Muonic hydrogen

Discussing light muonic atoms we will often speak about muonic hydrogen but almost ah results below are valid also for another phenomenologically interesting case, namely muonic helium. In the Sections on light muonic atoms, m is the muon mass, M is the proton mass, and rUe is the electron mass. 

The effects connected with the electron vacuum polarization contributions in muonic atoms were first quantitatively discussed in [4]. In electronic hydrogen polarization loops of other leptons and hadrons considered in Subsect. 3.2.5 played a relatively minor role, because they were additionally suppressed by the typical factors (mg/m). In the case of muonic hydrogen we have to deal with the polarization loops of the light electron, which are not suppressed at all. Moreover, characteristic exchange momenta mZa in muonic atoms are not small in comparison with the electron mass rUg, which determines the momentum scale of the polarization insertions m Za)jme 1.5). We see that even in the simplest case the polarization loops cannot be expanded in the exchange momenta, and the radiative corrections in muonic atoms induced by the electron loops should be calculated exactly in the parameter m Za)/me-... 

Numerically, contribution to the 2P — 2S Lamb shift in muonic hydrogen is equal to... 

The uncertainty here is due to the unknown nonlogarithmic terms. Calculation of these nonlogarithmic terms is one of the future tasks in the theory of muonic hydrogen. 

Contributions of order a Za) m in muonic hydrogen generated by the two-loop muon form factors have almost exactly the same form as the respective contributions in the case of electronic hydrogen. The only new feature is connected with the contribution to the muon form factors generated bj insertion of one-loop electron polarization in the radiative photon in Fig. 7.9. Respective insertion of the muon polarization in the electron form factors in electronic hydrogen is suppressed as (mg/m), but insertion of a light loop in the muon case is logarithmically enhanced. 

In the case of muonic hydrogen rur in (3.30) is the muon-proton reduced mass. 

All corrections to the energy levels obtained above in the case of ordinary hydrogen and collected in the Tables 3.2, 3.3, 3.7, 3.8, 3.9, 4.1, 5.1 are still valid for muonic hydrogen after an obvious substitution of the muon mass instead of the electron mass in all formulae. These contributions are included in Table 7.1. 

Nuclear size and structure corrections for the electronic hydrogen were considered in Chap. 6 and are collected in Table 7.1. Below we will consider what happens with these corrections in muonic hydrogen. The form of the main proton size contribution of order (Za) m (r ) from (6.3) does not change... 

Having in mind that the data from the muonic hydrogen Lamb shift experiment will be used for measurement of the rms proton charge radius [2] it is useful to write this correction in the form... 

The nuclear size correction of order Za) m in muonic hydrogen in the external field approximation is given by (6.13). Unlike ordinary hydrogen, in muonic hydrogen it makes a difference if we use mj or mmf in this expression (compare footnote after (6.13)). We will use the factor mj as obtained in [53]... 

Nuclear size corrections of order (Za) m to the S levels were calculated in [59, 53] and were discussed above in Subsect. 6.3.2 for electronic hydrogen. Respective formulae may be directly used in the case of muonic hydrogen. Due to the smallness of this correction it is sufficient to consider only the leading logarithmically enhanced contribution to the energy shift from (6.35) [21]... 

Radiative corrections to the leading nuclear finite size contribution were considered in Subsect. 6.4.1. Respective results may be directly used for muonic hydrogen, and numerically we obtain... 

This contribution is dominated by the diagrams with radiative photon insertions in the muon line. As usual in muonic hydrogen a much larger contribution is generated by the electron loop insertions in the external Coulomb... 

Radiative corrections to the nuclear polarizability a(Za) m to S -levels are described by the diagrams in Fig. 7.16 and in Fig. 7.17 (compare with the diagrams in Fig. 6.4). As usual for muonic hydrogen the dominant polarization operator contribution is connected with the electron loops, while heavier loops are additionally suppressed. The contribution of the diagrams in Fig. 7.16 was calculated in [52] on the basis of the experimental data on the proton structure functions... 

Theoretical results described above find applications in numerous high precision experiments with hydrogen, deuterium, helium, muonium, muonic hydrogen, etc. Detailed discussion of all experimental results in comparison with theory would require as much space as the purely theoretical discussion above. We will consider below only some applications of the theory, intended to serve as illustrations, their choice being necessarily somewhat subjective and incomplete (see also detailed discussion of phenomenology in the recent reviews [1, 2]). 

The current surge of interest in muonic hydrogen is mainly inspired by the desire to obtain a new more precise value of the proton charge radius as a result of measurement of the 2P — 25 Lamb shift [64]. As we have seen... 

The natural linewidth of the 2P states in muonic hydrogen and respectively of the 2P — 2S transition is determined by the linewidth of the 2P — IS transition, which is equal hP = 0.077 meV. It is planned [64] to measure 2P — 2S Lamb shift with an accuracy at the level of 10% of the natural linewidth, or with an error about 0.008 meV, which means measuring the 2P — 2S transition with relative error about 4 x 10 . 

We can write the 2P — 25 Lamb shift in muonic hydrogen as a difference of a theoretical number and a term proportional to the proton charge radius squared... 

A. Igarashi, M.P. Faifman, I. Shimamura, Nonadiabatically coupled hyperspherical equations applied to the collisional spin flip of muonic hydrogen, Hyperfine Interac. 138 (2001) 77. 

A. Dupays, B. Lepetit, J.A. Beswick, C. Rizzo, D. Bakalov, Nonzero total-angular-momentum three-body dynamics using hyperspherical elliptic coordinates Application to muon transfer from muonic hydrogen to atomic oxygen and neon, Phys. Rev. A 69 (2004) 062501. 

A. Igarashi, N. Toshima, Application of hyperspherical close-coupling method to antiproton collisions with muonic hydrogen, Eur. Phys. J. D 46 (2008) 425. 

With the recent advances in atomic theories and experimental techniques, the value of the information obtained from studies of atoms that are different from but similar to atomic hydrogen have increased. These studies include atomic helium, muonic hydrogen, positronium, muonium, antihydrogen, moderate Z ions, high Z ions, antiprotonic atoms and muonic atoms. 

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