The question of the conformal solution

There is currently a lot of activity to understand the behavior of the two-point functions in a Yang-Mills theory when quarks are not considered. The relevance of these results relies on the possibility to get working tools to manage low-energy phenomenology of QCD. E.g. we know quite well that the Nambu-Jona-Lasinio model is very successful to describe the behavior of hadronic matter and this model can be easily derived from full QCD if one knows the gluon propagator and the behavior of the ghost field. This idea dates back to a paper by Terry Goldman and Richard Haymaker (see here) on 1981. This can be summed up by saying that: given the gluon propagator one can get back a Nambu-Jona-Lasinio model. One could be able to derive it directly from QCD and this would be a great achievement explaining a lot of work done in several decades by a lot of smart researchers.

In these days it seems that we have to cope with a couple of scenarios and long lasting debate is still alive about them. One of this, the conformal solution, has been supported by a lot of researchers for a long period of time, making sometime very difficult for other scenarios to be commonly accepted and being published on archival journals. The conformal solution is easy to describe with the following sentences:

  • The gluon propagator goes to zero at lower momenta.
  • The ghost propagator goes to infinity at lower momenta faster than the free case.

Supporters of this solution also claimed that the running coupling reaches a fixed point at zero momenta. This fixed point was strongly lowered as lattice computations easily showed that things do not stay this way.

Indeed, lattice computations showed without doubt, increasing the volume, as the gluon propagator reaches a finite value at zero momenta and never converges toward zero. The ghost propagator is seen to behaves exactly as that of a free particle. But notwithstanding such an evidence, much effort is still devoted to understand why the conformal solution is not seen in such computations. A lot of hypotheses were put forward to explain why Yang-Mills propagators do not behave in such a well acquired way.

Axel Maas had a bright idea on this way to understand. He turned his attention to the D=2 model on the lattice (see here). Nobody did this before as this model is known to be trivial as it has no dynamics at all. This was proved by ‘t Hooft long ago. Maas showed that the 2D model on the lattice gives back the conformal solution. This means that a Yang-Mills model without dynamics has two-point functions behaving like those of the conformal solution. Most people saw this as a strong support to the conformal solution but things should be seen the other way round. Removing dynamics makes the conformal solution appear and lattice computations in 3 and 4d, that use the same code, are telling us the right behavior of the two-point functions. This is a serious setback for this approach and my personal view should be that any effort to further support this kind of solution should be  abandoned as is a loss of resources and precious time. But this could be a very difficult choice for a lot of people that for a long time worked on this and contributed to create this scenario.

In my opinion, the most severe drawback of this solution is that it goes against the original idea by Goldman and Haymaker that the gluon propagator should give back the Nambu-Jona-Lasinio model. Indeed, the conformal solution does not even give back a mass gap and it is practically useless to answer to a lot of open questions in the low-energy phenomenology. In some way it remembers to me the position  of the bootstrap model against the gauge paradigm that in the end proved to be the more fruitful approach. The last paper by Cucchieri and Mendes (see here) proves without doubt that the right form is a sum of Yukawa propagators and this gives back immediately a Nambu-Jona-Lasinio model.

As I cannot see reasons for people supporting the conformal solution to surrender, I think a lot of time will pass yet before truth will be acquired. Meantime, we stay here looking at the fight between such  strong contenders.


2 Responses to The question of the conformal solution

  1. rafael says:

    Hi Marco,

    I would like to mention two important papers:
    (1) : here authors show why there is a difference behavior among 2d, 3d & 4d gluon propagators in terms of auxiliary (and well controlled) lattice observables.
    (2) : it explains why a massive gluon propagator is preferred by phenomenology.


    • mfrasca says:

      Hi Rafael,

      I was aware of Cucchieri and Mendes’ paper. They have been so kind to cite my paper on Yang-Mills propagators. This paper is a cornerstone in the way to understand infrared behavior.

      About Natale (and I would add Aguilar) work I have something to say. Indeed, they wrote a cornerstone paper that went published (after some difficulties as Natale said to me) on the Journal of High-Energy Physics

      This paper shows, numerically, how Dyson-Schwinger equations indeed produce the right solution for infrared Yang-Mills. But they are stuck on Cornwell’s solution and did not catch in full what they were able to obtain on that paper. Cornwell’s solution is a beautiful hindsight on the right path to understand Yang-Mills theory in the infrared. But this solution has some severe shortcomings that make it quite difficult to support. One of the problems is that this solution requires a running coupling reaching a fixed point in the infrared and all the efforts Aguilar and others are doing is trying to get such a fixed point.

      Quite recently, Prosperi’s group at University of Milano (Italy), showed, in a paper published in Physical Review Letters, that the coupling tends to zero at lower momenta and they did this from experimental data. See

      and also (appeared on Physical Review D)

      My personal view is that no such fixed point exists and evidence is mounting in this direction. This result was also obtained, some time ago, with lattice computations, by Boucaud’s group. See

      This is a great paper supporting the conclusion that the running coupling goes to zero as the fourth power of momenta (this is in agreement with my propagator see but here I am doing self-promotion and someone may complain 🙂 ).

      Cornwell’s solution has another severe drawback and this is the form of the gluon propagator itself. This does not grant a proper understanding of low-energy phenomenology and, as such, is quite similar to the conformal solution.

      My view is that, sometime, one should pay more attention to her own results and try to pursue a personal way to understanding. Indeed, you can check my recent published papers. I get a propagator that matches perfectly the numerical results of Natale and Aguilar. See and also to appear shortly in International Journal of Modern Physics E.



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