Heavy quarkonium production in hadronic collisions provides valuable opportunities for exploring fundamental QCD dynamics with multiple scales. NRQCD factorization at NLO accuracy has successfully described many features of the data thus far. However, the quarkonium production mechanism is still not fully understood. In this talk, I will discuss hadronic quarkonium production at high $p_T$ in the QCD factorization approach, enabling us to resum $\ln(p_T^2/m^2)$-type enhanced contributions for high $p_T$ quarkonium production using quarkonium fragmentation functions at twist-2 and twist-4. Such logarithmically enhanced higher order contributions are not fully included in fixed order NRQCD calculations. I will show our recent numerical analysis of the quantum evolution of the coupled twist-2 and twist-4 fragmentation functions for quarkonium production and its application to phenomenology.
Ref: K.Lee, J.W.Qiu, G.Sterman and K.Watanabe, arXiv:2108.00305 [hep-ph].