A Nonperturbative Analogy

By Mark Trodden | July 31, 2006 8:35 pm

In the Fall semester, which is approaching at a far too rapid rate, I am teaching a new advanced graduate course PHY795: Modern Cosmology. Although this will certainly be a huge amount of work, I’m very much looking forward to exposing our graduate students to the topics I spend most of my life thinking about.

As part of my preparation, I have recently turned to explaining the question of baryogenesis – how did the measured excess of matter over antimatter in the universe arise dynamically as the universe evolved? I have discussed this question once before, and also described one of the possible and testable ways in which this may have occurred, through nonperturbative finite temperature dynamics in small extensions of the standard model of particle physics.

In explaining these ideas, I chose to avoid anything more than a passing reference to how baryon number violating transitions take place in such theories, writing

In the standard electroweak theory baryon number is an exact global symmetry. However, baryon number is violated at the quantum level through nonperturbative processes – it is an anomalous symmetry. This feature is closely related to the nontrivial vacuum structure of the electroweak theory.

At zero temperature, baryon number violating events are exponentially suppressed (this is most certainly a good thing, since we would like the protons making up our bodies to remain stable). However, at temperatures above or comparable to the critical temperature of the electroweak phase transition, B-violating vacuum transitions may occur frequently due to thermal activation.

I chose this route because the physics involved – that of nonperturbative physics in chiral quantum field theories – is a difficult one, the technicalities of which I thought would be beyond the level of the post.

However, in preparing for my class, I found myself going over a favorite analogy for some of this physics and thought I’d give it a shot here. I don’t quite recall where I first came across this analogy, although it may have been in a review article by either Emil Mottola or Peter Arnold, which I read when I was a graduate student. What follows will be a little more technical than usual, but I’m hoping that most people with some physics training will get something out of it.

The vacuum of the electroweak theory is degenerate – there are infinitely many vacua, related by large gauge transformations. The field theories constructed around these vacua are entirely equivalent, but transitions between these vacua result in the anomalous production of fermions, which is the method by which the baryon number may change.

Fortunately for us (who wants our protons to spontaneously decay away?), these baryon number violating transitions are forbidden classically and, in fact, even at the perturbative quantum level – baryon number is an exact global symmetry of the theory. Therefore, at zero temperature, the only way baryon number violating processes can occur is through quantum tunneling between the classical vacua of the theory. This in itself is nonperturbative physics, and the relevant calculation yields that if the universe were always close to zero temperature, not one event would have occurred within the present Hubble volume ever in the history of the universe. However, when we include the effects of nonzero temperature, classical transitions between vacua become possible due to thermal activation.

For an analogy to this heady mix of perturbative-nonperturbative physics and finite temperature field theory, it turns out we can lean on a physical system about which most physicists learn in high school or college – the simple pendulum!

This system is a mass m suspended at the end of an arm of length l, and confined to rotate ideally in the plane. The system possesses a periodic vacuum structure labeled by integer n since, measuring angles in radians as scientists do, the system is identical in a minimum energy state whenever the angle θ in the figure is given by 2nπ. It is in this sense analogous to the electroweak theory, and you are to imagine that fermions would be produced should the system make a transition between vacua labeled by different values of n.

If we say that the vacua have zero gravitational potential energy, then the potential energy for any value of the angle θ is simply given by mg[1-cos(θ)], where g is the acceleration due to gravity. Notice that, because there is a cosine in this expression, all the information about the multiple vacua I mentioned is right there in the energy.

If a physicist wanted to understand the classical or low energy quantum mechanics of this system, they might take this potential energy and make it simpler with the approximation that the angle is always small (really what one means by low energy). In that case, if we keep only up to second order in the angle (i.e. do perturbation theory), the potential energy becomes the simplest one we know of – the harmonic oscillator – and is easily solvable. (Actually, the truth is that the full problem is solvable, but anything more complicated probably won’t be, and the approximation is what I’m trying to explain here).

But we do play a price for this approximation – all information about the periodic vacua is lost.

If we think about raising the temperature of the system, we soon run into trouble. Suppose the pendulum is coupled to a thermal bath. Then it will be thermally excited to states of higher and higher energy as the temperature is raised. Obviously, before we start calculating, we can see that as the temperature becomes comparable with the height of the barrier preventing transitions between vacua, it becomes possible for the pendulum to make transitions between vacua, crossing the point θ = π randomly, at an unsuppressed rate. But note that these have to be nonperturbative transitions, since they probe the periodic structure of the vacuum, which is not captured at all by perturbation theory.

This situation is analogous to most familiar calculations in the electroweak theory, in which perturbation theory is usually a safe tool to use. Such an approximation scheme is only valid when the energy of the system is much less than the height of the barrier separating vacua (in the electroweak theory this is known as the sphaleron). In that limit, quantum tunneling between vacua is exponentially suppressed as expected. But in the early universe, when temperatures are much higher than the barrier height, perturbation theory must be abandoned, and these effects, and the associated violation of baryon number, are rampant.


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About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.


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