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 $Y\text{−}\mathrm{\Delta }$ 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, at any finite loop order, 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 $n=6$ 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.

Figure 1

Elementary circuit moves that preserve solution sets of the massless Landau equations: (a) series reduction, (b) parallel reduction, and (c) $Y\text{−}\mathrm{\Delta }$ 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, there is 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 $Y\text{−}\mathrm{\Delta }$ 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 LS of this graph exhaust all possible LS of six-particle amplitudes in any massless planar field theory, to any finite loop order.