August is vacation month in Italy and I am not an exception. This is the reason for my silence so far. But, of course, I cannot turn off my brain and physics has always been there. So, reading the daily from arxiv today , I have seen another beautiful paper by Francesco Sannino (you can find his page here) in collaboration with Joseph Schechter that has been his PhD thesis advisor. As you know, Sannino and Ryttov postulated an exact beta function for QCD starting from the exact result in the supersymmetric version of this theory (see here). The beta function Sannino and Schechter get has a pole. The form is

and, taken as is, this has no fixed point than the trivial one . We know that this seems in agreement with recent lattice computations even if, discussing with Valentin Zakharov at QCD10 (see here), he expressed some skepticism about them. They point out that the knowledge of this function permits a lot of interesting computations and what they do here is to get the mass gap of the Yang-Mills theory. They also point out as, for all the observables obtainable from such a beta function, the pole is harmless and the results appear really meaningful. Indeed, they get a consistent scenario from that guess.

So, let me point out the main results obtained so far by these people using this approach:

The beta function for Yang-Mills theory goes to zero with the coupling without displaying non-trivial fixed point but QCD has a non-trivial fixed point (see my paper here).

Yang-Mills theory has a mass gap.

Numerically their result for the mass gap seems to agree quite well with lattice computations. This should be also the mass for a possible observation of the lightest glueball. My view about is that the lightest glueball is the resonance and recent findings at KLOE-2 seems to point out in this direction. But the exact value of the mass gap is not so relevant. What is relevant is that these researchers have found a quite interesting exact form of the beta function for QCD that describes quite well the current understanding of this theory at lower energies that is slowly emerging.

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3 Responses to Sannino and the mass gap in Yang-Mills theory

“The beta function for Yang-Mills theory goes to zero with the coupling without displaying non-trivial fixed point but QCD has a non-trivial fixed point.”

I thought QCD was a YM theory… Why does that happen?
In this way, QCD is just like Asymptotic Gravity!

So, we have a kind of KLT relation, without susy, relating a QCD theory and gravity. Hmm. Sometimes I think that there are maps between some finite non perturbative non supersymmetric theories into finite perturbative supersymmetric theories.

I just skimmed through the paper and I saw no mention of a non trivial fixed point. And they identified QCD with N=3 YM, as it should be, which also means that it has a trivial point but not a non trivial.

QCD is not just a Yang-Mills theory but a Yang-Mills field coupled to a certain number of Fermion fields, quarks. This difference is not a trivial one and this is seen in the supersymmetric version of this theory. It happens that, a pure Yang-Mills field, has just a trivial fixed point in the infrared and this is clearly seen in lattice computations. When a number of quarks is introduced, this trivial fixed point moves toward a value different from zero. The guessed beta function by Ryttov and Sannino displays exactly this behavior. You can see my previous post about this matter that I linked here.

“The beta function for Yang-Mills theory goes to zero with the coupling without displaying non-trivial fixed point but QCD has a non-trivial fixed point.”

I thought QCD was a YM theory… Why does that happen?

In this way, QCD is just like Asymptotic Gravity!

So, we have a kind of KLT relation, without susy, relating a QCD theory and gravity. Hmm. Sometimes I think that there are maps between some finite non perturbative non supersymmetric theories into finite perturbative supersymmetric theories.

I just skimmed through the paper and I saw no mention of a non trivial fixed point. And they identified QCD with N=3 YM, as it should be, which also means that it has a trivial point but not a non trivial.

Dear Daniel,

QCD is not just a Yang-Mills theory but a Yang-Mills field coupled to a certain number of Fermion fields, quarks. This difference is not a trivial one and this is seen in the supersymmetric version of this theory. It happens that, a pure Yang-Mills field, has just a trivial fixed point in the infrared and this is clearly seen in lattice computations. When a number of quarks is introduced, this trivial fixed point moves toward a value different from zero. The guessed beta function by Ryttov and Sannino displays exactly this behavior. You can see my previous post about this matter that I linked here.

Marco