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A Feynman integral via higher normal functions

Part of: Families, fibrations Surfaces and higher-dimensional varieties $K$-theory in number theory Quantum field theory; related classical field theories

Published online by Cambridge University Press:  06 August 2015

Spencer Bloch ,
Matt Kerr  and
Pierre Vanhove
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.

Keywords

Feynman graphvariation of mixed Hodge structuresmotiveselliptic trilogarithmEisenstein serieshigher normal function

MSC classification

Primary: 81T18: Feynman diagrams
Secondary: 19F27: Étale cohomology, higher regulators, zeta and $L$-functions 14J28: $K3$ surfaces and Enriques surfaces 14D07: Variation of Hodge structures

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 12 , December 2015 , pp. 2329 - 2375
Copyright
© The Authors 2015 

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