# Particle Physicists Puzzle Over a New Duality

## Introduction

Last year, the particle physicist Lance Dixon was preparing a lecture when he noticed a striking similarity between two formulas that he planned to include in his slides.

The formulas, called scattering amplitudes, give the probabilities of possible outcomes of particle collisions. One of the scattering amplitudes represented the probability of two gluon particles colliding and producing four gluons; the other gave the probability of two gluons colliding to produce a gluon and a Higgs particle.

“I was getting a little confused because they looked kind of similar,” said Dixon, who is a professor at Stanford University, “and then I realized that the numbers were basically the same — it’s just that the [order] had gotten reversed.”

He shared his observation with his collaborators over Zoom. Knowing of no reason the two scattering amplitudes should correspond, the group thought perhaps it was a coincidence. They started calculating the two amplitudes at progressively higher levels of precision (the greater the precision, the more terms they had to compare). By the end of the call, having calculated thousands of terms that kept agreeing, the physicists were pretty certain they were dealing with a new duality — a hidden connection between two different phenomena that couldn’t be explained by our current understanding of physics.

Now, the antipodal duality, as the researchers are calling it, has been confirmed for high-precision calculations involving 93 million terms. While this duality arises in a simplified theory of gluons and other particles that does not quite describe our universe, there are clues that a similar duality might hold in the real world. Researchers hope that investigating the strange finding could help them make new connections between seemingly unrelated aspects of particle physics.

“This is a magnificent discovery because it is totally unexpected,” said Anastasia Volovich, a particle physicist at Brown University, “and there is still no explanation of why it should be true.”

**The DNA of Particle Scattering**

Dixon and his team discovered the antipodal duality by using a special “code” to compute scattering amplitudes more efficiently than they could with traditional methods. Typically, to figure out the probability of two high-energy gluons scattering to produce four lower-energy gluons, for example, you must consider all the possible pathways that might yield this outcome. You know the beginning and the end of the story (two gluons become four), but you also need to know the middle — including all the particles that can temporarily pop in and out of existence, thanks to quantum uncertainty. Traditionally, you must add up the probability of each possible middle event, taking them one at a time.

In 2010, these cumbersome calculations were circumvented by four researchers, including Volovich, who found a shortcut. They realized that many of the complicated expressions in an amplitude calculation could be eliminated by reorganizing everything into a new structure. The six basic elements of the new structure, called “letters,” are variables representing combinations of each particle’s energy and momentum. The six letters make up words, and the words combine to form terms in each scattering amplitude.

Dixon compares this new scheme to the genetic code, in which four chemical building blocks combine to form the genes in a strand of DNA. Like the genetic code, the “DNA of particle scattering,” as he calls it, has rules about which combinations of words are allowed. Some of these rules follow from known physical or mathematical principles, but others seem arbitrary. The only way to discover some of the rules is by looking for hidden patterns in the lengthy calculations.

Once found, these inscrutable rules have helped particle physicists calculate scattering amplitudes at much higher levels of precision than they could achieve with the traditional approach. The restructuring also allowed Dixon and his collaborators to spot the hidden connection between the two seemingly unrelated scattering amplitudes.

**Antipode Map**

At the heart of the duality is the “antipode map.” In geometry, an antipode map takes a point on a sphere and inverts the coordinates, sending you straight through the sphere’s center to a point on the other side. It’s the mathematical equivalent of digging a hole from Chile to China.

In scattering amplitudes, the antipode map that Dixon found is a bit more abstract. It inverts the order of the letters used to calculate the amplitude. Apply this antipode map to all the terms in the scattering amplitude for two gluons becoming four, and (after a simple change of variables) this yields the amplitude for two gluons becoming one gluon plus a Higgs.

In Dixon’s DNA analogy, the duality is like reading a genetic sequence backward and realizing that it encodes a totally new protein unrelated to the one encoded by the original sequence.

“We all used to be convinced that the antipode map was useless. … It didn’t seem to have any physical significance, or to do anything meaningful,” said Matt von Hippel, an amplitude specialist at the Niels Bohr Institute in Copenhagen who wasn’t involved in the research. “And now there’s this totally inexplicable duality using it, which is pretty wild.”

**Not Quite Our World**

There are now two big questions. First, why does the duality exist? And second, will a similar connection be found to hold in the real world?

The 17 known elementary particles that comprise our world abide by a set of equations called the Standard Model of particle physics. According to the Standard Model, two gluons, the massless particles that glue together atomic nuclei, easily interact with each other to double their own number, becoming four gluons. However, to produce one gluon and one Higgs particle, colliding gluons must first morph into a quark and an antiquark; these then transform into a gluon and a Higgs via a different force than the one governing gluons’ mutual interactions.

These two scattering processes are so different, with one involving an entirely different sector of the Standard Model, that a duality between them would be very surprising.

But the antipodal duality is also unexpected even in the simplified model of particle physics that Dixon and his colleagues were studying. Their toy model governs fictional gluons with extra symmetries, which enable more precise calculations of scattering amplitudes. The duality links a scattering process involving these gluons and one that requires an external interaction with particles described by a different theory.

Dixon thinks he has a very tenuous clue about where the duality comes from.

Recall those inexplicable rules found by Volovich and her colleagues that dictate which combinations of words are allowed in a scattering amplitude. Some of the rules seem to arbitrarily restrict which letters can appear next to each other in the two-gluon-to-gluon-plus-Higgs amplitude. But map those rules over to the other side of the duality, and they transform into a set of well-established rules that ensure causality — guaranteeing that the interactions between incoming particles occur before the outgoing particles appear.

For Dixon, this is a tiny hint at a deeper physical connection between the two amplitudes, and a reason to think something similar might hold in the Standard Model. “But it’s pretty weak,” he said. “It’s, like, secondhand information.”

Other dualities between disparate physical phenomena have already been found. The AdS-CFT correspondence, for example, in which a theoretical world without gravity is dual to a world with gravity, has fueled thousands of research papers since its 1997 discovery. But this duality, too, only exists for a gravitational world with a warped geometry unlike that of the actual universe. Still, for many physicists, the fact that multiple dualities almost hold in our world hints that they could be scratching the surface of an all-encompassing theoretical structure in which these surprising connections are manifest. “I think they’re all part of the story,” said Dixon.