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I’m spending this week in Waterloo, Canada, attending talks at the Perimeter Institute as part of the PASCOS conference (short for particles strings and cosmology). The videos and slides from all of the talks are posted here online each day.

It’s a pretty remarkable feature of the world we live in that the physics can change depending on how closely we look. More precisely, the strength of an interaction (such as electric force, or the strong force) can change with the energy scale used in experiments to probe the physics. Higher energy scales correspond to shorter distances, and so the physics of our world depends on how far we’ve zoomed in to nature.

For example, the charge of an electron e increases with the energy scale. The equation that describes how it changes is called the Callan-Symanzik equation (a special case of the renormalization group equation), which says \mu\frac{\partial g}{\partial \mu} = \beta (g) where g is a coupling constant like the electric charge and \mu is the energy scale. The key is \beta (g), the famous beta-function that can be calculated from quantum field theory.

In QED, \beta(e)=\frac{e^3}{12\pi^2}, which is positive, so the Callan-Symanzik equation tells us that charge increases with energy. The whole business of calculating Feynman diagrams amounts to doing perturbation theory in successively increasing orders of the coupling constant. In QED, you do perturbation theory in the fine structure constant \alpha = \frac{e^2}{\hbar c}. The fact that this number is so tiny (about 1 \over 137) is what makes QED so amenable to perturbation theory: a few diagrams quickly converge to the answer! The more complicated the diagram is (measured in terms of number of vertices), the less is contributes (as every vertex introduces a factor of \alpha). But if the charge, and thus \alpha increases with energy, then we have a potential calamity. At high energies, \alpha must become bigger than 1. If so then perturbation theory in QED must break down, with more complicated diagrams contributing more and more to the answer than simpler ones. To quote David Gross,

“In the case of QED this is only an academic problem, since the trouble shows up only at enormously high energy. However in the case of the strong interactions, it was an immediate catastrophe.”

Theories that have a negative sign on the beta-function are called asymptotically free, meaning the coupling constants become weaker at high energies. This would allow you to do perturbation theory at arbitrarily high energy scales. It was a huge deal when Gross and Wilcek showed QCD to be asymptotically free because people could now use it to do calculations at high energies, corresponding to short distances. QCD, however, has the opposite problem, the coupling constant is large at low energies, or large distances. Gross writes

“At large distances however perturbation theory was useless. In fact, even today after nineteen years of study we still lack reliable, analytic tools for treating this region of QCD. This remains one of the most important, and woefully neglected, areas of theoretical particle physics.

(Aside: Here’s a neat fact that I don’t understand – in four space time dimensions the only asymptotically free theories are non abelian gauge theories)

The beta-function depends only on the coupling constant of the theory. For some particular value of the coupling constants, if the beta-function is zero then it has the property of being scale invariant, that is the physics is the same no matter how close we look. For a trivial example in QED if we set e, the charge of the electron, to be zero, then \beta (e) = 0. This is called a trivial fixed point. If \beta = 0 at some non zero value of the coupling constants, we have what’s called a non-trivial fixed point. The significance of it is that if you reach this coupling constant, then the CS equation says \mu\frac{\partial g}{\partial \mu} = 0, that is at this point once you try to change the energy, the coupling constant stays the same. That’s scale invariance.

In all known examples, quantum field theories that are both unitary and scale invariant also exhibit a much stronger symmetry of conformal invariance, meaning that they are invariant under not just scale but also conformal transformations. A theory exhibiting this symmetry is called a conformal field theory (CFT).

In 2007 Howard Georgi published a paper in which he asks, what if there is a scale-invariant sector of the standard model that interacts so weakly with the rest of it that it hasn’t yet been detected? In other words, he asked the following question: If all you have is a normal particle detector and standard model particles at hand, can one see the effects of a scale invariant theory? That is to say, is there any stuff that you can detect that would lead you to confirm the existence of such a theory?

To investigate this idea, he consider an weakly coupled interaction between fields of a standard model particle and fields of a conformally invariant theory (the reason for the weakness of the coupling is that such exotic effects have not been detected as yet). For concreteness, he considered a theory discovered by Banks and Zaks with a known non-trivial fixed point. The interaction is mediated by a heavy particle.

CFT stuff

What he found was that the stuff that such a theory would predict could not be described in terms of particles. He called it unparticle stuff , because of it’s bizarre experimental signature: it would appear as a non-integer number of invisible particles. Such a theory is testable, to search for an invisible particle you look for missing energy in your particle detector that was carried away by it! The non-integer number will give the energy curve a characteristic shape.

Even if this turns out to have nothing to do with how our world actually works, I still think that this paper presented a neat and quick technique of extracting predictions out of a cute speculative idea. These are the kind of crazy ideas that make beyond the standard model phenomenology exciting! Georgi sure seems to be having fun with it.


I was introduced to this topic during a lecture by Benjamin Grinstein on ‘Unitary Representations of the Conformal Group for pedestrians’. His paper with others on the subject is here.

Here’s the first paper by Howard Georgi on Unparticles. The literature is growing fast.

And a Nature article on Unparticle Physics.


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  1. […] invisible to our detectors but can be detected by their missing energy. Such an idea seems to be a recurring theme in many of the talks in this conference, and is increasingly becoming an active area of […]

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