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 α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, which is O(α5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}H. We, respectively, obtain RZ(H)=1.010(9)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\textrm{Z}(\textrm{H}) = 1.010(9)$$\end{document} fm, RZ(μH)=1.040(33)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \text {H}$$\end{document} is 182.640(18)meV.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$182.640(18)\,\textrm{meV}.$$\end{document}

These future measurements hold the potential to extract the Zemach radius with a sub-percent uncertainty, thereby constraining the magnetic properties of the proton.
A precise theory prediction for the 1S hfs in µH is essential for the success of the experimental campaigns.Firstly, to narrow down the frequency search range, which is important given the limited beam time available to the collaborations at PSI, RIKEN-RAL and J-PARC.Secondly, for the interpretation of the results.One can either extract the Zemach radius given a theory prediction for the proton-polarizability effect in the µH 1S hfs, or vice versa, extract the protonpolarizability effect with input for the Zemach radius.Furthermore, one can combine the precise measurements of the 1S hfs in H and µH to disentangle the Zemach radius and polarizability effects, leveraging radiative corrections as explained in Ref. [13], and compare their empirical values to theoretical expectations.
The biggest uncertainty in the theory prediction comes from proton-structure effects, entering through the twophoton exchange (TPE).These contain the above-mentioned Zemach radius and polarizability effects.Presently, they are evaluated within a "data-driven" dispersive approach [14][15][16].While the dispersive method itself is rigorous, it requires sufficient experimental data to map out the proton spin structure functions g 1 (x, Q 2 ) and g 2 (x, Q 2 ) as full functions of the Bjorken variable x and the photon virtuality Q 2 .This has been the aim of a dedicated "Spin Physics Program" at Jefferson Lab [17][18][19][20][21] that recently extended the previously scarce data for g 2 [22,23].
In this work, we use an entirely different approach -the chiral perturbation theory (χPT) [24][25][26] -which has been successfully used to give a prediction for the proton-polarizability effect in the µH Lamb shift [27].To be precise, we work in the framework of baryon chiral perturbation theory (BχPT) -the manifestly Lorentz-invariant formulation of χPT in the baryon sector [26,28,29] (see also [30,31] for reviews).We show that the leading-order (LO) BχPT prediction for the polarizability effect in the hfs is effectively vanishing, thereby, in substantial disagreement with the data-driven evaluations.
The paper is organized as follows.In Sec.II, we discuss the forward TPE, and in particular, the polarizability effect in the hfs.A new formalism where one splits into contributions from the longitudinal-transverse and helicitydifference photoabsorption cross sections of the proton, σ LT and σ T T , is introduced in Eq. (12).It will be shown that this decomposition is advantageous for both the dispersive, as well as the effective field theory (EFT) calculations, as it gives a cleaner access to the uncertainties.More details are given in Appendix A. In Sec.III, we present our LO BχPT prediction for the polarizability effect in the hfs of H and µH, together with a detailed discussion of the uncertainty estimate.In Sec.IV, we compare our results to data-driven dispersive and heavy baryon effective field theory (HB EFT) calculations.In Sec.V, the Zemach radius is extracted from H and µH spectroscopy based on our prediction for the polarizability effect.In Sec.VI, we discuss the TPE effect in the µH hfs in view of the forthcoming experiments.Full details of the theoretical prediction for the 1S µH hfs are collected in Appendix C. We finish with an outlook and conclusions.

II. TWO-PHOTON EXCHANGE IN THE HYPERFINE SPLITTING
The (muonic-)hydrogen hfs receives contributions from QED-, weak-and strong-interaction effects: where the leading-order in α contribution is given by the Fermi energy: with α the fine-structure constant, Z the charge of the nucleus (in the following Z = 1 for the proton), m, M the lepton and proton masses, κ the anomalous magnetic moment of the proton, and a −1 = αm r the inverse Bohr radius, with m r = mM/(m + M ) the reduced mass.The strong-interaction effects arise from the composite structure of the proton.They begin to enter at O(α 5 ), see for instance Ref. [14], where they are split into the Zemach-radius, recoil, and polarizability contributions: which can all be attributed to the forward TPE shown in Fig. 1.For a first comprehensive theory summary of the Lamb shift, fine and hyperfine structure in µH, including proton-structure dependent effects, we refer to Ref. [32].
The Zemach and recoil terms (∆ Z and ∆ recoil ) are elastic contributions with a proton in the intermediate state, see ) that shall be evaluated in this work.
The forward TPE contribution to the hfs can be expressed through the spin-dependent forward doubly-virtual Compton scattering (VVCS) amplitudes, S 1 and S 2 , cf.Eq. (A2).The latter can be related to the proton structure functions g 1 and g 2 in a dispersive approach, cf.Eqs.(A11) and (A12).A full derivation of the well-known formalism for the TPE contribution to the hfs can be found in Appendix A.
The largest TPE effect is due to the Zemach radius contribution: The recoil contribution is one order of magnitude smaller [7], and will not be considered in this paper.It has been recently updated in Ref. [33].The hfs is therefore best suited for a precision extraction of the Zemach radius, defined as the following integral over the electric and magnetic Sachs form factors G E (Q 2 ) and G M (Q 2 ) [32]: where q 2 = −Q 2 is the photon virtuality.Equivalently, we can write: where the linear electric and magnetic radii are defined as: with Im G(t) the imaginary part of the normalized electric or magnetic Sachs form factor, G E,M (Q 2 )/G E,M (0).As one can see from Eqs. ( 7) and ( 8), a measurement of the Zemach radius gives access to the magnetic properties of the proton.
The polarizability effect in the hfs is fully constrained by empirical information on the proton spin structure functions g 1 (x, Q2 ) and g 2 (x, Q 2 ), and the Pauli form factor F 2 (Q 2 ), functions of Q 2 and the Bjorken variable x = Q 2 /2M ν, where ν is the photon energy in the lab frame.This is in contrast to the Lamb shift, where the knowledge of a subtraction function, with x 0 the inelastic threshold, , and the generalized Gerasimov-Drell-Hearn (GDH) integral: Here, Ī1 is the polarizability part of I 1 .For the origin of the Pauli form factor in the above equations, see discussion in Appendix A. As we will show in Sec.III B, instead of decomposing into ∆ 1 and ∆ 2 , it is convenient to decompose into contributions from the longitudinal-transverse and helicity-difference cross sections σ LT and σ T T : where we define: Or equivalently, in terms of the VVCS amplitudes, we can write: Here, Si denotes the non-Born part of the amplitudes.An advantage of the BχPT calculation in this work is that the non-Born amplitudes can be calculated directly, and need not be constructed through the dispersive formalism.Furthermore, at the present order of our calculation in the BχPT power counting, there are no contributions to the elastic form factors, and thus, I 1 in Eq. ( 11) is given by the polarizability part only.

III. CHIRAL LOOPS
Assuming BχPT is an adequate theory of low-energy nucleon structure, it should be well applicable to atomic systems, where the relevant energies are naturally small.In Ref. [27], the polarizability effect in the µH Lamb shift has been successfully predicted at LO in BχPT.Here, we extend this calculation to the polarizability effect in the hfs.This requires the spin-dependent non-Born VVCS amplitudes, S1 and S2 , at chiral O(p3 ) in the BχPT power counting.
Figure 1 in Ref. [27] shows the leading polarizability effect given by the TPE diagrams of elastic lepton-proton scattering with one-loop πN insertions.For the Compton-like processes, it is convenient to use the chirally-rotated leading BχPT Lagrangian for the pion π a (x) and nucleon N (x) fields [38]: where γ 5 = iγ 0 γ 1 γ 2 γ 3 , g A ≃ 1.27 [39] is the axial coupling of the nucleon, f π ≃ 92.21 MeV is the pion-decay constant, τ a are the Pauli matrices, M N ≃ 938.27MeV and m π ≃ 139.57MeV are the nucleon and pion masses. 3As described in Ref. [27], the Born part is separated from the O(p 3 ) VVCS amplitudes by subtracting the on-shell pion-loop γN Nvertex in the one-particle-reducible VVCS graphs, see diagrams (b) and (c) in Figure 1 of Ref. [27].For more details on the BχPT framework, we refer to Refs.[40][41][42], where the complete next-to-next-to-leading-order (NNLO) in the δ-expansion [43] BχPT calculation of the spin-independent and spin-dependent nucleon VVCS amplitudes can be found. 4n practice, most results here were obtained based on our BχPT prediction for the πN -production channel in the structure functions g i , given in Ref. [42,Appendix B].It has been verified that the results agree with the calculation based on the VVCS amplitudes Si .

A. Numerical results
Our LO BχPT prediction for the polarizability effect in the 1S hfs of H and µH amounts to: The error estimate will be described and motivated in the subsequent sections.The corresponding contributions to the nS hfs are trivially obtained through a 1/n 3 scaling, as can be seen from Eqs. ( 3) and ( 4).Splitting into contributions from the spin structure functions g 1 and g 2 , we obtain: Strikingly, the contributions from the longitudinal-transverse and helicity-difference cross sections σ LT and σ T T : are one order of magnitude larger than the total, and differ in their respective signs.This indicates a cancellation of LO contributions between ∆ LT and ∆ T T .Including in addition the correction due to electron vacuum polarization (eVP) in the TPE diagram, see Fig. 10 and discussion in Appendix B, gives a negligible effect within the present uncertainties: Nevertheless, it is important in view of the anticipated 1 ppm accuracy (corresponding to ∼ 0.2 µeV) of the µH 1S hfs measurement by the CREMA collaboration [8].We therefore include the additional ∆ eVP pol.
The amplitude S1(0, Q 2 ) at LO in BχPT (blue) and HBχPT (red).The dashed lines show the corresponding slope terms, i.e., the first terms in the expansion in powers of Q 2 .The BχPT slope has been calculated from the polarizabilities given in Ref. [55,Table I], the HBχPT slope is given in Eq. ( 21).
To understand why the contributions from σ LT and σ T T largely cancel in ∆ pol., we study the heavy-baryon (HB) limit of the spin-dependent VVCS amplitudes [48].Expanding the LO BχPT expression for the S1 amplitude in µ = m π /M N while keeping the ratio of the light scales τ π = Q 2 /4m 2 π fixed, one obtains: We then take a closer look at the first term in the low-energy polarizability expansion: The HBχPT predictions for the proton polarizabilities [49][50][51][52][53][54] entering Eq. ( 19) read: We can see that the leading terms in the chiral expansion are of O(1/m 2 π ).They cancel among the different polarizabilities, thus, Eq. ( 21) becomes a subleading contribution: Accordingly, one would expect the chiral loops in the hfs to be small.Indeed, the LO BχPT prediction in Eq. ( 15) is essentially vanishing, where the small number is mainly a remnant of higher orders in the HB expansion.This has to be taken into account in the uncertainty estimate.Note that the HB expansion above has been introduced for instructive purposes only, but is not entering our calculation of the polarizability effect.The HBχPT prediction of the S 1 (0, Q 2 ) amplitude, Eq. ( 18), raises with Q, thus, its contribution to the hfs will be divergent.This can be seen from Fig. 2, where we compare the chiral-loop contribution to S1 (0, Q 2 ) as predicted by BχPT and HBχPT, respectively.

B. Uncertainty estimate
BχPT is a low-energy EFT of QCD describing strong interactions in terms of hadronic degrees of freedom (pion, nucleon, ∆(1232) resonance).An important requirement for a reliable BχPT prediction is that the contribution from beyond the scale at which this EFT is safely applicable, i.e., Q max > m ρ = 775 MeV, has to be small.For the LO BχPT prediction of the polarizability effect in the µH Lamb shift [27], the contribution from beyond this scale was less than 15 %, thus, within the expected uncertainty.Comparing the TPE master formulas for Lamb shift and hfs, Eqs.(A8) and (A7), the weighting function in the former has a stronger suppression for large Q 2 .It is therefore important to verify that the same quality criterion still holds for the hfs prediction presented here.Let us consider the polarizability effect as a running integral with momentum cutoff Q max , as shown in Fig. 3.The convergence of the ∆ 1 (green line) contribution, as well as of the total ∆ pol.(black line), is poor.They display a sign change of the running integral at energies above Q max ≈ 2 GeV (µH) and ≈ 4 GeV (H), respectively.∆ 2 (red line) converges better.Its contributions from above Q max = m ρ amount to 42 % (H) and 26 % (µH), respectively.
The bad high-momentum asymptotics indicated above are merely an artefact of the conventional splitting into ∆ 1 and ∆ 2 .For the alternative splitting into ∆ LT and ∆ T T , introduced in Eq. ( 12), the cut-off dependence improves considerably.For ∆ T T (blue line), the contribution from above Q max = m ρ amounts to less than 4 % for both hydrogens.For ∆ LT (orange line), the high-energy contributions are less than 35 % (µH) and 32 % (H), respectively.In this way, our results are in agreement with the natural expectation of uncertainty for a LO prediction, 30 % [≃ (M ∆ − M )/GeV], in BχPT with inclusion of the ∆ resonance.Based on this analysis, we decided to assign errors of 30 % to the σ LT and σ T T contributions, and propagate them to ∆ 1 , ∆ 2 and ∆ pol. .It is interesting to note that in this way the uncertainty of ∆ 1 is larger than the uncertainty of ∆ pol. .This can be understood from the opposite signs of the ∆ i,j contributions, where i = 1, 2 and j = LT, T T , on the example of µH:

IV. COMPARISON WITH OTHER RESULTS
In this section, we compare our LO BχPT prediction for the polarizability effect in the H and µH hfs to other available evaluations.Furthermore, we study the contribution of the S1 (0, Q 2 ) subtraction function and the scaling of the polarizability effect with the lepton mass.

A. Heavy-baryon effective field theory
Let us start by comparing our BχPT prediction to other model-independent calculations using HB EFT [56][57][58]. 5irst results for the elastic and inelastic TPE effects on the hfs in H and µH have been obtained in Ref. [56], where the contribution of the leading chiral logarithms, O(m 3 α 5 /M 2 × [ln m π , ln ∆, ln m]), was calculated in HB EFT matched to potential NRQED.At this order in the chiral expansion the polarizability effects in the hfs from pion-nucleon and pion-delta loops cancel each other in the large-N c limit, while the ∆ exchange cancels part of the point-like corrections, see also Ref. [48].The analytical results presented in Ref. [56,59] motivate the relative size of the Zemach and polarizability corrections.Updated HB EFT predictions for the TPE effects on the hydrogen spectra can be found in Refs.[57,58,60].In Ref. [58], the difference between the pion-loop polarizability contributions in H and µH is quoted as where c 4 is a Wilson coefficient linked to the hfs in the following way: For comparison, we can evaluate the analogue of ∆c 4 from other theory predictions for the polarizability contribution.Within errors, our LO BχPT prediction agrees with this result: Here, the uncertainties of the H and µH predictions have been combined in quadrature to estimate the error on their difference.For comparison, from the data-driven dispersive evaluations of Carlson et al. [14], one can deduce: where we combined all errors quoted in Ref. [14] and estimated the error on ∆c 4 in the same way as done above.

B. Data-driven dispersive evaluations
There is a clear discrepancy between the BχPT prediction, presented here, and the conventional data-driven dispersive evaluations.The dispersive evaluations rely on empirical information for the inelastic proton spin structure functions, the elastic Pauli form factor and polarizabilities.The discrepancy can be seen from Fig. 4, where our LO BχPT prediction for the polarizability effect in the H and µH hfs is compared to the available dispersive evaluations.Adding an estimate for the next-to-leading-order (NLO) effect of the ∆(1232) resonance [61], obtained from large-N c relations for the nucleon-to-delta transition form factors, to the model-independent LO BχPT prediction will improve agreement for ∆ 2 but not for ∆ pol .
The origin of this discrepancy has to be understood in order to give a reliable prediction of the TPE effect in the µH hfs, needed for the forthcoming experiments.Part of the discrepancy might be due to underestimated uncertainties.An evaluation of the total polarizability effect suffers from cancellations in two places: firstly, between contributions from the cross sections σ LT and σ T T , secondly, between the elastic Pauli form factor F 2 and the inelastic structure functions in the low-Q region.Each of these cancellations reduces the result by an order of magnitude.In the calculation presented here, the former is taken into account by estimating the uncertainty due to higher-order corrections in the BχPT power counting based on the large σ LT and σ T T contributions, see discussion in Sec.III B. In the dispersive approach, it would be important to take into account correlations between parametrizations of the g 1 and g 2 structure functions, which both rely on measurements of σ LT and σ T T .The latter cancellations in the low-Q region will be discussed in the following subsection.
C. Low-Q region and contribution of the S1(0, Q 2 ) subtraction function One major drawback of the data-driven dispersive evaluations is that they require independent input for the inelastic spin structure functions or related polarizabilities, and the elastic Pauli form factor.Our notation in Eq. (10b) conveniently illustrates how the zeroth moment of the inelastic spin structure function g 1 and the elastic Pauli form factor F 2 combine in the subtraction function: At Q 2 = 0, this is zero, because the Pauli form factor, F 2 (0) = κ, and the generalized GDH integral, I 1 (0) = −κ 2 /4, so the two terms cancel exactly.A NLO BχPT prediction of the slope amounts to: [I 1 ] ′ (0) = 0.39(4) GeV −2 [42].It can be expressed through a combination of lowest-order spin [γ E1M 2 ] and generalized polarizabilities [P ′(M 1,M 1)1 (0) and P ′(L1,L1)1 (0)], see Eq. ( 19).In the HBχPT expansion, we showed that the leading O( 1 /m 2 π ) terms cancel among these individual polarizabilities, given in Eq. ( 20), turning the result subleading in O( 1 /mπ), see Eq. ( 21).We can conclude that there is a strong cancellation between the elastic and inelastic contributions, which continues for higher The contribution of S1 (0, Q 2 ) to the hfs is given by: Evaluations of this subtraction function contribution with empirical parametrizations for g 1 (x, Q 2 ) and F 2 (Q 2 ) tend towards larger values than the LO BχPT prediction.A partial calculation of the TPE effect at NLO in BχPT, considering only the one-loop box diagram with intermediate ∆(1232)-excitation, will lower the theoretical prediction for the polarizability contribution from BχPT further, and in fact, turn it into a negative contribution [61,62].Any imprecision in the empirical parametrizations, and thus in the cancellation between the elastic and inelastic moments, is enhanced by the 1/Q prefactor in the infrared region of the integral in Eq. (28).Therefore, the BχPT calculation, where the polarizability effect can be accessed directly through the non-Born part of the VVCS amplitudes and does not rely on input from separate measurements, has a clear advantage in this regard.
To illustrate this further, we reproduce the estimate for ∆ 1 in the low-Q region from Ref. [14] (see references therein for the details on the input).In this region, no experimental data from EG1 [63,64] exist and the integral is completed by interpolating data between higher Q 2 and Q 2 = 0, making use of empirical values for the static polarizabilities.For Q 2 ∈ 0, Q 2 max with Q 2 max = 0.0452 GeV 2 , the approximate formulas read [14]: where Note that the formulas for H and µH differ, because one sets m e = 0.The first terms are related to the elastic Pauli form factor, where is the Pauli radius.The other terms are related to the g 1 contribution.Considering the more general Eq.(29b), they are defined through: The strong cancellation between elastic and inelastic contributions, observed in Eq. ( 29), can be a source of uncertainty.
In addition, the quality of the low-Q approximation is rather poor.We can test it at LO in BχPT.Recall that at this order in the BχPT power counting, there is no contribution to the elastic form factors. Therefore, only the inelastic structure function g 1 enters.Our results are shown in Fig. 5.The approximate formulas in Eq. ( 29) give a 50 % (67 %) larger value for δ 1 in the region of Q 2 < 0.0452 GeV 2 in the case of µH (H).Therefore, in the data-driven dispersive approach one has to properly account for the uncertainty introduced by the approximate formulas, as well as from cancellations between elastic and inelastic contributions.  5.The polarizability contribution δ1 in the low-Q region for hydrogen (red) and muonic hydrogen (blue).The solid lines are the exact results according to Eq. ( 10b) with an upper cut on the Q integration.The dotted and dashed lines are evaluated with the approximate formulas for hydrogen and muonic hydrogen, respectively, see Eq. ( 29).

D. Scaling with lepton mass
It is customary to use the high-precision measurement of the 1S hfs in H [65,66]: 1S-hfs (H) = 1 420.405751 768(1) MHz, (32) to refine the prediction of the TPE in the µH hfs [16,58] or the prediction of the total µH hfs [13].We will do the same in Sec.VI.The strategies in Refs.[13,16,58] are slightly different, but all make statements about the scaling of various contributions to the hfs in a hydrogen-like atom when varying the lepton mass m ℓ .In Fig. 6, we study the scaling of the polarizability effect based on our LO BχPT prediction.In the left panel, we assume that the ∆ i (with i = 1, 2, LT, T T and pol.) are scaling with the reduced mass m r .In the right panel we assume that the δ i are independent of the lepton mass, thus, ∆ i would be scaling with m ℓ .The curves in the upper (lower) panel are normalized for H (µH), so they are fixed to 1 at m ℓ = m e (m ℓ = m µ ).If the polarizability effect would scale according to our assumptions, i.e., ∝ m r or ∝ m ℓ , all curves would be constantly 1.We can see that the scaling works best for the contributions from σ LT and σ T T , which are large in their absolute values.Considering the total, in which the contributions from σ LT and σ T T cancel by about one order of magnitude, the scaling violation is enhanced by about one order of magnitude in relative terms.The same enhancement of the scaling violation can be observed for the numerically small contributions from g 1 and g 2 .Comparing left and right panels, the BχPT predictions seems to support the assumption that ∆ LT and ∆ T T are scaling with m r .For ∆ LT , the scaling is nearly perfect.For ∆ T T , we observe a violation of the scaling that is increasing with lepton mass.The approximation ∆ T T (µH) ∼ m r (µH)/m r (H) ∆ T T (µH) holds at the level of 10%.The approximation holds on a similar level after including an estimate for the NLO effect of the ∆(1232) resonance [61].
Scaling of δi and ∆i/mr (with i = 1, 2, LT, T T and pol.), as a function of the lepton mass m ℓ .

V. EXTRACTION OF THE ZEMACH RADIUS FROM SPECTROSCOPY
The TPE, entering the hfs, can be decomposed into Zemach radius, polarizability and recoil contributions, as described in Eq. ( 5).On top of the O(α 5 ) TPE, we consider the leading radiative corrections given by eVP, see Fig. 10 and discussion in Appendix B. Our prediction for the polarizability effect in the hfs, which is smaller than the conventional results from data-driven dispersive evaluations, also implies a smaller proton Zemach radius as previously determined from spectroscopy, cf.Eq. (1).In the following, we will extract the Zemach radius from the precisley measured 1S hfs in H, see Eq. (32), and the 2S hfs in µH [5]: HFS (2S, µH) = 22.8089(51) meV.
We use the theory predictions for the 1S hfs in H [13]: and the 2S hfs in µH [13]: with the recently re-evaluated O(α 5 ) recoil correction [33]: up to a factor 3 more precise than the previous best determination [67] based on the electromagnetic form factors obtained from dispersion theory [68].An itemized list of contributions to the 2S hfs in µH is given in   Appendix C. From the LO BχPT prediction for the polarizability effect, including also the eVP in Eq. ( 17), we obtain: This can be compared to other determinations of the proton Zemach radius collected in Table I. 6 The radii we find are in agreement with the proton form factor analysis from Ref. [69], which uses the proton charge radius from the µH Lamb shift [5] as a constraint for their fit.
Figure 7 shows how the Zemach and charge radius of the proton are correlated.It suggests that a "smaller" charge radius, as seen initially in the µH Lamb shift by the CREMA collaboration [5] (red line), comes with a "smaller" Zemach radius.The dashed black curve is calculated with a dipole form, for the electric and magnetic Sachs form factors, by varying Λ.The light red and orange bands show R Z as extracted by us, Eq. ( 37), based on the LO BχPT prediction for the polarizability effect in the hfs.

VI. THEORY PREDICTION FOR THE GROUND-STATE HYPERFINE SPLITTING IN µH
The upcoming measurements of the 1S hfs in µH [8][9][10][11] crucially rely on a precise theory prediction.The limiting uncertainty is given by the TPE, which is conventionally split into Zemach radius, polarizability and recoil contributions [13]:  see Appendix C and Table II for an itemized list of the individual contributions.As explained in Sec.IV D, it is customary to refine the theory prediction of 1S hfs in µH with the help of the high-precision measurement of the 1S hfs in H.We do so by combining our BχPT prediction for the polarizability effect in the µH hfs, Eq. (17b), and the Zemach radius extracted from H spectroscopy, Eq. (37a), based on the same prediction for the polarizability effect in the H hfs. We arrive at: E hfs (1S, µH) = 182.640(18) meV, (39a) E TPE hfs (1S, µH) = −1.157(16) meV, where E TPE hfs corresponds to the TPE including radiative corrections and recoil corrections from Ref. [33], as indicated by the curly brace in Eq. (38).
In Figs. 8 and 9, we compare our predictions to results from data-driven dispersive evaluations [14,16] and HB EFT [58].While almost all available predictions for the total hfs in µH are in agreement after the H refinement procedure, further improvements of the theory are required in order to compete with the anticipated experimental accuracy.

VII. CONCLUSIONS AND OUTLOOK
We have presented the LO BχPT prediction for the O(α 5 ) polarizability effect on the hfs in H and µH, see Eq. (15).Contrary to the data-driven evaluations, the BχPT prediction is compatible with zero.This was expected from the HBχPT limit of the VVCS amplitudes, in particular S1 (0, Q 2 ), which partially display a cancellation of the leading order in the chiral expansion of small m π , see discussion in Sec.III A. The small polarizability effect is then mainly a remnant of higher orders in the HB expansion.
A new formalism where the polarizability effect is split into contributions from the longitudinal-transverse and helicity-difference cross sections, σ LT and σ T T , instead of contributions from the spin structure functions, g 1 and g 2 , has been introduced in Eq. (12).It was shown that these contributions, ∆ LT and ∆ T T , cancel by one order of magnitude when combined into ∆ pol. .Only ∆ LT and ∆ T T are good observables in the BχPT framework, for which the contributions from beyond the scale at which this EFT is safely applicable, Q max > m ρ = 775 MeV, are within the expected uncertainty.In addition, only ∆ LT and ∆ T T satisfy the conventionally assumed scaling with the reduced mass m r of the hydrogen-like system to 10% relative accuracy, while the cancellations in ∆ pol.enhance any violation in the scaling by one order of magnitude.
As shown in Fig. 4, our model-independent LO BχPT prediction is substantially smaller than the data-driven dispersive evaluations.An estimate for the effect of the ∆(1232)-resonance [61], obtained from large-N c relations for the nucleon-to-delta transition form factors, shows that the discrepancy is likely to increase at the NLO.The smaller polarizability effect, in turn, leads to a smaller Zemach radius as extracted from the experimental 1S hfs in H and the 2S hfs in µH, cf.Eq. (37).Therefore, resolving the present discrepancy for the polarizability effect is crucial for the analysis of the forthcoming measurements of the 1S hfs in µH and the extraction of the Zemach radius.
The data-driven approach relies on empirical information on the inelastic spin structure functions, or the measured cross sections to be precise, as well as the elastic form factors and polarizabilities at Q 2 = 0. Due to the large cancellations between σ LT and σ T T , as well as g 1 and F 2 , precise parametrizations of the former are needed, and the uncertainty of the TPE evaluation has to be estimated with great care, taking into account all correlations.Furthermore, due to a lack of data at low-Q, one uses an interpolation from Q 2 = 0 to the onset of data [14].As we showed in Sec.IV C based on LO BχPT, the quality of these approximations is rather poor and is yet another source of uncertainty.New data from the Jefferson Lab "Spin Physics Program" [17][18][19][20][21], including also the substantially extended dataset for g 2 [23], will allow for a re-evaluation of the polarizability effect on the hfs in H and µH.
An accurate theoretical prediction of the 1S hfs in µH is crucial for the future measurement campaigns, since it allows to reduce the search range for the resonance in experiment.Thus, one might find the resonance faster and acquire more statistics during the allocated beam time, see discussion in Ref. [13].The present discrepancy between predictions for the polarizability effect can be mended if the high-precision measurement of the 1S hfs in H is implemented as a constraint.Applying this procedure, good agreement is found between all theory predictions for the total 1S hfs in µH hfs, see Fig. 9.Eventually, after a successful measurement of the 1S hfs in µH, one can combine it with the 1S hfs in H to disentangle the Zemach radius and polarizability effects, leveraging radiative corrections as explained in Ref. [13].The empirical polarizability effect, obtained in this way, can reach a precision of ∼ 40 ppm [13].That is sufficient to discriminate between the presently inconsistent theoretical predictions.for the higher-order corrections of O(α(Zα) 6 ) [#h14].This includes higher-order muon vacuum polarization corrections.Previously included were only the logarithmically enhanced terms [84].The effect on the TPE from eVP corrections to the wave function is given in Ref. [87,Eq. (B3)].For the radius-independent term, we are keeping the error estimate from Ref. [58], which does take into account missing higher-order recoil corrections.
Appendix D: Expansions in terms of polarizabilities In the following, we will present two further low-energy expansions of the polarizability effect in the hfs.Due to the high-energy asymptotics of the TPE contribution to the hfs, these formulas will merely serve illustrative purposes, while their approximation of the full result is rather poor.Up to and including second moments of the structure functions, Eq. ( 10) can be written as [62]: where γ 0 (Q 2 ) and δ LT (Q 2 ) are the forward spin and longitudinal-transverse polarizabilities of the proton, The first term in this expansion, corresponds to the S1 (0, Q 2 ) subtraction term already discussed in Sec.IV C. Analogously to Ref. [27, Eq. ( 12)], we try to find an approximation for the hfs master formula assuming that the photon energy in the atomic system is small compared to all other scales.Thus, we expand the numerator of Eq. (A7) around ν = 0.The resulting approximate formula for the polarizability contribution to the hfs we call Ẽ: For BχPT, it gives: Ẽ(1S, H) = 3.0 peV, (D4a) Ẽ(1S, µH) = 19.0µeV. (D4b) In Fig. 11, we show Eq.(D3) as a running integral with cut-off Q max (gray line), this time for the 2S hfs in µH.In addition, we show the contributions of the longitudinal-transverse (orange line) and helicity-difference (blue line) cross sections to Eq. (D3), and the exact result from Eq. (10).One can easily see that the quality of the approximation is indeed rather poor.

FIG. 1 .
FIG. 1. Two-photon-exchange diagram in forward kinematics: (a) Elastic contribution; (b) polarizability contribution.The horizontal lines correspond to the lepton and the proton (bold), where the 'blob' represents all possible excitations.The crossed diagrams are not drawn.

FIG. 3 .
FIG. 3. Polarizability effect on the 1S hyperfine splitting in H (left panel) and µH (right panel): Cutoff dependence of the leading-order πN -loop contribution.The total results, Eqs.(15a) and (15b), are indicated by the black arrows.

TABLE I .
[13,16,58]ions of the proton Zemach radius RZ, in units of fm.Predictions for the 1S hyperfine splitting in µH[13,16,58], compared to the projected uncertainty of the planned CREMA measurement (red vertical line).