December 2015 BI-TP 2015/20 Strangeness Production in AA and pp Collisions Paolo Castorinaa and Helmut Satzb a: Dipartimento di Fisica ed Astronomia, Universita’ di Catania, 6 and INFN, Sezione de Catania, Catania, Italy 1 b: Fakult¨at fu¨r Physik, Universit¨at Bielefeld, Germany 0 2 Abstract n a J Boost-invariant hadron production in high energy collisions occurs in causally discon- 7 nected regions of finite space-time size. As a result, globally conserved quantum numbers ] (charge, strangeness, baryon number) are conserved locally in spatially restricted cor- h p relation clusters. Their size is determined by two time scales: the equilibration time - p specifying the formation of a quark-gluon plasma, and the hadronization time, specifying e the onset of confinement. The expected values for these scales provide the theoretical h [ basis for the suppression observed for strangeness production in elementary interactions (pp, e+e−) below LHC energies. In contrast, the space-time superposition of individual 1 v collisions in high energy heavy ion interactions leads to higher energy densities, resulting 4 in much later hadronization and hence much larger hadronization volumes. This largely 5 4 removes the causality constraints and results in an ideal hadronic resonance gas in full 1 chemical equilibrium. In the present paper, we determine the collision energies needed for 0 . that; we also estimate when pp collisions reach comparable hadronization volumes and 1 thus determine when strangeness suppression should disappear there as well. 0 6 1 : v i X The main aim in the study of high energy nucleus-nucleus collisions is the production r of the quark-gluon plasma predicted as a new state of matter by statistical QCD. It is a expected that nuclear collisions, in contrast to proton-proton interactions, will provide a much larger interaction volume and thus allow an investigation of bulk QCD features. The first step in such a study is therefore the determination of observables which show a different behavior in pp and in AA collisions. The three basic deviations in AA collisions so far established are – enhanced strangeness production [1], – supppressed quarkonium production [2], and – jet quenching [3,4] Our topic in the present paper is the issue of strangeness production. In pp collisions, one finds a suppression of strange hadron production relative to an equilibrium distribution of species abundances at the hadronization temperature. In nuclear collisions at sufficiently high energy (RHIC and above), this suppression disappears; now also strange hadrons are produced in accord with the ratios predicted by a grand canonical resonance gas at T . H 1 The strangeness suppression in pp collisions as well as in low energy nuclear collisions has been accounted for in terms of local strangeness conservation [5]. To conserve strangeness, a produced s quark has to becompensated by a corresponding s¯sufficiently nearby. If ina givenrapidityrangeonlyasinglestrangeparticlepairisproduced, theuseofanequivalent overall composition volume [6] for a resonance gas is not valid. Strangeness conservation then requires a smaller conservation volume, and this leads to an effective reduction of the strangeness production rates. In high energy nuclear collisions, the superposition of many nucleon-nucleon interaction volumes leads to abundant strangeness production and thus removes the need for a smaller conservation volume. In a recent paper [7], we had shown that in case of a boost-invariant production process, causality requirements lead to the existence of causally disconnected spatial production regions. Globally conserved quantum numbers (charge, strangeness, baryon number) must therefore be conserved within these regions, which are smaller than the effective overall global volume of a grand canonical description. This provides a theoretical basis for the smaller strangeness conservation volumes just mentioned. In high energy heavy ion collisions, the superposition of several such volumes from individual nucleon-nucleon collisions in the same rapidity region is expected to remove such constraints. In the present paper, we want to quantify these considerations somewhat more, and also show that at extreme collision energies, even proton-proton collisions are expected to lead to full chemical equilibrium for strangeness. The crucial quantity is the size d of the causally connected interaction region at the time of hadronization. It was found to be [7] d τ τ h h = 1 , (1) τ0 sτ0 (cid:18)τ0 − (cid:19) where τ and τ denote the equilibration time (quark-gluon plasma formation time) and 0 h the hadronization time (color confinement time), respectively. The resulting variation of the correlation scale d is shown in Fig. 1; with the canonical choice τ 1 fm, that gives 0 ∼ the scale in fm. 10 9 8 7 6 0 τ 5 / d 4 3 2 1 0 1 2 3 4 5 τ/τ 0 Figure 1: The correlation scale d as function of hadronization time τ /τ h 0 For one-dimensional isentropic expansion, corresponding to boost-invariant production, 2 the times are related to the corresponding entropy densities s, s τ = s τ . (2) 0 0 h h For ideal gas behavior, entropy and energy density are related by s ǫ3/4, ∼ so that we then have τ ǫ 3/4 h 0 = , τ ǫ 0 (cid:18) h(cid:19) where ǫ is the initial energy density of the collision, and ǫ is the energy density at 0 h the hadronization transition, for which lattice calculations give ǫ 0.4 GeV/fm3 [8,9]. h ≃ Combining this with eq. 1, we obtain in Fig. 2 the variation of the scale parameter with the initial energy density ǫ . 0 6 5 4 0 τ / d 3 2 1 1 2 3 4 5 6 ε / ε 0 h Figure 2: The correlation scale d as function of initial energy density ǫ 0 For the assumed boost-invariant production, the initial energy density ǫAA for AA colli- 0 sions is given in terms of the resulting hadron production through the well-known rela- tion [10] m dN m A1/3 dN ǫAA = T AA = T AA . (3) 0 τ0πRA2 dy !0 τ0πR02 dy !0 Here m 0.5 GeV is the average transverse momentum per hadron, R A1/3R T A 0 ≃ ≃ the average nuclear radius, with R = 1.25 fm, and (dN /dy) the average hadron 0 AA 0 multiplicity at central rapidity. We parametrize the latter quantity as (dN /dy) = AA 0 A(dN /dy) , with (dN /dy) denoting one-half of the hadron multiplicity per participant A 0 A 0 in the AA collision. This multiplicity is shown in Fig. 3, where it is compared to the corresponding quantity in proton-proton collisions [11]. In that case, we have m dN ǫpp = T pp , (4) 0 τ0πRp2 dy !0 3 Figure 3: The charged hadron multiplicity per participant in AA and in pp collisions as function of the collision energy √s [11]. where R 0.8 fm denotes the proton radius. p ≃ Using the results of Fig. 3, we can now compare the correlation scale d(√s) in AA and pp collisions. The result is shown in Fig. 4. 10 Au-Au ε = 0.5 GeV/fm3 9 Au-Au εh = 0.4 GeV/fm3 h 8 pp ε = 0.5 GeV/fm3 h 7 pp ε = 0.4 GeV/fm3 h 6 0 τ 5 / d 4 3 2 1 0 100 1000 10000 1/2 S Figure 4: The correlation scale d for pp and AA collisions, for different input values of ǫ h It is evident that the higher energy density in AA collisions leads at a given √s to much larger correlation volumes than that found for pp interactions. We can use these results to address two issues: At what incident energy has the correlation scale reached a value in AA collisions, • for which we should expect grand canonical considerations to be valid? An exper- imental indication is provided by the disappearence of strangeness suppression in such collisions. The issue has also been addressed in theoretical studies [12]. At what incident energy does the scale parameter reach a comparable value in pp • interactions? This would indicate when we should expect strangeness suppression to vanish also in such elementary collisions. 4 To address the first issue, we recall that a parametric way to take strangeness suppression into account in the statistical hadronization model is the introduction of a suppression factor γ , multiplying by γn the Boltzmann factor of each species containing n quarks [13]. s s Fits of low energy nuclear collision data lead to γ 0.6, increasing with collision energy. s ≃ In Fig. 5, the behavior of γ vs. √s for nucleus-nucleus collisions [14] shows that for s √s 30 40 Gev the strangeness suppression essentially disappears. ≃ − 1.2 1.1 1.0 SPS Pb-Pb 0.9 0.8 RHIC Au-Au S 0.7 γ 0.6 0.5 AGS Au-Au 0.4 0.3 0.2 1 10 100 √ s [GeV] NN Figure 5: γ for heavy ion collisions at different energies [14] s The mentioned alternative of local strangeness conservation [5] is based on exact conser- vation (“canonical” formulation) combined with a smaller volume of correlation radius R . It is within this volume that strangeness must be conserved, and with increasing R , c c one evidently recovers the grand canonical form. As shown in Fig. 6 [12], this occurs for a strangeness correlation radius of about 2 fm; note that d 2R . c ≃ 1 10-1 10-2 o ati10-3 R 10-4 Grand Canonical PbPb Canonical , R =2 fm 10-5 pp C Canonical , R =1 fm pp C 10-6 --ππp/p/ ++ππ++/K/K --ππ--/K/K ΛΛ / p / p --ΛΞΛΞ// ----ΞΩΞΩ// ----ΩΩ/K/K ----πΩπΩ// Figure 6: Comparison of canonical and grand-canonical results for different size of the strangeness correlation volume [12] 5 For the second issue, we have to determine at which collision energy the energy density in pp interactions reaches the value at which in AA collisions strangeness suppression vanishes. InFig.7weshowthecollisionenergyvaluesinAAandinppcollisions, forwhich the energy densities of the two interactions are equal. Wenote that to obtainthe values at which in AA collisions strangeness suppression vanishes, pp interaction require collision energies of some 5 - 7 TeV. It should be emphasized here, however, that a somewhat larger correlation volume than that used here remains definitely possible, and that would shift the necessary pp energy to higher values. The values we have obtained just reach the highest presently obtainable LHC energies, and there are already first indications showing a considerable increase of strangeness production in pp collisions at 900 GeV and at 7 TeV [15,16]. τ τ (pp) = (Au-Au) h h 80 v Ge60 n A) i A 2) (40 1/ (s 20 1000 10000 s(1/2) (pp) in GeV Figure 7: The collision energy values for which the initial energy density ǫ in pp is equal 0 to that in AA It would obviously be of great interest to see if high energy pp results approach the AA results also for the other two indicators mentioned, quarkonium suppression and jet quenching. References [1] B. Mueller and J. 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