A step toward defining the Standard Model on a lattice

The Standard Model of particle physics is an extraordinarily successful theory of nature, describing the known elementary particles and three of the four fundamental forces—the strong force, the weak force, and electromagnetism. The success of the theory rests on outstanding agreement between theoretical calculations and experiments across eighteen orders of magnitude, from the low energies of hydrogen’s 21-centimeter radio emission to the high energies of collisions at the Large Hadron Collider. And yet for all of its successes over the more than fifty years of its existence, a foundational problem remains: how to give the full theory a nonperturbative definition, one that does not rely on treating its interactions as small corrections. In an October preprint⁠(opens in a new window) prepared with the aid of GPT‑6 Astra, I proposed a candidate definition on a lattice—a grid representing spacetime.

I had a strategy for tackling this problem, but had not been able to prove that it worked in all the cases it needed to cover. Astra helped turn that strategy into a construction and took the lead in developing the detailed proof. To explain what we did, it helps to start with what is missing from our existing description of the Standard Model.

What does it mean to say that we lack a definition of the Standard Model? Quantum mechanics and special relativity tell us how the particles and forces of the Standard Model interact. Unfortunately, we don’t yet know how to make complete predictions based directly on these interactions. Instead, we often start by imagining that the interactions are turned off, so that electrons would no longer repel each other. We can solve that problem exactly, then put the interactions back in as a series of small corrections. When the corrections get progressively smaller, a few terms can give a remarkably precise answer—for example, it has allowed physicists to predict the strength of the electron’s magnetism to within a few parts in a trillion. This approach is known as perturbation theory.

However, much of the richness of the Standard Model lies outside the realm of perturbation theory. Take the mass of the proton: the proton is made up of three quarks, but its mass comes primarily from the motion and interactions of the quarks and the strong force-carrying gluons inside it. If you turn off the strong force, the proton falls apart entirely, so perturbation theory is not a useful guide to understanding the size of the proton’s mass. More powerful methods are needed to access information about the theory nonperturbatively, where the interactions are present from the start and not treated as small corrections.

For the full Standard Model in continuous spacetime, we don’t have a nonperturbative definition, or even an understanding of whether the interacting theory is mathematically well defined. A promising alternative is to discretize spacetime into a grid of points, known as a lattice. The spacing of the grid defines the shortest distance our theory can resolve, and studying a finite grid allows us to give a rigorous nonperturbative definition of the theory at a given spacing. It also enables us to put the theory on a computer, where we can calculate its predictions directly without resorting to perturbation theory. This approach, known as lattice field theory, has led to remarkable progress in our understanding of the strong force over the last few decades, including a quantitative understanding of the mass of the proton from first principles.

If lattice field theory has been so successful for the strong force, why haven’t we extended it to a nonperturbative lattice definition of the entire Standard Model? Here a crucial difference stands in the way: the Standard Model is an example of what is known as a chiral theory.

A chiral theory is one that looks different under reflection in space. Think of gloves in a mirror: the mirror image of a left-handed glove is a right-handed glove. If you have one of each kind, the mirror image does as well, so your collection of gloves looks the same under reflection in space. But if you only have a left-handed glove, the right-handed glove in the mirror is different.

This notion of handedness extends to many of the fundamental particles in the Standard Model, known as fermions, including quarks and electrons. Most forces in the Standard Model are not chiral: the strong force and electromagnetism, for example, treat left-handed and right-handed fermions in the same way. Such theories can be put on a lattice more readily.

But the third fundamental force of the Standard Model, the weak force, lacks this reflection symmetry. It treats left- and right-handed fermions differently, and so does not look the same under mirror reflections that exchange left and right. This treating of different-handed particles differently is what makes the Standard Model a chiral theory. And it turns out that putting chiral theories on a lattice is profoundly challenging. If we try to put a left-handed fermion on the lattice, the most natural rules give you a right-handed partner you never asked for. This is a surprising outcome, which we’ll unpack momentarily. Whatever the reason, this means a straightforward lattice formulation of the Standard Model gives the wrong particle content for the weak force. Doubling is one reason a complete nonperturbative definition of the Standard Model has been missing since the theory took shape.

To understand doubling, consider an analogy with light. Light has two polarizations, which describe the orientation of the electric field as the light moves through space. (This is familiar if you wear polarized sunglasses, which block the horizontally-polarized light that reflects off water or roads while letting through vertically-polarized light.) Although polarized sunglasses pick out a basis of horizontal and vertical polarizations, for our analogy it will be more useful to pick a basis where these two polarizations are left-circular and right-circular, meaning that the electric field is rotating either clockwise or counterclockwise around the direction of motion. As we’ve said, electromagnetism is reflection-symmetric, so its actual laws are the same for both left-circular and right-circular polarizations. (For the technically inclined, left and right refer to chirality when we talk about fermions. In our analogy with light, they refer to the photon’s helicity, which tells us whether its spin points along or against its direction of travel.)

But imagine you wanted to emulate a world with only left-circular light, akin to the way the weak interactions single out left-handed fermions. To study this in the lab, you could put a laser producing left-circular light into a box with a mirror at either end. The light could build up as it bounced back and forth, allowing you to study it carefully—kind of like putting the Standard Model on a lattice. In each case you are bounding the accessible length scales of the theory to gain control—the lattice bounds the shortest accessible distances, while the box bounds the longest distances.

If you built the box using ordinary mirrors (like putting a chiral theory on the lattice in the most straightforward way), you’d be in for a surprise: the left-circular light coming out of the laser becomes right-circular when it bounces off the mirror at the other end, so now the box contains both polarizations of light. This occurs because the mirror reverses the direction of the light without changing how the electric field is rotating (as seen from the same end of the box), which flips the polarization. This is a bit like the appearance of doublers when you put a left-handed fermion on the lattice. You supplied only left-handed light, and the box supplied the other hand.

There is a clever fix. You can make a chiral mirror using special films whose molecules line up in layers. The alignment turns a little from one layer to the next, tracing a spiral down into the film. For light whose wavelength and handedness match the spiral, the small reflections from different layers reinforce one another, flipping the rotation of the electric field in the light that bounces back. The film sends left-handed light back left-handed and lets right-handed light pass through. Such chiral mirrors have been around for more than a decade, and the phenomenon even occurs in nature: some scarab beetles have a similar spiral in their shiny shells. If you constructed your box using chiral mirrors, then the light traveling in both directions would be left-circular.

A similar trick works on the lattice: decades ago, Paul Ginsparg and Nobel laureate Ken Wilson¹ worked out the lattice rules analogous to the requirement that chiral mirrors reverse the rotation of the electric field upon reflection. Rajamani Narayanan and Herbert Neuberger² built the analog of the chiral mirrors by finding the necessary definition of fundamental particles on the lattice, known as overlap fermions. Finally, Martin Lüscher³ showed how to tell the difference between left-handed and right-handed overlap fermions on the lattice, while Yoshio Kikukawa⁴ and collaborators explored how this might be used to describe weak interactions in the Standard Model. However, one piece was still missing.

To identify that piece, let’s come back to our box with chiral mirrors. The circular polarization of light tells us which way the electric field turns as light propagates, but there’s another important property of the light in our experiment: the optical phase. Imagine water waves rolling past the pilings of a dock, and watch the water rise and fall at one piling. The wave’s phase is the stage of that rise and fall you catch it in when you look, perhaps just past its highest point and on its way down. For our circularly polarized light, the optical phase tracks the same kind of cycle at a fixed spot in the beam, with the electric field pointing up at the “crest” and down at the “trough.”

The relative phase of two waves determines how much they add or cancel at the same point. At one piling of a dock, two waves arriving in phase lift the water higher as their crests meet, while a crest arriving with a trough can leave the water close to its usual level. Similarly, light with the same phase in the same place builds up, while light with different phases can cancel out. In our box with chiral mirrors, left-circular light is now bouncing back and forth inside the box. After a lap, the returning light meets fresh light from the laser going in the same direction. If their electric fields are in step, they reinforce and the light builds up inside the box. If the fields are half a cycle apart, they point in opposite directions and the returning light cancels some of the fresh light. To build up a strong beam, we want the phase of each return to arrive in step with the source.

To get the phase right, we can rotate one chiral mirror relative to the other. Imagine a mark at twelve o’clock on the rim, and spin the mirror about the beam as though the beam were its axle. Moving the mark to three o’clock turns every layer of the film by a quarter turn and shifts the phase of the reflected light by half a cycle. We can set the angle to make up for the phase accumulated on the rest of the trip. Once the returning light arrives in step with the laser, we have a splendid laboratory for studying purely left-handed light.

On the lattice, overlap fermions give us a way of keeping only the left-handed fermions that we want, and getting rid of the doublers. But this doesn’t tell us how to set the phase. (The phase in question is quite different from the one we’ve encountered in our chiral mirror thought experiment, but the analogy is good.) On the lattice, this becomes tricky because the phase depends on what the forces of the Standard Model are doing at every grid point. Lüscher worked out the phase prescription for a simpler class of theories such as a chiral version of electromagnetism, but the alignment for the full Standard Model remained open for more than 25 years.

We can try to assign a phase by starting with one configuration of forces on the lattice and following the allowed left-handed fermions as we change the force fields in small steps. Now take a series of steps that bring the force fields back to where they started. The same states are allowed at the end, yet the way we carried them around the loop can leave us with a different phase. In the box, we adjusted the mirror to compensate for the phase the light picked up on a lap. On the lattice we have to work out how to compensate for the phase picked up around every loop.

My idea was to take the force fields on the grid where we want to define the theory and represent those same fields on a succession of finer grids. Comparing the allowed fermion states on two successive grids tells us one part of the phase correction for a small change in the forces. Adding up those parts gives us the phase rule on the original grid. The proof shows that it gives the right correction around every allowed loop. The correction for a change at one spot depends mainly on the fields nearby, and different mathematical descriptions of the same fields give the same answer. This supplies the measure for the Standard Model at finite lattice spacing, under a bound on field strengths at the scale of the grid.

Developing this refinement strategy into a proof required several rounds of work with Astra and discussions with outside experts. GPT‑6 Astra was essential to turning the refinement idea into a working construction. It suggested the right way to carry a given set of force fields onto finer grids, showed that the comparisons on finer grids would converge to a definite answer, and took the lead in developing the detailed proof. Human experts also played a key role; lattice field theorists Maarten Golterman, Andreas Kronfeld, and Yigal Shamir graciously spent time looking over Astra’s proof and asking probing questions that helped me to build confidence in the result. Their experience with the problem and the questions they asked were invaluable as the construction took shape.

The result is a candidate nonperturbative lattice definition for the Standard Model, potentially answering a question that had been open for more than fifty years. As with all scientific progress, however, this construction is much more a beginning than an end. Scientific dialogue among experts is required to decide if this prescription has all the properties we desire. The method that Astra and I proposed is far more an answer in principle than an answer in practice; much work lies ahead to make it useful for realistic lattice simulations. Right now I’m thinking a lot about whether this construction satisfies reflection positivity at finite lattice spacing, a condition that helps connect calculations in this lattice model to a quantum theory in which probabilities are conserved.

The construction also leaves open the question of defining the interacting theory in continuous spacetime. OpenAI’s recent proof of the existence of Yang-Mills theory (of which the strong and weak forces of the Standard Model are examples) provides the missing puzzle piece. This enabled them to prove the existence of the simplest unified Yang-Mills theory that includes the gauge fields and fermions of the Standard Model by starting with overlap fermions on the lattice and taking the continuum limit. I’m optimistic these developments will lead to new progress in our understanding of the Standard Model.