COMPUTING SCIENCE
g-OLOGY
Brian Hayes
g-Willikers
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).
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.
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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