Friday, June 03, 2005

The most accurate theory we have

It's often said that Quantum Electrodynamics is the most accurate theory we have in physics. This is based on a few things, first it agrees with a large number of experiments with good precision. It also reduces to standard classical electromagnetism in the classical limit, and classical E&M is a very well tested theory. Finally, there are a handful of things where the agreement between theory and experiment is truly spectacular. Of these, the crown jewel are the QED predications of the magnetic moments anomaly (MMA) of the muon and the electron.

For example, the best current experimental value for the muon MMA is

amu(exp) = 11 659 208 (6) x 10^{-10}.

First off, it's amazing that this can be measured so precisely. What's even more amazing is that the theory prediction for this quantity is

amu(theory) = 11 659 187(8) x 10^{-10}.

The difference is

2.1(1.0) X 10^{-9}.

Now, it's not zero, and that might be meaningful, but it's far more likely that the theory part has a larger error than suspected. In particular, the dominant part of the error in the theory is the low energy QCD contribution. That's something that can, in principle, bet computed using lattice QCD. But what's more remarkable than the small difference is the agreement. To a very high level of accuracy, the theory agrees with experiment. The same is true for the electron.

Obviously, to produce theoretical predictions like this requires a lot of work. This work started in the forties, when Schwinger computed the first approximation for the MMA. Since then, people have computed, order by order in perturbation theory, this quantity. And no person has done more than Tom Kinoshita to improve the prediction. Tom gave a talk last week on his work (with collaborators) in this field.

At the first order in perturbation theory, the exact solution (\alpha/(2\pi)) is easy to find. It's a standard exercise. As you go to higher and higher orders though, the calculation quickly becomes very hard. The second order calculation was completed a few years after the first, and the third was many years after that. The final results for the fourth order calculation weren't finalized until last year.

Tom is still at it. With the fourth order calculation done, he and his collaborators have moved on to the fifth order electron MMA. A new generation of experiments should make this calculation necessary. For those in the know, the fifth order calculation requires evaluation of around 12000 five loop Feynman diagrams. Currently the only practical way you can do this is to evaluate them numerically. Like my own work, much of this invovles figuring out ways to automate as much of the computation as possible.

This doesn't have much to do with lattice QCD, but it's certainly inspirational to know that theory and experiment agree so well. And it's nice to know that when I'm sweating over a two-loop calculation, somebody else is dealing with five-loops.

2 comments:

Anonymous said...

The reports of Kinoshita in the nineties come as "standard knowledge" included in a table of the Peskin-Schroeder book, comparing the measurement of alpha from a wide range of sources, from Josephson effect to, of course, g-2. By the way, the corresponding papers, or some of them, can be found scanned at KEK.

Anonymous said...

These reports are needed because coincidence from a single source can no be claimed as success. Is it possible to get five digits from random sources? I can not tell, but hep-ph/0503104 noticed that for instance
M_mu/M_Z=0.00115869, a "result" as good as alpha/2 pi.