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

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 $$\alpha $$α , which is $$O(\alpha ^5)$$O(α5) 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 $$\chi $$χ PT). At leading orders it provides a prediction for the proton polarizability effects in hydrogen (H) and muonic hydrogen ( $$\mu $$μ H). We find large cancellations among the various contributions leading to, within the uncertainties, a zero polarizability effect at leading order in the B $$\chi $$χ PT expansion. This result is in significant disagreement with the current data-driven evaluations. The small polarizability effect implies a smaller Zemach radius $$R_\textrm{Z}$$RZ , if one uses the well-known experimental 1Shfs in H or the 2Shfs in $$\mu $$μ H. We, respectively, obtain $$R_\textrm{Z}(\textrm{H}) = 1.010(9)$$RZ(H)=1.010(9) fm, $$R_\textrm{Z}(\mu \textrm{H}) = 1.040(33)$$RZ(μH)=1.040(33) fm. The total proton-structure effect to the hfs at $$O(\alpha ^5)$$O(α5) is then consistent with previous evaluations; the discrepancy in the polarizability is compensated by the smaller Zemach radius. Our recommended value for the 1Shfs in $$\mu \text {H}$$μH is $$182.640(18)\,\textrm{meV}.$$182.640(18)meV.