Model-independent determination of the two-photon exchange contribution to hyperfine splitting in muonic hydrogen

We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electron-proton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for the muon-proton sector, and consequently of the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen, which reads $\delta \bar E_{p\mu,\rm HF}^{\rm TPE}(nS)=-\frac{1}{n^3}1.161(20)$ meV. Together with the associated QED analysis, we obtain a prediction for the hyperfine splitting in muonic hydrogen that reads $E^{\rm th}_{p\mu,\rm HF}(1S)=182.623(27)$ meV and $E^{\rm th}_{p\mu,\rm HF}(2S)=22.8123(33)$ meV. The error is dominated by the two-photon exchange contribution.


Hyperfine splitting in Hydrogen
The hyperfine splitting of the ground state of hydrogen is one of the most accurate measurements made by mankind [1][2][3][4][5][6][7][8]. One has E exp hyd,HF (1S) = 1420.405751768(1) MHz , (1.1) where we take the average from Ref. [9]. Since then, there has been a continuous effort to derive such number from theory. The first five digits of this number can be reproduced by the theory of an infinitely massive proton (except for the 1/m p prefactor of the Fermi term incorporating the anomalous magnetic moment of the proton) and a non-relativistic lepton, systematically incorporating the relativistic corrections of the latter. A summary of these pure QED computations can be found in Refs. [9][10][11]. Particularly detailed is the account of Ref. [10], which we take for reference. Such computation has reached (partial) O(m e α 8 ) precision and reads (compared to the notation of Ref. [12] we set Z e = 1 and Z = Z p (=1 for numerics)) δE QED hyd,HF (1S) = where m r = m p m e /(m p + m e ). c (p) F ≡ Z + κ p = 2.792847356 and c (e) F = 1.00115965 are the magnetic moments of the proton and electron respectively, which we take exactly, i.e. they include the O(α) corrections. Besides those, there are pure QED recoil corrections of O m e α 6 me mp , computed in Ref. [11] (we take Z = 1 in this expression): On top of these, there are recoil corrections of higher orders, as well as corrections due to the hadronic structure of the proton. It would be helpful to organize such computations/results using effective field theories techniques designed for few-body atomic physics such as NRQED [13] and potential NRQED (see Refs. [14][15][16]). These theories profit from the hierarchy of scales that we have in the problem, which we name in the following way: • m e α 2 : ultrasoft scale.
• m e , m r : hard scale.
By considering ratios of these scales, the main expansion parameters are obtained: The effects produced by the hard, pion and chiral scales are encoded in the Wilson coefficient c pe 4 of the NRQED Lagrangian: (1.5) Note that we have rescaled c pe 4 by α 2 compared with the definition used in Refs. [12,[17][18][19], to which we refer for the contextual discussion.
The study of c pe 4 is one of the main subjects of this work. Like the magnetic moments, it has an expansion in powers of α: with δc pe 4 ∼ 1 + O(α), whereas c pe 4,TPE is strict O(1) (although the number could be large), and has a one-to-one correspondence with the two-photon exchange energy shift of the hyperfine splitting. The coefficient c pe 4,TPE can be written in a compact way (see  is logarithmically dependent on the ultraviolet cutoff scale of NRQED(e)/pNRQED, ν. Such ν-scale dependence cancels with the ν-scale dependence of the computation in pNRQED yielding finite results (see Eq. (1.3)). Similar considerations apply to subleading terms. Also, unlike δc pe,(−1) 4 , δc pe,(0) 4 may receive contributions from the pion and chiral scales, i.e. from the hadronic structure of the proton. They are at present unknown and set the precision with which we can determine c pe 4,TPE , which is of O(α) and so we estimate it to be of the order of 1%. Overall, the structure of δc pe,(0) 4 is the following (Z = 1) (1.14) Note that K pe is scheme dependent. One may also consider splitting this term in the following way: K pe = K pe hard + K pe had , where K pe hard encodes the effects associated to the hard scale exclusively. On the other hand K pe had encodes effects associated to the chiral and higher scales. Both coefficients will, in general, mix logarithmically: K hard ∼ ln ν pion /m e and K had ∼ ln Λ χ /ν pion (for the discussion we do not distinguish between the pion and chiral scale), such that K pe ∼ ln Λ χ /m e . Therefore, all these coefficients are factorizationscale and scheme dependent. c pe 4 is the Wilson coefficient that appears naturally in the computation of the hyperfine splitting: . In other words, the following energy shift associated to c pe 4 (in the following we take Z = 1) where the error comes from an estimate of the uncomputed O(α 8 , α 7 m e /m p ) corrections to hyperfine splitting, as well as from the uncertainty of the measured value of the proton magnetic moment. From this result we determine (a specific part of) the Wilson coefficient of the four fermion operator of the NRQED Lagrangian (a preliminary number was already obtained in Refs. [17,18] (see also [22])), which we will namec pe 4,TPE : 3 It differs from c pe 4,TPE by O(α) effects. Note also that at O(α) c pe 4 is factorization-scale and scheme dependent: c pe 4 → c pe 4,X (ν). The factorization-scale dependence cancels with the logarithmic term explicitly displayed in Eq. (1.18). The scheme dependence of c pe 4,X (ν) cancels with the scheme dependence of δK X hard . Therefore, even if the number we obtain is definition dependent (we choose which part of the hard term is subtracted out), it can be taken as factorization-scale and scheme independent once the definition has been fixed. Related to that, we do not need the explicit expression of δK hard , which depends on the renormalization scheme used for the computation of the potential loops in the effective theory computation, to determinec pe 4,TPE . Note also that the discussion is similar 4 for the definition of the proton radius (see [19], and, for instance, Eq. (2.15) of [12]): where one defines (at O(α)) the proton radius in terms of the factorization-scale and scheme dependent Wilson coefficient c D in the following way: where δc D,X = 0 for X = MS scheme. Finally, if we were specifically interested in c pe 4,TPE , the error is dominated by the unknown coefficient αK pe . Its natural size is α × c pe 4 . Assigning a 1% error yields c pe 4,TPE = −48.7 (5) . (1.20) 3 Note that effects associated to the diagram in Fig. 2, due to the muon vacuum polarization, are included in K pe , producing the correction K pe µ,VP = π 2 Zc (p) F mp mµ [23]. This is natural as they correspond to corrections associated to the muon mass scale. In practice the generated energy shift is small: ∼ 0.27 kHz. 4 For the proton radius the situation is somewhat worse, since one subtracts contributions at scales of the order of the proton mass. e p µ Figure 2. Contribution to K pe due to muon vacuum polarization effects.

Hyperfine splitting in muonic hydrogen
We now discuss the computation of the hyperfine splitting in muonic hydrogen. A summary of results for the 2S hyperfine can already be found in [24] (see also [25]). Here the computation is organized as in Ref. [12] (from which we also borrow the notation and refer for further details) but applied to the hyperfine splitting energy shift. Note also that the expressions below incorporate the muon magnetic moment, c (µ) F = 1.00116592, exactly wherever it appears (incorporating then higher order corrections in α), since it corresponds to the Wilson coefficient that appears in the bilinear muon sector of the effective theory (as the binding energy of muonic hydrogen is much smaller than the muon and electron masses).
The leading order correction reads (note that now where in both cases we pick up the part of V (2,1) and V (2,2) VP that contributes to the hyperfine (see Ref. [12] for notation). Note that for 2S these numbers disagree with Pachucki [26] but agree with Martynenko [27].
Eqs. (2.1), (2.2) and (2.3) produce O(m r α 5 ) energy shifts due to scales below the muon mass. In these results we have kept the exact mass dependence.
Similarly to the hydrogen case, effects associated to the muon mass or higher scales are encoded in c pµ 4 , the Wilson coefficient of the spin-dependent dimension six four-fermion operator. They produce the following correction which starts contributing at O(α 5 ). Following the discussion for hydrogen we write where c pµ 4,TPE is equal to Eq. (1.7) after changing m e → m µ . δc pµ 4 admits an expansion in powers of the muon mass as in the hydrogen case: The strict O(α 5 ) contribution is associated to the two-photon exchange. It reads This completes all the contributions to O(α 5 ). Note that, formally, δE TPE pµ,HF has an extra m µ /m p suppression factor.
At O(α 6 ) the dominant contributions are those obtained in the infinite proton mass limit. For the soft and ultrasoft scale computations, this limit makes the system equivalent to hydrogen but with an active light fermion. The Breit corrections are equal to the hydrogen case. They read (see [10] for instance) For the hard scale (which now we take to be of O(m µ )) there is an important change. The hard and pion scales are approximately equal, and, a priori, they should be computed at once. This means that the separation we made for the case of hydrogen between the hard and the pion scales has to be reanalyzed. Chiral counting suggests that chiral corrections always have a 1/Λ 2 χ suppression factor. Therefore, there is no contribution to δc pµ,(−1) 4 at any order in α associated to those scales, which is then completely determined by the point-like computation. Neglecting electron vacuum polarization effects, one has the same expression as for the hydrogen case (Eq. (1.13)): which produces the following correction to the energy shift: . (2.10) Eqs. (2.8) and (2.10) are all the O(m µ α 6 ) corrections for a hydrogen-like system in the infinite proton mass limit. Nevertheless, electron vacuum polarization effects also produce corrections to the energy at this order. At present the complete set of these corrections is unknown. Therefore, one cannot claim that the complete O(m µ α 6 ) correction to the hyperfine splitting of muonic hydrogen is known. Nevertheless, vacuum polarization effects seem to be suppressed compared with standard relativistic corrections for the same power of α. To test this idea we have performed a partial analysis by evaluating some higher order corrections generated by the electron vacuum polarization. We first take those corrections generated by second-order quantum mechanics perturbation theory of the spin-dependent Breit-Fermi potential with the two-loop correction to the static potential. The result reads  (2.12) We indeed observe that they are suppressed. For the 2S our numbers agree with Borie [28]. We add both results to our final numbers and use their size for the estimate of the error associated to the yet uncalculated other O(m µ α 6 ) electron vacuum polarization corrections. , due to the diagram in Fig. 3 (note that we only consider energy scales of the order of the mass of the muon or higher, lower scales are subtracted, as they are incorporated in the effective theory computation. See also the discussion in Ref. [27]). We have performed a numerical computation of that diagram in the point-like approximation, which does not appear to contribute to δc pµ,(−1) 4 . We have found a very tiny energy shift δE(2S) ∼ 4 × 10 −5 meV, which should be interpreted as a recoil correction. This connects us with the other set of corrections that we consider: The recoil corrections of O(mα 6 mµ mp ). For the ground state the leading correction corresponds to Eq. (1.3) changing the electron by the muon mass. This is the expression we use in Table 1 for entry ix). For the 2S state one can deduce the analogous expression from the 8E hyd,HF (2S) − E hyd,HF (1S) energy shift theoretical expression (see [9]) and Eq. (1.3). One obtains (for Z = 1) Note that, for instance, in Ref. [29] this contribution is obtained rescaling the 1S result by 1/2 3 . This is not correct, as it is a bound state computation, and the dependence on the principal quantum number, n, is more complicated. On the other hand the logarithmic correction, the O(mα 6 mµ mp ln α) term, is the same, i.e. it only has the 1/2 3 factor difference with respect to the 1S result. The difference in the finite piece introduces an error in their computation, which is, nevertheless, small. As this difference is associated to the point-like part of the computation, we may expect that its size is small compared to pure hadronic effects for the same power of α.
Even if Eq. (2.13) has a mµ mp suppression factor, it is also logarithmically enhanced, making this contribution roughly equivalent to the contribution due to Eqs. (2.8) and (2.10) (see Tables 1 and 2).
Finally, we compute the O(α) correction to the energy shift proportional to c pµ 4 produced by 2nd order perturbation theory with V (2.14) This produces a tiny correction. As in the hydrogen case (see Eq. (1.16)), there is some double counting in the above computations, and (in the following we take Z = 1) where the error in the first term is the expected size of the O(α 6 ) uncomputed corrections (either related to the electron vacuum polarization effects or to higher recoil corrections along the lines of item ix) in Table 1). For the 2S we obtain where again the error in the first term is the expected size of the O(α 6 ) uncomputed corrections (either related to the electron vacuum polarization effects or to higher recoil corrections along the lines of item ix) in Table 2).  This number is in good agreement with the number quoted in Ref. [27]. We have checked, for the 1S contributions related to the vacuum polarization, that we agree with the analytic expressions obtained in Ref. [30] but the numerical agreement is not very good.

Determination of c pµ 4,TPE from hyperfine splitting in muonic hydrogen
The possibility of measuring the hyperfine splitting in muonic hydrogen has become a reality in recent years: [31][32][33][34]. Actually an experimental number has been given for the 2S hyperfine splitting in Ref. [32] 6 . In addition, there is an ongoing effort at PSI to determine the hyperfine splitting of the ground state at around the 100 MHz level accuracy [35]. This i)  calls for a determination of c pµ 4 as precise as possible. One possibility is to fix it by direct comparison to the experimental results from muonic hydrogen physics: can be related to a large extent (see the discussion in Refs. [17,18] The first term can be determined from the hyperfine energy shift measurement with high precision. The constants K that appear in the above equation are unknown at present. As for the hydrogen case, they introduce an error of around 1%, which we will add to the error budget. The key point is that the second term within parenthesis can be determined using chiral perturbation theory. As we have already mentioned, c pe 4,TPE and c pµ 4,TPE encode the contribution associated to the diagram shown in Fig. 1 shifting the external lepton (l = e, µ). In this diagram all scales from the mass of the lepton, m l , up to infinity are incorporated, whereas smaller scales are cut off, as those effects are associated to the bound state dynamics which is taken into account in the computation in the pNRQED effective theory. These coefficients keep the complete dependence on m l and are valid both for NRQED(µ) and NRQED(e), i.e. for hydrogen and muonic hydrogen. Their computation can be organized in the following way The other terms (associated to energies of O(m π )) can be computed using chiral effective theories for heavy baryons [36]. Indeed they were computed with logarithmic accuracy: O(ln m 2 π , ln(M ∆ − M p ), ln m l )) in Refs. [17,18]. Even though they diverge logarithmically such divergence cancels in the difference c pµ 4,TPE − c pe 4,TPE .
A chiral computation would allow us to relate c pµ 4 and c pe 4 in a model independent way. Since c pl and compute each of these terms. Actually, such computation was already done in Ref. [22] at leading order in the chiral expansion. Here we further elaborate on that result by considering the leading corrections to that computation and adding an error analysis to the final results 7 . For the point-like contribution we obtain logarithmic term for the estimate of subleading terms setting the factorization scale to the proton mass. We observe a nice convergence pattern. 7 For the numerical values of the coefficients we use those of Ref. [22] except for gA = 1.2723 (23), which we take from Ref. [37], b1,F = 3κp/ √ 2 and gπN∆ = 3/(2 √ 2)gA, which we have fixed to the large Nc prediction.
The polarizability term was computed using the expression of the spin-dependent structure functions obtained in [22,38]. We obtain (note that this term vanishes in the large N c limit, except for the tree-level-like contribution) c pµ 4,pol − c pe 4,pol = 0.17(9) (π), 0.25(10) (π&∆) , where we use the same error analysis as in Ref. [22]. It is possible to compare this number with a determination using dispersion relations [29]. They compute both coefficients, for hydrogen and muonic hydrogen. For the difference they obtain c pµ 4,pol − c pe 4,pol = −0.3(1.4), where we have combined in quadrature the error of the individual coefficients. Note that there is a very strong cancellation between both terms, so small inaccuracies in the parameterization of the experimental data could get amplified, still, within errors, there is perfect agreement. We should also mention that a recent analysis using relativistic baryon chiral perturbation theory has challenged standard values for the polarizability term obtained from dispersion relations [39].
For the Born term we get zero at leading order (in other words, the Zemach correction [40] is the same for the hydrogen and muonic hydrogen). The first nontrivial term in the 1/m p , 1/F π expansion reads (see [26] for instance) where we have approximated the magnetic Sachs form factor to G M (−p 2 ), its one-loop chiral expression [41][42][43]. The integral in Eq. (2.31) is finite. The high energy behavior is cut by the cancellation between the integrands of the muon and electron in such a way that the ultraviolet behavior of the integrand scales as 1/p 2 (note that G  Even if formally subleading this contribution turns out to be more important than the polarizability contribution. There are reasons for that, based on large N c arguments, already discussed in Refs. [17,18] where we take α(K pµ − K pe ) = −αK pe µ,VP as the estimate of this term for the numerical evaluation. The error is the combination in quadrature of the error in Eq. (2.33), the error associated to α(K pµ −K pe ) (which we take of the order of α×(c pµ 4,TPE −c pe 4,TPE ) and K pe µ,VP ), and the error of Eq. (1.18). The last two errors are clearly subdominant compared to the error in Eq. (2.33). Note that Eq. (2.34) has no input from muonic hydrogen data.
Similarly to the two-photon exchange contribution c 4,TPE , the constants K also admit a computation in the chiral theory such that K pµ −K pe = −K pe µ,VP +K pµ χP T −K pe χP T +O( mµ mp ). In other words, the leading term in the chiral expansion can be completely determined from a chiral plus point-like computation without any new counterterm. Therefore, there is room to improve this analysis, were this necessary.
We would like to finish comparing these numbers with other estimates using experimental information of the Sachs form factors and dispersion relations. We summarize the comparison in Table 3 and Fig. 4. Within uncertainties all numbers are consistent. The value of Ref. [29] is slightly lower, even though we get similar values for c pµ 4,TPE − c pe 4,TPE . This can be traced back to the fact that they have a relatively big number (in absolute terms) forc pe 4,TPE . They obtainc pe 4,TPE = −50.3(1.1). This number is slightly more than one sigma away from the number we obtain in Eq. (1.18) from the hydrogen. The value of Ref. [27] uses the number for the polarizability term computed in Ref. [44], and also incorporates the shift mentioned in Ref. [45] (albeit this produces a very small change). [26] [27] [29] Eq.   If we focus strictly on the two-photon exchange contribution we should increase the error, as we expect the size of αK pµ to be of the order of α × c pµ 4 . Therefore we add in quadrature a 1% error to the error in Eq.   Table 3 and the main text.

Conclusions
We have obtained a model-independent expression for the two-photon-exchange contribution to the hyperfine splitting in muonic hydrogen. We have used the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operators of NRQED applied to the electron-proton and to the muon-proton sector. The difference can be reliably computed using chiral perturbation theory: where the error is the linear sum of the errors of the two-photon exchange contribution in Eq. (2.34), and of the perturbative contribution in Eqs. (2.19) and (2.20). A forthcoming PSI experiment is expected to reach a precision of the order of 100 MHz (∼ 0.4 µeV) for the hyperfine splitting of the muonic hydrogen [35]. Such precision may put into a test the universality of the lepton interactions.
Finally, we also give a prediction for the following energy difference: