  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.
