Recoil correction relativistic

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

Radiative-Recoil corrections are the expansion terms in the expressions for the energy levels which depend simultaneously on the parameters a, m/M and Za. Their calculation requires application of all the heavy artillery of QED, since we have to account both for the purely radiative loops and for the relativistic two-body nature of the bound states. 

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-... 

Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. 

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,... 

The contributions that have been considered in order to obtain precise theoretical expressions for hydrogenic energy levels are as follows the Dirac eigenvalue with reduced mass, relativistic recoil, nuclear polarization, self energy, vacuum polarization, two-photon corrections, three-photon corrections, finite nuclear size, nuclear size correction to self energy and vacuum polarization, radiative-recoil corrections, and nucleus self energy. 

Relativistic Recoil Corrections to the Atomic Energy Levels... 

Abstract. The quantum electrodynamic theory of the nuclear recoil effect in atoms to all orders in aZ and to first order in m/M is considered. The complete aZ-dependence formulas for the relativistic recoil corrections to the atomic energy levels are derived in a simple way. The results of numerical calculations of the recoil effect to all orders in aZ are presented for hydrogenlike and lithiumlike atoms. These results are compared with analytical results obtained to lowest orders in aZ. It is shown that even for hydrogen the numerical calculations to all orders in aZ provide most precise theoretical predictions for the relativistic recoil correction of first order in m/M. 

It is known that to the lowest order in aZ the relativistic recoil correction to the energy levels can be derived from the Breit equation. Such a derivation was made by Breit and Brown in 1948 [1] (see also [2]). They found that the relativistic recoil correction to the lowest order in aZ consists of two terms. The first term... 

First attempts to derive formulas for the relativistic recoil corrections to all orders in aZ were undertaken in [11,12]. As a result of these attempts, only a part of the desired expressions was found in [12] (see Ref. [13] for details). The complete aZ-dependence formula for the relativistic recoil effect in the case of a hydrogenlike atom was derived in [14]. The derivation of [14] was based on using a quasipotential equation in which the heavy particle is put on the mass shell [15,16]. According to [14], the relativistic recoil correction to the energy of a state a is the sum of a lower-order term ALL and a higher-order term A Eh ... 

As was shown in [13], to include the relativistic recoil corrections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron field interacting with the quantized electromagnetic field and with the Coulomb field of the nucleus Vc an additional term. In the Coulomb gauge, this term is given by... 

Table 2. The values of the relativistic recoil correction to hydrogen energy levels beyond the Salpeter contribution, in kHz. The values given in the second and third rows include the (aZ)6m2/M contribution and all the contributions of higher orders in aZ. In the last row the sum of the (otZ f m2/M and (otZ)7 og2 (aZ)m2/M contributions is given...

For low Z, in addition to the corrections considered here, the Coulomb inter-electronic interaction effect on the non-relativistic nuclear recoil correction must be taken into account. It contributes on the level of order (1 /Z)(aZ)2m2/M. 

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