Electron anomalous magnetic moment

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). 

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]. 

These three examples reflect various aspects of quantum electrodynamics theory. The electron anomalous magnetic moment follows from free-electron QED, the transition frequencies in hydrogen follow from bound-state QED, and, at least in principle, the relevant condensed matter theory follows from the equations of many-body QED. 

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... 

Quantum Mechanics has been the most spectacularly successful theory in the history of science. As is often mentioned the accuracy to which the anomalous magnetic moment of the electron can be calculated is a staggering nine decimal places. Quantum Mechanics has revolutionized the study of radiation and matter since its inception just over one hundred years ago. The impact of the theoiy has been felt in... 

The static anomalous magnetic moment of the electron to order a is, therefore, given by... 

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). 

We also have to consider the electron-loop contribution to the muon anomalous magnetic moment... 

The physical nature of these contributions is quite transparent. They correspond to the anomalous magnetic moment which is hidden in the two-loop electron factor. The true order in Za of these anomalous magnetic moment contributions is lower than their apparent order and they should be subtracted from the electron factor prior to calculation of the contributions to HFS. We... 

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 octahedral iodo or perchlorato compounds of MeNNAsEt have anomalous magnetic moments due to a high-spin low-spin equilibrium. The low-spin CoLI2 (L = NNAs or NNAsEt) have moments of 2.48—3.03 BM, and in conjunction with the electronic spectral data which are inconsistent with tetrahedral or five-co-ordinate geometry, this suggests that the compounds may be rare examples of planar cobalt(n). 

The fine structure constant a can be determined with the help of several methods. The most accurate test of QED involves the anomalous magnetic moment of the electron [40] and provides the most accurate way to determine a value for the fine structure constant. Recent progress in calculations of the helium fine structure has allowed one to expect that the comparison of experiment [23,24] and ongoing theoretical prediction [23] will provide us with a precise value of a. Since the values of the fundamental constants and, in particular, of the fine structure constant, can be reached in a number of different ways it is necessary to compare them. Some experiments can be correlated and the comparison is not trivial. A procedure to find the most precise value is called the adjustment of fundamental constants [39]. A more important target of the adjustment is to check the consistency of different precision experiments and to check if e.g. the bound state QED agrees with the electrical standards and solid state physics. 

The 1998 recommended value of a-1 based on all the available data, but which is primarily influenced by the comparison of theory and experiment for the anomalous magnetic moment of the electron, is... 

For the literature prior to 1990, see T. Kinoshita Theory of the anomalous magnetic moment of the electron — Numerical approach, in Quantum Electrodynamics, ed. by T. Kinoshita (World Scientific, Singapore, 1990), pp. 218-321... 

These seventeen integrals have been computed in the course of the calculation of the anomalous magnetic moment of the electron [10] and we can borrow them from that reference. In the next section we present the structure of the computer program in which this strategy is realized. 

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. 

Table 2. Comparison of theory and experiment. Here, go(e) stands for the magnetic moment of a free electron and it contains the anomalous magnetic moment a a/2n...

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 ... 

In chapter 3 we showed how the relativistic Breit Hamiltonian can be reduced to non-relativistic form by means of a Foldy Wouthysen transformation. We obtained equations (3.244) and (3.245) which represent the non-relativistic Hamiltonian for two particles of masses m, and nij and electrostatic charges and —ej and from this Hamiltonian we were able to derive the interelectronic interactions. We could, however, consider using (3.244) and (3.245) as the Hamiltonian for an electron of charge —e, = — e and mass m, = m, and a nucleus of mass nij = Ma and charge —ej = + 7. e. As before, we make the assumption that the nucleus has spin 1 /2, behaves like a Dirac particle and has an anomalous magnetic moment compared with that given by the Dirac theory. Consequently we may rewrite (3.245) by making the replacements... 

Other electron nuclear interaction terms involving 7ra rather than Ia arise from this treatment. However, these terms have all been dealt with in the previous chapter and we do not repeat them here.) The terms in (4.23) are the same as those obtained previously starting from the Dirac equation. Equation (3.244) will yield both the electron and nuclear Zeeman terms and a Breit equation for two nuclei, reduced to non-relativistic form, would yield the nuclear-nuclear interaction terms. Although many nuclei have spins other than 1/2, and even the proton with spin 1 /2 has an anomalous magnetic moment which does not fit the simple Dirac theory, the approach outlined here is fully endorsed by quantum electrodynamics provided that only terms involving M l are retained (see equation (4.23)). The interested reader is referred to Bethe and Salpeter [11] for further details. In our present application we see that the expressions for both... 

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