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Brian Hayes

The Muon's Story

The electron and the muon are twins (or triplets, since there is a third sibling called the tau). The only apparent difference between them is mass, the muon being 200 times as heavy. But mass matters mightily in the calculation of g. Because certain effects are proportional to the square of the mass, they are enhanced 40,000 times in the muon. As a result, the muon g factor depends not just on electromagnetic interactions but also on manifestations of the weak and the strong nuclear forces. The virtual particles that appear in muon Feynman diagrams include the usual photons and electrons and also heavier objects such as the W and Z (quanta of the weak force) and the strongly interacting particles known collectively as hadrons.

A theoretical framework called the Standard Model extends the ideas of QED to the strong and weak forces. Unfortunately, however, the theory does not always allow high-precision calculations from first principles in the way QED does. The strong-force contributions have to be computed on a more empirical basis; in effect, even the theoretical value of the muon g factor is based in part on experimental findings.

The muon g factor has attracted much attention lately because the theoretical and experimental values seem to be diverging. The latest measurements from the E821 group differ from accepted theoretical values by roughly two standard deviations. Physicists have not been reticent about speculating on the possible meaning of this discrepancy, suggesting it could be our first glimpse of physics beyond the Standard Model. Perhaps the muon is not truly an elementary particle but has some kind of substructure? Another popular notion is supersymmetry, which predicts that all particles have shadowy "superpartners," with names such as selectrons, smuons and photinos.

One of these adventurous interpretations of the muon results could well turn out to be true. On the other hand, it seems prudent to keep in mind that the g-factor experiments and calculations are fearfully difficult, and it's always possible an error has crept in somewhere along the way. It would not be the first time. Feynman, in his book QED: The Strange Theory of Light and Matter, tells the story of an early computation of the two-loop electron g factor:

It took two ‘independent' groups of physicists two years to calculate this next term, and then another year to find out there was a mistake—experimenters had measured the value to be slightly different, and it looked for a while that the theory didn't agree with experiment for the first time, but no: it was a mistake in arithmetic. How could two groups make the same mistake? It turns out that near the end of the calculation the two groups compared notes and ironed out the differences between their calculations, so they were not really independent.

The story has been re-enacted more recently. In the mid-1990s two groups independently calculated a small, troublesome contribution to the muon g factor called hadronic light-by-light scattering. Kinoshita's group and a European collaboration of Johan Bijnens, Elisabetta Pallante and Joaquím Prades got compatible results. Then, six years later, Marc Knecht and Andreas Nyffeler recalculated the effect by another method and came up with an answer that was the same in magnitude but opposite in sign. The other groups rechecked their work, and both found they had made essentially the same mistake entering formulas into a computer-algebra program. The correction slightly diminished the disagreement between theory and experiment.

In mentioning such incidents, my aim is certainly not to embarrass the participants. They are working far out on the frontier of computational science, where no maps or signposts show the way. But for that very reason a certain amount of caution is in order when evaluating the results. 

A definitive understanding of the muon g factor will have to await further refinements of both the experimental and the theoretical values. Incremental improvements can be expected soon, but major advances may be some time in coming. On the experimental side, the E821 project has been shut down by the Department of Energy, at least for the time being. As for theory, the next major stage will require serious attention to the five-loop Feynman diagrams. There are 12,672 of those. Don't hold your breath.

© Brian Hayes

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