$B \to χ_{c0,2} K$ decays: a model estimation

In this paper, we investigate the vertex corrections and spectator hard scattering contributions to $B \to χ_{c0,2}K$ decays, which has no leading contribution from naive factorization scheme. A non-zero binding energy $b=2m_c-M$ is introduced to regularize the infrared divergence of the vertex part. The spectator diagrams also contain logarithmic and linear infrared divergences, for which we adopt a model dependent parametrization. If we neglect possible strong phases in the hard spectator contributions, we obtain a too small branching ratio for $χ_{c0}K$ while too large one for $χ_{c2}K$, as can be seen from the ratio of the branching ratio of $B^+ \to χ_{c2}K^+$ to that of $B^+ \to χ_{c0} K^+$, which is predicted to be $2.15^{+0.63}_{-0.76}$ in our model, while experimentally it should be about 0.1 or even smaller. But a closer examination shows that, assuming large strong phases difference between the twist-2 and twist-3 spectator terms, together with a slightly larger spectator infrared cutoff parameter $Λ_h$, it is possible to accommodate the experimental data. This shows that, for $B\to χ_{c0,2}K$ decays with no factorizable contributions, QCDF seems capable of producing decay rates close to experiments, in contrast to the $B\to J/ψK$ decay which is dominated by the factorizable contributions.

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