We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively.We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively.We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively

Redlich, Krzysztof

Tounsi, Ahmed

arXiv

We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively.We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively.We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively.We demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively

CERN Document Server

arXiv:hep-ph/0111159v1  13 Nov 2001Strangeness Enhancement and Canonical SuppressionAhmed Tounsi(a), Krzysztof Redlich (b,c)(a) Laboratoire de Physique The´orique et Hautes Energies, Universite´ Paris 7, 2 pl Jussieu, F–75251 Cedex 05(b) Theory Division, CERN, CH-1211 Geneva 23, Switzerland(c) Institute of Theoretical Physics, University of Wroclaw, PL-50204 WroclawWe demonstrate the essential role of canonical suppression in strangeness enhancement. Thepattern of enhancement of strange and multistrange baryons observed by theWA97 collaboration canbe understood on this basis. Besides, it is shown that in canonical approach strangeness enhancementis a decreasing function of collision energy. It is the largest at√s = 8.7 GeV where the enhancementof Ω,Ξ and Λ is of the order of 100, 20 and 3 respectively.I. INTRODUCTIONUltra-relativistic heavy ion collisions provide a unique opportunity to study the properties of highly excited hadronicmatter under extreme conditions of high density and temperature [1–5]. From the analysis of rapidity distribution ofprotons and of their transverse energy measured in 158 A GeV/c Pb+Pb collisions an estimate of the initial conditionleads to energy density of 2-3 GeV/fm3 and a baryon density of the order of 0.7/fm3. Lattice QCD at vanishingbaryon density suggests that the phase transition from confined to the quark-gluon plasma (QGP) phase appears atthe temperature Tc = 173 ± 8 MeV which corresponds to the critical energy density [6] ǫc ∼ 0.6 ± 0.3 GeV/fm3.One could thus conclude that the required initial conditions for quark deconfinement are already reached in heavyion collisions at top SPS energy and RHIC energies. Thus, of particular relevance was to find experimental probesto check whether the produced medium in its early stage was indeed in the QGP phase. Different probes have beentheoretically proposed and studied in relation with CERN-SPS and more recently with BNL-RHIC experiments. Wewill concentrate on strange hadrons and particularly on multistrange baryons. More precisely we will examine theissue of enhancement of strange and multistrange baryons as a possible signature of QGP formation.II. STRANGENESS ENHANCEMENT AND QGPIt was long ago argued that enhanced production of strange particles in nucleus-nucleus (AA) collisions, relativelyto proton-proton (pp) or to proton-nucleus (pA) collisions, could be a signal of QGP formation [7]. In QGP theproduction and equilibration of strangeness is very efficient due to a large gluon density and a low energy thresholdfor dominant QCD processes of ss¯ production [7,8]:GG→ ss¯ (1)In contrast, in a hadronic system, e.g., in pp, the higher threshold for strangeness production was argued to make thestrangeness yield considerably smaller and the equilibration time much longer than in QGP. From these strangenessQGP characteristics one expects a global strangeness enhancement , which should increase from pp, pA to AA collisions,as well as enhancement of multistrange (anti)baryons . The global strangeness content in the collision fireball ismeasured by < ss¯ > /Npart, the total number of strange quarks per participant nucleon or by < ss¯ > / < uu¯+ dd¯ >,the total number of strange quarks per produced light quark. Furthermore, this (anti)hyperon enhancement waspredicted to depend on the strangeness content of the (anti)hyperons and to appear in a typical hierarchy:EΛ < EΞ < EΩThis hierarchy is observed by the WA97 and NA57 collaborations [9,10] which measure the yield per participantin Pb+Pb relative to p+Pb and p+Be collisions. In particular the WA97 data show a pattern with this hierarchyand a saturation of enhancement for a number of participant nucleons Npart > 100. Recent results of the NA57collaboration [10] are showing in addition an abrupt change of anti-cascade enhancement for lower Npart. Similarbehavior was previously seen on the K+ yield measured by the NA52 experiment in Pb-Pb collisions [11]. Theseresults are very interesting as they might be interpreted as an indication of the onset of new dynamics. However, a1more detailed experimental study and theoretical understanding are still required here. It is, e.g., not clear what isthe origin of different centrality dependence of Ξ and Ξ¯. Nevertheless, this abrupt change could be possibly accountedfor in canonical approach by assuming a very particular increase of volume parameter with centrality [12].Anyway, strangeness enhancement is also seen at low energies, and found to be a decreasing function of collisionenergy in a compilation of the data on K+/π+ ratio in A+A relative to p+p collisions, where the double ratio(K+/π+)(AA)/(K+/π+)pp can be considered as an enhancement measure [13,14]. Such a behaviour with collisionenergy could also be expected for multistrange baryons. It was indeed shown [15,16] that a statistical model (SM)implementing canonical strangeness conservation explains the WA97 pattern and predicts that enhancement is adecreasing function of collision energy. This, we summarize in the following sections.III. STRANGENESS ENHANCEMENT AND CANONICAL SUPPRESSIONThe enhancement E measured in experiments is the ratio of the yield of a given (anti)baryon per participant nucleonin the large AA system to the yield of the same (anti)baryon in a the small pp or pA system:E =(Y ield)|AA(Y ield)|pA(2)In a large system with a large number of produced particles, the conservation law of a quantum number, e.g.,strangeness, can be implemented on the average by using the corresponding chemical potential. This is the GrandCanonical formulation (GC). In a small system, however, with small particles multiplicities, conservation laws must beimplemented locally on an event-by-event basis [17,18]. This is the Canonical formulation (C). The (C) conservationof quantum numbers is known to severely reduce the phase space available for particle production [17]. This thecanonical suppression (CS). If in Eq. (2) the denominator is reduced by CS then E is increased. That is, in ourapproach, the essence of strangeness enhancement from pp, pA to AA collisions.To better understand CS, consider a gas of particles with strangeness s = +1, 0,−1 and with total strangenessS = 0 (this condition is imposed by the fact that in heavy ion collisions the initial state of the system is strangenessneutral). Group theory projection techniques [19,20] allow to construct the partition function of the gas, from whichall thermal physical quantities are derived. One obtains for the thermal kaon density in the (C) formulationnCK =Z1KVS1√S1S−1I1(x1)I0(x1)(3)whereZ1K = VgK2π2m2kTK2(mKT)(4)S1 = Z1K + Z1Λ¯ + Z1K⋆ + ... (5)x1 ≡ 2√S1S−1 ∝ V (6)V is the volume parameter which is assumed, as usual, to be linear in Npart:V =V02Npart (7)where V0 ≈ 7 fm3 is taken as the volume of the nucleon. S1 and S−1 are the sum over one-particle partition functionsfor particles carrying strangenes 1 and -1 respectively. I1, I0 and K2 are modified Bessel and Hankel functions. Thedensity nCK in Eq. (3) depends on the volume, but for x1 → ∞, I1(x1)/I0(x1) → 1 and nCK reaches its GC limit,nGCK , independent of the volume. For x1 → 0 I1(x1)/I0(x1)→ x1/2 and the density nCK is linearly dependent on thevolume. One clearly sees on Eq. (3) that the factorFCS1 =I1(x1)I0(x1)(8)2called canonical suppression factor, measures the deviation of the kaon density from its GC value nGCK .FIG. 1. Canonical suppression factor for threevalues of particle strangeness: s = 1, 2, 3, at topCERN-SPS energy.FIG. 2. dotted line:√s = 130GeV; short dashedline:√s = 17.3GeV; long dashed line:√s = 12.3GeV;dot-short dashed line:√s = 8.3GeVThe previous considerations can naturally be extended to a gas of particles with strangeness s = 0,±1,±2,±3,i.e., to a gas composed of all particles (antiparticles) and their resonances. With the condition that the gas has totalstrangeness S = 0, the canonical particle density of a particle i having strangeness s reads [21]nCi =1VZ1iZCS=0∞∑n=−∞∞∑p=−∞ap3an2a−2n−3p−s1 In(x2)Ip(x3)I−2n−3p−s(x1) (9)whereai =√Si/S−i (10)xi = 2√SiS−i ∝ V (11)ZCS=0 =∞∑n=−∞∞∑p=−∞ap3an2a−2n−3p1 In(x2)Ip(x3)I−2n−3p(x1) (12)In Eq. (10), Si =∑k Z1k where the sum is over all particles and resonances carrying strangeness i. For a particle of massmk, with spin-isospin degeneracy factor gk, carrying baryon number Bk and electric charge QK , with correspondingchemical potential µB and µQ, the one-particle partition function is given, in the Boltzmann, byZ1k ≡ Vgk2π2m2kTK2(mkT)exp(BkµB +QkµQ) (13)Here too one can show [15] that for small x1nCi ≃Z1iV(S1)s(S+1S−1)s/2Is(x1)I0(x1)(14)and the canonical suppression factor is nowFCSs =Is(x1)I0(x1)(15)In Eq. (8) and Eq. (15) one sees that strangeness content of the particle appears in the suppression factor as the orderof Bessel function Is(x1). Fig. 1 shows that the suppression factor, for a given value of Npart, is smaller for largervalues of s. At a given energy (temperature) Npart depends on the colliding system. In the small system for p-pcollisions Npart = 2. For p-Be (p-Pb) collisions Npart ≈ 2.5 (≈ 4.75). These values have been determined by theWA97 collaboration from a Wounded Nucleon Model in the framework of the Glauber Model [22]. In particular forsmall x1, FCSs ∼ (x1/2)s, and one expects that the larger the strangeness content of the particle the smaller the3suppression factor and hence the larger the enhancement. This explains [15] the enhancement hierarchy of the WA97pattern. Furthermore, one sees on Fig. 2 that for a given strangeness, e.g., s = 2, the suppression factor, for anynumber of participant nucleons, is decreasing with decreasing energy: this means that enhancement increases withdecreasing energy. Finally, enhancement saturation appears, as explained above, as the grand canonical limit for largenumber of participant nucleons (large volume).IV. STRANGENESS ENHANCEMENT ENERGY DEPENDENCEWe have studied strangeness enhancement at four energies:√s = 8.73, 12.3, 17.3 GeV (NA49 and WA97, SPS) and√s = 130 GeV (RHIC). To study the energy and centrality dependence of (multi)strange particle yields in terms of theabove model one needs to establish first the variation of thermal parameters with energy and centrality. Temperatureis to a good approximation [23] only a function of collision energy and is independent of the number of participatingnucleons. Baryonic chemical potential µB is weakly dependent on centrality [16]. Thermal parameters are, howeversensitive to collision energy. At top SPS energy√s = 17.3 GeV we use the parameters obtained [24] in experimentaldata analysis: T = 166 MeV and µB = 266 MeV. At RHIC energy we take [25] T = 175 MeV and µB = 51 MeV. At√s = 8.73, 12.3 GeV the parameters are fixed according to the method explained in [16]. We have T = 145 MeV,µB = 370 MeV and T = 152 MeV , µB = 280 MeV respectively.FIG. 3. Centrality dependence of the relative en-hancement of particle yields/participant in centralPb-Pb to p-p collisions at fixed energy√s = 8.73GeV.FIG. 4. Centrality dependence of the relativeenhancement of particle yields/participant in centralPb-Pb to p-p collisions at fixed energy√s = 130GeV.In Fig. 3 and Fig. 4 we show the results on relative (multi)strange baryon enhancement from p-p to Pb-Pb collisionsat√s = 8.73 GeV and at√s = 130 GeV respectively. It is clear that the same enhancement pattern as at SPS isexpected in SM to appear for all relevant energies. To see the dependence of the strength of the enhancement withenergy we show in Fig. 5 and Fig. 6 the relative enhancement of Ξ and (Ω + Ω¯) for different collision energies. Theenhancement is the largest at lowest energy, for Ω it can be even by a factor of almost ten larger at√s = 8.73 thanobserved at SPS top energy. At RHIC enhancement of Ω is smaller than at SPS. These predictions are in contrastwith UrQMD [26] which predicts an enhancement at RHIC larger by a factor of four than at SPS. Note that at allenergies all figures display the WA97 pattern: hierarchy and saturation, the latter indicating that the grand canonicallimit is reached. This pattern was predicted as a signal for quark-gluon plasma formation [7]. In the context ofthe considered SM the enhancement pattern of (multi)strange baryons should be observed at all SPS energies, withincreasing strength toward lower beam energy. Thus, the results of the above SM makes it clear that strangenessenhancement and enhancement pattern is not a unique signal of deconfinement as these features are expected to bealso there at energies where QGP is not expected to be formed.The quantitative results shown in Fig. 3 to Fig. 6 contain some uncertainties. The magnitude of the enhancementis very sensitive to temperature taken at given collision energy. Changing T by 5 MeV, a typical error on T in thermalanalysis, can change, e.g., the enhancement of Ω shown in Fig. 3 by a factor of two. The Npart dependence of strangebaryon enhancement seen in Figs. 3-6 was obtained assuming linear dependence of volume parameter V in Eq. (7) withNpart. In general V could have a weaker dependence with centrality which could be reflected in only moderate increase4of the enhancement with centrality and saturation appearing at larger volume. In addition, including variation ofthermal parameters, in particular the baryon-chemical potential, with centrality or extending the model to canonicaldescription of baryon number conservation [27] or, finally, including a possible asymmetry between strangeness under-saturation factor in pp and AA collisions [27,28], could change our numerical values. However, independently of theseuncertainties, the main results: (i) enhancement decreasing with increasing collision energy and (ii) enhancementpattern being preserved at all SPS and RHIC energies, are always valid.FIG. 5. Centrality dependence of relative enhance-ment of Ξ− yields/participant in central Pb-Pb to p-preactions at different collision energies.FIG. 6. Centrality dependence of relative enhance-ment of (Ω + Ω¯) yields/participant in central Pb-Pbto p-p reactions at different collision energies.V. CONCLUDING REMARKSWe have shown that in terms of statistical model the relative enhancement of (multi)strange baryons from proton-proton to nucleus-nucleus collisions is a decreasing function of collision energy. Experimentally this fact was alreadyobtained for kaon yields and is shown to be expected for multistrange baryons. In addition, an increase of theenhancement with strangeness content of the particle is a generic feature of our model, independent of collisionenergy. On the qualitative level the only input being required in the model to make the above predictions is theinformation that freezeout temperature is increasing with collision energy and that the chemical potential showsthe opposite dependence. The above required conditions are well confirmed by a very detailed analysis of particleproduction at different collision energies.We have presented the quantitative predictions for relative enhancement of Λ, Ξ and Ω yields in the energy rangeform√s = 8.7 up to√s = 130 GeV. We have discussed possible uncertainties of presented results. The relativeenhancement at RHIC was found to be lower than at the SPS. This is in contrast with UrQMD [26] predictionsand with the previous qualitative predictions of observed enhancement as being entirely due to quark-gluon plasmaformation [7]. Even an abrupt change in the enhancement of Ξ¯/Npart versus Npart, recently reported by the NA57,could be possibly accounted for in terms of canonical model when assuming a very particular centrality dependence ofthe correlation volume [12]. It would correspond to a sudden jump of the volume, as in a first order phase transition,which can be taken into account in the canonical suppression factor.ACKNOWLEDGMENTSInteresting discussions with B. Mu¨ller and H. Satz are acknowledged. (K.R.) also acknowledges a partial supportof the Polish Committee for Scientific Research (KBN-2P03B 03018). (A.T.) also wishes to thank M. Gonin andthe organizing committee of CIPPQG for a good atmosphere at Ecole Polytechnique and for an excellent Conferencedinner at the Palais du Luxembourg.5REFERENCES[1] Proceedings of Quark Matter ’99 , Nucl. Phys. A 661 (1999).[2] H. Satz, Rept. Prog. Phys. 63, 1511 (2000).[3] R. Stock, Phys. Lett. 456, 277 (1999); Prog. Part. Nucl. Phys. 42, 295 (1999).[4] J. Stachel, Nucl. Phys. A 654, 119c (1999).[5] U. Heinz, Nucl. Phys. A 685, 414 (2001); Nucl. Phys. A 661, 349 (1999).[6] F. Karsch, Nucl. Phys. B (Proc. Suppl.) 83-84, 14 (2000) and Proceedings of Quark Matter 2001.[7] J. Rafelski, and B. Mu¨ller, Phys. Rev. Lett. 48, 1066 (1982); P. Koch, B. Mu¨ller, and J. Rafelski, Phys. Rept. 142, 167(1986) ; J. Rafelski, Phys. Lett. B 262, 333 (1991).[8] J. Rafelski, J. Letessier, and A. Tounsi, Acta Phys. Pol. B 27.[9] E. Andersen, et al., WA97 Collaboration, Phys. Lett. B 449, 401 (1999).[10] N. Carrer, NA57 Collaboration, Proceedings of Quark Matter 2001[11] S. 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Strangeness Enhancement and Canonical Suppression

6 pages, 6 figuresWe demonstrate the essential role of canonical suppression in strangeness enhancement. The pattern of enhancement of strange and multistrange baryons observed by the WA97 collaboration can be understood on this basis. Besides, it is shown that in canonical approach strangeness enhancement is a decreasing function of collision energy. It is the largest at $\sqrt{s}=8.7$ GeV where the enhancement of $\Omega,\Xi$ and $\Lambda$ is of the order of 100, 20 and 3 respectively

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Strangeness Enhancement and Canonical Suppression

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