## 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.