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 α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, which is O(α5)\documentclass[12pt]{minimal}
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\begin{document}$$O(\alpha ^5)$$\end{document} 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 (Bχ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document}PT). At leading orders it provides a prediction for the proton polarizability effects in hydrogen (H) and muonic hydrogen (μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}H). We find large cancellations among the various contributions leading to, within the uncertainties, a zero polarizability effect at leading order in the Bχ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document}PT expansion. This result is in significant disagreement with the current data-driven evaluations. The small polarizability effect implies a smaller Zemach radius RZ\documentclass[12pt]{minimal}
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\begin{document}$$R_\textrm{Z}$$\end{document}, if one uses the well-known experimental 1S hfs in H or the 2S hfs in μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document}H. We, respectively, obtain RZ(H)=1.010(9)\documentclass[12pt]{minimal}
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\begin{document}$$R_\textrm{Z}(\textrm{H}) = 1.010(9)$$\end{document} fm, RZ(μH)=1.040(33)\documentclass[12pt]{minimal}
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\begin{document}$$R_\textrm{Z}(\mu \textrm{H}) = 1.040(33)$$\end{document} fm. The total proton-structure effect to the hfs at O(α5)\documentclass[12pt]{minimal}
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\begin{document}$$O(\alpha ^5)$$\end{document} 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 μH\documentclass[12pt]{minimal}
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\begin{document}$$\mu \text {H}$$\end{document} is 182.640(18)meV.\documentclass[12pt]{minimal}
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\begin{document}$$182.640(18)\,\textrm{meV}.$$\end{document}
