The pion source as seen through HBT correlations at RHIC energies is investigated within the UrQMD approach. We find that the calculated transverse momentum, centrality, and system size dependence of the Pratt-HBT radii R_L and R_S are reasonably well in line with experimental data. The predicted R_O values in central heavy ion collisions are larger as compared to experimental data. The corresponding quantity sqrt R_O^2-R_S^2 of the pion emission source is somewhat larger than experimental estimates

Bleicher, Marcus

Li, Qingfeng

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/0602032 v1   10 Feb 2006Transport model analysis of femtoscopy data at RHIC energiesQingfeng Li 1∗ , Marcus Bleicher 2, and Horst St¨ ocker 1,21) Frankfurt Institute for Advanced Studies (FIAS),Johann Wolfgang Goethe-Universit¨ at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, Germany2) Institut f¨ ur Theoretische Physik,Johann Wolfgang Goethe-Universit¨ at,Max-von-Laue-Str. 1,D-60438 Frankfurt am Main, GermanyAbstractThe pion source as seen through HBT correlations at RHIC energies is investigated within theUrQMD approach. We ﬁnd that the calculated transverse momentum, centrality, and system sizedependence of the Pratt-HBT radii RL and RS are reasonably well in line with experimental data.The predicted RO values in central heavy ion collisions are larger as compared to experimentaldata. The corresponding quantityqR2O − R2S of the pion emission source is somewhat larger thanexperimental estimates.PACS numbers: 25.75.Gz,25.75.-q,25.75.Dw∗ Fellow of the Alexander von Humboldt Foundation.1In the quest to discover the high temperature phase of Quantum Chromodynamics(QCD), the Quark Gluon Plasma (QGP), the beam energies of accelerators have beenboosted upwards from SIS, AGS, SPS, to RHIC. However, it is well known that the phasetransition from hadrons to quarks might only occur in a small volume part of the systemand within a rather short timespan in heavy ion collisions (HICs). This implies that theQGP drops formed at RHIC might be represented only by a few, locally thermally equili-brated drops of matter, in which quarks and gluons are de-conﬁned. Thus, it is essential toprobe the space-time structure of the (equilibrated?) source – the ”region of homogeneity”.Unfortunately, the small size and transient nature of the reactions preclude direct measure-ment of the time and/or position. Instead, correlations of two ﬁnal-state particles at smallrelative momenta provide the most direct link to the size and lifetime of subatomic reac-tions. A well-established technique, the so-called ”femtoscopy” or ”HBT” in the heavy-ioncommunity (in reference to Hanbury-Brown and Twiss’s original work with photons) hasbeen extensively used for HICs with energies from SIS, AGS, SPS, to RHIC [1, 2, 3, 4, 5, 6,7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].Numerous HBT-measurements with various two-particle species have been pursued (seee.g. [1, 16] and references therein). Identically charged pion interferometry has been mostextensively investigated. Basic, but important systematics of femtoscopic measurementsfrom the AGS, SPS, and RHIC have been discovered [1, 7, 14, 16], such as the dependenceof the HBT radii on system size, collision centrality, rapidity, transverse momentum, andparticle mass. However, the existence of the so-called HBT-puzzle (i.e., the fact that modelcalculations that incorporate a phase transition to a new state of matter with many degreesof freedom signiﬁcantly over-predict the observed source sizes) [1, 4, 19, 20, 21, 27, 28, 32, 33]drives us to a deeper and more systematical theoretical exploration. The Ultra-relativisticQuantum Molecular Dynamics (UrQMD, v2.2) transport model (employing hadronic andstring degrees of freedom) (for details, the reader is referred to Refs.[34, 35, 36, 37]) andthe analyzing program CRAB (v3.0β) [38, 39, 40] are employed here as tools to analyzethe two-particle interferometry. With this equipment, the excitation functions of the HBTradii of negatively charged pions are calculated systematically. In this paper, we focus onRHIC energies, where the biggest challenge is faced by the current theoretical models. Thefemtoscopy results at lower energies will be presented in a further study [41].The correlation function of two particles is decomposed in Pratt’s (so-called longitudinal2co-moving system ”Out-Side-Long”) three-dimensional convention (Pratt-radii). The three-dimensional correlation function is ﬁt with the standard Gaussian form:C(qO,qS,qL) = 1 + λexp(−R2Lq2L − R2Oq2O − R2Sq2S − 2R2OLqOqL), (1)in which qi and Ri are the components of the pair momentum diﬀerence q = p2 − p1and the homogeneity length (Pratt-radii) in the i direction, respectively. The λ is theincoherence factor, which lies between 0 (complete coherence) and 1 (complete incoherence)in realistic HICs. ROL represents the cross-term and is supposed to vanish at mid-rapidityunder the assumption of a longitudinally boost invariant expansion and is also found tobe negligible in our present calculations. Furthermore, in the present UrQMD calculationsat RHIC energies, the Coulomb and other potential interactions are not considered (the”cascade mode” is used) due to the excessive computing times which would have been usedotherwise. The Coulomb ﬁnal state interactions are not taken into account in the analyzingprogram CRAB.Fig. 1 gives the transverse momentum kT dependence (kT = (p1T +p2T)/2) of the Pratt-radii RL (left plots), RO (middle plots), and RS (right plots) at nucleon-nucleon center-of-mass energies√sNN = 30, 62.4, 130, and 200 GeV (plots from top to bottom) in Au+Aureactions. The experimental results at energies√sNN = 62.4, 130, and 200 GeV for centralcollisions (< 15%, < 10%, and < 5% of the total cross section σT, respectively) and atmid-rapidities (|ηcm| < 0.5) are taken from Refs. [22, 23, 24, 25, 26]. The experimental errorbars are shown as the sum of both statistical and systematic errors. The correspondingcalculations with the same trigger- and acceptance- conditions as in the experiments areshown, as well as the lower energy case√sNN = 30 GeV for central collisions (< 15% ofσT).Both the absolute values and the decrease of the Pratt-radii RL and RS with transversemomentum is reproduced by the present model calculations very well. The origin of thedecrease of the Pratt-radii with the increase of transverse momentum is still under discussion:it may be caused by the strong underlying transverse ﬂow [24], or, by the temperatureinhomogeneities within the hadron source (point of view of the hydrodynamics model) [31].Here, it is also seen that the calculated kT-dependence of RS is somewhat ﬂatter than thatof RL, which implies that ﬂow eﬀects on the kT-dependence of the Pratt-radii can at leastnot be excluded. The UrQMD calculations of RL and RS reproduce the experimental data3024681015%T02468exp.:  ;     th.: 024680 200 400 600 80002468kT (MeV/c)O S L(62.4 GeV)(30 GeV)(130 GeV)sNN(200 GeV)Ri (fm)0 200 400 600 800 0 200 400 600 800FIG. 1: Transverse momentum kT dependence (at midrapidity) of the Pratt-radii RL, RO, andRSin Au+Au collisions at√sNN = 30, 62.4, 130, and 200 GeV. Experimental data for the latterthree cases are also shown [22, 23, 24, 25, 26]. The experimental errors are the sums of bothstatistical and systematic errors.well within the error bars, while the calculated RO’s are larger than the experimental data— the RO is about 25% too large. We must conclude that at RHIC, larger ratios of RO andRS are seen from hadron transport model than expected. Similar observations have alsobeen reported from other model calculations (c.f. [1, 19, 19, 20, 21, 42, 43, 44]). Ref. [28]has argued that the origin of this HBT-puzzle might be multifaceted.Fig. 2 shows the kT-dependence of the Pratt-radii in Au+Au reaction at√sNN = 200GeV for four centralities: 0 −5%, 10 −20%, 30 −50%, and 50− 80% of total cross section.Here, a pseudo-rapidity cut |ηcm| < 0.5 has been chosen. For better visibility, we haveshifted in the ﬁgure the values of the radii by 0, 5, 10, and 15 fm for the four centralities. Itis very interesting to see that our calculations for the centrality dependence of Pratt-radiiare in reasonably good agreement with the experimental data. The important observation,however, is that the calculated RO values tend to deviate from the data for central reactions40510152025RO (fm)th.                      exp.   0-5%          10-20%      30-50%      50-80%   +15 fm+10 fm+5 fm+0 fm100 200 300 400 500 60005101520RS (fm)+15 fm+10 fm+5 fm+0 fm200 300 400 500 600 RL (fm)+15 fm+10 fm+5 fm+0 fmkT (MeV/c)FIG. 2: Midrapidity kt-dependence of Pratt-radii in Au+Au reaction at√sNN = 200 GeV for fourcentralities: 0−5%, 10−20%, 30−50%, and 50−80% of the total cross section, which are shiftedby 15, 10, 5, and 0 fm, respectively. Experimental data are taken from Ref. [24].by ≈ 20%, while they agree at midcentral and peripheral collisions.The centrality dependence of the Pratt-radii can be seen more clearly from Fig. 3 ,which shows the Pratt-radii at kT = 250 ∼ 350MeV/c as a function of the number ofparticipants Npart. The quantityqR2O − R2S is also shown for comparison. In spite of thereasonable results on the centrality dependence of the Pratt-radii, the calculated quantityqR2O − R2S obviously deviates from that extracted from data for the most central collisions:it is about twice as large as measured by experiments. The pion freeze-out volume Vf hasbeen investigated thoroughly experimentally [7]. Vf can be expressed as Vf = (2π)3/2RLR2S.The linear increase of Vf with Npart is expected. It can be explained reasonably well by thepresent model, although a smaller ”thermal ellipse” is predicted due to a little shorter RLand RS values, as shown.The calculations for central Cu+Cu collisions are shown in Fig. 3 (solid dots), which are inline with the centrality dependence of the HBT space-time structure calculated for Au+Au50 50 100 150 200 250 300 350 40002468024680246802468050010001500200025003000(a)RL (fm)Npart(b) RO (fm)(c) RS (fm)sNN=200 G eVkT=250~350 M eV/c(d)  (RO2-RS2)1/2 (fm)(e)  Vf (fm3)FIG. 3: Centrality dependence of the Pratt-radii ((a)-(c)), the quantityqR2O − R2S (in (d)), andthe freeze-out volume Vf ((e)) at kT = 250 ∼ 350MeV/c, in Au+Au collisions at√sNN = 200 GeV.Experimental data are taken from Ref. [24]. The calculated results for central Cu+Cu collisionsshown with solid dots are perfectly located on the Au+Au systematic curves.collisions. This implies that the participant multiplicity is a very good scaling variable, whichdrives the geometry (HBT radii) at mid-rapidity, at least for mid-size to heavy systems. Inorder to check this, the kT-dependence of the ratios of the Pratt-radii between diﬀerentsystems is shown in Fig. 4 . The radius ratios shown are from (a), Cu+Cu vs. p+p, (b),Au+Au vs. p+p, and (c), Au+Au vs. Cu+Cu. In order to read the ﬁgure more conveniently,the RO and RS ratios are shifted by 5 and 10, respectively. In the p+p calculation, the non-femtoscopic correlations at large relative momenta are also seen, that is, the pion correlationsfunction saturates at large relative momentum, but the value is not equal to 1, which was6100 200 300 400 500 60002468200 300 400 500 600048121620(c)  Au+Au/Cu+CuRi ratiosNN=200 GeV          th.            exp.RL:             RO:             RS:             kT (MeV/c]+5(b) Au+Au/p+p+0+10(a)Cu+Cu/p+p+0+5+10FIG. 4: kT-dependence of the ratios of Pratt-radii between diﬀerent systems: (a), Cu+Cu vs. p+p,(b), Au+Au vs. p+p, and (c), Au+Au vs. Cu+Cu. Central collisions at√sNN = 200 GeV and amid-rapidity cut are chosen. In (a) and (b), the ratios are shifted by 5 and 10, respectively. Thepreliminary experimental data are taken from Ref. [14].also implied by the preliminary data reported recently by Ref. [14]. We eliminate this eﬀectby multiplying a constant into the parametrization of the correlation function. The RL andRS values in p+p collisions can be reproduced well, while the calculated RO values are againlarger than the experimental data, similar to the nucleus-nucleus collisions. This might bethe origin of the whole puzzle, namely that the HBT-correlations are somewhat incorrectlyput into the model in the elementary p+p dynamics. Figs. 4 (a) and (b) show that thecalculated RL and RS ratios, reproduce the experimental data reasonably well. They arealmost ﬂat as a function of kT, which means the kT-dependence of the Pratt-radii still existsin the elementary p+p collisions at RHIC energies. Since the UrQMD model gives too largea RO value in p+p collisions, the calculated RO-ratio between Au+Au (or Cu+Cu) andp+p is smaller than the experimental data, in particular at low transverse momenta. Thisphenomenon disappears when we consider the Pratt-radii-ratios between two heavy systems,for examples, between Au+Au and Cu+Cu, in Fig. 3 (c). It is interesting to see that for7the Pratt-radii-ratios between Au+Au and Cu+Cu collisions, all radii-ratios are ﬂat withkT and approach ∼ 1.4, which is equal to the ratio between the initial radii of nuclei.To summarize, by using the CRAB program, we analyzed the evolution of the Pratt-radii RL, RO, and RS at RHIC energies in collisions simulated by the UrQMD transportmodel. The calculated transverse momentum-, centrality-, and system dependence of thePratt-radii are in reasonable agreement with the experimental data. The calculated ROvalues for central collisions are ∼ 25% larger as compared to experimental data. As aconsequence, the extracted quantityqR2O − R2S of the pion emission source is somewhatlarger than experimental estimates.AcknowledgmentsWe would like to thank S. Pratt for providing the CRAB program and acknowledgesupport by the Frankfurt Center for Scientiﬁc Computing (CSC). We thank H. Appelsh¨ auser,T. J. Humanic, S. Pratt and M.A. Lisa for valuable discussions. Q. Li thanks the Alexandervon Humboldt-Stiftung for a fellowship. This work is partly supported by GSI, BMBF,DFG, and Volkswagenstiftung.[1] M. A. Lisa, S. Pratt, R. Soltz and U. Wiedemann, arXiv:nucl-ex/0505014.[2] M. A. Lisa [the STAR Collaboration], Acta Phys. Polon. B 35, 37 (2004)[3] T. J. Humanic, Int. J. Mod. Phys. E 15, 197 (2006)[4] E. Frodermann, U. Heinz and M. A. Lisa, arXiv:nucl-th/0602023.[5] M. A. Lisa, U. W. Heinz and U. A. Wiedemann, Phys. Lett. B 489, 287 (2000)[6] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 93, 012301 (2004)[7] D. Adamova et al. [CERES Collaboration], Phys. Rev. Lett. 90, 022301 (2003)[8] I. G. Bearden et al., Phys. Rev. C 58, 1656 (1998).[9] D. Adamova et al. [CERES collaboration], Nucl. Phys. A 714, 124 (2003)[10] P. Chung et al., Phys. Rev. Lett. 91, 162301 (2003)[11] L. Ahle et al. [E802 Collaboration], Phys. Rev. C 66, 054906 (2002)8[12] P. F. Kolb and U. W. Heinz, Invited review for ’Quark Gluon Plasma 3’. Editors: R.C. Hwaand X.N. Wang, World Scientiﬁc, Singapore. arXiv:nucl-th/0305084.[13] B. Tomasik and U. A. Wiedemann, Invited review for Quark Gluon Plasma 3, eds. R.C. Hwaand X.-N. Wang, World Scientiﬁc. In *Hwa, R.C. (ed.) et al.: Quark gluon plasma* 715-777.arXiv:hep-ph/0210250.[14] Z. Chajecki, arXiv:nucl-ex/0511035.[15] R. Nouicer, arXiv:nucl-ex/0512044.[16] M. Lisa, arXiv:nucl-ex/0512008.[17] D. Adamova et al. [CERES Collaboration], Nucl. Phys. A 698 (2002) 253.[18] H. Appelshauser, J. Phys. G 30, S935 (2004)[19] S. Soﬀ, S. A. Bass and A. Dumitru, Phys. Rev. Lett. 86, 3981 (2001)[20] S. Soﬀ, S. A. Bass, D. H. Hardtke and S. Y. Panitkin, Phys. Rev. Lett. 88, 072301 (2002)[21] D. Zschiesche, H. St¨ ocker, W. Greiner, and S. Schramm, Phys. Rev. C 65, 064902 (2002)[22] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 93, 152302 (2004)[23] B. B. Back et al. [PHOBOS Collaboration], arXiv:nucl-ex/0409001.[24] J. Adams et al. [STAR Collaboration], Phys. Rev. C 71, 044906 (2005)[25] C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 87, 082301 (2001)[26] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. 88, 192302 (2002)[27] U. W. Heinz and P. F. Kolb, arXiv:hep-ph/0204061.[28] S. Pratt and D. Schindel, arXiv:nucl-th/0511010.[29] M. A. Lisa et al. [E895 Collaboration], Phys. Rev. Lett. 84, 2798 (2000).[30] S. Kniege et al. [NA49 Collaboration], J. Phys. G 30, S1073 (2004)[31] T. Csorgo, M. Csanad, B. Lorstad and A. Ster, Acta Phys. Hung. A 24, 139 (2005)[32] D. H. Rischke and M. Gyulassy, Nucl. Phys. A 597, 701 (1996)[33] D. H. Rischke and M. Gyulassy, Nucl. Phys. A 608, 479 (1996)[34] S. A. Bass et al., [UrQMD-Collaboration], Prog. Part. Nucl. Phys. 41, 255 (1998).[35] M. Bleicher et al., [UrQMD-Collaboration], J. Phys. G: Nucl. Part. Phys. 25, 1859 (1999).[36] E. L. Bratkovskaya, M. Bleicher, M. Reiter, S. Soﬀ, H. St¨ ocker, M. van Leeuwen, S. A. Bass,W. Cassing, Phys. Rev. C 69, 054907 (2004).[37] X. Zhu, M. Bleicher and H. St¨ ocker, Phys. Rev. C 72, 064911 (2005)[38] S. E. Koonin, Phys. Lett. B 70 (1977) 43.9[39] S. Pratt et al., Nucl. Phys. A 566 (1994) 103C.[40] S. Pratt, CRAB version 3, http://www.nscl.msu.edu∼pratt/freecodes/crab/home.html[41] Q. Li, M. Bleicher, and H. St¨ ocker, in preparation, 2006.[42] D. Molnar and M. Gyulassy, Phys. Rev. Lett. 92, 052301 (2004)[43] Z. Lin, C. M. Ko and S. Pal, Phys. Rev. Lett. 89, 152301 (2002)[44] T. Hirano and K. Tsuda, Phys. Rev. C 66, 054905 (2002)10

Transport model analysis of femtoscopy data at RHIC energies

Abstract

Similar works

Full text

Available Versions

Hochschulschriftenserver - Universität Frankfurt am Main