Two-photon exchange corrections in elastic lepton-proton scattering

In recent years, two experimental approaches, with and without polarized protons, gave strikingly different results for the ratio of the electric to magnetic proton form factors. Even more recently, a mysterious discrepancy ("the proton radius puzzle") has been observed in the extraction of the proton charge radius from the muonic hydrogen versus regular hydrogen and electron-proton scattering. Two-photon exchange (TPE) contributions are the largest source of the hadronic uncertainty in these experiments. To determine TPE corrections to the S level in muonic hydrogen, the forward virtual Compton scattering is calculated within dispersion relation (DR) formalism. Comparing a box graph model with the DRs at fixed low momentum transfer, we develop and test the subtracted DR formalism for TPE in electron-proton scattering. Its relative effect on the cross section is in the $1-2~\%$ range. We include the inelastic states both in the approximation of the {\it near-forward} unpolarized virtual Compton scattering as well as using the empirical information on the $\pi N$ states contribution. We compare the resulting TPE with MAMI, VEPP-3 and CLAS data, and make predictions for the OLYMPUS and the forthcoming MUSE experiments.