Magnetic moment, anomalous

We may combine an electric field E(x) and a magnetic field B x) into the following potential 

The last expression gives the potential matrix in the standard representation. The transformation law (89) gives precisely the Poincare transformation of the electromagnetic field strengths E and B, which can be combined into a tensor field on Minkowski space. 

Physically, the potential matrix shown here describes a particle with an additional anomalous magnetic moment. The magnitude of the anomalous moment is given by the coupling constant in units of the Bohr-magneton le /2mc. 

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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... 

Barefield, Busch and Nelson (1968) Iron, cobalt and nickel complexes having anomalous magnetic moments [202]. 

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 muon mass is only slightly lower than the pion mass, and we should expect that insertion of hadronic vacuum polarization in the radiative photon in Fig. 7.13 will give a contribution to the anomalous magnetic moment comparable with the contribution induced by insertion of the muon vacuum polarization. 

Respective corrections are written via the slope of the Dirac form factor and the anomalous magnetic moment exactly as in Subsect. 7.3.4. The only difference is that the contributions to the form factors are produced by the hadronic vacuum polarization. 

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 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 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... 

The recoil part of the proton size correction of order Za)Ep was first considered in [9, 10]. In these works existence of the nontrivial nuclear form factors was ignored and the proton was considered as a heavy particle without nontrivial momentum dependent form factors but with an anomalous magnetic moment. The result of such a calculation is most conveniently written in terms of the elementary proton Fermi energy Ep which does not include the contribution of the proton anomalous magnetic moment (compare (10.2) in the muonium case). Calculation of this correction coincides almost exactly... 

The ultraviolet divergence is generated by the diagrams with insertions of two anomalous magnetic moments in the heavy particle line. This should be expected since quantum electrodynamics of elementary particles with nonvanishing anomalous magnetic moments is nonrenormalizable. 

Really the original works [9, 10] contain just the elementary proton ultraviolet divergent result in (11.12) which turns into the ultraviolet finite muonium result in (10.5) if the anomalous magnetic moment k is equal zero. 

For a vanishing anomalous magnetic moment of the heavy particle (k = 0) this correction turns into the muonium result in (10.6) and (10.7). 

The theoretical error of the sum of all nonrecoil contributions is about 1 Hz, at least an order of magnitude smaller than the uncertainty introduced by the proton anomalous magnetic moment k, and we did not write it explicitly in (12.23). In relative units this theoretical error is about 2 x 10 °, to be compared with the estimate of the same error 1.2 x 10 made in [67]. Reduction of the theoretical error by three orders of magnitude emphasizes the progress achieved in calculations of nonrecoil corrections during the last years. 

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