Landau Singularities of the 7-Point Ziggurat II

TL;DR

This paper analyzes the singularities of specific three-loop 7-point Feynman graphs by solving Landau equations, revealing new singularities outside known alphabets and comparing residues to super-Yang-Mills theory.

Contribution

It identifies new Landau singularities for nine three-loop 7-point graphs and examines their relation to known mathematical structures and amplitude cancellations.

Findings

Found singularities outside the heptagon alphabet.

Established failure of $Y{-} riangle$ equivalence for certain solutions.

Compared residues to $ ext{N}=4$ super-Yang-Mills theory to study cancellations.

Abstract

We solve the Landau equations to find the singularities of nine three-loop 7-point graphs that arise as relaxations of the graph studied in arXiv:2211.16425. Along the way we establish that $Y{-}\Delta$ equivalence fails for certain branches of solutions to the Landau equations. We find two graphs with singularities outside the heptagon symbol alphabet; in particular they are not cluster variables of ${\rm Gr}(4,7)$. We compare maximal residues of scalar graphs exhibiting these singularities to those in $\mathcal{N}=4$ super-Yang-Mills theory in order to probe their cancellation from its amplitudes.