Abstract
We compute the gluon transverse-momentum-dependent fragmentation function (TMDFF) at next-to-leading order (NLO) into heavy quarkonium in the color-octet \( {}^3{S}_1^{\left[8\right]} \) channel, based on the NRQCD factorization approach. The spurious rapidity divergences are explicitly shown to cancel in a well-defined TMDFF, which incorporates the needed soft factor. We also compute the integrated gluon FF at NLO in the same \( {}^3{S}_1^{\left[8\right]} \) channel, and show that the matching coefficient of the TMDFF onto the FF at large transverse momentum is the expected one. These results are relevant to perform precise and sensible phenomenological studies of transverse-momentum spectra of quarkonium production, for which the production mechanism through fragmentation plays a relevant role, like in the future Electron-Ion Collider.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Angeles-Martinez et al., Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects, Acta Phys. Polon. B 46 (2015) 2501 [arXiv:1507.05267] [INSPIRE].
E. Chapon et al., Prospects for quarkonium studies at the high-luminosity LHC, Prog. Part. Nucl. Phys. 122 (2022) 103906 [arXiv:2012.14161] [INSPIRE].
A. Arbuzov et al., On the physics potential to study the gluon content of proton and deuteron at NICA SPD, Prog. Part. Nucl. Phys. 119 (2021) 103858 [arXiv:2011.15005] [INSPIRE].
M.G. Echevarria, T. Kasemets, P.J. Mulders and C. Pisano, QCD evolution of (un)polarized gluon TMDPDFs and the Higgs qT-distribution, JHEP 07 (2015) 158 [Erratum ibid. 05 (2017) 073] [arXiv:1502.05354] [INSPIRE].
P.J. Mulders and J. Rodrigues, Transverse momentum dependence in gluon distribution and fragmentation functions, Phys. Rev. D 63 (2001) 094021 [hep-ph/0009343] [INSPIRE].
D. Boer and C. Pisano, Polarized gluon studies with charmonium and bottomonium at LHCb and AFTER, Phys. Rev. D 86 (2012) 094007 [arXiv:1208.3642] [INSPIRE].
J.P. Ma, J.X. Wang and S. Zhao, Transverse momentum dependent factorization for quarkonium production at low transverse momentum, Phys. Rev. D 88 (2013) 014027 [arXiv:1211.7144] [INSPIRE].
G.-P. Zhang, Probing transverse momentum dependent gluon distribution functions from hadronic quarkonium pair production, Phys. Rev. D 90 (2014) 094011 [arXiv:1406.5476] [INSPIRE].
D. Boer, Linearly polarized gluon effects in unpolarized collisions, PoS QCDEV2015 (2015) 023 [arXiv:1510.05915] [INSPIRE].
R. Bain, Y. Makris and T. Mehen, Transverse Momentum Dependent Fragmenting Jet Functions with Applications to Quarkonium Production, JHEP 11 (2016) 144 [arXiv:1610.06508] [INSPIRE].
A. Mukherjee and S. Rajesh, Probing Transverse Momentum Dependent Parton Distributions in Charmonium and Bottomonium Production, Phys. Rev. D 93 (2016) 054018 [arXiv:1511.04319] [INSPIRE].
A. Mukherjee and S. Rajesh, Linearly polarized gluons in charmonium and bottomonium production in color octet model, Phys. Rev. D 95 (2017) 034039 [arXiv:1611.05974] [INSPIRE].
J.-P. Lansberg, C. Pisano and M. Schlegel, Associated production of a dilepton and a ϒ(J/ψ) at the LHC as a probe of gluon transverse momentum dependent distributions, Nucl. Phys. B 920 (2017) 192 [arXiv:1702.00305] [INSPIRE].
A. Bacchetta, D. Boer, C. Pisano and P. Taels, Gluon TMDs and NRQCD matrix elements in J/ψ production at an EIC, Eur. Phys. J. C 80 (2020) 72 [arXiv:1809.02056] [INSPIRE].
U. D’Alesio, F. Murgia, C. Pisano and P. Taels, Azimuthal asymmetries in semi-inclusive J/ψ + jet production at an EIC, Phys. Rev. D 100 (2019) 094016 [arXiv:1908.00446] [INSPIRE].
M.G. Echevarria, Proper TMD factorization for quarkonia production: pp → ηc,b as a study case, JHEP 10 (2019) 144 [arXiv:1907.06494] [INSPIRE].
S. Fleming, Y. Makris and T. Mehen, An effective field theory approach to quarkonium at small transverse momentum, JHEP 04 (2020) 122 [arXiv:1910.03586] [INSPIRE].
F. Scarpa et al., Studies of gluon TMDs and their evolution using quarkonium-pair production at the LHC, Eur. Phys. J. C 80 (2020) 87 [arXiv:1909.05769] [INSPIRE].
M. Grewal, Z.-B. Kang, J.-W. Qiu and A. Signori, Predictive power of transverse-momentum-dependent distributions, Phys. Rev. D 101 (2020) 114023 [arXiv:2003.07453] [INSPIRE].
D. Boer et al., J/ψ meson production in SIDIS: matching high and low transverse momentum, JHEP 09 (2020) 040 [arXiv:2004.06740] [INSPIRE].
U. D’Alesio et al., Process dependence of the gluon Sivers function in p↑p → J/ψ + X within a TMD scheme in NRQCD, Phys. Rev. D 102 (2020) 094011 [arXiv:2007.03353] [INSPIRE].
D. Boer, C. Pisano and P. Taels, Extracting color octet NRQCD matrix elements from J/ψ production at the EIC, Phys. Rev. D 103 (2021) 074012 [arXiv:2102.00003] [INSPIRE].
U. D’Alesio et al., J/ψ polarization in semi-inclusive DIS at low and high transverse momentum, JHEP 03 (2022) 037 [arXiv:2110.07529] [INSPIRE].
R. Kishore, A. Mukherjee, A. Pawar and M. Siddiqah, cos 2ϕt azimuthal asymmetry in back-to-back J/ψ-jet production in ep → eJ/ψ jet X at the EIC, Phys. Rev. D 106 (2022) 034009 [arXiv:2203.13516] [INSPIRE].
J. Bor and D. Boer, TMD evolution study of the cos 2ϕ azimuthal asymmetry in unpolarized J/ψ production at EIC, Phys. Rev. D 106 (2022) 014030 [arXiv:2204.01527] [INSPIRE].
D. Chakrabarti, R. Kishore, A. Mukherjee and S. Rajesh, Azimuthal asymmetries in J/ψ-photon production at the EIC, Phys. Rev. D 107 (2023) 014008 [arXiv:2211.08709] [INSPIRE].
D. Boer et al., Transverse momentum dependent shape function for J/ψ production in SIDIS, JHEP 08 (2023) 105 [arXiv:2304.09473] [INSPIRE].
G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D 51 (1995) 1125 [Erratum ibid. 55 (1997) 5853] [hep-ph/9407339] [INSPIRE].
H. Fritzsch, Producing Heavy Quark Flavors in Hadronic Collisions: A Test of Quantum Chromodynamics, Phys. Lett. B 67 (1977) 217 [INSPIRE].
F. Halzen, Cvc for Gluons and Hadroproduction of Quark Flavors, Phys. Lett. B 69 (1977) 105 [INSPIRE].
M. Gluck, J.F. Owens and E. Reya, Gluon Contribution to Hadronic J/ψ Production, Phys. Rev. D 17 (1978) 2324 [INSPIRE].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B → Xsγin effective field theory, Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, Nucl. Phys. A 1026 (2022) 122447 [arXiv:2103.05419] [INSPIRE].
D.P. Anderle et al., Electron-ion collider in China, Front. Phys. (Beijing) 16 (2021) 64701 [arXiv:2102.09222] [INSPIRE].
M.G. Echevarria, Y. Makris and I. Scimemi, Quarkonium TMD fragmentation functions in NRQCD, JHEP 10 (2020) 164 [arXiv:2007.05547] [INSPIRE].
M. Copeland et al., Polarized TMD fragmentation functions for J/ψ production, arXiv:2308.08605 [INSPIRE].
M. Copeland et al., Polarized J/ψ production in semi-inclusive DIS at large Q2: Comparing quark fragmentation and photon-gluon fusion, arXiv:2310.13737 [INSPIRE].
J.P. Ma, Gluon fragmentation into P wave triplet quarkonium, Nucl. Phys. B 447 (1995) 405 [hep-ph/9503346] [INSPIRE].
E. Braaten and J. Lee, Next-to-leading order calculation of the color octet 3S1 gluon fragmentation function for heavy quarkonium, Nucl. Phys. B 586 (2000) 427 [hep-ph/0004228] [INSPIRE].
Y.-Q. Ma, J.-W. Qiu and H. Zhang, Heavy quarkonium fragmentation functions from a heavy quark pair. I. S wave, Phys. Rev. D 89 (2014) 094029 [arXiv:1311.7078] [INSPIRE].
M.G. Echevarria, I. Scimemi and A. Vladimirov, Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order, JHEP 09 (2016) 004 [arXiv:1604.07869] [INSPIRE].
A. Vladimirov, V. Moos and I. Scimemi, Transverse momentum dependent operator expansion at next-to-leading power, JHEP 01 (2022) 110 [arXiv:2109.09771] [INSPIRE].
M.G. Echevarria, I. Scimemi and A. Vladimirov, Universal transverse momentum dependent soft function at NNLO, Phys. Rev. D 93 (2016) 054004 [arXiv:1511.05590] [INSPIRE].
A.V. Manohar and I.W. Stewart, The Zero-Bin and Mode Factorization in Quantum Field Theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [INSPIRE].
I. Scimemi and A. Vladimirov, Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum, JHEP 06 (2020) 137 [arXiv:1912.06532] [INSPIRE].
M.G. Echevarria, A. Idilbi, A. Schäfer and I. Scimemi, Model-Independent Evolution of Transverse Momentum Dependent Distribution Functions (TMDs) at NNLL, Eur. Phys. J. C 73 (2013) 2636 [arXiv:1208.1281] [INSPIRE].
V. Moos, I. Scimemi, A. Vladimirov and P. Zurita, Extraction of unpolarized transverse momentum distributions from fit of Drell-Yan data at N4LL, arXiv:2305.07473 [INSPIRE].
Acknowledgments
This project is supported by the State Agency for Research of the Spanish Ministry of Science and Innovation through the grants PID2019-106080GB-C21, PCI2022-132984, PID2022-136510NB-C31, PID2022-136510NB-C33 and CNS2022-135186, as well as by the European Union Horizon 2020 research and innovation program under grant agreement Num. 824093 (STRONG-2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2308.12356
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Echevarria, M.G., Romera, S.F. & Scimemi, I. Gluon TMD fragmentation function into quarkonium. J. High Energ. Phys. 2023, 181 (2023). https://doi.org/10.1007/JHEP12(2023)181
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2023)181