Constituent Quark Scaling of Strangeness Enhancement in Heavy-Ion Collisions

In the frame work of a nuclear overlap model, we estimate the number of nucleon and quark participants in proton-proton, proton-nucleus and nucleus-nucleus collisions. We observe the number of nucleon ($N_{N-part}$)-normalized enhancement of multi-strange particles which show a monotonic increase with centrality, turns out to be a centrality independent scaling behavior when normalized to number of constituent quarks participating in the collision ($N_{q-part}$). In addition, we observe that the $N_{q-part}$-normalized enhancement, when further normalized to the strangeness content, shows a strangeness independent scaling behavior. This holds good at top RHIC energy. However, the corresponding SPS data show a weak $N_{q-part}$-scaling with strangeness scaling being violated at top SPS energy. This scaling at RHIC indicates that the partonic degrees of freedom playing an important role in the production of multi-strange particles. Top SPS energy, in view of the above observations, shows a co-existence of hadronic and partonic phases. We give a comparison of data with HIJING, AMPT and UrQMD models to understand the particle production dynamics at different energies.


I. INTRODUCTION
Relativistic heavy-ion collisions aim at creating matter at extreme conditions of energy density and temperature which is governed by the partonic degrees of freedom called Quark-Gluon Plasma (QGP). The main focus of these studies is the observation of quark-hadron phase transition and exploring the Quantum Chromodynamics (QCD) phase diagram. In the early phases of ultrarelativistic heavy-ion collisions, when a hot and dense region is formed in the core of the reaction zone, different quark flavors are produced. Then the produced matter undergoes transverse expansion and multiple scattering among the produced particles. The formation of the hadrons from the partonic phase is accomplished through further expansion and cooling of the system. In protonproton (p + p) collisions, the formation of QGP is not expected, whereas a possible formation of QGP is expected in nucleus-nucleus (A+A) collisions. Hence, a comparative study of produced particles in A+A collisions, with that of p+p collisions, could give better understanding of the properties of the medium formed in A+A collisions.
In the mid-rapidity region, strangeness enhancement has been proposed as a potential signature of QGP [1,2]. Strange baryons are produced in strong interaction processes and decay through weak interaction. It has been observed that multi-strange baryons, i.e. Ω(sss), Ξ(ssd) and Λ(uds), are formed and decouple from the system earlier in time [3]. Due to their different reaction rates * Corresponding Author, Email: Raghunath.Sahoo@cern.ch in the medium, particles with different strangeness decouple at different times. Relativistic Quantum Molecular Dynamics (RQMD) results suggest that the multistrange baryons freeze-out at energy densities more than 1 GeV /f m 3 [3] which corresponds to the critical energy density predicted by lattice QCD calculations [4]. The study of multi-strange baryons are of paramount importance in high energy heavy-ion collisions because of their dominant strangeness content (s-quark). As the colliding species in nuclear collisions do not contain any strange valence quark, particles with non-zero strange quarks can only be produced out of the collision process. The production of strange particles is enhanced in a QGP phase compared to a hadronic system. This is because, the production rate of gg → ss (gluon fusion) is high in a QGP medium [5], which is absent in hadronic phase. In addition, multi-strange baryons are less suffered by hadronic rescatterings in the later stage of the evolution of the fireball because of their small hadronic interaction cross sections. That is why multi-strange hadrons are good probes to carry the early stage information [6][7][8][9][10]. Hydrodynamic model estimations on hadron p T spectra suggest that the thermal freeze-out temperature of multi-strange baryons are close to their chemical freeze-out temperature (T ch ∼ 160 MeV), which is around the critical temperature, T c , for deconfinement transition. This indicates that multi-strange baryons are almost not affected by hadronic rescatterings at the later stage of the heavyion collisions [6,9,11]. Hence, multi-strange baryons could carry the information of possible formation of a QGP phase. It could be envisaged that the multi-strange baryons are formed out of partonic interactions rather than nucleonic interactions. We shall be justifying this in the following sections.
In this paper, both nuclei and nucleons are considered as superposition of constituent or "dressed" quarks (partons or valons). Baryons are composed of three and mesons are of two such quarks. The concept of constituent quarks is very well known [12,13] and established in the realm of the discovery of constituent quark scaling of identified particles elliptic flow at RHIC [14]. The constituent quark approach is successful in explaining many features of hadron-hadron, hadron-nucleus [15] and nucleus-nucleus collisions. These include global properties like the charged particle and transverse energy density per participant pair [16,17]. QCD calculations support the presence of three objects of size 0.1-0.3 fm inside a nucleon [18]. Furthermore, it has been seen that nucleus-nucleus collisions and p + p collisions have similar initial states if the results are scaled by the number of constituent quark participants [19]. These observations also indicate that the particle production is essentially controlled by number of constituent quarks pairs participating in the collision. In a constituent quark picture, nucleon-nucleon (N N ) collision looks like a collision of two light nuclei with essentially one qq pair interacting in the collision, leaving other quarks as spectators. These quark spectators form hadrons in the nucleon fragmentation region with a part of the entire nucleon energy being used for the particle production ( √ s qq ∼ √ s N N /3). But in A+A collisions, due to the large size of the nucleus compared to the nucleon, there is a higher probability of q − q interaction between the projectile and target nucleons. It has been observed at RHIC energies that the production rate of strange and multi-strange baryons in Au+Au collisions at √ s N N = 200 GeV, when scaled by the number of nucleon participants, are different (get enhanced) when compared to similar measurements in p+p collisions at same energy [20]. The observed enhancement increases with strangeness content of the baryons and also as a function of collision centrality, while going from peripheral to central collisions. Similar observations have been made at SPS energy for Pb+Pb collisions at √ s N N = 17.3 GeV, when compared with the corresponding measurements for p+Be collisions at the same energy [21]. In this paper, we have compared the centrality dependence of number of nucleon participant normalized multi-strange baryon enhancement at SPS and RHIC energies with the expectations from HIJING, AMPT and UrQMD models. The linear rise of the enhancement seen in the data, gets converted into a spectacular quarkparticipant scaling behavior when normalized to number of quark-participants. Furthermore, we explore the strangeness scaling, where quark-participant normalized enhancement for different multi-strange baryons, when divided by the strangeness content show a strangenessindependent scaling behavior.

II. CALCULATION OF THE NUMBER OF PARTICIPANTS
The calculations of the mean number of nucleon/quark participants are done in the following way. In the nuclear overlap model, the mean number of participants, i.e. N N −part , in the collisions of a nucleus A and a nucleus B with impact parameter b is given by where is the thickness function, defined as the probability of having a nucleonnucleon (N N ) collision within the transverse area ele- is the probability for a nucleon to pass through the nucleus without any collision. A and B are the mass numbers of two nuclei participating in the collision process. We use the Woods-Saxon nuclear density profile, with parameters, the normal nuclear density n 0 = 0.17 f m −3 , the nuclear radius R = (1. 3 GeV . Considering proton as a point particle and nucleus as an extended object in a p + A collision, the number of participating nucleons is given by In order to calculate the number of quark participants, N q−part , in nucleus-nucleus collisions, the density for quarks inside the nucleus is changed to three times that of the nucleon density (n q 0 = 3n 0 = 0.51 f m −3 ). Instead of nucleon-nucleon cross section, quark-quark cross section is used which is 4.67 mb and 3.3 mb at √ s N N = 200 GeV and 17.3 GeV, respectively [16,22]. In p + p(p) collisions, the quark participants are calculated by considering the proton and antiproton as hard sphere of radius 0.8 f m [23]. And for asymmetric collisions, like p+A collisions, proton is considered as a hard sphere of radius 0.8 f m and nucleus as extended objects with a Woods-Saxon density profile.

III. RESULTS AND DISCUSSION
At lower center of mass energies, it has been found that the particle production scales with the number of participating nucleons, contrary to the case of high energies where hard processes dominate. Hard processes have much smaller cross-section than the soft processes. However, the number of binary collisions increase with increase in collision centrality faster than the number of participants. As a result, the particle production per participant nucleon increases with centrality. By using constituent quark approach, we are going to show how the particle production at higher energies depends on the participating quarks. For this, multi-strange particles are chosen because of the proposed signature of strangeness enhancement in QGP medium. Few of the mechanisms for strangeness enhancement include: (i) The chemical and flavor equilibration time in gluon rich plasma have been predicted to be shorter than a thermally equilibrated hadronic matter of T ∼ 160 MeV [5]. Gluon fusion (gg → ss) is a dominant production mechanism of strangeness in an equilibrated gluon rich plasma [5]. This might allow for strangeness equilibration within the lifetime of QGP and hence resulting in strong enhancement of strangeness compared to a hadronic system. This enhancement in strangeness can also be explained in the context of statistical mechanics. In p + p system, the strangeness enhancement is not expected as the available volume is much smaller compared to nucleus-nucleus system. The p + p system can be treated as canonical system and the net strangeness number should be conserved. So the ss pairs are created at the same point and are annihilated as well, resulting in the suppression of strange hadrons. But in Au+Au system, strange and anti-strange hadrons are created independently and statistically distributed over the entire nuclear fireball, which could be treated in a grand canonical ensemble approach (GC) [24]. As the system thermalizes, the phase space suppression disappears as the volume available is more. The volume here is linearly proportional to the number of participating nucleons, i.e. N N −part . So a relative strangeness enhancement is observed in Au+Au collisions with respect to p + p collisions [20]. (ii) Early stage multiple scattering in heavy-ion collisions, may lead to an increase of the color electric field strength. In string-hadronic model, particles are produced through fragmenting color field (string) in Schwinger mechanism [25]. In high energy heavy-ion collisions, the string density could be very high so that color flux tubes overlap leading to a superposition of color electric fields. This results in enhanced production of multi-strange particles. This mechanism has been verified in the framework of UrQMD model with Ω showing an enhancement factor up to 100 [26].
To study the constituent quarks dependence of strangeness enhancement, we need to estimate the number of participating quarks, which has been done in the framework of nuclear overlap model. It is very essential to check how good is our estimation of number of participating nucleons in the collision. In order to do that, the mean number of participating nucleons, calculated in overlap model, are compared with the number estimated by the STAR experiment [27]. A very good agreement of nuclear overlap model calculations with that of STAR estimations has been observed. We have then estimated the number of quark participants within the prescription described in the previous section. The ratio of quark partipants and nucleon participants has been found to increase monotonically with collision centrality [16]. This ratio shows a sharp increase for N N −part ≤ 100, which follows a type of linear rise while going from peripheral to central collisions.
The yield enhancement factor, E(i), for particle species i is given by where, Y ield AA (i) and Y ield N N (i) are the yields of strange particles, and N AA N −part and N N N N −part are the number of nucleon participants, in nucleus-nucleus and nucleon-nucleon collisions, respectively. The number of nucleon participants, i.e. N N −part , is used to characterize the collision centrality. However, in the constituent quark framework, the nucleon participants no longer bear the meaning of sources of particle production. It is assumed that in this picture the constituent quarks are participating in the reaction and are the sources of interest for particle production. To understand the collision dynamics, the collision data are compared with models like HIJING-1.35, AMPT (default version) and UrQMD-3.3p1. HIJING is a Heavy Ion Jet Interaction Generator which includes multiple minijet production, nuclear shadowing of parton distribution functions and the mechanism of jet interaction with dense matter. This model is based on perturbative QCD [28,29]. Motivated by perturbative QCD, AMPT (A Multi-Phase Transport) model includes both initial partonic and final state hadronic interactions with quark-gluon to hadronic matter transition and their space-time evolution. Here, the partons are allowed to undergo scattering before they hadronize [30,31]. UrQMD (Ultra-relativistic Quantum Molecular Dynamics) model is based on microscopic transport theory, where hadronic interactions play an important role describing the evolution of the system [32].
In Figure 1, N N −part -normalized enhancement of (a) Λ and (b)Λ-baryons for Au+Au collisions at √ s N N = 200 GeV are shown as a function of N N −part . This collision data are from the STAR experiment [20] and are compared with the corresponding estimates of HIJING, AMPT and UrQMD models. In this paper, the p+p yield is taken from experimental data for the enhancement factor calculations. It is observed that none of these models correctly describes the collision data. All the models, i.e. HIJING, AMPT and UrQMD, show a centrality independent behavior. It is observed that the N N −partnormalized enhancement, obtained from AMPT model, is higher than the data obtained from HIJING and UrQMD models. This is because of the initial state partonic interactions in AMPT model. In Figure 2, N N −part -normalized enhancement of (a) Ξ − and (b)Ξ + -baryons at √ s N N = 200 GeV for Au+Au collisions are shown as a function of N N −part . The collision data points, obtained from STAR experiment [20], are compared with HIJING, AMPT and UrQMD models. The enhancement observed from the model estimates are independent of centrality and are much less than the data (enhancement factor being close to one). Although AMPT model has partonic degrees of freedom, the enhancement of Ξ − andΞ + being close to one, like HI-JING and UrQMD models, indicate that AMPT model doesn't describe the enhancement of baryons of higher strangeness content.
The collision data obtained from STAR experiment [20] on N N −part -normalized enhancement of Ω +Ωbaryons at √ s N N = 200 GeV for Au+Au collisions are shown in Figure 3 as a function of N N −part and are compared with the corresponding estimates of HI-JING, AMPT and UrQMD models. The enhancement observed from HIJING and AMPT models are independent of centrality and the enhancement factor is close to one. Whereas the corresponding estimates from UrQMD show a rise with centrality like experimental data, only difference being the enhancement factor is higher in case of data compared to UrQMD. A close observation of UrQMD estimation of N N −part -normalized enhancement of strange-baryons at √ s N N = 200 GeV for Au+Au collisions (shown in Figure 1 -3) shows a strangeness dependent increase in the enhancement. This means that the N N −part -normalized enhancement increases monotonically while going from Λ to Ω-baryons. This monotonic behavior of strangeness enhancement seems interesting in view of UrQMD being a microscopic trasport model with hadronic scattering playing a vital role in describing the space-time evolution of the system. Multistrange baryons are produced early in time and also their freeze-out time is very small [3]. With hydro+UrQMD calculations and taking hadronic rescattering into account, the mean freeze-out times for Ω, Ξ and Λ at top RHIC energy are, 17.3, 32.2 and 27.4 f m/c, respectively [11]. The hadronic rescattering cross sections for Ω and Ξ particles and also their mean freeze-out time are much less compared to Λ particles [33]. This shows that Ω comes out of the fireball almost without interacting with the hadronic medium and hence the enhancement is much less affected compared to Ξ and Λ particles. Additionally, UrQMD calculations with a color flux tube break up mechanism having strong color fileds predict a dramatic enhancement in the multi-strange hadron production compared to p+p interactions, e.g. Λ's are enhanced by a factor of 7, whereas Ω's are enhanced by a factor of 60 [26].
The enhancement ratio of baryons and anti-baryons is usually affected by the net-baryon content or baryon stopping at mid-rapidity. This difference decreases with increase in collision energy. For this reason, the enhancement of baryons and anti-baryons are different at SPS energies which is not that much at RHIC energies. In Figure 4 and Figure 5, N N −part -normalized enhancement of Λ andΛ-baryons for Pb+Pb collisions at √ s N N = 17.3 GeV are plotted as a function of N N −part , respectively. The collision data at √ s N N = 17.3 GeV are from SPS NA57 experiment [21] and are compared with the corresponding estimates of HIJING, AMPT and UrQMD models, as shown in Figure 4 and Figure 5. The HI-JING and AMPT data show a centrality independent behavior, whereas UrQMD shows a weak centrality dependence. But the collision data of Λ show a remarkable centrality dependence, as shown in production mechanism ofΛ has been assigned to the following reasons. Firstly, the production threshold for antihyperons is larger than hyperons. Secondly, while going from higher to lower collision energy, baryon density of the system increases. This doesn't favor the production of particles without having a common valence quark with nucleons [21]. This difference in yields go away as one moves from SPS energy to top RHIC energy.
In Figure 6 and Figure 7, N N −part -normalized enhancement of Ξ − andΞ + -baryons at √ s N N = 17.3 GeV for Pb+Pb collisions for SPS data have been shown as a function of N N −part . These data points are from NA57 experiment and are compared with the corresponding estimates of HIJING, AMPT and UrQMD models [21]. At SPS energies, both Ξ − andΞ + show strangeness enhancement which rises with centrality, unlike the models under discussion. However, all the models with partonic have been shown as a function of N N −part . These data points are from NA57 experiment and are compared with the corresponding estimates of HIJING, AMPT and UrQMD models [21]. A similar observation has been made for Ω +Ω-baryons like the Ξ-baryons. To understand different enhancement profiles of multi-strange baryons at SPS energies, it is important to study their production dynamics. However, it could be observed that the shape of the enhancements for (anti) baryons are similar, which goes inline with the predictions of a grand canonical ensemble approach (GC) [34]. In Figure 9, N q−part -normalized enhancement for 200 GeV have been shown as a function of collision centrality. These data points are from the STAR experiment at RHIC [20]. Here we observe that when the enhancement is normalized to the number of constituent quarks, it turns out to be a centrality independent scaling behavior. This enhancement, however, depends on the particle mass and shows an increase with higher mass number. The former observation is very interesting in view of partonic degrees of freedom playing a crucial role in particle production especially for the multi-strange particles, the enhancement of which has been conjectured to be a signal of formation of a partonic phase. In other words, at top RHIC energy the collision could be described at partonic collectivity at RHIC [35]. This supports present observation of centrality independent scaling behavior of constituent quarks normalized multi-strange baryon enhancement. However, the top SPS energy shows a weak quark participant scaling towards higher strangeness content of the particles, which is shown in Figure 10. Hence, it could be inferred here that an onset of deconfinement phase transition might have taken place already at top SPS energies leading to a mixed phase of partons and hadrons. This goes inline with other experimental findings related to the onset of deconfinement or saturation of different observables starting from SPS energies [36,37]. In Figure 10, N q−part -normalized enhancement for multi-strange baryons at √ s N N = 17.3 GeV for Pb+Pb collisions have been shown as a function of collision centrality. These data points are from NA57 experiment [21]. Here, we observe that for lower strangeness content particles, the quark participant normalized multi-strange enhancement turns out to be a centrality independent scaling-behavior. However, this is not true specifically for the Ω baryons, for which we observe a linear rise. This makes a difference between SPS and RHIC energies so far the constituent quark scaling of multi-strange baryons goes.
Statistical model estimation of strangeness enhancement in a broad range of energies spanning from √ s N N = 8.73 − 130 GeV suggests that (i) the absolute value of the enhancement decreases with increasing collision energy and increases with strangeness content, (ii) the enhancement pattern as a function of centrality seems to be preserved at all energies [34]. The fact that the chemical freeze-out temperature is almost constant at higher energies and there is little effect of baryon chemical potential on the strength and pattern of strangeness enhancement, going from 130 to 200 GeV collision energy won't change these observations. At lower collision energies, the initial conditions don't favor a deconfinement transition. Hence, as discussed in Ref. [34], strangeness enhancement and the pattern of enhancement are not a unique signal of deconfinement, which is a consequence of canonical suppression of strangeness in p + p collisions. However, a quark participant scaling of strangeness enhancement could be considered as a signal of deconfinement transition and formation of a QGP phase, which is seen for the top RHIC energy. A weak quark participant scaling of strangeness enhancement seen at the top SPS energy in turn indicates a co-existence of partonic and hadronic phases.
The here predicts any enhancement for Ωs except UrQMD. The reasons for this are discussed earlier in this paper. It is evident from these plots that the absolute enhancement number decreases with collision energy. This could be understood in a grand canonical ensemble approach, which predicts a significant decrease in (anti) baryon enhancements with collision energy [20,34]. We do observe a monotonic rise in the number of constituent quarks normalized strangeness enhancement as a function of strangeness content for both the energies (i.e. E(Ω) > E(Ξ) > E(Λ)). This could be understood as follows. The strangeness enhancement in A+A collisions is always measured relative to p+p or p+A collisions after appropriately dividing by the number of participants. In p+p collisions, even at higher energies, the available ther-mal or energy phase space is very less compared to A+A collisions. The absolute yields of strange particles in p+p collisions go down with increase in strangeness content because of canonical suppression. Compared to A+A collisions, the gluon density in p + p collisions is very less, which contributes to strangeness production (gg → ss) more abundantly. In A+A collisions, there is enough phase space available for the formation of multi-strange particles. When we look at strangeness enhancement, particles with higher strangeness content are enhanced more, compared to the enhancement of particles with lower strangeness content [34]. This also talks about the effect of the medium produced in A+A collisions, where quark coalescence and gluon fusion play important roles in multi-strange particle production. However, when the enhancement factor is further normalized with respect to the strangeness content and is plotted as a function of the strangeness content, a monotonic rise is observed for top SPS energy. Whereas it remains almost flat for the top RHIC energy. This means at top most RHIC energy both constituent quark and strangeness scaling of multi-strange baryons have been observed. It can be observed from Figures 9 and 10 that the N q−part scaling is more prominent at RHIC compared to SPS. In view of the fact that onset of deconfinement starts at SPS energies and there are several observations of a mixed phase of partons and hadrons at SPS energies [36,38], a weak N q−part scaling and strangeness scaling at these energies are expected. However, RHIC shows a strangeness scaling along with N q−part scaling. This gives an evidence of a purely partonic phase. These are shown in Figures  13 and 14 . We expect that these scaling laws will hold good at LHC energies.

IV. SUMMARY AND CONCLUSIONS
In this work, we have calculated the number of nucleon and quark participants in a proton-proton, protonnucleus and nucleus-nucleus collisions in the framework of a nuclear overlap model. The data for the enhancement of multi-strange baryons (in A+A collisions compared to p + p or p + Be) at top SPS and RHIC energies have been compared with HIJING, UrQMD and AMPT models. We observe the N N −part -normalized yields of multi-strange particles show a monotonic increase with centrality. This turns out to be a centrality independent scaling behavior when normalized to number of constituent quarks participating in the collision. This geometrical scaling indicates the partonic degrees of freedom playing important role in the production of multi-strange particles. Furthermore, we see the constituent quark normalized yield when further divided by the strangeness content, it shows a strangeness independent scaling behavior at the top RHIC energy, whereas this scaling is not seen at the top SPS energy. This goes in line with other observations at RHIC towards the formation of a partonic phase and a mixed phase of partons and hadrons at SPS energies. These features are reproduced neither by explicit hadronic kinetic models like UrQMD, HIJING nor by AMPT model which treats the partonic phase on the basis of pQCD with massless partons and a noninteracting equation-of-state. A quark participant scaling of strangeness enhancement, in view of the above observations, seems to signal a deconfined state of quark-gluon plasma.