04-29-09 - QCD

Someone made me think briefly about QCD (Quantum Chromo Dynamics).

I never really got my head around the standard model in grad school. I think I understood QED pretty well, and Weak isn't really that bad either, but then you get into QCD and the maths gets really tough and there's this sea of particles and I had no idea what's going on. Part of the problem is that a lot of the texts go through a historical perspective and teach you the stages of understanding and the experiments that led to modern QCD. I think that's a big mistake and I discourage anyone from reading that. I was always really confused by all the talk of the various mesons and baryons. Often the classes would start with talking about K+ transitions or pion decay or scattering coefficients for "Omegas" and I'd be like "WTF are these particles and who cares what they do?".

I think it's way better just to say "we have quarks and gluons". And yes, the quarks can combine together into these various things, but we don't even really need to talk about them because nobody fucking cares about what exactly the meson made from (strange-antistrange) is called.

I much prefer a purely modern approach to QFT based on symmetry. In particular I really like Weinberg's approach in his textbook which is basically - we expect to observe every phenomenon in the universe which is *possible* to exist. If something is possible but doesn't ever happen, that is quite strange and we should wonder why. In particular with QFT - every possible Lagrangian which leads to a consistent theory should correspond to something in nature. When you start to write these down it turns out that very few are actually possible (given a few constraints, such as the postulate that relativity is required, etc.).

Anyway, I was never really happy with my intuition for QFT. Part of the problem is the math is just so hard, you can't do a lot of problems and get really comfortable with it. (David Politzer at Caltech once gave me a standard model homework problem to actually compute some real scattering coefficients that had been experimentally tested. It took me about 50 pages and I got it horribly wrong).

The whole gauge-field symmetry-group idea seems like it should be very elegant and lead to some intuition, but I just don't see it. You can say hand wavey things, like : electromagnetism is the presence of an extra U(1) symmetry; you can think of this as an extra circular dimension that's rolled up tiny so it has no spatial size, or if you like you can do the Feynman way and say that everything flying around is a clock that is pointing in some direction (that's the U(1) angle). In this picture, the coupling of a "charge" to the field is the fact that the charge distorts the U(1) dimension. If you're familiar with the idea of general relativity where masses distort spacetime and thus create the gravity force, it's the same sort of thing, but instead of distorting spacetime, charge distorts the U(1) fiber. As charges move around in this higher-D space, if they are pushed by variation of the U(1) fiber clock angle, that pushes them in real space, which is how they get force. Charges are a pole in the curvature of the fiber angle; in a spacetime sense it's a pinched spot that can't be worked out by any stretching of the space fabric. Okay this is sort of giving us a picture, but it's super hand wavey and sort of wrong, and it's hard to reconcile with the real maths.

Anyway, the thing I wanted to write about QCD is the real problem of non-perturbative analysis.

When you're taught QED, the thing people latch onto are the simple Feynman diagrams where two electrons fly along and exchange a photon. This is appealingly classical and easy to understand. The problem is, it's sort of a lie. For one thing, the idea that the photon is "thrown" between the electrons and thus exchanges momentum and forces them apart is a very appealing picture, but kind of wrong, since the photon can actually have negative momentum (eg. for an electron and positron, the photon exchanged between them pulls them together, so the sort of spacemen playing catch kind of picture just doesn't work).

First of all, let's back up a bit. QFT is formulated using the sum of all complex exponential actions mechanism. Classically this would reduce to "least action" paths, which is equivalent to Lagragian classical mechanics. There's a great book which teaches ordinary Quantum Mechanics using this formulation : Quantum Mechanics and Path Integrals by Feynman & Hibbs (this is a serious textbook for physics undergrads who already know standard QM ; it's a great bridge from standard QM to QFT, because it introduces the sum-on-action formalism in the more familiar old QM). Anyway, the math winds up as a sum of all possible ways for a given interaction to happen. The Feynman diagram is a nice way to write down these various ways and then you still integrate over all possible ways each diagram can happen.

Now let's go back to the simple QED diagram that I mentioned. This is often shown as your first diagram, and you can do the integral easily, and you get a nice answer that's simple and cute. But what happened? We're supposed to sum on *all* ways that the interaction can happen, and we only did one. In fact, there are tons of other possibilities that produce the same outcome, and we really need to either sum them all, or show that they are small.

One thing we need to add is all the ways that you can add vacuum -> vacuum graphs. You can make side graphs that start from nothing, particles pop out of the vacuum, interact, then go back to the vacuum. These are conveniently not mentioned because if you add them all up they have an infinite contribution, which would freak out early students. Fortunately we have the renormalization mechanism that sweeps this under the rug just fine, but it's quite complex.

The other issue is that you can add more and more complex graphs; instead of just one photon exchange, what about two? The more complex graphs have higher powers of the coupling constant (e in this case). If the coupling constant is small, this is like a Taylor expansion, each term is higher powers of e, and e is small, so we can just go up to 3rd order accuracy or whatever we want. The problem with this is that even when e is small, as the graphs get more complex there are *more* of them. As you allow more couplings, there are more and more ways to make a graph of N couplings. In order for this kind of Taylor expansion to be right, the number of graphs must go up more slowly than 1/e. Again it's quite complex to prove that.

Starting with a simple problem that we can solve exactly, and then adding terms that make us progressively more accurate is the standard modus operandi in physics. Usually the full system is too hard to solve analytically, and too hard to get intuition for, so we rely on what's called a perturbation expansion. Take your complex system that you can't solve, and expand it into Simple + C * Complex1 + C^2 * Complex2 + ... - higher and higher powers of C, which should be small.

And with QCD we get a real problem. Again you can start with a simple graph of quarks flying along passing gluons. First of all, unlike photons, there are gluon-gluon couplings which means we need to add a bunch more graphs where gluons interact with other gluons. Now when we start adding these higher order terms, we have a problem. In QCD, the coupling constant is not small enough, and the number of graphs that are possible for each order of the coupling constant is too high - the more complex terms are not less important. In fact in some cases, they're *more* important than the simpler terms.

This makes QCD unlike any other field theory. Our sort of classical intuition of particles flying around exchanging bosons completely breaks down. Instead the quarks live in a foaming soup of gluons. I don't really even want to describe it in hand wavey terms like that because any kind of picture you might have like that is going to be wrong and misleading. Even the most basic of QCD problems is too hard to do analytically; in practice people do "lattice QCD" numerical computations (in some simple cases you can do the summations analytically and then take the limit of the lattice size going to zero).

The result is that even when I was doing QFT I never really understood QCD.

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