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

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