  We study inclusive processes involving two heavy quarkonia in
nonrelativisitic QCD (NRQCD) and demonstrate that, in the presence of two
P-wave Fock states, NRQCD factorization breaks down, leaving uncanceled
infrared singularities. As phenomenologically important examples, we consider
the decay $\Upsilon \to \chi_{cJ}+X$ via $b\bar{b}({}^3P_{J_b}^{[8]})\to
c\bar{c}({}^3P_J^{[1]})+gg$ and the production process $e^+e^-\to
J/\psi+\chi_{cJ}+X$ via $e^{+}e^{-}\to
c\bar{c}({}^3P_{J_1}^{[8]})+c\bar{c}({}^3P_J^{[1]})+g$. 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.
