Correction terms higher-order radiative corrections

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. 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

Predictions of the values of and 34 from standard theory - dominantly the QED terms - requires values for many atomic constants including m, and Av as well as the calculation of higher order QED radiative corrections. The... 

Higher order terms in the perturbation series lead to more complex expressions which have been analyzed in detail, in particular for highly ionized atoms. Terms appear which reproduce the expressions of nonrelativistic many-body perturbation theory (MBPT) [66,70,71] together with further radiative corrections. Generally speaking, radiative correction terms, even in second order, include contributions from positron states. Apart from hinting how such higher order terms can be included in the theory, we shall not need to discuss them in this chapter. 

Table 4.1 summarizes the various contributions to the energy, expressed as a double expansion in powers of a 1/137.036 and the electron reduced mass ratio ji./M 10 . Since all the lower-order terms can now be calculated to very high accuracy, including the QED terms of order Ry, the dominant source of uncertainty comes from the QED corrections of order Ry or higher. The comparison between theory and experiment is therefore sensitive to these terms. For the isotope shift, the QED terms independent of /x/M cancel out, and so it is only the radiative recoil terms of order a fx/M 10 Ry ( 10 kHz) that contribute to the uncertainty. Since this is much less than the finite nuclear size correction of about 1 MHz, the comparison between theory and experiment clearly provides a means to determine the nuclear size. 

---