Nonperturbative numerical calculation of the fine and hyperfine structure of muonic hydrogen by Breit potential including the effects from the proton size

TL;DR

This paper employs a high-precision nonperturbative numerical approach to calculate the fine and hyperfine structures of muonic hydrogen, revealing small but significant differences from perturbative methods when proton structure effects are included.

Contribution

It introduces a nonperturbative numerical method for calculating muonic hydrogen energy levels, improving accuracy over traditional perturbative approaches.

Findings

Differences between NPnum and perturbative results are small for 2P states.

Significant differences (~0.009 meV and 0.08 meV) are found for 2S hyperfine splittings.

Results highlight the importance of nonperturbative calculations in precise atomic physics.

Abstract

By solving the two-body Schordinger equation in a very high precise nonperturbative numerical (NPnum) way, we reexamine the contributions of fine, hyperfine structure splittings of muonic hydrogen based on the Breit potential. The comparison of our results with those by the first order perturbative theory ($^{1st}$PT) in the literature shows, when the structure of proton is considered, the differences between the results by the $^{1st}$PT and NPnum methods are small for the fine and hyperfine splitting of $2P$ state, while are about $-0.009$ meV and $0.08$ meV for the $F=1$ and total hyperfine splitting of $2S$ state of muonic hydrogen, respectively. These differences are larger than the current experimental precision and would be significant to be considered in the theoretical calculation.