All-Loop Singularities of Scattering Amplitudes in Massless Planar Theories

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, at any finite loop order, is a subset of those of a certain n-particle ⌊(n-2)^{2}/4⌋-loop "ziggurat" graph. We determine this singularity locus explicitly for n=6 and find that it corresponds precisely to the vanishing of the symbol letters familiar from the hexagon bootstrap in supersymmetric Yang-Mills (SYM) theory. Further implications for SYM theory are discussed.