A sound confirmation of Yang-Mills scenario

Today arxiv contains a very interesting paper by Attilio Cucchieri and Tereza Mendes (see here). They do a significant lattice computation for a SU(2) Yang-Mills theory in 3 and 4 dimensions. Quarks are absent (quenched approximation). Their aim is to verify the current emerging scenario in a particular situation where the coupling on the lattice is taken to go to infinity (\beta=0). This case, as pointed out by the authors, is unphysical but is a quite interesting test-bed for the behavior of the two-point functions of the theory (gluon and ghost propagators). They do not aim to check the running coupling as this, currently, is matter of hot debate yet.

I would like to remember what is at stake today about this question.  For some years there has been the strong conviction that the gluon propagator should go to zero and ghost propagator should go to infinity faster than the free propagator (conformal solution). This was needed to confirm two confining scenarios that were commonly accepted by the community. Most of these results emerged from an idea due to Gribov that there remains an ambiguity in the gauge also after it is fixed. Taking into account Gribov ambiguity provoked a flourishing of papers all reaching similar conclusions. At that time, due to the small achievable volumes, lattice computations were not able to clarify the situation even if the gluon propagator was never seen to tend to zero in a significant way. As increasing volumes made available a completely new scenario emerged. People obtained that the gluon propagator indeed reaches a finite value at zero momentum while the ghost propagator is seen to behave as that of a free particle (decoupling solution). A commonly accepted definition of running coupling was seen to converge to zero making the theory trivial.

This emerging data prompted several explanations. People started to criticize these lattice computations as maybe there was an accumulation of Gribov copies that modify the right results into the observed ones. Maybe the fixing gauge algorithms should be better analyzed and so on. People from Australia (see here and here) claimed that low energy data should not be trusted. Discarding them one finds again the conformal solution. Cucchieri and Mendes give a sound answer to all these doubts. Indeed, it is not clear why in 2 dimensions one gets the conformal solutions but not in 3 and 4 dimensions, notwithstanding the code used to do these computations is always the same. Further, any reason adduced by Australian group to remove low energy data is proved substantially unfounded and the results obtained by Cucchieri and Mendes represent a correct picture of the case \beta=0 for 3 and 4 dimensions. Indeed, Cucchieri and Mendes show that Gribov copies play no role in the scenario seen at low energies for Yang-Mills theory. This is a crucial point that has been source for misleading research for a lot of years.

So, let us take a look at the scenario found by Cucchieri and Mendes. These authors consider a maximum lattice dimensions of 100^3 for 3 dimensions and 64^4 for 4 dimensions. They show without any doubt that one gets the decoupling solution: The gluon propagator reaches a finite value at zero momentum and the ghost propagator is that of a free particle. What is more interesting here are the fits. For the propagator they fit to a sum of Stingl-forms

D(x)=\sum_{i=1,2}c_i\cos(b_i+\lambda_i x)e^{-\lambda_i x}

and for the ghost propagator


being m the gluon mass. The authors tried to avoid to fix the values of their computations with experimental data. As you know, the relevant parameter here is \sqrt{\sigma}, the string tension. Notwithstanding this operative choice, they get for the gluon mass the following values

m=0.499 GeV

using only data with p^2<4 GeV and

m=0.466 GeV

using all data. I hope that now some bell is ringing for you as this is the mass of the \sigma resonance. This resonance is not seen by people doing quenched computations to obtain the spectrum of a pure Yang-Mills theory. Why? What are they missing with respect to Cucchieri and Mendes? This should not become a longstanding question. We need an answer right now.

Now, take a look at the fit of the gluon propagator. Try to do a Fourier transform and you will get back something like


and this is shockingly similar to my propagator having the general form


that is a sum of free particle propagators!

I should say that I am somewhat impressed by Cucchieri and Mendes results. They showed that the decoupling scenario is the right one in the physical case of 4 dimensions. My view is that we should move on from the current position and try to find the theoretical framework that better fits the data. It goes without saying what is the one I am supporting.


6 Responses to A sound confirmation of Yang-Mills scenario

  1. Daniel de França MTd2 says:

    Why excluding quarks doesn’t change the result? And shouldnt the group be SU(3)?

    • mfrasca says:

      Hi Daniel,

      I proved quite recently that quark effects are higher order in a 1/g series, see


      But, even if things would not stay that way, to know the exact gluon propagator gives immediately a Nambu-Jona-Lasinio model that describes quite well low-energy phenomenology.

      SU(2) or SU(3) does not change anything as has been proved in studies with Dyson-Schwinger equations. It is just more convenient from a computational standpoint.



  2. Daniel de França MTd2 says:

    If from SU(2) to SU(3) doesn’t change anything, why not going to SU(N), N->infinity, and using one of those ADS/CFT dualities?

  3. mfrasca says:

    AdS/CFT has presently the status of a conjecture and is looking to people working on lattice for confirmations, not the other way round. Anyhow, the limit N\rightarrow\infty is generally studied on the lattice to see how far it differs from the physical case N=3. But when you turn to the gluon and ghost propagators, the only meaningful object you work with is ‘t Hooft coupling, Ng^2, and so, you can judge by yourself how relevant can be varying N with respect to the behavior of the propagators at low momenta.

  4. rafael says:

    Hi Marco,

    yes, it clearly shows how hot the debate about scalling/decoupling solution is getting!
    I still have some disturbing points, which lattice data failled to probe:

    (1) 2d data (from Maas) used \beta much bigger than the values used in 3d and 4d (by Adelaide and São Carlos groups). So, 2d simulations are nearer to continuum… so, why is it contradictory to 3d and 4d?
    (2) There is no consensus about how well-controlled systematics on gauge-fixing, continuum and thermodynamical limits have to be to set a real answer.
    (3) If the fundamental-modular-Gribov reagion is really important for confinement, well, it is not yet proved that Canonical-ensemble answers (from lattice) will be even able to probe that (i.e. canonical and micro-canonical ensembles may be inequivalent for V->infinity).

    I think there is a puzzling state nowadays on this area, much more rigorous work has to be performed before lattice comunity gets a consensus.


  5. mfrasca says:

    Hi Rafael,

    I have an answer to your point 1) and if you have something about it please, let me know. Yang-Mills theory in D=1+1 is trivial and has no dynamics. This is a fact known since ’70s after ‘t Hooft solved QCD in this case. You know that, even if there is no propagating degree of freedom for Yang-Mills, full QCD in this case is interesting yet. The conformal solution, obtained with a kind of truncation of the Dyson-Schwinger hierarchy, corresponds exactly to this case: No propagating degrees of freedom. People that worked in this way just removed any dynamics from the theory in D=3 and 4 making it identical to the two-dimensional case. Currently, people is missing this elementary point about all this matter. There are other severe criticisms for the conformal solution but let me stop here.

    About point 2), this is too technical and a matter for lattice specialists. I cannot help.

    About point 3), I should say that Gribov ideas, on which derivations of the conformal solution are based, proved till now to be a blind alley. I am convinced that, here, lattice computations are saying the right thing. The right thing is that we can live happily without considering Gribov copies and horizons both in the infrared and in the ultraviolet.



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