The symmetries of the standard model

With the previous couple of entries a number of basic concepts have been introduced. It is now about time to make use of them in terms of the standard model.

The standard model from the theoreticians point of view is a set of local and global symmetries, which constraint the overall form of the theory. This skeleton is then fleshed out by adding to the symmetries particles such that they respect the symmetries. Furthermore, interactions between the particles are added, which superficially respect the at least the local symmetries, i.e. they do not break them explicitly. This then gives the set-up of the standard model (the procedure is quite similar if one is looking for a theory beyond the standard model, though there is not (yet) coercive experimental guidance how to choose the ingredients). And then...we let the system run, and see what comes out. This may actually break some of the symmetries, there may appear interactions which have not been there before, or we can observe new particles, which are somehow constructed from those we have put in. The proton is an example of the latter case.

So what are the symmetries in the standard model?

First, there are three local symmetries, which are at the heart of the theory. Each of them is associated with an interaction.

There is first a very simple symmetry, called electromagnetic or U(1) symmetry, which is associated with electromagnetism and the photon. It tells us that we can modify the electromagnetic field locally to some extent without altering the physics.

The next in line is the one associated with the strong interactions, the gluons, and the quarks, the so-called color symmetry or SU(3). It tells us that the interaction among quarks and gluons can locally be changed to some extent, again without changing anything measurable.

Finally, there is the one associated with the weak force, the so-called weak symmetry or SU(2). Except for the gluons, everything in the standard model is in one way or the other associated with this symmetry. This implies we can change a lot of how the standard model looks without changing the measurements.

These three symmetries, also called together SU(3)xSU(2)xU(1), are at the very heart of standard model. Everything else is build around it. However, the interactions change this structure considerable, and when looking just at measurements, it appears at first sight that the weak local symmetry is gone. However, in fact it is still there, but very well hidden by the interactions. I will come back to this in the future.

Then there are a number of global symmetries. First, there is a so-called chiral symmetry associated with the quarks and leptons. I.e., there is a special relation between particles spinning in direction of their movement and those spinning in the opposite direction. Because you can visualize them with either left or right hand, this is associated with the word chiral, which in a loose sense means handedness (precisely, it means hand). This symmetry is not left intact by the interactions, and this can be associated with how the particles become a mass. The second is that the number of each type of quarks and leptons are individually conserved. Also this symmetry is not surviving when interactions are turned on. However, the total number of quarks and leptons is actually almost conserved, and their change in number is, at the current time, essentially negligible. For a quark to turn into a lepton, experiments found that this needs at least 10000000000000000000000000000000000 years. The next symmetry counts the total number of quarks and leptons. This number is conserved in the standard model. Finally, there is also a rather obscure symmetry, which relates things which have a very distinct property when looking at them or at their mirror image, called axial symmetry. Again, this symmetry is broken. In contrast to the previous cases, this symmetry is actually not broken by the interactions, but enforcing the theory to describe quantum effects. Because that is so different from the rest, this is called an anomalous breaking, and the effect itself is called an anomaly.

On top of these local and global symmetries, there are three more symmetries, which have to do with fundamental properties of a physical system. One is related about what happens if you look at things and then again look at them in a mirror. That is called parity. The next connects to what happens when you replace every particle by its anti-particle and vice versa. This is called (charge) conjugation. And the last one is a statement what happens if you reverse all movements, and thus is called time reversal. All the three individual symmetries are broken by the interactions. However, if you combine all three together, this is a single symmetry, and this is still obeyed.

So, you see, the standard model is essentially a zoo of symmetries, and they again become very much modified by interactions. This is one of the reasons which yields many technical problems when one tries to answer even simple questions in the standard model.