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

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