  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.
