Cluster adjacency and the four-loop NMHV heptagon
James Drummond, Jack Foster, Ömer Gürdoğan, Georgios Papathanasiou
TL;DR
The paper computes the symbol of the $4$-loop NMHV heptagon amplitude in planar ${\cal N}=4$ SYM by exploiting cluster adjacency in the ${\rm Conf}_7({\mathbb P}^3)$ cluster algebra. A manifestly cluster-adjacent ansatz, combined with integrability, spurious-pole cancellation, and collinear constraints, fixes all coefficients, yielding a unique, physically consistent symbol. In the multi-Regge limit, the result agrees with known BFKL-based results up to next-to-leading logarithm and provides new predictions up to $N^3$LLA, enabling deeper tests of the all-loop structure and potential all-loop insights into the central emission block. This work demonstrates cluster adjacency as a powerful bootstrap constraint for high-loop amplitudes and sets the stage for extensions to more particles and other theories.
Abstract
We exploit the recently described property of cluster adjacency for scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory to construct the symbol of the four-loop NMHV heptagon amplitude. We use a manifestly cluster adjacent ansatz and describe how the parameters of this ansatz are determined using simple physical consistency requirements. We then specialise our answer for the amplitude to the multi-Regge limit, finding agreement with previously available results up to the next-to-leading logarithm, and obtaining new predictions up to (next-to)$^3$-leading-logarithmic accuracy.