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 , 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 . For SU(N), . 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

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