Correction terms radiative-recoil corrections

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 theoretical accuracy of hyperfine splitting in muonium is determined by the still uncalculated terms which include single-logarithmic and nonlogarithmic radiative-recoil corrections of order oP Za) m/M)EF, as well as by the nonlogarithmic contributions of orders Za) m/M)EF and a Za) m/M)EF- We estimate all these unknown corrections to hyperfine splitting in muonium as about 70 Hz. Calculation of all these contributions and reduction of the theoretical uncertainty of the hyperfine splitting in muonium below 10 Hz is the current task of the theory. 

An additional radiative corrections that depends on the finite mass of the nucleus, referred to as a radiative recoil correction , was given in [25] and [26]. The leading terms in the radiative recoil correction are... 

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. 

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. 

All the symbols have their usual meanings. In the non-recoil limit, the motion of the nucleus is neglected and its finite mass enters only as a reduced mass of the electron. The additional terms arising from the dynamical effects of the nucleus, namely the recoil corrections and radiative-recoil corrections, have been omitted from equation 1 and will not be considered here. For more detailed discussions of the theory, see the review by Sapirstein and Yennie [3] and more recently [4,5,6], The expansion in (Za) is now carried out by expressing F and H as power series in (Za) and ln(Za) 2, as shown below in equations 2 and 3, where a is the ratio of the electron mass to its reduced mass. 

Rydberg constant, where c is the speed of light and h the Planck constant. QED corrections for radiative effects e ad are of order a due to the lepton magnetic anomalies, recoil contributions e ec are of order ame/rtin and combined radiative and recoil terms Brad-rec start at The strong interaction adds... 

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. 

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