Centrality dependence of chemical freeze-out parameters and strangeness equilibration in RHIC and LHC

We have estimated chemical freeze-out parameters from yield data of identified spectra of $\pi^\pm$, $k^\pm$ and $p$, $\bar{p}$ at BES, RHIC and LHC and studied their variation with the centrality. We have considered a simple hadron resonance gas model and employed a formalism involving conserved charges ($B, Q, S$) of QCD for parameterization. Along with temperature and three chemical potentials ($T,\mu_B,\mu_Q,\mu_S$), a strangeness undersaturation factor ($\gamma_S$) has been used to incorporate the partial equilibrium in the strangeness sector. Our obtained freeze-out temperature does not vary much with centrality, whereas chemical potentials and $\gamma_S$ seem to have a significant dependence. For the peripheral collisions, the strange hadrons are found to deviate from a complete chemical equilibrium at freeze-out. This deviation appears to be more prominent as the collision energy decreases toward lower BES energies.


INTRODUCTION
In the last few decades, several ion collider experiment collaborations have come up to explore the phase diagram of quantum chromodynamics (QCD). Relativistic Heavy Ion Collider (RHIC) started to investigate the signatures of deconfined quark-gluon plasma whereas the Beam Energy Scan (BES) program became motivated in searching the QCD critical point [1]. The Large Hadron Collider (LHC) has been set up to investigate the medium created in zero baryon density where a crossover transition from hadronic state to a deconfined state of quarks and gluons may happen [2].
The hot and dense fireball, created in these collisions, experiences a fast expansion due to the initial pressure gradient. Assuming that the system starts from a state of strongly interacting quarks and gluons, a fast thermalization can occur. The thermodynamic parameters like temperature and chemical potentials can describe this thermalized medium. The matter and energy density dilute with the expansion and the temperature declines. As the energy density(temperature) drops below the hadronization threshold, the matter evolves as a state of hadrons and their resonances. The mean free path increases with further expansion and various collisions among particles abate. In this context, one can define freeze-out as the boundary, onwards which no interaction is supposed to happen among hadrons. In the standard description, two freeze-out surfaces are described, depending on the type of interaction. The Chemical Freeze-Out(CFO) happens when inelastic scattering stops and the particle abundances become fixed. The kinetic freeze-out is described by the point where elastic collisions cease. In this free non-interacting limit, the ideal hadron resonance gas * deeptak@jcbose.ac.in model may give a reasonable description of the hadrons at freeze-out.
Yields of strange hadrons help to understand the extent of chemical equilibrium achieved in these collisions. Being a massive quark, strange quarks are expected to equilibrate later than the u and d quarks [3]. Equilibrated strangeness spectra may be a suitable signature to understand the existence of a deconfined partonic phase [4]. In a recent result from LHC [5], a charged particle multiplicity (dN ch /dη) dependent production of strangeness has been observed. This strangeness production can be related to the centrality via its connection with the system size and centrality (N part ) of the collision system [6,7]. Here, N part denotes the number of participating nucleons of a particular collision system.
Following the success of the statistical picture in describing the yield of produced hadronic species in collisions of elementary particles and heavy-ions [8], studies have been performed to determine freeze-out parameters, considering the Hadron Resonance Gas (HRG) model. In this context, a χ 2 fitting with the available yield data is well-practiced [9][10][11][12][13][14][15][16][17][18][19]. Generally, one extracts the temperature T and the baryon chemical potential µ B by the minimization procedure, whereas the charge chemical potential µ Q and the strange chemical potential µ S are determined from the constraints of the colliding nuclei. To scale the possible non-equilibration of strange hadrons, a strangeness under saturation factor γ S can be introduced [13,[20][21][22][23][24][25][26]. This parameter scales the deviation of strange hadrons from a Grand Canonical Ensemble (GCE). In a recent work [27], we have shown that in χ 2 analysis, a larger systematic variation can arise depending on the chosen set of ratios. So a conserved-charge dependent extraction of thermal parameters has been proposed [28], which seems to suitably estimate thermal parameters and predict equilibration in the most central collision. It will be interesting to check the centrality variation of thermal parameters and the equilibration of strange particles in this framework.
In this manuscript, we have tried to study the centrality variation of freeze-out parameters, with an emphasis on the saturation of strangeness [5] in heavy nuclei collision. We have observed a similar saturation with centrality (N part ) for collision energies ranging from 7.7 GeV BES to LHC (2.76 TeV). We have employed a recently developed parameter extraction process [28], this relies on net conserved charges rather than the standard chisquare minimization with particle ratios. We have used a strangeness suppression factor γ S to measure the possible deviation of strange hadrons from respective equilibrium yield. We have found the kaons to deviate from equilibrium at chemical freeze-out for the peripheral and semi-peripheral collision, though the temperature does not change much with centrality. Further, we have studied the scaling behavior of all freeze-out parameters with N part . The parameters attain a saturation and γ S nearly becomes 1 onwards N part = 150. This flattening indicates that the system created in the heavy-ion collision reaches a grand canonical limit, in which thermodynamic description becomes independent of the system size. Finally, the efficacy of our parametrization has been verified by predicting various hadron yields ratio and comparing them with experimental data.
We have organized the manuscript as follows. A short description of the Hadron resonance gas model (HRG) is given in section II. In section III, we have briefly discussed the parameter extraction method in our approach. IV describes our results followed by discussion. We summarise our results in section V.

II. HADRON RESONANCE GAS MODEL
The hadron resonance gas (HRG) model describes the system as a mixture of hadrons and their resonances. It is a standard exercise to incorporate all available hadron yields for obtaining a good description of the medium. In recent years various studies have been performed using HRG model [11,12,14,. This model has successfully described hadron yields from AGS to LHC energies [11,12,32,33,[35][36][37]41]. Bulk properties of hadronic matter have also been studied in this model [14,39,40].
In the present work, we have considered the ideal HRG model in which hadrons are treated as point-like particles. A grand canonical ensemble can describe the partition function of hadron resonance gas as [14], The sum runs over all hadrons and resonances. In the idealistic scenario of a chemical freeze-out, we can neglect all dissipative interactions and finite volume corrections. The thermodynamic potential for i'th species is given as, where the upper sign is for baryons and lower for mesons.
Here V is the volume and T is the temperature of the system. For the i th species of hadron, g i , E i and m i are respectively the degeneracy factor, energy, and mass, while µ i = B i µ B + S i µ S + Q i µ Q is the chemical potential, with B i , S i and Q i denoting the baryon number, strangeness and the electric charge respectively. For a thermalized system, the number density n i can be calculated from partition function as,

III. APPLICATION TO FREEZE-OUT
We first outline the usual application of the HRG model for characterizing the freeze-out temperature and chemical potentials in the context of heavy-ion collision experiments. The rapidity density for i'th hadron may be related to the corresponding number density as [13], where the subscript Det denotes the detected hadrons. Here, where the summation is over the heavier resonances (j) that decay to the i th hadron. This number density n i is calculated using Eq.3. In this context, it is also important to consider the constraints regarding the conserved charges. Following the assumption of an isentropic evolution, one can employ conservation conditions like strangeness neutrality (Eq.6) and baryon density to charge density (Eq.7) to restrict the values of chemical potentials [16].
where r is net-charge to net-baryon number ratio of the colliding nuclei. For example, in Au + Au collisions r = N p /(N p + N n ) = 0.4, with N p and N n denoting the number of protons and neutrons in the collidng nuclei. In proton proton collision this ratio is 1.
The usual approach should be solving Eq.4 to extract thermal parameters. The freeze-out description will be more reasonable if we include data for a larger number of detected particles in our solving mechanism. That is why a χ 2 minimization is performed with all the available yields. One may avoid the volume systematics by taking ratios of two hadrons. Further, performing a minimization procedure with available yield ratios one should be able to parameterize the chemical freeze-out surface. In a recent work Ref. [27], we have shown that a larger systematic variation may arise in χ 2 analysis due to variation in the chosen set of ratios.

Our approach
Following the complicacy regarding the chosen set of ratios, we tried to come up with an alternative method in Ref. [28]. It is essential to note that individual hadrons are not a conserved quantity in the strong interaction, whereas net charges like B, Q, S are. So we opted to introduce ratios regarding net charges. Along with the constraints Eq. (6−7) we have proposed two new independent equations, the net baryon number normalized to the total baryon number and the net baryon number normalized to the total hadron yield, as given below.
We want to mention that, the left-hand side consists of the particle yields data from the heavy-ion collision and those on the right are the number densities calculated in the thermal HRG model. i runs only over detected(Det) hadrons with available experimental yield data. Finally, we solve Eq.[ 6−9] with desired accuracy.

Application to centrality
The geometric information of the collision system is important to understand as different final observables like eccentricity, elliptical flow, charge particle multiplicity are directly dependent on the initial conditions like impact parameter (b), the number of participating nucleons (N part ) [54]. We employ the centrality bins to differentiate collision events according to their impact parameters. As there is no direct method to measure b, the centrality bins can be calibrated from charge particle multiplicity with the Glauber model [6,54,55]. Each centrality bin is represented by a corresponding N part . Most central (0% − 5% centrality) collisions correspond to the events with the lowest value of impact parameters (Highest value of N part ) whereas, the most peripheral (70% − 80%) are with the largest impact parameter and smallest N part . The degree of equilibration of the created medium should strongly depend on centralities as the system's initial volume and initial energy, baryon deposition directly depend on these initial specifications.

Strangeness with centrality
Introducing a strangeness suppression factor for performing the analysis in the central collision of heavy nuclei is optional [27,28], whereas this appears essential when one deals with peripheral or semi-central collision. Complete chemical equilibrium may not be achieved in the strangeness sector due to the higher mass threshold of strange particles and their hadronic counterpart [3]. Initially, this suppression factor (γ S ) was introduced to be a factor considering the phase space undersaturation [21]. Ref.
[ [22,23]] has discussed this undersaturation as an effect of the canonical ensemble consideration of strangeness, where exact conservation of strangeness should be considered for a smaller collision system. In ref. [56] a core corona dependent model has also tried to discuss this suppression of strangeness. Irrespective of the reason for this undersaturation, considering this factor γ S , gives rise to a better agreement to the thermal description of heavy-ion data. It seems that the strange sector may have a deviation from the respective grand canonical description, and this parameter is a measure of that departure [13,[20][21][22][23][24][25][26]. In the presence of this factor, the number density is modified in the following manner [13], Here, n s i denotes the number of valence strange quarks or antiquarks in the i'th hadron. In this work, we have calculated number densities following 10. So, γ S = 1 for all non-strange particles. A smaller value of γ S denotes a larger deviation from the grand canonical limit and equilibrium.
As we have introduced one added parameter γ S , an extra equation is needed to close our system of equations. This parameter is not related to any conserved quantity, rather it is used to describe the possible non-equilibrium of the strange sector. Keeping in mind, that we have used only yields of kaons among the strange particles, we have utilized kaon to pion ratio to evaluate the value of γ S in Eq.11.

IV. RESULTS AND DISCUSSIONS
In this analysis we have used the yields of, π ± (139.57 MeV), k ± (493.68 MeV) and p,p (938.27 MeV) as mostly these data are available in all centrality bins for collision energies ranging from BES (7.7 GeV) to LHC (2.76 TeV). For convenience, we have represented the centrality bins by their corresponding number of participants (N part ). Data have been used following RHIC [6,[57][58][59][60][61][62][63][64][65][66][67][68][69][70][71], STAR BES [72] and LHC [73][74][75][76]. Data for p-p collision is available in RHIC for √ s = 200Gev. This is included in our analysis for completeness. In the present analysis, we have only taken mid-rapidity data. The details of the experimental yields used in the analysis are listed in the Ref. [6,72,76]. In our HRG spectrum, we have used all confirmed hadrons up to 2 GeV, with masses and branching ratios following the Particle Data Group [77] and THERMUS [78]. Finally, we solve Eq. (6−9, 11) numerically using Broyden's method with a minimum convergence criterion of 10 −6 . The variances of thermal parameters have been estimated by repeating the analysis at the given extremum value of hadrons yields.
A. Freeze-out Parameters We have described the variation of our extracted freeze-out parameters with number of participants(N part ) for various collision energies in Fig.[1]. In plots, the horizontal-axis is the number of participants. Results for collision energy LHC−2.76 TeV to BES−7.7 GeV have been shown in different columns in descending order from left to right. For the clarity of discussion, we shall discuss variation concerning √ s first and then try to understand the changes with centrality.
The variation of the freeze-out temperatures in Fig. [1a] has good agreement with general understanding [28]. The temperature increases with collision energy and near √ s = 39 GeV saturates around the value of 160 MeV as it reaches the Hagedorn limit [79]. At LHC, T CF O is lower than the expected value as the proton yield is lower than their preceding RHIC energies [76]. There is a similarity in the variation of T with N part for all √ s. It starts from a slightly higher value for the most peripheral collision which is denoted by the least number of participants(N part ) and then decreases with centrality. This increase of temperature toward peripheral bins has been explained with the presence of an afterburning stage [80]. This stage has to depend on the number of participants as that will increase the hadron multiplicity. The system is expected to spend a longer time in the hadronic stage for more central collisions, giving a larger shift from the hadronization temperature.
We have plotted the baryon chemical potential as a functions of N part for all √ s in Fig.[1b]. The general expectation is that at lower collision energies, a large number of nucleons would be deposited in the collision region due to stopping. But at very high collision energies, they may almost pass through each other due to a weaker strong force. So at higher RHIC and LHC energies, the medium is created having almost zero net baryon number. Therefore the net baryon density and hence the estimated chemical potential µ B would decrease with increasing √ s. In the same manner, one should expect a rise of µ B for higher N part . With the number of participants, the value of the deposited net baryon number increases due to baryon stopping among a large number of the participating nucleons. Contrarily, in a peripheral collision less number of nucleons get deposited in the collision zone, creating a dilute system of net baryon. This produces a smaller value of µ B . We have observed this trend in all √ s. There is a significant difference between µ B values of peripheral and central collisions in lower BES energy. As the energy increases, µ B approaches zero and the rate of change with N part decreases. This indicates that at higher √ s, energy becomes more effective than the number of participants due to increased transparency among colliding nucleons.
Strangeness chemical potential µ S shows a similar trend as µ B in Fig.[1c]. It decreases as collision energy increases and becomes zero at LHC energy. On the other hand, µ S escalates as one goes from peripheral to the central collision. This correlation between the behavior of µ S and µ B can be described in the following way. A higher baryon density demands hyperons to be produced more than anti-hyperons. To maintain the strange neutrality, this excess amount of strangeness from the baryonic sector has to be nullified from the mesonic sector. So in the mesonic sector, k + is more abundant than k − . Being the lightest strange particles, this difference between charged kaons determines the sign and trend of µ S .
The general trend of charge chemical potential µ Q is the same as other µs except for the sign. As N part increases, it becomes more negative and the magnitude decreases with √ s in Fig.[1d]. However, µ Q is more negative for large baryon densities. We can understand this as following. The neutrons are more abundant than protons in the colliding nuclei. This abundance generates a net negative isospin value in the collision system and produces more π − than π + , to conserve the isospin. As the lightest charged particle, these pions determine the negative µ Q . This reasoning will be more clear if we look into the value of µ Q for √ s = 200 GeV at N part = 2. In this case of p-p collision, the isospin dominance should not act in favor of π − , as there is no neutron in the colliding particles. So one should expect the µ Q to be positive for this case. Indeed we have observed a positive value of µ Q for the p-p collision of 200 GeV RHIC energy. The net value of isospin increases with N part , thus increases asymmetry between the yield of charged pions. So the magnitude of µ Q rises following the µ B .
If strange particles achieve chemical equilibrium, then the thermal abundance of kaons should be described by equilibrium thermal parameters (T, µs) of a grand canonical ensemble. This is not observed in cases of a small collision system like p-p, p-A and even in A-A with a smaller N part [13]. Several models [22-24, 81, 82] have tried to describe this source of strangeness undersaturation in the smaller system and advocated the use of γ S . The common perception from all this work is that γ S scales the deviation of strange particles from their respective equilibrium thermal yield of a grand canonical    ensemble, while γ S = 1 denotes the equilibration in the strange sector.
We have shown the variation of γ S in Fig.[1e]. It is interesting to notice that even in LHC and high RHIC energies, the γ S has an increasing trend from lower peripheral to a central collision, though the temperature and other chemical potentials do not change much. Initially, it starts from a lower value in the case of peripheral collisions and increases with participants. Around N part = 150, γ S tends to saturate near 1. It appears that the strangeness tends to attain its grand canonical limit as the system size increases. The saturation of γ S with the colliding system size for central A-A collisions suggests that the strangeness suppression may be independent of the hadronic scatterings, which happens in the later time of the evolution [81]. From the pattern, one can also conclude that the strangeness equilibration has a sharp dependence on the number of participants and system volume. The difference between the values between peripheral and central is larger for lower BES energies. The general understanding from the above study is that the strangeness sector may be closer to equilibrium in a peripheral collision of higher √ s whereas the deviation from equilibrium is larger for peripheral cases in lower collision energy. We have found that strangeness equilibration (γ S = 1) happens in collisions with a N part more than 150. This may be the threshold N part for the creation of a deconfined phase in a A-A collision, which drives the strangeness towards equilibrium [3]. Further studies with other heavy ions and data of hyperons may help to understand this.

B. Scaling Nature of CFO parameters
The scaling behavior with N part is important to calibrate the chemical composition at freeze-out with system size. To simplify the discussion, we have normalized the obtained parameters by their corresponding value for the most central collision of individual √ s. As an example, to understand the scaling of temperature at 200 GeV, we have divided the extracted T of each centrality bin (N part ) with that of the most central collision (maximum N part ). This variation has been shown in Fig.[2] for all five freezeout parameters. For simplicity, we have plotted for two collision energies from both RHIC (200, 62.4) and BES (27, 11.5). If parameters do not vary much with centrality (N part ), then this scaled quantity should lie near around 1. The scaled freeze-out temperature shows this pattern for all collision energies. It seems that for given incident energy the freeze-out temperature does not vary much with the system size, whereas the scaled baryon chemical potential (µ B ) has an increasing trend as it becomes maximum at most central collisions. In the case of equilibrium among all the charges, all the chemical potentials have to commensurate with each other. Scaled µ Q and µ S should follow the pattern of µ B with both the number of participants and collision energy, which we have already discussed in section [IV A]. We have indeed observed a similar trend for all three µs. There is a trend of saturation near 1 around N part = 150. Onward this point, the system may achieve a thermodynamical state which is independent of the system size. Further analysis with other colliding ions in these √ s may shed light on this issue. Nontriviality could have appeared in the case of γ S as it is a non-equilibrium parameter. But the observed trend is similar to the chemical potentials. It starts from a smaller magnitude and saturates near 1 from N part = 150. The system may have enough energy and number density for strangeness equilibration onward this centrality bin [83] and we can employ a complete grand canonical description to describe the yield at freeze-out. The deviation of scaled γ S from 1 is larger for peripheral collisions in lower BES energy, which indicates that colliding energy has a crucial contribution to decide the equilibration of strangeness.

C. Particle Yield Ratios
In this section, we shall discuss ratios regarding detected particles to check the efficiency of our parameterization. We have predicted particle ratios from our extracted freeze-out parameters and plotted alongside their experimental values. Variances in the experimental yield ratios have been obtained using the standard error propagation method [84], considering both the systematic and statistical errors of data. Whereas, we have calculated the variance of predicted ratios by evaluating them at the extremums of the obtained freeze-out parameters.
In Fig.[3a] we have plotted the variation of the particle to anti-particle ratios for pions and kaons. There is good agreement between model prediction and experimental data for both the ratios. No notable variation has been observed for π − /π + with N part and √ s. The abundance of pions is determined by the temperature and µ Q . For the ratio of negatively charged to positively charged pions, the variation should depend on µ Q only. Here we want to mention that there is no prominent variation of µ Q /T with respect to √ s and N part , as the value of µ Q is much smaller(around .5 M eV ) than the value of T (about 150 M eV ). On the other hand, µ Q is much lower than the mass of pion itself. So it does not differentiate between the thermal yield of π − and π + and the ratio lies near unity for all collisional energies and centrality classes. As we have discussed earlier, the asymmetry between k − and k + should depend on net baryon density due to the contribution of net strangeness from the hyperon sector. So one should expect a larger yield of k + than k − at higher baryon density(µ B ). We have observed this pattern with both centrality and √ s. At lower BES energies the ratio is far from unity due to higher net baryon density. This approaches 1 as the collision energy increases. At LHC energy, the yields of particle and anti-particle become equal as no net baryon is deposited in the collision region. With N part , a commensurating trend has
The charge independent k/π ratios k + /π + , k − /π − are important observables for understanding the strangeness production in high energy collisions. As the lightest mass hadrons, pions may act as the proxy of entropy whereas the kaons carry the signature of strangeness. Being produced at the later stage of evolution, strange particles are important in studying chemical equilibrium in heavy-ion collisions [3]. A charged particle multiplicity (dN ch /dη) dependent saturation of strangeness normalized to pions has already been observed in LHC [5], which can be utilized to investigate the system size dependent strangeness production. Ref. [83] has related this saturation to equilibration with a threshold dN ch /dη. In heavy nuclei collisions, the overlap region and dN ch /dη both are related to N part [6]. We have observed the same saturation trend with N part here for all √ s in Fig.[3b] and have suitably reproduced it with our parametrization. This saturation starts around N part = 150 in higher RHIC and LHC.
Here we want to mention that, there is no variation of π − /π + with centrality and collision energy, but k − /k + has a strict dependence on both. At lower collision energy, k + /π + is much higher than k − /π − due to excess yield of k + . The difference between both the ratio decreases with increasing collision energies and they become equal at LHC, as the particle-antiparticle yields become the same. The pattern of γ S has a close resemblance to both the ratios. It seems that as N part decreases, the kaon yields deviate far from their equilibrium yield. So a non-equilibrium parameter γ S had to be introduced in our thermal model. Lower the value of γ S , larger is the deviation from equilibrium for kaons.
From the discussion of freeze-out parameters, it appears that the variation of anti-proton(anti baryon) to proton(baryon) is a guideline to understand the variation of µ B . This ratio becomes 1 at upper RHIC and LHC energies as the colliding nuclei pass through each other and the hadrons are created out of a medium having zero net baryon density. In lower collision energy, baryon stopping motivates a larger net baryon density. As a result, protons are more abundant than anti-proton and advocates a smallerp/p at lower collision energy. Our thermal model prediction has good agreement with the experimental data in Fig.[4a]. One can notice that in peripheral collisions, this ratio tends to increase, which means the dominance of baryons over anti-baryons decreases. Initial net baryon number density decreases as one goes from central to peripheral collision due to nuclear distribution [85] of the colliding nuclei. This induces a smaller baryon anti-baryon asymmetry in their yield.
We have discussed proton to positively charged pion ratio in Fig. [4b]. As we have already discussed, pions may act as the measure of entropy. So the ratio p/π + will describe the variation of baryon production with entropy. If the particles are produced only from deposited energy, then pions will be highly abundant than massive protons. But if the medium starts to evolve from a finite baryon density, then per pion proton production will be larger to conserve the net baryon density. That is why a clear increasing trend for p/π + has been observed as the collision energy decreases. This same increment is expected with N part as more baryon deposition happens in the more central collisions. This variation is prominent in lower √ s, due to the higher efficacy of baryon stopping.

V. SUMMARY AND CONCLUSION
The equilibration of the system created in a heavyion collision should strongly depend on the system size and number of participants. A comparison among the chemical freeze-out conditions of high and low multiplicity A-A collisions may shed light in that direction. In this study, we have tried to explore the freeze-out parameters of various centrality bins of A-A collision for LHC (2.76 TeV), RHIC (200 GeV, 130 GeV, 62.4 GeV) and BES (39 GeV, 27 GeV, 19.6 GeV, 11.5 GeV, and 7.7 GeV).
We have applied a conserved charge-dependent freeze-out parameterization procedure to various extract temperature and chemical potentials. We have incorporated a strangeness suppression factor (γ S ) to estimate the possible non-equilibrium in strange hadrons in the peripheral collisions. We have discussed the variation of these chemical freeze-out parameters with both centrality and collision energy.
The variation of parameters with collision energies has good agreement with general understanding, whereas there are significant variations with the number of participants. The extracted freeze-out temperature does not have a strong dependence on the number of participants (centrality), whereas the chemical potentials show a wide variation. To have a better understanding of variation with N part , we have also presented the behavior of scaled parameters. These parameters have been normalized with the value obtained in most central collisions and compared along with the other collision energies. Scaled µ Q and µ S appear to follow the scaling behavior of µ B , which may be a signal of equilibration among three conserved charges. The strangeness suppression factor (γ S ) deviates from the equilibrium value at peripheral collisions and tends to saturate near unity at central collisions. In the peripheral collisions, γ S starts around .7 and increases towards central. This means, the strange hadrons are deviated from equilibrium at low multiplicity collision, whereas there is a sign of equilibration as the centrality increases. We have also inferred that the flattening of the scaled parameter and γ S appears around a threshold of N part = 150. So we can apply a grand canonical description for the systems created out of A-A collisions with more participants than 150. Analysis with other colliding ions and yields data of other strange hyperons may help to understand this further. Further, we have estimated different particle ratios to cross-check the effectiveness of our parameterization. Our predicted ratios seem to have good agreement with experimental data. We have only reproduced ratios regarding pions, kaons, and protons as they are used in our analysis. A saturating trend with N part has been observed for kaon to pion ratio also, which can be well explained with the help of γ S .
To summarize, we have studied the centrality variation of chemical freeze-out parameters. We have used the number of participants (N part ) dependent calibration of the centrality bins. Instead of the χ 2 minimization, a conserved charge-dependent parametrization process has been followed utilizing the rapidity spectra of the pion, kaon, proton at LHC, RHIC and BES. The extracted freeze-out parameters, along with a strangeness suppression factor, appear to suitably reproduced the data of various yield ratios. All three chemical potentials appear to have good agreement among each other in a central collision, which asserts the essence of equilibration in the high multiplicity heavy-ion collision. The strange sector deviates from the respective equilibrium value in a peripheral collision. Our results also show that this  √ s. Experimental data (Red) are from [6,72,76]. Thermal prediction (Black) are estimated from freeze-out parametrization. departure from equilibrium decreases and strange equilibration may happen after a particular centrality in A-A collision. This result can be a guideline for applying the grand canonical ensemble in explaining heavy-ion collision data for various collision system.