Inclusive Photoproduction of Polarized $^3P_1$ Quarkonium

We analyse inclusive photoproduction of polarized $^3P_1$ quarkonium in the framework of QCD. To separate nonperturbative and perturbative parts in the density matrix of the produced quarkonium we use a method , which is equivalent to the diagramatic expansion widely used in analysing deeply inelastic scatterings. A systematic expansion in a small velocity $v$, with which a heavy quark moves inside the quarkonium in its rest frame, is performed for the nonperturbative parts, and they are expressed as matrix elements in nonrelativistic QCD. At the leading order of $v$ there are four matrix elements representing nonperturbative physics. The perturbative parts are calculated at the leading order of coupling constants. Some numerical results, especially, numerical results for HERA are given.