Bhabha Scattering and a special pencil of K3 surfaces

TL;DR

This paper investigates a specific family of K3 surfaces linked to Bhabha scattering, revealing its identity with the Apéry–Fermi pencil and highlighting its unexpected appearance in various physical theories.

Contribution

It identifies a particular K3 surface pencil with the Apéry–Fermi pencil by analyzing its Picard lattice, connecting physics and number theory.

Findings

Identified the pencil with the Apéry–Fermi pencil through Picard lattice analysis

Connected the K3 pencil to Apéry's proof of ζ(3) irrationality

Revealed the pencil's appearance in diverse physical contexts

Abstract

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 Ap\'ery--Fermi pencil, that was related to Ap\'ery'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.