Cluster Adjacency Properties of Scattering Amplitudes in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

We conjecture a new set of analytic relations for scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. They generalize the Steinmann relations and are expressed in terms of the cluster algebras associated to $\mathrm{Gr}(4,n)$. In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture.

Figure 1

Seven-point, three-loop, massless integral.

Figure 2

The seven-point, two-loop integrals $I^{(2)}$ and ${\tilde{I}}^{(2)}$.

Figure 3

The one-loop hexagon integrals $I^{(1)}$ and ${\tilde{I}}^{(1)}$.