Proof of NRQCD factorization at all orders in the coupling constant in heavy quarkonium production

Recently the proof of the factorization in heavy quarkonium production in the NRQCD color octet mechanism is given at next-to-next-to-leading order (NNLO) in the coupling constant by using diagrammatic method of QCD. In this paper we prove factorization in heavy quarkonium production in the NRQCD color octet mechanism at all orders in the coupling constant by using the path integral method of QCD. Our proof is valid to all powers in the relative velocity of the heavy quark. We find that the gauge invariance and the factorization at all orders in the coupling constant require gauge-completed non-perturbative NRQCD matrix elements that were introduced previously to prove factorization at NNLO.