The recoil correction to the proton-finite-size contribution to the Lamb shift in muonic hydrogen

The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the scattering, which may produce an internal inconsistency of theory. Recently the leading proton-finite-size contribution to the two-photon exchange was found within the external field approximation. The recoil part of the two-photon-exchange has not been considered. Here we revisit calculation of the external-field part and take the recoil correction to the finite-size effects into account.