  Quarkonium production at low transverse momentum in hadron collisions can be
used to extract Transverse-Momentum-Dependent(TMD) gluon distribution
functions, if TMD factorization holds there. We show that TMD factorization for
the case of P-wave quarkonium with $J^{PC}=0^{++}, 2^{++}$ holds at one-loop
level, but is violated beyond one-loop level. TMD factorization for other
P-wave quarkonium is also violated already at one-loop.
