Heptagons from the Steinmann Cluster Bootstrap

TL;DR

This paper enhances the bootstrap approach for seven-point amplitudes in planar N=4 SYM by incorporating Steinmann relations, significantly reducing complexity and enabling determination of multi-loop amplitudes with minimal ambiguities.

Contribution

It reformulates the heptagon bootstrap using Steinmann relations, enabling the computation of higher-loop seven-point amplitudes with fewer ambiguities and greater efficiency.

Findings

Steinmann relations reduce the function space needed for bootstrap.

Unique determination of three-loop NMHV and four-loop MHV amplitudes.

Minimal ambiguity remains when imposing symmetries and physical constraints.

Abstract

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal ($\bar{Q}$) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations.