Improved theory of the muonium hyperfine structure.

Terms contributing to the hyperfine structure of the muonium ground state at the level of few tenths of kHz have been evaluated. The α(Zα) radiative correction has been calculated numerically to the precision of 0.02 kHz. Leading ln(Zα) terms of order α(Zα), n = 1, 2, 3, and some relativistic corrections have been evaluated analytically. The theoretical uncertainty is now reduced to 0.17 kHz. At present, however, it is not possible to test QED to this precision because of the 1.34 kHz uncertainty due to the muon mass. PACS numbers: 36.10.Dr, 12.20.Ds, 31.30.Jv, 06.20.Jr Typeset using REVTEX 1 The hyperfine splitting of the muonium ground state is one of very precisely measured quantities [1]: ∆ν(exp) = 4 463 302.88 (16) kHz (0.036 ppm). (1) Currently new experiment is in progress to improve the measurement of ∆ν(exp) and muon mass by a factor of five or more [2]. This is very important for testing the validity of quantum electrodynamics (QED) since ∆ν can be calculated very precisely in QED, being relatively free from the effect of hadronic interaction. The precision of such a test is limited at present by the uncertainty in theoretical calculation, which may exceed 1 kHz. This paper reports our result in which we have reduced this uncertainty by nearly an order of magnitude. As is well known, the bulk of the hyperfine splitting is given by the Fermi formula EF = 16 3 (Zα)2cR∞ me mμ [ 1 + me mμ ] −3 , (2) where Z is the charge of the muon in units of the electron charge, R∞ is the Rydberg constant for infinite nuclear mass, and me and mμ are the electron and muon masses, respectively. Of course Z = 1 for the muon, but it is kept in the formula in order to distinguish the contribution of binding effect (Zα) from that of radiative correction (α). Many correction terms of both α and Zα type have been calculated over 40 years. It is customary to classify them into three types: radiative non-recoil correction, pure recoil correction, and radiative-recoil correction. In addition there is a small weak interaction contribution. Thus one may write ∆ν(theory) = ∆ν(rad) + ∆ν(recoil) + ∆ν(rad-recoil) + ∆ν(weak). (3) Conventionally, the effect of hadronic vacuum polarization is included in ∆ν(rad-recoil). Purely radiative terms of orders α(Zα) and α(Zα) have been known for some time [3]: ∆ν(rad) = (1 + aμ) ( 1 + 3 2 (Zα) + ae + α(Zα)(ln 2− 5 2 ) − 2 3π ln(Zα) [ ln(Zα)− ln 4 + 281 480 ] + α(Zα) π (15.38± 0.29) )