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A Feynman integral via higher normal functions

Published online by Cambridge University Press:  06 August 2015

Spencer Bloch
Affiliation:
5765 S. Blackstone Ave., Chicago, IL 60637, USA email spencer_bloch@yahoo.com
Matt Kerr
Affiliation:
Department of Mathematics, Campus Box 1146, Washington University in St. Louis, St. Louis, MO 63130, USA email matkerr@math.wustl.edu
Pierre Vanhove
Affiliation:
Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 route de ChartresF-91440 Bures-sur-Yvette, France Institut de physique théorique, Université Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France email pierre.vanhove@cea.fr
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Abstract

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We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne’s conjectures relating values of $L$-functions inside the critical strip to periods.

Type
Research Article
Copyright
© The Authors 2015 

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