Friday, March 21, 2008

Introducing "color"

Last week, I raised two objections to the quark model: (1) it seems to violate the exclusion principle (at least in some cases), and (2) it doesn’t explain why quarks only come in certain combinations (three quarks, three antiquarks, or a quark-antiquark pair). As it turns out, both problems can be handled by introducing a new quantum property, exclusively for quarks, called color.

Our idea is to suppose that every quark actually comes in one of three types, which we’ll call red, blue, and green (R, B, G), by way of analogy with the primary colors of light. Oh, and by the way, I probably don’t need to mention this, but of course these labels have absolutely no connection to real colors; it’s just that there happen to be three of them. (Particles have no color at all; in fact, they don’t “look” like anything in the first place.)

Anyway, adding the new color label immediately fixes our problem with the exclusion principle. Why? The key point is that exclusion applies only to identical particles. If the quarks have different colors, they’re no longer identical, so they’re perfectly welcome to be in the same state. Take the Delta baryon from last week:

Δ++ = uRuBuG

This particle is a legitimate problem if all the up-quarks are the same, but they are now distinguished by their color, so there’s no conflict. Simple!

Well, not quite so simple, actually. This may solve one problem, but it immediately brings up another: if quarks can now have a color quantum number, then why can’t we have lots of different proton states? That is, we know protons are made of two up-quarks and one down-quark, but now we can color them too, so why can’t I make up lots of different color combinations, like

p = uRuBdG
= uRuGdG
= uBuRdR
… etc.

By my count (you might want to check me) there could be a total of 18 different proton states! And we’d be able to tell from experiments, too: for example, the Δ++ would be eighteen times more likely to decay into a proton; the neutron would decay about eighteen times faster, etc.

So this is a problem, because we don’t have 18 kinds of protons, we just have one. We now have to find a way of introducing this new “color” property of quarks without proliferating all our hadrons. But how do we do it?

The usual way to solve these kinds of problems is to take a theorist out to dinner, buy her a few drinks, and then ask what she would do. She’ll likely say something like this: “Now that you’ve introduced this color thing, what you need is a rule telling you how to use it.” Actually, she’d probably say that you needed a symmetry, because it turns out that every rule in physics is generated by a symmetry principle, but that’s a story for a different day. She’d then go on to say, “Why not require that your hadrons be invariant under rotations in color-space?” In simpler terms, she’s saying we should insist that our hadrons be colorless (or “white”, to stick to the analogy).

Take a look at this Venn diagram of color, which is what we’d get if we shined the three primary lights at a screen and let them overlap.* The overlap areas are important in this analogy, because we need them for the antiquarks: if quarks are given one of three possible colors, then antiquarks must be given one of three possible anticolors. (You may remember a couple of homeworks ago when I mentioned that antiparticles have all opposite quantum numbers, and this applies to color, too.) Physicists usually just refer to these as antired (R with a bar over it), antiblue (B with a bar over it), and antigreen (G with a bar over it), but if you like you can call them cyan, yellow, and magenta, to stick with the analogy.

Also, a key thing to notice here is that on the color wheel, an anticolor is exactly the same as a combination of the two other colors: for example, antired is exactly the same as blue plus green, and antiblue is the same as red plus green. This holds (mostly) for quantum color as well: the theory does not distinguish between, say, antiblue and RG, or antigreen and RB.

What does this mean for us? Well, assuming we trust our theorist, we can now get rid of our huge number of proton states by requiring the proton to be colorless: that is, an equal mixture of R, B, and G. Now instead of 18 proton states, we have only one.** While we’re at it, our new rule also fixes problem (2) from the beginning of the blog: there are now a unique set of ways to obtain colorless hadrons by mixing different color quarks and antiquarks:

(1) Equal mixture of red, blue, and green (RBG): baryon (qqq)

(2) Equal mixture of antired, antiblue, and antigreen: antibaryon (anti-q anti-q anti-q)

(3) Equal mixture of color and anticolor (R anti-R, B anti-B, G anti-G): meson (q anti-q)

And that’s that! These are the only ways to make a colorless hadron, and hence they are the only types of particles allowed. This is why we never see, for instance, a two-quark pair, or a single antiquark: any other combination would have to be colored, and so our rule says it’s out of bounds. Thus, our new color quantum number (along with our rule for using it) solves our problem with the exclusion principle without proliferating the number of hadrons we have, and it also explains why hadrons come only in these three varieties.

Well, okay, so that’s great, and you guys have been good sports about this so far, but I’m guessing at least a few of you are really irritated by this color nonsense. I can hear it now: “Is this seriously how science works? The quark model had a problem, so all we do is patch it together by introducing a new quantum number? Honestly, what’s to stop us from doing this every time we have a problem? A theory seems to violate uncertainty, or relativity, or energy conservation? No problem, don’t throw it away, more quantum numbers should fix it right up.”

If you were thinking this, then good for you. You’re thinking like a responsible scientist should, and you’re even being a little sarcastic about it. In point of fact, when the color idea came out in the mid-60s, a lot of very smart folks thought that the whole thing was just the last gasp of the dying theory of quarks. Even so, I ask you to reserve judgment for another couple of weeks, at which point I’ll talk about what color really means. But the point is well taken, and it’s a good idea to maintain a healthy dose of skepticism about all this stuff.

Speaking of which, you guys aren’t busting my chops nearly enough. You have to make me work for it. If you have questions or comments, let me hear them! Meanwhile, here are a few questions for you:

*In case it helps, here’s a quick review from Rebecca of how colors of light combine. It may help for the questions this week, but remember that it’s only an analogy to our “quantum color”: When you shine all three primary colors of light (red, green, and blue) together, you get white light (look at the center of the Venn diagram). If you just shine red and blue light on a piece of paper, it looks magenta. Likewise, you can make the paper look cyan by shining green and blue light on it, or make it look yellow by shining red and green light on it. You can tell this from the Venn diagram where the circles of red, green, and blue overlap. That’s cool, but what if the paper isn’t white? A red book looks red because it reflects the red (and only the red) light into your eye. But the light shining on the book from the light bulb or the sun is white light, composed of all three primary colors of light. So what happens to the green and blue parts of the white light? The red book only reflects red, so it must absorb the other two colors. The bottom line is that a red book absorbs green and blue (which make up cyan light), so a red book is kinda like anti-cyan.

**Actually, that’s not entirely true. We really have three proton states left: uRuBdG, uBuGdR, and uRuGdB. Thus, we would expect, say, the decay Δ++p π+ to be three times as likely as it would be otherwise, and it turns out this is the case: when computing the probability for this decay using quantum field theory, you must multiply by three to get the right answer.

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