Chiral perturbation theory of the hyperfine splitting in (muonic) hydrogen

Abstract

The ongoing experimental efforts to measure the hyperfine transition in muonic hydrogen prompt an accurate evaluation of the proton-structure effects. At the leading order in $\alpha$, which is $O({\alpha }^5)$ in the hyperfine splitting (hfs), these effects are usually evaluated in a data-driven fashion, using the empirical information on the proton electromagnetic form factors and spin structure functions. Here we perform a first calculation based on the baryon chiral perturbation theory (B$\chi$PT). At leading orders it provides a prediction for the proton polarizability effects in hydrogen (H) and muonic hydrogen ($\mu$H). We find large cancellations among the various contributions leading to, within the uncertainties, a zero polarizability effect at leading order in the B$\chi$PT expansion. This result is in significant disagreement with the current data-driven evaluations. The small polarizability effect implies a smaller Zemach radius $R_{\mathrm{Z}}$, if one uses the well-known experimental $1S$ hfs in H or the $2S$ hfs in $\mu$H. We, respectively, obtain $R_{\mathrm{Z}}(\mathrm{H})=1.010(9)$ fm, $R_{\mathrm{Z}}(\mu \mathrm{H})=1.040(33)$ fm. The total proton-structure effect to the hfs at $O({\alpha }^5)$ is then consistent with previous evaluations; the discrepancy in the polarizability is compensated by the smaller Zemach radius. Our recommended value for the $1S$ hfs in $\mu \text{H}$ is $182.640(18)\mathrm{meV}.$

Comments: 22 pages, 11 figures, 3 tables, typo in Eq. (13) corrected, references updated