Quarkonia and Dirac spectra

28/08/2008

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

V(r)=-\frac{\alpha_s}{r}+0.8762499705\alpha_s\sqrt{\sigma}

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:

M(n,j)=\frac{3}{2}m_q+\frac{m_q}{2}\frac{1}{\sqrt{1+\frac{\alpha_s^2}{(n-\delta_j)^2}}}+0.8762499705\alpha_s\sqrt{\sigma}

being

\delta_j=j+\frac{1}{2}-\sqrt{(j+\frac{1}{2})^2-\alpha_s^2}.

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(n,j)=m_c+m_b-\frac{m_cm_b}{m_c+m_b}+\frac{m_cm_b}{m_c+m_b}\frac{1}{\sqrt{1+\frac{\alpha_s^2}{(n-\delta_j)^2}}}

+0.8762499705\alpha_s\sqrt{\sigma}

obtaining

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

27/08/2008

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

G(p^2)=\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}

being

B_n=(2n+1)\frac{\pi^2}{K^2(i)}\frac{(-1)^{n+1}e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}

and

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.


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