Improved perturbative QCD approach to the bottomonium spectrum

Abstract

Recently it has been shown that the gross structure of the bottomonium spectrum is reproduced reasonably well within the nonrelativistic bound state theory based on perturbative QCD. In that calculation, however, the fine splittings and the $S--P$ level splittings are predicted to be considerably narrower than the corresponding experimental values. We investigate the bottomonium spectrum within a specific framework based on perturbative QCD, which incorporates all the corrections up to $\mathcal{O}({\ensuremath{\alpha}}_{S}^{5}{m}_{b})$ and $\mathcal{O}({\ensuremath{\alpha}}_{S}^{4}{m}_{b}),$ respectively, in the computations of the fine splittings and the $S--P$ splittings. We find that the agreement with the experimental data for the fine splittings improves drastically due to an enhancement of the wave functions close to the origin as compared to the Coulomb wave functions. The agreement of the $S--P$ splittings with the experimental data also becomes better. We find that natural scales of the fine splittings and the $S--P$ splittings are larger than those of the bound states themselves. On the other hand, the predictions of the level spacings between consecutive principal quantum numbers depend rather strongly on the scale $\ensuremath{\mu}$ of the operator $\ensuremath{\propto}{C}_{A}{/(m}_{b}{r}^{2}).$ The agreement of the whole spectrum with the experimental data is much better than the previous predictions when $\ensuremath{\mu}\ensuremath{\simeq}3--4\mathrm{GeV}$ for ${\ensuremath{\alpha}}_{S}{(M}_{Z})=0.1181.$ There seems to be a phenomenological preference for some suppression mechanism for the above operator.