Heptagons from the Steinmann Cluster Bootstrap

TL;DR

This work leverages Steinmann relations to refine the seven-point (heptagon) cluster bootstrap, dramatically shrinking the space of allowed generalized polylogarithm symbols and enabling higher-loop determinations in planar N=4 SYM. By combining the 42-letter cluster alphabet with integrability, first-entry constraints, and dihedral/parity symmetries, the authors obtain a unique symbol for the three-loop NMHV and four-loop MHV seven-point amplitudes within the BDS-like normalization, with only a single ambiguity remaining when certain final-entry constraints are relaxed. The analysis reveals a tension between the collinear behavior and Steinmann constraints, and shows that multi-particle factorization limits are consistent with all-known lower-point data and OPE-inspired predictions. The results demonstrate the power of Steinmann constraints to push the bootstrap program to previously inaccessible loop orders and point toward future functional-level extensions and higher-point generalizations.

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.

Figures (2)

• Figure 1: The figure on the left (right) shows the discontinuity of an amplitude in the $s_{345}$ ($s_{234}$) channel due to the respective intermediate states. These two channels overlap, which implies that the states that cross the first cut cannot produce a discontinuity in the second channel (or vice versa).
• Figure 2: Factorization of a seven-point amplitude in the limit $s_{345} {\rightarrow} 0$. Notice that the collinear limit $p_7 \parallel p_1$ can be taken "inside" the factorization limit.