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Article
Report number	arXiv:1910.08358 ; CERN-TH-2019-168
Title	From positive geometries to a coaction on hypergeometric functions
Author(s)	Abreu, Samuel (Louvain U., CP3) ; Britto, Ruth (Trinity Coll., Dublin ; Hamilton Math. Inst., Dublin ; IPhT, Saclay) ; Duhr, Claude (CERN) ; Gardi, Einan (U. Edinburgh, Higgs Ctr. Theor. Phys.) ; Matthew, James (U. Edinburgh, Higgs Ctr. Theor. Phys.)
Publication	2020-02-20
Imprint	2019-10-18
Number of pages	45
In:	JHEP 2002 (2020) 122
DOI	10.1007/JHEP02(2020)122
Subject category	math.NT ; Mathematical Physics and Mathematics ; math.MP ; Mathematical Physics and Mathematics ; math-ph ; Mathematical Physics and Mathematics ; hep-th ; Particle Physics - Theory
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.
Copyright/License	preprint: (License: arXiv nonexclusive-distrib 1.0) publication: © 2020-2025 The Authors (License: CC-BY-4.0)

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