The transverse mass spectra of J/psi and psi' mesons and Omega hyperons produced in central Au+Au collisions at RHIC energies are discussed within a statistical model used successfully for the interpretation of the SPS results. The comparison of the presented model with the future RHIC data should serve as a further crucial test of the hypothesis of statistical production of charmonia at hadronization. Finally, in case of validity, the approach should allow to estimate the mean transverse flow velocity at the quark gluon plasma hadronization

Bugaev, Kyrill A.

Gazdzicki, Marek

Gorenstein, Mark I.

English

Gaździcki, Marek

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:hep-ph/0206109 v2   8 Aug 2002Ω, J/ψ and ψ′ Transverse Mass Spectra at RHICK.A. Bugaeva,b, M. Ga´ zdzickic and M.I. Gorensteina,da Bogolyubov Institute for Theoretical Physics, Kiev, Ukraineb Gesellschaft f¨ ur Schwerionenforschung (GSI), Darmstadt, Germanyc Institut f¨ ur Kernphysik, Universit¨ at Frankfurt, Germanyd Institut f¨ ur Theoretische Physik, Universit¨ at Frankfurt, GermanyAbstractThe transverse mass spectra of J/ψ and ψ′ mesons and Ω hyperons pro-duced in central Au+Au collisions at RHIC energies are discussed within astatistical model used successfully for the interpretation of the SPS results.The comparison of the presented model with the future RHIC data shouldserve as a further crucial test of the hypothesis of statistical productionof charmonia at hadronization. Finally, in case of validity, the approachshould allow to estimate the mean transverse ﬂow velocity at the quarkgluon plasma hadronization.The concepts of chemical (hadron multiplicities) and kinetic (hadron momentum spec-tra) freeze–outs were introduced to interpret data on hadron production in relativisticnucleus–nucleus (A+A) collisions. The equilibrium hadron gas (HG) model describes re-markably well the light hadron multiplicities measured in A+A collisions at the SPS [1]and RHIC [2] energies, where the creation of a quark–gluon plasma (QGP) is expected.Recently it was found [3] that also charmonium yield systematics in nuclear collisions atthe SPS [4] follow the pattern predicted by the hadron gas model. The hadronizationtemperature parameter extracted from the ﬁts to the hadron multiplicities is similar for1both energies: TH = 170 ± 10 MeV. It is close to an estimate of the temperature TC forthe QGP–HG transition at zero baryonic density.Experimental results on inclusive hadron spectra and correlations show evidence fora hydrodynamic expansion of the matter created in heavy ion collisions. Strong trans-verse ﬂow eﬀects in Pb+Pb collisions at the SPS (average collective transverse velocityis approximately 0.5 ÷ 0.6) are ﬁrmly established from the combined analysis [5,6] of thepion transverse mass spectra and correlations. The kinetic (‘thermal’) freeze–out of pionsand nucleons seems to occur at a rather late stage of an A+A reaction. The thermalfreeze–out temperature parameter of pions measured by the NA49 Collaboration [7] forcentral Pb+Pb collisions at the SPS is Tf ∼ = 120 MeV.Further exploration of idea of the statistical J/ψ production [3] led to the formulationof the hypothesis that the kinetic freeze–out of J/ψ and ψ′ mesons takes place directly athadronization [8]. This means that chemical and thermal freeze–outs occur simultaneouslyfor those mesons and they, therefore, carry information on the ﬂow velocity of stronglyinteracting matter just after the transition to the HG phase. A possible inﬂuence of theeﬀect of rescattering in the hadronic phase on the transverse momentum (pT) spectrawas recently studied within a “hydro + cascade” approach [9,10]. A+A collisions areconsidered there to split into three stages: hydrodynamic QGP expansion (“hydro”),transition from QGP to HG (“switching”) and the stage of hadronic rescattering andresonance decays (“cascade”). The switching from hydro to cascade takes place at T = TC,where the spectrum of hadrons leaving the surface of the QGP–HG transition is takenas an input for the subsequent cascade calculations. The results [9,10] suggest that thepT spectrum of Ωs is only weakly aﬀected during the cascade stage. The correspondingcalculation for charmonia are not yet performed within this model, but a similar resultmay be expected due to their very high masses and low interaction cross sections.In previous work [11] devoted to the analysis of the SPS data it was demonstrated2that the measured transverse mass (mT =qp2T + m2) spectra of Ω± hyperons [12] andJ/ψ and ψ′ mesons [13] produced in Pb+Pb at 158 A GeV collisions can be reproducedwithin a hydrodynamical approach using the same freeze–out parameters: hadronizationtemperature T ∼ = 170 MeV and the mean transverse ﬂow velocity vT ∼ = 0.2. Within ourapproach the value of the vT parameter extracted in this way should be interpreted asthe mean ﬂow velocity of the hadronizing QGP.In the present letter we discuss the transverse mass spectra of J/ψ and ψ′ mesons andΩ hyperons in central Au+Au collisions at RHIC energies within the statistical approachto charmonium production successfully used for the interpretation of the SPS data [3,11].Assuming kinetic freeze–out of matter at constant temperature T, the transverse massspectrum in cylindrically symmetric and longitudinally boost invariant ﬂuid expansion canbe approximated as [14]:dNimTdmT∝ mTZ R0rdr K1 mT coshyTT!I0 pT sinhyTT!, (1)where yT = tanh−1 vT is the transverse ﬂuid rapidity, R is the transverse system size,K1 and I0 are the modiﬁed Bessel functions. The spectrum (1) is obtained under theassumption that the freeze–out occurs at constant longitudinal proper time τ =√t2 − z2(t is the time and z is the longitudinal coordinate), i.e. the freeze–out time t is independentof the transverse coordinate r. The analysis of the numerical calculations done within ahydrodynamical model [10] shows that the latter is approximately fulﬁlled. The quality ofthe approximation made gets better for heavy particles considered here because a possibledeviation from Eq.(1) decreases with increasing particle mass at constant pT. In order tocalculate (1) the function yT(r) has to be given. A linear ﬂow proﬁle, yT(r) = ymaxT  r/R,is often assumed in phenomenological ﬁts [14] and we used it as well. Recent numericalhydrodynamical calculations [10] show that deviations from this simpliﬁed proﬁle aresmaller than 10% for both SPS and RHIC energies. Their importance for our results is3discussed later in the paper.In Figs. 1 – 3 the measured mT–spectra of Ω− [12], J/ψ and ψ′ [13] produced inPb+Pb collisions at 158 A GeV are shown together with the ﬁt performed using Eq.(1).The model parameters are: T = 170 MeV, ymaxT = 0.28 [11]. The value of ymaxT foundfrom ﬁtting the SPS data appears to be close to the numerical estimate done within ahydrodynamical approach (ymaxT ≈ 0.3) [10]. This encourages us to use a correspondingestimate of ymaxT ∼ = 0.55 made for Au+Au collisions at√sNN = 130 GeV [10] for the ﬁrstguess of the mT–spectra of charmonia at RHIC energies within our approach. As followsfrom the analysis [2] of hadron yields within the HG model, in our calculations we usea hadronization temperature T = 170 MeV, the same as in the case of the SPS data.The resulting mT–spectra for Ω, J/ψ and ψ′ are shown in Figs. 1, 2 and 3, respectively.The normalization of the SPS and RHIC spectra is arbitrary. Due to large transverseﬂow velocity of hadronizing QGP at RHIC energies, the particle spectra are expected tosigniﬁcantly deviate from the simple exponential form of the mT–distribution:dNmTdmT= C m1/2T exp￿−mTT ∗￿, (2)where C and T ∗ are normalization and inverse slope parameters, respectively.In order to quantify the deviations of the RHIC spectra from Eq. (2) we plotted inFig. 4 the local inverse slope deﬁned as:T∗ ≡ −"ddmTln m−1/2TdNdm2T!#−1, (3)versus mT − m for J/ψ, ψ′ and Ω. The T ∗ parameter was calculated using Eq.(1) withthe values of parameters used previously for the SPS and RHIC energies. No signiﬁcantdependence of T ∗ is observed in the SPS case ymaxT = 0.28, whereas a very strong decreaseof T ∗ with increasing mT in the region mT −m < 1 GeV is seen for J/ψ and ψ′ at RHIC(ymaxT = 0.55) and the moderate one is expected for Ω hyperons. Thus the shape of themT–spectrum is strongly dependent on the magnitude of the transverse ﬂow and the mass4of the particle. These characteristic “hydrodynamical” features of the spectrum shouldallow for a clear test of our approach in the near future.To be more speciﬁc, we follow the suggestion of Ref. [10] and consider two domains0 < mT − m < 0.6 GeV and 0.6 < mT − m < 1.6 GeV of mT–spectra. Within thesemT domains the Ω, J/ψ and ψ′ spectra at RHIC calculated according to Eq. (1) wereapproximated by Eq. (2) and the values of C and T ∗ were found by the maximumlikelihood method. The inverse slopes in low mT domain (long dashed lines in Figs. 1 –3) are larger than the high mT ones (dashed–dotted lines in Figs. 1 – 3). However, thediﬀerence between low and high mT inverse slopes strongly depends on the particle mass– for Ω the inverse slopes diﬀer by 13 per cent only, whereas for ψ′ by about 40 per cent.It is important to check the sensitivity of the above results to the assumed ﬂow proﬁles(density and velocity) of hadronic matter at freeze–out. We ﬁrst note that because of thebasic assumption of our model (Ω, J/ψ and ψ′ freeze–out at QGP hadronization) thefreeze-out density at the boundary between the mixed and hadronic phases is determinedby the hadronization temperature T = 170 MeV and hence the density proﬁles of theseparticles are constant as the function of r. Thus the uncertainty is due to the velocityproﬁle only. The numerical hydrodynamical calculations [10] indicate that the transverserapidity proﬁle may deviate from the linear dependence assumed above by less than 10%.In order to estimate a possible inﬂuence of these deviations we calculated the spectra ofΩ, J/ψ and ψ′ using a proﬁle of the form yT(r) = ymaxT  (r/R)α for α = 0.9 and 1.1. In thecalculations we keep the average transverse velocity as for ymaxT = 0.55 and α = 1 case.It turns out that the shape of the spectra is almost insensitive to the assumed value ofα parameter in the whole range of transverse mass for Ω and for mT − m > 0.6 GeV forJ/ψ and ψ′ mesons. In the low mT domain (mT − m < 0.6 GeV) the ﬁtted inverse slopeparameters for J/ψ and ψ′ mesons vary by 3 per cent and 5 per cent, respectively, underconsidered variations of the α parameter. Therefore we conclude - the principal results of5our analytical model are weakly aﬀected by the approximations made.The transverse ﬂow velocity of hadronizing matter depends on the energy and central-ity of the collision. Thus, in general, we consider the parameter ymaxT as a free parameter,who’s value should be extracted from the ﬁt of Eq.(1) to the measured spectrum. Provid-ing the ﬁt is successful, the ﬁtted ymaxT may be treated as an estimate of the transverseﬂow velocity of hadronizing matter.The mT spectra in Au+Au collisions at RHIC were also calculated within a hydrody-namical ansatz in Ref. [15]. There is however a principal diﬀerence between our approach[11] and the one used in Ref. [15]. We consider the early kinetic freeze–out for Ω, J/ψand ψ′ only, whereas in [15] the simultaneous (chemical and thermal) freeze–out was pos-tulated for all hadrons. This generates a substantial diﬀerence in the model predictionsfor the RHIC freeze–out parameters.The results on charmonium production in central Au+Au collisions at RHIC energiesare expected to be available very soon. They will, obviously, allow to test various ap-proaches [16–20] developed in the course of the analysis of the corresponding SPS data.In particular, a statistical approach to charmonium production in nuclear collisions yieldswell deﬁned and characteristic predictions for charmonia multiplicities [3] and the trans-verse mass spectra discussed in this letter. The analysis of the mT–spectra of J/ψ andψ′ mesons and Ω hyperons within our approach should yield a unique information on thetransverse ﬂow velocity of hadronizing QGP in A+A collisions at RHIC energies.Acknowledgments. We thank P. Braun–Munzinger, W. Broniowski, W. Florkowski andR. Renfordt for discussions and comments. The ﬁnancial supports from the HumboldtFoundation and INTAS grant 00–00366 are acknowledged.6REFERENCES[1] J. Cleymans and H. Satz, Z. Phys. C57, 135 (1993),J. Sollfrank, M. Ga´ zdzicki, U. Heinz and J. Rafelski, Z. Phys. C61, 659 (1994),F. Becattini, M. Ga´ zdzicki and J. Sollfrank, Eur. Phys. J C5, 129 (1998),P. 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Teaney, J. Lauret and E. V. Shuryak, nucl–th/0110037.[11] M. I. Gorenstein, K. A. Bugaev and M. Ga´ zdzicki, Phys. Rev. Lett. 88, 132301 (2002).[12] F. Antinori et al. (WA97 Collab.), J. Phys. G27, 375 (2001).7[13] M.C. Abreu et al. (NA50 Collab), Phys. Lett. B499, 85 (2001).[14] E. Schnedermann, J. Sollfrank and U. Heinz, Phys. Rev. C48, 2462 (1993).[15] W. Broniowski and W. Florkowski, nucl–th/0112043.[16] T. Matsui and H. Satz, Phys. Lett. B178, 416 (1986).[17] P. Braun–Munzinger and J. Stachel, Phys. Lett. B490, 196 (2000).[18] M. I. Gorenstein et al., Phys. Lett. B509, 277 (2001) and B524, 256 (2002).[19] R. Thews, M. Schroedter, J. Rafelski, Phys. Rev. C63, 054905 (2001).[20] L. Grandchamp and R. Rapp, Phys. Lett. B523, 60 (2001) and hep–ph/0205305.8FIGURES0 1 2 3 mT−m (GeV/c2)10−810−610−410−210 010 2 dN/(mTdmT) (a.u.)T* = 362 MeVT* = 315 MeVSPSRHICFIG. 1. The transverse mass spectra (in arbitrary units) of Ω− hyperons produced inPb+Pb (Au+Au) collisions. The points indicate experimental results measured for Pb+Pbcollisions at 158 A GeV. The dashed line shows the ﬁt of the hydrodynamical model to thesedata (T = 170 MeV and ymaxT = 0.28). The prediction of our approach for central Au+Aucollisions at√sNN = 130 GeV is indicated by the solid line (T = 170 MeV and ymaxT = 0.55).The long dashed and dashed–dotted lines correspond to the exponential spectrum ﬁtted in theintervals (mT − m) ∈ [0, 0.6] GeV and (mT − m) ∈ [0.6, 1.6] GeV), respectively.90 1 2 3 mT−m (GeV/c2)10−810−610−410−210−1010−12 dN/(mTdmT) (a.u.)T* = 589 MeVT* = 399 MeVSPSRHICFIG. 2. The same as in Fig. 1 but for J/ψ mesons.100 1 2 3 mT−m (GeV/c2)10−410−6 10−610−810−1010−12 dN/(mTdmT) (a.u.)T* = 722 MeVT* = 440 MeVSPSRHICFIG. 3. The same as in Fig. 1 but for ψ′ mesons.110 1 2 3 4  mT − m (GeV)0200400600800100012001400  T* (MeV)SPS0 1 2 3 4  mT − m (GeV)RHICΩJ/ψψ’FIG. 4. The dependence of the inverse slope parameter T∗ of J/ψ (solid line) and ψ′ (longdashed line) mesons and Ω hyperons (short dashed line) on mT −m expected within the modelof statistical production at QGP hadronization. The left panel indicates the behaviour given bythe ﬁt of the hydrodynamical ansatz to the data on Pb+Pb collisions at 158 A GeV, whereasthe right panel shows the predictions for central Au+Au collisions at√sNN = 130 GeV.120 1 2 3 4  mT − m (GeV)0200400600800100012001400  T* (MeV)SPS0 1 2 3 4  mT − m (GeV)RHICΩJ/ψψ’

Omega, J/psi and psi' transverse mass spectra at RHIC

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