B-meson semi-inclusive decay to 2-+ charmonium in nonrelativistic QCD and X(3872)

The semi-inclusive B-meson decay into spin-singlet D-wave 2(-+) charmonium, B -> eta(c2) + X, is studied in nonrelativistic QCD. Both color-singlet and color-octet contributions are calculated at next-to-leading order in the strong coupling constant alpha(s). The nonperturbative long-distance matrix elements are evaluated using operator evolution equations. It is found that the color-singlet D-1(2) contribution is tiny, while the color-octet channels make dominant contributions. The estimated branching ratio B(B -> eta(c2) + X) is about 0.41 x 10(-4) in the naive dimensional regularization scheme and 1.24 x 10(-4) in the 't Hooft-Veltman scheme, with renormalization scale mu = m(b) = 4.8 GeV. The scheme sensitivity of these numerical results is due to cancellation between S-1(0)vertical bar 8 vertical bar and P-1(1)vertical bar 8 vertical bar contributions. The mu-dependence curves of next-to-leading order branching ratios in both schemes are also shown, with mu varying from m(b)/2 to 2m(b) and the nonrelativistic QCD factorization or renormalization scale mu(Lambda) taken to be 2m(c). Comparison of the estimated branching ratio of B -> eta(c2) + X with the observed branching ratio of B -> X(3872) + K may lead to the conclusion that X(3872) is unlikely to be the 2(-+) charmonium state eta(c2).