# The Fine Structure Constant is Probably Constant

*Cross-posted to Sarah Kavassalis’s blog, The Language of Bad Physics.*

A few weeks ago there was a bit of media excitement about a somewhat surprising experimental result. Observations of quasar spectra indicated that the fine structure constant, the parameter in physics that describes the strength of electromagnetism, seems to be slightly different on one side of the universe than on the other. The preprint is here.

Remarkable, if true. The fine structure constant, usually denoted α, is one of the most basic parameters in all of physics, and it’s a big deal if it’s not really constant. But how likely is it to be true? This is the right place to trot out the old “extraordinary claims require extraordinary evidence” chestnut. It’s certainly an extraordinary claim, but the evidence doesn’t really live up to that standard. Maybe further observations will reveal truly extraordinary evidence, but there’s no reason to get excited quite yet.

Chad Orzel does a great job of explaining why an experimentalist should be skeptical of this result. It comes down to the figure below: a map of the observed quasars on the sky, where red indicates that the inferred value of α is slightly lower than expected, and blue indicates that it’s slightly higher. As Chad points out, the big red points are mostly circles, while the big blue points are mostly squares. That’s rather significant, because the two shapes represent different telescopes: circles are Keck data, while squares are from the VLT (“Very Large Telescope”). Slightly suspicious that most of the difference comes from data collected by different instruments.

But from a completely separate angle, there is also good reason for theorists to be skeptical, which is what I wanted to talk about. Theoretical considerations will always be trumped by rock-solid data, but when the data are less firm, it makes sense to take account of what we already think we know about how physics works.

The crucial idea here is the notion of a scalar field. That’s just fancy physics-speak for a quantity which takes on a unique numerical value at every point in spacetime. In quantum field theory, scalar fields lead to spinless particles; the Higgs field is a standard example. (Other particles, such as electrons and photons, arise from more complicated geometric objects — spinors and vectors, respectively.)

The fine structure constant is a scalar field. We don’t usually think of it that way, since we usually reserve the term “field” for something that actually varies from place to place rather than remaining constant, but strictly speaking it’s absolutely true. So, while it would be an amazing and Nobel-worthy result to show that the fine structure constant were varying, it wouldn’t be hard to fit it into the known structure of quantum field theory; you just take a scalar field that is traditionally thought of as constant and allow it to vary from place to place and time to time.

That’s not the whole story, of course, When a field varies from point to point, those variations carry energy. Think of pulling a spring, or twisting a piece of metal. For a scalar field, there are three important contributions to the energy: kinetic energy from the field varying in time, gradient energy from the field varying in space, and potential energy associated with the value of the field at every point, unrelated to how it is changing.

For the fine structure constant, the observations imply that it changes by only a very tiny bit from one end of the universe to the other. So we really wouldn’t expect the gradient energy to be very large, and there’s correspondingly no reason to expect the kinetic energy to matter much.

The potential energy is a different matter. The potential is similar to the familiar example of a ball rolling in a hill; how steep the potential is near its minimum is related to the *mass* of the field. For most scalar fields, like the Higgs field, the potential is extremely steep; this means that if you displace the field from the minimum of its potential by just a bit, it will tend to immediately roll back down. The Higgs is quite massive.

*A priori*, we don’t know ahead of time what the potential should look like; specifying it is part of defining the theory. But quantum field theory gives us clues. At heart, the world is quantum, not classical; the “value” of the scalar field is actually the expectation value of a quantum operator. And such an operator gets contributions from the intrinsic vibrations of all the other fields that it couples to — in this case, every kind of charged particle in the universe. What we actually observe is not the “bare” form of the potential, but the renormalized value, which takes into account the accumulated effects of various forms of virtual particles popping in and out of the quantum vacuum.

The basic effect of renormalization on a scalar field potential is easy to summarize: it makes the mass large. So, if you didn’t know any better, you would expect the potential to be as steep as it could possibly be — probably up near the Planck scale. The Higgs boson probably has a mass of order a hundred times the mass of a proton, which sounds large — but it’s actually a big mystery why it isn’t enormously larger. That’s the hierarchy problem of particle physics.

So what about our friend the fine structure constant? If these observations are correct, the field would have to have an extremely tiny mass — otherwise it wouldn’t vary smoothly over the universe, it would just slosh harmlessly around the bottom of its potential. Plugging in numbers, we find that the mass has to be something like 10^{-42} GeV or less, where 1 GeV is the mass of the proton. In other words: extremely, mind-bogglingly small.

But there’s no known reason for the mass of the scalar field underlying the fine structure constant to be anywhere near that small. This was established in some detail by Banks, Dine, and Douglas. They affirmed our intuition, that a tiny change in the fine structure constant should be associated with a huge change in potential energy.

Now, there are loopholes — there are always loopholes. In this case, you could possibly prevent those quantum fluctuations from renormalizing your scalar-field potential simply by shielding the field from interactions with other fields. That is, you can impose a symmetry that forbids the field from coupling to other forms of matter, or only lets it couple in certain very precise ways; then you could at least imagine keeping the mass small. That’s essentially the strategy behind the supersymmetric solution to the hierarchy problem.

Problem is, that route is a complete failure when we turn to the fine structure constant, for a very basic reason: we can’t prevent it from coupling to other fields, it’s the parameter that governs the strength of electromagnetism! So like it or not, it will couple to the electromagnetic field and all charged particles in nature. I talked about this in one of my own papers from a few years ago. I was thinking about time-dependent scalars, not spatially-varying ones, but the principles are precisely the same.

That’s why theorists are skeptical of this claimed result. Not that it’s impossible; if the data stand up, it will present a serious challenge to our theoretical prejudices, but that will doubtless goad theorists into being more clever than usual in trying to explain it. Rather, the point is that we have good reasons to suspect that the fine structure constant really is constant; it’s not just a fifty-fifty kind of choice. And given those good reasons, we need really good data to change our minds. That’s not what we have yet — but what we have is certainly more than enough motivation to keep searching.

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