Quantum Field Theory

Gyromagnetic Ratio and Anomalous Magnetic Moment

The anomalous magnetic moment for the electron and muon illustrates the progress of our understanding in particle physics from classical mechanics to quantum theory, quantum field theory, the Standard Model, and beyond.

A classical electron moving around a nucleus in a circular orbit has an orbital angular momentum, L=mevr, and a magnetic dipole moment, = -evr/2, where e, me, v, and r are the electron´s charge, mass, velocity, and radius, respectively. A classical electron of homogeneous mass and charge density rotating about a symmetry axis has an angular momentum, L=(3/5)meR2, and a magnetic dipole moment, = -(3/10)eR2, where R and are the electron´s classical radius and rotating frequency, respectively. The classical gyromagnetic ratio of an orbiting or a spinning electron is defined as the ratio of the

Figure 07 Classical g-ratio [view large image]

magnetic moment to the angular momentum. In both cases one finds cl = /L= -e/(2me). The minus sign indicates that is in the opposite direction to L (see Figure 07).

In quantum theory, the interaction between an electron and a magnetic field can be portrayed by the Feynman diagram in Figure 08a, which shows a photon from the magnetic field is absorbed by the electron and thus altered its trajectory. The gyromagnetic ratio derived from the Dirac equation takes the form: e = /L= -g e/(2me), where g = 2 is related to the fact
that the spin of the electron is equal to /2. If the vertex correction as shown in Figure 08 is taken into account, then g = 2 ( 1 + /2), where = e2/(4c) ~ 1/137.036 is the fine structure constant giving g - 2 = 0.002322814. The extra term arises from the electron self-interaction, in which it emits and reabsorbs a virtual photon, making a loop in the Feynman diagram as shown in Figure 08b. The same process also applies to the muon.

Figure 08b Vertex Cor-rection [view large image]

The more accurate calculation including higher loop diagrams up to the 4th order term yields the following expression:
g - 2 = 2 ( /2 - 0.328 (/2)2 + 1.181 (/2)3 - 1.510 (/2)4) = 0.0023193042800. The experimental value is: g - 2 = 0.0023193043768 (in agreement with the calculated value to ten significant figures).

In calculating the effects of the cloud of virtual particles, we need to include not just the effects of virtual photons and virtual electron-positron pairs, but also virtual quarks, virtual Higgs particles, and, in fact, all the particles of the Standard Model. It turns out, though, that because of the larger muon mass, any such heavy particles would affects the muon magnetic moment more than the electron magnetic moment. The muon g - 2 has been calculated with the Standard Model to a precision of 0.6 ppm (parts per million). The calculated value with the combined effect is g - 2 = 0.0023318360. A remarkable fact is that the muon g - 2 factor not only can be predicted to high precision, but also measured to equally high precision. The measurements at Brookhaven National Laboratory in 2001/2004 (Figure 09) yields an average value of g - 2 = 0.0023318416. Thus, the comparison of measurement and theory provides a sensitive test of the Standard Model. If there is physics not included in the
current theory, and such new physics is of a nature that will affect the muon's spin, then the measurement would differ from the theory. This is what appears to have been observed, although there are several interpretations of the result that must be considered. One of the missing pieces in the theoretical calculation is the exotic particles predicted by the theory of supersymmetry. Although these particles are rare and unstable their mere existence in the vacuum would modify observable quantities such as the muon magnetic moment.

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