Of quarks and gluons: quantum chromodynamics

In this series of posts describing my MSc work for non-specialists, we’ve discussed the standard model for particle physics, how to draw pictures of it, and some of its properties. This week I’ll talk about the Feynman rules for the fundamental particles of the atomic nucleus: quarks, and the gluons through which they interact. The force mediated by gluons is called the colour force (roughly speaking, because quarks come in threes and there are three primary colours, so colour-coding the quarks works pretty well). If we want to sound fancy we can translate “colour-force” into Greek to get, more-or-less directly, “chromodynamics”. So what are the rules for these quantum chromodynamic — or QCD — particles?

I snuck one in at the end of last week’s post, where the electron lines in the QED vertex were replaced with quarks, but QCD proper concerns itself with the vertices between quarks and gluons. While QED has just one prettyvertex, QCD has three. (This is one of the reasons that problems in QCD are generally harder to solve than their counterparts in QED.) The first vertex looks somewhat familiar. (Placing a bar over a label indicates an antiparticle.)

The other two vertices come about because the gluon carries colour charge (in contrast, the photon is electrically neutral). This means that gluons can interact amongst themselves:

free-gluons-momentum-not-includedOne consequence of this is that it’s very easy to produce gluons if energy is available to do so (the way the maths works out, the three-gluon vertex is particularly important for this). In general, the energy required to produce a particle is enough energy to give the particle its mass (using Einstein’s famous E=mc2 equation) plus a little extra to provide the new particle’s energy of motion. But the mass of the gluon happens to be zero, so all that’s needed is that little extra. At sufficiently high energies, this means that one should expect gluons everywhere. This gluons-everywhere situation can be described by a model called the colour glass condensate (CGC). This is what I used in my MSc work, and I’ll discuss it in more detail next week. Before that, let’s talk a little more about Feynman diagrams in QCD.

Some features of QCD don’t show up in the pictures until we start doing calculations. For instance, last week we saw that by adding extra vertices (and virtual particles) we can get from A to B in more way than one. How important is each of these diagrams?

photon   versus   photon-with-loop-feynman

It turns out that the number of vertices in a diagram has a lot to say about that diagram’s importance. Broadly speaking, for every vertex in a diagram, it’s importance is multiplied by a quantity called the vertex factor. In QED, the vertex factor is very small. Very complicated diagrams, with many vertices, therefore have a very small importance. Of course, other considerations also affect the calculations made for each diagram, but in general we can safely ignore very complicated diagrams — just using the simple ones gives us a decent idea of what’s going on. Unfortunately, things don’t look so pretty in QCD.

In QCD, under ordinary conditions, the vertex factor is not small. This means that more complicated diagrams are more important. In theory, an infinitely complicated diagram would be infinitely important (instead of infinitely unimportant, as in QED). This is a problem. To date, the problem has not been solved. Some physicists think this means we need an entirely new theory, not based on Feynman diagrams (and the associated perturbation theory) to describe what goes on inside the atomic nucleus. In this work, I simply avoided the problem.

The QCD vertex factor depends on a value called the QCD coupling constant which (roughly speaking) describes the strength of interactions between QCD particles. This turns out to be closely related to the energy involved:

The parameter αs determines the coupling constant — and the vertex factor. We see here (by taking lots of measurements and producing the graph) that αs decreases as the energy goes up. That means, if energy is high enough, the vertex factor will be small after all. If we’re willing to work in the very high energy region — and with modern particle accelerators, that isn’t unreasonable — we can still get some use out of perturbative QCD. (The term “perturbative” essentially means that we’re assuming more complicated diagrams are only small changes or perturbations to their simpler counterparts.) This is why the virtual photon in the DIS diagram always has to have very high energy.

Of course, now that we’ve restricted ourselves to working at very high energies, we can expect the case of gluons everywhere to become rather relevant. Next week, I’ll talk about the gluon-saturated state called the colour glass condensate.


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