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We present several classes of constraints on the discontinuities of Feynmanintegrals that go beyond the Steinmann relations. These constraints follow froma geometric formulation of the Landau equations that was advocated by Pham, inwhich the singularities of Feynman integrals correspond to critical points ofmaps between on-shell spaces. To establish our results, we review elements ofPicard-Lefschetz theory, which connect the homotopy properties of the space ofcomplexified external momenta to the homology of the combined space of on-shellinternal and external momenta. An important concept that emerges from thisanalysis is the question of whether or not a pair of Landau singularities iscompatible-namely, whether or not the Landau equations for the twosingularities can be satisfied simultaneously. Under conditions we describe,sequential discontinuities with respect to non-compatible Landau singularitiesmust vanish. Although we only rigorously prove results for Feynman integralswith generic masses in this paper, we expect the geometric and algebraicinsights that we gain will also assist in the analysis of more general Feynmanintegrals.

Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces