All-loop singularities of scattering amplitudes in massless planar theories

TL;DR

This paper proves that all first-type Landau singularities in massless planar quantum field theories are contained within those of a specific 'ziggurat' graph, explicitly characterizing these singularities in four dimensions for six particles.

Contribution

It establishes a graph-theoretic framework linking Landau singularities to electrical circuit operations, providing explicit singularity loci for certain cases in massless planar theories.

Findings

Singularity set is a subset of a 'ziggurat' graph's singularities.

Explicit singularity locus determined for D=4, n=6 case.

Identifies correspondence with symbol letters in hexagon bootstrap for SYM.

Abstract

In massless quantum field theories the Landau equations are invariant under graph operations familiar from the theory of electrical circuits. Using a theorem on the $Y$-$\Delta$ reducibility of planar circuits we prove that the set of first-type Landau singularities of an $n$-particle scattering amplitude in any massless planar theory, in any spacetime dimension $D$, at any finite loop order in perturbation theory, is a subset of those of a certain $n$-particle $\lfloor{(n{-}2)^2/4}\rfloor$-loop "ziggurat" graph. We determine this singularity locus explicitly for $D=4$ and $n=6$ and find that it corresponds precisely to the vanishing of the symbol letters familiar from the hexagon bootstrap in SYM theory. Further implications for SYM theory are discussed.