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Brian Hayes


If an electron can get away with spontaneously tossing around a virtual photon, what's to stop it from juggling two or three of them? Nothing at all: A Feynman diagram decorated with a single photon loop can just as well be festooned with two loops. Furthermore, it turns out there are seven distinct two-loop diagrams (see Figure 3).

Figure 3. Two-loop Feynman diagramsClick to Enlarge Image

Drawing the seven two-loop Feynman diagrams is actually the easy part of understanding their effect; the hard part is calculating each diagram's contribution to the value of g. The mathematical expression associated with a diagram takes the form of an integral, summing up the amplitudes of an infinite family of particle paths. Some of the two-loop integrals are complicated, and early attempts to evaluate them went astray; the task was not completed until 1957. The result is again expressed in terms of α, but—reflecting the much lower probability of a two-loop event—the α term is now squared. And it is multiplied by a curious coefficient that combines various rational fractions, logarithms and the Riemann zeta function—this last item being familiar in number theory but an exotic interloper in physics.

Figure 4. Seventy-two three-loopClick to Enlarge Image

What comes next is no surprise: If two loops are good, three must be better. However, for an electron-photon event with three loops there are 72 Feynman diagrams, representing integrals of daunting difficulty (see Figure 4). When work on evaluating those integrals got under way in the 1960s, it soon became clear that the methods of pencil-and-paper algebra had reached their limits. In this way Feynman-diagram calculations became a major impetus to the development of computer-algebra systems—programs that can manipulate and simplify symbolic expressions.

Despite such computational power tools, some of the three-loop diagrams resisted analytic solution for 30 years. To fill in the gaps, physicists tried numerical methods of evaluating the integrals—an even more computer-intensive task. A simple example of numerical integration is estimating the area of a geometric figure by randomly throwing darts at it and counting the hits and misses. The same basic idea can be applied to a Feynman integral, but the object being measured is now a complicated volume in a high-dimensional space; this makes the dart-throwing process painfully inefficient. Merely deciding whether or not a dart has hit the target becomes time-consuming. It was not until 1995 that a reliable, high-precision value of the three-loop contribution was published, by Toichiro Kinoshita of Cornell University. He evaluated all 72 diagrams numerically, comparing and combining his results with analytic values that were then known for 67 of the diagrams. A year later the last few diagrams were solved analytically by Stefano Laporta and Ettore Remiddi of the University of Bologna.

The three-loop correction is proportional to 3, which makes its order of magnitude less than one part per million. Even so, to match the precision of the experimental measurement it's necessary to go on to the four-loop diagrams, of which there are 891. Attacking all those intricately tangled diagrams by analytic methods is hopeless for now. Numerical computations have been under way since the early 1980s. A thousandfold increase in the computer time invested in the task has brought a thirtyfold improvement in precision—but the best results still amount to only a few significant digits.

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