Strangeness Production in Light and Intermediate size Nucleus-Nucleus Collisions

Within the statistical model, the net strangeness conservation and incomplete total strangeness equilibration lead to the suppression of strange particle multiplicities. Furthermore, suppression effects appear to be stronger in small systems. By treating the production of strangeness within the canonical ensemble formulation we developed a simple model which allows to predict the excitation function of $K^+/\pi^+$ ratio in nucleus-nucleus collisions. In doing so we assumed that different values of $K^+/\pi^+$, measured in p+p and Pb+Pb interactions at the same collision energy per nucleon, are driven by the finite size effects only. These predictions may serve as a baseline for experimental results from NA61/SHINE at the CERN SPS and the future CBM experiment at FAIR.


I. INTRODUCTION
The multiplicity of pions per participating nucleon is known to be similar in nucleus-nucleus (A+A) and in inelastic proton-proton (p+p) interactions at the same collision energy per nucleon. This is in line with the Wounded Nucleon Model [1] (WNM) in which the final states in A+A collisions are treated as a superposition of independent nucleon-nucleon collisions.
Similar picture emerges from the hadron statistical models within the grand canonical ensemble (GCE) formulation. At fixed temperature and chemical potentials all hadron multiplicities are proportional to the system volume V . Taking V to be proportional to the number of wounded nucleons N W in A+A collisions, one restores the WNM results for hadron multiplicities.
Production of strange hadrons appears to be quite different in p+p and heavy-ion collisions.
In particular, the ratio of K + to π + multiplicities is significantly larger in collisions of heavy ions. It was advocated to interpret this strangeness enhancement as a possible signature for the quark-gluon plasma creation [2]. A non-monotonic dependence of the K + to π + ratio as function of the collision energy (the horn) was predicted [3] as a fingerprint of the deconfinement phase transition. The predicted behavior was indeed observed by the NA49 Collaboration in central Pb+Pb collisions [4] at the SPS energies (for more details cf. Ref. [5]). Moreover, these findings have been recently confirmed by the RHIC and LHC data [6]. The experimental data on K + /π + ratio in p+p and Pb+Pb (Au+Au in the AGS energy range) collisions are presented in Fig. 1 as function of the center-of-mass energy of the nucleon pair √ s N N (for details see [7] and references therein).
Numbers of strange quarks N s and antiquarks N s in a final state of p+p or A+A collisions are equal to each other due to the net strangeness conservation in strong interactions. In the SPS energy range strange quarks are essentially carried by K − , K 0 mesons and Λ hyperons. On the other hand, almost all N s created in the collision process are finally revealed in K + and K 0 particles. For the event averages one obtains an approximate relation < K + > ∼ = 0.5 < Ns >.
This explains the choice of the K + multiplicity as an estimator for the total strangeness [5].
Conservation of strangeness in large statistical systems can be treated within the GCE formulation, in which all hadron multiplicities are proportional to the system volume V . In small systems, however, one has to follow the canonical ensemble (CE) treatment [8]. The multiplicities of (anti)strange hadrons in CE decrease with decreasing volume faster than the GCE multiplicities.
A comparison of the statistical model results with hadron multiplicity data, within both CE and GCE, evidences an incomplete strangeness equilibration. For reasonable fit of the data one has to introduce the strangeness suppression factor γ S [9]. Note that in p+p interactions the γ S factor is smaller than in central Pb+Pb collisions [10].
In the present study the difference of the K + /π + ratio in p+p and Pb+Pb collisions is In addition, a beam energy scan of Pb+Pb collisions, with much higher statistics than that performed by the NA49 Collaboration, is planned. We hope that the atomic number dependence of the K + /π + ratio from p+p to Pb+Pb collisions in the SPS energy range may reveal new and important physical information.
The Letter is organized as follows. In Section II the strangeness suppression effects in the statistical systems are considered in the CE formulation. In Section III the model parameters are extracted from the data on p+p and Pb+Pb collisions. The model predictions of the K + /π + ratio for light and intermediate nucleus-nucleus collisions are calculated. Finally, Section IV summarizes the paper. Appendix A includes details of the calculations.

II. STRANGENESS SUPPRESSION
We first introduce the following notations: where . . .   The π + multiplicity and the quantity z (see Appendix A, Eq. (15)) can be presented as: where i=p, A, or Pb. The π + ii and z i correspond, respectively, to the GCE π + multiplicity and N s gce = N s gce in i + i collisions. Note that strange (anti)quark multiplicity N s gce corresponds to the complete strangeness equilibration and does not yet take into account the CE suppression effects. We assume that the values of the pion number density n π + = π + /V and the strange (anti)quark number density n s = N s gce /V are not sensitive to the type of reactions, i.e. they have the same values in p+p, A+A, and Pb+Pb collisions at the same collision energy. The volumes V i are, however, different in each of these i + i reactions, and they are assumed to be proportional to the number of wounded nucleons N W (N W = 2 in inelastic p+p collisions). The GCE formulation will be adopted for pion multiplicity in all types of i + i collisions. The total number of negatively charged particles is larger than one (even in p+p collisions) at the SPS energies. Therefore, the CE effects of electric charge conservation are small and can be neglected. To calculate N s = N s both the CE effects and the incomplete strangeness equilibration are considered. This is discussed in Appendix A (see Eq. (18)). For the K + multiplicity it then follows: where the relation K + ∼ = 0.5N s has been used.
Finally, we obtain the following expressions for K + pp and η p in p+p collisions: The above equations assume: (i) the same n s and n π + GCE values of the particle number densities, as defined in Eq. (3) in p+p, A+A, and Pb+Pb collisions; (ii) the incomplete strangeness equilibration regulated by γ i S in i + i collisions (i = p, A, and Pb); (iii) the relation I 1 /I 0 ∼ = 1 is adopted in central Pb+Pb collisions, as γ P b S z P b 1.

III. PREDICTIONS FOR LIGHT ION COLLISIONS
The left-hand-sides of Eqs. (5) and (6) involve quantities which have been experimentally measured. The energy dependences of K + pp and η p are shown in Fig. 2. For the K + pp we used the fit function a · ( √ s N N ) b with a=0.028 and b=0.736 presented by the solid line in the right panel of Fig. 2. All in all there are 3 unknowns, γ p S , z p , and γ P b S , entering to the right-hand-sides of Eqs. (5) and (6). However, they can be combined as The solution of the transcendental Eq. (8), X = X( √ s N N ), is shown in the left panel of Fig. 3.
On the other hand, Eq. (9) gives the value of Y = η p I 0 (2X)/I 1 (2X) presented in the right panel of Fig. 3. Assuming now z A = z p · N W /2, where N W is the average number of wounded nucleons in A+A collisions, one can calculate the K + to π + ratio as: Next, following the prescription of Ref. [13], we used the following expression for the dependence of γ A S on N W and √ s N N : panel of Fig. 3).
Furthermore, taking γ A S = γ p S and γ A S = γ P b S , we obtain the lower (R low A ) and upper (R up A ) limits for R A defined in Eq. (10): In Fig. 4 and Fig In Fig. 6 we illustrate with green boxes the system size dependence (expressed in terms of wounded nucleons) of the K + / π + ratio at fixed energy of √ s N N = 7.6 GeV. The upper limit (full circles) corresponds to γ A S = γ P b S and the lower limit (open circles) to γ A S = γ p S . Interestingly, the K + / π + ratio becomes approximately independent of the number of wounded nucleons for N W > 40.

IV. SUMMARY
In summary, the K + /π + ratio in p+p and Pb+Pb collisions is considered within the statisti-

APPENDIX
The GCE partition function for strange quarks and antiquarks reads where the quantity z is the so-called one-particle partition function In Eqs. (14), (15), V and T are the system volume and temperature, respectively, m s is the mass of strange (anti)quark and K 2 is the modified Bessel function. Furthermore, the Boltzmann approximation is used because the quantum statistics effects are negligible. The λ and λ in Eq. (14) are auxiliary parameters introduced to calculate N s and N s averages: The parameter γ S regulates the strangeness equilibration [9]. It is used to fit the average value of the total strangeness measured by experiments: γ S < 1 corresponds to an incomplete strangeness equilibration, whereas γ S = 1 means a complete chemical equilibrium.
The GCE partition function (see Eq. (14)) leads to the equal average values of N s and N s .
However, the terms with N s = N s contribute to Z gce . On the other hand, the CE partition function requires N s = N s in each microscopic state of the system: The average numbers of strange quarks and antiquarks become: The ratio of Bessel functions I 1 and I 0 in Eq. (18) describes the suppression effect due to conservation of the net strangeness in each microscopic state of the CE. The CE suppression