Relativistic corrections to gluon fragmentation into the $^3P_{J}^{[1,8]}$ states

We compute relativistic corrections to the gluon fragmentation functions to ${}^3P_J^{[1,8]}$ Fock states of heavy quarkonium within non-relativistic QCD factorization framework. We find that, at $\mathcal{O}(v^2)$ sub-leading order, the $S$-$D$ mixing effect must be taken into account to absorb the infrared divergence of spin-triplet $P$-wave production within full QCD into the NRQCD long-distance matrix elements. Unlike the $S$-wave case, we find that the short-distance coefficients of the fragmentation functions at leading and sub-leading order are no longer proportional to each other. However, upon convolution with the gluon production cross section, their ratios are almost constant across the whole $p_T$ region. We find the relativistic corrections to be negative and substantial, which makes them a non-negligible ingredient in the study of $J/\psi$ production at the LHC.