**Anomalous Magnetic Dipole Moments**

The most precise measurement of α comes from the anomalous magnetic dipole moment, or g−2 ("g minus 2"), of the electron. To make this measurement, two ingredients are needed:

- 1) A precise measurement of the anomalous magnetic dipole moment, and
- 2) A precise theoretical calculation of the anomalous magnetic dipole moment in terms of α.

As of February 2007, the best measurement of the anomalous magnetic dipole moment of the electron was made by Gabrielse et al. using a single electron caught in a Penning trap. The difference between the electron's cyclotron frequency and its spin precession frequency in a magnetic field is proportional to g−2. An extremely high precision measurement of the quantized energies of the cyclotron orbits, or *Landau levels*, of the electron, compared to the quantized energies of the electron's two possible spin orientations, gives a value for the electron's spin g-factor:

- g/2 = 1.001 159 652 180 85 (76),

a precision of better than one part in a trillion. (The digits in parentheses indicate the uncertainty in the last listed digits of the measurement.)

The current state-of-the-art theoretical calculation of the anomalous magnetic dipole moment of the electron includes QED diagrams with up to four loops. Combining this with the experimental measurement of g yields the most precise value of α:

- α−1 = 137.035 999 070 (98),

a precision of better than a part in a billion. This uncertainty is ten times smaller than the nearest rival method involving atom-recoil measurements.

A value of α can also be extracted from the anomalous magnetic dipole moment of the muon. The g-factor of the muon is extracted using the same physical principle as for the electron above – namely, that the difference between the cyclotron frequency and the spin precession frequency in a magnetic field is proportional to g−2. The most precise measurement comes from Brookhaven National Laboratory's muon g−2 experiment, in which polarized muons are stored in a cyclotron and their spin orientation is measured by the direction of their decay electrons. As of February 2007, the current world average muon g-factor measurement is,

- g/2 = 1.001 165 920 8 (6),

a precision of better than one part in a billion. The difference between the g-factors of the muon and the electron is due to their difference in mass. Because of the muon's larger mass, contributions to the theoretical calculation of its anomalous magnetic dipole moment from Standard Model weak interactions and from contributions involving hadrons are important at the current level of precision, whereas these effects are not important for the electron. The muon's anomalous magnetic dipole moment is also sensitive to contributions from new physics beyond the Standard Model, such as supersymmetry. For this reason, the muon's anomalous magnetic moment is normally used as a probe for new physics beyond the Standard Model rather than as a test of QED.

Read more about this topic: Precision Tests Of QED, Measurements of The Fine-structure Constant Using Different Systems, Low-energy Measurements

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