But that luck ended when physicists tried to tackle the strong force. The strong force is around 100 times stronger than the electromagnetic force, with an alpha analogue of about 1, and it refuses to be ignored. Squaring or cubing 1 doesn’t create any shrinking effect whatsoever, so the perturbative series heads straight toward infinity from the earliest terms. Physicists have spent decades developing an alternative way to handle the strong force using supercomputers, achieving spectacular results along the way. But the numerical calculations don’t give much insight into how the strong force does what it does.

Ünsal and Dunne recognized that resurgence, with its power to tame divergent series, could take them a step toward the dream of understanding the strong force with pencil and paper. In particular, they set out to solve a mystery that had plagued the theory of the strong force for 40 years.

In 1979, the physicists Gerard ’t Hooft and Giorgio Parisi inferred the existence of tiny, bizarre terms in strong force calculations. They called them renormalons, and no one knew what to make of them. Renormalons didn’t seem to correspond to any specific ripple or other concrete field behavior. But there they were, messing up the calculations nonetheless.

Ünsal and Dunne tackled renormalons with resurgence. Even though they were working in a 2D analogue of the strong force, it took them roughly a year. But in 2012, they showed that — at least in their simplified model — ’t Hooft and Parisi’s renormalons matched behaviors that physicists understood.

They “solved the mystery and could find what it was that the renormalons corresponded to,” said Jordan Cotler, a physicist at Harvard University, although he adds that other mysteries still remain. He’s currently mounting a similar attempt to understand some of those mysteries in a more realistic theory of the strong force.

Last year, however, researchers used resurgence to add a further wrinkle. Mariño and his collaborators carried out a more rigorous calculation (although also in a simplified theory) and discovered new renormalons beyond what the group calls “the standard lore” of ’t Hooft and Parisi. Mariño now suspects that renormalons are just the tip of a nonperturbative iceberg. Resurgence and other nonperturbative methods may reveal that physicists have been spoiled by their historical success in matching individual mathematical terms to specific events. If he’s right, the quantum world may someday become even harder to visualize than it already is.

“I have doubts that this picture — one exponential [to] one object — is going to go through in general field theories,” he said. “It might happen that the world of exponential corrections is really wild.”

Mariño has also been a key player in the discovery of a new nonperturbative effect in string theory, the speculative and unproved notion that the universe is not made of pointlike particles but is composed of extended objects such as strings. The wiggling of such strings would determine the properties of the particles we observe.

String theory, like quantum theory, is usually treated as a perturbative series of Feynman-like diagrams representing strings merging and splitting in increasingly complicated ways. But unlike quantum theorists, string theorists lack even the faintest of guides to the theory’s nonperturbative effects. They assume that, just as quantum theory contains tunneling and renormalons, the full nonperturbative formulation of string theory also contains dragons.

One striking example of non-perturbative phenomena in string theory — sheetlike objects known as D-branes — was discovered in the 1990s. D-branes would later spur some of string theory’s biggest developments.

Mariño wondered what else might be out there.

He was part of a group that in 2010 noticed a series of negative counterparts hiding in the shadow of the D-brane terms. It wasn’t clear what physical phenomenon these partner terms might describe.

A clue came six years later, when Cumrun Vafa of Harvard and his collaborators explored a generalized string theory where certain quantities could go negative. They found D-branes with negative tension — the brane version of having negative mass. These exotic beasts warped the structure of reality around them, creating multiple dimensions of time and violating the fundamental principle that probabilities must always add up to 100%. But the group found no indication that these objects should escape from their bizarro world and show up in standard string theory.

Now Ricardo Schiappa, a friend of Mariño’s and a theoretical physicist at the University of Lisbon, believes he’s found evidence otherwise. In recent months, Schiappa and his collaborators  used resurgence to scrutinize a handful of simple string theory models. They found that Vafa’s negative-tension D-branes exactly matched the exponentially small terms that Mariño had found in 2010. Negative D-branes are unavoidable partners of D-branes, the group argued in a January preprint. “What we have discovered now is that they are fundamental for perturbation theory,” Schiappa said.

Other theorists aren’t yet sure what to make of the fresh finding. Vafa notes that Schiappa’s crew did their calculations in stripped-down string models, and that the result isn’t guaranteed to hold in more sophisticated formulations. But if it does, and if string theory actually describes our universe, it must contain some other way of stopping negative D-branes from forming.

“They shouldn’t be there as a regular object in that theory,” Vafa said. Otherwise, “this opens a whole Pandora’s box of puzzles.”

Black Swans and Other Anomalies

Despite their progress in spotting renormalons and negative branes, physicists cite two formidable obstacles to crowning resurgence the official successor to perturbation theory.

First, not all theories have been proved to have resurgent structure. The question is particularly acute for quantum field theories, which physicists have been checking on a case-by-case basis. It’s a painstaking process, a bit like studying mammals one species at a time. After observing humans, dolphins and cats, you might start to feel confident that live birth is a universal mammalian feature. But there’s always the chance that around the next corner you’ll find a platypus laying an egg.

That’s why Serone has devoted the last three years to stress-testing resurgence in certain quantum field theories. In 2021, he and his collaborators studied a theory that shares key features with the strong force but is still simple enough to allow them to calculate the many a’s needed to perform resurgence. They calculated the energy of empty space in such a universe using resurgence and two other methods, showing that all three agreed. There have been qualitative arguments that resurgence should hold in quantum field theory, but this was one of the first concrete calculations, kindling further optimism.

“In most of the cases it has been tested so far, either resurgence works, or we have good reasons to believe we understand when it doesn’t,” Serone said.

The graver problem is that to spot nonperturbative pieces, you need to know a frightening number of perturbative terms. In his recent research, for instance, Serone picked quantum field theories with mathematical backdoors that let him generate thousands of terms. But for the strong force, calculating just eight or nine is currently out of the question. Even pioneers of the method don’t mince their words about when they expect to see it produce a real number like the mass of the proton (a mathematical feat worth a million-dollar prize).

“It’s extremely difficult,” Ünsal said, sighing. “I don’t see an immediate way.”

“What Écalle was saying is that the answer is rigorously there in principle. But to actually get the answer is really, really hard,” Bender said. “My advice would be, don’t stand on one foot while you’re waiting.”

A New Hope

But the daunting difficulty hasn’t killed the dream of trying to get real predictions out of resurgence. For one thing, the technique has already produced otherwise unobtainable results in quantum mechanics. Back in the 1980s, the French mathematical physicists at Saclay used proto-resurgent methods to make an exact prediction for particle tunneling — a problem that physicists had previously only been able to approximate. Dunne and Ünsal have done similar pen-and-paper calculations using the more refined tools of Écalle. Another group has checked these results using standard methods. They were only able to get as far as six decimal places — a Herculean effort that took months of time and substantial computer power.

Such dramatic examples have motivated Dunne to develop hyper-efficient ways of practicing resurgence, in the hopes of someday porting them to quantum field theories. Over the last five years, together with Ovidiu Costin, a mathematician at Ohio State University, he has found techniques that get more bang for the perturbative buck. In some cases (which are still far from the real-world theories), they’ve found that just 10 to 15 terms suffice. “That number could have come out to be 1,000, and I would have given up and gone somewhere else,” he said. “It’s kind of tantalizing.”

Dunne and Costin’s work has even managed to catch the eye of Écalle himself. The founder of resurgence hasn’t closely followed the waves his work set off, calling himself “an accomplished ignoramus in theoretical physics.” Nevertheless, while worrying that any work on speculative models such as string theory may be “built on quicksand,” he praises the researchers’ efforts to give resurgence a mathematical tune-up.

“Even if the physical ground gives way, the impressive math results of, say, O. Costin and G. Dunne are there to stay,” he said.

For Écalle, resurgence is something of a past chapter. Nearly 40 years have passed since his original trilogy. He continued to develop alien calculus until around 2000, and he has spent the last 20 years exploring a more algebraic offshoot. Should he ever decide to publish a sequel trilogy gathering all his findings in one place, who knows what treasures physicists will find within.

“I think he has discovered many tools that are still to be explored,” Mariño said.

Correction: April 7, 2023
Jean Écalle is 76 years old, not 73.