Recall a Prediction on J/psi Nuclear Modification Factor for STAR Measurements
aa r X i v : . [ h e p - ph ] J un Recall a prediction on
J/ψ nuclear modification factorfor STAR measurements
Xiao-Ming Xu
Department of Physics, Shanghai University, Baoshan, Shanghai 200444, China
Abstract
STAR collaboration has offered eminent nuclear modification factor of
J/ψ athigh p T and midrapidity produced in Cu-Cu collisions at √ s NN = 200 GeV. Recall-ing a prediction we can understand that the feature of high- p T nuclear modificationfactor is related to c ¯ c produced by 2 → → c and¯ c . The nuclear modification factor at high p T is sensitive to the earliest form ofdeconfined matter that does not have a temperature. Keywords:
J/ψ nuclear modification factor; the prethermal stage; recombinationmechanism. 1ecently, STAR collaboration has measured the midrapidity ratio R AA of J/ψ pro-duced in Cu-Cu collisions to p - p collisions at √ s NN = 200 GeV [1]. The ratio is a functionof the transverse momentum p T . The ratio increases with increasing p T and arrives at0.9 ± p T > c . Error bars at p T > c are large. If J/ψ , χ c and ψ ′ undergo only dissociation processes due to the interaction with gluons of quark-gluonplasma [2], the ratio must be smaller than 1 [3]. On the other hand, the transverse mo-mentum larger than 5 GeV/ c is very much higher than the average momentum of quarksand gluons of quark-gluon plasma in thermal equilibrium. If the J/ψ nuclear modificationfactor R AA at p T > c is taken to be larger than 1, how can we understand themeasured p T dependence? This is can be understood from my prediction [4] as follows.The prediction in Ref. [4] is about the ratio of momentum distribution of J/ψ producedin central Au-Au collisions at √ s NN = 200 GeV to nucleon-nucleon collisions. Thepredicted ratio is shown by the solid curve in Fig. 1. The J/ψ production includes thecontributions of direct
J/ψ , the radiative feeddown from direct χ cJ and the decay of direct ψ ′ . The theoretical ratio is larger than 1 at the transverse momentum between 2.5 and 7GeV at rapidity y = 0. This enhancement as stated below is caused by c ¯ c yielded through2 → → c ¯ c pair is produced in the initial nuclear collision, in the prethermal stage and inthe thermal stage. The produced c ¯ c pair is a pointlike color singlet or a color octet pairfrom 2 → a + b → c ¯ c and 2 → a + b → c ¯ c + x where a , b and xdenote partons. The pointlike c ¯ c pair expands and may get into free states when it travelsthrough the deconfined matter.A charm quark and a charm antiquark can recombine into a bound state with aprobability. The probability is proportional to the nonperturbative matrix elements innonrelativistic QCD [13], < O H ( S ) >, < O H ( S ) >, < O H ( P ) > O H ( S ) = χ + ~σT a ψ · ( a + H a H ) ψ + ~σT a χ , O H ( S ) = χ + T a ψ ( a + H a H ) ψ + T a χ , and O H ( P ) = χ + ( − i ↔ D · ~σ ) T a ψ ( a + H a H ) ψ + ( − i ↔ D · ~σ ) T a χ with ψ as the Pauli spinor fieldthat annihilates a heavy quark, χ as the Pauli spinor field that creates a heavy antiquarkand a + H as the operator that creates the quarkonium H in the out state. The nonper-turbative matrix elements are constants. In the recombination mechanism proposed inRef. [5], the probability for c ¯ c to form a bound state is a constant.When c ¯ c penetrates through deconfined matter, it is broken into a free charm quarkand a free charm antiquark by reactions g + c ¯ c [ n S +1 L (1) J ] → c + ¯ c, g + c ¯ c [ S +1 L (8) J ] → c + ¯ c where n S +1 L (1 , J is the spectroscopic notation for quantum numbers and for singlet oroctet by the superscript. Cross sections for the reactions were obtained [4] with a formulain Ref. [2, 14]. In hadronic matter charmonia are dissociated by mesons into charmedmesons via the reactions q ¯ q + c ¯ c [ n S +1 L (1) J ] → q ¯ c + c ¯ q Cross sections for the reactions were calculated [4] with a formula of Ref. [15]. Based onthese cross sections c ¯ c survival probability can be obtained.Momentum distribution of direct charmonium consists of five terms dN direct dyd p T = dN → dyd p T ( S a/A = 1) + dN → dyd p T + dN → dyd p T + dN → dyd p T + dN → dyd p T (1)where the five terms result from c ¯ c pairs produced in the initial nuclear collision via the2 → → → → → J/ψ , dN J/ψ prompt dyd p T , includes the contributions of direct J/ψ , the radiativefeeddown from direct χ cJ and the decay of direct ψ ′ . Let dN J/ψ dyd p T = dN → dyd p T ( S a/A = 1) be themomentum distribution of prompt J/ψ while the cross sections for charmonia dissociatedby gluons and hadrons are set as zero. The nuclear modification factor is R AA = dN J/ψ prompt dyd p T / dN J/ψ dyd p T (2)The R AA has been shown in Fig. 1. R AA < S a/A = 1, dN → dyd p T = dN → dyd p T = dN → dyd p T = dN → dyd p T = 0, and the charmoniumdissociation cross sections are taken into account. Therefore, R AA > N → dyd p T = 0, dN → dyd p T = 0, dN → dyd p T = 0 and dN → dyd p T = 0 indicates that charmonia yielded inthe prethermal stage and in the thermal stage overcome the loss of charmonia due to thedissociation of charmonia in collisions with gluons and hadrons. Now the question left iswhy the ratio R AA > p T ?Momentum and space distributions of partons in the prethermal stage were studiedin detail in Refs. [16–18]. The Fig. 5 given by Eskola and Wang [16] showed the vari-ation of transverse momentum distribution dN/d p T with time. Before hard scatteringspartons are in Gaussian distribution due to the initial state radiation. However, the largemomentum transfer in the hard scatterings considerably increases the parton numbers atlarge p T and an approximate exponential distribution comes with a larger p T tail. Theabundance of partons with transverse momenta greater than 5 GeV is exactly a requisitewhat we want for getting the enhancement of J/ψ production at large p T as the 2 → a + b → c ¯ c explicitly lead to large- p T c ¯ c pairs. The dashed, dot-dashed anddotted curves in Figs. 2-3 stand for direct charmonia from the initial nuclear collision,the prethermal stage and the thermal stage, respectively. We found that the yield ofcharmonia resulting from c ¯ c pairs produced in the prethermal stage can be larger thanthat in the initial nuclear collision and can be much larger than that in the thermal stage.Therefore, quark-gluon matter in the perthermal stage dominates the contributions to R >
R > . < p T < y = 0 is a result of c ¯ c yielded from the prethermal stage and by means of the recombination mechanism. InCu-Cu collisions at √ s NN = 200 GeV the thermal stage is shortened or disappeared andthe number density of deconfined matter gets smaller. Hence, charmonium dissociationgets weaker and less c ¯ c pairs are produced. But the two factors compete. Since quarksand gluons at high p T in the prethermal stage are still abundant, we can expect R AA ∼ . J/ψ at high p T at midrapidity is related to theearliest form of deconfined matter, i.e., quark-gluon matter in the prethermal stage. Thenuclear modification factor R AA ∼ . c ¯ c produced in deconfinedmatter in the prethermal stage and the recombination of c and ¯ c . Acknowledgements
This work was supported by National Natural Science Foundation of China underGrant No. 10675079. 4 eferences [1] Z. Tang, for the STAR Collaboration, arXiv:0804.4846, talk given at the QuarkMatter 2008 conference, Jaipur, India, February 4-10, 2008.[2] D. Kharzeev, H. Satz, Phys. Lett. B334 (1994) 155.[3] X.-M. Xu, D. Kharzeev, H. Satz, X.-N. Wang, Phys. Rev. C53 (1996) 3051.[4] X.-M. Xu, Nucl. Phys. A697 (2002) 825.[5] X.-M. Xu, Nucl. Phys. A658 (1999) 165.[6] P. Braun-Munzinger, J. Stachel, Phys. Lett. B490 (2000) 196.[7] R.L. Thews, M. Schroedter, J. Rafelski, Phys. Rev. C63 (2001) 054905.[8] L. Grandchamp, R. Rapp, Phys. Lett. B523 (2001) 60.[9] M.I. Gorenstein, A.P. Kostyuk, H. St¨ocker, W. Greiner, Phys. Lett. B509 (2001) 277.[10] Z.W. Lin, D. Molnar, Phys. Rev. C68 (2003) 044901.[11] V. Greco, C.M. Ko, R. Rapp, Phys. Lett. B595 (2004) 202.[12] L. Yan, P. Zhuang, N. Xu, Phys. Rev. Lett. 97 (2006) 232301.[13] W. E. Caswell, G. P. Lepage, Phys. Lett. B167 (1986) 437; G. P. Lepage, L. Magnea,C. Nakhleh, U. Magnea, K. Hornbostel, Phys. Rev. D46 (1992) 4052; G. T. Bodwin,E. Braaten, G. P. Lepage, Phys. Rev. D51 (1995) 1125.[14] M.E. Peskin, Nucl. Phys. B156 (1979) 365; G. Bhanot, M.E. Peskin, Nucl. Phys.B156 (1979) 391.[15] T. Barnes, E.S. Swanson, Phys. Rev. D46 (1992) 131; E.S. Swanson, Ann. Phys. 220(1992) 73.[16] K.J. Eskola, X.-N. Wang, Phys. Rev. D49 (1994) 1284.[17] P. L´evai, B. M¨uller, X.-N. Wang, Phys. Rev. C51 (1995) 3326.[18] Z. Lin, M. Gyulassy, Phys. Rev. C51 (1995) 2177.5 T (GeV/c)0.60.811.21.41.6 r a t i o Figure 1: Ratio versus transverse momentum at rapidity y = 0 for prompt J/ψ productionin central Au-Au collisions at √ s NN = 200 GeV.6igure 2: J/ψ momentum distributions versus rapidity at p T = 4 GeV in the left paneland transverse momentum at y = 0 in the right panel for central Au-Au collisions at √ s NN = 200 GeV. The dashed, dot-dashed, dotted and lower solid curves correspond to c ¯ c productions in the initial collision, the prethermal stage, the thermal stage and theall three stages (direct J/ψ ), respectively. The upper solid curves (prompt
J/ψ ) are thesum of all contributions including the radiative feeddown from direct χ cJ and the decayof direct ψ ′ . 7igure 3: Direct ψ ′ momentum distributions versus rapidity at p T = 4 GeV in the leftpanel and transverse momentum at y = 0 in the right panel for central Au-Au collisionsat √ s NN = 200 GeV. The dashed, dot-dashed, dotted and solid curves correspond to c ¯ c productions in the initial collision, the prethermal stage, the thermal stage and the allthree stages (direct ψ ′ ), respectively. 8igure 4: Direct χ c momentum distributions versus rapidity at p T = 4 GeV in the leftpanel and transverse momentum at y = 0 in the right panel for central Au-Au collisionsat √ s NN = 200 GeV. The dashed, dot-dashed, dotted and solid curves correspond to c ¯ c productions in the initial collision, the prethermal stage, the thermal stage and the allthree stages (direct χ cc