Hyperfine Splitting in Muonium

Being a purely electrodynamic bound state, muonium is the best system for comparison between the hyperfine splitting theory and experiment. Unlike the case of hydrogen the theory of hyperfine splitting in muonium is free from uncertainties generated by the hadronic nature of the proton, and is thus much more precise. The scale of hyperfine splitting is determined by the Fermi energy in (8.2)... 

The theoretical accuracy of hyperfine splitting in muonium is determined by the still uncalculated terms which include single-logarithmic and nonlogarithmic radiative-recoil corrections of order oP Za) m/M)EF, as well as by the nonlogarithmic contributions of orders Za) m/M)EF and a Za) m/M)EF- We estimate all these unknown corrections to hyperfine splitting in muonium as about 70 Hz. Calculation of all these contributions and reduction of the theoretical uncertainty of the hyperfine splitting in muonium below 10 Hz is the current task of the theory. 

The weak interaction contribution to hyperfine splitting in hydrogen is easily obtained by generalization of the muonium result in (10.38)... 

As in the case of the Lamb shift, QED provides the framework for systematic calculation of numerous corrections to the Fermi formula for hyperfine splitting (see the scheme of muonium energy levels in Fig. 8.3). We again... 

Here we call the heavy particle the muon, having in mind that the precise theory of hyperfine splitting finds its main application in comparison with the highly precise experimental data on muonium hyperfine sphtting. However, the theory of nonrecoil corrections is valid for any hydrogenhke atom. 

The agreement between theory and experiment is excellent. However, the error bars of the theoretical value are apparently about an order of magnitude larger than respective error bars of the experimental result. This is a deceptive impression. The error of the theoretical prediction in (12.31) is dominated by the experimental error of the value of the electron-muon mass ratio. As a result of the latest experiment [83] this error was reduced threefold but it is still by far the largest source of error in the theoretical value for the muonium hyperfine splitting. 

Another reason to improve the HFS theory is provided by the perspective of reducing the experimental uncertainty of hyperfine splitting below the weak interaction contribution in (10.38). In such a case, muonium could become the first atom where a shift of atomic energy levels due to weak interaction would be observed [85]. 

Fig. 5. The /rSR spectra from fused quartz at room temperature and silicon at 77 K, each in a magnetic field of 10 mT. For quartz, the two high-frequency lines result from muonium with a hyperfine parameter close to that in vacuum. The two high-frequency lines in Si result from Mu, and their larger splitting arises because the hyperfine parameter is less than the vacuum value (0.45 Afree). The lowest line in each sample comes from muons in diamagnetic environments. The lines from 40 to 50 MHz in Si arise from Mu. From Brewer et al. (1973).

Figure 2 Breit-Rabi diagram for muonium with transverse-field transitions indicated by arrows. The zero-field splitting corresponds to the muon hyperfine coupling constant (/A = 4463 MHz for Mu). In high fields, the four eigenstates become pure Zeeman states ( 1 >= a a >, 2 >= P a >, 3 > = P P >, 4>= a P .

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