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We calculate the two-photon exchange (TPE) corrections in the region wherethe kinematical variables describing the elastic $ep$ scattering are moderatelylarge momentum scales relative to the soft hadronic scale. For such kinematicswe use the QCD factorization approach formulated in the framework of thesoft-collinear effective theory (SCET). Such technique allows us to develop adescription for the soft-spectator scattering contribution which is found to beimportant in the region of moderately large scales.  Together with the hard-spectator contribution we present the completefactorization formulas for the TPE amplitudes at the leading power and leadinglogarithmic accuracy. The momentum region where both photons are hard isdescribed by only one new nonperturbative SCET form factor. It turns out thatthe same form factor also arises for wide-angle Compton scattering which isalso described in the framework of the SCET approach. This allows us toestimate the soft-spectator contribution associated with the hard photons in amodel independent way.  The main unknown in our description of the TPE contribution is related withthe configuration where one photon is soft. The nonperturbative dynamics inthis case is described by two unknown SCET amplitudes. We use a simple model inorder to estimate their contribution.  The formalism is then applied to a phenomenological analysis of existing datafor the reduced cross section as well as for the transverse and longitudinalpolarization observables.

Two-photon exchange corrections to elastic electron-proton scattering at large momentum transfer within the SCET approach