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

Abstract

AbstractThe 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 ( α 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$$\chi $$ 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 $$R_\textrm{Z}$$ R Z , if one uses the well-known experimental 1S hfs in H or the 2S hfs in $$\mu $$ μ H. We, respectively, obtain $$R_\textrm{Z}(\textrm{H}) = 1.010(9)$$ R Z ( H ) = 1.010 ( 9 ) fm, $$R_\textrm{Z}(\mu \textrm{H}) = 1.040(33)$$ R Z ( μ H ) = 1.040 ( 33 ) fm. The total proton-structure effect to the hfs at $$O(\alpha ^5)$$ O ( α 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 $$\mu \text {H}$$ μ H is $$182.640(18)\,\textrm{meV}.$$ 182.640 ( 18 ) meV .