The $\sigma$ exchange effect in the Lamb shift of muonic hydrogen by two photons and two pions exchange

TL;DR

This paper estimates the sigma exchange effect on the Lamb shift in muonic hydrogen, finding a significant correction that should be considered in experimental analysis, based on phenomenological interactions involving photons and pions.

Contribution

It introduces a novel calculation of the sigma exchange effect in muonic hydrogen's Lamb shift using phenomenological interactions, providing a quantitative correction estimate.

Findings

The sigma exchange correction to the 2S energy level is about -14 μeV.

This correction constitutes approximately 44% of the two-photon exchange contribution.

The correction exceeds current experimental precision, impacting data interpretation.

Abstract

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.