Two-loop QED hadronic corrections to Bhabha scattering

Theoretical predictions for Bhabha scattering at the two-loop level require the inclusion of hadronic vacuum polarization in the photon propagator. We present predictions for the contributions from reducible amplitudes which are proportional to the vacuum polarization $π(q^2)$ and from irreducible ones where the vacuum polarization appears in a loop representing vertex or box diagrams. The second case can be treated by using dispersion relations with a weight function proportional to the $R$-ratio as measured in electron-positron annihilation into hadrons and kernels that can be calculated perturbatively. We present simple analytical forms for the kernels and, using two convenient parametrizations for the function $R(s)$, numerical results for the quantities of interest. As a cross check we evaluate the corresponding corrections resulting from light and heavy lepton loops and we find perfect agreement with previous calculations. For the hadronic correction our result are in good agreement with a previous evaluation.