Hidden symmetries

By Sean Carroll | October 24, 2005 12:24 pm

Symmetries, you may have heard, play a crucial role in modern physics. (Leon Lederman and Chris Hill wrote a whole popular book about the subject, if you’re interested.) But one of the things that makes them so interesting is that they can be hidden — the symmetry is secretly there, even though you don’t easily notice. And sometimes you may be interested in the converse situation — it looks like there is an obvious symmetry of nature, but in fact there are tiny violations of it, which we haven’t yet detected.

To physicists, a “symmetry” is a situation where you can rearrange things a bit (values of quantum fields, positions in space, any of the characteristics of some physical state) and get the same answer to any physical question you may want to ask. An obvious example is, in fact, position in space: it doesn’t matter where in the world you set up your experiment to measure the charge of the electron, you should get the same answer. Of course, if your experiment is to measure the Earth’s gravitational field, you might think that you do get a different answer by moving somewhere else in space. But the rules of the game are that everything has to move — you, the experiment, and even the Earth! If you do that, the gravitational field should indeed be the same.

How do such symmetries get hidden? The classic example here is in the weak interactions of particle physics: the interactions by which, for example, a neutron decays into a proton, an electron, and an anti-neutrino. It turns out that a very elegant understanding of the weak interactions emerges if we imagine that there is actually a symmetry (labeled “SU(2)”) between certain particles; examples include the up and down quarks, as well as the electron and the electron neutrino. (This is the insight for which Glashow, Salam and Weinberg won the Nobel Prize in 1979.) If this electroweak symmetry were manifest (or “unbroken” or “linearly realized,” depending on one’s level of fastidiousness), that means that it would be impossible to tell the difference between ups and downs, or between electrons and their neutrinos.

Of course, in reality it’s not so hard to tell. These purportedly-indistinguishable particles have some similar properties, but they have different masses, and even different electric charges. Nobody would ever mistake an electron for an electron neutrino. (They would mistake a red quark for a green quark or a blue quark, as those are related by an unbroken symmetry — the SU(3) of quantum chromodynamics, for which the Nobel came much more recently.)

The reason is that the SU(2) symmetry of the weak interactions is spontaneously broken (or “nonlinearly realized”). The symmetry is firmly embedded in the laws of physics, but is hidden from our view because the particular state in which we find the universe is not invariant under this symmetry. There is something about the vacuum — empty space itself — which knows the difference between an up quark and a down quark, and it’s the influence of the vacuum on these particles that makes them look different to us.

This idea of spontaneous symmetry breaking has a long history in physics — it was elucidated in condensed matter physics by Philip Anderson (Nobel 1977) and others, and in particle physics by my colleague Yochiro Nambu and erstwhile colleague Jeffrey Goldstone (no Nobel yet, which is a shame). And here’s an interesting thing — if the vacuum is not invariant under some symmetry, there must be some field that is making it not invariant, by taking on a “vacuum expectation value.” In other words, this field likes to have a non-zero value even in its lowest-energy state. That’s not what we’re used to; the electromagnetic field, for example, has its minimum energy when the field itself is zero. But “zero” doesn’t break any symmetries; it’s only when a field has a nonzero value in the vacuum that it can affect different particles in different ways.

Mexican hat potential The way to do that, in turn, is to imagine that the symmetry-breaking field has a “Mexican-Hat Potential,” as illustrated at right. (Image swiped from The Official String Theory Web Site, which also has a nice discussion at a more technical level.) This is a graph of the potential energy of a set of two fields φ1 and φ2. Fields like to sit at the minimum of their potentials; notice that in this example, the minimum is not at zero, but along a circle at the brim of the hat. Notice also that there is a symmetry — we can rotate the hat, and everything looks the same. But in reality the field would actually be sitting at some particular point in the brim of the hat. The point is that you should imagine yourself as sitting there along with the field, in the brim of the hat. If you were at the peak in the center of the potential, the symmetry would be manifest — spin around, and everything looks the same. But there in the brim, the symmetry is hidden — spin around, and things look dramatically different in different directions. The symmetry is still there, but it’s nonlinearly realized.

Now comes more fun. Fields can oscillate back and forth, and in quantum field theory, what you see when you look at an oscillating field is a set of particles. Furthermore, the amount of curvature in the potential tells you the mass of the particle. Sitting at the brim of the hat, there are two directions in which you can oscillate — a flat direction along the brim, and a highly-curved direction moving radially away from the center. That’s one massless particle (motion along the brim of the hat) and one quite massive particle (radial motion). (See the little ball oscillating in either direction in the figure?) The fact that there will always be a massless particle when you have spontaneous symmetry breaking is Goldstone’s theorem, and the particle itself is a Nambu-Goldstone boson.

This looks like a problem for the idea of spontaneous symmetry breaking in the weak interactions — where is the massless particle? The answer was first figured out by — well, I won’t say who, because it was by about a dozen people at once, and physicists from different countries love to give impassioned speeches about how someone from their home country discovered it first. Suffice it to say that the idea itself is now called the Higgs mechanism. The point is that not all symmetries are created equal. Sometimes you have a “global” symmetry, which is an honest equivalence between two or more different fields. Breaking global symmetries really does give rise to Nambu-Goldstone bosons. But other times you have gauge symmetries, which aren’t really symmetries at all — they are just situations in which it’s useful to introduce more fields than really exist, along with a symmetry between them, to make a more elegant description of the physics. Gauge symmetries come along with gauge bosons, which are massless force-carrying particles like the photon and the gluons.

Here’s the secret of the Higgs mechanism: when you spontaneously break a gauge symmetry, the would-be Nambu-Goldstone boson gets “eaten” by the gauge bosons! What you thought would be a massless spin-1 gauge boson and a massless spin-0 NG-boson shows up as a single particle, a massive spin-1 gauge boson. In the case of the weak interactions, these massive gauge bosons are the two charged W particles and the neutral Z particle, discovered at CERN in the early 1980’s (Nobel 1984 to Carlo Rubbia and Simon van der Meer — I’m telling you, if you want to win the Nobel Prize, you could do worse than studying spontaneous symmetry breaking). Note that Glashow, Salam, and Weinberg won the Nobel for predicting the existence of these particles before they were actually discovered — that’s because their model made other predictions which were verified to high accuracy.

But there is a missing piece — a prediction that has not yet been verified. While it’s true that the gauge bosons eat the would-be massless NG-boson, what about the massive particle corresponding to radial oscillations in the Mexican hat potential? That should be there, and we call it the Higgs boson. No Nobel prizes yet, because we haven’t found it! It’s a major goal of the currently-running Tevatron accelerator at Fermilab, as well as the coming-soon LHC accelerator at CERN. Of course, the fun thing about physics is that you can’t be absolutely sure that the thing is there until you’ve actually found it. But most people are willing to bet that we’ll find at least the Higgs, probably much more, at the energies where the electroweak symmetry is broken.

Why am I telling you this? The original idea was to work my way up to consider spontaneous violation of Lorentz invariance — symmetry under changes of orientation and changes of velocity — and how it could help explain the matter/antimatter asymmetry of the universe. (The Wikipedia entry on Lorentz invariance is sadly bizarre, consisting mostly of a discussion of loop quantum gravity, which is interesting but beside the point.) But I’ve gone on, and had better get going — I’m sitting at a pleasant outdoor cafe in Berkeley (Caffe Strada at the corner of College and Bancroft, for those in the know), and at some point I should actually trudge up to the Physics department to give my talk this afternoon. Stay tuned!

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .

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