The on-shell expansion: from Landau equations to the Newton polytope

We study the application of the method of regions to Feynman integrals with massless propagators contributing to off-shell Green's functions in Minkowski spacetime (with non-exceptional momenta) around vanishing external masses, $p_i^2\to 0$. This on-shell expansion allows us to identify all infrared-sensitive regions at any power, in terms of infrared subgraphs in which a subset of the propagators become collinear to external lightlike momenta and others become soft. We show that each such region can be viewed as a solution to the Landau equations, or equivalently, as a facet in the Newton polytope constructed from the Symanzik graph polynomials. This identification allows us to study the properties of the graph polynomials associated with infrared regions, as well as to construct a graph-finding algorithm for the on-shell expansion, which identifies all regions using exclusively graph-theoretical conditions. We also use the results to investigate the analytic structure of integrals associated with regions in which every connected soft subgraph connects to just two jets. For such regions we prove that multiple on-shell expansions commute. This applies in particular to all regions in Sudakov form-factor diagrams as well as in any planar diagram.