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Quarkonium Production and Polarization in an Improved Color Evaporation Model
Available online at www.sciencedirect.com
Nuclear Physics A 982 (2019) 751–754 www.elsevier.com/locate/nuclphysa
XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018)
Quarkonium Production and Polarization in an Improved Color Evaporation Model R. Vogt Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Physics Department, Univerity of California at Davis, Davis, CA 95616, USA
Abstract An improved version of the color evaporation model (ICEM) has been introduced to describe heavy quarkonium production. In contrast to the traditional color evaporation model, the constraint was imposed that the invariant mass of the intermediate heavy quark-antiquark pair must be larger than the mass of produced quarkonium. A momentum shift between heavy quark-antiquark pair and the quarkonium was also introduced. Calculations show that the model can describe the charmonium yields as well as ratio of ψ over J/ψ better than the traditional color evaporation model. The ICEM has been extended to calculate the polarization of prompt J/ψ and Υ(1S) production for the first time in the color evaporation approach. The first calculations were made at leading order. The polarization parameter λϑ was calculated as a function of center of mass energy and rapidity in p + p collisions. The xF dependence of the polarization was also calculated and compred to experimental results in p+Cu and π+W collisions. The next step of calculating the pT dependence of the polarization has been taken, with a calculation in the kT -factorization approach. We compare the unpolarized pT distributions for the J/ψ and ψ , as well as the ψ to J/ψ ratio, in p + p collisions at RHIC and the LHC for several energies. We also present the polarization results and make comparison to polarization data. Keywords: quarkonium, polarization
1. Introduction After 40 years, the production mechanism of quarkonium is still uncertain. Nonrelativistic QCD (NRQCD) [1], the most widely used model for quarkonium prduction, encounters serious challenges in both the apparent non-universality of the long distance matrix elements (LDMEs) and general inability to predict the quarkonium polarization. The production cross sections in NRQCD, based on an expansion in the strong coupling constant and the QQ velocity [2], is factorized into hard and soft contributions and divided into different color and spin states. The LDMEs, which weight the contributions from each color and spin state, are fit to the data above some minumum transverse momentum, pT , and are thus very sensitive to this cut. While the LDMEs are conjectured to be universal, they depend on the collision system. They also fail to describe both the yields and the polarization for pT cuts less than twice the quarkonium mass. In addition,
https://doi.org/10.1016/j.nuclphysa.2018.08.003 0375-9474/© 2018 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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R. Vogt / Nuclear Physics A 982 (2019) 751–754
the LHCb ηc pT distributions [3] disagrees with calculations [4] using J/ψ LDMEs using heavy quark spin symmetry [5, 6, 7]. The color evaporation model (CEM) [8, 9, 10], which considers all QQ (Q = c, b) production regardless of the quark color, spin, and momentum, is able to predict quarkonium yields with only a single normalization parameter [11]. However, it was not previously applied to the polarization. We have developed an improved color evaporation model [12] and applied it to study the quarkonium polarization [13, 14, 15]. 2. Improved Color Evaporation Model In the traditional CEM, all quarkonium states are treated the same as QQ below the HH threshold. The invariant mass of the heavy QQ pair is restricted to be less than twice the mass of the lowest mass meson with a constituent heavy quark. The distributions for all quarkonium family members are assumed to be identical. An improved CEM (ICEM) [12] was recently introduced. The improved CEM describes the charmonium yields as well as the ratio of ψ over J/ψ better than the traditional CEM. It is necessary to produce a cc pair in the hard collision to form a charmonium state ψ, because the mass of the cc pair is much larger than the QCD nonperturbative scale ΛQCD . Before the cc pair hadronizes, it will exchange many soft gluons between various color sources, as well as emit soft gluons. To separate the hard part from the other parts, the scale λ with mc λ ΛQCD was introduce and the hard part defined as encompassing all particles that are off shell by more than λ2 [12]. Soft gluons exchanged between the cc pair and other color sources (with momentum denoted by PS ) are distinguished from soft gluons emitted by the cc pair (with momentum denoted by PX ). These two kinds of soft gluons are significantly different. The total energy of exchanged gluons can be either positive or negative. However, the emitted gluons will eventually evolve to experimentally observable particles. Thus their total momentum must be time-like and their total energy must be positive. A relationship is constructed between P and Pψ , the average momentum of ψ that has hadronized from a cc pair with fixed momentum P. In the rest frame of P, P = (M, 0, 0, 0). For each event, we have P = Pψ + PS + PX . In the spirit of the traditional CEM, the distributions of PS and PX are assumed to be rotation invariant in this frame, implying PS = (mS , 0, 0, 0) and PX = (mX , 0, 0, 0). Because exchanged gluons can flow in either direction, mS ≈ 0 is expected. Thus Pψ = (M − mX , 0, 0, 0) with mX > 0 and Mψ < M − m X < M ,
(1)
where P0ψ must be larger than Mψ . Equation (1) sets a lower limit on M that is significantly different from and larger than the lower limit 2mc on the traditional CEM. As both PS and PX are on the order of λ, power counting of Pψ gives (O(mc ), O(λ), O(λ), O(λ)). Combined with the on-shell condition P2ψ = Mψ2 , P0ψ = Mψ + O(λ2 /mc ). Thus Pψ =
Mψ P + O(λ2 /mc ) , M
(2)
which differs from the relation used in the traditional CEM where Pψ is identified with P. Note that the proportionality between the momenta of the mother and daughter particles in Eq. (2) was first proposed in Ref. [16] to relate the momentum of the χcJ to the J/ψ produced by its decay. It has since been used in many calculations of quarkonium production in the NRQCD framework. By combining Eqs. (1) and (2), the cross section in the improved color evaporation model is 2MD 2MD dσψ (P) Mψ dσc¯c (M, P = (M/Mψ )P) dσc¯c (M, P ) 3 3 P d P dM δ (P − ) = F dM = F . (3) ψ ψ 3 3 M d P dMd P dMd3 P Mψ Mψ Figure 1, from Ref. [12] compares the ICEM result to the LHCb data from Refs. [17, 18]. The mass and scale parameters from Ref. [11] are used. The normalization parameter Fψ was adjusted by a factor of 1.4 because the modified mass range reduces the total cross section. Note that in the ratio shown on the right-hand side of Fig. 1 the theoretical uncertainties are assumed to be correlated.
R. Vogt / Nuclear Physics A 982 (2019) 751–754
753
Fig. 1. (Color online) Results from the ICEM for J/ψ (left) and ψ (center) production as well as the ψ to J/ψ ratio (right) at √ s = 7 TeV in the LHCb forward acceptance, 2.5 < y < 5. The LHCb J/ψ data are taken from Ref. [17] while the ψ data as well as the ratio are from Ref. [18]. From Ref. [12].
3. Polarization in the ICEM Until recently, no calculation of quarkonium polarization in the CEM existed. It was typically stated that there was either no polarization or that the CEM must produce unpolarized quarkonium. The truth was that no calculation had ever been made of the polarization in this model until recently. These calculations were done in collaboration with UC Davis student Vincent Cheung. We started with the assumption that the final state spin, like the kinematics of the ICEM, is ‘frozen in’ and thus unchanged by the color evaporation process. The contributions are now sorted according to their final-state spin so that results may be found for individual spin states while color is averaged over, as usual. Because the first calculations were leading order assuming collinear factorization [13, 14], only the dependence on rapidity and center of mass energy could be obtained. The first work simply separated the polarization into longitudinal and transverse components using the traditional CEM [13]. The follow on paper further separated the spin states to obtain the polarization of the S and P states independently. Thus it was possible to take feed down contributions to the prompt J/ψ yield into account. To do so properly, as well as produce separate results for J/ψ and ψ , the ICEM was used in this work [14]. In addition, we could calculate the rapidity and energy dependence of the polarization parameter λθ . We could make only limited comparison to data, however, since the transverse momentum of the quarkonium state is defined to be zero in the LO ICEM assuming collinear factorization. We compared to the Feynman x, xF , dependence of Υ polarization from the E866 fixed-target experiment since these data were integrated over low pT , pT < MΥ . Rather good agreement was found [14]. Note also that, at LO, only the helicity frame is available. More recently we have worked on obtaining the pT dependence of quarkonium production in the kT factorization approach [15]. The approach differs from the collinear factorization because off-shell matrix elements and an unintegrated gluon distribution are used in the calculations. Instead of using the full contingent of available diagrams, only the gg-initiated QQ pair production diagrams, three at leading order, are employed. Now the four-momenta and polarization vectors of the initial gluons depend on kT , so that k1 = (x1 s, k1T , x1 s) k2 = (x2 s, k2T , −x2 s), (k1 ) = (0, k1T /|k1T |, 0), and (k2 ) = (0, k2T /|k2T |, 0). The expression for the ICEM cross section is then 4m2D dφ2 dφ1 2 2 dk2T σ = Fψ d sˆ dx1 dx2 dk1T (4) 2π 2π m2Q × Φ1 (x1 , k1T , μ1 )Φ2 (x2 , k2T , μ2 )σ(R ˆ + R → QQ)δ( sˆ − x1 x2 s + | k1T + k2T |2 ) ,
(5)
where the cross section σ ˆ is proportional to the amplitude, A(R+R → QQ) = μ (k1 ) ν (k2 )Aμ,ν (g+g → QQ). See Ref. [15] for more details. The work presented at this conference and in Ref. [15] will be followed by a calculation of Υ production and polarization in the kT -factorization approach and, in the further future, a fully NLO calculation of the ICEM with polarization.
R. Vogt / Nuclear Physics A 982 (2019) 751–754
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Fig. 2. (Color √ online) Results from the LO ICEM in the kT -factorization approach for the J/ψ pT distribution(left) and polarization (right) at s = 7 TeV in the LHCb forward acceptance, 2.5 < y < 5. The LHCb J/ψ data are taken from Ref. [17] while the polarization data are from Ref. [19]. From Ref. [15].
Figure 2(left) shows the LO ICEM J/ψ pT distribution obtained in the kT -factorization approach compared to the LHCb forward data from Ref. [17]. The uncertainty band here is obtained by taking the mass and scale ranges 1.2 ≤ m ≤ 1.5 GeV, 0.5 ≤ μR /mT ≤ 2, and μF /mT = 1 [15] with the differences added in quadrature around the central value m = 1.27 GeV, μR /mT = μF /mT = 1. Note that the factorization scale is not varied because lowering the factorization scale pushes up against the limit of the unintegrated gluon distribution at high pT for e.g. μF /mT = 0.5. The right-hand side of Fig. 2 shows the result for the J/ψ polarization parameter, λθ , from LHCb [19] in the Collins-Soper (CS) frame. Even though the calculation is leading order, the normalization is similar to that found in Ref. [11]. Acknowledgments I would like to thank V. Cheung and Y.-Q. Ma for collaboration on the papers this talk was based on. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics (Nuclear Theory) under Contract No. DE-SC-0004014. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
W. E. Caswell and G. P. Lepage, Phys. Lett. B 167 (1986) 437. G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51 (1995) 1125; Phys. Rev. D 55 (1997) 5853. R. Aaij et al. [LHCb Collaboration], JHEP 1511 (2015) 103. M. Butensch¨on, Z. G. He and B. A. Kniehl, Phys. Rev. Lett. 114 (2015) 092004. M. Neubert, Phys. Rep. 245 (1994) 259. F. De Fazio, in At the Frontier of Particle Physics/Handbok of QCD, edited by M. A. Shifman (World Scientific, Singapore, 2001) p. 1671, arXiv:hep-ph/0010007. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phys. Rep. 281 (1997) 145. V. D. Barger, W. Y. Keung and R. J. Phillips, Phys. Lett. B 91 (1980) 253. V. D. Barger, W. Y. Keung and R. J. Phillips, Z. Phys. C 6 (1980) 169. R. Gavai, D. Kharzeev, H. Satz, G. A. Schuler, K. Sridhar and R. Vogt, Int. J. Mod. Phys. A 10 (1995) 3043. R. E. Nelson, R. Vogt and A. D. Frawley, Phys. Rev. C 87 (2013) 014908. Y. Q. Ma and R. Vogt, Phys. Rev. D 94 (2016) 114029. V. Cheung and R. Vogt, Phys. Rev. D 95 (2017) 074021. V. Cheung and R. Vogt, Phys. Rev. D 96 (2017) 054014. V. Cheung and R. Vogt, in preparation. Y. Q. Ma, K. Wang and K. T. Chao, Phys. Rev. D 83 (2011) 111503. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 71 (2011) 1645. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 72 (2012) 2100. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 73 (2013) 2631.
Nuclear Physics A 982 (2019) 751–754 www.elsevier.com/locate/nuclphysa
XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018)
Quarkonium Production and Polarization in an Improved Color Evaporation Model R. Vogt Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Physics Department, Univerity of California at Davis, Davis, CA 95616, USA
Abstract An improved version of the color evaporation model (ICEM) has been introduced to describe heavy quarkonium production. In contrast to the traditional color evaporation model, the constraint was imposed that the invariant mass of the intermediate heavy quark-antiquark pair must be larger than the mass of produced quarkonium. A momentum shift between heavy quark-antiquark pair and the quarkonium was also introduced. Calculations show that the model can describe the charmonium yields as well as ratio of ψ over J/ψ better than the traditional color evaporation model. The ICEM has been extended to calculate the polarization of prompt J/ψ and Υ(1S) production for the first time in the color evaporation approach. The first calculations were made at leading order. The polarization parameter λϑ was calculated as a function of center of mass energy and rapidity in p + p collisions. The xF dependence of the polarization was also calculated and compred to experimental results in p+Cu and π+W collisions. The next step of calculating the pT dependence of the polarization has been taken, with a calculation in the kT -factorization approach. We compare the unpolarized pT distributions for the J/ψ and ψ , as well as the ψ to J/ψ ratio, in p + p collisions at RHIC and the LHC for several energies. We also present the polarization results and make comparison to polarization data. Keywords: quarkonium, polarization
1. Introduction After 40 years, the production mechanism of quarkonium is still uncertain. Nonrelativistic QCD (NRQCD) [1], the most widely used model for quarkonium prduction, encounters serious challenges in both the apparent non-universality of the long distance matrix elements (LDMEs) and general inability to predict the quarkonium polarization. The production cross sections in NRQCD, based on an expansion in the strong coupling constant and the QQ velocity [2], is factorized into hard and soft contributions and divided into different color and spin states. The LDMEs, which weight the contributions from each color and spin state, are fit to the data above some minumum transverse momentum, pT , and are thus very sensitive to this cut. While the LDMEs are conjectured to be universal, they depend on the collision system. They also fail to describe both the yields and the polarization for pT cuts less than twice the quarkonium mass. In addition,
https://doi.org/10.1016/j.nuclphysa.2018.08.003 0375-9474/© 2018 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
752
R. Vogt / Nuclear Physics A 982 (2019) 751–754
the LHCb ηc pT distributions [3] disagrees with calculations [4] using J/ψ LDMEs using heavy quark spin symmetry [5, 6, 7]. The color evaporation model (CEM) [8, 9, 10], which considers all QQ (Q = c, b) production regardless of the quark color, spin, and momentum, is able to predict quarkonium yields with only a single normalization parameter [11]. However, it was not previously applied to the polarization. We have developed an improved color evaporation model [12] and applied it to study the quarkonium polarization [13, 14, 15]. 2. Improved Color Evaporation Model In the traditional CEM, all quarkonium states are treated the same as QQ below the HH threshold. The invariant mass of the heavy QQ pair is restricted to be less than twice the mass of the lowest mass meson with a constituent heavy quark. The distributions for all quarkonium family members are assumed to be identical. An improved CEM (ICEM) [12] was recently introduced. The improved CEM describes the charmonium yields as well as the ratio of ψ over J/ψ better than the traditional CEM. It is necessary to produce a cc pair in the hard collision to form a charmonium state ψ, because the mass of the cc pair is much larger than the QCD nonperturbative scale ΛQCD . Before the cc pair hadronizes, it will exchange many soft gluons between various color sources, as well as emit soft gluons. To separate the hard part from the other parts, the scale λ with mc λ ΛQCD was introduce and the hard part defined as encompassing all particles that are off shell by more than λ2 [12]. Soft gluons exchanged between the cc pair and other color sources (with momentum denoted by PS ) are distinguished from soft gluons emitted by the cc pair (with momentum denoted by PX ). These two kinds of soft gluons are significantly different. The total energy of exchanged gluons can be either positive or negative. However, the emitted gluons will eventually evolve to experimentally observable particles. Thus their total momentum must be time-like and their total energy must be positive. A relationship is constructed between P and Pψ , the average momentum of ψ that has hadronized from a cc pair with fixed momentum P. In the rest frame of P, P = (M, 0, 0, 0). For each event, we have P = Pψ + PS + PX . In the spirit of the traditional CEM, the distributions of PS and PX are assumed to be rotation invariant in this frame, implying PS = (mS , 0, 0, 0) and PX = (mX , 0, 0, 0). Because exchanged gluons can flow in either direction, mS ≈ 0 is expected. Thus Pψ = (M − mX , 0, 0, 0) with mX > 0 and Mψ < M − m X < M ,
(1)
where P0ψ must be larger than Mψ . Equation (1) sets a lower limit on M that is significantly different from and larger than the lower limit 2mc on the traditional CEM. As both PS and PX are on the order of λ, power counting of Pψ gives (O(mc ), O(λ), O(λ), O(λ)). Combined with the on-shell condition P2ψ = Mψ2 , P0ψ = Mψ + O(λ2 /mc ). Thus Pψ =
Mψ P + O(λ2 /mc ) , M
(2)
which differs from the relation used in the traditional CEM where Pψ is identified with P. Note that the proportionality between the momenta of the mother and daughter particles in Eq. (2) was first proposed in Ref. [16] to relate the momentum of the χcJ to the J/ψ produced by its decay. It has since been used in many calculations of quarkonium production in the NRQCD framework. By combining Eqs. (1) and (2), the cross section in the improved color evaporation model is 2MD 2MD dσψ (P) Mψ dσc¯c (M, P = (M/Mψ )P) dσc¯c (M, P ) 3 3 P d P dM δ (P − ) = F dM = F . (3) ψ ψ 3 3 M d P dMd P dMd3 P Mψ Mψ Figure 1, from Ref. [12] compares the ICEM result to the LHCb data from Refs. [17, 18]. The mass and scale parameters from Ref. [11] are used. The normalization parameter Fψ was adjusted by a factor of 1.4 because the modified mass range reduces the total cross section. Note that in the ratio shown on the right-hand side of Fig. 1 the theoretical uncertainties are assumed to be correlated.
R. Vogt / Nuclear Physics A 982 (2019) 751–754
753
Fig. 1. (Color online) Results from the ICEM for J/ψ (left) and ψ (center) production as well as the ψ to J/ψ ratio (right) at √ s = 7 TeV in the LHCb forward acceptance, 2.5 < y < 5. The LHCb J/ψ data are taken from Ref. [17] while the ψ data as well as the ratio are from Ref. [18]. From Ref. [12].
3. Polarization in the ICEM Until recently, no calculation of quarkonium polarization in the CEM existed. It was typically stated that there was either no polarization or that the CEM must produce unpolarized quarkonium. The truth was that no calculation had ever been made of the polarization in this model until recently. These calculations were done in collaboration with UC Davis student Vincent Cheung. We started with the assumption that the final state spin, like the kinematics of the ICEM, is ‘frozen in’ and thus unchanged by the color evaporation process. The contributions are now sorted according to their final-state spin so that results may be found for individual spin states while color is averaged over, as usual. Because the first calculations were leading order assuming collinear factorization [13, 14], only the dependence on rapidity and center of mass energy could be obtained. The first work simply separated the polarization into longitudinal and transverse components using the traditional CEM [13]. The follow on paper further separated the spin states to obtain the polarization of the S and P states independently. Thus it was possible to take feed down contributions to the prompt J/ψ yield into account. To do so properly, as well as produce separate results for J/ψ and ψ , the ICEM was used in this work [14]. In addition, we could calculate the rapidity and energy dependence of the polarization parameter λθ . We could make only limited comparison to data, however, since the transverse momentum of the quarkonium state is defined to be zero in the LO ICEM assuming collinear factorization. We compared to the Feynman x, xF , dependence of Υ polarization from the E866 fixed-target experiment since these data were integrated over low pT , pT < MΥ . Rather good agreement was found [14]. Note also that, at LO, only the helicity frame is available. More recently we have worked on obtaining the pT dependence of quarkonium production in the kT factorization approach [15]. The approach differs from the collinear factorization because off-shell matrix elements and an unintegrated gluon distribution are used in the calculations. Instead of using the full contingent of available diagrams, only the gg-initiated QQ pair production diagrams, three at leading order, are employed. Now the four-momenta and polarization vectors of the initial gluons depend on kT , so that k1 = (x1 s, k1T , x1 s) k2 = (x2 s, k2T , −x2 s), (k1 ) = (0, k1T /|k1T |, 0), and (k2 ) = (0, k2T /|k2T |, 0). The expression for the ICEM cross section is then 4m2D dφ2 dφ1 2 2 dk2T σ = Fψ d sˆ dx1 dx2 dk1T (4) 2π 2π m2Q × Φ1 (x1 , k1T , μ1 )Φ2 (x2 , k2T , μ2 )σ(R ˆ + R → QQ)δ( sˆ − x1 x2 s + | k1T + k2T |2 ) ,
(5)
where the cross section σ ˆ is proportional to the amplitude, A(R+R → QQ) = μ (k1 ) ν (k2 )Aμ,ν (g+g → QQ). See Ref. [15] for more details. The work presented at this conference and in Ref. [15] will be followed by a calculation of Υ production and polarization in the kT -factorization approach and, in the further future, a fully NLO calculation of the ICEM with polarization.
R. Vogt / Nuclear Physics A 982 (2019) 751–754
754
Fig. 2. (Color √ online) Results from the LO ICEM in the kT -factorization approach for the J/ψ pT distribution(left) and polarization (right) at s = 7 TeV in the LHCb forward acceptance, 2.5 < y < 5. The LHCb J/ψ data are taken from Ref. [17] while the polarization data are from Ref. [19]. From Ref. [15].
Figure 2(left) shows the LO ICEM J/ψ pT distribution obtained in the kT -factorization approach compared to the LHCb forward data from Ref. [17]. The uncertainty band here is obtained by taking the mass and scale ranges 1.2 ≤ m ≤ 1.5 GeV, 0.5 ≤ μR /mT ≤ 2, and μF /mT = 1 [15] with the differences added in quadrature around the central value m = 1.27 GeV, μR /mT = μF /mT = 1. Note that the factorization scale is not varied because lowering the factorization scale pushes up against the limit of the unintegrated gluon distribution at high pT for e.g. μF /mT = 0.5. The right-hand side of Fig. 2 shows the result for the J/ψ polarization parameter, λθ , from LHCb [19] in the Collins-Soper (CS) frame. Even though the calculation is leading order, the normalization is similar to that found in Ref. [11]. Acknowledgments I would like to thank V. Cheung and Y.-Q. Ma for collaboration on the papers this talk was based on. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics (Nuclear Theory) under Contract No. DE-SC-0004014. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
W. E. Caswell and G. P. Lepage, Phys. Lett. B 167 (1986) 437. G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51 (1995) 1125; Phys. Rev. D 55 (1997) 5853. R. Aaij et al. [LHCb Collaboration], JHEP 1511 (2015) 103. M. Butensch¨on, Z. G. He and B. A. Kniehl, Phys. Rev. Lett. 114 (2015) 092004. M. Neubert, Phys. Rep. 245 (1994) 259. F. De Fazio, in At the Frontier of Particle Physics/Handbok of QCD, edited by M. A. Shifman (World Scientific, Singapore, 2001) p. 1671, arXiv:hep-ph/0010007. R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phys. Rep. 281 (1997) 145. V. D. Barger, W. Y. Keung and R. J. Phillips, Phys. Lett. B 91 (1980) 253. V. D. Barger, W. Y. Keung and R. J. Phillips, Z. Phys. C 6 (1980) 169. R. Gavai, D. Kharzeev, H. Satz, G. A. Schuler, K. Sridhar and R. Vogt, Int. J. Mod. Phys. A 10 (1995) 3043. R. E. Nelson, R. Vogt and A. D. Frawley, Phys. Rev. C 87 (2013) 014908. Y. Q. Ma and R. Vogt, Phys. Rev. D 94 (2016) 114029. V. Cheung and R. Vogt, Phys. Rev. D 95 (2017) 074021. V. Cheung and R. Vogt, Phys. Rev. D 96 (2017) 054014. V. Cheung and R. Vogt, in preparation. Y. Q. Ma, K. Wang and K. T. Chao, Phys. Rev. D 83 (2011) 111503. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 71 (2011) 1645. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 72 (2012) 2100. R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C 73 (2013) 2631.