## Another success at SLAC

25/08/2008

Yesterday I was reading a copy of August of Physics World, that I receive being member of Institute of Physics, and come to a small piece about a recent measure at SLAC. BABAR collaboration was able to identify and measure the mass of the ground state of $\bar b b$ meson also called bottomonium and identified as $\eta_b(1S)$ (their paper is here). In order to reach their aims, they used the process $\Upsilon(3S)\rightarrow \gamma\eta_b(1S)$ and they were successful in obtaining a very precise value of the mass, $9388.9 ^{+3.1}_ {-2.3} (stat) +/- 2.7(syst)$ MeV, collecting about 20000 photons produced in the process.

We want to build on this beautiful result at SLAC by deriving the mass of the ground state of bottomonium. We already know, since the studies of charmonium, that a Coulomb-like potential does most of the job but not all. This is a great intuition by Politzer, one of the discoverers of asymptotic freedom (the others being Gross and Wilczek). The reason why this approximation works so well is that these quarks are really massive and so the interaction happens at very short distances and also a non-relativistic approximation does hold.

In order to verify how good is this approximation we consider our gluon propagator. This 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. At this stage we can derive the potential between quarks setting $p_0=0$ and Fourier transforming in space coordinates giving

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

So, using the small distance approximation we get finally

$V(r)\approx -\frac{\alpha_s}{r}+\alpha_s\sum_{n=0}^\infty B_nm_n$

and we can estimate the constant to be $\epsilon=0.876\alpha_s\sqrt{\sigma}$. Finally, from PDG we have $\alpha_s(m_b)=0.22$ being $m_b=4.68$ GeV the mass of the bottom quark. $\eta_b(1S)$ is a singlet state and so no spin-orbit effect is present. Using standard formula for hydrogen atom we have finally

$m_{\eta_b}=2m_b-\frac{1}{4}\alpha_s^2 m_b+\epsilon=9.388$ GeV

that is the mass measured at SLAC. We just note the relevance of the constant term $\epsilon=0.0848$ GeV to reach the agreement, a very nice confirmation of our gluon propagator and the approximations used for heavy quark bound states.

As a final consideration we note as a good theory permits to do calculations to be compared with experiments. Bad theories do not have such a property proving themselves ugly already at the start.