Proof of NRQCD Factorization at All Order in Coupling Constant in Heavy Quarkonium Production

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