# The proton and perturbation problem

[Draft] [“Building a ‘verse” series]

I’ve cited this physicist’s video elsewhere, but Perimeter Institute’s overview of her lecture includes a helpful characterization of perturbation theory in the context of understanding the proton better: “Phiala Shanahan builds the universe – with a new approach to calculations and the aid of supercomputers, Emmy Noether Visiting Fellow Phiala Shanahan is rebuilding nuclear physics from the bottom up.” [1]

Low-energy QCD [quantum chromodynamics] is in such a snarl because, in the parlance of the field, its non-perturbative. Most everyday physics is perturbative. Practically speaking, that means we can make rough calculations based on approximations, and then dial in on more correct answers through a series of small corrections.

If you bumped, or “perturbed,” a swinging pendulum, the resulting motion would probably be a swing with a wobble in it. If you wanted to describe the motion mathematically, you could get close by writing down the well-known equations for the basic swing and then adding a bit of math to represent the wobble. The pendulum has been perturbed, and perturbative methods can be applied.

A non-perturbative pendulum would be a different story. The wobble would likely be bigger than the swing, and the resulting motion would be a complex mixture of both. You could only describe it by including both wobble and swing in the calculations from the beginning, which is much more difficult. Many non-perturbative physics problems, in fact, have not been solved analytically at all.

The proton, which in its simplest description is three quarks and some gluons, gets non-perturbative when you look a little closer. Quark and antiquark pairs appear from the vacuum, gluons are emitted and absorbed, and the result, says Shanahan, is a “bubbling, boiling, dynamically complicated structure.”

These calculations are extremely demanding. … So, Shanahan uses the world’s largest supercomputers – and is always looking for ways to move beyond even that.

… Shanahan’s approach to discretizing QCD is already paying off. She’s been able to confirm the mass of the proton, map the pressures inside it, and begin to study the distributions of its gluons.

So, how do you go from analytically modeling a single proton to a helium atom and then to heavier elements? As I mentioned here, it helps to know how to think like a physicist:

Chad Orzel’s article “In Physics, Infinity Is Easy But Ten Is Hard” (March 14, 2017) discusses mathematical approximations (smoothing by abstraction) in physics: “This all comes back to my half-joking description of physics as the science of knowing when to approximate cows as spheres. We’re always looking for the simplest possible models of things, dealing with only the most essential features, and it turns out that an effectively infinite number of interacting objects can be abstracted away much more easily than you can handle a countable number of interacting objects.”

The Hardest Thing To Grasp In Physics? Thinking Like A Physicist” (August 29, 2016): “… so what do I mean by, ‘Think like a physicist?‘ It’s something I’ve been intermittently talking about since I started blogging here at Forbes, a very particular approach to problem-solving that involves abstracting away a lot of complication to get to the simplest possible model that captures the essential elements of the problem. … This kind of simplified model-building isn’t completely unique to physics, but we seem to rely on it more heavily than other sciences. … This kind of quick-and-dirty mental modeling is efficient and also extremely versatile. When you find that the simplest approximation doesn’t quite match reality, it’s often very straightforward to make a small tweak that improves the prediction without needing to start over from scratch (moving from point cows to spherical cows, as it were). And you can iterate this process, slowly building up a more and more accurate model from lots of steps that are individually pretty simple. Of course, while it’s a powerful method for thinking about the world, it also has its pitfalls.”

Notes

[1] How many “building blocks” are there? Why is there a difference in mass between the proton and neutron? Her lecture is a good recap of the Standard Model before ~20′ in the video, which then gets into dark matter. She also discusses the computational challenges of calculating properties of atomic nuclei (protons, neutrons), when even the world’s most powerful supercomputers are not practical for even light nuclei! How “big” is the proton? — a deeper understanding than just the CODATA root mean square charge radius. So, there’s an interplay of theory and experimental predictions for future colliders.

Here’s a fascinating quote from her lecture:

We [analytically] calculated the pressure distribution inside the proton … we find a positive pressure in the inside, that’s a repulsive pressure, and as you move further away a negative, a confining pressure inside the proton. What’s quite remarkable is that the pressure inside the proton at its peak value is greater than the inside of a neutron star, the most dense known objects in the universe. So unlike the proton-neutron mass difference which we’ve measured extremely precisely experimentally, this is an example where theory is more precise than the current experiments. We have a prediction to be tested at future experiments. So there really is a synergy and an interplay between theory and experiment as we strive to learn more about the universe. This is a particularly exciting time for thinking about the structure of the proton. In the next 10 years or so we expect a new particle collider — an electron ion collider — to be built in the United States, and this machine is designed precisely to study the internal dynamics inside a proton, and will allow measurements of things like the pressure distribution to a much higher precision than we’ve ever had before. So we really are looking at a new frontier of understanding of the smallest things in the universe.