Meson exchange in lepton-nucleon scattering and proton radius puzzle

We study two-photon contribution to the elastic lepton-proton scattering, associated with sigma meson exchange, with special attention paid to the low-Q2 region. We show that the corresponding amplitude grows sharply but remains finite at Q2->0. The analytical formula for the amplitude at Q2=0 is obtained. We also estimate the shift of the muonic hydrogen energy levels, induced by sigma meson exchange. For the 2S level the shift is approximately 30 times smaller than needed to resolve the proton radius puzzle, but still exceeds the precision of the muonic hydrogen measurements.