Gluon fragmentation into 3 P_J^[1,8] quark pair and test of NRQCD factorization at two-loop level

The next-to-leading order (NLO) ( 𝒪 ( α_s^3 )) corrections for gluon fragmentation functions to a heavy quark-antiquark pair in 3 P_J^[1,8] 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 ( α_s^2 ) perturbative NRQCD matrix elements ⟨ 𝒪^3P_J^[1,8] ( 3 S_1^[8] )⟩ are obtained simultaneously. The NLO short distance coefficients are found to have significant corrections comparing with the LO ones.