Scattering amplitudes are essential ingredients in theoretical predictions for collider experiments. In some cases, symmetries and other constraints can fix amplitudes completely, and hence conventional Feynman diagram computations are circumvented. The traditional cutting rules relate discontinuities across branch cuts of amplitudes to cuts through the corresponding Feynman diagrams. Here we probe the analytic structure further by generalizing the cutting rules, relating sequential discontinuities (discontinuities of discontinuities) to multiple cuts. As a corollary, we present a new proof in perturbation theory of the Steinmann relations, which forbid sequential discontinuities in partially overlapping momentum channels, in the case where all external particles are massive. These types of formulas are crucial in determining amplitudes to high loop order using modern bootstrapping methods, suggesting that our new relations could provide useful constraints for such programs.
