Upsilon Production as a Probe for Early State Dynamics in High Energy Nuclear Collisions at RHIC
aa r X i v : . [ nu c l - t h ] J a n Υ Production as a Probe for Early State Dynamics in High EnergyNuclear Collisions at RHIC
Yunpeng Liu a , Baoyi Chen a , Nu Xu b , Pengfei Zhuang a a Physics Department, Tsinghua University, Beijing 100084, China b Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Abstract Υ production in heavy ion collisions at RHIC energy is investigated. While the transverse mo-mentum spectra of the ground state Υ (1 s ) are controlled by the initial state Cronin e ff ect, theexcited b ¯ b states are characterized by the competition between the cold and hot nuclear mattere ff ects and sensitive to the dissociation temperatures determined by the heavy quark potential.We emphasize that it is necessary to measure the excited heavy quark states in order to extractthe early stage information in high energy nuclear collisions at RHIC. Keywords:PACS: + pQuarkonium production in relativistic heavy ion collisions is widely accepted as a probe ofdeconfinement phase transition at finite temperature and density. The J /ψ suppression[1] hasbeen observed at SPS[2, 3] and RHIC[4] energies and considered as a signature of the creatednew state of matter, the so-called quark-gluon plasma (QGP). At RHIC energy, however, the J /ψ production and suppression mechanisms are complicated, there are primordial productionand nuclear absorption in the initial state and regeneration[5, 6, 7, 8] and anomalous suppressionduring the evolution of the hot medium. Υ mesons, the bound states of bottom quarks, may o ff er a relatively cleaner probe into thehot and dense medium[9, 10, 11]. At RHIC energy, there are three important advantages instudying Υ production compared with J /ψ : (i) Υ regeneration in the hot medium can be safelyneglected[12] due to the small production cross section of bottom quarks in nucleon + nucleoncollisions. The initial creation becomes the only production mechanism and the perturbativeQCD calculations are more reliable for estimating the production. (ii) Υ s are so heavy, there isalmost no feed-down for them. (iii) From the Υ measurement in d + Au collisions where thereis no hot nuclear matter e ff ect, the nuclear modification factor is about unity, implying that thecold nuclear absorption is negligible[13]. Therefore, the heavy and tightly bound Υ s can onlybe dissociated in the QGP phase. It is interesting to note that the recent experimental result inAu + Au collisions at RHIC[14] has already indicated a strong medium e ff ect on Υ production.The quarkonium dissociation temperature in the hot medium can be determined by solvingthe Schr¨odinger equation for c ¯ c or b ¯ b system with potential V between the two heavy quarks[15].The potential depends on the dissociation process in the medium. For a rapid dissociation wherethere is no heat exchange between the heavy quarks and the medium, the potential is just theinternal energy U , while for a slow dissociation, there is enough time for the heavy quarks toexchange heat with the medium, and the free energy F which can be extracted from the lattice Preprint submitted to Physics Letters B July 24, 2018 alculations is taken as the potential[16, 17]. From the thermodynamic relation F = U − T S < U where S is the entropy density, the surviving probability of quarkonium states with potential V = U is larger than that with V = F . In the literatures, a number of e ff ective potentials in between F and U have been used to evaluate the charmonium evolution in QCD medium[15, 17, 18].In this Letter we investigate Υ mid-rapidity production at RHIC energy ( √ s NN =
200 GeV),by solving a classical Boltzmann equation for the phase space distribution of Υ states movingin a hydrodynamic background medium. Since the Υ transverse momentum distribution shouldbe more sensitive to the dynamic evolution of the system, compared with the global Υ yield, wewill calculate not only the centrality dependence of the nuclear modification factor, but also itstransverse momentum dependence and the averaged transverse momentum for the ground andexcited Υ states. We will also discuss the dependence of these observables on the heavy quarkpotential and the similarity between Υ production at RHIC and J /ψ production at SPS.From the experimental data in p + p collisions, 51% of the observed ground state Υ (1 s ) is fromthe direct production, and the decay contributions from the excited b ¯ b states Υ (1 p ), Υ (2 s ), Υ (2 p )and Υ (3 s ) are respectively 27% , ,
10% and 1%[19]. The other two states η b (1 s ) and η b (2 s )are scaler mesons with typical width of strong interaction and therefore not considered here. Tosimplify the numerical calculation, we will not distinguish Υ (2 p ) from Υ (1 p ) and Υ (3 s ) from Υ (2 s ) and take the contribution fractions from the directly produced Υ (1 s ) , Υ (1 p ) and Υ (2 s ) tothe observed Υ (1 s ) in the final state to be a Υ = ,
37% and 12%.Like the description for J /ψ [20], the Υ motion in a hot medium is characterized by theclassical transport equation p µ ∂ µ f Υ = − C f Υ , (1)where f Υ ( ~ p , ~ x , t ) is the Υ distribution function in the phase space with Υ = s , p , s , and theloss term C is responsible to Υ suppression in the hot medium. We have neglected here the Υ regeneration at RHIC energy, as discussed above. Taking into account the gluon dissociationprocess Υ + g → b + ¯ b as the suppression source, the loss term can be written as C = Z d ~ k (2 π ) E g F ( k , p ) f g ( k , T , u ) σ ( k , p , T ) , (2)where ~ k is the gluon momentum, E g = | ~ k | the gluon energy, F ( k , p ) = k µ p µ the flux factor, f g ( k , T , u µ ) the gluon thermal distribution as a function of the local temperature T and velocity u µ of the medium, and σ ( k , p , T ) the dissociation cross section at finite temperature. The crosssection in vacuum can be calculated with the Operator Production Expansion method[21, 22, 23,24] and is often used for J /ψ suppression and should be better for Υ suppression. The mediume ff ect on the cross section is reflected in the temperature dependence of the Υ binding energy ǫ Υ .By solving the Schr¨odinger equation for the b ¯ b system with the heavy quark potential V at finitetemperature, one obtains ǫ Υ ( T ) and the wave function ψ ( ~ x , T ) and in turn the average size ofthe system h r i ( T ). With increasing T , ǫ Υ decreases and vanishes at the dissociation temperature T Υ and h r i increases and goes to infinity at T Υ . From the lattice simulation[25] on the J /ψ spectral function at finite temperature, the shape of the spectral function changes only a littlefor T < T J /ψ but suddenly collapses around the dissociation temperature T J /ψ . To simplify ournumerical calculation, we replace the temperature dependence of the binding energy in the crosssection by a step function, σ ( k , p , T ) = σ ( k , p ) / Θ ( T Υ − T ). Under this approximation, whilethe cross section becomes temperature independent at T < T Υ , the dissociation rate α = C / E Υ depends still on the hot medium, since the gluon density is sensitive to the temperature.2or V = U , H.Satz and his collaborators[15, 26] solved the Schr¨odinger equation and foundthe dissociation temperatures T Υ / T c = , . , . Υ = s , p , s , respectively, where T c =
165 MeV[27] is the critical temperature for the deconfinement. Since the lattice calculatedpotential is mainly in the temperature region T / T c < T Υ / T c = V = F and the dissociation temperatures T Υ / T c = , . , Υ = s , p , s , respectively. For all Υ states, the dissociation temperatures are about 30% lowerin case of V = F compared to that of V = U . In Fig.1, we show the Υ dissociation rate α as afunction of transverse momentum at fixed temperature T c < T < T Υ in the case of V = U . In thecalculation here we have chosen a typical medium velocity v QGP = . ~ v QGP and Υ momentum ~ p have the same direction. All the rates are large at low momentum and dropo ff at high momentum. When the temperature increases from 200 MeV (left panel) to 250 MeV(right panel), the rates for all Υ states increase by a factor of about 2. This can be understoodqualitatively by the temperature dependence of the gluon density, n g ( T =
250 MeV) / n g ( T = = (250 / = . (GeV/c) t p ) - (f m α (1s) Υ (1p) Υ× Υ× (GeV/c) t p (1s) Υ (1p) Υ× Υ× Figure 1: The Υ dissociation rate α as a function of transverse momentum at temperature T =
200 MeV(left panel) and250 MeV(right panel) for the potential V = U . The medium velocity is fixed as v QGP = . Υ momentum. Υ (1 s ) , Υ (1 p ) and Υ (2 s ) are respectively shown by dashed, dot-dashed and dotted lines.The rates for Υ (1 p ) and Υ (2 s ) are multiplied by a factor 0 . In our following numerical calculations, the local temperature T ( ~ x , t ) and medium velocity u µ ( ~ x , t ) which appear in the gluon distribution function f g ( k , T , u ) and step function Θ ( T Υ − T )are controlled by the ideal hydrodynamic equations ∂ µ T µν = , ∂ µ ( n B u µ ) = , (3)where T µν is the energy-momentum tensor, and n B the baryon density. Taking into account theHubble-like expansion assumption for the longitudinal motion[28], the above hydrodynamicsdescribes the transverse evolution of the medium in the central rapidity region. To close the3ydrodynamic equations, we take the equation of state[27, 29] for ideal parton gas with partonmasses m u = m d = m g = m s =
150 MeV and hadron gas with hadron masses up to 2 GeV.Here we did not consider the back-coupling of the Υ states to the medium evolution. The initialcondition for the hydrodynamic equations is determined by the corresponding nucleon + nucleoncollisions and colliding nuclear geometry[30], leading to an initial temperature T i =
340 MeV.The transport equation (1) can be solved analytically[30]. The initial transverse momentumdistribution f Υ can be described by the Monte Carlo event generator PYTHIA with MSEL = + nucleon collisions and the Glauber model for the nuclear geometry. Since thecollision time for the two nuclei to pass through each other at RHIC energy is less than thestarting time of the hot medium, all the cold nuclear matter e ff ects can be included in the initialdistribution. Considering the fact that there is almost no Υ suppression in d + Au collisions[13],we neglect the nuclear absorption and take into account only the Cronin e ff ect[32, 33], namelythe gluon multi-scattering with nucleons before the two gluons fuse into an Υ . The Cronin e ff ectleads to a transverse momentum broadening, the averaged transverse momentum square h p t i NN in nucleon + nucleon collisions is extended to h p t i NN + a gN l , where l is the path length that thetwo gluons travel in the cold nuclear medium and a gN is a constant determined by the p + A data.With the known distribution function f Υ ( ~ p , ~ x , t ), we can calculate the yield N Υ AA in the finalstate by integrating f Υ over the hypersurface in the phase space determined by the critical tem-perature T c . The nuclear modification factors defined by R AA = N Υ AA / (cid:16) N coll N Υ NN (cid:17) are shown inFig.2 as functions of the number of participant nucleons N part in Au + Au collisions at √ s = N coll is the number of binary collisions and N Υ NN is the Υ yield in the correspond-ing nucleon + nucleon collisions. Since the heavy quark potential at finite temperature is not yetclear, we take the two limits of V = U and V = F to determine the Υ dissociation temperature T Υ . For the ground state Υ (1 s ), the binding energy is about 1.1 GeV which is much larger thanthe typical temperature of the fireball T ∼
300 MeV at RHIC energy. Therefore, Υ (1 s ) is un-likely to be destroyed in heavy ion collision at RHIC. Considering the fact that the dissociationtemperatures T Υ = T c for V = U and 3 T c for V = F are both far above the temperature of thefireball, the suppression due to the gluon dissociation is very small and independent of the heavyquark potential. As to the excited states, the situation is di ff erent as their binding energies aremuch smaller and thus their yields are strongly suppressed. Especially, at V = F which leads tothe lowest dissociation temperatures around the phase transition for Υ (1 p ) and Υ (2 s ), almost allof them are eaten up by the fireball in semi-central and central collisions. Including the decaycontribution from the excited states to the ground state, the finally observed nuclear modifica-tion factor for Υ (1 s ), R AA = P Υ= s , p , s a Υ N Υ AA / (cid:16) N coll N Υ NN (cid:17) is controlled by both the ground andexcited states. In central Au + Au collisions, as one can see in the figure, R AA ∼ a s = R AA in minimum bias events is 0.53 for V = F and 0.63 for V = U . Both values are in reasonable agreement with the preliminaryPHENIX data R AA < . R AA shows limitedsensitivity to the heavy quark potential, the excited states display di ff erent dependence on thepotential used in the calculation.In our numerical calculations, the momentum parameters h p t i NN and a gN are respectivelytaken to be 7.1 (GeV / c) from the PYTHIA simulation[31] and 0.2 (GeV / c) / fm to best describethe available data for p + A collisions[34]. In order to further address the Υ production dynamics,we calculated the nuclear modification factor R AA as a function of transverse momentum p t incentral Au + Au collisions at RHIC energy, see Fig.3. There are three nuclear matter e ff ectsthat may a ff ect the Υ p t dependence: the Cronin e ff ect in the initial state, the leakage e ff ect[1,4 Υ (1p) Υ (2s) Υ (1s) Total Υ V=U AA ca t i on F ac t o r R part N (1s) Υ (1p) Υ (2s) Υ (1s) Total Υ V=F N u c l ea r M od i f i Figure 2: Centrality dependence of the nuclear modification factors R AA in Au + Au collisions at top RHIC energy √ s NN =
200 GeV. Upper and lower panels are for the heavy quark potential V = U and V = F , respectively. Thedirectly produced Υ (1 s ) , Υ (1 p ) , Υ (2 s ) and the total Υ (1 s ) are respectively shown by dashed, dot-dashed, dotted and solidlines.
30] for higher p t particles, and the suppression mechanism in the hot medium. While both theCronin e ff ect and leakage e ff ect lead to a p t broadening, i.e. reduction of low p t particles andenhancement of high p t particles, a strong enough suppression may weaken the broadening. Thereason is the following: When the gluon traveling length l which characterizes the Cronin e ff ectis long, the temperature which controls the suppression becomes high, the competition betweenthe Cronin e ff ect and the suppression may reduce the p t broadening. For the ground state, there isonly a weak suppression, a very strong p t broadening due to the Cronin e ff ect and leakage e ff ectis expected. Since most of the excited states are dissociated in central collisions at both V = U and V = F , their p t broadening is strongly suppressed and not sensitive to the heavy quarkpotential. The shape of the finally observed total R AA is determined by the directly produced Υ (1 s ), as shown in Fig.3.To obtain more dynamic information on the nuclear medium through survived excited states,we calculated the averaged transverse momentum square h p t i AA as a function of centrality for5 Υ (1p) Υ (2s) Υ (1s) Total Υ V=U AA ca t i on F ac t o r R (GeV/c) t p (1s) Υ (1p) Υ (2s) Υ (1s) Total Υ V=F N u c l ea r M od i f i - ca t i on
0 1 2 3 4 5
Figure 3: Transverse momentum dependence of the nuclear modification factors R AA in central Au + Au collisions at topRHIC energy √ s NN =
200 GeV. The peak caused by the Cronin e ff ect is clear at p t ∼ / c for both potentials. Au + Au collisions, the results are shown in Fig.4. To reduce the theoretical uncertainty in h p t i NN and focus on the nuclear matter e ff ect, we considered the di ff erence between nucleus + nucleusand nucleon + nucleon collisions, ∆ h p t i ≡ h p t i AA − h p t i NN . For the directly produced groundstate Υ (1 s ), the suppression is weak and the Cronin e ff ect plays the dominant role. As a result, ∆ h p t i increases monotonously with collision centrality. For the excited states, however, the e ff ectof initial Cronin e ff ect is overwhelmed by the disassociation especially in central collisions. Inother words, in the most central collisions, the high temperature region is larger than that inperipheral collisions, most of the excited Υ s are destroyed. Therefore, as one can see, the valueof ∆ h p t i goes up in peripheral collisions due to the Cronin e ff ect, then becomes saturated in semi-central collisions from the competition between the Cronin e ff ect and the increased suppression,and finally starts to decrease when the suppression becomes strong. In addition, as one cansee in the figure, in case of V = F , the decrease is remarkable due to the stronger suppressione ff ect. Di ff erent from the total R AA ( N part ) and R AA ( p t ) which are approximately one half of thecorresponding values for the directly produced Υ (1 s ), see Figs.2 and 3, the total ∆ h p t i is very6lose to the one for the ground state, as shown in Fig.4. at SPS ψ J/ × ( G e V / c ) (1s) Υ (1p) Υ (2s) Υ (1s) Total Υ V=U part N (1s) Υ (2s) Υ (1p) Υ (1s) Total Υ V=F 〉 t p 〈 ∆ Figure 4: Centrality dependence of the di ff erence ∆ h p t i ≡ h p t i AA − h p t i NN between the averaged transverse momentumsquare from Au + Au and p + p collisions at top RHIC energy √ s NN =
200 GeV. For comparison, we showed in the upperpanel also the J /ψ data at SPS energy[2] multiplied by a factor 2 . It is interesting to compare the Υ production at RHIC energy and the J /ψ production at SPSenergy. Assuming there is no Υ regeneration at RHIC and no J /ψ regeneration at SPS, theproduction mechanism in the two cases is then characterized by the initial creation. In addition,from the relationship between the fireball temperature and the dissociation temperature for theground state at V = U , T Υ = T c ≫ T RHIC and T J /ψ = T c ≫ T SPS , the suppression of Υ (1 s ) atRHIC and J /ψ at SPS is negligible. Therefore, the averaged transverse momentum squares for Υ in Au + Au collisions at RHIC and J /ψ in Pb + Pb collisions at SPS are related to each otherthrough the Cronin e ff ect, ∆ h p t i| RHIC Υ = a RHIC gN R Au a SPS gN R Pb ∆ h p t i| SPS J /ψ = . ∆ h p t i| SPS J /ψ , (4)where we have taken a SPS gN = .
08 GeV / fm[32, 33], and R Au and R Pb are the nuclear radii. This7elation in fact predicts that the centrality dependence of ∆ h p t i for Υ at RHIC is proportional tothat for J /ψ at SPS. In the upper panel of Fig.4 we showed the J /ψ data at SPS[2] multipliedby the factor 2 .
4. It is clear that the relation (4) works well. For R AA ( N part ) and R AA ( p t ), theirbehavior depends on the details of the hot medium, it becomes di ffi cult to obtain similar relationsbetween Υ at RHIC and J /ψ at SPS. Υ (1p) Υ (2s) Υ (1s) Total Υ V=U AA ca t i on F ac t o r R c / T i T (1s) Υ (1p) Υ (2s) Υ (1s) Total Υ V=F N u c l ea r M od i f i - ca t i on Figure 5: Initial temperature dependence of the nuclear modification factor R AA in central Au + Au collisions at top RHICenergy √ s NN =
200 GeV.
How hot is the fireball formed in relativistic heavy ion collisions? This is a crucial questionthat will have an influence on all signatures for QGP formation. In all above calculations, theinitial temperature T i =
340 MeV is determined by the initial energy density and baryon densitywhich are controlled by the nucleon + nucleon collisions and the nuclear geometry. To extractthe initial temperature of the system from Υ production, we now take T i as a free parameterand calculate the momentum integrated R AA as a function of T i in central Au + Au collisions atRHIC energy, the result is shown in Fig.5. While in case of V = F , see bottom plot of Fig.5,the values of R AA are almost constants in the temperature region 1 . < T i / T c < V = U , see top plot of Fig.5, the8xcited states and the finally observed ground state are sensitive to the temperature. Therefore,the experimental results of R AA for any state will allow us to extract the information on the initialtemperature of the system.In summary, we studied Υ production in high energy nuclear collisions at RHIC in a transportmodel. The observed Υ (1 s ) is mainly from the direct production, and the contribution from thefeed down of the excited states is small. The transverse momentum distribution of Υ (1 s ) isnot sensitive to the hot medium, but characterized by the Cronin e ff ect in the initial stage. Theabove conclusion is almost independent of the heavy quark potential. However, the behavior ofthe excited Υ states is controlled by the competition between the cold and hot nuclear mattere ff ects and sensitive to the heavy quark potential. Therefore, the yield and transverse momentumdistribution for the excited states should be measured in the future experiments, in order to probethe dynamic properties of the formed fireball. The initial state Cronin e ff ect can be studied via thecentrality dependence of h p t i in A + A collisions. We did not address the influence of the partondistribution in A + A collisions[35, 36]. Although it will a ff ect the details of the predictions madein this letter, the qualitative trends, especially the nature of the high sensitivity of the excited Υ states to the potential and initial temperature, will remain to be true. The consideration on thevelocity dependence of the dissociation temperature and the influence of reduced binding energyin low temperature region will be discussed in the future. Acknowledgement:
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