Two photon exchange corrections at large momentum transfer revised

Motivated by experimental data at large momentum transfer we update the analysis of the two-photon exchange effect in the electron-nucleon scattering using the effective field theory formalism. Our approach is suitable for describing the hard region, where the hadronic model calculations are not accurate enough. We improve the estimates of various long-range matrix elements and discuss the obtained numerical effects for the unpolarised elastic cross section. Assuming a linear behaviour of the reduced cross section with respect to the photon polarisation, we show that the obtained description allows us to resolve the form factor discrepancy for $Q^2=2.5-3.5$GeV$^2$. However, the effect obtained is quite small for higher values of $Q^2$. It is possible that nonlinear effects may be important in understanding the discrepancy in this region. Estimates of the elastic electron-neutron cross section in the region are also performed. The obtained TPE effects are sufficiently large and must be taken into account.