is written in a new, manifestly supersymmetric and [subscript R]-invariant form: it is expressed as a sum of very simple SUSY and SUR -invariant Grassmann polynomials, each multiplied by a “basis amplitude”. For N [superscript 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 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 =4 gauge theory. We also analyze the more significant reduction that occurs in =8 supergravity because gravity amplitudes are not ordered. All results are valid at both tree and loop level