Breakdown of QCD factorization for P-wave quarkonium production at low transverse momentum

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 JPC=0++,2++JPC=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 level.