Running coupling is an important quantity in quantum field theory. It identifies the behavior of an interacting theory in the given limit. Indeed, its importance emerged in a full glory after Gross, Wilczek and Politzer showed that QCD becomes a free theory at high energies (asymptotic freedom). For this reason, they were awarded a well deserved Nobel prize in 2004 (see here). Their result implies that the strong interaction coupling goes to zero as the energy involved in the particle interactions becomes larger. Significant experimental confirmations emerged in the course of time but this result already explained the success of Bjorken scaling. The work of Gross, Wilczek and Politzer served also to convince the community that QCD was the right theory to explain strong interactions. This was a long sought result. This matter is so well-acquired today that people is able to work out higher order corrections to this result and found it in close agreement with experimental evidence. So, you can see that this is the story of a great success in physics.
Then, one may ask what happens in the low energy limit. Here we are in troubles as there are no acquired techniques to manage QCD than working with a computer on a lattice. But the situation is not yet at our hands also using computers. The reason relies on the fact that no generally accepted definition of a running coupling constant exists for the low energy limit. But a lot of prejudices exist about. A firm conviction of a large part of the community is that a meaningful running coupling for strong interactions should reach a fixed point as the energy becomes smaller. The reason for this is that, being these interactions so strong, a coupling cannot be zero. Of course, this is wishful thinking and we have no proof whatsoever that things stay that way. So, people work the other way round looking for a definition that grants the existence of a fixed point. An example of this can be found in this paper.
When I read something like this “I reach for my gun” as Hawking would say. The reason is that, in the course of time, we have gathered a lot of expectations and would-be about Yang-Mills theory and QCD in the infrared that we are no more able to distinguish between what is proved and what is not and, mostly, when we are just trying to give a chance to our wishful thinking to be reality. Till now, nobody was able to let us know what are the right excitations of the theory in the low energy limit and so, there is no reason on Earth to believe that the running coupling do not reach zero value lowering the energy. So, your theory could admit bound states with zero charge but surely you do not need a fixed point running coupling here.
But if you look back to all that people believing that glueballs are just bound states of gluons, they are in strong need for an infrared fixed point of the theory. Clearly, they hope that describing the theory with two wrong concepts may turn it into a rigth one. Things do not stay that way and evidence is coming out that such bound states that diagonalize the theory are indeed with zero charge. Bound states anyway.
Some time passed by since Terry Tao was so kind to take a look to my work. His concern about a main theorem in my paper, the so called mapping theorem, was motivated by the fact that no proof exists that there are common solutions between Yang-Mills equations and the one of the quartic scalar field. This point is quite crucial as, if such solutions do not exist, I cannot do any claim about Yang-Mills theory.
Some people are in confusion yet about this matter and I find occasionally someone, e.g. the Czech guy, claiming that my paper is false also after I have proved that such solutions exist.
Of course, Terry meant to point out a weakness in the proof given in my paper as I gave no evidence whatsoever of the form of these solutions and so the proof is, at least, incomplete. My next preprint proved that such solutions indeed exist and my argument is true already at level of perturbation theory. The conclusion is straightforward: Smilga’s choice select a class of common solutions between Yang-Mills equations and a quartic scalar field. I have not presented them explicitly in my paper and this is the reason why all this arguing was started. Terry’s suggestion was to complete the proof and this I have done.
Curiously enough, I was able to see such solutions only in the Smilga’s book. I think this was Smilga’s idea and was also my source of inspiration. I was in need of these solutions to treat classical Yang-Mills equations with a gradient expansion against a lot of unmanageable chaotic solutions. I would like to remember here that this approach is quite common in physics. For interested readers, I invite them to look at this beautiful Wikipedia entry about BKL solution. This is the way this approach is used in general relativity with a widespread example as the Kasner solution. This is an exact solution of Einstein equations that depends solely on time. Exactly as happens to the solutions obtained by a Smilga’s choice from Yang-Mills equations. Indeed, I suspect that Kasner solution may be helpful to quantize Einstein equations in the infrared limit. Currently I have no time to exploit this but I have given a hint about here.
Dmitry Podolsky (see his blog here) hit correctly the point when asked for the fate of chaotic solutions in the infrared quantum field theory. Presently, the fact that they are not relevant has the status of a conjecture: No quantum field theory can be built out of classical chaotic solutions. I do not even know how to face this kind of question as no closed form chaotic solutions exist to start from.
Finally, this gives the current situation about this matter. My paper that started all this is correct and in agreement with current lattice results. People’s mood about lattice computations range from fully convinced to skeptical. My view is that they represent correctly the infrared physics at hand but I am a supporter of these people working on lattice computations and so, my judgement should not be counted.
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.
Two lines to point out this paper, appeared in the latest number of Nature, coming from PAMELA Collaboration. This project is directed by Piergiorgio Picozza. He has been my professor of nuclear physics at Rome University La Sapienza (but all my mistakes about are my own responsability…). PAMELA is a satellite that is orbiting Earth since 1023 days to date (you can find a counter on PAMELA’s site) and is gathering data about cosmic radiation. They have found an enhanced production of positrons. In this last paper they reach the relevant conclusion that this excess may come from dark matter through an annihilation process. This appears a significant evidence for the existence of this exotic matter that we expect to be produced at LHC in the next months as the consequences of the accident happened last year will be definitively fixed. Let me say that I find PAMELA exceedingly sexy as what she is saying to us is really exciting.
Criticisms to present management of science are recurrent claiming that only well-founded research is pursued while the search for new and risky avenues is generally dismissed as there is no revenue, at least in short time, and, in the worst case, any investment may be lost.
About this matter I have found an article on Physics World’s blog (see here). Of course, one can disagree about writer’s arguments but the feeling that we are livng a time of stall is somehow pervasive in some communities. My personal view is that we have recurring periods of hype and a lot of work for preparing them. In a period of hype giant figures emerge but to recognize giants that, nevertheless, prepared the field for the coming revolution era is surely more difficult. It is the same situation we find in soccer where there is a player doing a decisive pass but, in the end, we only remember the one that realized the goal.