Running coupling and Yang-Mills theory


Forefront research, during its natural evolution, produces some potential cornerstones that, at the end of the game, can prove to be plainly wrong. When one of these cornerstones happens to form, even if no sound confirmation at hand is available, it can make life of researchers really hard. It can be hard time to get papers published when an opposite thesis is supported. All this without any certainty of this cornerstone being a truth. You can ask to all people that at the beginning proposed the now dubbed “decoupling solution” for propagators of Yang-Mills theory in the Landau gauge and all of them will tell you how difficult was to get their papers go through in the peer-review system. The solution that at that moment was generally believed the right one, the now dubbed “scaling solution”, convinced a large part of the community that it was the one of choice. All this without any strong support from experiment, lattice or a rigorous mathematical derivation. This kind of behavior is quite old in a scientific community and never changed since the very beginning of science. Generally, if one is lucky enough things go straight and scientific truth is rapidly acquired otherwise this behavior produces delays and impediments for respectable researchers and a serious difficulty to get an understanding of the solution of  a fundamental question.

Maybe, the most famous case of this kind of behavior was with the discovery by Tsung-Dao Lee and Chen-Ning Yang of parity violation in weak interactions on 1956. At that time, it was generally believed that parity should have been an untouchable principle of physics. Who believed so was proven wrong shortly after Lee and Yang’s paper. For the propagators in the Landau gauge in a Yang-Mills theory, recent lattice computations to huge volumes showed that the scaling solution never appears at dimensions greater than two. Rather, the right scenario seems to be provided by the decoupling solution. In this scenario, the gluon propagator is a Yukawa-like propagator in deep infrared or a sum of them. There is a very compelling reason to have such a kind of propagators in a strongly coupled regime and the reason is that the low energy limit recovers a Nambu-Jona-Lasinio model that provides a very fine description of strong interactions at lower energies.

From a physical standpoint, what does it mean a Yukawa or a sum of Yukawa propagators? This has a dramatic meaning for the running coupling: The theory is just trivial in the infrared limit. The decoupling solution just says this as emerged from lattice computations (see here)

What really matters here is the way one defines the running coupling in the deep infrared. This definition must be consistent. Indeed, one can think of a different definition (see here) working things out using instantons and one see the following

One can see that, independently from the definition, the coupling runs to zero in the deep infrared marking the property of a trivial theory. This idea appears currently difficult to digest by the community as a conventional wisdom formed that Yang-Mills theory should have a non-trivial fixed point in the infrared limit. There is no evidence whatsoever for this and Nature does not provide any example of pure Yang-Mills theory that appears always interacting with Fermions instead. Lattice data say the contrary as we have seen but a general belief  is enough to make hard the life of researchers trying to pursue such a view. It is interesting to note that some theoretical frameworks need a non-trivial infrared fixed point for Yang-Mills theory otherwise they will crumble down.

But from a theoretical standpoint, what is the right approach to derive the behavior of the running coupling for a Yang-Mills theory? The answer is quite straightforward: Any consistent theoretical framework for Yang-Mills theory should be able to get the beta function in the deep infrared. From beta function one has immediately the right behavior of the running coupling. But in order to get it, one should be able to work out the Callan-Symanzik equation for the gluon propagator. So far, this is explicitly given in my papers (see here and refs. therein) as I am able to obtain the behavior of the mass gap as a function of the coupling. The relation between the mass gap and the coupling produces the scaling of the beta function in the Callan-Symanzik equation. Any serious attempt to understand Yang-Mills theory in the low-energy limit should provide this connection. Otherwise it is not mathematics but just heuristic with a lot of parameters to be fixed.

The final consideration after this discussion is that conventional wisdom in science should be always challenged when no sound foundations are given for it to hold. In a review process, as an editorial practice, referees should be asked to check this before to kill good works on shaky grounds.

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3

Ph. Boucaud, F. De Soto, A. Le Yaouanc, J. P. Leroy, J. Micheli, H. Moutarde, O. Pène, & J. Rodríguez-Quintero (2002). The strong coupling constant at small momentum as an instanton detector JHEP 0304:005,2003 arXiv: hep-ph/0212192v1

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory PoS FacesQCD:039,2010 arXiv: 1011.3643v3


Igor Suslov and the beta function of the scalar field


I think that blogs are a very good vehicle for a scientist to let his/her work widely known and can be really helpful also for colleagues doing research in the same field. This is the case of Igor Suslov at Kapitza Institute in Moscow. Igor is doing groundbreaking research in quantum field theory and, particularly, his main aim is to obtain the beta function of the scalar field in the limit of a very large coupling. This means that the field of research of Igor largely overlaps mine. Indeed, I have had some e-mail exchange with him and we cited our works each other. Our conclusions agree perfectly and he was able to obtain the general result that, for very large bare coupling \lambda one has


where d is the number of dimensions. This means that for d=4 Igor recovers my result. More important is the fact that from this result one can draw the conclusion that the scalar theory is indeed trivial in four dimensions, a long sought result. This should give an idea of the great quality of the work of this author.

On the same track, today  on arxiv Igor posted another important paper (see here). The aim of this paper is to get higher order corrections to the aforementioned result. So, he gives a sound initial explanation on why one could meaningfully take the bare coupling running from 0 to infinity and then, using a lattice formulation of the n components scalar field theory, he performs a high temperature expansion.  He is able to reach the thirteenth order correction! This is an expansion of \beta(\lambda)/\lambda in powers of \lambda^{-\frac{2}{d}} and so, for d=4, one gets an expansion in 1/\sqrt{\lambda}. Again, this Igor’s result is in agreement with mine in a very beautiful manner. As my readers could know, I have been able to go to higher orders with my expansion technique in the large coupling limit (see here and here). This means that my findings and this result of Igor must agree. This is exactly what happens! I was able to get the next to leading order correction for the two-point function and, from this, with the Callan-Symanzik equation, I can derive the next to leading order correction for \beta(\lambda)/\lambda that goes like 1/\sqrt{\lambda} with an opposite sign with respect to the previous one. This is Igor’s table with the coefficients of the expansion:

So, from my point of view, Igor’s computations are fundamental for all the understanding of infrared physics that I have developed so far. It would be interesting if he could verify the mapping with Yang-Mills theory obtaining the beta function also for this case. He did some previous attempt on this direction but now, with such important conclusions reached, it would be absolutely interesting to see some deepening. Thank you for this wonderful work, Igor!

I. M. Suslov (2011). Renormalization Group Functions of \phi^4 Theory from High-Temperature
Expansions J.Exp.Theor.Phys., v.112, p.274 (2011); Zh.Eksp.Teor.Fiz., v.139, p.319 (2011) arXiv: 1102.3906v1

Marco Frasca (2008). Infrared behavior of the running coupling in scalar field theory arxiv arXiv: 0802.1183v4

Marco Frasca (2010). Mapping theorem and Green functions in Yang-Mills theory arxiv arXiv: 1011.3643v2

Sannino and the mass gap in Yang-Mills theory


August is vacation month in Italy and I am not an exception. This is the reason for my silence so far. But, of course, I cannot turn off my brain and physics has always been there. So, reading the daily from arxiv today , I have seen another beautiful paper by Francesco Sannino (you can find his page here) in collaboration with Joseph Schechter that has been his PhD thesis advisor. As you know, Sannino and Ryttov postulated an exact beta function for QCD starting from the exact result in the supersymmetric version of this theory (see here).  The beta function Sannino and Schechter get has a pole. The form is


and, taken as is, this has no fixed point than the trivial one g=0. We know that this seems in agreement with recent lattice computations even if, discussing with Valentin Zakharov at QCD10 (see here), he expressed some skepticism about them.  They point out that the knowledge of this function permits a lot of interesting computations and what they do here is to get the mass gap of the Yang-Mills theory. They also point out as, for all the observables obtainable from such a beta function, the pole is harmless and the results appear really meaningful. Indeed, they get a consistent scenario from that guess.

So, let me point out the main results obtained so far by these people using this approach:

  • The beta function for Yang-Mills theory goes to zero with the coupling without displaying non-trivial fixed point but QCD has a non-trivial fixed point (see my paper here).
  • Yang-Mills theory has a mass gap.

Numerically their result for the mass gap seems to agree quite well with lattice computations. This should be also the mass for a possible observation of the lightest glueball. My view about is that the lightest glueball is the \sigma resonance and recent findings at KLOE-2 seems to point out in this direction. But the exact value of the mass gap is not so relevant. What is relevant is that these researchers have found a quite interesting exact form of the beta function for QCD that describes quite well the current understanding of this theory at lower energies that is slowly emerging.



One of the questions that is not that easy to answer is: When is a quantum field theory exactly solved? Of course, we have the example of a free theory. When one is able to put the generating functional into a Gaussian form, the spectrum of the theory is that of a harmonic oscillator and when the coupling is zero, one is left with a possibly solved theory. But this case is trivial and does not say anything about the case of an exactly solved but interacting quantum field theory. An immediate answer to this question is: When one is able to get all the n-point functions. This implies that, if you are able to solve all the hierarchy of Dyson-Schwinger equations, you are done. Solving this set of equations is practically impossible in almost all the interesting case. But there is an exception and a notable one. So, consider the case of a massless quartic scalar field theory. Lattice computations in d=3+1 strongly hint toward triviality in the low-energy limit. Better, for d>3+1 there is a beautiful proof by Michael Aizenman that went published here. In this case the hierarchy is exactly solved (see here) and this is true also for d=3+1. So far, triviality and exact solution indeed are the same thing. But why does an interacting theory become trivial? The reason is in the behavior of the running coupling as the energy varies. We have learned from quantum field theory that couplings have not always the same value. Rather, their value is varying depending on the energy scale they are measured. In a trivial theory, couplings happen to go to zero in the given limit and an interacting theory becomes free!

For the scalar theory in the low-energy limit (infrared) in d=3+1, evidence is becoming wider that the beta function, the function that determines the behavior of the running coupling, goes like


being \lambda the coupling and d space-time dimension. I have proved this firstly here for d=3+1 but other authors arrived to an identical conclusion by different means (see here and here). But there is a surprise here: Some authors, a few years ago, proved an identical result for Yang-Mills theory (see here) with lattice computations. So, this is again a striking proof of the correctness of my mapping theorem but an indirect one. Then, we can conclude this post by stating a shocking result: Yang-Mills theory is trivial in the infrared even if QCD is not. But this result is enough to make QCD manageable at very low-energies.

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