The $\sigma$ exchange effect in the Lamb shift of muonic hydrogen by two   photons and two pions exchange
  Based on the simple phenomenological $\sigma \pi \pi$, $\sigma N N$ and $\pi \pi \gamma$ interactions, we estimate the $\sigma$ exchange effect in the Lamb shift of muonic hydrogen. We at first calculate the effective couplings of $\sigma\mu\mu$ by two photons and two pions exchange, then calculate the corresponding corrections to the energy shift of the 2S/2P states of muonic hydrogen. We find the correction to the energy shift of 2S state is about -14$\mu$eV by the current used parameters, which is about $44\%$ of the usual two photons exchange contribution, and is larger than the current experimental precision, and should be considered in the experimental analysis. 
Hai-Qing Zhou
nucl-th hep-ph nucl-ex