Infrared divergences of B meson exclusive decays to P-wave charmonia in QCD factorization and nonrelativistic QCD

In the framework of QCD factorization, we study the $B$ meson exclusive decays $B\to \chi_{cJ}K$ where the spin-triplet P-wave charmonium states $\chi_{cJ}(J=0,1,2)$ are described by the color-singlet non-relativistic wave functions. We find that for these decays (except $\chi_{c1}$) there are infrared divergences arising from nonfactorizable vertex corrections as well as logarithmic end-point singularities arising from nonfactorizable spectator interactions at leading-twist order. The infrared divergences due to vertex corrections will explicitly break down QCD factorization within the color-singlet model for charmonia. Unlike in the inclusive decays where the higher Fock states with color-octet $c \bar c$ pair and soft gluon can make contributions to remove the infrared divergences, their contributions can not be accommodated in the exclusive two body decays. As a result, the infrared divergences encountered in exclusive processes involving charmonia may raise a new question to the QCD factorization and NRQCD factorization in B exclusive decays.