Abstract
In the paper, we calculate the fragmentation functions for c → ηc and b → ηb up to next-to-leading-order (NLO) QCD accuracy. The ultraviolet divergences in the real corrections are removed through operator renormalization under the modified min- imal subtraction scheme. We then obtain the fragmentation functions \( {D}_{c\to {\eta}_c} \)(z, μF) and \( {D}_{b\to {\eta}_b} \)(z, μF) up to NLO QCD accuracy, which are presented as figures and fitting functions. The numerical results show that the NLO corrections are significant. The sensitives of the fragmentation functions to the renormalization scale and the factorization scale are analyzed explicitly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
N. Brambilla et al., Heavy Quarkonium: Progress, Puzzles, and Opportunities, Eur. Phys. J. C 71 (2011) 1534 [arXiv:1010.5827] [INSPIRE].
Quarkonium Working Group collaboration, Heavy quarkonium physics, hep-ph/0412158 [INSPIRE].
M. Butenschoen and B.A. Kniehl, J/ψ polarization at Tevatron and LHC: Nonrelativistic-QCD factorization at the crossroads, Phys. Rev. Lett. 108 (2012) 172002 [arXiv:1201.1872] [INSPIRE].
K.-T. Chao, Y.-Q. Ma, H.-S. Shao, K. Wang and Y.-J. Zhang, J/ψ Polarization at Hadron Colliders in Nonrelativistic QCD, Phys. Rev. Lett. 108 (2012) 242004 [arXiv:1201.2675] [INSPIRE].
B. Gong, L.-P. Wan, J.-X. Wang and H.-F. Zhang, Polarization for Prompt J/ψ and ψ(2s) Production at the Tevatron and LHC, Phys. Rev. Lett. 110 (2013) 042002 [arXiv:1205.6682] [INSPIRE].
M. Butenschoen and B.A. Kniehl, World data of J/ψ production consolidate NRQCD factorization at NLO, Phys. Rev. D 84 (2011) 051501 [arXiv:1105.0820] [INSPIRE].
G.T. Bodwin, H.S. Chung, U.-R. Kim and J. Lee, Fragmentation contributions to J/ψ production at the Tevatron and the LHC, Phys. Rev. Lett. 113 (2014) 022001 [arXiv:1403.3612] [INSPIRE].
G.T. Bodwin, K.-T. Chao, H.S. Chung, U.-R. Kim, J. Lee and Y.-Q. Ma, Fragmentation contributions to hadroproduction of prompt J/ψ, χcJ, and ψ(2S) states, Phys. Rev. D 93 (2016) 034041 [arXiv:1509.07904] [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1 [hep-ph/0409313] [INSPIRE].
Z.-B. Kang, J.-W. Qiu and G. Sterman, Factorization and quarkonium production, Nucl. Phys. B Proc. Suppl. 214 (2011) 39 [INSPIRE].
Z.-B. Kang, J.-W. Qiu and G. Sterman, Heavy quarkonium production and polarization, Phys. Rev. Lett. 108 (2012) 102002 [arXiv:1109.1520] [INSPIRE].
S. Fleming, A.K. Leibovich, T. Mehen and I.Z. Rothstein, The Systematics of Quarkonium Production at the LHC and Double Parton Fragmentation, Phys. Rev. D 86 (2012) 094012 [arXiv:1207.2578] [INSPIRE].
S. Fleming, A.K. Leibovich, T. Mehen and I.Z. Rothstein, Anomalous dimensions of the double parton fragmentation functions, Phys. Rev. D 87 (2013) 074022 [arXiv:1301.3822] [INSPIRE].
C.-H. Chang and Y.-Q. Chen, The Production of Bc or \( {\overline{B}}_c \) meson associated with two heavy quark jets in Z0 boson decay, Phys. Rev. D 46 (1992) 3845 [Erratum ibid. 50 (1994) 6013] [INSPIRE].
E. Braaten, K.-m. Cheung and T.C. Yuan, Perturbative QCD fragmentation functions for Bc and Bc * production, Phys. Rev. D 48 (1993) R5049 [hep-ph/9305206] [INSPIRE].
E. Braaten, K.-m. Cheung and T.C. Yuan, Z0 decay into charmonium via charm quark fragmentation, Phys. Rev. D 48 (1993) 4230 [hep-ph/9302307] [INSPIRE].
E. Braaten and T.C. Yuan, Gluon fragmentation into heavy quarkonium, Phys. Rev. Lett. 71 (1993) 1673 [hep-ph/9303205] [INSPIRE].
E. Braaten and T.C. Yuan, Gluon fragmentation into P wave heavy quarkonium, Phys. Rev. D 50 (1994) 3176 [hep-ph/9403401] [INSPIRE].
E. Braaten and T.C. Yuan, Gluon fragmentation into spin triplet S wave quarkonium, Phys. Rev. D 52 (1995) 6627 [hep-ph/9507398] [INSPIRE].
Y.-Q. Chen, Perturbative QCD predictions for the fragmentation functions of the P wave mesons with two heavy quarks, Phys. Rev. D 48 (1993) 5181 [INSPIRE].
T.C. Yuan, Perturbative QCD fragmentation functions for production of P wave mesons with charm and beauty, Phys. Rev. D 50 (1994) 5664 [hep-ph/9405348] [INSPIRE].
J.P. Ma, Calculating fragmentation functions from definitions, Phys. Lett. B 332 (1994) 398 [hep-ph/9401249] [INSPIRE].
J.P. Ma, Gluon fragmentation into P wave triplet quarkonium, Nucl. Phys. B 447 (1995) 405 [hep-ph/9503346] [INSPIRE].
J.P. Ma, Quark fragmentation into p wave triplet quarkonium, Phys. Rev. D 53 (1996) 1185 [hep-ph/9504263] [INSPIRE].
P.L. Cho, M.B. Wise and S.P. Trivedi, Gluon fragmentation into polarized charmonium, Phys. Rev. D 51 (1995) R2039 [hep-ph/9408352] [INSPIRE].
M. Beneke and I.Z. Rothstein, Psi-prime polarization as a test of color octet quarkonium production, Phys. Lett. B 372 (1996) 157 [Erratum ibid. 389 (1996) 769] [hep-ph/9509375] [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].
W.-l. Sang, L.-f. Yang and Y.-q. Chen, Relativistic corrections to heavy quark fragmentation to S-wave heavy mesons, Phys. Rev. D 80 (2009) 014013 [INSPIRE].
G. Hao, Y. Zuo and C.-F. Qiao, The Fragmentation Function of Gluon Splitting into P-wave Spin-singlet Heavy Quarkonium, arXiv:0911.5539 [INSPIRE].
Y. Jia, W.-L. Sang and J. Xu, Inclusive hc Production at B Factories, Phys. Rev. D 86 (2012) 074023 [arXiv:1206.5785] [INSPIRE].
G.T. Bodwin, H.S. Chung, U.-R. Kim and J. Lee, Quark fragmentation into spin-triplet S-wave quarkonium, Phys. Rev. D 91 (2015) 074013 [arXiv:1412.7106] [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].
Y.-Q. Ma, J.-W. Qiu and H. Zhang, Fragmentation functions of polarized heavy quarkonium, JHEP 06 (2015) 021 [arXiv:1501.04556] [INSPIRE].
D. Yang and W. Zhang, Relativistic corrections of the fragmentation functions for a heavy quark to Bc and \( {B}_c^{\ast } \), Chin. Phys. C 43 (2019) 083101 [arXiv:1905.02923] [INSPIRE].
P. Artoisenet and E. Braaten, Gluon fragmentation into quarkonium at next-to-leading order, JHEP 04 (2015) 121 [arXiv:1412.3834] [INSPIRE].
R. Sepahvand and S. Dadfar, NLO corrections to c- and b-quark fragmentation into j/ψ and γ, Phys. Rev. D 95 (2017) 034012 [INSPIRE].
P. Artoisenet and E. Braaten, Gluon fragmentation into quarkonium at next-to-leading order using FKS subtraction, JHEP 01 (2019) 227 [arXiv:1810.02448] [INSPIRE].
F. Feng and Y. Jia, Next-to-leading-order QCD corrections to gluon fragmentation into \( {}^1{S}_0^{\left(1,8\right)} \) quarkonia, arXiv:1810.04138 [INSPIRE].
P. Zhang, C.-Y. Wang, X. Liu, Y.-Q. Ma, C. Meng and K.-T. Chao, Semi-analytical calculation of gluon fragmentation into \( {}^1{S}_0^{\left(1,8\right)} \) quarkonia at next-to-leading order, JHEP 04 (2019) 116 [arXiv:1810.07656] [INSPIRE].
X.-C. Zheng, C.-H. Chang and X.-G. Wu, NLO fragmentation functions of heavy quarks into heavy quarkonia, Phys. Rev. D 100 (2019) 014005 [arXiv:1905.09171] [INSPIRE].
X.-C. Zheng, C.-H. Chang, T.-F. Feng and X.-G. Wu, QCD NLO fragmentation functions for c or \( \overline{b} \) quark to Bc or \( {B}_c^{\ast } \) meson and their application, Phys. Rev. D 100 (2019) 034004 [arXiv:1901.03477] [INSPIRE].
F. Feng, S. Ishaq, Y. Jia and J.-Y. Zhang, Fragmentation function of gluon into spin-singlet P-wave quarkonium, Phys. Rev. D 102 (2020) 014038 [arXiv:1712.09986] [INSPIRE].
P. Zhang, C. Meng, Y.-Q. Ma and K.-T. Chao, Gluon fragmentation into \( {}^3{P}_J^{\left[1,8\right]} \) quark pair and test of NRQCD factorization at two-loop level, arXiv:2011.04905 [INSPIRE].
X.-C. Zheng, Z.-Y. Zhang and X.-G. Wu, Fragmentation functions for a quark into a spin-singlet quarkonium: Different flavor case, Phys. Rev. D 103 (2021) 074004 [arXiv:2101.01527] [INSPIRE].
J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys. B 194 (1982) 445 [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
F. Feng, Apart: A Generalized Mathematica Apart Function, Comput. Phys. Commun. 183 (2012) 2158 [arXiv:1204.2314] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
T. Hahn and M. Pérez-Victoria, Automatized one loop calculations in four-dimensions and D-dimensions, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565] [INSPIRE].
J.G. Korner, D. Kreimer and K. Schilcher, A Practicable γ5 scheme in dimensional regularization, Z. Phys. C 54 (1992) 503 [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
A.H. Mueller, Cut Vertices and their Renormalization: A Generalization of the Wilson Expansion, Phys. Rev. D 18 (1978) 3705 [INSPIRE].
E.J. Eichten and C. Quigg, Quarkonium wave functions at the origin, Phys. Rev. D 52 (1995) 1726 [hep-ph/9503356] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, Chin. Phys. C 40 (2016) 100001 [INSPIRE].
Y.L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+e− Annihilation by Perturbation Theory in Quantum Chromodynamics., Sov. Phys. JETP 46 (1977) 641 [INSPIRE].
V.N. Gribov and L.N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
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: 2105.14580
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
Zheng, XC., Wu, XG. & Huang, XD. NLO fragmentation functions for a quark into a spin-singlet quarkonium: same flavor case. J. High Energ. Phys. 2021, 14 (2021). https://doi.org/10.1007/JHEP07(2021)014
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)014