## Confinement revisited

27/09/2012

Today it is appeared a definitive updated version of my paper on confinement (see here). I wrote this paper last year after a question put out to me by Owe Philipsen at Bari. The point is, given a decoupling solution for the gluon propagator in the Landau gauge, how does confinement come out? I would like to remember that a decoupling solution at small momenta for the gluon propagator is given by a function reaching a finite non-zero value at zero. All the fits carried out so far using lattice data show that a sum of few Yukawa-like propagators gives an accurate representation of these data. To see an example see this paper. Sometime, this kind of propagator formula is dubbed Stingl-Gribov formula and has the property to have a fourth order polynomial in momenta at denominator and a second order one at the numerator. This was firstly postulated by Manfred Stingl on 1995 (see here). It is important to note that, given the presence of a fourth power of momenta, confinement is granted as a linear rising potential can be obtained in agreement with lattice evidence. This is also in agreement with the area law firstly put forward by Kenneth Wilson.

At that time I was convinced that a decoupling solution was enough and so I pursued my analysis arriving at the (wrong) conclusion, in a first version of the paper, that screening could be enough. So, strong force should have to saturate and that, maybe, moving to higher distances such a saturation would have been seen also on the lattice. This is not true as I know today and I learned this from a beautiful paper by Vicente Vento, Pedro González and Vincent Mathieu. They thought to solve Dyson-Schwinger equations in the deep infrared to obtain the interquark potential. The decoupling solution appears at a one-gluon exchange level and, with this approximation, they prove that the potential they get is just a screening one, in close agreement with mine and any other decoupling solution given in a close analytical form. So, the decoupling solution does not seem to agree with lattice evidence that shows a linearly rising potential, perfectly confining and in agreement with what Wilson pointed out in his classical work on 1974. My initial analysis about this problem was incorrect and Owe Philipsen was right to point out this difficulty in my approach.

This question never abandoned my mind and, with the opportunity to go to Montpellier this year to give a talk (see here), I presented for the first time a solution to this problem. The point is that one needs a fourth order term in the denominator of the propagator. This can happen if we would be able to get higher order corrections to the simplest one-gluon exchange approximation (see here). In my approach I can get loop corrections to the gluon propagator. The next-to-leading one is a two-loop term that gives rise to the right term in the denominator of the propagator. Besides, I am able to get the renormalization constant to the field and so, I also get a running mass and coupling. I gave an idea of the way this computation should be performed at Montpellier but in these days I completed it.

The result has been a shocking one. Not only one gets the linear rising potential but the string tension is proportional to the one obtained in d= 2+1 by V. Parameswaran Nair, Dimitra Karabali and Alexandr Yelnikov (see here)! This means that, apart from numerical factors and accounting for physical dimensions, the equation for the string tension in 3 and 4 dimensions is the same. But we would like to note that the result given by Nair, Karabali and Yelnikov is in close agreement with lattice data. In 3 dimensions the string tension is a pure number and can be computed explicitly on the lattice. So, we are supporting each other with our conclusions.

These results are really important as they give a strong support to the ideas emerging in these years about the behavior of the propagators of a Yang-Mills theory at low energies. We are even more near to a clear understanding of confinement and the way mass emerges at macroscopic level. It is important to point out that the string tension in a Yang-Mills theory is one of the parameters that any serious theoretical approach, pretending to go beyond a simple phenomenological one,  should be able to catch. We can say that the challenge is open.

Marco Frasca (2011). Beyond one-gluon exchange in the infrared limit of Yang-Mills theory arXiv arXiv: 1110.2297v4

Kenneth G. Wilson (1974). Confinement of quarks Phys. Rev. D 10, 2445–2459 (1974) DOI: 10.1103/PhysRevD.10.2445

Attilio Cucchieri, David Dudal, Tereza Mendes, & Nele Vandersickel (2011). Modeling the Gluon Propagator in Landau Gauge: Lattice Estimates of Pole Masses and Dimension-Two Condensates arXiv arXiv: 1111.2327v1

M. Stingl (1995). A Systematic Extended Iterative Solution for QCD Z.Phys. A353 (1996) 423-445 arXiv: hep-th/9502157v3

P. Gonzalez, V. Mathieu, & V. Vento (2011). Heavy meson interquark potential Physical Review D, 84, 114008 arXiv: 1108.2347v2

Marco Frasca (2012). Low energy limit of QCD and the emerging of confinement arXiv arXiv: 1208.3756v2

Dimitra Karabali, V. P. Nair, & Alexandr Yelnikov (2009). The Hamiltonian Approach to Yang-Mills (2+1): An Expansion Scheme and Corrections to String Tension Nucl.Phys.B824:387-414,2010 arXiv: 0906.0783v1

## QCD is confining

12/10/2011

At Bari Conference , after I gave my talk, Owe Philipsen asked to me about confinement in my approach. The question came out also in the evening, drinking a beer at a pub in the old Bari. Looking at my propagator, it is not so straightforward to see if the theory is confining or not. But we know, from lattice computations, that this must be so. You can realize this from the following figure (see here)

The scale is given by $r_0=0.5\ fm$, the so called Sommer’s scale, We note a clear linear rising till about 1.5 fm. A linear rising potential is an evidence of confinement as showed about forty years ago by Kenneth Wilson (see here) with his famous area law. Due to this clear evidence coming from lattice computations, any attempt to explain mass gap must show confinement through a linear rising potential.

Indeed, this is not all the story and going to 1.5 fm cannot be enough to display all the behavior of a Yang-Mills theory. As showed quite recently on the lattice Philippe de Forcrand and Slavo Kratochvila (see here), increasing distance, the potential must saturate. This is an effect of the mass gap that causes screening. This means that, at larger distances, the potential sets on an asymptote becoming horizontal. The linear approximation holds on a finite range.

This is indeed what I observe with my approach. I can prove that the potential has a Yukawa form with a form factor dependent on the distance. The mass scale entering into it is just the mass gap. So, you get a linear fit like the following (see here)

that shows confinement with the area law till 10 fm! If one increases the distance the fit worsens and saturation appears as expected. From this we can easily derive the string tension that is given by $(g^2/4\pi)C_2 0.000507/r_0^2$. For SU(N), $C_2=(N^2-1)/2N$. This is a fine proof of confinement for a Yang-Mills theory and so, for QCD too. This also means that my approach is again consistent with lattice data. Just for completeness, and to give a great thank to Arlene Aguilar and Daniele Binosi, I show the fit of my propagator with the one obtained numerically solving Dyson-Schwinger equations (see here)

The agreement is almost perfect.

Gunnar S. Bali (2000). QCD forces and heavy quark bound states Phys.Rept.343:1-136,2001 arXiv: hep-ph/0001312v2

Wilson, K. (1974). Confinement of quarks Physical Review D, 10 (8), 2445-2459 DOI: 10.1103/PhysRevD.10.2445

Slavo Kratochvila, & Philippe de Forcrand (2003). Observing string breaking with Wilson loops Nucl.Phys. B671 (2003) 103-132 arXiv: hep-lat/0306011v2

Marco Frasca (2011). QCD is confining arXiv arXiv: 1110.2297v1

A. C. Aguilar, D. Binosi, & J. Papavassiliou (2008). Gluon and ghost propagators in the Landau gauge: Deriving lattice
results from Schwinger-Dyson equations Phys.Rev.D78:025010,2008 arXiv: 0802.1870v3

## Low-energy effective Yang-Mills theory

22/03/2011

As usual I read the daily from arxiv and often it happens to find very interesting papers. This is the case for a new paper from Kei-Ichi Kondo. Kondo was in Ghent last year (here his talk) and I have had the chance to meet him. His research is on very similar lines as mine. A relevant paper by him is about the derivation of the Nambu-Jona-Lasinio model from QCD (see here) with a similar hindsight I exposed in recent papers (see here and here). This new paper by Kondo presents a relevant attempt to derive a consistent low-energy effective Yang-Mills theory from the full Lagrangian. The idea is to decompose the gauge field into two components and integrate away the one that just contributes to the high-energy behavior of the theory. Kondo shows how a mass term could be introduced at the expenses of BRST symmetry breaking. This symmetry can be recovered at the cost of nilpotency. But this mass term is gauge invariant and gives rise to a meaningful propagator for the theory. Then, the computations show how Wilson’s area law is satisfied granting quark confinement and positivity reflection for the gluon propagator is violated granting gluon confinement too. The gluon propagator is then given in a Gribov-Stingl form

$D(p)=\frac{1+d_1p^2}{c_0+c_1p^2+c_2p^4}$

but this form is only recovered if the mass term is introduced in the original Yang-Mills Lagrangian as said above. It is interesting to note that, if this is the right propagator, a Nambu-Jona-Lasinio model could anyhow be derived taking the small momenta limit. A couple of observations are in order here. Firstly, Cucchieri and Mendes fits often recover this functional form (e.g. see here). Last but not least, this functional form is acausal but produces a confining potential increasing with the distance. But even if the mass term would be zero and no Gribov-Stingl form is obtained, Kondo shows that area law still holds and one has confinement yet. As a final conclusion, Kondo shows that his effective model describes confinement through a dual Meissner effect, a hypothesis that come out at the dawn of the studies in QCD.

This paper represents a fine piece of work. A point to be clarified is, given from other studies and lattice computations that gluon mass arises dynamically, how this approach should change and, most important, how the form of the propagator changes. I just suspect that my conclusions about this matter would be recovered.

Kei-Ichi Kondo (2011). A low-energy effective Yang-Mills theory for quark and gluon confinement arxiv arXiv: 1103.3829v1

Kei-Ichi Kondo (2010). Toward a first-principle derivation of confinement and
chiral-symmetry-breaking crossover transitions in QCD Phys. Rev. D 82, 065024 (2010) arXiv: 1005.0314v2

Marco Frasca (2010). Glueball spectrum and hadronic processes in low-energy QCD Nucl.Phys.Proc.Suppl.207-208:196-199,2010 arXiv: 1007.4479v2

Marco Frasca (2008). Infrared QCD International Journal of Modern Physics E 18, (2009) 693-703 arXiv: 0803.0319v5

Attilio Cucchieri, & Tereza Mendes (2009). Landau-gauge propagators in Yang-Mills theories at beta = 0: massive
solution versus conformal scaling Phys.Rev.D81:016005,2010 arXiv: 0904.4033v2

## A confirmation again

08/01/2009

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.