Effects of vacuum polarization by hadronic and heavy-fermion insertions were the last unknown two-loop QED corrections to high-energy Bhabha scattering and have been first announced in cite{Actis:2007fs}. Here we describe the corrections in detail an
d explore their numerical influence. The hadronic contributions to the virtual O(alpha^2) QED corrections to the Bhabha-scattering cross-section are evaluated using dispersion relations and computing the convolution of hadronic data with perturbatively calculated kernel functions. The technique of dispersion integrals is also employed to derive the virtual O(alpha^2) corrections generated by muon-, tau- and top-quark loops in the small electron-mass limit for arbitrary values of the internal-fermion masses. At a meson factory with 1 GeV center-of-mass energy the complete effect of hadronic and heavy-fermion corrections amounts to less than 0.5 per mille and reaches, at 10 GeV, up to about 2 per mille. At the Z resonance it amounts to 2.3 per mille at 3 degrees; overall, hadronic corrections are less than 4 per mille. For ILC energies (500 GeV or above), the combined effect of hadrons and heavy-fermions becomes 6 per mille at 3 degrees; hadrons contribute less than 20 per mille in the whole angular region.
Virtual hadronic contributions to the Bhabha process at the NNLO level are discussed. They are substantial for predictions with per mil accuracy. The studies of heavy fermion and hadron corrections complete the calculation of Bhabha virtual effects at the NNLO level.
Using dispersion relations, we derive the complete virtual QED contributions to Bhabha scattering due to vacuum polarization effects in photon propagation. We apply our result to hadronic corrections and to heavy lepton and top quark loop insertions.
We give the first complete estimate of their net numerical effects for both small and large angle scattering at typical beam energies of meson factories, LEP, and the ILC. The effects turn out to be smaller, in most cases, than those corresponding to electron loop insertions, but stay, with amounts of typically one per mille, of relevance for precision experiments. Hadronic corrections themselves are typically about 2-3 times larger than those of intermediate muon pairs (the largest heavy leptonic terms).
