All-loop singularities of scattering amplitudes in massless planar theories

TL;DR

The paper addresses the all-loop structure of Landau singularities in massless planar scattering by proving that all first-type singularities are contained in the $n$-particle ziggurat graph with $L=\lfloor{(n-2)^2/4}\rfloor$ loops, independent of spacetime dimension $D$. It leverages circuit-invariance of the Landau equations and Gitler’s planar-Y$\Delta$ reducibility to reduce the problem to ziggurat graphs, and provides an explicit analysis for $D=4$, $n=6$ showing the leading singularities lie on the hexagon locus $\mathcal{S}_6$ defined by $S_6=\{u,v,w,1-u,1-v,1-w,1/u,1/v,1/w\}$. The work further argues that second-type singularities are absent in dual-conformal invariant theories like planar SYM, so first-type loci capture the dual-conformal part of the singularity structure; it also discusses implications for symbol alphabets and the amplituhedron, including conjectures about saturation of singularities across helicity sectors and loop orders. Overall, the results provide an all-order, graph-theoretic framework tying Landau singularities to ziggurat graphs and offering concrete, testable connections to hexagon bootstrap and symbol-analysis programs in 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 $Y$-$Δ$ 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.

Figures (4)

• Figure 1: Elementary circuit moves that preserve solution sets of the massless Landau equations: (a) series reduction, (b) parallel reduction, and (c) $Y$-$\Delta$ reduction.
• Figure 2: The four-, six-, five- and seven-terminal ziggurat graphs. The open circles are terminals and the filled circles are vertices. The pattern continues in the obvious way, but note an essential difference between ziggurat graphs with an even or odd number of terminals in that only the latter have a terminal of degree three.
• Figure 3: The six-terminal ziggurat graph can be reduced to a three loop graph by a sequence of three $Y$-$\Delta$ reductions and one FP assignment. In each case the vertex, edge, or face to be transformed is highlighted in gray.
• Figure 4: The three-loop six-particle wheel graph. The leading first-type Landau singularities of this graph exhaust all possible first-type Landau singularities of six-particle amplitudes in any massless planar field theory, to any finite loop order.