Inclusive $S-$wave charmonium productions in $B$ decays

The inclusive $S-$wave charmonium production rates in $B$ decays are considered using the Bodwin-Braaten-Lepage (BBL) approach, including the relativistic corrections and the color-octet mechanism suggested as a possible solution to the $\psi^{'}$ puzzle at the Tevatron. We first consider relativistic and radiative corrections to $J/\psi \rightarrow e^{+} e^{-}$ and $J/\psi \rightarrow$ Light Hadrons (LH), in order to determine two nonperturbative parameters, $\langle J/\psi | O_{1} (^{3}S_{1}) | J/\psi \rangle$, $\langle J/\psi | P_{1} (^{3}S_{1}) | J/\psi \rangle$, in the factorization formulae for these decays. Using these two matrix elements and including the color-octet $c\bar{c}(^{3}S_{1})$ state contribution, we get a moderate increase in the decay rates for $B$ decays into $J/\psi ~({\rm or}~\psi^{'}) ~+ X$. Our results, $B(B \rightarrow J/\psi~({\rm or}~\psi^{'}) + X) = 0.58~(0.23) \%$ for $M_{b} = 5.3$ GeV, get closer to the recent CLEO data. As a byproduct, we prefer a larger decay rate for $\eta_{c} \rightarrow$ LH compared to the present data.