Strong force interactions are complex. This new trick makes them easier to understand

Humans intuitively understand gravity. The closer we get to massive body, the more we feel the gravitational pull. The further away we get from that mass, the less grip it has.

But at the heart of an atom, where the “strong force” shapes the behaviour of quarks and gluons, something counterintuitive happens.

The closer quarks and gluons are to one another, the more freely they move about. Yet if they try to move further apart from each other, the force gets dramatically stronger. 

In that sense, it is the opposite of the way gravity works. And it’s the reason your table, you chair, your computer — and you! — don’t fly apart. Despite the positive charges in the nuclei of atoms, these particles remain tightly bound together. 

It was clear in the early 20th century that the atom consisted of electrons orbiting a positively charged nucleus, and that there must be a strong force holding the nucleus together and overcome the electromagnetic repulsion. 

But it wasn’t until the deep inelastic scattering experiments at the Stanford Linear Accelerator (SLAC) in the late 1960s and early 1970s that scientists had the first experimental evidence that protons are made of smaller, point-like constituents (quarks and gluons, collectively called partons).  As crucially, these experiments showed that particles in nuclei can move freely around when they are close to each other yet become tightly bound if they move away from one another.

But still, the question was: Why?

Our understanding deepened in the 1970s, when scientists David Gross, Frank Wilczek, and independently David Politzer, discovered how “asymptotic freedom” works in the theory of the strong interactions. They described clearly how the strength of the strong force changes with distance or energy (what physicists call scale dependence), a discovery that won them the Nobel Prize in 2004.

Coming out of their equations was the (mathematically) famous 11/3 factor in the one loop beta function. It comes from detailed calculations of the quantum fluctuations involving gluon loops in the nucleus. It is at the heart of the modern day version of quantum chromodynamics, our current best description of the strong force in the nuclei of atoms.

A new mathematical tool

Nevertheless, the calculations for interacting gluons and quarks are notoriously difficult, especially when using tools like Feynman diagrams. 

But recently, two mathematical physics researchers at Perimeter Institute — faculty member and Krembil William Rowan Hamilton Chair Kevin Costello, and postdoctoral researcher Roland Bittleston —developed a new tool for doing this, using an “index theorem on twistor space.” 

Twistor theory is a mathematical framework which recasts  complicated physical fields on a space-time in terms of simple algebraic data. Meanwhile, index theory bridges analysis and topology. It can answer analytic questions, for example, ‘how many solutions does a certain differential equation have?’ in terms of the topology, or shape, of a space. 

Costello says twister theory provides a way of taking complicated equations and making them simpler. Bittleston adds that scale symmetries do not usually have an interesting topology, but on twistor space scale symmetry wraps into a circle, which does have an interesting topology.

The new mathematical tool that Costello and Bittleston have devised makes the process of understanding what is happening in the nuclei of atoms computationally easier. The paper on their approach, One-Loop QCD 𝛽 Function as an Index, recently received an editor’s highlight in Physical Review Letters (PRL).

Their method is based on a “very old” theorem and could have been discovered much sooner had people looked, Costello and Bittleston say. In fact, even they were not specifically looking for it. Their discovery happened quite accidentally when they were looking at something else.

“For a while now, we have been interested in some special hidden symmetries — much, much deeper, and harder to see symmetries, and how those might be violated at the quantum level. Then, sort of by accident, as we were doing those kinds of computations, this famous 11/3 formula popped out,” Bittleston says.

“At first, it was confusing,” he adds. “We didn’t know how and why this was appearing. “It was clear that there should be some explanation, but it took us a little while to figure out the right way of thinking about it.”

Evolution from an old theorem

Their new tool is the latest development in an old chapter of mathematical history that starts with the Grothendieck–Riemann–Roch Theorem, developed in the 1950s by Alexander Grothendieck, and based on the previous work of 19th century mathematicians Bernhard Riemann and Gustav Roch. 

The theorem tells you how geometric quantities that count solutions, dimensions, or “degrees of freedom,” can change when you map one space into another. 

Grothendieck is a fascinating character in the history of mathematics. He was a German-born French mathematician who was deeply into abstract mathematics and wasn’t that interested in theoretical physics at all. 

But he pushed mathematics to a whole new level, because instead of just studying objects, he studied the relationships between spaces. That ultimately does have utility for theoretical physicists who are studying how the fragments of spacetime that make up reality come together. 

But then Grothendieck just walked away from academia entirely. In 1971, as he advanced the Grothendieck–Riemann–Roch Theorem, he left a satirical note titled Hexenküche (German for "Witches Kitchen") featuring doodles of little devils with pitchforks, exclaiming: “The diagram is commutative!”

He felt the Grothendieck-Riemann-Roch theorem was a prime example of how pure mathematics had become a kind of “delirium.”  He wanted his mathematics to unfold naturally and was frustrated that his proof required hundreds of pages of explanation and didn’t reveal the hidden structures he so deeply craved. 

He decided that he would be better off focusing on political activism and caring about the world’s ills. The translation of his note says in part: “A gripping example of how our thirst for knowledge and discovery indulges itself more and more in a logical delirium far removed from life, while life itself is going to Hell in a thousand ways and is under the threat of final extermination. High time to change our course!” 

It seems Grothendieck despaired too soon, though, as it turns out his theorem left the next generation of mathematical physicists, like Costello and Bittleston, a mathematical gift. A gift that provides an easier way to figure out what’s happening with the fluctuating interactions that create the strong force.

Adding the instanton

Costello says the mathematical trick they are using is related to the “instanton,” a knotted gluon field introduced by physicists Alexander Polyakov and collaborators around 1970 to describe quantum tunnelling in quantum chromodynamics, where the vacuum is filled with complex quantum fluctuations. 

Initially it was hoped that instantons could explain the confinement of quarks at low energies, but to date this dream has not been realized.  Nevertheless instantons play an important role in our understanding of the strong force.

Bittleston says the scale dependence of QCD would not normally be visible from twistor space, but it became detectable indirectly in the presence of an instanton.

While their technique is useful for understanding the strong force, Costello and Bittleston say it might also be useful in conformal field theories that can be used to study phase transitions and quantum gravity, for example.

Costello and Bittleston are thrilled with their “accidental” discovery. Their technique was mostly available to theoretical physicists in the 1970s, had anyone looked. But they are happy that they have stumbled into it now, and that the pure abstract mathematics that Grothendieck longed for can be used to “predict something fundamental about the real world” after all.