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, 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-Z, if one uses the well-known experimental 1S hfs in H or the 2S hfs in mu H. We, respectively, obtain R-Z(H)=1.010(9) fm, R-Z(mu 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 H is 182.640(18) meV.