The question of the running coupling


Today I was reading a PhD thesis about matters we frequently discuss in this blog (see here). This is a very good work. But when I have come to the question of the running coupling I was somewhat perplexed. Indeed, there is a recurring wishful thinking about running coupling in a Yang-Mills. This prejudice claims that coupling in the low momenta limit should reach a non-trivial fixed point for the theory to be meaningful. Then, if you read the literature since the inception of the success of gauge theories you will read a myriad of papers claiming this “fact” that is not a fact having been never proved.

In this case we have two kind of evidences: lattice and experimental. These evidences show that the coupling at low momenta goes to zero, that is the theory is free also in the infrared! This is a kind of counterintuitive result as are all the results that are coming out from lattice computations. The reason for this relies on the fact that Yang-Mills theory is a scalar theory in disguise and so shares the same fate. But maybe, the most interesting result comes from Giovanni Prosperi and his group at University of Milan. They studied the meson spectrum and showed how the running coupling derived from measurements bends clearly toward zero. Their work has been published on Physical Review Letters (see here and here). They do this studying quarkonium spectra, a matter we discussed extensively in this blog. Their paper has been enlarged and published on Physical Review D (see here and here).

On the lattice the question is linked to the behavior of the gluon and ghost propagators. We have seen that the gluon propagator reach a non-null constant as momentum goes to zero and the ghost propagator behaves as that of a free particle. This means that if we write


for the gluon propagator and


for the ghost propagator, being Z(p^2) and F(p^2) the dressing function, following Alkofer and von Smekal we can define a running coupling as


but the gluon propagator reaches a non-null value for p\rightarrow 0 and so Z(p^2)\sim p^2 and the ghost propagator goes like that of a free particle and so F(p^2)\sim 1. This means nothing else that \alpha(p^2)\rightarrow 0 at low momenta. This is lattice response.

So, why with all this cumulating evidence people does not yet believe it? The reason relies on the fact that is very difficult to remove prejudices and truth takes some time to emerge. We have to live with them for some time to come yet.

Quarkonia and Dirac spectra


In these days we are discussing at length the question of heavy quarkonia, that is bound states of heavy quark-antiquark and we have got a perfect agreement for their ground states assuming a potential in the form


being \sigma=(0.44GeV)^2 the string tension for Yang-Mills theory. This potential was derived here and here. We derived it in the limit of small distances and this means that excited states and states with higher angular momentum can fail to be recovered and the full potential without any approximation should be used instead. Anyhow, our derivation of ground states was in the non-relativistic approximation. We want to check here the solution of Dirac equation to get a complete confirmation of our results and, as an added bonus, we will derive also the mass of  B_c that is a bottom-charm meson. As said we cannot do better as to go higher excited states we need to solve Dirac equation with the full potential, an impossible task unless we recur to numerical computations.

So, let us write down the Dirac spectrum for a heavy quark-antiquark state:




We apply this formula to charmonium, bottomonium and toponium obtaining

m_{\eta_c}=2977 MeV

against the measured one m_{\eta_c}=2979.8\pm 1.2 MeV and

m_{\eta_b}=9387.5 MeV

against the measured one 9388.9 ^{+3.1}_ {-2.3} (stat) +/- 2.7(syst) MeV and, finally

m_{\eta_t}=344.4 GeV

that confirms our preceding computation. The agreement is absolutely striking. But we can do better. We consider a bottom-charm meson B_c and the Dirac formula




m_{B_c}=6.18 GeV

against the PDG average value 6.286\pm 0.005 GeV the error being about 2%!

Our conclusion is that, at least for the lowest states, our approximation is extremely good and confirms the constant originating from our form of gluon propagator that was the main aim of all these computations. The implications are that quarkonia could be managed with our full potential and Dirac equation on a computer, a task surely easier than solving full QCD on a lattice.

The interquark potential


In our initial post about quarkonia we have derived the interquark potential from the gluon propagator. In this post we want to deepen this matter being this central to all hadronic bound states. The gluon propagator is given by





m_n = (2n+1)\frac{\pi}{2K(i)}\sqrt{\sigma}

with \sqrt{\sigma}=0.44 GeV being \sigma the string tension that is just an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally. We have obtained this propagator in a series of papers starting from a massless scalar theory. The most relevant of them is here. It is immediate to recognize that this propagator is just an infinite superposition of Yukawa propagators. But the expectations from effective theories are quite different (see Brambilla’s CERN yellow report here). Indeed, a largely used interquark potential is given by

V(r)=-\frac{a}{r}+\sigma r +b

but this potential is just phenomenological and not derived from QCD. Rather, as pointed out by Gocharov (see here) this potential is absolutely not a solution of QCD. We note that it would be if the linear term is just neglected as happens at very small distances where

V_C(r)\approx -\frac{a}{r}+b.

We can derive this potential from the gluon propagator imposing p_0=0 and Fourier transforming in space obtaining

V(r)=-\frac{\alpha_s}{r}\sum_{n=0}^\infty B_n e^{-m_n r}

and we can Taylor expand the exponential in r obtaining

V(r)\approx -\frac{\alpha_s}{r}+Ar+b

but we see immediately that A=0 and so no linear term exists in the potential for heavy quarkonia! This means that we can formulate a relativistic theory of heavy quarkonia by solving the Dirac equation for the corresponding Coulombic-like potential whose solution is well-known and adding the b constant. We will discuss such a spectrum in future posts.

For lighter quarks the situation is more involved as we have to take into account the full potential and in this case no solution is known and one has to use numerical computation. But solving Dirac equation on a computer should be surely easier than treating full QCD.

Ground state of toponium


Following the series of posts I started after the beautiful result of BABAR collaboration, now I try to get a prevision for a new resonance, i.e. the ground state of \bar t t quarkonium that is known in literature as toponium. This resonance has a large mass with respect to the others due to t quark being about 37 times more massive than b quark. In this case we have a theoretical reference by Yuri Goncharov (see here) published in Nuclear Physics A (see here). Goncharov assumes a mass for the top quark of 173.25 GeV and gets \alpha_s(m_t)\approx 0.12. He has a toponium ground state mass of 347.4 GeV. How does our formula compare with these values?

Let us give again this formula as


being \sqrt{\sigma}=0.44 GeV. We obtain

m_{\eta_t}=345.9 GeV

that is absolutely good being the error of about 0.4% compared to Goncharov’s paper! Now, using the value of PDG m_t=172.5 GeV we get our final result

m_{\eta_t}=344.4 GeV.

This is the next quarkonium to be seen. Alhtough we can theoretically do this computation, we want just to point out that no toponium could be ever observed due to the large mass and width of the t quark (see here for a computation).

Ground state of charmonium


After the satisfactory derivation of the bottomonium ground state mass (see here) we would like to apply similar concepts to charmonium. Before we go on I would like to mention here the beautiful paper by Nora Brambilla and a lot of other contributors that any serious researcher in the field of heavy quark physics should read (see here). This paper has been published as a yellow report by CERN. What we want to prove here is that the knowledge of the gluon propagator can give a nice understanding of the ground state of quarkonia. Anyhow, for charmonium we could not be that lucky as relativistic effects are more important here than for bottomonium. Besides, if we would like to expand to higher order in r the quark potential we would be no more able to treat the Schroedinger equation unless we treat these terms as a perturbation but this approach is not successful giving at best slowing convergence of the series for bottomonium and an useless result for charmonium.

PDG gives us the data for the ground state of charmonium \eta_c(1S):

m_{\eta_c}=2979.8\pm 1.2 MeV

m_c=1.25\pm 0.09 GeV (\bar{MS} scheme)


and then, our computation gives

m_{\eta_c}=2m_c-\frac{1}{4}\alpha_s^2m_c+0.876\alpha_s\sqrt{\sigma}\approx 2602.8 MeV

that has an error of about 13%. With a quark mass of 1.44 GeV we would get a perfect agreement with \eta_c(1S) mass that makes this computation quite striking together with the analogous computations for the ground state of the bottomonium.

As said at the start, heavy quarkonia are a well studied matter and whoever interested to deepen the argument should read the yellow report by Brambilla and others.

Update:I would like to point out the paper by Stephan Narison (see here and here) that obtains the pole masses of c and b quarks being these the ones I use to obtain the right ground state of charmonium and bottomonium. Striking indeed!

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