We’re solving the fundamental mystery of how reality is glued together

As you read this, every atom in your body is desperately trying to tear itself apart. In fact, that goes for every atom, everywhere, since the beginning of time. Thankfully, those efforts have failed.

These self-destructive tendencies relate to the nucleus, a tiny knot of matter at the centre of every atom. Inside, protons are packed shoulder to shoulder, each one bristling with positive charge and frantic to get away from its companions. If atoms obeyed only electricity and magnetism, the universe would have been a brief, bright firework.

Instead, something else intervenes, a force so strong it makes electromagnetism look feeble. This maintains the solid furniture of reality by keeping the building blocks of atoms glued together.

But the deeper physicists have probed this force, the stranger it has seemed. The equations that describe it look disarmingly simple, yet follow them through and something puzzling happens: a theory built from weightless ingredients somehow produces particles that are unmistakably heavy.

Sweeping away this inconsistency wouldn’t just tidy up our understanding of the force that binds atoms together and cement one of the most successful theories in modern physics. It could also illuminate the mysterious nature of mass in the visible universe and its even more elusive origins.

After more than 20 years of stalled progress, physicists and mathematicians now think they are finally starting to prise the problem open. “It feels like an exciting time,” says Ajay Chandra at Purdue University in Indiana.

The mystery of atomic glue

Atoms are mostly empty space. Yet, at the centre of each one, there lies something extraordinarily dense: the atomic nucleus, made up of protons and neutrons crammed together. That arrangement poses an obvious problem. Protons carry positive charge and so should violently repel one another. Why, then, doesn’t the nucleus simply fly apart?

By the 1930s, physicists suspected the answer must be a new force of nature – one stronger than electromagnetism and capable of keeping the nucleus together despite the protons’ mutual repulsion.

Over the following decades, experiments that smashed particles together began to reveal what was really going on inside atoms. Protons and neutrons turned out to be made of smaller particles called quarks, and something was binding these quarks together. By the early 1950s, physicists were beginning to close in on the nature of this mysterious nuclear glue.

At Brookhaven National Laboratory in Upton, New York state, physicists Chen-Ning Yang and Robert Mills wondered if the mathematics behind electromagnetism and quantum mechanics could be extended to describe it. In 1954, they wrote down a new set of equations.

Those equations implied that the force would be carried by a particle. Later christened the gluon, it would transmit what became known as the strong nuclear force. And like the photon that carries light, it was expected to be massless – at least at first.

Nearly two decades later, experiments at the Stanford Linear Accelerator in California began smashing protons apart. Physicists expected to see quarks tightly bound by this powerful force. Instead, they behaved as if they were almost free. “The big surprise was that the quarks were not trapped inside a proton at all. They were just sort of moving around like they didn’t have a care in the world,” says David Tong at the University of Cambridge. “And yet, somehow, they’re bound inside a proton, where you would imagine that they’re feeling very, very strong forces.”

For a while, this seemed inexplicable. But in 1973, three physicists – Frank Wilczek, David Gross and, independently, David Politzer – showed that this behaviour was exactly what the Yang-Mills equations predicted. At vanishingly small distances deep inside a proton, the strong force weakens, allowing quarks to jiggle about. But zoom out just a little and the opposite happens. Pulling quarks apart causes the force to strengthen dramatically, like a rubber band resisting being stretched.

Crucially, though, this force doesn’t extend far beyond the nucleus, disappearing entirely outside this tiny realm. However, that raises an issue. In quantum physics, short-range forces are usually conveyed by mass-carrying particles, like the W and Z bosons that drive a form of radiation known as the weak nuclear force. Yet Yang and Mills had built their theory from massless ones. Something in the mathematics seemed to be generating mass from nowhere.

The result isn’t just a mathematical artefact; we appear to have seen this mysterious mass. In 2024, experiments at the Beijing Spectrometer III in China presented the strongest evidence yet of the existence of “glueballs”, or particles made entirely of gluons that nevertheless possess mass. Though not definitive, other experiments have pointed to the same conclusion since the 1970s.

So, where does this mass come from? You might have heard that the Higgs boson, discovered at the CERN particle physics laboratory near Geneva, Switzerland, in 2012, gives particles their mass. However, the Higgs mechanism actually accounts for less than 2 per cent of the mass of protons and neutrons. The rest, we think, arises from the restless energy of quarks and gluons interacting inside atoms – behaviour described by the Yang-Mills theory.

But exactly how this happens is a puzzle. The mismatch between the massless ingredients of the equations and the heavy particles that emerge from them is known as the Yang-Mills mass gap. For many physicists, it isn’t an urgent crisis. Yang-Mills theory does an excellent job of describing the behaviour of quarks. But confidence in a theory isn’t the same as proof that its equations truly hold together.

A real proof would require a watertight chain of logic showing how mass emerges from a theory built entirely from massless ingredients. In 2000, the Clay Mathematics Institute in Massachusetts named it as one of seven Millennium Prize Problems – and offered a $1 million reward for a solution. In principle, the challenge is mathematical, rather than physical. But solving it would deepen our understanding of one of nature’s strangest properties: why matter has mass at all.

Physics’ million-dollar question

So why is a proof so difficult to come by? Firstly, Yang-Mills equations are “non-Abelian”. In simple terms, that means the order in which you do things matters. A familiar example comes from geometry: rotate an object, like a picture of a top hat, by 90 degrees and then flip it left to right, and you get a different result from flipping first and then rotating. In physics, this property means gluons can interact strongly with one another, creating a sort of chaotic feedback loop.

Each gluon alters the field that carries the strong force, which, in turn, alters the behaviour of other gluons, which then reshape the field again. Instead of a smooth, linear system, you get one that is intensely self-coupled. Inside a nucleus, where these interactions are strongest, the gluon field fluctuates violently. “At the smallest scales, they become incredibly rough and wildly oscillating,” says Chandra.

That turbulence makes the equations almost impossible to tame with pen-and-paper mathematics. So, physicists have taken a different tack. Rather than treating space-time as perfectly smooth, they chop it into a four-dimensional grid – known as a lattice – and let supercomputers approximate how gluons and quarks behave on each tiny patch. By summing over vast numbers of possible field configurations, they can extract physical quantities from the chaos.

As computing power has grown, this approach has led to calculations that have converged with real-world measurements with impressive accuracy. “There’s no doubt at all that the theory’s right: it agrees beautifully with experiments,” says Tong. “We can now predict the mass of the proton and the neutron from first principles, just by doing numerical computations.”

But herein lies a problem. These approximations don’t amount to a proof – an exact, analytical demonstration that a mass gap emerges from the equations themselves. Without that kind of rigour, it is difficult to know how far the lessons of Yang-Mills theory can be trusted or extended to other areas of physics.

Producing that rigour – the kind that would claim the Clay Mathematics Institute’s million-dollar prize – means confronting the mathematical chaos directly. For 20 years, that has proved beyond reach.

Taming chaos

Stubborn equations tend to attract stubborn mathematicians, though. Few are more persistent or inventive than Martin Hairer at the Swiss Federal Institute of Technology in Lausanne. In 2014, he won the Fields medal, the highest accolade for mathematicians, for his work on a class of equations that most researchers had quietly given up on, known as stochastic differential equations. These describe a system buffeted by randomness – financial markets, the ragged edge of a flicking flame and, crucially, roiling quantum fields.

On paper, the equations we use to describe such systems often dissolve into infinities. Hairer found a way to make sense of them. His breakthrough was to build what he called regularity structures, a mathematical toolkit for handling equations that are too rough for ordinary calculus. Even wildly irregular systems, he showed, could be split into contributions from different length scales, each analysed in isolation before being recombined.

Picture a storm. At the smallest scales are microscopic gusts; at larger scales, rolling swells; larger still, the broad atmospheric patterns that steer the whole. Instead of trying to tame it all at once, Hairer’s technique would involve building a separate mathematical description at each level before stacking the descriptions back together so that the worst local turbulences cancel in a controlled way. Some joke that Hairer’s insights are so far removed from how most mathematicians think that he can only have got them from alien civilisations.

Alien-sourced or not, people have started to run with them. A few years ago, Hao Shen at the University of Wisconsin-Madison told Hairer about some success he had had, using tools they had developed together, in attacking quantum field theories that are simpler than Yang-Mills. That set Hairer wondering if those same tools could be used to tackle the questions about what glues reality together.

It turned out they could. In 2022, Hairer, Shen, Chandra and Ilya Chevyrev at the International School for Advanced Studies in Trieste, Italy, revealed the results of turning the tools to the non-Abelian Yang-Mills equations in two dimensions. They showed that, beneath the apparent chaos, the evolving gluon field can be defined rigorously, its fluctuations controlled and its behaviour at tiny scales made precise. In technical terms, the equations can be “renormalised” and solved, at least for a time, in 2D.

“We were dealing with objects that are much rougher than what you see in calculus, but have some kind of probabilistic structure that gives us a fighting chance to handle the roughness,” says Chandra.

Two years later, the team pushed the analysis into three dimensions. That was no small feat. But it still falls short of the million-dollar challenge, which concerns Yang-Mills in four-dimensional space-time. Here, the terrain shifts dramatically.

“When you go to four dimensions, it’s very different,” says Chandra. He likens solving the 3D Yang-Mills equations to climbing a mostly smooth mountain that has occasional roughness you can grab onto. “You can kind of anchor yourself to the rough objects,” he says. But for 4D, those handholds aren’t there: “You’re just not going to leave the ground.”

Hairer agrees. “Dimension four is really special for these kinds of theories, especially Yang-Mills,” he says. That’s because the 4D Yang-Mills equations are “scale-invariant”, meaning they look essentially the same no matter how closely you zoom in or out. Hairer’s method relies on teasing apart behaviour at different scales before surgically stitching them back together. But if every scale behaves identically, that strategy loses its leverage.

Even so, Hairer and his collaborators have shown that modern mathematical techniques could prise open once-impossible problems. And the fact that a Fields medallist has trained his sights on the Yang-Mills mass gap is giving everyone new hope. “It does make a difference,” says Chandra. “When the strongest people in the field are working on the most important problems in the field, it says that the field is healthy.”

Hairer is cautious about predicting a swift capture of the million-dollar prize. Even if that caution proves justified, his work – and that of others – could still drive progress on other formidable mathematical challenges (see “Two Millennium Prize Problems for the price of one”, below). Others, however, are more optimistic.

Harnessing quantum correlation

“It seems doable,” says statistician Sourav Chatterjee at Stanford University in California. He approaches Yang-Mills from a different angle, via probability. That isn’t as strange as it may sound. Quantum theory is, at heart, already probabilistic. The Schrödinger equation, for instance, doesn’t tell you exactly what will happen when you measure a particle. It tells you the odds of different outcomes.

In the 1960s, physicists began reformulating quantum field theories in probabilistic terms. Instead of thinking of particles as tiny beads flying through space, these theories describe reality as a field – something that exists everywhere at once. “Once you build a stochastic object, a probabilistic object, you can convert it into quantum theory,” says Chatterjee.

Yang-Mills is such a theory. Take the temperature in a room. At every point in space, the air has some temperature. If you could zoom in finely enough, you could assign a number to each location. A quantum field works in a similar way, except the value at each point fluctuates, governed by probabilities. And unlike temperature readings, these quantum probabilities aren’t independent, but “correlated”. If you measure the field in one region, that measurement gives you information about the field a short distance away.

Temperature can be correlated in a loose sense – hot regions, say, sit near other hot regions – but that depends on the specific conditions of the system. In a quantum field, the correlations are more fundamental.

The strength of this connection – how quickly it fades with distance – encodes physical information about our nuclear glue, like mass. If correlations die away slowly, the corresponding gluon is massless; its influence stretches far. If correlations decay exponentially fast, it is massive. “The idea is that the rate of decay will tell you the mass,” says Chatterjee. In other words, by proving that the quantum links of gluons fade quickly, mathematicians could prove that they have mass.

Rather than grappling with smooth space-time from the outset, Chatterjee begins his approach the way many physicists do, by breaking space-time into a grid. Replacing the continuous fabric of space-time with a vast but finite network of points and links turns the Yang-Mills theory into a gigantic probability model, one that mathematicians can rigorously analyse by summing over all the small bits of the grid.

The key question is what happens as the grid becomes finer and finer. Do the sums stabilise instead of blowing up to infinity? Then, if they settle, does the strength of quantum correlations decay quickly, implying the existence of a mass?

In a paper published in 2024, Chatterjee showed that, no matter the number of dimensions, the answer can be yes to both questions. In other words, starting with a lattice that grows increasingly fine can yield something that resembles smooth space-time, without losing the mass. This makes it a promising path for 4D solutions, he says.

Chatterjee’s work isn’t yet the full, physical Yang-Mills theory of quarks and gluons. His result applies to a closely related version known as Yang-Mills-Higgs theory, which includes an additional Higgs field – the field associated with the Higgs boson – and is mathematically more tractable. Even so, it is significant, he says. It gives reason to hope that these probability-based approaches are worth pursuing. “It shows you can pass to the continuum limit and still have the mass,” he says.

These small, slow gains from applying probability theory excite physicist Michael Douglas at Harvard University. In recent years, work using probabilistic methods has made many of the arguments simpler and more elegant, he says. Although the Yang-Mills problem remains challenging, it also looks accessible to present-day mathematics, he says. “Something new has to be discovered, but it’s not mysterious. You know what sort of thing you might need to do.”

So maybe we will soon, finally, find answers to the biggest problems of physics. Solving Yang-Mills in all dimensions of space-time would reveal the true origins of mass. It would put the theory of our nuclear glue on a firm foundation, allowing us to relax about the unsatisfactory numerical solutions that, though useful, have felt like a frustrating, hand-waving avoidance of our most important questions about what holds our universe together.

The many approaches mathematicians have attempted thus far have cleared much of the undergrowth. Since the mass mystery was first posed, we have developed and tested 30 or so methods that may one day pave the road for a full proof, says Douglas. “There’s quite a bit of information for the mathematicians to work with now, and no clearly insurmountable obstacle,” he says. “We might just need method 31.”

The Navier-Stokes problem is one of the Clay Mathematics Institute’s seven Millennium Prize Problems. It concerns equations describing the flow of fluids, from water in a pipe to air swirling in a storm.

These equations are non-linear, meaning the motion of a fluid field feeds back on itself: eddies create other eddies, and small disturbances can cascade across scales. Mathematicians can show that fluid motion remains smooth for short periods of time. The open question is whether, in three dimensions, this smooth behaviour always persists – or whether it can instead “blow up”, with quantities such as the fluid’s velocity becoming infinite.

The puzzle echoes the Yang-Mills mass gap problem, another system in which a field interacts with itself (see main copy). Both sets of equations must grapple with extremely rough fluctuations. For this reason, fluid equations are often used as a “toy model” for ideas that also arise in Yang-Mills theory. A breakthrough in the mathematics of one could inspire progress in the other.