The standard model of particle physics, which included quantum electrodynamics theory, predicts the relationship between the value of the electron magnetic moment that we measure in Bohr magnetons (called g/2), and the measured value of the fine structure constant, α . The most accurate calculated values for the constants are tabulated in a recent review (of determinations of α). A very small value of the additional standard model correction ahadronic must be included, but it is small enough that this relation between g/2 and α comes essentially from QED theory.
The difference between the g/2 that we measure, and the g(α)/2 value that we calculate using an independently measured α in the QED formula, is very small.. The most up-to-date result, from a review of electron g/2 measurements, is
at the one standard deviation level. The standard model of particle physics, which includes QED theory, thus predicts the electron magnetic moment in Bohr magnetons to a remarkable level of precision.
If the electron is composed of constituent particles bound together by some unknown attraction then we would expect that the standard model formula displayed above would not accurately predict the measured magnetic moment. Antiprotons and protons, for example, are not at all well described by this equation. As is well known, this is because antiprotons and protons are not the point particles with no size that are assumed in deriving the formula. They instead have a measurable size as a consequence of being the spatially extended bound state of three quarks.
The established limit on δa given above sets a limit on the size of the electron (radius R) and upon the rest energy (m*) of the particles out of which the electron is made. A "chirally invariant" model** in which the electron mass (m) is made smaller than the typical nuclear mass by suppressing the lowest order, suggests any remaining difference between g/2 and g(α)/2 would be of second order in the small ratio m/m*. The result
shows that mass of the constituents of the electron, if there are any, must be remarkable large compared to the 0.0005 GeV/c2 rest mass of the electron. The binding energy would need to be spectacularly large. Equivalently the limit can be written as an extremely small limit upon the radius R of the electron. The electron is smaller than the extremely small size that this measurement would be able to detect.
If the measurement accuracy with which we measure the electron magnetic moment was the limit on the electron radius and constituent mass, then we could set a much more stringent limit, namely that m* > 1 TeV/c2. This limit precision will only be attained, of course, if someone figures out how to independently measure the fine structure constant as accurately as we do. Hopefully this will happen over the next years.
These are surprising limits for a measurement done with no large accelerator and carried out at a temperature that is only 100 mK above zero. However, a search for a contact interaction at the LEP storage ring*** probes for electron structure at the 10 TeV energy scale in which case R < 2 x 10-20 m.
The electron is a remarkable particle indeed! The electron's ingredients, if any, must be unbelievably massive, and the electron's incredibly small size still remains undetectable.
**S.J. Brodsky and S.D. Drell, Phys. Rev. D 22, 2236 (1980)
***D. Bourilkov, Phys. Rev. D 64, 071701R (2001)