The last couple of weeks have been busy beyond belief, with administrative duties, research, teaching, some travel, and a relative visiting. One of my trips was to Lehigh University, to give a colloquium. Lehigh is the home of Intelligent Design nut job Michael Behe but, since I was visiting the Physics Department, I was able to avoid his nonsense altogether (although the physicists there – wonderful rational people that they are – seem rightly embarrassed to be at the same institution as him).
Yesterday afternoon I drove up to Montreal, to participate in a workshop on The Stabilization of Embedded Defects, at McGill University. This topic (I’ll take a shot at explaining it below) is one in which I have dabbled in the past, but which hasn’t been a part of my research program for some time. However, it is an interesting area and, since the program is being organized by my Ph.D. advisor, Robert Brandenberger (who very recently left Brown to head to McGill), and since good friends like Tanmay Vachaspati and Ana Achucarro (people I haven’t seen in ages) were going to be here, I thought it would make for a delightful trip. So far, I was right!
Topological defects are extended solutions to field theories that can arise when the vacuum structure of the field theory is topologically nontrivial. As a somewhat simple example (and, I admit, a clumsy one, but its the best I can do right now), let us model a field theory by standing many pencils on their ends on a table top, and connecting the pencils to their nearest neighbor pencils by springs. What is the vacuum configuration of these pencils? Obviously, because of gravity, each individual pencil would like to lie down on the table, but doesn’t care which direction it is pointing as long as it is lying down. So the vacuum configuration of the theory is all the pencils lying down, facing in the same direction, because if any pencil faces in a direction different from that of its neighbor, then there will be energy bound up in the spring which is stretched between them, which can be reduced by the two pencils aligning. Obviously, there are an infinite number of equivalent vacua, corresponding to all the pencils aligning in any of the possible directions in the plane of the table.
Now suppose that the table is very big (maybe even infinite), and pencils that are very far apart from each other fall down into different vacuum states, because causality doesn’t permit information to travel between them so that they can align. You could imagine, in fact, that pencils that trace out a very large circle all fall down pointing outwards in different directions along that circle. All other pencils inside that circle will try to align with the pencil closest to them, in order to reduce the energy in the springs. However, if you think through this for a moment, you’ll see that there will always be one pencil, the one at the center, which is equally pulled in all directions and so will remain standing up – now stably. In fact, a few pencils on each side of this one will be partially standing up, because of the spring tension.
These few pencils, and particularly the one at the center, represent what is meant by a topological defect. It is a small region of space, in which the field configuration is out of the vacuum manifold, but which remains metastably in that configuration because the topological properties of the vacua chosen by the field at infinity are nontrivial. I won’t harp on about how these topological properties are defined, because it’s more technical than the sketch above and won’t buy us much clarity here.
In the model system above, the topological defect is point-like – it is just a single point in space. In three spatial dimensions one can have either point-like defects (monopoles), line-like defects (strings) or membrane-like defects (domain walls). Which, if any, of these exist depends on the particular particle physics model one considers.
So what are embedded defects? Well, suppose that one takes the above structure, and imposes a new, larger symmetry on the theory, so that the symmetry is large enough that the topological defect can now be smoothly relaxed to the vacuum of the theory. In this case, we have transformed the theory into one with no topological defects, and one might think that nothing of interest remains. However, one is still allowed to write down the original defect configuration, by setting to zero, by hand, the extra fields required to impose the new symmetry. The question is: what happens to this configuration?
It turns out that, for a range of values of the couplings of the theory, although the defect solution is no longer topologically stable, it is dynamically metastable – meaning that once the configuration is there, the field realignments that are required to settle the fields into the ultimate vacuum must go through configurations that are of higher energy than the defect configuration, and so the defect will not decay immediately, as one might have thought from topological considerations alone. Such metastable configurations are embedded defects.
A fascinating fact is that the standard model itself has the requisite vacuum structure for embedded defects – the so-called Z-strings. However, for physical values of the parameters, these configurations are unstable, and so will not form as the universe cools. The purpose of this workshop is to discuss physical mechanisms through which such defects may be again rendered metastable, and thus may contribute to cosmology.
Another goal of the workshop is to understand what kinds of symmetry breaking schemes give rise to such defects – a question that is of pressing importance given the upcoming start of the Large Hadron Collider (LHC) experiments, which may reveal new particle physics structures beyond the electroweak scale.
Today was fun, with some excellent talks and some discussion sessions, one of which – on new cosmological applications – I moderated. Tomorrow there will be more talks, including one by Ana Achucarro on numerical simulations that should cast light on one of the suggested stabilization mechanisms.
The meeting ends on Sunday, after which I’ll head back to Syracuse and hopefully get back to regular blogging.