Hyperfine Splitting in Muonium: Accuracy of the Theoretical Prediction

Last twenty years theory of hyperfine splitting in muonium developed without any experimental input. Finally, this situation is changing and a new experiment on measuring hyperfine splitting in muonium is now in progress at J-PARC. The goal of the MuSEUM experiment is to improve by an order of magnitude experimental accuracy of the hyperfine splitting and muon-electron mass ratio. Uncertainty of the theoretical prediction for hyperfine splitting will be crucial for comparison between the forthcoming experimental data and the theory in search of a possible new physics. In the current literature estimates of the error bars of the theoretical prediction differ by a factor of two. We explain the origin of this discrepancy and present an estimate of these error bars.


I. INTRODUCTION
Calculations of hyperfine splitting (HFS) in one-electron atoms have a long and distinguished history starting with the classic works by Fermi [1] and Breit [2]. Modern state of the HFS theory in muonium was reviewed in every detail in [3,4]. Small corrections to HFS calculated after publication of these reviews are collected in [5]. High precision measurements of HFS in muonium for a long time were considered as a test of the high precision QED and a source for precise values of the fine structure constant α and the muon-electron mass ratio m µ /m e . While the role of muonium HFS in determining the fine structure constant was made obsolete by the highly precise α obtained from the measurements of the electron anomalous magnetic moment a e [6] and the recoil frequency of the 133 Cs atoms [7], it remains the best source for the precise value of the muon-electron mass ratio.
After a twenty years lull a new MuSEUM experiment on measuring the muonium HFS and the muon-electron mass ratio is now in progress at J-PARC, see, e.g., [8]. The goal of the experiment is to reduce the experimental uncertainties of the muonium HFS and muon-electron mass ratio by an order of magnitude. As a byproduct the experimental team hopes to obtain limits on possible new physics contributions to muonium HFS. A proper estimate of the uncertainty of the theoretical prediction is critical in comparison between theory and experiment and figuring out the limits on new physics. At the Osaka, December 2018, International workshop on Physics of Muonium and Related Topics it turned out [9] that now for almost twenty years CODATA adjustments of the fundamental physical constants [5,[10][11][12][13] claim that the uncertainty of the theoretical prediction of the HFS splitting in muonium is roughly two times lower than this uncertainty in [3,4] and some other theoretical papers on muonium. The CODATA adjustments of the fundamental physical constants is a highly respected and reliable source, and the two times lower error bars cited in [5,[10][11][12][13] found their way in experimental and theoretical papers on muonium HFS, too numerous to cite them here.
Below we will derive the uncertainty of the current theoretical prediction of the HFS in muonium and slightly improve its estimate in [3,4]. This improvement is made possible by the new theoretical contributions and more accurate values of the fundamental physical constants that were obtained after the reviews [3,4] were published. We trace out the origin of two times lower error bars in [5,[10][11][12][13] and explain why they cannot be used for comparison between theory and experiment.

II. ZEEMAN SPLITTING AND EXPERIMENTAL MEASUREMENTS OF MUO-NIUM HFS
Let us describe schematically how muonium HFS and the muon-electron mass ratio were measured in the up to the present moment most precise LAMPF experiments [14,15]. Measurements were done at nonzero magnetic field and two transition frequencies ν 12 and ν 34 between the Zeeman energy levels were measured. An elementary quantum mechanical calculation leads to the Breit-Rabi formulae for these frequencies (see, e.g., [15,16]) where x = (µ µ − µ e )B/(h∆ ν ) 1 is proportional to the external magnetic field B. This field B is calibrated by measuring the Larmor spin-flip frequency hν p = 2µ p B, where µ p is the proton magnetic moment. We represent all magnetic moments in terms of total magnetic moments and do not write them as products of the respective Bohr magnetons and g-factors as in [5,[10][11][12][13][14][15] to make the formulae more transparent. We can always restore the g-factors that we swallowed in magnetic moments later if we wish. Transition frequencies ν 12 and ν 34 and the spin-flip frequency ν p were measured in the LAMPF experiments [14,15]. All other parameters in Eq. (1) except the hyperfine splitting at zero field ∆ν and the muon magnetic moment µ µ are known with a high accuracy. Then Eq. (1) turns into a system of two equations with two unknowns. Solving these equations we obtain . (2) These ∆ ν and µ µ /µ p are the experimental values of HFS at zero field and of the ratio of muon and proton magnetic moments obtained in the LAMPF experiments [14,15] (we skip here all hard experimental problems). The ratio of the electron and proton magnetic moments is measured with very high accuracy and combining it with the quantum electrodynamics (QED) theory of electron and muon g-factors one can obtain an experimental value of electron-muon mass ratio from the ratio of magnetic moments µ µ /µ p in the standard way, see e.g., [15] and/or [5]. Combining the results of the two LAMPF experiments [14,15] one obtains ∆ν ex HF S (Mu) = 4 463 302 776 (51) Hz, δ = 1.

III. THEORETICAL PREDICTION OF MUONIUM HFS AND ITS UNCERTAINTY
Theoretical QED formula for HFS in muonium has the form where the Fermi frequency is R ∞ is the Rydberg constant, c is the speed of light, Z = 1 is the muon charge in terms of the positron charge, m r = m e m µ /(m e + m µ ) is the reduced mass, function F (α, Zα, m e /m µ ) is a sum of all known QED contributions, ∆ν weak is the weak interaction contribution, and ∆ν th is the estimate of all yet uncalculated terms. Explicit expressions for all terms on the right hand side (RHS) in Eq. (5) are collected in [3][4][5].
To obtain a theoretical prediction for HFS and its uncertainty we plug the values of all constants known independently of this very theoretical formula on the RHS hand side of Eq. (5). Currently the relative uncertainty of the Rydberg constant δR ∞ = 5.9 × 10 −12 [5], and the relative uncertainty of the fine structure constant δ α = 2.3 × 10 −10 [5]. Nothing would change in the discussion below if we would use the relative uncertainty of α obtained from measurements of a e [6] and/or recoil frequency of 133 Cs [7]. The less precisely known constant on the RHS in Eq. (5) is the experimental electron-muon mass ratio from Eq. (4) that respectively introduces the largest contribution to the uncertainty of the theoretical prediction for HFS. We also need to take into account the uncertainty ∆ν th that is due to the uncalculated contributions to the theoretical formula in Eq. (5). The estimate of this uncertainty is relatively subjective, we consider 70 Hz to be a fair estimate [17][18][19]. In [5] uncertainty due to the uncalculated terms is assumed to be 85 Hz. We will use 70 Hz as an estimate of the uncalculated terms, but our conclusions below would not change if we would adopt the estimate from [5]. After simple calculations we obtain the theoretical prediction for the muonium HFS The first uncertainty is due to the uncertainty of (m µ /m e ) ex , the second one is due to the uncalculated theoretical terms, and third is due to the uncertainty of α. This last uncertainty is too small for any practical purposes and can be safely omitted. We see that the uncertainty of the theoretical prediction is dominated by the uncertainty of the experimental mass ratio m e /m µ , and to reduce it one should measure the mass ratio with a higher accuracy. The second largest contribution to the uncertainty is due to the uncalculated terms in the theoretical formula for HFS. Combining uncertainties we obtain We can compare this theoretical prediction for HFS with the result of the experimental measurements [14,15] in Eq. (3). Theory and experiment are compatible but the theoretical error bars are too large due to relatively large experimental uncertainty of the mass ratio (m µ /m e ) ex . In this situation it is reasonable to invert the problem and use the QED theoretical formula for muonium HFS in Eq.
where the first uncertainty is due to the uncertainty of ∆ν ex HF S and the second uncertainty is due to uncalculated terms in ∆ν th HF S (Mu) in Eq. (5). Combining uncertainties we obtain m µ m e = 206.768 281 (4), δ = 2 × 10 −8 .
This value of the mass ratio is compatible but an order of magnitude more accurate than the experimental mass ratio (m µ /m e ) ex in Eq. (4). Hyperfine splitting in muonium is the best source for a precise value of the electron-muon mass ratio. It is not by chance that the uncertainty in Eq. (10) practically coincides with the uncertainty of the mass ratio obtained as a result of the CODATA adjustment [5]. The QED formula in Eq. (5) together with the experimentally measured HFS was used in the adjustment, and since the procedure described above produces by far the most precise value of the mass ratio, the result of the adjustment and its uncertainty should practically coincide with the value of the mass ratio in Eq. (10).

IV. CODATA ESTIMATE OF THE THEORETICAL UNCERTAINTY
The uncertainty of the theoretical prediction for muonium HFS in Eq. (8) is roughly two times larger than the respective uncertainty in 2014 CODATA adjustment of the fundamental physical constants (see eq.(216) in [5]). Identical uncertainty can be found in all 1998-2014 CODATA adjustments [10][11][12][13] and should be explained.
The difference between the uncertainties of the theoretical prediction for muonium HFS in the adjustments and in this work is due exclusively to the estimate of the experimental error of the mass ratio in Eq. (6). It looks like as if the uncertainty of the mass ratio used in adjustments to calculate the muonium HFS and its uncertainty according to Eq. (5) and Eq. (6) is roughly two times lower than the uncertainty of the experimental mass ratio in Eq. (4). Let us figure out how this could happen. The trick is connected with eq.(223) in [5] and similar equations in [10][11][12][13]. This equation (223) in [5] is just another form of Eq. (2) for the ratio of the muon and proton magnetic moments. Let us transform Eq. (2) to the form used in the adjustments. We notice that the product ν 12 ν 34 in the numerator of the RHS in Eq. (2) can be identically written as Substituting this representation in Eq. (2) we obtain To comply with the notation in [5] we introduce f p = 2ν p , ν(f p ) = ν 34 − ν 12 and ∆ν = ν 12 + ν 34 .
In this notation Eq. (12) has the form (unlike in [5] magnetic moments below include all relevant QED corrections, see the discussion after Eq. (1)) and coincides with eq.(223) from the 2014 CODATA adjustment [5]. Let us emphasize that Eq. (13), as well as the equivalent Eq. (2), contains only the experimentally measured frequencies on the RHS. We already used Eq. (2) to obtain the experimental value of the mass ratio in Eq. (4). The symbol ∆ν on the RHS in Eq. (13) is nothing but the sum of two measured frequencies and it coincides with the experimental HFS at zero field in Eq. (2). No QED theory for HFS is used in Eq. (13). As we already explained (see discussion after Eq. (2)) it is easy to convert the LHS of Eq. (13) into the mass ratio. We will assume below that such transformation is already made.
In the adjustments the theoretical QED formula for the muonium HFS from Eq. (5) where the function f is quadratic in the mass ratio and parametrically depends on some other constants, see Eq. (13). One can solve this equation and obtain a theoretical prediction for the mass ratio and its uncertainty based on the theoretical QED formula for HFS from Eq. (5), the Breit-Rabi formula for the Zeeman energy levels and the experimentally measured transition frequencies ν 12 and ν 34 . This is what effectively was done in CODATA adjustments [5,[10][11][12][13]. The theoretical prediction for the mass ratio one obtains in this way has roughly two times lower error bars than the experimental mass ratio in Eq. (4) and is compatible with it. One can consider this comparison as a test of the theoretical formula for HFS splitting that was used to obtain this prediction for the mass ratio. Obviously this is not the best way to obtain the prediction for the mass ratio and test the theoretical QED formula for HFS splitting. As we already discussed much more precise value of the mass ratio may be obtained using the theoretical QED formula for HFS from Eq. (5) and the experimental number for HFS from Eq. (3), as discussed in the end of the previous section. Let us return to the discussion of the uncertainty of the theoretical prediction for muonium HFS in 1998-2014 adjustments [5,[10][11][12][13]. There the solution of Eq. (14) obtained with the help of the theoretical QED formula for HFS is plugged back in this very formula [20] and the obtained result together with its uncertainty is declared to be the theoretical prediction for the muonium HFS and its uncertainty. This is clearly a circular logic, one cannot use a value of a parameter obtained with the help of a theoretical formula in this very formula with the goal to test it. To illustrate this point let us mention that using the same logic one could plug a more precise theoretical prediction for the mass ratio from Eq. (10) obtained with the help of the theoretical QED formula in this very formula and claim that the uncertainty introduced by the mass ratio in the theoretical prediction of HFS is effectively an order of magnitude lower. This obviously makes no sense.

V. CONCLUSIONS
We have shown above that the uncertainty of the current theoretical QED prediction for the muonium HFS is about 523 Hz (relative uncertainty is 1.2 × 10 −7 ), see Eq. (8). By far the largest contribution to this uncertainty is due to the experimental uncertainty of the muonelectron mass ratio in Eq. (4), it exceeds the uncertainty due to the uncalculated terms in the theoretical formula by about a factor of seven. The uncertainty of the theoretical prediction for muonium HFS in Eq. (8) is roughly two times larger than in the 2014 adjustment (see eq.(216) in [5]) and in other 1998-2014 adjustments [10][11][12][13]. All these years the underestimation of the error bars of HFS was not practically important because there were no experimental activity on measuring muonium HFS and the muon-electron mass ratio, and the adjustments produced the value of the muon-electron mass ratio with the correct error bars. Now the situation is rapidly changing. The MuSEUM experiment [8] at J-PARC is going on and its result will be obtained in a not so far future. It is expected that the muonium HFS and the electron-muon mass ratio will be measured with an order of magnitude higher accuracy than in the old experiments [14,15]. One of the goals of the MuSEUM experiment is to compare the theoretical prediction for the muonium HFS with the experimental results in search of new physics. A discrepancy between theory and experiment could be interpreted as a new physics effect. The proper magnitude of the error bars of the theoretical prediction for the muonium HFS is crucial for such comparison. An underestimation of these error bars could lead to an erroneous claim of a new physics discovery. I hope that the discussion above convincingly resolves the discrepancy in the literature on the magnitude of the error bars in the theoretical prediction for the muonium HFS.