## QCD is confining

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

### 5 Responses to QCD is confining

1. Philip Gibbs says:

Very nice.
In the first plot if r0 is 0.5fm doesn’t the axis go to 1.5fm rather than 6fm?

• mfrasca says:

Fixed. Thank you very much, Phil.

2. ohwilleke says:

Interesting paper, although, to be honest, the non-ideomatic English constructions make it challenging to read in a number of places. For the most part it follows the formal grammatical rules, but in phrasings that a native speaker would never use or with sentences that really need to be broken up for clarity. I would be happy to suggest some edits of phrasing and usage to make it read more naturally if you wouldn’t find this offensive and this would be a benefit to you. But, I wouldn’t want to make those kinds of suggestions if you would only find them annoying or distracting from the business of actually doing physics.

My one other suggestion is that it would be helpful to add to the introduction a paragraph or two that would help to frame your discussion more clearly by formally defining the term “infrared trivial fixed point” and explaining briefly why this has been an issue in low energy QCD scholarship and hence why a reader should care whether there is one or not, with a reference or two to the literature in support of these background matters. Obviously, most of your readers will be familiar with the terminology and context, but an introductory framing paragraph along that line would make the paper start more smoothly, instead of abruptly marching onward in the middle of the conversation. It would be particularly useful to include that kind of material in this paper, because very general title that you have given it invites a broader audience than some of your more technical work.

None of this, of course, takes away from the substance, which is a reassuring one, elegantly presented, that confirms that the different people trying to describe the elephant by touching only parts of it are really describing the same elelphant.

3. Daniel de França MTd2 says:

Hi Marco!

I don’t want to sound dismissive or arrogant, but your paper sounds like claiming the 1 million dollar prize…

• mfrasca says:

Hi Daniel,

Are you fine? No, I don’t. I am just pushing my approach to its logical consequences. Nothing more. The agreement with numerical data is astonishing good and so, it is my view that all this is worthwhile to be pursued.

Best,

Marco