Leading Recoil Correction

The leading recoil correction of order Za m/M)Ep is generated by the graphs with two exchanged photons in Fig. 10.1, similar to the case of the recoil 

Eides et al. Theory of Light Hydrogenic Bound States, STMP 222, 193-215 (2007) 

The subtracted heavy pole (Fermi) contribution is generated by the exchange of a photon with a small (atomic scale mZa) momentum and after subtraction of this contribution only high loop momenta k (m k M) contribute to the integral for the recoil correction. Then the exchange loop momenta are comparable to the virtual momenta determining the anomalous magnetic moment of the muon and there are no reasons to expect that the 

Let us emphasize that, unlike the other cases where we encountered the logarithmic contributions, the result in (10.5) is exact in the sense that this is a complete contribution of order Za m/M)Ep. There are no nonlogarithmic contributions of this order. 

Leading terms logarithmic in Za were first considered in [7], and the complete logarithmic contribution was obtained in [8, 9] 

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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. 

Calculation of the leading recoil corrections of order a Za) becomes now almost trivial. One has to take into account that in our approximation the analogue of the Breit Hamiltonian in (3.3) has the form [20]... 

Recoil contributions in (10.6), and (10.7) are symmetric with respect to masses of the light and heavy particles. As in the case of the leading recoil correction, they were obtained without expansion in the mass ratio, and hence an exact dependence on the mass ratio is known (not just the first term in the expansion over m/M). Let us mention that while for the nonrecoil nonlogarithmic contributions of order Za), both to HFS and the Lamb shift, only numerical results were obtained, the respective recoil contributions are known anal3dically in both cases (compare discussion of the Lamb shift contributions in Subsect. 4.2.3). 

As was noted in [28] this contribution may be obtained without any calculations at all. It is sufficient to realize that with logarithmic accuracy the characteristic momenta in the leading recoil correction in (10.3) are of order M and, in order to account for the leading logarithmic contribution generated by the polarization insertions, it is sufficient to substitute in (10.5) the running value of a at the muon mass instead of the fine structure a. This algebraic operation immediately reproduces the result above. 

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,... 

In the case of states with nonvanishing angular momenta the small distance contributions are effectively suppressed by the vanishing of the wave function at the origin, and the perturbation theory becomes convergent in the nonrelativistic region. Then this nonrelativistic approach leads to an exact result for the recoil correction of order (Zo ) (m/M)m for the P states [30]... 

The leading logarithm squared contribution to the recoil correction of order Zay m/M) was independently obtained with the help of these methods in [32, 31]... 

Let us start systematic discussion of such corrections with the recoil corrections to the leading contribution to the Lamb shift. The most important observation here is that the mass dependence of all corrections of order a." Za.Y obtained above is exact, as was proved in [1, 2], and there is no additional mass dependence beyond the one already present in (3.7)-(3.24). This conclusion resembles the similar conclusion about the exact mass dependence of the contributions to the energy levels of order (Za) m discussed above, and it is valid essentially for the same reason. The high frequency part of these corrections is generated only by the one photon exchanges, for which we know the exact mass dependence, and the only mass scale in the low frequency part, which depends also on multiphoton exchanges, is the reduced mass. 

The radiative-recoil correction to the Lamb shift induced by the polarization insertions in the exchanged photons was also calculated in [9]. The result of that work contradicts the results in [8, 4]. The calculations in [9] are made in the same way as the calculation of the recoil correction of order (Za) (m/M)m in [10], and lead to a wrong result for the same reason. 

This calculation of the leading logarithm squared term [30] (see Fig. 9.11) also produces a recoil correction to the nonrecoil logarithm squared contribution. We will discuss this radiative-recoil correction below in the Subsect. 10.2.11 dealing with other radiative-recoil corrections, and we will consider in this section only the nonrecoil part of the logarithm squared term. 

Leading logarithmic recoil correction, relative order Za Lepage (1977) [8] Bodwin, Yennie (1978) [9] 2 ZaY HZay 11.179... 

Radiative-recoil corrections of order a Za)" m/M)Ep were never calculated completely. As we have mentioned in Subsect. 9.4.1.1 the leading logarithm squared contribution of order a Za) EF may easily be calculated if one takes as one of the perturbation potentials the potential corresponding to the electron electric form factor and as the other the potential responsible for the main Fermi contribution to HFS (see Fig. 10.16). Then one obtains the leading logarithm squared contribution in the form [18]... 

Radiative-recoil corrections of order a Za) m/M)Ep are similar to the radiative corrections to the Zemach contribution, and in principle admit a straightforward calculation in the framework of the skeleton integral approach. Leading logarithmic contributions of this order were considered in [6, 7]. The logarithmic estimate in [7] gives... 

Only the leading logarithm squared contribution to the recoil correction of order Za) m/M) is known now [6, 7]. Numerically the contribution in (4.24) is below 1 kHz. Due to linear dependence of the recoil correction on the electron-nucleus mass ratio, the respective contribution to the hydrogen-deuterium isotope shift (see Subsect. 12.1.7 below) is phenomenologically... 

The radiative-recoil corrections are known only in the leading order [28]... 

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