Quarkonium Production in an Improved Color Evaporation Model
aa r X i v : . [ h e p - ph ] S e p Quarkonium Production in an Improved Color Evaporation Model
Yan-Qing Ma , , ∗ and Ramona Vogt , † School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA Physics Department, University of California at Davis, Davis, CA 95616, USA (Dated: September 21, 2016)We propose an improved version of the color evaporation model to describe heavy quarkoniumproduction. In contrast to the traditional color evaporation model, we impose the constraint thatthe invariant mass of the intermediate heavy quark-antiquark pair to be larger than the mass ofproduced quarkonium. We also introduce a momentum shift between heavy quark-antiquark pairand the quarkonium. Numerical calculations show that our model can describe the charmoniumyields as well as ratio of ψ ′ over J/ψ better than the traditional color evaporation model.
PACS numbers: 12.38.Bx, 12.39.St, 14.40.Pq
I. INTRODUCTION
The study of heavy quarkonium production is one ofthe best ways to understand hadronization in QCD. Cur-rently, the most widely used theory for heavy quarko-nium production is the nonrelativistic QCD (NRQCD)approach [28] proposed in 1994. By introducing a system-atic velocity expansion, this theory can naturally solvethe infrared divergence problem encountered in the colorsinglet model (CSM) [29–31]. In this sense, NRQCD fac-torization can be thought of as a generalized version ofCSM. Furthermore, it also successfully explained the ψ ′ surplus found at Tevatron [32] by including color octetcontributions.Nevertheless, recent studies have shown that NRQCDfactorization encouters serious difficulties [33]. First,naive power counting implies that ψ ( nS ) and Υ( nS ) pro-ductions at hadron colliders are dominated by the S color octet channel which results in transverse polariza-tion at high transverse momentum, p T . However, exper-imental measurements found these states to be almostunpolarized. Current explanations of J/ψ polarizationinclude S color octet dominance [34–36] and cancela-tion of transverse polarization between the S and P J color octet channels [34, 37, 38]. Whether these expla-nations can be generalized to other quarkonium states isstill in question. Second, the nonperturbative color octetlong-distance matrix elements (LDMEs) extracted fromhadron colliders [39–41] are inconsistent with the upperbound set by e + e − collisions [42]. Thus the LDMEs arenot universal. Finally, there is still no convincing proofof NRQCD factorization to all orders in α s . The state-of-art proof is only to next-to-next-to-leading order forspecial cases [43].Considering the above difficulties, one should definitelystudy NRQCD factorization in more detail, but, at the ∗ [email protected] † [email protected] same time, one may need to turn to other theories ofquarkonium production. A theory which is known to sat-isfy all-order factorization is the color evaporation model(CEM) [44, 45]. In this model, to produce a charmoniumstates ψ , one first produces a charm quark-antiquark pair cc with invariant mass smaller than the D -meson thresh-old. The pair then hadronizes to the ψ by randomlyemitting soft particles . The production cross section isexpressed as dσ ψ ( P ) d P = F ψ Z M D m c dM dσ c ¯ c ( M, P ) dM d P , (1)where m c ( M D ) is the mass of charm quark ( D meson)and M is the invariant mass of the c ¯ c pair. In this model,it is assumed that the ψ momentum, P , is approximatelythe same as the momentum of the cc pair. The predictivepower of the CEM is based on the assumption that thehadronization factor F ψ is universal and thus indepen-dent of the kinematics and spin of the ψ , as well as theproduction process.Although CEM is intuitive, simple, and successful toexplain J/ψ production data, it has very fatal flaw. Astraightforward conclusion from the CEM is that the ra-tio of differential cross sections of two charmonia statesis independent of the kinematics and independent of thecolliding species. However, it has long known that exper-imental results of ratio of production cross section of ψ ′ over that of J/ψ depend on their transverse momentum(recent experimental data see Refs. [46, 47]). This dis-agreement is regarded as the main evidence that CEM isa wrong.Considered the advantages of CEM mentioned above,we may need to study whether a modification of CEMcan provide a correct theory for quarkonium production.In this paper, by taking into account physical effectsoverlooked in the original CEM, we propose an improved We refer to them as soft gluons here.
FIG. 1. An illustration of charmonium production in a highenergy collision. See text for details. color-evaporation model (ICEM). On the one hand, thenice features of CEM are retained in the ICEM, includ-ing having only one parameter for each quarkonium stateand satisfying all-order factorization. On the other hand,the ICEM can correctly describe charmonium productioncross section ratios.
II. THE IMPROVED COLOR-EVAPORATIONMODEL
Our picture of heavy quarkonium (say charmonium)production is as follows. To produce a charmonium state ψ , it is necessary to produce a cc pair in the hard col-lision, because the mass of the cc pair is much largerthan the QCD nonperturbative scale Λ QCD . Before the cc pair hadronizes to charmonium, it will exchange manysoft gluons between various color sources, as well as emitsoft gluons. An illustration of this picture is given inFig. 1. In this figure, the blob marked by ‘ H ’ denotesthe hard collision kernel, the blob marked by ‘ S ’ de-notes soft interactions, and the thick double lines de-notes the cc pair with momentum P . To separate thehard part from the other parts, we introduce a scale λ with m c ≫ λ ≫ Λ QCD , and define the hard part as allparticles that are off shell by more than λ .We emphasize that we distinguish soft gluons ex-changed between the cc pair and other color sources (withmomentum denoted by P S ) from soft gluons emitted bythe cc pair (with momentum denoted by P X ). Indeed,these two kinds of soft gluons are significantly differ-ent. The total energy of exchanged gluons can be eitherpositive or negative. However, the emitted gluons willeventually evolve to experimentally observable particles.Thus their total momentum must be time-like and theirtotal energy must be positive.In our model, we construct a relationship between P and h P ψ i , the average momentum of ψ that hashadronized from a cc pair with fixed momentum P . Therelationship is easy to obtain in the rest frame of P , with P = ( M, , , . For each event, we have P = P ψ + P S + P X . (2)In the spirit of the traditional CEM, we assume the dis-tributions of P S and P X are rotation invariant in thisframe, which implies h P S i = ( m S , , , and h P X i =( m X , , , . Because exchanged gluons can flow in ei-ther direction, we may expect m S ≈ . Thus h P ψ i =( M − m X , , , with m X > . Therefore, M ψ < M − m X < M , (3)where we use the fact that h P ψ i must be larger than M ψ .Equation (3) sets a lower limit on M that is significantlydifferent from the lower limit m c of the traditional CEM.As both P S and P X are order of λ , power counting of P ψ gives ( O ( m c ) , O ( λ ) , O ( λ ) , O ( λ )) . Combining with theon-shell condition P ψ = M ψ , we arrive at P ψ = M ψ + O ( λ /m c ) . Thus we have h P ψ i = M ψ M P + O ( λ /m c ) , (4)which again differs from the relation used in the tradi-tional CEM where P ψ is identified with P . Note thatthe proportionality between the momenta of the motherand daughter particles in Eq. (4) was first proposed inRef. [48] to relate the momentum of the χ cJ and the J/ψ produced by its decay. It has since been used in manycalculations of quarkonium production in the NRQCDframework. In this paper, we prove the relation rigor-ously with clear assumptions. By combining Eqs. (3) and(4), we arrive at the improved color evaporation model(ICEM): dσ ψ ( P ) d P = F ψ Z M D M ψ d P ′ dM dσ c ¯ c ( M, P ′ ) dM d P ′ δ ( P − M ψ M P ′ )= F ψ Z M D M ψ dM dσ c ¯ c ( M, P ′ = ( M/M ψ ) P ) dM d P , (5)with correction at O ( λ /m c ) . If one is only interested inthe transverse momentum distribution, we have dσ ψ ( P ) dp T = F ψ Z M D M ψ dM MM ψ dσ c ¯ c ( M, P ′ ) dM dp ′ T | p ′ T =( M/M ψ ) p T . (6)Before performing any numerical calculations, we canalready expect some advantages of the ICEM. First, be-cause there is an explicit charmonium mass dependencein Eq. (5), the ratio of differential cross sections of twocharmonia is no longer p T -independent in the ICEM.Thus it is possible to explain data such as dσ ψ (2 S ) /dσ J/ψ .Second, by making a distinction between the momentumof the cc pair and that of charmonium, the predicted p T spectra will be softer and thus may explain the high p T data better.We emphasize that the ICEM Eq. 5 does not meanthat c ¯ c pair with invariant mass smaller than M ψ has nopossibility to hadronize to ψ . In fact, this kind of c ¯ c paircan absorb energy by interacting with other color source,and thus can have larger invariant mass and hadronize to ψ . At the same time, even if the invariant mass of c ¯ c pairis larger than M ψ , it may loss energy by interacting withother color source, and eventually cannot hadronize to ψ because of invariant mass being too small. By assuming m S ≈ , we effectively approximate that the two effectscancel each other. As a result, the Eq. 5 should be onlyinterpreted at the integration level.An exception for the above argument is for the groundstate particle production, say η c for charmonium. Basedon the quark-hadron duality, c ¯ c pair with invariant masssmaller than D meson threshold must hadronize to char-monium, therefore it is not possible for a c ¯ c pair with M c ¯ c > M η c to emit too much energy so that its invariantmass becomes smaller than M η c . This means that theapproximation m S ≈ is not reasonable here and thusICEM is not good for η c production. However, for η c production, as the condition Eq. (3) is not needed, theoriginal CEM should be good. III. NUMERICAL RESULTS
To confront our model with experimental data, we up-dated the CEM parameters determined in Ref. [50]. Inthat work, in an attempt to reduce the uncertainty onthe total charm cross section, the charm mass was fixedat . ± . GeV while the factorization and renor-malization scales were fit to a subset of the measuredtotal charm cross section data. The values found were µ F /m = 2 . +2 . − . and µ R /m = 1 . +0 . − . employing theCT10 proton parton densities [51].The central open charm parameter set ( m, µ F /m, µ R /m ) = (1 . , . , . was used to cal-culate the energy dependence of the forward J/ψ crosssection, σ ( x F > , in the CEM using the exclusive cc production code described in Ref. [52]. Because theNLO cc code is an exclusive calculation, the mass cut ison the invariant average over kinematic variables of the c and c . Thus, in this calculation µ F and µ R are definedrelative to the transverse mass of the charm quark, µ F,R ∝ m T = p m + p T where p T = 0 . p T c + p T c ) .The normalization F ψ is the scale factor that adjustedthe fraction of the total charm cross section in the massrange m < M < m D to the forward cross section data.To determine the uncertainty on the J/ψ calcula-tion, the charm mass was varied between the up-per and lower limits, 1.36 and 1.18 GeV respec-tively, for the central values of µ F /m and µ R /m ,and the scales were varied around their central values while the charm mass was held fixed atits central value of 1.27 GeV: ( µ F /m, µ R /m ) =( C, L ) , ( L, C ) , ( L, L ) , ( C, H ) , ( H, C ) , ( H, H ) where H ( L ) is the upper (lower) limit of the factorization and renor-malization scales determined from the charm fits. Usingthe same value of F ψ in all cases, the uncertainty band onthe J/ψ cross section was calculated by finding the up-per and lower limits of the mass and scale variations andadding them in quadrature, as discussed in Refs. [50, 53].To calculate the charmonium p T dependence, a Gaus-sian transverse momentum broadening is added to thefinal state. The value of the average k T kick applied wastaken to be h k T i = 1+(1 /
12) ln( √ s/ GeV [50], giving1.19 GeV at RHIC and 1.49 GeV at 7 TeV.Since the ICEM calculation discussed here reduces thecross section relative to the calculation in Ref. [50], thevalue of F ψ had to be increased by 40% to retain agree-ment with the data. The ψ ′ cross section and its uncer-tainty was calculated with the same parameters but witha value of F ψ ′ scaled to the ψ ′ data.To obtain the uncertainty on the ψ ′ /ψ ratio, the massand scale uncertainties were assumed to be correlated.The resulting uncertainty band is dominated by the scaleuncertainty, the mass uncertainty is small.Our results for J/ψ production cross section as a func-tion of p T are shown in Fig. 2, where we compare withdata at hadron colliders for center of mass energies of0.2 TeV and 7 TeV. The 0.2 TeV RHIC data are mea-sured by the PHENIX Collaboration [46] at central ra-pidities, | y | < . , and the 7 TeV LHC data are mea-sured by the LHCb Collaboration [49] at forward rapid-ity, . < y < . The largest discrepancy between themodel and the data is in the RHIC data at intermediate p T , < p T < GeV. However, since the experimentaluncertainty is rather large in this region, our results arein general agreement with the data.We now turn to the ψ ′ production cross section as afunction of p T in Fig. 3. We again compare with themidrapidity PHENIX data [46] at 0.2 TeV and the for-ward LHCb data [47] at 7 TeV. Since the ψ ′ rates are gen-erally lower, the measured uncertainty is larger. Giventhis, the agreement of the calculation with the data isalso good.The ratio of the production cross sections of ψ ′ to thatof J/ψ as a function of p T is given in Fig. 4. The 0.2 TeVRHIC data and 7 TeV LHC data are taken from Ref. [46]and Ref. [47], respectively. Although the original CEMpredicts a constant for this ratio, in contradiction withthe data, our ICEM calculations are in good agreementwith all data. IV. SUMMARY AND DISCUSSION
By distinguishing between exchanged and emitted softgluons and considering some physical constraints, we pro-pose an improved color evaporation model for charmo-nium production. Comparison with data shows that the
FIG. 2. Results for
J/ψ production. The 0.2 TeV PHENIX data and 7 TeV LHCb data are taken from Ref. [46] and Ref. [49],respectively.FIG. 3. Results for ψ ′ production. The 0.2 TeV PHENIX data and 7 TeV LHCb data are taken from Ref. [46] and Ref. [47],respectively. ICEM can nicely reproduce the p T dependence of the ra-tio of the ψ ′ to J/ψ production cross sections. Thus, thisimproved model overcomes one of the main obstacles ofthe original CEM. The success of the ICEM calculationconfirms our picture of charmonium production.We note that the question of polarization in the ICEMas well as the original CEM has not yet been addressed.As seen in the NRQCD approach, the polarization is animportant test of models. The prediction of the final-state charmonium polarization depends on whether softgluons change spin and angular momentum of the c ¯ c pair. A preliminary study of charmonium polarization in theCEM will be presented elsewhere [54]. ACKNOWLEDGMENTS
We thank Kuang-Ta Chao, Raju Venugopalan andHong-Fei Zhang for useful discussions. The work of RVwas performed under the auspices of the U.S. Departmentof Energy by Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344and and supportedby the U.S. Department of Energy, Office of Science, Of-fice of Nuclear Physics (Nuclear Theory) under contractnumber DE-SC-0004014. .
FIG. 4. Results for ratio of the ψ ′ production cross section to that of J/ψ . The 0.2 TeV PHENIX data and 7 TeV LHCb dataare taken from Ref. [46] and Ref. [47], respectively.[1] G. T. Bodwin, E. Braaten, and G. P. Lepage,
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Hadronic Production of ψ/J
Mesons , Phys.Rev.
D14 (1976) 3115 [
InSPIRE ].[4] C.-H. Chang,
Hadronic Production of
J/ψ
AssociatedWith a Gluon , Nucl.Phys.
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InSPIRE ].[5] E. Braaten and S. Fleming,
Color octet fragmentationand the ψ ′ surplus at the Tevatron , Phys. Rev. Lett. (1995) 3327 [ hep-ph/9411365 ][ InSPIRE ].[6] N. Brambilla, S. Eidelman, B. Heltsley, R. Vogt,G. Bodwin, et al. , Heavy quarkonium: progress, puzzles,and opportunities , Eur. Phys. J. C (2011) 1534[ arXiv:1010.5827 ] [ InSPIRE ].[7] K.-T. Chao, Y.-Q. Ma, H.-S. Shao, K. Wang, and Y.-J.Zhang,
J/ψ
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Fragmentation contributions to
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Quarkonium production in the LHC era: apolarized perspective , Phys. Lett. B (2014) 98–109[ arXiv:1403.3970 ] [
InSPIRE ].[10] H. Han, Y.-Q. Ma, C. Meng, H.-S. Shao, and K.-T.Chao, η c Production at LHC and Implications for theUnderstanding of
J/ψ
Production , Phys. Rev. Lett. (2015) 092005 [ arXiv:1411.7350 ][ InSPIRE ].[11] H.-F. Zhang, Z. Sun, W.-L. Sang, and R. Li,
Impact of η c hadroproduction data on charmonium production andpolarization within NRQCD framework , Phys. Rev. Lett. (2015) 092006 [ arXiv:1412.0508 ][ InSPIRE ].[12] Y.-Q. Ma, K. Wang, and K.-T. Chao,
J/ψ ( ψ ′ ) production at the Tevatron and LHC at O ( α s v ) innonrelativistic QCD , Phys. Rev. Lett. (2011) 042002 [ arXiv:1009.3655 ][ InSPIRE ].[13] M. Butenschoen and B. A. Kniehl,
Reconciling
J/ψ production at HERA, RHIC, Tevatron, and LHC withNRQCD factorization at next-to-leading order , Phys. Rev. Lett. (2011) 022003 [ arXiv:1009.5662 ][ InSPIRE ].[14] B. Gong, L.-P. Wan, J.-X. Wang, and H.-F. Zhang,
Polarization for Prompt
J/ψ , ψ (2 S ) production at theTevatron and LHC , Phys.Rev.Lett. (2013) 042002[ arXiv:1205.6682 ] [
InSPIRE ].[15] Y.-J. Zhang, Y.-Q. Ma, K. Wang, and K.-T. Chao,
QCD radiative correction to color-octet
J/ψ inclusiveproduction at B Factories , Phys. Rev. D (2010) 034015 [ arXiv:0911.2166 ][ InSPIRE ].[16] G. C. Nayak, J.-W. Qiu, and G. Sterman,
NRQCDFactorization and Velocity-dependence of NNLO Polesin Heavy Quarkonium Production , Phys. Rev. D (2006) 074007 [ hep-ph/0608066 ][ InSPIRE ].[17] H. Fritzsch,
Producing Heavy Quark Flavors inHadronic Collisions: A Test of QuantumChromodynamics , Phys.Lett.
B67 (1977) 217 [
InSPIRE ].[18] F. Halzen,
Cvc for Gluons and Hadroproduction ofQuark Flavors , Phys.Lett.
B69 (1977) 105 [
InSPIRE ].[19]
PHENIX Collaboration , A. Adare et al. , Groundand excited charmonium state production in p + p collisions at √ s = 200 GeV , Phys. Rev. D (2012) 092004 [ arXiv:1105.1966 ][ InSPIRE ].[20]
LHCb Collaboration , R. Aaij et al. , Measurement of ψ (2 S ) meson production in pp collisions at sqrt(s)=7TeV , Eur. Phys. J. C (2012) 2100[ arXiv:1204.1258 ] [ InSPIRE ].[21] Y.-Q. Ma, K. Wang, and K.-T. Chao,
QCD radiativecorrections to χ cJ production at hadron colliders , Phys. Rev. D (2011) 111503 [ arXiv:1002.3987 ][ InSPIRE ].[22]
LHCb Collaboration , R. Aaij et al. , Measurement of
J/ψ production in pp collisions at √ s = 7 TeV , Eur. Phys. J. C (2011) 1645 [ arXiv:1103.0423 ][ InSPIRE ].[23] R. Nelson, R. Vogt, and A. Frawley,
Narrowing theuncertainty on the total charm cross section and itseffect on the
J/ψ cross section , Phys. Rev. C (2013) 014908 [ arXiv:1210.4610 ][ InSPIRE ].[24] H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P. M. Nadolsky,J. Pumplin, and C. P. Yuan,
New parton distributionsfor collider physics , Phys. Rev. D (2010) 074024[ arXiv:1007.2241 ].[25] M. L. Mangano, P. Nason, and G. Ridolfi, Heavy quarkcorrelations in hadron collisions at next-to-leadingorder , Nucl. Phys. B (1992) 295.[26] M. Cacciari, P. Nason, and R. Vogt,
QCD predictionsfor charm and bottom production at RHIC , Phys. Rev. Lett. (2005) 122001 [ hep-ph/0502203 ].[27] H. S. Cheung and R. Vogt in progress (2016) .[28] G. T. Bodwin, E. Braaten, and G. P. Lepage, Rigorous QCDanalysis of inclusive annihilation and production of heavyquarkonium , Phys. Rev. D (1995) 1125 [ hep-ph/9407339 ][ InSPIRE ].[29] S. Ellis, M. B. Einhorn, and C. Quigg,
Comment onHadronic Production of Psions , Phys.Rev.Lett. (1976) 1263 [ InSPIRE ].[30] C. Carlson and R. Suaya,
Hadronic Production of ψ/J
Mesons , Phys.Rev.
D14 (1976) 3115 [
InSPIRE ].[31] C.-H. Chang,
Hadronic Production of
J/ψ
Associated Witha Gluon , Nucl.Phys.
B172 (1980) 425–434 [
InSPIRE ].[32] E. Braaten and S. Fleming,
Color octet fragmentation andthe ψ ′ surplus at the Tevatron , Phys. Rev. Lett. (1995) 3327 [ hep-ph/9411365 ][ InSPIRE ].[33] N. Brambilla, S. Eidelman, B. Heltsley, R. Vogt, G. Bodwin, et al. , Heavy quarkonium: progress, puzzles, andopportunities , Eur. Phys. J. C (2011) 1534[ arXiv:1010.5827 ] [ InSPIRE ].[34] K.-T. Chao, Y.-Q. Ma, H.-S. Shao, K. Wang, and Y.-J.Zhang,
J/ψ
Polarization at Hadron Colliders inNonrelativistic QCD , Phys. Rev. Lett. (2012) 242004[ arXiv:1201.2675 ] [
InSPIRE ].[35] G. T. Bodwin, H. S. Chung, U.-R. Kim, and J. Lee,
Fragmentation contributions to
J/ψ production at theTevatron and the LHC , Phys. Rev. Lett. (2014) 022001[ arXiv:1403.3612 ] [
InSPIRE ].[36] P. Faccioli, V. Knunz, C. Lourenco, J. Seixas, and H. K.Wohri,
Quarkonium production in the LHC era: a polarizedperspective , Phys. Lett. B (2014) 98–109[ arXiv:1403.3970 ] [
InSPIRE ].[37] H. Han, Y.-Q. Ma, C. Meng, H.-S. Shao, and K.-T. Chao, η c Production at LHC and Implications for the Understandingof
J/ψ
Production , Phys. Rev. Lett. (2015) 092005[ arXiv:1411.7350 ] [
InSPIRE ].[38] H.-F. Zhang, Z. Sun, W.-L. Sang, and R. Li,
Impact of η c hadroproduction data on charmonium production andpolarization within NRQCD framework , Phys. Rev. Lett. (2015) 092006 [ arXiv:1412.0508 ][ InSPIRE ].[39] Y.-Q. Ma, K. Wang, and K.-T. Chao,
J/ψ ( ψ ′ ) production atthe Tevatron and LHC at O ( α s v ) in nonrelativistic QCD , Phys. Rev. Lett. (2011) 042002 [ arXiv:1009.3655 ][ InSPIRE ].[40] M. Butenschoen and B. A. Kniehl,
Reconciling
J/ψ production at HERA, RHIC, Tevatron, and LHC withNRQCD factorization at next-to-leading order , Phys. Rev. Lett. (2011) 022003 [ arXiv:1009.5662 ][ InSPIRE ].[41] B. Gong, L.-P. Wan, J.-X. Wang, and H.-F. Zhang,
Polarization for Prompt
J/ψ , ψ (2 S ) production at theTevatron and LHC , Phys.Rev.Lett. (2013) 042002[ arXiv:1205.6682 ] [
InSPIRE ].[42] Y.-J. Zhang, Y.-Q. Ma, K. Wang, and K.-T. Chao,
QCDradiative correction to color-octet
J/ψ inclusive productionat B Factories , Phys. Rev. D (2010) 034015[ arXiv:0911.2166 ] [ InSPIRE ].[43] G. C. Nayak, J.-W. Qiu, and G. Sterman,
NRQCDFactorization and Velocity-dependence of NNLO Poles inHeavy Quarkonium Production , Phys. Rev. D (2006) 074007 [ hep-ph/0608066 ] [ InSPIRE ].[44] H. Fritzsch,
Producing Heavy Quark Flavors in HadronicCollisions: A Test of Quantum Chromodynamics , Phys.Lett.
B67 (1977) 217 [
InSPIRE ].[45] F. Halzen,
Cvc for Gluons and Hadroproduction of QuarkFlavors , Phys.Lett.
B69 (1977) 105 [
InSPIRE ].[46]
PHENIX Collaboration , A. Adare et al. , Ground andexcited charmonium state production in p + p collisions at √ s = 200 GeV , Phys. Rev. D (2012) 092004[ arXiv:1105.1966 ] [ InSPIRE ].[47]
LHCb Collaboration , R. Aaij et al. , Measurement of ψ (2 S ) meson production in pp collisions at sqrt(s)=7 TeV , Eur. Phys. J. C (2012) 2100 [ arXiv:1204.1258 ][ InSPIRE ].[48] Y.-Q. Ma, K. Wang, and K.-T. Chao,
QCD radiativecorrections to χ cJ production at hadron colliders , Phys. Rev. D (2011) 111503 [ arXiv:1002.3987 ][ InSPIRE ].[49]
LHCb Collaboration , R. Aaij et al. , Measurement of
J/ψ production in pp collisions at √ s = 7 TeV , Eur. Phys. J. C (2011) 1645 [ arXiv:1103.0423 ][ InSPIRE ].[50] R. Nelson, R. Vogt, and A. Frawley,
Narrowing theuncertainty on the total charm cross section and its effect onthe
J/ψ cross section , Phys. Rev. C (2013) 014908[ arXiv:1210.4610 ] [ InSPIRE ].[51] H.-L. Lai, M. Guzzi, J. Huston, Z. Li, P. M. Nadolsky,J. Pumplin, and C. P. Yuan,
New parton distributions forcollider physics , Phys. Rev. D (2010) 074024[ arXiv:1007.2241 ].[52] M. L. Mangano, P. Nason, and G. Ridolfi, Heavy quarkcorrelations in hadron collisions at next-to-leading order , Nucl. Phys. B (1992) 295.[53] M. Cacciari, P. Nason, and R. Vogt,
QCD predictions forcharm and bottom production at RHIC , Phys. Rev. Lett. (2005) 122001 [ hep-ph/0502203 ].[54] H. S. Cheung and R. Vogt in progressin progress