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### World of Glue

I’m a bit overdue for my next post introducing the Standard Model through Feynman diagrams, so let’s continue our discussion of the theory the subnuclear “strong” force, quantum chromodynamics, or QCD. This is the force that binds quarks into protons and neutrons, and the source of many theoretical and experimental headaches at the LHC.

For a listing of all the posts in this series, see the original post here.

Last time we met the quarks and “color,” a kind of charge that can take three values: red, green, and blue. Just as the electromagnetic force (which has only one kind of charge) pulls negatively charged electrons to positively charged protons to form electrically neutral atoms, QCD forces the quarks to be confined into color-neutral bound states:

• Three quarks, one of each color. These are called baryons. Common examples are the proton and neutron.
• A quark and an anti-quark of the same color (e.g. a red and anti-red quark). These are called mesons.

Collectively baryons and mesons are called hadrons. Because the strong force is so strong, these color-neutral bound states are formed very quickly and they are all we see in our detectors. Also like the electromagnetic force, the strong force is mediated by a particle: a boson called the gluon, represented in plush form below (courtesy of the Particle Zoo)

Gluons are so named because they glue together mesons and baryons into bound states. In Feynman diagrams we draw gluons as curly lines. Here’s our prototypical gluon-quark vertex:

We see that the gluon takes an incoming quark of one color and turns it into an outgoing quark of another color. According to the rules associated with Feynman diagrams, we can move lines around (always keeping the orientation of arrows relative to the vertex the same!) and interpret this vertex as

• A red quark and a blue anti-quark (with color anti-blue) annihilate into a gluon
• A red quark emits a gluon and turns into a blue quark
• A red quark absorbs a gluon and turns into a blue quark
• A gluon decays into a blue quark and a red anti-quark

As a simple homework, you can come up with the two interpretations that I’ve left out.

Let us make the following caveats:

1. The quarks needn’t have different colors, but one arrow has to be pointing in while the other is pointing out. (The quarks carry electric charge, and remember that one way to understand the arrows is as the flow of electric charge.)
2. The quarks involved in the vertex can have any flavor (up, down, strange, charm, bottom, top), but both must have the same flavor. This is because QCD is “flavor blind,” it treats all of the flavors equally and doesn’t mix them up. (Compare this to the W boson!)
3. The interpretations for the vertex above are all correct, except that none of them are allowed kinematically. This is because the gluon is massless. In other words, conservation of energy and momentum are violated if you consider those processes. This is for precisely the same reason that we couldn’t have single-vertex photon diagrams in QED.

Homework: up an down quark scattering by gluons. Draw all the diagrams for the following processes that are mediated by a single gluon (i.e. only contain a single internal gluon line)

1. uu uu (one diagram)
2. u anti-u u anti-u (two diagrams)
3. u anti-u d anti-d (one diagram)
4. u anti-d u anti-d (one diagram)

You may assign colors as necessary (explain why it matters or does not matter). Why is it impossible to draw a u anti-d d anti-u diagram (note that this process is allowed if you replace the gluon by a W)? [Hint: you might want to review some of our discussions about QED and muons; the diagrams are all very similar with photons replaced by gluons!]

We can continue to make analogies to QED. We explained that the QED vertex had to be charge neutral: an arrow pointing inwards carries some electric charge, while the arrow pointing outward must carry the same electric charge. The gluon vertex above is electrically neutral in this sense, but does not seem to be color neutral! It brings in some red-charge while splitting out some blue charge.

The resolution is that gluons themselves carry color charge! This is now very different from QED. It’s a little bit like the W boson, which carried electric charge and so could interact with photons. We can see from the vertex above that the gluon must, in fact, carry two charges: in that example the gluon carries an incoming blue charge and and outgoing red charge; or, in other words charge blue and charge anti-red. These are the charges that it must carry in order for the vertex to be color neutral.

Thus there are actually many types of gluons which we can classify according to the color and anti-color which they can carry. Since there are three colors (and correspondingly three anti-colors), we expect there to be nine types of gluons. However, for somewhat mathematical reasons, it turns out that there are only eight.

The mathematical reason is that the gluons are in the adjoint representation of SU(3) and so number only 32-1 = 8. They are associated with the space of traceless Hermitian 3×3 matrices. The “missing” gluon is the quantum superposition of “red/anti-red + blue/anti-blue + green/anti-green.” If that’s all gibberish to you, then that’s okay—these are little details that we won’t need.

The fact that gluons themselves carry color charge means something very important: gluons feel the strong force, which means that gluons interact with other gluons! In other words, we can draw three- and four- gluon vertices:

There are no five or higher vertices, but as homework you can convince yourself that from these vertices you can draw diagrams with any number of external gluons. In fancy schmancy mathematical language, we say that QCD is non-Abelian because the force mediators interact with themselves. (In fact, the weak force is also non-Abelian, but its story is one of broken dreams which we will get to when we meet the Higgs.)

Now, you might wonder: if the strong force is so strong that gluons bind quarks together, and if gluons also interact with themselves, is it possible for gluons to bind each other into some kind of bound state? The answer is yes, though we have yet to confirm this experimentally. The bound states are called glueballs and can be pretty complicated objects. Theoretically we have good reasons to believe that they should exist (and eventually decay into mesons and baryons), and very sophisticated simulations of QCD have also suggested that they should exist… but experimentally they are very hard to see in a detector and we have yet to confirm any glueball signature. Very recently some theorists have suggested that there might have been hints at the BES collider in Beijing.

Gluon hunting, however, is something of a lower energy frontier since our predictions for the lightest glueball masses are around 1.7 GeV; so don’t expect anything from the LHC on this. It’s worth remarking, on the other hand, about a mathematical issue regarding a world of glue. (This is a bad pun for a computer game that I like.) The question is: if the universe had no matter particles and only gluons—which would then form glueball bound states—are there any massless particles observable in nature? Sure, the gluons themselves are massless, but they’re not observable states; only glueballs are observable. Everything we know about QCD—which isn’t the whole story—suggests that glueballs always have some non-zero mass, but we don’t know how to prove this. This question, in fact, is one of the Clay Mathematics Millennium Prize Problems, making it literally a million dollar question.

Next time: the wonderful world of hadrons and what we can actually detect at the LHC.

Cheers,
Flip, US LHC blog