Bmeson semi-inclusive decay to spin-tripletD-wave charmonium in NRQCD

Abstract

We study the $B$ meson semi-inclusive decays into spin-triplet $D$-wave chamonium states ${\ensuremath{\psi}}_{J}(J=1,2,3)$ based on the nonrelativistic QCD factorization formula at next-to-leading order in ${\ensuremath{\alpha}}_{s}$ and leading order in $v$ (the relative velocity of charm quark and antiquark in charmonium). The finite short-distance coefficients for ${^{3}D}_{J}^{[1]}$ channels are obtained for the first time. The long-distance matrix elements are estimated with the help of potential model and QCD evolution equations. The branching ratios $\mathrm{Br}(B\ensuremath{\rightarrow}{\ensuremath{\psi}}_{1}X)$ and $\mathrm{Br}(B\ensuremath{\rightarrow}{\ensuremath{\psi}}_{2}X)$ are, respectively, predicted to be about $6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ and $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, with about 50% relative errors mainly coming from the uncertainties of long-distance matrix elements. The branching ratio $\mathrm{Br}(B\ensuremath{\rightarrow}{\ensuremath{\psi}}_{3}X)$ can be very small due to a possible cancellation between the ${^{3}S}_{1}^{[8]}$ and ${^{3}P}_{2}^{[8]}$ channels. As an optimistic estimate, we may use the ${^{3}S}_{1}^{[8]}$ channel alone to set up the upper limit for $\mathrm{Br}(B\ensuremath{\rightarrow}{\ensuremath{\psi}}_{3}X$), which is about $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ and may be used in searching for the missing state ${\ensuremath{\psi}}_{3}$.