Bhabha Scattering and a special pencil of K3 surfaces

TL;DR

This paper identifies a K3 pencil arising from 2-loop Bhabha scattering and shows it is birational to a double-sextic pencil whose generic fibre is a K3 surface with Picard lattice $U\oplus E_8(-1)^{\oplus2}\oplus\langle-12\rangle$ and transcendental lattice $U\oplus\langle12\rangle$. By analyzing singular and special fibres, the authors compute the full geometric Picard lattice (rank 19) of the generic fibre and demonstrate maximal-rank (20) Picard lattices for certain special fibres, which are non-isometric among themselves. They then connect the studied pencil to the Apéry–Fermi family by a precise birational map, showing that the two pencils share the same Picard lattice up to a parameter change, thus unifying appearances of this geometry across physics and number theory. The work highlights deep links between Feynman integral structures, Apéry-type irrationality proofs, and the geometry of K3 surfaces, including a precise description of how the Apéry–Fermi lattice arises in this context.

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 Apéry--Fermi pencil, that was related to Apéry's proof of the irrationality of $ζ(3)$ through the work of F. Beukers, C. Peters and J. Stienstra. The same pencil appears miraculously in different and seemingly unrelated physical contexts.

Figures (2)

• Figure :
• Figure :

Theorems & Definitions (31)

• Remark 1
• Remark 1.1
• Proposition 1.2
• Corollary 2.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 4.1
• proof