| Summary: | Abstract The next-to-leading order (NLO) ( O $$ \mathcal{O} $$ ( α s 3 $$ {\alpha}_s^3 $$ )) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in 3 P J 1 8 $$ {P}_J^{\left[1,8\right]} $$ states are calculated within the NRQCD factorization. We use the integration-by-parts reduction and differential equations to semi-analytically calculate the 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 the matching procedure, infrared-safe short distance coefficients and O $$ \mathcal{O} $$ ( α s 2 $$ {\alpha}_s^2 $$ ) perturbative NRQCD matrix elements ⟨ O 3 P J 1 8 $$ {\mathcal{O}}^3{P}_J^{\left[1,8\right]} $$ (3 S 1 8 $$ {S}_1^{\left[8\right]} $$ )⟩ are obtained simultaneously. The NLO short distance coefficients are found to have significant corrections comparing with the LO ones.
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