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We study the algebraic and analytic structure of Feynman integrals byproposing an operation that maps an integral into pairs of integrals obtainedfrom a master integrand and a corresponding master contour. This operation is acoaction. It reduces to the known coaction on multiple polylogarithms, butapplies more generally, e.g. to hypergeometric functions. The coaction alsoapplies to generic one-loop Feynman integrals with any configuration ofinternal and external masses, and in dimensional regularization. In this case,we demonstrate that it can be given a diagrammatic representation purely interms of operations on graphs, namely contractions and cuts of edges. Thecoaction gives direct access to (iterated) discontinuities of Feynman integralsand facilitates a straightforward derivation of the differential equations theyadmit. In particular, the differential equations for any one-loop integral aredetermined by the diagrammatic coaction using limited information about theirmaximal, next-to-maximal, and next-to-next-to-maximal cuts.

The algebraic structure of cut Feynman integrals and the diagrammatic coaction