External field approximation

Employing this simplification (known as the external field approximation - EFA) in Eq. (4.58) and replacing summation over l with integration over the impact parameters, we obtain... 

We will first discuss corrections to the basic Dirac energy levels which arise in the external field approximation. These are leading relativistic corrections with exact mass dependence and radiative corrections. 

Let us turn now to the discussion of radiative corrections which may be calculated in the external field approximation. 

The external field approximation is clearly inadequate for calculation of the recoil corrections and, in principle, one needs the machinery of the relativistic two-particle equations to deal with such contributions to the energy levels. The first nontrivial recoil corrections are generated by kernels with two-photon exchanges. Naively one might expect that all corrections of order Za) m/M)m are generated only by the two-photon exchanges in Fig. 4.1. However, the situation is more complicated. More detailed consideration shows that the two-photon kernels are not sufficient and irreducible kernels in Fig. 4.2 with arbitrary number of the exchanged Coulomb pho-... 

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]... 

Relativistic and radiative corrections depend on the electron and muon masses only via the explicit mass factors in the electron and muon magnetic moments, and via the reduced mass factor in the Schrodinger wave function. All such corrections may be calculated in the framework of the external field approximation. 

In the external field approximation the heavy particle magnetic moment factorizes and the relativistic and radiative corrections have the form... 

Fig. 9.2. Skeleton two-photon diagram for HFS in the external field approximation...

In the external field approximation the skeleton integral with the muon polarization insertion coincides with the respective integral for muonium (compare (9.12) and the discussion after this equation) and one easily obtains [33]... 

Hadronic vacuum polarization in the external field approximation for the pointlike proton also was calculated in [33]. Such a calculation may serve only as an order of magnitude estimate since both the external field approximation and the neglect of the proton form factor are not justified in this case, because the scale of the hadron polarization contribution is determined by the same yo-meson mass which determines the scale of the proton form factor. Again a more accurate calculation is feasible but does not seem to be warranted, and only an estimate of the hadronic polarization contribution appears in the literature [7]... 

The book is organized as follows. In the introductory part we briefly discuss the main theoretical approaches to the physics of weakly bound two-particle systems. A detailed discussion then follows of the nuclear spin independent corrections to the energy levels. First, we discuss corrections which can be calculated in the external field approximation. Second, we turn to the essentially two-particle recoil and radiative-recoil corrections. Consideration of the spin-independent corrections is completed with discussion of the nuclear size and structure contributions. A special section is devoted to the spin-independent... 

First we consider the external-field approximation. For a point-like nucleus, they can be compactly parameterized by the equation... 

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