Bhabha scattering and a special pencil of K3 surfaces

We study a pencil of K3 surfaces that appeared in the $2$-loop diagrams in Bhabha scattering. By analysing in detail the Picard lattice of the general and special members of the pencil, we identify the pencil with the celebrated Apéry–Fermi pencil, that was related to Apéry’s proof of the irrationality of $\zeta (3)$ through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts.

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We would like to thank Johannes Henn for asking the original question that led to this paper. We also thank Marco Besier, Claude Duhr, Alice Garbagnati, Bert van Geemen, Davide Cesare Veniani, Stefan Weinzierl and the anonymous referee for helpful comments and discussions. The first author was supported by SFB/TRR 45. The authors also would like to express a special thanks to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support, and to Oliver Labs for his software for visualization of algebraic surfaces, “Surfex”.

Received revised 20 September 2018

Accepted 7 January 2019

Published 26 April 2019