Nuclear Size Corrections of Order Za

Let us consider first the contribution generated only by the elastic intermediate nuclear states. This means that we will treat the nucleus here as a particle which interacts with the photons via a nontrivial experimentally measurable form factor G (A ), i.e. the electromagnetic interaction of our nucleus is nonlocal, but we will temporarily ignore its excited states. 

As usual we start with the skeleton integral contribution in (3.33) corresponding to the two-photon skeleton diagram in Fig. 3.8. Insertion of the factor GE —k ) — 1 in the proton vertex corresponds to the presence of a nontrivial proton form factor . 

We have already considered these corrections together with other radiative-recoil corrections above, in Subsect. 5.1.3. This discussion will be partially reproduced here in order to make the present section self-contained. 

The low momentum integration region in the integral in (6.10) produces a linearly divergent infrared contribution, which simply reflects the presence of the correction of order calculated in Sect. 6.1. We need to subtract 

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Fig. 6.2. Diagrams for elastic nuclear size corrections of order Za) m with one form factor insertion. Empty dot corresponds to factor Gb(—fc ) — 1...

Nuclear size corrections of order (Za) may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form factor and the field of the pointlike charge as a perturbation operator [16, 17]. 

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]... 

The nuclear size correction of order Za) m to P levels from (6.39) gives an additional contribution 4 x 10 meV to the 2Pi — 2Si energy splitting and may safely be neglected. 

Using the dipole form factor one can connect the third Zemach moment with the proton rms radius, and include the nuclear size correction of order Za) m in (7.62) on par with other contributions in (7.58) and (7.74) depending on the proton radius. Then the total dependence of the Lamb shift on Vp acquires the form [3, 25]... 

Fig. 11.2. Elastic nuclear size correction of order Za)Ep with two form factor insertions. Empty dot corresponds either to Gsi—k ) — 1 or Gm(—A )/(l + /t) — 1...

The leading nuclear size correction of order Za) m r )EF may easily be calculated in the framework of nonrelativistic perturbation theory if one takes as one of the perturbation potentials the potential corresponding to the main proton size contribution to the Lamb shift in (6.3). The other perturbation potential is the potential in (9.28) responsible for the main Fermi contribution... 

The logarithmic nuclear size correction of order Za) EF may simply be obtained from the Zemach correction if one takes into account the Dirac correction to the Schrodinger-Coulomb wave function in (3.65) [7]... 

The description of nuclear structure corrections of order Za) m in terms of nuclear size and nuclear polarizability contributions is somewhat artificial. As we have seen above the nuclear size correction of this order depends not on the charge radius of the nucleus but on the third Zemach moment in (6.15). One might expect the inelastic intermediate nuclear states in Fig. 6.4 would... 

Respective corrections to the energy levels in deuterium are even much larger than in hydrogen due to the large radius of the deuteron. The nuclear size contribution of order (Za) to the 2S — IS splitting in deuterium is equal to (we have used in this calculation the value of the deuteron charge radius obtained in [44] from the analysis of all available experimental data)... 

The recoil part of the proton size correction of order Za)Ep was first considered in [9, 10]. In these works existence of the nontrivial nuclear form factors was ignored and the proton was considered as a heavy particle without nontrivial momentum dependent form factors but with an anomalous magnetic moment. The result of such a calculation is most conveniently written in terms of the elementary proton Fermi energy Ep which does not include the contribution of the proton anomalous magnetic moment (compare (10.2) in the muonium case). Calculation of this correction coincides almost exactly... 

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... 

In the Schrodinger-Coulomb approximation the expression in (6.33) reduces to the leading nuclear size correction in (6.3). New results arise if we take into account Dirac corrections to the Schrodinger-Coulomb wave functions of relative order (Za). For the nS states the product of the wave functions in (6.33) has the form (see, e.g, [17])... 

Electron-loop radiative corrections to the leading nuclear finite size contribution in light muonic atoms were considered in [60, 20]. Two diagrams in Fig. 7.15 give contributions of order a Za) m r ). The analytic expression for the first diagram up to a numerical factor coincides with the expression for the mixed electron and muon loops in (7.48), and we obtain... 

At present, contributions from two-photon corrections and finite nuclear size introduce the largest uncertainty in the theoretical expressions for energy levels. Corrections from two virtual photons, of order a2, have been calculated as a power series in Za ... 

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