Tuesday, March 4, 2008

A quark problem

If you take a look back at the table I showed in last week’s blog, you’ll see that the only quantum number I showed for the fundamental particles was electric charge, Q. (By “quantum number” here I just mean a property of the particles, like spin or energy, for example.) Anyway, I’d like to reproduce the table again this week, but add a few other quantum numbers:

For starters, these new numbers, baryon number and lepton number, are a clever labeling system, in that they match the properties of the particles we talked about in the homework. For instance, the proton, a uud bound state (two up quarks and a down quark), is indeed a baryon (B = 1 = 1/3 + 1/3 + 1/3) and not a lepton (L = 0) and it has a total charge of +1. By contrast, the electron is a lepton (L = 1) and not a baryon (B = 0).

We can go further with the labeling. It turns out that antiparticles have all opposite quantum numbers from their corresponding particle; for example, an antimuon has B = 0, L = -1 (since it’s an antilepton), Q = +1. As a final example, a positively charged pion (π+) is a meson composed of an up and an antidown quark. It’s not a baryon or a lepton (B = 0 = 1/3 – 1/3), (L = 0) and its charge is +1 (Q = 1 = 2/3 + 1/3).

Okay, so I think it’s at least mildly nifty that you can make such a labeling system, but it turns out these properties of particles are much more useful than that: they are actually conserved values, so any particular physical process can only occur if it leaves these quantum numbers unchanged. For instance, we could have the interaction

because there is exactly one baryon, no leptons, and no net electric charge both before and after. Another valid interaction is

Once again, we have one baryon in both the initial and final states, no leptons (remember the antineutrino counts as -1 leptons!), and no net charge. However, we would never expect to see interactions like

(doesn’t conserve baryon number)

(doesn’t conserve lepton number)

(doesn’t conserve charge)

Historically speaking, the theory that hadrons were made of quarks was starting to look like a reasonably good idea at this point. It explained the proliferation of hadrons, it sorted them neatly into mesons and baryons, and best of all it made predictions about what kinds of interactions they could and couldn’t undergo. Unfortunately, it ran into some embarrassing trouble at the next step: accounting for Pauli’s exclusion principle.

I haven’t really talked about Pauli’s exclusion principle, and for the most part I’d like to save it until the end of the semester, because it’s one of the most magnificent ideas in physics, and it has a lot to do with my research in particular. But I’ll say a couple things about it now, since it’s important to understand our quark problem.

As it relates to chemistry, Pauli’s exclusion principle mostly just says that two electrons can’t be in exactly the same state. This is why electrons in atoms aren’t all sitting at the bottom, in the 1s state; instead they have to fill out the other energy levels and the other orbitals (a friend of mine calls this the “bus seat rule”: once someone has taken a seat, that’s it, the next guy just has to find a different seat). Actually, as you probably remember from chemistry, it’s okay for two electrons to be at the same energy level and in the same orbital; they’re not really in the same state, because electrons can have two different spins.

Why does this matter to us? Well, it turns out the exclusion principle also applies to quarks; that is, no two identical quarks can be in exactly the same state. (Notice, by the way, that this rule only applies to identical quarks. It’s perfectly fine for an up quark and a down quark to be in the same state, because they’re two distinct particles.) Now, bearing this in mind, let’s take another look at the proton and neutron:

p = uud

n = udd

So far, so good. Sure, the proton has two up quarks in the same state, and the neutron has two down quarks in the same state, but we can get around this the same way we did in chemistry: the quarks also have two possible spins, so at most two identical quarks can be in the (otherwise) same state. The real problem is this guy:

Δ++ = uuu

In 1951, Fermi and his collaborators found this “delta” baryon, and sure enough, it had all the properties the quark model predicted it would have: it got the charge right, the mass right, it even got the lifetime right. But it figured out all these things by assuming the delta was made up of three identical up quarks in exactly the same state! This is absolutely forbidden by Pauli’s exclusion principle. What gives?

Even setting this exclusion fiasco aside for the moment, we still have cracks starting to show in our naive quark theory. Certainly it’s true that the quark combinations qqq, anti-q anti-q anti-q, and q anti-q neatly fit the observed sequence of baryons, antibaryons, and mesons, respectively, but what about all the other possible combinations, like qq, anti-q anti-q, etc. For that matter, why can’t we just have single quarks by themselves?

I’ll let you mull over these puzzles until next week, but if you have ideas or suggestions, by all means write me a comment and let me know. In the meantime, here are this week’s questions:

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