Meson exchange in lepton-nucleon scattering and the proton radius puzzle

I study two-photon contribution to the elastic lepton-proton scattering, associated with $\sigma$ meson exchange, with special attention paid to the low-$Q^2$ region. It is shown that the corresponding amplitude grows sharply but remains finite at $Q^2\to 0$. The analytical formula for the amplitude at $Q^2=0$ is obtained. I 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.

Figure 1

Box, crossed-box, and $\sigma$ meson exchange diagrams.

Figure 2

$Q^2$ dependence of calculated $f_s$ form factor for muon and electron scattering.

Figure 3

$f_s\left(Q^2\right)$ for muon scattering and very small $Q^2$.