I will present a combinatorial operation on one-loop Feynman diagrams, described by cutting and pinching edges, that corresponds to the Hopf-algebraic coaction on the multiple polylogarithms resulting from their integration. A generalization of this operation, expressed simply in terms of master integrands paired with master contours, can be applied directly to larger classes of functions including hypergeometric functions. I will discuss applications to analyses of discontinuities of and differential equations for Feynman diagrams.

Trinity College Dublin