We analyze longitudinal pion spectra from E_lab= 2AGeV to \sqrt s_NN=200GeV
within Landau's hydrodynamical model. From the measured data on the widths of
the pion rapidity spectra, we extract the sound velocity c_s in the early stage
of the reactions. It is found that the sound velocity has a local minimum
(indicating a softest point in the equation of state, EoS) at E_beam=30AGeV.
This softening of the EoS is compatible with the assumption of the formation of
a mixed phase at the onset of deconfinement.Comment: 3 pages, 2 figure

Bleicher, Marcus

English

arXiv

We analyze longitudinal pion spectra from E_lab= 2AGeV to sqrt s_NN=200GeV within Landau's hydrodynamical model. From the measured data on the widths of the pion rapidity spectra, we extract the sound velocity c_s in the early stage of the reactions. It is found that the sound velocity has a local minimum (indicating a softest point in the equation of state, EoS) at E_beam=30AGeV. This softening of the EoS is compatible with the assumption of the formation of a mixed phase at the onset of deconfinement

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arXiv:hep-ph/0509314 v1   27 Sep 2005Evidence for the Onset of Deconﬁnementfrom Longitudinal Momentum Distributions?Observation of the Softest Point of the Equation of StateMarcus Bleicher11Institut f¨ ur Theoretische Physik, J. W. Goethe Universit¨ at, 60438 Frankfurt am Main, GermanyWe analyze longitudinal pion spectra from Elab = 2A GeV to√sNN = 200 GeV within Landau’shydrodynamical model. From the measured data on the widths of the pion rapidity spectra, weextract the sound velocity c2s in the early stage of the reactions. It is found that the sound velocityhas a local minimum (indicating a softest point in the equation of state, EoS) at Ebeam = 30A GeV.This softening of the EoS is compatible with the assumption of the formation of a mixed phase atthe onset of deconﬁnement.Over the last years, a wealth of detailed data in the20A − 160A GeV energy regime has become available.The systematic study of these data revealed surpris-ing (non-monotonous) structures in various observablesaround 30A GeV beam energy. Most notable irregularstructures in that energy regime include,• the sharp maximum in the K+/π+ ratio [1, 2],• a step in the transverse momentum excitation func-tion (as seen through  m⊥  − m0 ) [2, 3],• an apparent change in the pion per participant ratio[2] and• increased ratio ﬂuctuations (due to missing data atlow energies it is unknown if this is a local max-imum or an ongoing increase of the ﬂuctuations)[4].It has been speculated, that these observation hint to-wards the onset of deconﬁnement already at 30A GeVbeam energy. Indeed, increased strangeness production[5] and enhanced ﬂuctuations have long been predictedas a sign of QGP formation [6, 7, 8, 9, 10, 11] within dif-ferent frameworks and observables. The suggestion of anenhanced strangeness to entropy ratio (∼ K/π) as indica-tor for the onset of QGP formation was especially advo-cated in [12]. Also the high and approximately constantK± inverse slopes of the mT spectra above ∼ 30A GeV- the ’step’ - was also found to be consistent with the as-sumption of a parton ↔ hadron phase transition at lowSPS energies [13, 14]. Surprisingly, transport simulations(supplemented by recent lattice QCD (lQCD) calcula-tions) have also suggested that partonic degrees of free-dom might already lead to visible eﬀects at ∼ 30A GeV[15, 16, 17]. Finally, the comparison of the thermody-namic parameters T and  B extracted from the transportmodels in the central overlap region [18] with the experi-mental systematics on chemical freeze-out conﬁgurations[19, 20, 21] in the T− B plane do also suggest that a ﬁrstglimpse on a deconﬁned state might be possible around10A − 30A GeV.In this letter, we explore whether similar irregu-larities are also present in the excitation function oflongitudinal observables, namely rapidity distributions.Here we will employ Landau’s hydrodynamical model[22, 23, 24, 25, 26, 27, 28]. This model entered thefocus again after the most remarkable observation thatthe rapidity distributions at all investigated energies canbe well described by a single Gaussian at each energy.The energy dependence of the width can also be rea-sonably described by the same model. For recent appli-cations of Landau’s model to relativistic hadron-hadronand nucleus-nucleus interactions the reader is referred to[29, 30, 31, 32, 33] (and Refs. therein).The main physics assumptions of Landau’s picture areas follows: The collision of two Lorentz-contracted nucleileads to full thermalization in a volume of size V/√sNN.This justiﬁes the use of thermodynamics and establishesthe system size and energy dependence. Usually, a simpleequation of state p = c2sǫ with c2s = 1/3 (cs denotes thespeed of sound) is assumed. For simplicity, chemical po-tentials are not taken into account. From these assump-tions follows a universal formula for the distribution ofthe produced entropy, determined mainly by the initialLorentz contraction and Gaussian rapidity spectrum fornewly produced particles. Under the condition that cs isindependent of temperature, the rapidity density is givenby [25, 26]:dNdy=Ks1/4NN q2πσ2yexp￿−y22σ2y￿(1)withσ2y =83c2s1 − c4sln(√sNN/2mp) , (2)where K is a normalisation factor and mp is the protonmass. The model relates the observed particle multiplic-ity and distribution in a simple and direct way to theparameters of the QCD matter under consideration.Let us now analyze the available experimental dataon rapidity distributions of negatively charged pions interms of the Landau model. Fig. 1 shows the measuredroot mean square σy of the rapidity distribution of neg-atively charged pions in central Pb+Pb (Au+Au) reac-tions as a function of the beam rapidity. The dotted lineindicates the Landau model predictions with the com-monly used constant sound velocity c2s = 1/3. The full20 1 2 3 4 5 6yp0.00.51.01.52.02.53.0yLandau, cs2=1/3linear ﬁtRHICSPSAGSFIG. 1: The root mean square σy of the rapidity distributionsof negatively charged pions in central Pb+Pb (Au+Au) reac-tions as a function of the beam rapidity yp. The dotted lineindicates the Landau model prediction with c2s = 1/3, whilethe full line shows a linear ﬁt through the data points. Data(full symbols) are taken from [3, 33, 34, 35]. The statisticalerrors given by the experiments are smaller than the symbolsizes. Systematic errors are not available.line shows a linear ﬁt through the data points, while thedata points [3, 33, 34, 35] are depicted by full symbols.At a ﬁrst glance the energy dependence looks struc-tureless. The data seem to follow a linear dependenceon the beam rapidity yp without any irregularities. How-ever, the general trend of the rapidity widths is also wellreproduced by Landau’s model with an equation of statewith a ﬁxed speed of sound. Nevertheless, there seemto be systematic deviations. At low AGS energies andat RHIC, the experimental points are generally under-predicted by Eq. (2), while in the SPS energy regimeLandau’s model overpredicts the widths of the rapiditydistributions. Exactly these deviations from the simpleLandau picture do allow to gain information on the equa-tion of state of the matter produced in the early stage ofthe reaction. By inverting Eq. (2) we can express thespeed of sound c2s in the medium as a function of themeasured width of the rapidity distribution:c2s = −43ln(√sNN/2mp)σ2y+s￿43ln(√sNN/2mp)σ2y￿2+ 1 .(3)Let us now investigate the energy dependence of thesound velocities extracted from the data. Figure 2 showsthe speed of sound as a function of beam energy forcentral Pb+Pb (Au+Au) reactions as obtained from the1002 5 1012 5 1022 5 1032 5 1042Ebeam (GeV)0.20.250.30.350.40.450.50.550.6cs2RHICSPSAGSFIG. 2: Speed of sound as a function of beam energy forcentral Pb+Pb (Au+Au) reactions as extracted from the datausing Eq. (3). The statistical errors (not shown) are smallerthan 3%.data using Eq. (3). The sound velocities exhibit a clearminimum (usually called the softest point) around abeam energy of 30A GeV. A localized softening of theequation of state is a long predicted signal for the mixedphase at the transition energy from hadronic to partonicmatter [36, 37, 38]. Therefore, we conclude that the mea-sured data on the rapidity widths of negatively chargedpions are indeed compatible with the assumption of theonset of deconﬁnement at the lower SPS energy range.However, presently we can not rule out that also an in-creased resonance contribution may be the cause of thesoftening [39].In conclusion, we have explored the excitation func-tions of the rapidity widths of negatively charged pionsin Pb+Pb (Au+Au) collisions.• The rapidity spectra of pions produced in centralnucleus-nucleus reactions at all investigated ener-gies can be well described by single Gaussians.• The energy dependence of the width of the pionrapidity distribution follows the prediction of Lan-dau’s hydrodynamical model if a variation of thesound velocity is taken into account.• The speed of sound excitation function extractedfrom the data has a pronounced minimum (softestpoint) at Ebeam = 30A GeV.• This softest point might be due to the formationof a mixed phase indicating the onset of deconﬁne-ment at this energy.3Further explorations of this energy domain is needed andcan be done at the future FAIR facility and by CERN-SPS and BNL-RHIC experiments.AcknowledgementsThe author thanks C. Blume and M. Gazdzicki forfruitful and stimulating discussions. This work was sup-ported by GSI, DFG and BMBF. This work used com-putational resources provided by the Center for ScientiﬁcComputing at Frankfurt (CSC).References[1] S. V. Afanasiev et al. [The NA49 Collaboration], Phys.Rev. C 66 (2002) 054902 [arXiv:nucl-ex/0205002].[2] M. Gazdzicki et al. [NA49 Collaboration], J. Phys. G 30(2004) S701 [arXiv:nucl-ex/0403023].[3] C. Blume, J. Phys. G: Nucl. Part. Phys. 31, S57 (2005)[4] C. Roland [NA49 Collaboration], J. Phys. G 31 (2005)S1075.[5] P. Koch, B. Muller and J. Rafelski, Phys. Rept. 142(1986) 167.[6] M. Bleicher, S. Jeon and V. Koch, Phys. Rev. C 62 (2000)061902 [arXiv:hep-ph/0006201].[7] E. V. Shuryak and M. A. Stephanov, Phys. Rev. C 63(2001) 064903 [arXiv:hep-ph/0010100].[8] H. Heiselberg and A. D. Jackson, Phys. Rev. C 63 (2001)064904 [arXiv:nucl-th/0006021].[9] B. Muller, Nucl. Phys. A 702 (2002) 281[arXiv:nucl-th/0111008].[10] M. Gazdzicki, M. I. Gorenstein and S. Mrowczynski,Phys. Lett. B 585, 115 (2004) [arXiv:hep-ph/0304052].[11] M. I. Gorenstein, M. Gazdzicki and O. S. Zozulya, Phys.Lett. B 585, 237 (2004) [arXiv:hep-ph/0309142].[12] M. Gazdzicki and M. I. Gorenstein, Acta Phys. Polon. B30, 2705 (1999).[13] M. I. Gorenstein, M. Gazdzicki and K. A. Bugaev, Phys.Lett. B 567, 175 (2003) [arXiv:hep-ph/0303041].[14] Y. Hama, F. Grassi, O. Socolowski, T. Kodama,M. Gazdzicki and M. Gorenstein, Acta Phys. Polon. B35 (2004) 179.[15] H. Weber, C. Ernst, M. Bleicher et al., Phys. Lett. B442, 443 (1998).[16] E. L. Bratkovskaya et al., Phys. Rev. Lett. 92, 032302(2004)[17] E. L. Bratkovskaya et al., Phys. Rev. C 69, 054907 (2004)[18] L. V. Bravina et al., Phys. Rev. C 60, 024904 (1999),Nucl. Phys. A 698, 383 (2002).[19] P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 606,320 (1996).[20] P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 638,3 (1998).[21] J. Cleymans and K. Redlich, Phys. Rev. C 60, 054908(1999).[22] E. Fermi, Prog. Theor. Phys. 5, 570 (1950).[23] L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953).[24] S. Z. Belenkij and L. D. Landau, Usp. Fiz. Nauk 56, 309(1955).[25] E. V. Shuryak, Yad. Fiz. 16, 395 (1972).[26] P. Carruthers, Annals N.Y.Acad.Sci. 229, 91 (1974).[27] P. Carruthers and M. Doung-van, Phys. Rev. D 8, 859(1973).[28] P. Carruthers, LA-UR-81-2221[29] E. L. Feinberg, Z. Phys. C 38 (1988) 229.[30] J. Stachel and P. Braun-Munzinger, Phys. Lett. B 216,1 (1989).[31] P. Steinberg, arXiv:nucl-ex/0405022.[32] M. Murray, arXiv:nucl-ex/0404007.[33] G. Roland, Talk presented at Quark Matter 2004, seeproceedings.[34] J.Klay et al. [E895 Collaboration], Phys. Rev. C 68,054905 (2003)[35] I.G. Bearden et al. [Brahms Collaboration], Phys. Rev.Lett. 94, 162301 (2005)[36] C. M. Hung and E. V. Shuryak, Phys. Rev. Lett. 75,4003 (1995) [arXiv:hep-ph/9412360].[37] D. H. Rischke, Y. Pursun, J. A. Maruhn, H. Stoeckerand W. Greiner, Heavy Ion Phys. 1, 309 (1995)[arXiv:nucl-th/9505014].[38] J. Brachmann, A. Dumitru, H. Stoecker and W. Greiner,Eur. Phys. J. A 8, 549 (2000) [arXiv:nucl-th/9912014].[39] R. Hagedorn, Nuov. Cim. Suppl. 3, 147 (1965); J. Rafel-ski and R. Hagedorn, Bielefeld Symp., ed. H. Satz, pp.253 (1980)

Evidence for the Onset of Deconfinement from Longitudinal Momentum Distributions? Observation of the Softest Point of the Equation of State

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