Leading Relativistic Corrections with Exact Mass Dependence

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. 

1 Leading Relativistic Corrections with Exact Mass Dependence 

The solution of the problem of the proper mass dependence of the relativistic corrections of order (Za) may be found in the effective Hamiltonian framework. In the center of mass system the nonrelativistic Hamiltonian for a system of two particles with Coulomb interaction has the form 

In a nonrelativistic loosely bound system expansion over (Za) corresponds to the nonrelativistic expansion over Hence, we need an effective 

Eides et al. Theory of Light Hydrogenic Bound States, STMP 222, 19-80 (2007) 

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Relativistic Corrections to the Leading Polarization Contribution with Exact Mass Dependence... 

The mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

The recoil correction in (4.19) is the leading order (Za) relativistic contribution to the energy levels generated by the Braun formula. All other contributions to the energy levels produced by the remaining terms in the Braun formula start at least with the term of order (Za) [17]. The expression in (4.19) exactly reproduces all contributions linear in the mass ratio in (3.5). This is just what should be expected since it is exactly Coulomb and Breit potentials which were taken in account in the construction of the effective Dirac equation which produced (3.5). The exact mass dependence of the terms of order Za) m/M)m and Za) m/M)m is contained in (3.5), and, hence,... 

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