Insights into Freezeout Dynamics in Symmetric Heavy Ion Collisions with Changing Event Centrality

Abstract

This study presents the investigation of freezeout parameters, namely the kinetic freezeout temperature (T) and transverse flow velocity (${\beta }_T$), in different centrality intervals with fixed as well as with variable flow profile ($n_0$) in the blast-wave model (using Boltzmann Gibbs statistics). The model is used to fit the experimental data of transverse momentum spectra of ${\pi }^{+}$, $K^{+}$, and p in $Au$–$Au$ and $Pb$–$Pb$ collisions at 200 GeV and 2.76 TeV, respectively. In our observation, when the parameter $n_0$ is considered as a free parameter, the parameter T decreases from head-on to peripheral collisions, while it increases towards the periphery if $n_0$ is fixed. In addition, parameter ${\beta }_T$ decreases from central to peripheral collisions in both cases. These findings provide valuable insights into the dynamics of quark-gluon plasma formation and expansion in high-energy nuclear collisions. Moreover, the kinetic freezeout temperature T and the transverse flow velocity ${\beta }_T$ are mass-dependent; while the former becomes larger for massive particles, the latter becomes larger for light particles, showing the mass differential kinetic freezeout scenario.

1. Introduction

Identifying the novel realms of strongly interacting matter and analyzing the Quark Gluon Plasma (QGP) [1,2,3,4], which is intuitively predicted by Quantum Chromodynamics (QCD) [5,6,7,8,9,10], are two of the most recent advances in relativistic heavy ion collisions. Under extreme circumstances of high temperatures and/or high net baryon densities, the QGP is considered to be a state of strongly interacting matter. When choosing higher energies and selecting different masses of the colliding nuclei, the QGP can now be produced in lab settings during high-energy nucleus–nucleus collisions, such as Au-Au, Cu-Cu, Pb-Pb, and Xe-Xe collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC), respectively.

The creation of the Quark Gluon Plasma (QGP) requires a very high temperature and/or density. The temperature is a crucial term in thermal and subatomic particles [11], as it is widely used in both theoretical and experimental research. A variety of temperatures at different points in the collision can be found in the literature. When collisions first occur, the initial temperature represents the level of excitation of the system. At the stage of stopping of the inelastic scattering and degradation of particle identities, the chemical freezeout temperature defines the level of excitation at the chemical freezeout stage. This is directly followed up with the kinetic freezeout temperature (T), which represents the level of excitation at the kinetic freezeout stage [12,13]. The primary goal of this study is to check the behavior of the kinetic freezeout temperature (T); therefore, other temperatures are not discussed here. However, for further information, see [14,15].

Even though the kinetic freezeout temperature has been extensively studied in the literature, its behavior is still questionable in a number of ways. For instance, some studies assume that the particles should have the same kinetic freezeout temperature [16], while some other studies assert that the time at which different particles freeze out varies based on their masses [17,18,19]. Other studies show the freezeout temperature is affected by the cross-sectional interaction of the particles [13,20], or how the kinetic freezeout temperature behaves with centrality [21,22,23,24,25]. Furthermore, details regarding the kinetic freezeout temperature can be found using a variety of theoretical models. Therefore, the traditional blast-wave model with both a fixed and variable flow profile is employed in the current study. The kinetic freezeout parameters have an impact on the flow profile. Therefore, using both the fixed and variable flow profiles, we try to explore the behavior of (T) and (${\beta }_T$) with shifting centrality intervals.

In the current study, the blast-wave model with Boltzmann–Gibbs statistics was utilized to extract the parameters T, ${\beta }_T$ from the transverse momentum ($p_T$) spectra of protons, kaons, and pions. While the results and their interpretation are covered in Section 3, the technique and theoretical framework are described in detail in Section 2. These sections form the core of the work, providing a comprehensive analysis of the data and the application of the model. The key findings, along with their implications and significance, are summarized in Section 4.

2. The Method and Formalism

Experimental collaborations frequently employ the phenomenological model known as blast-wave [26] to fit the hadron and light (anti-)nuclei spectra [27,28]. The model is based on the assumption that particles are emitted thermally for an expanding source with variables: temperature (T) and velocity profile (${\beta }_T$ [26,29]). When employing Boltzmann–Gibbs statistics in the blast-wave model, the probability density function of $p_T$ distributions yields

$$\begin{array}{cc}\hfill f_1\left(p_T\right)=& {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}=C{p}_T{m}_T{\int }_0^{R}rdr\hfill \\ \hfill & \times I_0\left[{\displaystyle \frac{p_{T}sinh\left(\rho \right)}{T}}\right]K_1\left[{\displaystyle \frac{m_{T}cosh\left(\rho \right)}{T}}\right],\hfill \end{array}$$

(1)

The equation for $m_T=\sqrt{p_T^2+m_0^2}$ can be used to denote the transverse mass, $m_T$. T reflects the internal thermal source temperature from where the particles radiate. Considering boost invariance for the system’s longitudinal expansion, the heat source’s velocity profile is described by the following equation:

$$\begin{array}{c}\hfill \rho ={tanh}^{-1}\left[\beta \left(r\right)\right]={tanh}^{-1}{\beta }_S{(r/R)}^{n_0},\end{array}$$

(2)

where $\beta \left(r\right)$ is the transverse expansion velocity at 0 $\le r$ $\le R$ and r represents the radial distance from the fireball’s center. When 0 $\le r\le R$, the transverse expansion velocity is $\beta \left(r\right)$. ${\beta }_S$ describes the transverse expansion velocity at the surface and $n_0$ is the velocity profile’s exponent. Moreover, the average radial flow velocity is given by $\langle {\beta }_T\rangle$ = $2{\beta }_S/(n_0+2)$ [30].

In this study, we address the soft excitation process employing the blast-wave fit with Boltzmann–Gibb’s statistics (BGBW). The two-component fit, in which the second component describes the hard scattering process, can be used in situations where the fit in the high $p_T$ area is not good. The second component can be represented by the inverse power law [31,32,33] or the Hagedorn function [34], which is

$$f_2\left(p_T\right)={\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}=A{p}_T{\left(1+{\displaystyle \frac{p_T}{p_0}}\right)}^{-n}.$$

(3)

A modification of the Hagedorn function is applied in [35,36,37,38,39,40] as follows:

$$\begin{array}{ccc}\hfill f\left(p_T\right)& =& {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}={\displaystyle \frac{A{p}_T^2}{m_T}}{\left(1+{\displaystyle \frac{p_T}{p_0}}\right)}^{-n}\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}=A{p}_T{\left(1+{\displaystyle \frac{p_T^2}{p_0^2}}\right)}^{-n}\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}=A{\left(1+{\displaystyle \frac{p_T^2}{p_0^2}}\right)}^{-n},\hfill \end{array}$$

(6)

whereas in Equations (4)–(6), $p_0$ and n are the free parameters, and they are sequentially different from one another. Since analyzing their differences in detail is outside the scope of this study, we refrain from doing so. The superposition of hard scattering and soft excitation processes can be used to explain the broad $p_T$ range. If Equation (1) describes the soft process, Equations (4)–(6) indicate the hard scattering process, and the superposition of the two components can describe the broad $p_T$ range.

$$f\left(p_T\right)={\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}=k{f}_1\left(p_T\right)+(1-k)f_2\left(p_T\right),$$

(7)

In the equation above, k signifies the soft process’s contribution fraction to the hard process, while $f_1$ and $f_2$ stand for the corresponding soft and hard processes, respectively. The hard scattering contributes to the entire $p_T$ region, whereas the soft region is the $p_T$ region from 0∼2 or 3 GeV/c. It is important to highlight that while there is the superposition of soft and hard components, the former is the main component of the particle production, and the contribution of the latter is negligible.

3. Results and Discussion

The $p_T$ spectra of identified particles in different centrality intervals in $Au$–$Au$ and $Pb$–$Pb$ collisions at 200 GeV and 2.76 TeV, respectively, are shown in Figure 1. The symbols respresent the experimental data, while the curves over them are for the fit results of BGBW model considering $n_0$ as a fixed parameter as well as a free parameter. The solid curves show the results of the fit when $n_0$ is a fixed parameter, and the dashed curves show the fit results when $n_0$ is a free parameter. The upper panels in Figure 1 show the $p_T$ spectra of the particles created in collisions between $Au$ and $Au$, while the lower panels demonstrate the $p_T$ spectra of the particles from $Pb$–$Pb$ collisions.

The result of T with respect to centrality is shown in Figure 2. The upper two panels show the results obtained from $Au$–$Au$ collisions, while the lower ones present the behavior of T with fluctuating centrality in $Pb$–$Pb$ collisions. The trend of the symbols from left to right shows the variation of T with centrality, and various symbols with various colors stand for distinct particles. In addition, the panels on the left side show the results of T with centrality when $n_0$ is one, whereas the panels on the right side present the behavior of T with centrality when $n_0$ is regarded as a free parameter. One can see that T is monotonously increasing from central to peripheral collisions when $n_0$ is fixed, which may express a decrease in fireball lifetimes from central to peripheral collisions, which is also consistent with [21,22]. This could be explained as the system producing a longer-lived and more extended QGP droplet in central collisions, where the nuclear overlap is maximum. The longer lifespan offers more thorough cooling and hydrodynamic expansion before freezeout. As the system expands and cools over an extended period of time, particles decouple at a later stage when the temperature has dropped considerably, resulting in a lower T. Furthermore, the increased interaction period allows for more extensive hadronic rescattering, which helps in cooling the system prior to freezeout once more. The fireball, on the other hand, has a shorter lifespan in peripheral collisions with smaller overlap zones. When the temperature is still quite high early in the evolution, the particles experience freezeout, allowing the system to disintegrate faster. Larger T values in these collisions are observed as a result of the faster expansion and cooling times. Hydrodynamic models predict this tendency consistently, with longer evolution times causing systems to naturally reach lower temperatures before decoupling. In addition, it is noticed that larger T is associated with heavier particles, revealing the mass differential freezeout scenario. This result is consistent with [41,42]. Heavier hadrons decouple from the system sooner than lighter ones because of their smaller elastic scattering cross-sections in the hadronic medium. For example, after hadronization, protons encounter fewer final-state rescatterings and freezeout at an earlier, hotter stage of the system’s evolution. Nevertheless, lighter particles, such as pions, have a smaller extracted T because they keep interacting and thermalizing until the system has cooled down even more. We also notice that T in $Pb$–$Pb$ collisions is larger than in $Au$–$Au$ collisions, which shows a correlation with the size of the system. Furthermore, we notice that T is monotonously increasing from peripheral to central collisions when $n_0$ is treated as a free parameter. Such behavior can be interpreted when the central collisions experience a violent reaction, where the energy deposition is larger and the degree of excitation function of the system rises, which naturally raises the temperature. This is consistent with [25,43]. The difference between the two distinct trends of T with centrality indicates how sensitive the parameter $n_0$ is to T. It is important to mention that the correlation of T is a known and open issue in the community till today, but the observed sensitivity of $n_0$ to the parameter T in the present work is a very significant achievement. In addition, the parameter T is larger in $Pb$–$Pb$ collisions and for heavier particles.

Figure 1. Distributions of the transverse momentum of different light flavoured identified particles in Au-Au and Pb-Pb collisions at $\sqrt{s_{NN}}=200$ GeV and $\sqrt{s_{NN}}=2.76$ TeV, respectively. Experimental data are obtained from the PHENIX and ALICE collaboration [44,45].

Figure 1. Distributions of the transverse momentum of different light flavoured identified particles in Au-Au and Pb-Pb collisions at $\sqrt{s_{NN}}=200$ GeV and $\sqrt{s_{NN}}=2.76$ TeV, respectively. Experimental data are obtained from the PHENIX and ALICE collaboration [44,45].

Results of ${\beta }_T$ are presented in Figure 3, in the same way as in Figure 2. We perceived that ${\beta }_T$ consistently decreases from central to peripheral collisions. The cause of this behavior is that in central collisions, more participants are involved in interaction, depositing a considerable amount into the system, which results in a larger pressure gradient. Due to the large pressure gradient in central collisions, the system expands quickly, and it corresponds to larger ${\beta }_T$ which indicates a strong collective flow in such collisions. As the collision becomes peripheral, the pressure gradient becomes lesser due to lesser energy deposition in the system, and the parameter ${\beta }_T$ becomes lesser as well. The trend of ${\beta }_T$ is the same in both cases when $n_0$ is a fixed as well as a free parameter. Similar to T, ${\beta }_T$ is also mass-dependent; however, it shows an inverse relation to that of T. The heavier the particle, the smaller the parameter ${\beta }_T$. The heavier particles have greater inertia, which resists their motion, and it is consistent with the hydrodynamical behavior [46]. The parameter ${\beta }_T$ also tends to increase for a large system ($Pb$–$Pb$).

From the above observation, it is obvious that the dependence of $T_0$ and ${\beta }_T$ is negative for a fixed $n_0$, while this correlation is positive when $n_0$ is considered a free parameter. Both the negative and positive correlations between T and ${\beta }_T$ are correct and have their own explanations. The positive correlation of T and ${\beta }_T$ can be explained as the result of a higher degree of excitation, which corresponds to higher temperature and quick expansion. On the other hand, the negative correlation between them indicates a short lived fireball in peripheral collisions and less developed transverse flow due to weaker collective expansion. In the observation of the current results, we can say that the results of T is dependent on the parameter $n_0$.

It is important to highlight that if $n_0$ is set to one, the model cannot be adjusted to change the initial normalization of particle production; therefore, changes in the freezeout temperature and flow velocity are the only ways to account for all centrality dependency. In this limited situation, T bears the entire weight of centrality-based particle spectrum change description, resulting in a specific trend that solely captures the thermal freezeout dynamics. The model can redistribute the centrality dependence between the initial-state particle production (encoded in $n_0$) and the freezeout conditions (specified by T) when $n_0$ is viewed as a free parameter. Here, some of the variance that would otherwise be imposed on T can be absorbed by $n_0$.

It is crucial to point out that the study of T and ${\beta }_T$ is connected to the Equation of State ($EoS$). The dynamics of the expanding fireball and $EoS$ can be inferred from the rising pattern of T (decreasing ${\beta }_T$) from center to periphery collisions. According to this sequence, central collisions produce stronger radial flow, which effectively transforms thermal energy into collective motion and lowers the system’s freezeout temperature. The enhanced expansion corresponds with a stiffer $EoS$, where more active hydrodynamic development is driven by a sharp increase in pressure with energy density. Peripheral collisions, which have less cooling and weaker flow, hence show a greater T, indicating their diminished collective effects.

On the other hand, it signifies that the $EoS$ is comparatively soft if T and ${\beta }_T$ drops from central to peripheral collisions. In this particular case, the system paradoxically maintains a larger thermal energy (higher T) at freezeout even if central collisions actually produce larger ${\beta }_T$ than peripheral ones. This suggests that the system is not as effectively cooled by the expansion process as we would anticipate in typical hydrodynamic situations. The primary finding here is that, in contrast to the case of a stiff $EoS$, the link between flow and cooling efficiency is acting differently.

Despite flow, there are two potential physical explanations for the maintenance of increased thermal energy. A less effective conversion of thermal energy into collective motion may occur from the EoS softening, which causes the pressure to rise less rapidly with energy density. The second possibility is that a phase transition or critical phenomenon in the QCD phase diagram could interfere with the hydrodynamic evolution.

This softer $EoS$ behavior could be found in specific regions of the QCD phase diagram, particularly at a hypothetical critical point where the equation of state becomes extremely sensitive to changes in energy density or when the system switches between quark and hadron degrees of freedom. The decreased cooling efficiency demonstrates that more of the system’s energy exists in thermal form during freezeout, even though collective motion still happens but does not dissipate thermal energy as efficiently as it would in the case of a rigid $EoS$.

The study of T in the two cases of constraint and non-constraint $n_0$ in the present work is very important. The results obtained are useful for objectives beyond improving our knowledge of QGP characteristics. By examining initial state effects and final-state interactions, for example, they can shed light on how variations in initial energy density correspond to quantifiable flow patterns. By investigating if comparable freezeout patterns appear when collective effects are weaker, the results could also help close the gap between heavy-ion collisions and smaller systems, such as proton–proton or proton–nucleus collisions. Furthermore, by evaluating how well the models reproduce the actual T trends under various flow circumstances, this behavior can be utilized to improve transport models, specifically hybrid techniques that combine hydrodynamics with hadronic cascades. Finally, the centrality dependency of the kinetic freezeout temperature could provide indirect evidence for critical phenomena, which would help in the quest for the QCD critical point if it is sensitive to the QCD phase transition. For instance, in heavy-ion collisions, centrality reflects the overlap area of colliding nuclei and relates to the system’s initial energy density. If a non-monotonic trend emerges in T versus centrality (or energy), such as a sudden dip, peak, or saturation at centralities (or energy) [19,41,47,48], it might signal the system passing through a first-order phase transition or the critical point during expansion. At the critical point, the system would exhibit enhanced fluctuations and slower dynamics due to critical slowing down, potentially altering the freezeout conditions. This could manifest as follows:

(1)

Anomalous centrality dependence of T (deviating from smooth participant scaling);

(2)

Correlated fluctuations in other observables (e.g., particle ratios, flow patterns);

(3)

Non-trivial system-size dependence.

All in all, these findings improve phenomenological models, broaden our knowledge of QGP evolution, and govern future experimental investigations at the LHC, RHIC, and planned experiments at FAIR and NICA.

4. Conclusions

The main observations and conclusions are summarized here.

The analysis of transverse momentum ($p_T$) spectra of identified particles (${\pi }^{+}$, $K^{+}$ and p) produced in both collisions of $Au$–$Au$ and $Pb$–$Pb$ at energies of 200 GeV and 2.76 TeV, respectively, is performed by the blast wave model with Boltzmann–Gibbs statistics. The freezeout parameters (T and ${\beta }_T$) are extracted, which provides key insights into heavy-ion collision dynamics. We investigated the behavior of the freezeout parameters when $n_0$ is fixed as well as a free parameter.

The kinetic freezeout temperature (T) shows a decreasing trend from peripheral to central collisions, representing a short fireball’s life time in peripheral collisions when $n_0$ is fixed, whereas it is decreasing from central to peripheral collisions when $n_0$ is treated as a free parameter, reflecting higher energy deposition and a stronger degree of excitation in central collisions. This trend is consistent across both systems, with T being larger in $Pb$–$Pb$ collisions due to a larger size of the system.

Conversely, the parameter ${\beta }_T$ decreases from central to peripheral collisions, attributed to larger pressure gradients and stronger collective flow in central collisions. ${\beta }_T$ exhibits inverse mass dependence, with heavier particles having smaller ${\beta }_T$ values due to their greater inertia, aligning with hydrodynamic behavior. ${\beta }_T$ is also noticed to be larger in $Pb$–$Pb$ collisions compared to $Au$–$Au$ collisions.