The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)

Appendix B. Asymptotic Freedom

Chapter 8 set out the claim that the color force decreases in strength as we go to higher energy and shorter distances, a property known as asymptotic freedom. This appendix will reveal how asymptotic freedom comes about. This will give me a chance to show you how physicists make use of Feynman diagrams. The color red is represented by a thick line, green by a medium line, and blue by a thin line.

The basic gluon-exchange interaction between two quarks looks like this:


Let’s now try to include the effects of virtual particles on this interaction, which is to say, include the Feynman diagrams involving interactions more complicated than simple gluon exchange. Remember, though, that the more interactions there are in a diagram, the smaller the contribution of that diagram to the final result. Physicists classify the Feynman diagrams by the number of loops that appear in them. To calculate the corrections to the basic gluon-exchange interaction, we need to include two one-loop diagrams. The first is this:


Here a red-antiblue gluon turns into a virtual quark-antiquark pair, denoted by q116. The virtual quark-antiquark pair then annihilates to re-form the gluon before it hits the second quark. Notice that the quark-antiquark pair that gets created can be any flavor: up, down, strange, charm, top, or bottom. So there really should be six diagrams that look exactly the same except for the type of quark in the central loop. Also, notice that the color of the quarks in the loop is fixed: one must be red and the other must be antiblue.

The second process is this:


Here the exchanged gluon splits into two gluons, as allowed by the Feynman rules for the color force (see Chapter 8 and Appendix C). The two gluons then recombine before hitting the second quark. Notice that the color of the gluons in the loop is only partially determined by the colors of the incoming quarks. The left-hand gluon in the loop, for instance, could be red-antigreen, red-antiblue, or red-antired. So, there are really three diagrams like this one: one diagram for each color.

The first of these two processes is the same sort of thing that occurs in QED; for instance, when a photon splits into an electron-positron pair. This process is the origin of the screening effect in QED. Its effect in QCD is the same: it increases the strength of the quark-quark interaction. The second process, though, has the opposite effect: it reduces the strength of the interaction. It is like a tug-of-war with the flavors on one team and the colors on the other. There are as many interaction-increasing diagrams as there are flavors, and as many interaction-reducing diagrams as there are colors. A rather long and messy calculation (the calculation that won Gross, Politzer, and Wilczek the Nobel Prize) reveals that each diagram of the second type contributes an amount five and a half times larger than a diagram of the first type. That is, each player on the color team can pull five and a half times harder than each player on the flavor team. Even though there are only three colors, the color team wins. Taken together, the whole set of diagrams will have the effect of reducing the strength of the quark-quark interaction.

Now, recall that, as the collision energy gets higher, the more complicated Feynman diagrams become more important. That means that the reduction caused by the one-loop diagrams is greater at higher energy than it is at lower energy. This is exactly what we set out to prove: at higher energy the quark-quark interaction gets weaker. Physicists call this asymptotic freedom; the larger the energy of the electron beam used to probe the proton, the more the quarks inside the proton behave like free particles. This property is ultimately responsible for the scaling behavior observed in high-energy inelastic collisions.

For any particle, larger energy means larger momentum. The Heisenberg uncertainty principle (Ax △p > ) implies that larger momentum means shorter distance. That is, a high-energy electron beam probes the short-distance behavior of the color force. On the other hand, when we try to pull quarks apart we are probing the long-distance behavior of the color force. By the same reasoning as before, we should expect that at long distances the strength of the color force increases. Now, however, we run into a problem. The whole Feynman diagram method was based on the idea that the more complicated diagrams contribute less to the final result, and so can be ignored. This is only true as long as the interactions are small, though. If the interactions are large, then the more complicated diagrams could contribute as much as, or more than, the basic gluon exchange diagram. When that happens, all bets are off. The one-loop calculation is no longer reliable, even for an approximate answer. If we were able to include two-, three-, or four-loop diagrams, we still wouldn’t know if we had done enough. Maybe the five-loop contribution is larger than any of those we have already calculated. There is just no way of knowing when it’s safe to stop calculating.

To sum up the discussion, the one-loop approximation becomes increasingly accurate as we go to higher energy collisions and probe shorter distances. This means that asymptotic freedom is a rigorous result of the theory: the higher the energy, the more confidence we have in the approximation we are using. On the other hand, quark confinement is not a rigorous result of the theory. At longer distances, the Feynman diagram method fails. We know that the color force should get stronger for a while, but when it gets too strong the approximation no longer works. There’s no guarantee that the force won’t get weaker again at some even longer distance. Of course, we expect that it just keeps getting stronger, and this is why we’ve never seen a free quark. However, there’s no known way to derive this result from the theory of QCD. As a result, the bound states of quarks and gluons remain a great mystery.