In a recent post of mine (see here) I have pointed out to you a beautiful paper by Dan Pirjol and Carlos Schat. This paper is now appeared on Physical Review Letters (see here) but you can find also the preprint on arxiv (see here). I think its content is really important as it gives a serious clue toward our understanding of low-energy QCD. Dan agreed to publish here a contribution about his work and I am glad to post it.
We find that, within experimental errors on the hadron masses, the
so-called gluon-exchange model (OGE) is disfavored by data. This is probably not very surprising, since at low energies the real degrees of freedom of QCD should include, in addition to the gluons, also pions (the Goldstone bosons of the spontaneously broken chiral symmetry). The OGE model does not include the pion exchange effects; an alternative to the OGE model which includes their effects is the so-called GBE (Goldstone boson exchange) model.
There has been a long-standing debate about the most appropriate model
of quark forces in the framework of the constituent quark model, in particular about their spin-flavor dependence. The main candidates are
the OGE and GBE models (see e.g. the second paper in Ref. for a
discussion of this controversy). Our letter attempts to resolve this controversy using only minimal assumptions about the orbital dependence
of the hadronic wave functions. More precisely, we assume only isospin
symmetry, but no other assumption is made about the form of the wave
functions. The novel mathematical tool which makes our analysis possible is the application of the permutation group , which allows one to study the implications of the most general spin-flavor structure of the quark forces.
In a short time, Physical Review Letters will publish a shocking paper by Dan Pirjol and Carlos Schat with a proof of the fact that a simple gluon exchange model for bound states of QCD does not work. The preprint is here. The conclusions drawn by the authors imply that one cannot expect a simple idea of free gluons exchanged by quarks to work. I think the readers of this blog may be aware of the reason why this conclusion is correct and PRL will publish an important paper.
The idea is that, in the low energy limit, the nonlinearities in the Yang-Mills equations modify completely the properties of the glue. We can call these excitations gluons only in the high energy limit were asymptotic freedom grants that nonlinearities can be treated as small perturbations and gluons are what we are acquainted with. But when we have to cope with bound states, we are in a serious trouble as our knowledge of this regime of QCD is really very few helpful for our understanding.
This result of Pirjol and Schat should be taken together with the measurements of the COMPASS Collaboration (see my post) about the spin of the protons. They proved that glue does not contribute to form the spin of the proton. Collecting together all this a conclusion to be drawn is that the high energy excitations of a Yang-Mills theory cannot be the same of the excitations in the low energy limit.
So, let’s move on and take a better look at our equations.
Recently I have read a post in Dmitry’s blog by Fabien Buisseret claiming the following conclusion:
“In the present post were summarized various arguments showing that the glueballs and gluelumps currently observed in lattice QCD can be understood in terms of bound states of a few transverse constituent gluons. In this scheme, the lowest-lying glueballs can be identified with two-gluon states, while the lightest negative-C glueballs are compatible with three-gluon states.”
Indeed he considers free gluons interacting each other through a given potential forming bound states. Of course, as all of you may be aware, nobody in the Earth was able to prove that, in the low energy limit, gluons are the right states entering into a quantum Yang-Mills theory. So, this view appears as a well rooted prejudice in the community.
Let me explain what I mean with a classical example. I take the following quartic theory
In the small coupling limit you will get plane waves plus higher order corrections. Assume these plane waves are gluons as we all of us is aware from high-energy QCD. Indeed, these plane waves describe massless excitations. Now I claim that these solutions should hold also when the coupling becomes increasingly large. But here I have the exact solution
being sn a Jacobi snoidal function and an arbitrary constant. But now
and I am describing massive excitations that are not resembling at all my plane wave solutions given above. The claim is blatantly wrong already at a classical level with this very simple example.
This proves without any doubt that the view of glueballs as bound states of gluons is plainly wrong as nobody knows the behavior of a Yang-Mills theory in the infrared limit and so, nobody knows what are the right glue excitations for the theory here. As you may have realized, if you would know this you will be just filed for a Millenium Prize. This means that, unless we learn how to treat the theory at low energies, all this kind of approaches are doomed.
In my preceding post I discuss the question of the proton spin, experimental measurements and the relevant conclusion that these measurements are consistent with the fact that there is no glue contribution to spin. Rather, we have a partial contribution by quarks and the remanent should be due to orbital angular momentum. I concluded from this that the infrared carriers of the strong force are not properly gluons but rather spin 0 excitations. In this post I would like to expand on this matter to make clear that all this is plainly obtained from QCD and so, fully consistent with the theory. Anyhow, it should be remembered that the quark sea for nucleons plays a relevant role in this case making quarks dressed.
The question is rather simple. At higher energy QCD tends to become a free theory, that is the coupling becomes increasingly small and the gluon propagator one uses at the tree level is that of a free particle. This in turn means that the non-linear contributions from Yang-Mills theory are small and small perturbation theory applies. In this limit we can identify as the excitations of Yang-Mills theory with ordinary gluons carrying spin one.
In the infrared limit, the case of low energies, the behavior of Yang-Mills theory changes radically. The reason is that in this case the non-linear terms in the equations become so strong that ordinary gluons are no more the fundamental excitations of the theory. In this case one has glueballs and the lower end of the spectrum of the glueballs carries spin zero. This is the reason why COMPASS Collaboration see no spin contributions from glue.
One should see such things in the same way quasi-particles are seen in condensed matter. The free particles that make the theory are not the one of the free theory but will depend upon the way interactions act on them. So, in the ultraviolet limit we can safely call them gluons and in the infrared limit things are quite different. In this light, it would be interesting to try to see similar measurements for charmonium and see in this case what is the contribution of glue. It may happen that this physical situation is radically different from that of the nucleon.
There is currently a beautiful puzzle to be answered that relies on sound and beautiful experimental results. The question is how the components of a proton, that is quarks and gluons, concur to determine the value one half for the spin of the particle. During the conference QCD 08 at Montpellier I listened to a beatiful presentation of Joerg Pretz of the COMPASS Collaboration (see here and here). Hearing these results was stunning for me. I explain the reasons in a few words. The spin of the proton should be composed by the spin of the quarks, the contributions of gluons (gluons???) and orbital angular momentum. What happens is that the spin of quarks does not contribute too much. People then thought that the contribution of gluons (gluons again???) should have been decisive. The COMPASS Collaboration realized a beautiful experiment using charmed mesons. This experiment has been described by Pretz at QCD 08. They proved in a striking way that the contribution of the glue to proton spin can be zero and cannot be used to account for the particle spin. Of course, there are beautiful papers around that are able to explain how the proton spin comes out. I have found for example a paper by Thomas and Myhrer at Jefferson Lab (see here and here) that describes quite well an understanding of the puzzle and surely is worthwhile reading. But my question is another: Why the glue does not contribute?
From our preceding posts one should have reached immediately an answer, the same that come out to my mind when I listened Pretz’s talk. The reason is that, in the infrared, gluons that have spin one are not the true carriers of the strong force. The true carriers have no spin unless higher excited states are considered. This explains why COMPASS experiment did not see any contribution consistently with previous expectations.
This is again a strong support to our description of the gluon propagator (see here). No other theory around shows this.