Proof of Factorization of $J\mathrm{/}\mathrm{\Psi }$ Production in Non-Equilibrium QCD at RHIC and LHC

$J\mathrm{/}\psi$ suppression/production is one of the main signature of quark-gluon plasma detection at RHIC and LHC. In order to study $j\mathrm{/}\psi$ suppression/production in high energy heavy-ion collisions at RHIC and LHC, one needs to prove the factorization theorem of $j\mathrm{/}\psi$ production in non-equilibrium QCD medium, otherwise one will predict infinite cross section of $j\mathrm{/}\psi$. In this paper we prove factorization theorem of $j\mathrm{/}\psi$ production in non-equilibrium QCD at RHIC and LHC at all order in coupling constant.