Anomalous electron moment correction

Anomalous electron moment correction Atomic mass unit Avogadro constant Bohr magneton Bohr radius Boltzmann constant Charge-to-mass ratio for electron Compton wavelength of electron... 

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. 

For an elementary proton r )p = 0, g = 2, and only the first term in the square brackets survives. This term leads to the well known local Darwin term in the electron-nuclear effective potential (see, e.g., [1]) and generates the contribution proportional to the factor Sio in (3.4). As was pointed out in [2], in addition to this correction, there exists an additional contribution of the same order produced by the term proportional to the anomalous magnetic moment in (6.6). 

Electron Anomalous Magnetic Moment Contributions (Corrections of Order a Ep)... 

We have omitted here higher order electron-loop contributions as well as the heavy particle loop contributions to the electron anomalous magnetic moment (see, e.g., [11]) because respective corrections to HFS are smaller than 0.001 kHz. Let us note that the electron anomalous magnetic moment contributions to HFS do not introduce any additional uncertainty in the theoretical expression for HFS (see also Table 9.2). 

We used in (9.17) the subtracted electron factor. However, it is easy to see that the one-loop anomalous magnetic moment term in the electron factor generates a correction of order a Za)Ep in the diagrams in Fig, and also should be taken into account. An easy direct calculation of the anomalous magnetic moment contribution leads to the correction... 

This correction is induced by the gauge invariant set of diagrams in Fig. 9.8(d) with the polarization operator insertions in the radiative photon. The two-loop anomalous magnetic moment generates correction of order a Ep to HFS and the respective leading pole term in the infrared asymptotics of the electron factor should be subtracted to avoid infrared divergence and double counting. 

The contribution due to the three-loop slope of the Dirac form factor was the last unknown contribution to the hydrogen energy levels at order a3(Za)4. The two other contributions come from the three-loop electron anomalous magnetic moment and the three-loop vacuum polarization correction to the Coulomb propagator. These contributions can be extracted from the literature [10,13]. 

We have displayed the contributions due to the three-loop slope of the Dirac form factor, the three-loop anomalous magnetic moment of the electron and the three-loop photon vacuum polarization separately. Thanks to the cancellation between these contributions, the correction turns out to be quite small numerically. 

In the case of the recent experiment with hydrogen-like carbon the nontrivial QED effects contribute an observable amount (see Table 1). We need to mention that, due to some delay of the final publications of the experimental result [1] and theoretical calculations [10], no actual theoretical predictions have been published. Most of the presentations (conference and seminar talks and posters) dealt with unaccurate theoretical predictions because it was believed that nothing had been known on the two-loop corrections. However, that was not the case, because from the beginning of the theoretical calculations up to recent re-calculations it was clearly stated ed [6] that the (Za)2 term in Eq. (4) is of pure kinematic origin and so the result is valid in any order of the expansion in a for the anomalous magnetic moment of a free electron, and in particular... 

An interesting problem is the precise calculation and measurement of the Lamb shift 6 which we describe here, commenting on the main points of interest. First, there is a disparity - not yet accounted for - both between the at present most precisely known theoretical values of S, as well as between experiment and theory. Another important point is the opportunity provided to obtain information on the structure and properties of corrections which are not given directly by QED. In contrast to the anomalous magnetic moment, the Lamb shift characterizes the properties of bound electrons, i.e. it takes account of not only the QED effects but the effects arising from the nuclear structure. If the corrections independent of QED are far beyond the error limits of measurements for an anomalous magnetic moment, the corrections... 

In order to complete our derivation of the molecular Hamiltonian we must consider the nuclear Hamiltonian in more detail. A thorough relativistic treatment analogous to that for the electron is not possible within the limitations of quantum mechanics, since nuclei are not Dirac particles and they can have large anomalous magnetic moments. However, the use of quantum electrodynamics [18] shows that we can derive the correct Hamiltonian to order 1 /c2 by taking the non-relativistic Hamiltonian ... 

Taking these corrections into account, the value of the fine-structure constant for which the theoretical and experimental values of the electron anomalous magnetic moment are equal is... 

Ar)H, corrected for the anomalous magnetic moment of the electron, is 1420 53 0 03 Mc/s, a value much closer to the experimental value, and capable, as we shall see, of further refinement by the new theory. 

Corrections to the above formulae are introduced by the new quantum electrodynamics. There is no spectacular splitting of levels since there is no degeneracy, but in so far as AW depends on it is clear that the anomalous magnetic moment of the electron will influence the result. Calculations to order a3, relative to the gross structure, have been made by Karpins and Klein [68]. These authors obtain the result... 

The anomalous contribution to the magnetic moment of an electron has been explained by the quantum electrodynamic theory. The additional contribution—the radiative correction —arises from the interaction of the electron-positron virtual pair emitted and absorbed by the real electron. A theoretical expression in terms of the fine structure constant a is... 

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