The Algorithm That Lets Particle Physicists Count Higher Than Two

One day 20 years ago, Thomas Gehrmann had a lot of mathematical expressions on his computer screen.

He was trying to figure out the odds of three jets of elementary particles bursting from two particles smashing together. Physicists often do bread-and-butter calculations to check whether their theories match the results of their experiments. Gehrmann was going big, though, because of the longer calculations.

He had sketched diagrams of hundreds of possible ways the colliding particles could interact before shooting out three jets. Adding up the individual probabilities of those events would give a better idea of the outcome.

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Gehrmann needed software to tally the terms in his formula. Is it possible to computing it? He said to raise the flag of surrender and talk to your colleagues.

Fortunately for him, one of his colleagues knew of a still-unpublished technique for shortening this kind of formula. Gehrmann saw terms merging together and disappearing by the thousands. The future of particle physics was glimpsed in the 19 computable expressions that remained.

The Laporta algorithm is the main tool for generating precise predictions about particle behavior. Matt von Hippel is a particle physicist at the University of Copenhagen.

The inventor, Stefano Laporta, remains obscure despite the spread of the algorithm. He doesn't command a lot of researchers and rarely attends conferences. A lot of people assumed he was dead. Laporta is living in Bologna, Italy, which is where he came up with the method for assessing how the electron moves through a magnetic field.

One, two, many.

The challenge in making predictions is that there are many things that can happen. An electron that is just minding its own business can spontaneously emit and return a photon. There can be more particles in the photon. The busybodies interfere with the electron's affairs.

Particles that are before and after an interaction become lines leading in and out of a cartoon sketch, while those that briefly appear and then disappear form loops in the middle. The diagrams were translated into mathematical expressions where loops became summing functions. There are more likely events with fewer loops. Physicists need to look for subtle signs of novel elementary particles that may be missing from their calculations when making the kinds of precise predictions that can be tested in experiments. With more loops comes more integrals.

By the late 1990s theorists had mastered predictions at the one-loop level. The number of possible sequence of events explodes at two loops, the level of precision of Gehrmann's calculation. A quarter century ago, most two-loop calculations were hard to say nothing of three or four. The advanced counting system used by elementary particle theorists is called one, two, many.

They would count higher thanks to Laporta's method.

Using machines to predict events caught the imagination of Stefano Laporta. He was a student at the University of Bologna in the 1980s, and he taught himself to program a TI-58 calculator to forecast eclipses. He learned how theorists used the diagrams to predict how ephemeral particles affect an electron's path through a magnetic field. Laporta said it was a sort of love at first sight.

After a couple of years writing software for the Italian military, he returned to Bologna for his doctorate, and was joined by Remiddi in working on a three-loop calculation of the electron's magnetic moment.

Physicists have known for a long time that they could apply the opposite mathematical function to the integrals to create new equations called identities. They could change the terms into master integrals with the right identities.

You could spend a lifetime searching for the right way to collapse the calculation because of the infinite number of ways of producing identities. The three-loop electron calculation, which was finally published in 1996, was a long time in the making.

When he saw the hundreds of integrals they had started with, he felt the inefficiency of the rules. He changed the calculation. He developed a recipe for zeroing in on the right identities by studying the pattern of which derivatives contributed to the final integrals. He published a description of his strategy in 2001 after years of trial and error.

Physicists built on it. Laporta had a problem that involved 500 million Feynman integrals, and he was pushed by a particle physicist to use a different technique. The number of integrals was reduced by about 1,000. The simplification of terms was made more transparent by using a technique from applied mathematics. Their method is standard.

The man himself continued to plug away at the problem of the electron's magnetic moment, by including all possible four-loop events. After more than a decade of work, Laporta published his masterpiece, the electron's magnetic moment to 1,100 digits of precision. The prediction is in line with recent experiments.

He said it was a liberation. It lifted some weight from my shoulders.

A different path.

If the answer lies in a few master integrals, why must they go through heaps of intermediate Feynman integrals? Is there a straighter path?

The predictions that come out of the diagrams inexplicably feature certain types of numbers. The pattern in the outputs of nave models of quantum theory was spotted by researchers. They were able to find the same pattern in the electron's magnetic moment thanks to Laporta. Researchers are looking for a new way to get master integrals from the diagrams.

Laporta is affiliated with the University of Padua, where he works with a group of researchers trying to make his algorithm obsolete. He hopes that the fruits of their labor will aid his current project.

He said the number of calculations was staggering.