Abstract
It has been shown previously that the transverse momentum dependent (TMD) factorization of heavy quarkonium production requires a TMD shape function. Its perturbative tail can be extracted by matching the cross sections valid at low and high transverse momenta. In this article we compare the order-αs TMD expressions with the order-\( {\alpha}_s^2 \) collinear ones for J/ψ production in semi-inclusive deep inelastic scattering (SIDIS), employing nonrelativistic QCD in both cases. In contrast to previous studies, we find that the small transverse momentum limit of the collinear expressions contains discontinuities. We demonstrate how to properly deal with them and include their finite contributions to the TMD shape functions. Moreover, we show that soft gluon emission from the low transverse momentum Born diagrams provide the same leading order TMD shape functions as required for the matching. Their revised perturbative tails have a less divergent behavior as compared to the TMD fragmentation functions of light hadrons. Finally, we investigate the universality of TMD shape functions in heavy quarkonium production, identify the need for process dependent factorization and discuss the phenomenological implications.
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Acknowledgments
We thank Miguel Echevarría for helpful discussions and feedback. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824093 (STRONG 2020) and is part of its JRA4-TMD-neXt Work-Package. This project has also received funding from the French Agence Nationale de la Recherche via the grant ANR-20-CE31-0015 (“PrecisOnium”) and was also partly supported by the French CNRS via the IN2P3 project GLUE@NLO. C.P. also acknowledges financial support by Fondazione di Sardegna under the project “Proton tomography at the LHC”, project number F72F20000220007 (University of Cagliari). This project has also received funding from the LDRD program of LBNL, and the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract numbers DE-AC02-05CH11231.
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Boer, D., Bor, J., Maxia, L. et al. Transverse momentum dependent shape function for J/ψ production in SIDIS. J. High Energ. Phys. 2023, 105 (2023). https://doi.org/10.1007/JHEP08(2023)105
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DOI: https://doi.org/10.1007/JHEP08(2023)105