Abstract
In the paper, we present QCD predictions for γ + ηc production at an electron-positron collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for e+ + e− → γ + ηc using the conventional scale-setting approach has large renormalization scale ambiguities, usually estimated by choosing the renormalization scale to be the e+e− center-of-mass collision energy \( \sqrt{s} \). The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal β contributions into the QCD coupling αs(Q2). The renormalization group equation then sets the value of αs for the process. The PMC renormalization scale reflects the virtuality of the underlying process, and the resulting predictions satisfy all of the requirements of renormalization group invariance, including renormalization scheme invariance. After applying the PMC, we obtain a renormalization scale-and-scheme independent prediction, σ|NNLO,PMC ≃ 41.18 fb for \( \sqrt{s} \)=10.6 GeV. The resulting pQCD series matches the series for conformal theory and thus has no divergent renormalon contributions. The large K factor which contributes to this process reinforces the importance of uncalculated NNNLO and higher-order terms. Using the PMC scale-and-scheme independent conformal series and the Padé approximation approach, we predict σ|NNNLO,PMC+Pade ≃ 18.99 fb, which is consistent with the recent BELLE measurement \( {\sigma}^{\mathrm{obs}}={16.58}_{-9.93}^{+10.51} \) fb at \( \sqrt{s} \) ≃ 10.6 GeV. This procedure also provides a first estimate of the NNNLO contribution.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
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].
Belle collaboration, Observation of e+e− → γχc1 and search for e+e− → γχc0, γχc2, and γηc at \( \sqrt{s} \) near 10.6 GeV at Belle, Phys. Rev. D 98 (2018) 092015 [arXiv:1810.10291] [INSPIRE].
M.A. Shifman and M.I. Vysotsky, Form-factors of heavy mesons in QCD, Nucl. Phys. B 186 (1981) 475 [INSPIRE].
W.-L. Sang and Y.-Q. Chen, Higher Order Corrections to the Cross Section of e+e− → Quarkonium + γ, Phys. Rev. D 81 (2010) 034028 [arXiv:0910.4071] [INSPIRE].
D. Li, Z.-G. He and K.-T. Chao, Search for C= charmonium and bottomonium states in e+e− → γ + X at B factories, Phys. Rev. D 80 (2009) 114014 [arXiv:0910.4155] [INSPIRE].
H.S. Chung, J.-H. Ee, D. Kang, U.-R. Kim, J. Lee and X.-P. Wang, Pseudoscalar Quarkonium+gamma Production at NLL+NLO accuracy, JHEP 10 (2019) 162 [arXiv:1906.03275] [INSPIRE].
L.-B. Chen, Y. Liang and C.-F. Qiao, NNLO QCD corrections to γ + ηc(ηb) exclusive production in electron-positron collision, JHEP 01 (2018) 091 [arXiv:1710.07865] [INSPIRE].
X.-G. Wu, S.J. Brodsky and M. Mojaza, The Renormalization Scale-Setting Problem in QCD, Prog. Part. Nucl. Phys. 72 (2013) 44 [arXiv:1302.0599] [INSPIRE].
X.-G. Wu et al., Renormalization Group Invariance and Optimal QCD Renormalization Scale-Setting, Rept. Prog. Phys. 78 (2015) 126201 [arXiv:1405.3196] [INSPIRE].
X.-G. Wu, J.-M. Shen, B.-L. Du, X.-D. Huang, S.-Q. Wang and S.J. Brodsky, The QCD renormalization group equation and the elimination of fixed-order scheme-and-scale ambiguities using the principle of maximum conformality, Prog. Part. Nucl. Phys. 108 (2019) 103706 [arXiv:1903.12177] [INSPIRE].
M. Gell-Mann and F.E. Low, Quantum electrodynamics at small distances, Phys. Rev. 95 (1954) 1300 [INSPIRE].
S.J. Brodsky and X.-G. Wu, Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant at Four Loops, Phys. Rev. D 85 (2012) 034038 [Erratum ibid. 86 (2012) 079903] [arXiv:1111.6175] [INSPIRE].
S.J. Brodsky and X.-G. Wu, Eliminating the Renormalization Scale Ambiguity for Top-Pair Production Using the Principle of Maximum Conformality, Phys. Rev. Lett. 109 (2012) 042002 [arXiv:1203.5312] [INSPIRE].
S.J. Brodsky and L. Di Giustino, Setting the Renormalization Scale in QCD: The Principle of Maximum Conformality, Phys. Rev. D 86 (2012) 085026 [arXiv:1107.0338] [INSPIRE].
M. Mojaza, S.J. Brodsky and X.-G. Wu, Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in Perturbative QCD, Phys. Rev. Lett. 110 (2013) 192001 [arXiv:1212.0049] [INSPIRE].
S.J. Brodsky, M. Mojaza and X.-G. Wu, Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations, Phys. Rev. D 89 (2014) 014027 [arXiv:1304.4631] [INSPIRE].
S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, On the Elimination of Scale Ambiguities in Perturbative Quantum Chromodynamics, Phys. Rev. D 28 (1983) 228 [INSPIRE].
S.J. Brodsky and H.J. Lu, Commensurate scale relations in quantum chromodynamics, Phys. Rev. D 51 (1995) 3652 [hep-ph/9405218] [INSPIRE].
S.J. Brodsky and P. Huet, Aspects of SU(Nc) gauge theories in the limit of small number of colors, Phys. Lett. B 417 (1998) 145 [hep-ph/9707543] [INSPIRE].
S.J. Brodsky and X.-G. Wu, Self-Consistency Requirements of the Renormalization Group for Setting the Renormalization Scale, Phys. Rev. D 86 (2012) 054018 [arXiv:1208.0700] [INSPIRE].
X.-G. Wu, J.-M. Shen, B.-L. Du and S.J. Brodsky, Novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the C-scheme coupling, Phys. Rev. D 97 (2018) 094030 [arXiv:1802.09154] [INSPIRE].
J.-M. Shen, X.-G. Wu, B.-L. Du and S.J. Brodsky, Novel All-Orders Single-Scale Approach to QCD Renormalization Scale-Setting, Phys. Rev. D 95 (2017) 094006 [arXiv:1701.08245] [INSPIRE].
T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun. 140 (2001) 418 [hep-ph/0012260] [INSPIRE].
R. Mertig, M. Böhm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
F. Feng and R. Mertig, FormLink/FeynCalcFormLink: Embedding FORM in Mathematica and FeynCalc, arXiv:1212.3522 [INSPIRE].
F. Feng, Apart: A Generalized Mathematica Apart Function, Comput. Phys. Commun. 183 (2012) 2158 [arXiv:1204.2314] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090 [arXiv:1312.3186] [INSPIRE].
R. Cools and A. Haegemans, Algorithm 824: Cubpack: a package for automatic cubature; framework description, ACM Trans. Math. Softw. 29 (2003) 287.
HCubature web site, https://github.com/stevengj/cubature.
F. Feng, Y. Jia and W.-L. Sang, Next-to-next-to-leading-order QCD corrections to e+e− → J/ψ + ηc at B factories, arXiv:1901.08447 [INSPIRE].
D.J. Broadhurst, N. Gray and K. Schilcher, Gauge invariant on-shell Z2 in QED, QCD and the effective field theory of a static quark, Z. Phys. C 52 (1991) 111 [INSPIRE].
K. Melnikov and T. van Ritbergen, The Three loop on-shell renormalization of QCD and QED, Nucl. Phys. B 591 (2000) 515 [hep-ph/0005131] [INSPIRE].
A.H. Hoang and P. Ruiz-Femenia, Heavy pair production currents with general quantum numbers in dimensionally regularized NRQCD, Phys. Rev. D 74 (2006) 114016 [hep-ph/0609151] [INSPIRE].
H.S. Chung, \( \overline{MS} \) renormalization of S-wave quarkonium wavefunctions at the origin, JHEP 12 (2020) 065 [arXiv:2007.01737] [INSPIRE].
F. Feng, Y. Jia and W.-L. Sang, Can Nonrelativistic QCD Explain the γγ* → ηc Transition Form Factor Data?, Phys. Rev. Lett. 115 (2015) 222001 [arXiv:1505.02665] [INSPIRE].
X.-C. Zheng, X.-G. Wu, S.-Q. Wang, J.-M. Shen and Q.-L. Zhang, Reanalysis of the BFKL Pomeron at the next-to-leading logarithmic accuracy, JHEP 10 (2013) 117 [arXiv:1308.2381] [INSPIRE].
J. Zeng, X.-G. Wu, S. Bu, J.-M. Shen and S.-Q. Wang, Reanalysis of the Higgs-boson decay H → gg up to \( {\alpha}_s^6 \)-order level using the principle of maximum conformality, J. Phys. G 45 (2018) 085004 [arXiv:1801.01414] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
G.T. Bodwin, H.S. Chung, D. Kang, J. Lee and C. Yu, Improved determination of color-singlet nonrelativistic QCD matrix elements for S-wave charmonium, Phys. Rev. D 77 (2008) 094017 [arXiv:0710.0994] [INSPIRE].
H.S. Chung, J. Lee and C. Yu, NRQCD matrix elements for S-wave bottomonia and Γ[ηb(nS) → γγ] with relativistic corrections, Phys. Lett. B 697 (2011) 48 [arXiv:1011.1554] [INSPIRE].
K.G. Chetyrkin, J.H. Kühn and M. Steinhauser, RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. 133 (2000) 43 [hep-ph/0004189] [INSPIRE].
X.-K. Dong, L.-L. Wang and C.-Z. Yuan, Derived Born cross sections of e+e− annihilation into open charm mesons from CLEO-c measurements, Chin. Phys. C 42 (2018) 043002 [arXiv:1711.07311] [INSPIRE].
B.-L. Du, X.-G. Wu, J.-M. Shen and S.J. Brodsky, Extending the Predictive Power of Perturbative QCD, Eur. Phys. J. C 79 (2019) 182 [arXiv:1807.11144] [INSPIRE].
J.L. Basdevant, The Pade approximation and its physical applications, Fortsch. Phys. 20 (1972) 283 [INSPIRE].
M.A. Samuel, G. Li and E. Steinfelds, Estimating perturbative coefficients in quantum field theory using Pade approximants. 2., Phys. Lett. B 323 (1994) 188 [INSPIRE].
M.A. Samuel, J.R. Ellis and M. Karliner, Comparison of the Pade approximation method to perturbative QCD calculations, Phys. Rev. Lett. 74 (1995) 4380 [hep-ph/9503411] [INSPIRE].
E. Gardi, Why Pade approximants reduce the renormalization scale dependence in QFT?, Phys. Rev. D 56 (1997) 68 [hep-ph/9611453] [INSPIRE].
G. Cvetič, Improvement of the method of diagonal Pade approximants for perturbative series in gauge theories, Phys. Rev. D 57 (1998) 3209 [hep-ph/9711487] [INSPIRE].
Q. Yu, X.-G. Wu, J. Zeng, X.-D. Huang and H.-M. Yu, The heavy quarkonium inclusive decays using the principle of maximum conformality, Eur. Phys. J. C 80 (2020) 362 [arXiv:1911.05342] [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: 2007.14553
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
Yu, HM., Sang, WL., Huang, XD. et al. Scale-fixed predictions for γ + ηc production in electron-positron collisions at NNLO in perturbative QCD. J. High Energ. Phys. 2021, 131 (2021). https://doi.org/10.1007/JHEP01(2021)131
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2021)131