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The two-photon-exchange (TPE) effect plays a key role to extract the formfactors (FFs) of the proton. In this work, we present some exact properties onthe TPE effect in the elastic $ep$ scattering based on four types of typicaland general interactions. The possible kinematical singularities, theasymptotic behaviors and the branch cuts of the TPE amplitudes are analyzed.The analytic expressions clearly indicate some exact relations between thedispersion relation (DR) method and the hadronic model (HM) method. It suggeststhat the two methods should be modified to general forms, respectively. Afterthe modifications the new forms give the same results. Furthermore, theyautomatically and correctly include the contributions due to the seagullinteraction, the meson-exchange effect, the contact interactions and theoff-shell effect. To analyze the elastic $e^{\pm}p$ scattering data sets, thenew forms should be used.

Exact Relations for Two-Photon-Exchange Effect in Elastic $ep$ Scattering by Dispersion Relation and Hadronic Model