The Algorithm That Lets Particle Physicists Count Higher Than 2

Thomas Gehrmann remembers the deluge of mathematical expressions that came cascading down his computer screen one day 20 years ago.

He was trying to calculate the odds that three jets of elementary particles would erupt from two particles smashing together. It was the type of bread-and-butter calculation physicists often do to check whether their theories match the results of experiments. Sharper predictions require lengthier calculations, though, and Gehrmann was going big.

Using the standard method devised more than 70 years ago by Richard Feynman, he had sketched diagrams of hundreds of possible ways the colliding particles might morph and interact before shooting out three jets. Adding up the individual probabilities of those events would give the overall chance of the three-jet outcome.

But Gehrmann needed software just to tally the 35,000 terms in his probability formula. As for computing it? That’s when “you raise the flag of surrender and talk to your colleagues,” he said.

Fortunately for him, one of those colleagues happened to know of a still-unpublished technique for dramatically shortening just this kind of formula. With the new method, Gehrmann saw terms merge together and melt away by the thousands. In the 19 computable expressions that remained, he glimpsed the future of particle physics.

Today the reduction procedure, known as the Laporta algorithm, has become the main tool for generating precise predictions about particle behavior. “It’s ubiquitous,” said Matt von Hippel, a particle physicist at the University of Copenhagen.

While the algorithm has spread across the globe, its inventor, Stefano Laporta, remains obscure. He rarely attends conferences and doesn’t command a legion of researchers. “A lot of people just assumed he was dead,” von Hippel said. On the contrary, Laporta is living in Bologna, Italy, chipping away at the calculation he cares about most, the one that spawned his pioneering method: an ever more precise assessment of how the electron moves through a magnetic field.

One, Two, Many

The challenge in making predictions about the subatomic world is that infinitely many things can happen. Even an electron that’s just minding its own business can spontaneously emit and then reclaim a photon. And that photon can conjure up additional fleeting particles in the interim. All these busybodies interfere slightly with the electron’s affairs.

In Feynman’s calculation scheme, particles that exist 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. Feynman worked out how to translate these diagrams into mathematical expressions, where loops become summing functions known as Feynman integrals. More likely events are those with fewer loops. But physicists must consider rarer, loopier possibilities when making the kinds of precise predictions that can be tested in experiments; only then can they spot subtle signs of novel elementary particles that may be missing from their calculations. And with more loops come exponentially more integrals.

Illustration: Quanta Magazine