Massive One-loop Conformal Feynman Integrals and Quadratic Transformations of Multiple Hypergeometric Series

Abstract

The computational technique of $N$-fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. This shows the great simplicity and efficiency of the method in nonresonant cases (generic propagator powers) as well as some of its subtleties in the resonant ones (for unit propagator powers). We confirm certain results in the physics and mathematics literature and provide many new results, some of them dealing with the more general massive one-loop conformal $n$-point case. In particular, we prove two recent conjectures that give the massive one-loop conformal $n$-point integral (for generic propagator powers) in terms of multiple hypergeometric series. We show how these conjectures, that were deduced from a Yangian bootstrap analysis, are related by a tower of new quadratic transformations in Hypergeometric Functions Theory. Finally, we also use our MB method to identify spurious contributions that can arise in the Yangian approach.

arXiv commentAccepted in Phys. Rev. D. One section added. Typos corrected as well as several equations (mainly in Section 5). 30 pages, 6 figures

References in corpus (9)

• Yangian Bootstrap for Conformal Feynman Integrals
• Massive Conformal Symmetry and Integrability for Feynman Integrals
• Multiple Mellin-Barnes Integrals as Periods of Calabi-Yau Manifolds With Several Moduli
• The Double Box and Hexagon Conformal Feynman Integrals
• MultivariateResidues: a Mathematica package for computing multivariate residues
• Yangian Bootstrap for Massive Feynman Integrals
• The three-loop QED contributions to the $g-2$ of charged leptons with two internal fermion loops and a class of Kampé de Fériet series
• New Series Representations for the Two-Loop Massive Sunset Diagram
• New analytic continuations for the Appell $F_4$ series from quadratic transformations of the Gauss ${}_2{F}_1$ function

Cited by in corpus (7)

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• Yangian Ward Identities for Fishnet Four-Point Integrals
• Multiple Mellin-Barnes integrals with straight contours
• Analytic continuations of the Horn $H_1$ and $H_5$ functions
• Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model
• On the evaluation of the Appell $F_2$ double hypergeometric function
• Closed Form Expressions for Certain Improper Integrals of Mathematical Physics