A Quantum Diaries Survivor

This morning at the plenary session of the QCHS7 conference M.Shifman presented a talk on the 1/N expansion of QCD. I did try my best to follow the presentation, and actually grasped something, partially thanks to the nature of the speaker, who was careful to keep technicalities at a minimum and discuss the matter on general grounds.

So what is the 1/N expansion of QCD ? First of all let me explain what QCD is.

Basically, QCD is the theory of strong interactions. It was developed in the early seventies by similarity with QED, the theory of electromagnetic interactions that was by then very well understood and confirmed by a huge number of precise experiments. But QCD is different, because of the property of the carriers of the interaction, the gluons. There are eight gluons as opposed to a single carrier in QED, the photon. The gluons are sources of the strong field themselves, and because of that the theory has a much more complex nature. In fact, despite the more than thirty years of hard work, it is still far from being completely understood.

The most basic reason why QCD is difficult to handle – particularly in the prediction of low energy phenomena – is that it is a “strongly coupled” theory: that is to say, the theory describes a strong interaction, and because of that the calculations involving perturbation series (the computational method that allows to determine the observable phenomena) cannot be worked out at low energy, where the coupling constant is too large. It is as if you were trying to compute a certain quantity by first producing a rough approximation, and then adding small corrections to improve it: this procedure works if the corrections are small! That is the case of QED, but in QCD the corrections are too large and you are prevented from computing things to a reasonable accuracy.

Ok, now let’s go to Shifman’s talk. I have taken notes during his discussion, and I will report them here. Things are a bit too technical for the casual reader, which is excused at this point if he leaves. And my summary of Shifman’s talk is probably full of mistakes, which are entirely my own – do not blame Shifman for them!

A new development was from Gervais and Sakita in the 80ies, then Dashen-Manohar in the 90ies. According to them, you do not need the Skyrme models. Good predictions may come from algebra for N large.  They could obtain many predictions in one-to-one agreement with skyrme models. All are predictions to the LO, and coincide with old days SU(6) models of spin-flavor symmetry. For instance, g_(pND)/g_(pNN)=3/2+O(1/N^2), and experimentally it is measured at 1.5, and m_n/m_p = -2/3 and is measured at -0.68. Another way to expand in N is a Supersymmetry-motivated high N limit. At N=3 when we have three colors you can consider the quark being either in fundamental SU(3) representation, or in two index antisymmetric representation of SU(3), because it is identical, so in SU(3) it does not matter. But one can see what happens by going to large N, where you can analytically continue by keeping quarks in the antisymmetrical two index representation. Because this representation has Casimir operators which do not get suppressed with 1/N, you see that the mechanism eliminates the suppression of the quark loops. So, is that good or bad ? In some problem it is better to use one expansion (phenomena of mesons which are scalar or pseudoscalar mesons). All flavors are mixed, and the ‘t Hooft expansion predicts this. On the other hand, the new expansion improves the situation with the widths, for instance widths of quarkonia are expected to be narrow but glueballs even more in the ‘t Hooft expansion, and in the new expansion the widths of the glueballs and quarkonia become of the same order of magnitude and this helps understanding the empirical situation in which we are now.