From positive geometries to a coaction on hypergeometric functions

Souto Gonçalves de abreu, Samuel [UCL] Britto, Ruth [Trinity College] Duhr, Claude [CERN] Gardi, Einan [University of Edinburgh] Matthew, James [University of Edinburgh]

It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally- regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter ϵ. We show that the coaction defined on this class of integral is consistent, upon expansion in ϵ, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric p+1Fp and Appell functions.