Amplitudes from superconformal Ward identities

TL;DR

We address how finite ${\cal N}=1$ matter amplitudes can be determined from superconformal Ward identities, showing that on-shell collinear configurations induce a calculable anomaly. The authors derive first-order differential equations for the amplitudes, reducing the five-particle case to a single bosonic function $C(p)$ constrained by a twistor collinearity operator, and illustrate the method with a one-loop off-shell box and a non-planar two-loop five-particle integral. Boundary data and analyticity fix holomorphic ambiguities, yielding results whose symbols agree with bootstrap analyses and numerical integral representations. This framework provides a powerful, general approach to finite supersymmetric integrals that applies to both planar and non-planar topologies and offers avenues for extension to gauge multiplets and Wilson-loop connections.

Abstract

We consider finite superamplitudes of N=1 matter, and use superconformal symmetry to derive powerful first-order differential equations for them. Due to on-shell collinear singularities, the Ward identities have an anomaly, which is obtained from lower-loop information. We show that in the five-particle case, the solution to the equations is uniquely fixed by the expected analytic behavior. We apply the method to a non-planar two-loop five-particle integral.

Figures (2)

• Figure 1: One-loop $\mathcal{N} = 1$ matter supergraph. Black and grey blobs denote antichiral and chiral vertices, respectively.
• Figure 2: Non-planar two-loop $\mathcal{N} = 1$ matter supergraph.