Communications in Number Theory and Physics

Volume 12 (2018)

Number 1

Wick rotations, Eichler integrals, and multi-loop Feynman diagrams

Pages: 127 – 192



Yajun Zhou (Program in Applied and Computational Mathematics (PACM) Princeton University, Princeton, New Jersey, U.S.A.; and Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing, China)


Using contour deformations and integrations over modular forms, we compute certain Bessel moments arising from diagrammatic expansions in two-dimensional quantum field theory. We evaluate these Feynman integrals as either explicit constants or critical values of modular $L$-series, and verify several recent conjectures of Broadhurst.

This research was supported in part by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).

Received 14 July 2017

Accepted 25 November 2017

Published 27 April 2018