B-meson Semi-inclusive Decay to $2^{-+}$ Charmonium in NRQCD and X(3872)

Abstract

The semi-inclusive B-meson decay into spin-singlet D-wave $2^{-+}$ charmonium, $B\to {\eta }_{c2}+X$, is studied in nonrelativistic QCD (NRQCD). Both color-singlet and color-octet contributions are calculated at next-to-leading order (NLO) in the strong coupling constant ${\alpha }_s$. The non-perturbative long-distance matrix elements are evaluated using operator evolution equations. It is found that the color-singlet ${}^1{D}_2$ contribution is tiny, while the color-octet channels make dominant contributions. The estimated branching ratio $B(B\to {\eta }_{c2}+X)$ is about $0.41\times 10^{-4}$ in the Naive Dimensional Regularization (NDR) scheme and $1.24\times 10^{-4}$ in the t'Hooft-Veltman (HV) scheme, with renormalization scale $\mu =m_b=4.8$\,GeV. The scheme-sensitivity of these numerical results is due to cancelation between ${}^1{S}_0^{[8]}$ and ${}^1{P}_1^{[8]}$ contributions. The $\mu$-dependence curves of NLO branching ratios in both schemes are also shown, with $\mu$ varying from $\frac{m_b}{2}$ to $2m_b$ and the NRQCD factorization or renormalization scale ${\mu }_{\Lambda }$ taken to be $2m_c$. Comparison of the estimated branching ratio of $B\to {\eta }_{c2}+X$ with the observed branching ratio of $B\to X(3872)+K$ may lead to the conclusion that X(3872) is unlikely to be the $2^{-+}$ charmonium state ${\eta }_{c2}$.

Comments: Version published in PRD, references added, 26 pages, 9 figures