Gluon fragmentation into ${^{3}hspace{-0.6mm}P_{J}^{[1,8]}}$ quark pair and test of NRQCD factorization at two-loop level

The next-to-leading order (NLO) ($mathcal{O}(alpha_s^3)$) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in ${^{3}hspace{-0.6mm}P_{J}^{[1,8]}}$ states are calculated within the NRQCD factorization. We use integration-by-parts reduction and differential equations to semi-analytically calculate fragmentation functions in full-QCD, and find that infrared divergences can be absorbed by the NRQCD long distance matrix elements. Thus, the NRQCD factorization conjecture is verified at two-loop level via a physical process, which is free of artificial ultraviolet divergences. Through matching procedure, infrared-safe short distance coefficients and $mathcal{O}(alpha_s^2)$ perturbative NRQCD matrix elements $langle{mathcal O}^{{^{3}hspace{-0.6mm}P_{J}^{[1,8]}}}({^{3}hspace{-0.6mm}S_{1}^{[8]}})rangle_{mathrm{NLO}}$ are obtained simultaneously. The NLO short distance coefficients are found to have significant corrections comparing with the LO ones.