Running coupling in QCD

For a quantum field theory it is essential to know the form of the beta function. The reason for this is that this function gives us immediately an understanding on how the theory will behave in different regimes of the energy range. Currently, there is a lot of activity to obtain the full beta function of QCD, the theory of quarks and gluons. The reason for this is that we want to understand the behavior of this theory when the energy becomes lower while we know that at higher energy asymptotic freedom sets in moving the coupling toward zero so that perturbation theory applies. QCD appears as the perfect theory. Indeed, in this case we have no Landau pole or, possibly, no triviality at lower energies and so this appears as the only example in physics of a theory that holds on the full range of energy. I have read three papers about this matter recently (see here, here and here) and all of them give a clear insight about the question of the beta function for Yang-Mills theory and QCD. As my readers know, lattice computations for a pure Yang-Mills theory show clearly that the running coupling goes to zero at lower energies so one may ask if the same can happen when quarks are involved.

Running coupling from lattice computations

From supersymmetric QCD it has been shown that the beta function should have the form

where, apart for some costants here and there, is the number of quark flavors. So, we immediately realize that supersymmetric QCD can develop a fixed point in the infrared limit even if supersymmetric Yang-Mills theory has none! Sannino and Ryttov have been inspired by this beautiful result and proposed a similar beta function for ordinary QCD and the conclusion is the same: Even if Yang-Mills theory has no fixed point in the infrared, QCD has one due to the presence of quarks. So far, lattice computations for fully QCD confirm this scenario and we can be confident that this theory is the most beautiful one being meaningful for all the energy range.

I would like to add a final comment by noting that Ryttov and Sannino give for Yang-Mills theory in the infrared the same functional form I have got in my latest paper (see here)  and that agrees with lattice results. The picture of low-energy QCD is slowly emerging providing to us quite unexpected results and a deeper comprehension of the world as we perceive it.

Pioneers

Jeffrey Mandula is a well-knwon theoretical physicist whose main result, Coleman-Mandula theorem, opened the road to the discovery of supersymmetry. But Mandula is also known for his pioneering works on lattice QCD. Looking back to old papers on the question of the Yang-Mills propagators, I have found two beautiful papers by Mandula and Ogilvie (see here and here) published in the eighties, where they arrive to the following conclusion

From the behavior of the gluon propagator reported here, it appears that in pure Yang-Mills theory, a dynamical Higgs phenomenon occurs. Our best estimate of the effective gluon mass, as determined at large distances, is about 600 MeV, with finite size effects, possible scaling violations, and statistical uncertainties of at least . In analogy with the concept of a constituent quark mass, it may be useful to think of the mass in the gluon propagator as a constituent gluon mass. The massiveness of the gluon may be connected to the apparent suppression of many-gluon intermediate states in decay, and the relative absence of the mixing between the lowest quark model states and those with gluonic excitations.

They used small lattices due to the computer limitations at that time and ideas about infrared behavior of Yang-Mills theory were just beginning to flourish. But, with our hindsight, we should emphasize the deep intuition that these authors put forward when such analysis were just starting. Besides, resonance was not yet seen and their estimate of the gluon mass appears really good.

As you may know, after these works, things took a different turn and for a long time since now we have been coping with a different scenario from that devised by Mandula and Ogilvie that took the scenes and not yet left them. This scenario appears today to be in a serious difficulty against lattice computations but people do not generally agree about what the right view should be, making painfully slow truth achievement .

A beautiful paper on arxiv

Keeping on their way of producing sound work, Bogolubsky, Ilgenfritz, Mueller-Preussker and Sternbeck have got their paper (see here) published on Physics Letters B. This is a collaboration between people working in Russia, Germany and Australia. The main aim of this work is the computation on a lattice of the two-point functions and the running coupling of a pure Yang-Mills theory. They carry on lattice computations from to points entering into a deep enough infrared limit to get a meaningful behavior of the lattice theory in this case. I give below their main results

Gluon propagator

Dressing function of the ghost propagator

Running coupling

These results confirm completely the decoupling solution. The definition of the running coupling is the one proposed by Alkofer and von Smekal and it is my personal conviction that it conveys the right physical behavior of the theory. This is exactly the scenario I have derived in my paper (see here) that has been published on Physics Letters B too and has arisen a lot of rumors around. You will not find this paper cited in this work as these authors have concerns about gauge invariance in my computations. As you may know from my dispute with Terry Tao, gauge invariance is not a problem here. One could ask why a mathematical technique, like a gradient expansion is, should not work for Yang-Mills equations but it does for all other equations of mathematical physics. Anyhow, I am here ready to listen to whoever is able to prove this. With this proof in hand one should also warn all general relativists that use this technique and put it in their handbooks.

The authors conclude their paper by pointing out weaknesses in lattice computations that may bring in discussion their results. Finally, they ask if the other solution, the one with a scaling behavior, can emerge from lattice computations. The understanding of this question is surely of relevant interest. We stay tuned to hear news about.

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 (). 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 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 for 3 dimensions and 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

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 , the string tension. Notwithstanding this operative choice, they get for the gluon mass the following values

using only data with and

using all data. I hope that now some bell is ringing for you as this is the mass of the 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.

Yang-Mills theory in D=2+1

There is a lot of work about the pursuing of a deep understanding of Yang-Mills theory in the low energy limit. The interesting case is in four dimensions as our world happens to have such a property. But we also know that a Yang-Mills theory in D=2+1 is not trivial at all and worthwhile to be studied. In this area there has been a lot pioneering work mostly due to V. Parameswaran Nair and Dimitra Karabali . These authors proved that a Hamiltonian formulation may be truly effective to manage this case. Indeed, they obtained a formula for the string tension that works quite well with respect to lattice computations. We would like to remember that, in D=2+1, coupling constant is such that its dimension is while, in D=3+1, is dimensionless.

Quite recently, some authors showed how, from such a formulation, a functional can be given from which one can obtain the spectrum (see here, here and here). These papers went all published on archival journals. Now, these spectra are quite good with respect to lattice computations, after some reinterpretation. We do not know if this is due to some problems in lattice computations or in the theoretical analysis. I leave this to your personal point of view. My idea is that this quenched lattice computations are missing the true ground state of the theory. This happens to be true both for D=3+1 and D=2+1. I do not know why things stay in this way but in this kind of situations are always theoreticians to lose. On the other side, being a physicist means that one should not have a blind faith in anything.

Finally, one may ask how my work performs with respect all this. Yesterday, I spent a few time to try to figure this out. The results I obtain agree fairly well with those of the theoretical papers. I obtain the zero Lionel Brits gets at being m a mass proportional to ‘t Hooft coupling. Brits wrote the third of the three papers I cited above. The string tension I get is in agreement with lattice computations. This zero is the problem on lattice computations and the same problem is seen in D=3+1. This fact is at the root of our presenting difficulty to understand what resonance is. We know that people working on a quenced lattice computation for the propagator do see this resonance. This difference between this two approaches should be understood and an effort in this direction must be made.

The question of the running coupling

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.

A confirmation again

One of my main activities in the morning is reading the daily coming from arxiv. Sometime it happens to find significant papers to be put in a post like this. This morning I have found a beautiful paper by a cooperation of people from Germany, Russia and Australia working on lattice QCD (see here). This paper has been written by Igor Bogolubsky, Ernst-Michael Ilgenfritz, André Sternbeck and Michael Mueller-Preussker. I put here the following picture representing one of the main conclusions

This picture gives the gluon propagator with a number of points (96)^4 and shows clearly that it reaches a finite value at smaller momenta implying a massive gluon. Indeed, the authors of the paper extended the lattice computations moving from (80)^4 to (96)^4 points and add some other improvement in the computation itself. The value of beta is quite high being 5.7. The agreement with previous computations of Cucchieri and Mendes is excellent (see here). These latter authors worked with a number of points of (128)^4 while beta was taken to be 2.2.

The other two important conclusions they reach is that the ghost propagator goes like that of a free particle and the running coupling goes to zero at lower momenta. For the running coupling we emphasize that there is no common agreement about its definition in the infrared and the authors properly point out this. But a running coupling that goes to zero does not mean at all that there is no confinement. Quite the contrary as proved by Kazuhiko Nishijima (see here): It gives a proof of confinement.

So, we obtain again a clear proof of the scenario we have already obtained from a theoretical standpoint (see here and here) and we have discussed at length in this blog. I think that evidence of existence of the mass gap both on lattice and from theory are becoming overwhelming. We are just wating the dust to settle down and textbooks reporting these findings.

Update: After an email exchage with Andre Sternbeck he gave further clarifications about his group work correcting something not correct in the post. I post here his corrigenda:

“Our study was for the gauge group SU(3) and not for SU(2). That is
the reason why the Beta-Value is larger than that used for SU(2) by
Cucchieri et al. and by myself et al. in 2007. The lattice spacings are
roughly of the same order, but the numerical effort spent for a 96^4
lattice in SU(3) is much bigger than what had been necessary in SU(2).”

I take this chance to thank him a lot for his comments.

A shocking step

Lorenz von Smekal has been one of the proponents of the functional approach to the understanding of infrared Yang-Mills theory. He is currently working at University of Adelaide where a lot of important work on lattice computations is performed. Today on arxiv appeared a paper by him (see here). I would like to report here his words in the introduction

“Without infrared enhancement of the ghosts in Landau gauge, the global gauge charges of covariant gauge theory are spontaneously broken. Within the language of local quantum field theory the decoupling solution can thus only be realised if and only if it comes along with a Higgs mechanism and massive physical gauge bosons. The Schwinger mechanism can in fact be described in this way, and it can furthermore be shown that a non-vanishing gauge boson mass, by whatever mechanism it is generated, necessarily implies the spontaneous breakdown of global symmetries.”

The “decoupling solution” cited here is the one currently seen on lattice computations having a finite gluon propagator at zero momenta and a ghost propagator behaving like a free particle without any fixed point in the running coupling. The point here is that, in this paragraph, the truth about the real situation of Yang-Mills theory in the infrared is simply exposed. Classical solutions exist that display such dynamical generation of mass for the massless scalar field theory and Yang-Mills theory (see here) and a quantum field theory can be built with them making the above argument truly consistent.

But the point to be emphasized here is the proposal of von Smekal arriving to present a modification of lattice computations. His proposal relies on a recent work done with Andre Sternbeck (see here) where they study the limit of the Yang-Mills theory. Indeed, in this limit they recover the results obtained by functional methods that disagree with lattice computations. Again, they simply freeze the dynamics and get meaningless results as also happens when one compares D=1+1 Yang-Mills theory with no dynamics and the D=4 case. Besides, recent QCD computations on the lattice, relying on known formulations of the Yang-Mills side, give too striking results to go to look for reformulations (see my post).

My view is that functional methods are generally useless. Also when the right solution is hit, of course numerically, one is not able to do any kind of real calculation in QCD. In physics this means that no true understanding is reached. One of the points that should have warned people working with functional methods is that no mass gap is ever obtained and there is no way to recover the low-energy phenomenology of QCD. But having a mass gap produces immediately a Nambu-Jona-Lasinio model from QCD from the ratio to the square of the mass gap itself and this is a real understanding as Nambu-Jona-Lasinio model gives a lot of comprehension of low energy phenomenology.

I think this paper is worth an in depth reading as it contains several pieces of true awareness. My criticisms should not be of any concern for such a good work.

Wonderful QCD!

On Science this week appeared a milestone paper showing two great achievements by lattice QCD:

• QCD gives a correct description of low energy phenomenology of strong interactions.
• Most of the ordinary mass (99%) is due to the motion of quarks inside hadrons.

The precision reached has no precedent. The authors are able to get a control of involved  errors in such a way to reach an agreement of about 1% into the computation of nucleon masses. Frank Wilczek gives here a very readable account of these accomplishments and is worthwhile reading. These results open a new era into this kind of method to extract results to be compared with experiments for QCD and give an important confirmation to our understanding of strong interactions. But I would like to point out Wilczek’s concern: Until we will not have a theoretical way to obtain results from QCD in the low energy limit, we will miss a great piece of understanding of physics. This is a point that I discussed largely with my posts in this blog but it is worthwhile repeating here coming from such an authoritative voice.

An interesting point about these lattice computations can be made by observing that again no resonance is seen. I would like to remember that in these computations entered just u, d and s quarks as the authors’ aims were computations of bound states of such quarks. Some authoritative theoretical physicists are claiming that this resonance should be a tetraquark, that is a combination of u and d quarks and their antiparticles. What we can say about from our point of view? As I have written here some time ago, lattice computations of the gluon propagator in a pure Yang-Mills theory prove that this can be fitted with a Yukawa form

being . This is given in Euclidean form. This kind of propagators says to us that the potential should be Yukawa-like, that is

if this is true no tetraquark state can exist for lighter quarks. The reason is that a Yukawa-like potential heavily damps any van der Waals kind of residual potential. But, due to asymptotic freedom, this is no more true for heavier quarks c and b  as in this case the potential is Coulomb-like and, indeed, such kind of states could have been seen at Tevatron.

We expect that the glueball spectrum should display itself in the observed hadronic spectrum. This means that a major effort in lattice QCD computations should be aimed in this direction now that such a deep understanding of known hadronic states has been reached.

Screening masses in SU(3) Yang-Mills theory

Thanks to a useful comment by Rafael Frigori (see here) I become aware of a series of beautiful papers by an Italian group at Universita’ della Calabria. I was mostly struck by a recent paper written by R. Fiore, R. Falcone, M. Gravina and A. Papa (see here) that appeared in Nuclear Physics B (see here). This paper belongs to a long series of works about the behavior of Yang-Mills theory at non-null temperature and its critical behavior. Indeed, using high-temperature expansion and Polyakov loops one arrives at the main conclusion that the ratio between the lowest and the higher state of the theory must be 3/2. This ratio depends on the universality class the theory belongs to and so, on the kind of effective theory one has in the proper temperature limit (below or above ). It should be said that, in order to get a proper verification of the above prediction, people use lattice computations. Fiore et al. use a lattice of points and, as all this kind of computations are done on lattices having such a dimension, one can cast some doubt about the fact that the true ground state of the theory is really hit. Indeed, this happens in all this kind of computations done to get a glueball spectrum that seem at odd with those giving the gluon propagator producing a lower screening mass at about 500 MeV (see my post here). A state at about 500 MeV is seen at accelerator facilities as resonance or f0(600) but is not predicted by any lattice computation. One of the reasons to reduce lattice volume is that one can reach higher values of granting the reaching of a non-perturbative regime, the one interesting for us.

What can we say about this ratio with our theory? We have put on arxiv a paper that answer this question (see here). These results were also presented at QCD 08 in Montpellier (see here). We assume that the cannot be seen at such small volumes but its excited state can be obtained. This implies that one can exchange the with the lowest state and as the higher one. Then this ratio gives exactly 3/2 as expected. We can conclude on the basis of this analysis that this ratio is the same independently on the temperature but, the one to be properly measured is given in the paper of Craig McNeile (see here) that gives close agreement between lattice and theoretical predictions.

So, we would like to see lattice computations of Yang-Mills spectra at lower lattice spacing and increased volumes granting in this way the proper value of the ground state. This is overwhelming important in view of the fact that no real understanding exists of the existence of the resonance with lattice computations. This will implies, as discussed above, a deeper understanding of the spectrum of the theory also at higher temperatures.