P波底夸克偶素遍举衰变到双J/Ψ
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摘要
重夸克偶素物理[1]是高能物理一个重要的前沿领域,历史上对确立量子色动力学(QCD)作为正确描述强相互作用的基本理论发挥了重要作用。我国高能物理研究所的北京谱仪,是世界上最重要的研究粲夸克偶素物理的平台之一[2]。本文尝试用非相对论性的量子色动力学因子化方法(Nonrelativistic QCD factorization或者NRQCD factorization)[3]来系统探讨P波底夸克偶素三重态(χ_(bJ),J=0,1,2)遍举衰变到两个矢量粲夸克偶素(J/ψ)的过程。
近十余年来,重夸克偶素物理领域的重要进展是其单举衰变过程(Inclusivedecay ofheavy quarkonium)能够在NRQCD的框架下被系统、严格地描述[3]。但是相应的关于利用该理论框架来研究夸克偶素遍举衰变的工作并不多。本论文拟处的部分过程已被Braguta,Likhoded和Luchinsky用基于光锥(light-cone)微扰QCD的方法计算过[4]。然而,对于这类初末态均牵涉到重夸克偶素的过程用NRQCD因子化方法来处理可能会更自然些[5]。
B-工厂的Belle和Babar实验组在质心能为10.6 GeV处对双粲夸克偶素的产生进行了研究。他们发现(?)。的产生截面[6-8]比理论上在非相对论极限下QCD领头阶的计算[9,10]大了将近一个量级。而S波态的J/ψ加P波态的χ_(bO)产生过程中,理论和实验之间也存在较大差距。为了解决B-工厂中J/ψ产生过程中理论与实验之间存在的巨大差距,文献[9-14]等做了诸多尝试。计算表明对于一些参量的数值选择比较敏感,这也说明了理论计算的不确定性。但是经过众多理论尝试后发现如果考虑微扰修正和辐射修正可以使理论和实验取得很好的一致。B-工厂中双粲夸克偶素产生的研究对理解粲夸克偶素的硬遍举过程、量子色动力学的动力学以及介子的内部结构有着很大的帮助。类似的,底夸克偶素衰变到双粲夸克偶素的过程也可以用来研究硬遍举过程和重夸克偶素的内部结构。这些过程是非常类似于B-工厂的双粲夸克偶素产生,因为底夸克偶素的质量非常接近B-工厂的能量中心值。我们的计算在非相对论量子色动力学的框架下考虑相对论修正到v~2阶以及电磁碎裂的领头阶。因为没有χ_(bJ)的总宽度实验数据,我们用两种方法对其总宽度做了理论计算,由这些给出分支比的数值结果。经过计算我们得出的分支比如下同时我们也发现计算到v~2阶的相对论修正对(?)的影响约为11.2%,对(?)的影响约为10.5%。电磁碎裂对(?)的影响约为0.06%,对(?)的影响约为0.05%。可以看出电磁贡献很小。但是对于(?)因其领头阶的强作用贡献为0,相对论修正到v~2阶也为0,但是它的电磁过程不为0,所以电磁贡献就很大。
Heavy Quarkonium Physics [1] is an important frontier area in high-energy physics. Historically it played an important role in establishing QuantumChromodynamics as a correct description of the basic theory of stronginteraction. The Beijing Spectrometer at the Institute of High Energy Physicsin China is one of the most important platforms researching charmoniumphy- sics [2]. This thesis attempts to use non-relativistic quantumchromodynamics factorization approach [3] (NRQCD) to systematicallyinvestigate the P-wave bottomonium triplet decay to double vectorcharmonium process.
In recent years, one important progress in heavy quarkonium physics isthat its inclusive decay can be systematically and strictly described in theframework of NRQCD. However, there are few application of this method tostudy exclusive decay of heavy quarkonium. This thesis is intended to dealwith parts of the processes having been studied by Braguta, Likhoded andLuchinsky [4] using light-cone perturbation QCD method. However, forprocesses including heavy quarkonium both in initial and final states,NRQCD factorization is a more natural choice [5].
Belle and BaBar Collabortions at B-factories have studied exclusivedouble charmonium production processes at (?) = 10.6 GeV. They find thatthe experimental cross section of (?)[6-8] at B-factories isabout an order of magnitude larger than the NRQCD prediction at leadingorder in bothα_s and v~2 [9, 10]. There is also a large discrepancy for theprocess (?). In order to resolve the great discrepancy betweenthe theory and the experiment for the J/ψproduction processes, a lot ofefforts have been made [9-14] . Calculations show that the predicted crosssection is very sensitive to some input parameters, which also explains the uncertainty of theoretical calculations. After a series of theoretical attempts, itis found that the theoretical prediction and experimental data can obtain agood agreement if we include the perturbative and radiative correction.Studying double charmonium production processes at B-factories are helpfulto our understanding of the hard exclusive processes, quantumchromodynamics and the internal structure of quarkonium. Similarly, thebottomonium decay into double charmonium process can also be used tostudy the hard exclusive processes and internal structure of heavyquarkonium. These processes are similar to the processes of doublecharmonium production at B-factories since the masses of bottomonium arevery close the center-of-mass energy of B-factories.
Our calculation contains the relativistic correction at the order of v~2 ofJ/ψand leading-order of electromagnetic fragmentation in the framework ofnon-relativistic QCD. The total decay width ofχ_(bJ) is not availableexperimentally, so we use two approaches to deduce the total decay width ofχ_(bJ). The numerical values of the branching ratios ofχ_(bJ) decays to doubleJ/ψareWe find the relativitic correction is about 11.2% for (?), andabout 10.5% for (?), We also find the QED contribution is verysmall, only about 0.06% for (?), and about 0.05% for (?)J/ψ, but very important for (?), since the leading-order QCDcontribution vanishes for this decay channel.
近十余年来,重夸克偶素物理领域的重要进展是其单举衰变过程(Inclusivedecay ofheavy quarkonium)能够在NRQCD的框架下被系统、严格地描述[3]。但是相应的关于利用该理论框架来研究夸克偶素遍举衰变的工作并不多。本论文拟处的部分过程已被Braguta,Likhoded和Luchinsky用基于光锥(light-cone)微扰QCD的方法计算过[4]。然而,对于这类初末态均牵涉到重夸克偶素的过程用NRQCD因子化方法来处理可能会更自然些[5]。
B-工厂的Belle和Babar实验组在质心能为10.6 GeV处对双粲夸克偶素的产生进行了研究。他们发现(?)。的产生截面[6-8]比理论上在非相对论极限下QCD领头阶的计算[9,10]大了将近一个量级。而S波态的J/ψ加P波态的χ_(bO)产生过程中,理论和实验之间也存在较大差距。为了解决B-工厂中J/ψ产生过程中理论与实验之间存在的巨大差距,文献[9-14]等做了诸多尝试。计算表明对于一些参量的数值选择比较敏感,这也说明了理论计算的不确定性。但是经过众多理论尝试后发现如果考虑微扰修正和辐射修正可以使理论和实验取得很好的一致。B-工厂中双粲夸克偶素产生的研究对理解粲夸克偶素的硬遍举过程、量子色动力学的动力学以及介子的内部结构有着很大的帮助。类似的,底夸克偶素衰变到双粲夸克偶素的过程也可以用来研究硬遍举过程和重夸克偶素的内部结构。这些过程是非常类似于B-工厂的双粲夸克偶素产生,因为底夸克偶素的质量非常接近B-工厂的能量中心值。我们的计算在非相对论量子色动力学的框架下考虑相对论修正到v~2阶以及电磁碎裂的领头阶。因为没有χ_(bJ)的总宽度实验数据,我们用两种方法对其总宽度做了理论计算,由这些给出分支比的数值结果。经过计算我们得出的分支比如下同时我们也发现计算到v~2阶的相对论修正对(?)的影响约为11.2%,对(?)的影响约为10.5%。电磁碎裂对(?)的影响约为0.06%,对(?)的影响约为0.05%。可以看出电磁贡献很小。但是对于(?)因其领头阶的强作用贡献为0,相对论修正到v~2阶也为0,但是它的电磁过程不为0,所以电磁贡献就很大。
Heavy Quarkonium Physics [1] is an important frontier area in high-energy physics. Historically it played an important role in establishing QuantumChromodynamics as a correct description of the basic theory of stronginteraction. The Beijing Spectrometer at the Institute of High Energy Physicsin China is one of the most important platforms researching charmoniumphy- sics [2]. This thesis attempts to use non-relativistic quantumchromodynamics factorization approach [3] (NRQCD) to systematicallyinvestigate the P-wave bottomonium triplet decay to double vectorcharmonium process.
In recent years, one important progress in heavy quarkonium physics isthat its inclusive decay can be systematically and strictly described in theframework of NRQCD. However, there are few application of this method tostudy exclusive decay of heavy quarkonium. This thesis is intended to dealwith parts of the processes having been studied by Braguta, Likhoded andLuchinsky [4] using light-cone perturbation QCD method. However, forprocesses including heavy quarkonium both in initial and final states,NRQCD factorization is a more natural choice [5].
Belle and BaBar Collabortions at B-factories have studied exclusivedouble charmonium production processes at (?) = 10.6 GeV. They find thatthe experimental cross section of (?)[6-8] at B-factories isabout an order of magnitude larger than the NRQCD prediction at leadingorder in bothα_s and v~2 [9, 10]. There is also a large discrepancy for theprocess (?). In order to resolve the great discrepancy betweenthe theory and the experiment for the J/ψproduction processes, a lot ofefforts have been made [9-14] . Calculations show that the predicted crosssection is very sensitive to some input parameters, which also explains the uncertainty of theoretical calculations. After a series of theoretical attempts, itis found that the theoretical prediction and experimental data can obtain agood agreement if we include the perturbative and radiative correction.Studying double charmonium production processes at B-factories are helpfulto our understanding of the hard exclusive processes, quantumchromodynamics and the internal structure of quarkonium. Similarly, thebottomonium decay into double charmonium process can also be used tostudy the hard exclusive processes and internal structure of heavyquarkonium. These processes are similar to the processes of doublecharmonium production at B-factories since the masses of bottomonium arevery close the center-of-mass energy of B-factories.
Our calculation contains the relativistic correction at the order of v~2 ofJ/ψand leading-order of electromagnetic fragmentation in the framework ofnon-relativistic QCD. The total decay width ofχ_(bJ) is not availableexperimentally, so we use two approaches to deduce the total decay width ofχ_(bJ). The numerical values of the branching ratios ofχ_(bJ) decays to doubleJ/ψareWe find the relativitic correction is about 11.2% for (?), andabout 10.5% for (?), We also find the QED contribution is verysmall, only about 0.06% for (?), and about 0.05% for (?)J/ψ, but very important for (?), since the leading-order QCDcontribution vanishes for this decay channel.
引文
[1] N. Brambilla et al. [Quarkonium Working Group], Heavy quarkonium physics. arXiv:0412158[hep-ph].
[2] D. M. Asner et al., Physics at BES-Ⅲ, arXiv:0809.1869 [hep-ex].
[3] G. T. Bodwin, E. Braaten and G. P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium. Phys Rev D, 1995, 51: 1125. [Erratum-ibid. D 55,5853 (1997)].
[4] V. V. Braguta, A. K. Likhoded and A. V. Luchinsky, Observation potential for χ_b at the Tevatron and LHC, Phys Rev D, 2005, 72: 094018.
[5] Y. Jia, Which hadronic decay modes are good for η_b searching: double J/ψ or something else?. Phys Rev D, 2008, 78: 054003.
[6] K. Abe et al. [Belle Collaboration], Observation of double c(?) production in e~+e~- annihilation at (?) GeV, Phys Rev Lett, 2002, 89: 142001.
[7] K. Abe et al. [Belle Collaboration], Study of double charmonium production in e~+e~- annihilation at (?) GeV, Phys Rev D, 2004, 70: 071102.
[8] B. Aubert et al. [BABAR Collaboration], Measurement of double charmonium production in e~+e~- annihilations at (?) GeV, Phys Rev D, 2005,72:031101.
[9] E. Braaten and J. Lee, Exclusive double-charmonium production in e~+e~- annihilation, Phys Rev D, 2003, 67: 054007. [Erratum-ibid. D, 2005, 72:099901]
[10] K. Y. Liu, Z. G. He and K. T. Chao, Problems of double charm production in e~+e~- annihilation at (?) GeV, Phys Lett B, 2003, 557: 45.
[11] G. T. Bodwin, J. Lee and E. Braaten, e~+e~- Annihilation into J/ψ + J/ψ, Phys Rev Lett, 2003, 90: 162001.
[12] Y. J. Zhang, Y. J. Gao and K. T. Chao, Next-to-leading order QCD correction to (?) at (?) GeV, Phys Rev Lett, 2006, 96: 092001.
[13] G. T. Bodwin, J. Lee and C. Yu, Resummation of Relativistic Corrections to (?), Phys Rev D, 2008, 77: 094018.
[14] Z. G. He, Y. Fan and K. T. Chao, Relativistic corrections to J/ψ exclusive and inclusive double charm production at B factories, Phys Rev D, 2007, 75: 074011.
[15]章乃森.粒子物理学,科学出版社,(1986).
[16] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Charmonium: The Model, Phys Rev D, 1979, 17: 3090. [Erratum-ibid. D, 1980, 21: 313]
[17] W. Buchmuller and S. H. H. Tye, Quarkonia And Quantum Chromodynamics, Phys Rev D, 1981, 24: 132.
[18] A. Kuhn, A fit of upsilon and charmonium spectra, Phys Lett B, 1980, 93: 338.
[19] C. Quigg and J. L. Rosner, Quarkonium level spacings, Phys Lett B, 1977, 71: 153.
[20] J. H. Kuhn, Weak Interactions Of Quarkonia, Acta Phys Polon B, 1981, 12: 347.
[21] I.I. Y. Bigi, Y. L. Dokshitzer, V. A. Khoze, J. H. Kuhn and P. M. Zerwas, Production and Decay Properties of Ultraheavy Quarks, Phys Lett B, 1986, 181: 157.
[22] S. K. Choi et al. [BELLE collaboration], Observation of the η_c (2S) in Exclusive (?) Decays, Phys Rev Lett, 2002, 89: 102001. [Erratum-ibid. 2002,89: 129901.]
[23] D. M. Asner et al. [CLEO Collaboration], Observation of η_c production in γγ Fusion at CLEO, Phys Rev Lett, 2004, 92: 142001.
[24] B. Aubert et al. [BABAR Collaboration], Measurements of the mass and Width of the η_c meson and of an η_c(2S) candidate, Phys Rev Lett, 2004, 92: 142002.
[25] J. L. Rosner et al. [CLEO Collaboration], Observation of hc(~1P_1) state of charmonium, Phys Rev Lett, 2005, 95: 102003.
[26] M. Andreotti et al., Results of a search for the hc(~1P_1) state of charmonium in the η_cγ and J/ψ π~0 decay modes, Phys Rev D, 2005, 72: 032001.
[27] S. K. Choi et al. [Belle Collaboration], Observation of a new narrow charmonium state in Exclusive (?) Decays, Phys Rev Lett, 2003, 91: 262001.
[28] D. E. Acosta et al. [CDF Ⅱ Collaboration], Observation of the narrow state X(3872) (?) in p(?) collisions at (?) TeV, Phys Rev Lett, 2004, 93:072001.
[29] V. M. Abazov et al. [DO Collaboration], Observation and properties of the X(3872) decaying to (?) in p(?) collisions at (?) TeV, Phys Rev Lett, 2004,93: 162002.
[30] B. Aubert et al. [BABAR Collaboration], Study of the B→ J /ψ K~- π~+π~- decay and measurement of the B→ X(3872)K~- branching fraction, Phys Rev D, 2005, 71: 071103.
[31] K. Abe et al. [Belle Collaboration], Observation of a near-threshold J /ψ mass enhancement in exclusive B→ K(?) J /ψ decays, Phys Rev Lett, 2005, 94: 182002.
[32] S. Uehara et al. [Belle Collaboration], Observation of a χ'_(c2) candidate in γγ→ D(D|-) production at Belle, Phys Rev Lett, 2006, 96: 082003.
[33] B. Aubert et al. [BABAR Collaboration], Observation of the bottomonium ground state in the decay γ(3S)→ γη_b, Phys Rev Lett, 2008, 101: 071801.[Erratum-ibid. 2009, 102: 029901]
[34] H. Fritzsch, Producing Heavy Quark Flavors In Hadronic Collisions: A Test Of Quantum Chromodynamics, Phys Lett B, 1977, 67: 217.
[35] F. Halzen, Cvc For Gluons And Hadroproduction Of Quark Flavors, Phys Lett B, 1977, 69: 105.
[36] J. H. Kuhn, J. Kaplan and E. G. O. Safiani, Electromagnetic Annihilation Of e~+e~- Into Quarkonium States With Even Charge Conjugation, Nucl Phys B,157, 125 (1979).
[37] G. T. Bodwin and A. Petrelli, Order v~4 corrections to S-wave quarkonium decay, Phys. Rev.D, 2002, 66: 094011.
[38] H. E. Haber, Spin formalism and applications to new physics searches, arXiv:9405376 [hep-ph].
[39] G. T. Bodwin, E. Braaten, D. Kang and J. Lee, Inclusive Charm Production in χ_b Decays, Phys Rev D, 2007, 76: 054001.
[40] E. J. Eichten and C. Quigg, Quarkonium Wave Functions At The Origin, Phys Rev D, 1995, 52: 1726.
[41] C. Amsler et al. [Particle Data Group], Review of particle physics, Phys Lett B, 2008, 667: 1.
[42] A. Petrelli, M. Cacciari, M. Greco, F. Maltoni and M. L. Mangano, NLO production and decay of quarkonium, Nucl Phys B, 1998, 514: 245.
[43] R. Barbieri, M. Ca(?)o, R. Gatto and E. Remiddi, Strong QCD Corrections To P Wave Quarkonium Decays, Phys Lett B, 1980, 95: 93; Nucl Phys B, 1981, 192: 61.
[2] D. M. Asner et al., Physics at BES-Ⅲ, arXiv:0809.1869 [hep-ex].
[3] G. T. Bodwin, E. Braaten and G. P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium. Phys Rev D, 1995, 51: 1125. [Erratum-ibid. D 55,5853 (1997)].
[4] V. V. Braguta, A. K. Likhoded and A. V. Luchinsky, Observation potential for χ_b at the Tevatron and LHC, Phys Rev D, 2005, 72: 094018.
[5] Y. Jia, Which hadronic decay modes are good for η_b searching: double J/ψ or something else?. Phys Rev D, 2008, 78: 054003.
[6] K. Abe et al. [Belle Collaboration], Observation of double c(?) production in e~+e~- annihilation at (?) GeV, Phys Rev Lett, 2002, 89: 142001.
[7] K. Abe et al. [Belle Collaboration], Study of double charmonium production in e~+e~- annihilation at (?) GeV, Phys Rev D, 2004, 70: 071102.
[8] B. Aubert et al. [BABAR Collaboration], Measurement of double charmonium production in e~+e~- annihilations at (?) GeV, Phys Rev D, 2005,72:031101.
[9] E. Braaten and J. Lee, Exclusive double-charmonium production in e~+e~- annihilation, Phys Rev D, 2003, 67: 054007. [Erratum-ibid. D, 2005, 72:099901]
[10] K. Y. Liu, Z. G. He and K. T. Chao, Problems of double charm production in e~+e~- annihilation at (?) GeV, Phys Lett B, 2003, 557: 45.
[11] G. T. Bodwin, J. Lee and E. Braaten, e~+e~- Annihilation into J/ψ + J/ψ, Phys Rev Lett, 2003, 90: 162001.
[12] Y. J. Zhang, Y. J. Gao and K. T. Chao, Next-to-leading order QCD correction to (?) at (?) GeV, Phys Rev Lett, 2006, 96: 092001.
[13] G. T. Bodwin, J. Lee and C. Yu, Resummation of Relativistic Corrections to (?), Phys Rev D, 2008, 77: 094018.
[14] Z. G. He, Y. Fan and K. T. Chao, Relativistic corrections to J/ψ exclusive and inclusive double charm production at B factories, Phys Rev D, 2007, 75: 074011.
[15]章乃森.粒子物理学,科学出版社,(1986).
[16] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T. M. Yan, Charmonium: The Model, Phys Rev D, 1979, 17: 3090. [Erratum-ibid. D, 1980, 21: 313]
[17] W. Buchmuller and S. H. H. Tye, Quarkonia And Quantum Chromodynamics, Phys Rev D, 1981, 24: 132.
[18] A. Kuhn, A fit of upsilon and charmonium spectra, Phys Lett B, 1980, 93: 338.
[19] C. Quigg and J. L. Rosner, Quarkonium level spacings, Phys Lett B, 1977, 71: 153.
[20] J. H. Kuhn, Weak Interactions Of Quarkonia, Acta Phys Polon B, 1981, 12: 347.
[21] I.I. Y. Bigi, Y. L. Dokshitzer, V. A. Khoze, J. H. Kuhn and P. M. Zerwas, Production and Decay Properties of Ultraheavy Quarks, Phys Lett B, 1986, 181: 157.
[22] S. K. Choi et al. [BELLE collaboration], Observation of the η_c (2S) in Exclusive (?) Decays, Phys Rev Lett, 2002, 89: 102001. [Erratum-ibid. 2002,89: 129901.]
[23] D. M. Asner et al. [CLEO Collaboration], Observation of η_c production in γγ Fusion at CLEO, Phys Rev Lett, 2004, 92: 142001.
[24] B. Aubert et al. [BABAR Collaboration], Measurements of the mass and Width of the η_c meson and of an η_c(2S) candidate, Phys Rev Lett, 2004, 92: 142002.
[25] J. L. Rosner et al. [CLEO Collaboration], Observation of hc(~1P_1) state of charmonium, Phys Rev Lett, 2005, 95: 102003.
[26] M. Andreotti et al., Results of a search for the hc(~1P_1) state of charmonium in the η_cγ and J/ψ π~0 decay modes, Phys Rev D, 2005, 72: 032001.
[27] S. K. Choi et al. [Belle Collaboration], Observation of a new narrow charmonium state in Exclusive (?) Decays, Phys Rev Lett, 2003, 91: 262001.
[28] D. E. Acosta et al. [CDF Ⅱ Collaboration], Observation of the narrow state X(3872) (?) in p(?) collisions at (?) TeV, Phys Rev Lett, 2004, 93:072001.
[29] V. M. Abazov et al. [DO Collaboration], Observation and properties of the X(3872) decaying to (?) in p(?) collisions at (?) TeV, Phys Rev Lett, 2004,93: 162002.
[30] B. Aubert et al. [BABAR Collaboration], Study of the B→ J /ψ K~- π~+π~- decay and measurement of the B→ X(3872)K~- branching fraction, Phys Rev D, 2005, 71: 071103.
[31] K. Abe et al. [Belle Collaboration], Observation of a near-threshold J /ψ mass enhancement in exclusive B→ K(?) J /ψ decays, Phys Rev Lett, 2005, 94: 182002.
[32] S. Uehara et al. [Belle Collaboration], Observation of a χ'_(c2) candidate in γγ→ D(D|-) production at Belle, Phys Rev Lett, 2006, 96: 082003.
[33] B. Aubert et al. [BABAR Collaboration], Observation of the bottomonium ground state in the decay γ(3S)→ γη_b, Phys Rev Lett, 2008, 101: 071801.[Erratum-ibid. 2009, 102: 029901]
[34] H. Fritzsch, Producing Heavy Quark Flavors In Hadronic Collisions: A Test Of Quantum Chromodynamics, Phys Lett B, 1977, 67: 217.
[35] F. Halzen, Cvc For Gluons And Hadroproduction Of Quark Flavors, Phys Lett B, 1977, 69: 105.
[36] J. H. Kuhn, J. Kaplan and E. G. O. Safiani, Electromagnetic Annihilation Of e~+e~- Into Quarkonium States With Even Charge Conjugation, Nucl Phys B,157, 125 (1979).
[37] G. T. Bodwin and A. Petrelli, Order v~4 corrections to S-wave quarkonium decay, Phys. Rev.D, 2002, 66: 094011.
[38] H. E. Haber, Spin formalism and applications to new physics searches, arXiv:9405376 [hep-ph].
[39] G. T. Bodwin, E. Braaten, D. Kang and J. Lee, Inclusive Charm Production in χ_b Decays, Phys Rev D, 2007, 76: 054001.
[40] E. J. Eichten and C. Quigg, Quarkonium Wave Functions At The Origin, Phys Rev D, 1995, 52: 1726.
[41] C. Amsler et al. [Particle Data Group], Review of particle physics, Phys Lett B, 2008, 667: 1.
[42] A. Petrelli, M. Cacciari, M. Greco, F. Maltoni and M. L. Mangano, NLO production and decay of quarkonium, Nucl Phys B, 1998, 514: 245.
[43] R. Barbieri, M. Ca(?)o, R. Gatto and E. Remiddi, Strong QCD Corrections To P Wave Quarkonium Decays, Phys Lett B, 1980, 95: 93; Nucl Phys B, 1981, 192: 61.