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arXiv · 1910.08358

From positive geometries to a coaction on hypergeometric functions

Samuel Abreu·Ruth Britto·Claude Duhr·Einan Gardi·James Matthew
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Abstract

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 $\epsilon$. We show that the coaction defined on this class of integral is consistent, upon expansion in $\epsilon$, 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 ${}_{p+1}F_p$ and Appell functions.

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Samuel Abreu, Ruth Britto, Claude Duhr, Einan Gardi, James Matthew. 2019-10-18. From positive geometries to a coaction on hypergeometric functions. https://doi.org/10.1007/jhep02(2020)122

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