We study inclusive processes involving two heavy quarkonia in nonrelativistic QCD (NRQCD) and demonstrate that, in the presence of two P-wave Fock states, NRQCD factorization breaks down, leaving uncanceled infrared singularities. We infer that such singularities will appear for double quarkonium hadroproduction at next-to-leading order. As a solution to this problem, we introduce to NRQCD effective field theory new types of operators whose quantum corrections absorb these singularities.
