Study of Thermodynamic Properties of ${\mathit{K}}_{\mathit{s}}^{\mathbf{0}}$, $\Lambda$, ${\Xi }^{-}$, and $\mathit{d}\mathbf{/}\stackrel{\mathbf{\_}}{\mathit{d}}$ Produced in Symmetric Proton–Proton Collisions at $\sqrt{{\mathit{S}}_{\mathit{N}\mathit{N}}}$ = 0.9 TeV and 7 TeV

Abstract

We study the thermodynamic properties produced in symmetric p–p collisions at $\sqrt{s_{NN}}=0.9\mathrm{TeV}$ and $7\mathrm{TeV}$, based on experimental data by the ALICE collaboration at CERN. Particularly, we analyze the initial temperature $T_i$, effective temperature T, freeze-out temperature $T_0$, chemical potential $\mu$, mean transverse momentum $⟨p_T⟩$, freeze-out volume V, and transverse flow velocity ${\beta }_T$ of different hadrons such as $K_S^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$. To effectively use the transverse momentum $p_T$ distributions of these hadrons, and to extract the thermodynamic parameters, the Single-Slope Standard Distribution with and without the chemical potential $\mu$, the Double-Slope Standard Distribution, and the modified Standard Distribution Functions are applied separately to fit the experimental data. The Modified Standard Distribution Function provides the most accurate description of the ALICE experimental data as compared to the Single-Slope (with and without $\mu$) and Double-Slope Standard Distribution Function. We have investigated the correlation between the extracted thermodynamic parameters and the measurements of mass and energy of particles of the collision, and we observed that the increase in $\sqrt{s_{NN}}$ is positively correlated with $T_i$, T, $T_0$, $⟨p_T⟩$, V, and negatively correlated with $\mu$. The comparison of p–p collisions with heavy-ion collisions (Au–Au collisions) suggests the possibility of collective-like dynamics even in small systems, which supports the hypothesis of thermalization and partial de-confinement in high-energy p–p collisions, indicating a transition towards a quark-gluon plasma (QGP)-like medium.

1. Introduction

Understanding the nature of strongly interacting matter under extreme conditions is a central goal in high-energy particle physics. Heavy-ion collisions at the Relativistic Heavy-Ion Collider (RHIC) and Large Hadron Collider (LHC) have long been used to create and study the quark-gluon plasma (QGP), a deconfined state of quarks and gluons. However, recent findings suggest that even in small systems such as proton–proton symmetric collisions (p–p) and p–$Pb$ collisions at high energies, signatures typically associated with QGP such as radial flow, collective behavior, and thermalization, have been discussed in the literature [1,2,3,4,5].

High-energy symmetric p–p collisions provide a vital framework for exploring the fundamental aspects of Quantum Chromodynamics (QCD) and the theory describing the strong interaction among quarks and gluons. At the extreme energies achieved at the Large Hadron Collider (LHC), such as $\sqrt{\left(s_{NN}\right)}$ = 0.9 TeV and 7 TeV, it becomes possible to recreate, on a microscopic scale, the conditions that existed microseconds after the Big Bang, enabling the study of strongly interacting matter at high temperatures and densities. During the formation of QGP (similar to thermalized media), the system experiences rapid expansion driven by strong pressure gradients in the collision region. This expansion leads to cooling and eventual hadronization. From collisions to hadronization, there are different temperature stages. At the initial stage, when the nuclei collide and the QGP begins to form, the temperature of the system is known as the initial temperature $T_i$, which is considered the greatest temperature, as it evolves and reaches the chemical freeze-out stage, where inelastic collisions cease. The temperature at this stage is called the chemical freeze-out temperature $T_{ch}$ [6,7,8]. At last, the system reaches the thermal or kinetic freeze-out stage; at this stage, the temperature of the system is labeled as the thermal or kinetic freeze-out temperature $T_0$ [9,10,11]. At the stage of $T_0$, interactions become so sparse that even elastic collisions are no longer significant. Additionally, there is another temperature at these stages of temperature called the effective temperature T, which accounts for both thermal motion and collective flow within the system, written mathematically as $T=T_0+m_0{\beta }_T$ [12], where $T_0$ is the freeze-out temperature, $m_0$ is the mass of the observed particle (hadron), and ${\beta }_T$ is the transverse flow velocity. By the evolutionary stages of the system, the hierarchy of these temperatures is found in the literature as $Ti\ge T_{ch}\ge T\ge T_0$ [13,14].

Although heavy-ion collisions ($Au-Au$ collisions) have traditionally been used to investigate the formation of a deconfined Quark Gluon Plasma (QGP) [15,16], recent findings from high-multiplicity p–p and p–$Pb$ collisions suggest that small systems can also exhibit collective behavior and partial thermalization at high energies [17,18]. Such observations have motivated renewed interest in studying the thermodynamic properties of hadronic matter produced in p–p collisions. Strange hadrons, such as $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and light (anti) nuclei, including $d/\overline{d}$, are of particular interest because their production yields and transverse momentum $p_T$ distributions provide sensitive probes of the medium’s thermal and dynamical evolution [5,19]. The analysis of these distributions allows the extraction of key thermodynamic quantities, including the initial temperature $T_i$, effective temperature T, kinetic freeze-out temperature $T_0$, chemical potential $\mu$, mean transverse momentum $<p_T>$, freeze-out volume V, and transverse flow velocity ${\beta }_T$. These observables collectively characterize the degree of thermalization, collective expansion, and freeze-out dynamics of the system [6,16,18].

This research is the continuation of our research papers [20,21,22,23] in which we have studied the centrality dependence and transverse momentum spectra of proton, deuteron, and triton temperatures in $Au-Au$ and $Xe-Xe$ collisions at 200 GeV and 54.4 GeV [20,21,22,23]. In this work, we perform a comprehensive study of these thermodynamic quantities for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{\left(s_{NN}\right)}$ = 0.9 TeV and 7 TeV, using statistical approaches. We introduce some new methods where the Standard Distribution Function (SDF) is applied for the first time as a Single-Slope SDF with and without chemical potential $\mu$ form, a Double-Slope SDF form, and then a modified SDF form to the experimental transverse momentum distributions. Additionally, we check the evolution of the fit of the results. Moreover, to study system size effects on the particles’ production, we present a comparison of some results with the data of heavy-ion ($Au-Au$) collisions [24]. Such analyses can provide insights about collective flow and thermal-like behavior, even in small collision systems, bridging the phenomenological gap between elementary (p–p) and heavy-ion collisions, and shedding light on the onset of QGP-like features in high-energy p–p interactions [6,15,17,19]. The motivation of this work is to demonstrate that the Modified SDF provides the best description of the “soft” and “hard” sectors of the spectra simultaneously, supporting the hypothesis of collective flow in small systems.

2. The Method and Formalism

The experimental data describing the transverse momentum ($p_T$) distributions of identified particles ($K_s^0$, $\Lambda$, ${\Xi }^{-}$, $d, \overline{d}$) produced in symmetric p–p collisions at $\sqrt{\left(s_{NN}\right)}$ = 0.9 TeV and 7 TeV are taken from the ALICE collaboration at the CERN Large Hadron Collider (LHC) [25,26,27,28]. The ALICE detector is optimized for studying the properties of strongly interacting matter at extreme energy densities. Data were collected using a minimum-bias trigger based on signals from the VZERO scintillator hodoscopes and the Silicon Pixel Detector (SPD). Charged particle tracking and momentum determination were performed using the Time Projection Chamber (TPC) and Inner Tracking System (ITS) in a 0.5 T solenoidal magnetic field. Particle identification relied on $dE/dx$ measurements in the TPC and time-of-flight information from the TOF detector. This precise setup allowed reconstruction of invariant mass spectra and extraction of accurate $p_T$ distributions for all identified particles [25].

The standard distribution function is rooted in classical and quantum statistics, incorporating Boltzmann–Gibbs, Bose–Einstein, and Fermi–Dirac statistics. The Two-Component Standard Distribution Function of the probability density, based on transverse momentum $p_T$, is given by [29].

$$f_{p_T}(p_T,T)={\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}=C{p}_T\sqrt{p_T^2+m_0^2}\times {\left[exp\left({\displaystyle \frac{\sqrt{p_T^2+m_0^2}-\mu }{T}}\right)+S\right]}^{-1},$$

(1)

where

• $f{p}_T$ ($p_T,T$) is the Two-Component Standard Distribution Function, in which the two components are the transverse momentum $p_T$ and the effective temperature T. The energy–temperature relation is given by; $E=k_{B}T$, where $k_B$ is the Boltzmann constant. In natural units, we set $k_B$ = 1 [30]; as a result, the temperature becomes equivalent to energy $E\sim T$. Therefore, we can measure the temperature in eV, MeV, GeV, etc., in particle physics.
• N is the number of events.
• C is the normalization constant (shows the number of quantum states or distributions per unit momentum volume).
• $m_0$ is the mass of produced particles. As eV/${\mathrm{c}}^2$, MeV/${\mathrm{c}}^2$, GeV/${\mathrm{c}}^2$, etc., are used as the units of mass, according to $E=m{c}^2$ or $m=E/c^2$. Conventionally, natural units are often used in particle physics, where the speed of light c is set to 1 [30,31]. Therefore, eV/${\mathrm{c}}^2$, MeV/${\mathrm{c}}^2$, GeV/${\mathrm{c}}^2$, etc., become simply eV, MeV, and GeV. Therefore, here, we used MeV and GeV as the units of mass of the particles ($K_s^0$, $\Lambda$, ${\Xi }^{-}$. d, $\overline{d}$).

  −

  For $K_s^0$, $m_0$ = 0.4976 GeV or 498.6 MeV [32].

  −

  For $\Lambda$, $m_0$ = 1.11568 GeV or 1115.68 MeV [32].

  −

  For ${\Xi }^{-}$, $m_0$ = 1.32171 GeV or 1321.71 MeV [32].

  −

  For $d/\overline{d}$, $m_0$ = 1.87561 GeV or 1875.61 MeV [32].

• S is the index of the Two-Component Standard Distribution Function (for fermions S = 1 and for bosons S = −1).
• $\mu$ is the chemical potential of the baryon [33].

The normalization constant C (also called the multiplicity parameter $N_0$) is mathematically given by:

$$C={\displaystyle \frac{gV}{8{\pi }^3}}.$$

(2)

where V is the volume (physical space) of the produced particle, also referred to in this model as the effective emission volume. g is the degeneracy factor (the number of internal states that the particle can occupy due to spin).

Proof. 

In quantum statistical mechanics, let the number of available quantum states or distributions in a volume V for the integral over phase space be as follows:

• Number of states = ${\displaystyle \frac{gV}{{(2\pi \hslash )}^3}}$.
• ℏ = 1 (neutral units).
• Number of states = ${\displaystyle \frac{gV}{8{\pi }^3}}\int d^{3}p$.
• Number of states per unit momentum volume $C={\displaystyle \frac{gV}{8{\pi }^3}}$.

□

From Equation (2), the volume of the produced system can be calculated as follows:

$$V=8{\pi }^3{\displaystyle \frac{C}{g}}$$

(3)

By inputting the value of C, Equation (1) becomes:

$${\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}={\displaystyle \frac{gV}{8{\pi }^3}}{p}_T\sqrt{p_T^2+m_0^2}{\left[exp\left({\displaystyle \frac{\sqrt{p_T^2+m_0^2}-\mu }{T}}\right)+S\right]}^{-1},$$

(4)

This equation is the Single-Slope Standard Distribution Function. Without chemical potential ($\mu$ = 0), Equation (4) can be written as follows:

$${\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}={\displaystyle \frac{gV}{8{\pi }^3}}{p}_T\sqrt{p_T^2+m_0^2}{\left[exp\left({\displaystyle \frac{\sqrt{p_T^2+m_0^2}}{T}}\right)+S\right]}^{-1},$$

(5)

The Single-Slope SDF is included primarily as a baseline reference. Due to the contribution of hard-scattering processes at high pT, this function is expected to yield higher chi-square values.

Another form of Equation (4) is the Double-Slope Standard Distribution Function, given as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}& ={\displaystyle \frac{gV}{8{\pi }^3}}{p}_T(\sqrt{p_T^2+m_0^2}{\left[exp\left({\displaystyle \frac{\sqrt{p_T^2+m_0^2}-\mu }{T_1}}\right)+S\right]}^{-1}\hfill \\ \hfill & +\sqrt{p_T^2+m_0^2}{\left[exp\left({\displaystyle \frac{\sqrt{p_T^2+m_0^2}-\mu }{T_2}}\right)+S\right]}^{-1}),\hfill \end{array}$$

(6)

Equation (4) shows the special form of the Two-Component Standard Distribution Function, which remains constant with the Ideal Gas Model [34]. Regardless of the form of the transverse momentum distribution of the particle, the probability density function based on the transverse momentum $p_T$ is given as:

$$f\left(p_T\right)={\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}.$$

(7)

Naturally, this equation is normalized to 1 (unity):

$${\int }_0^{\infty }f\left(p_T\right)d{p}_T=1.$$

(8)

We can directly obtain the mean (average) transverse momentum $<p_T>$ from the fit function using the probability density function:

$$\langle p_T\rangle ={\displaystyle \frac{{\int }_0^{\infty }{p}_{T}f\left(p_T\right)d{p}_T}{{\int }_0^{\infty }f\left(p_T\right)d{p}_T}}.$$

(9)

Using Equation (8),

$$\langle p_T\rangle ={\int }_0^{\infty }{p}_{T}f\left(p_T\right)d{p}_T.$$

(10)

$$\sqrt{\langle p_T^2\rangle }=\sqrt{{\int }_0^{\infty }{p}_T^{2}f\left(p_T\right)d{p}_T},$$

(11)

where $\sqrt{\langle p_T^2\rangle }=p_{T\left(\mathrm{rms}\right)}$ is the equation of the root mean square value of the transverse momentum $p_T$.

Therefore, Equation (8) can be written as follows:

$$p_{T\left(\mathrm{rms}\right)}=\sqrt{{\int }_0^{\infty }{p}_T^{2}f\left(p_T\right)d{p}_T}.$$

(12)

The initial temperature $T_i$ of sources of emission can be calculated from the String Percolation Model [35], using the following equation:

$$T_i={\displaystyle \frac{p_{T\left(\mathrm{rms}\right)}}{\sqrt{2}}},$$

(13)

where $p_{T(\mathrm{rms}})=\sqrt{\langle p_T^2\rangle }$ is the root mean square value of the transverse momentum.

To prove that $T=T_0+m_0{\beta }_T$ [12], we fit the data with the same function in modified form (in which we have used Lorentz Transformation) using $m_T=\langle {\gamma }_T\rangle (m_T-p_T\langle {\beta }_T\rangle )$, then one can find the $T_0$ and ${\beta }_T$ along with the system volume and chemical potential $\mu$. Therefore, the modified form of Equation (4) in terms of ${\mu }_T$ is given by:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}& ={\displaystyle \frac{gV}{8{\pi }^3}}{p}_T\langle {\gamma }_T\rangle \left(m_T-p_T\langle {\beta }_T\rangle \right)\hfill \\ \hfill & \times {\left[exp\left({\displaystyle \frac{\langle {\gamma }_T\rangle (m_T-p_T\langle {\beta }_T\rangle )-\mu }{T_0}}\right)+S\right]}^{-1}.\hfill \end{array}$$

(14)

Or:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{d{p}_T}}& ={\displaystyle \frac{gV}{8{\pi }^3}}{p}_T{\displaystyle \frac{1}{\sqrt{1-{\beta }_T^2}}}\left(\sqrt{p_T^2+m_0^2}-p_T\langle {\beta }_T\rangle \right)\hfill \\ \hfill & \times {\left[exp\left({\displaystyle \frac{\frac{1}{\sqrt{1-{\beta }_T^2}}\left(\sqrt{p_T^2+m_0^2}-p_T\langle {\beta }_T\rangle \right)-\mu }{T_0}}\right)+S\right]}^{-1}.\hfill \end{array}$$

(15)

This is the modified Standard Distribution Function, where

• $\langle {\gamma }_T\rangle ={\displaystyle \frac{1}{\sqrt{1-{\beta }_T^2}}}$.
• $m_T=\sqrt{p_T^2+m_0^2}$.
• $T_0$ is the freeze-out temperature.
• ${\beta }_T$ is the transverse flow velocity.

The standard formula to calculate the reduced Chi-square ${\chi }^2$ value is given as follows:

$${\chi }^2=\sum _i{\left({\displaystyle \frac{y_{\exp ,i}-y_{\mathrm{model},i}}{{\sigma }_i}}\right)}^2$$

3. Results and Analysis

The transverse momentum $p_T$ distributions of different particles (hadrons), such as $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in symmetric p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV, are described by experimental data taken from the ALICE collaboration at the CERN Large Hadron Collider (LHC) [25,26,27,28,29] and displayed in Figure 1, which displays all the particles at both energies on a single plot to demonstrate the hierarchy of yields and the scale difference between 0.9 TeV and 7 TeV. At 7 TeV, there are significantly higher yields (upper distributions) than in the case of 0.9 TeV (lower distributions). The yield decreases as $p_T$ increases, and the spectra are generally harder for the higher mass particles (i.e., the decrease with $p_T$ is slower at high $p_T$, like $\Lambda$ or ${\Xi }^{-}$ compared to $K_s^0$). The distinct ordering of the yields of these strange hadrons (e.g., $K_s^0>\Lambda >{\Xi }^{-}$) provides crucial information about strangeness production mechanisms in p–p collisions, which is essential as a baseline for comparison with heavy-ion collisions. The invariant yields generally follow an approximate exponential decay with increasing $p_T$, and a transition to a power-law decay at high $p_T$ is observed. Furthermore, Figure 2 visualizes the distributions of both datasets (0.9 TeV and 7 TeV) independently. It separates the particles to allow for a clearer view of the spectral shapes of individual species without visual overcrowding. The individual plots are made for the purpose of comparison, allowing us to analyze the trends within each dataset. In Figure 2a, the normalized invariant yield (${\displaystyle \frac{1}{N_{\mathrm{inel}}}}{\displaystyle \frac{d^{2}N}{2\pi p_{T}dyd{p}_T}}$) is presented as a function of transverse momentum $p_T$ for p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV. At low $p_T$ values, the particle yields are ordered as $K_s^0>\Lambda >{\Xi }^{-}$. This ordering reflects the strangeness content of the particles: the single-strange $K_s^0$ or $\Lambda$ is produced at a greater rate than the double-strange ${\Xi }^{-}$. The yields exhibit a sharp, approximately exponential fall-off as $p_T$ increases. In addition, the behavior at low $p_T$ is primarily governed by soft physics processes like thermal production and collective effects. These distributions serve as a baseline for hadronic production in symmetric p–p collisions at a moderate Large Hadron Collider (LHC) energy. The yields for the presumed $d/\overline{d}$ are several orders of magnitude lower than the strange hadrons, indicating their production is far less common. The data points are relatively scattered at high $p_T$, particularly for the multi-strange ${\Xi }^{-}$, suggesting that the production cross-section becomes very low at large transverse momenta for these low-energy collisions. In Figure 2b, the normalized invariant yield (${\displaystyle \frac{1}{N_{\mathrm{inel}}}}{\displaystyle \frac{d^{2}N}{2\pi p_{T}dyd{p}_T}}$) is also presented as a function of transverse momentum $p_T$ for p–p collisions at $\sqrt{s_{NN}}$ = 7 TeV. Similar to 0.9 TeV, the low $p_T$ particle yield ordering is observed as $K_s^0>\Lambda >{\Xi }^{-}$. The slopes of the $p_T$ distributions are less steep compared to the 0.9 TeV distributions, meaning the yields do not drop quickly as the $p_T$ increases. Additionally, the comparison between the two plots highlights the strong energy dependence of particle production in p–p collisions. The invariant yields at 7 TeV are significantly higher than those at 0.9 TeV by factors ranging from tens to hundreds across all $p_T$ values. This increase in particle yields is expected due to the larger phase space available for particle production at higher center-of-mass energy. Furthermore, the $p_T$ distributions are significantly harder at 7 TeV, while they fall more steeply with $p_T$ at 0.9 TeV. This confirms the dominance of hard scattering (jet-like) processes as the collision energy increases. The ordering of strange hadron yields ($K_s^0>\Lambda >{\Xi }^{-}$) is maintained at both energies, indicating that the fundamental mechanisms for strangeness production are qualitatively similar in p–p collisions across this energy range, although the rates are much higher. These p–p results are essential as a reference for Heavy-Ion (A–A) collisions (e.g., $Pb$–$Pb$). For instance, any deviation in the yield or the spectral shape in A–A data (such as the strangeness enhancement) is interpreted relative to these p–p baseline measurements to diagnose the formation of the Quark-Gluon Plasma (QGP).

Figure 3 presents the transverse momentum $p_T$ distributions of $K_s^0$, $\Lambda$, $X{i}^{-}$, d, and $\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV, plotted as subfigures a, b, c, d, and e, respectively. The distributions are plotted on a semi-log scale, with the y-axis showing the differential yield and the x-axis showing $p_T$ in GeV/c. The Single-Slope Standard Distribution Function (Equation (5)) is applied as a fit function (where $\mu$ = 0) and compared to the experimental data. In Figure 3a, the yield of $K_s^0$ mesons is significantly higher at 7 TeV as compared to 0.9 TeV throughout the $p_T$ range. This is consistent with the expectation from quantum chromodynamics (QCD) that the production cross-section increases at higher collision energies for all particle species. In both energies, the distribution exhibits a steep drop at low $p_T$ followed by a shallower slope at intermediate $p_T$. The data clearly deviates from the Single-Slope SDF fit for $p_T\gtrsim 1–2$ GeV/c. This hardening of the spectrum at higher $p_T$ is also a common feature in p–p collisions, indicative of the transition from soft-physics (low $p_T$) to hard-scattering processes (high $p_T$), which is better described by power-law functions like the Hagedorn and Tsallis distributions or by pQCD calculations. In Figure 3b, similar to $K_s^0$, the 7 TeV distribution is consistently higher than the 0.9 TeV distribution. This confirms the increased production with higher center-of-mass energy. The $\Lambda$ spectrum also shows a clear deviation from the Single-Slope SDF fit at high $p_T$; a deviation that is even more pronounced than for the $K_s^0$. This deviation is particularly important for baryons (such as $\Lambda$) because their production at intermediate $p_T$ is strongly influenced by baryon-meson chemistry and collective effects (if any, though less expected in p–p collisions than in heavy-ion collisions), which can lead to a phenomenon known as baryon-to-meson enhancement at intermediate $p_T$ relative to the exponential fit. In Figure 3c, the production yield of ${\Xi }^{-}$ at 7 TeV is notably higher than at 0.9 TeV. The data points at the two energies appear quite close, suggesting a less steep rise in production yield with energy compared to the lighter particles, especially at low $p_T$. The overall ${\Xi }^{-}$ spectrum is steeper than for $K_s^0$ and $\Lambda$, indicating that the average $p_T$ is lower. The Single-Slope SDF fit performs reasonably well over the measured range up to $p_T\approx 1.5$ GeV/c, but the data still show a slight deviation or hardening at the highest measured $p_T$ values. Figure 3d,e shows that both distributions of $d/\overline{d}$ are very steep, and the production yield is substantially lower than for the primary hadrons. The $d/\overline{d}$ distributions are nearly identical, particularly at low $p_T$, which is an expected outcome due to charge-conjugation (C) symmetry in p–p collisions. The Single-Slope SDF fit provides a reasonable description of these steep, low-yield distributions over the limited measured range. The fit parameters, such as volume V, effective temperature T, and chi-square ${\chi }^2$ values, are extracted from each subfigure to describe the data best at each center-of-mass energy, given in

Table 1. Thermodynamic parameters ($V,T,{\chi }^2$) for different hadrons in p–p collisions at various center-of-mass energies, extracted using the Single-Slope Standard Distribution Function without Chemical Potential ($\mu$ = 0), where $\Delta V$ and $\Delta T$ represent the statistical uncertainty (standard error) associated with the extracted parameters V and T, respectively, derived from the fitting procedure.

Particle	$\sqrt{{\mathit{s}}_{\mathit{NN}}}$ (TeV)	V (${\mathrm{GeV}}^{-3}$)	$\Delta \mathit{V}$	T (MeV)	$\Delta \mathit{T}$	${\mathit{\chi }}^2$
$K_s^0$	0.9	9989	67	177	8	585.0453
	7	12,445	84	219	9	699.4664
$\Lambda$	0.9	9183	58	211	9	559.5871
	7	11,356	79	258	11	498.279
${\Xi }^{-}$	0.9	8752	53	240	16	427.7322
	7	10,071	78	249	15	637.2076
d	0.9	2936	34	255	9	15.0141
	7	4364	39	270	13	0.6900
$\overline{d}$	0.9	2718	33	253	10	12.1237
	7	4791	51	274	11	0.2367

. The data demonstrates a clear dependence on both volume and effective temperature on center-of-mass energy, with elevated values at higher center-of-mass energies; i.e., higher energy collisions create a hotter and denser medium. This increment is indicative of higher effective thermal excitation and greater phase space occupancy at higher energies.

In Figure 4, the Single-Slope Standard Distribution Function (Equation (4), considering chemical potential ($\mu$)), is used to fit the transverse momentum $p_T$ distributions of $K_s^0$, $\Lambda$, ${\Xi }^{-}$, d, and $\overline{d}$, then compared to the experimental data. Similar to Figure 3, the distributions are plotted on a semi-log scale, with the y-axis showing the differential yield and the x-axis showing $p_T$ in GeV/c. In Figure 4a, the $p_T$ distributions of $K_s^0$ show a significant increase in production yield as the collision energy rises from 0.9 TeV to 7 TeV. This reflects the larger phase space and higher hard-scattering probability at higher energy in Quantum Chromodynamics (QCD). There is the clear deviation of the data from the Single-Slope SDF fit for $p_T$ values beyond ~2 GeV/c. This hardening of the distributions at intermediate-to-high $p_T$ is a signature of the transition from soft particle production mechanisms, which are well-modeled by the exponential fit, to hard-scattering processes where partonic interactions dominate and the cross-section follows a power law. In Figure 4b, $\Lambda$ exhibits a similar energy dependence, with the 7 TeV yield consistently exceeding the 0.9 TeV yield across the measured $p_T$ range. The distribution is also visually steeper than the $K_s^0$ distribution at low $p_T$. However, this Single-Slope SDF fit fails to describe the data at high $p_T$ (e.g., $p_T>3$ GeV/c). This pronounced deviation suggests the presence of mechanisms that favor the production of baryons over mesons at intermediate $p_T$, a phenomenon often linked to hadronization via coalescence/recombination or specific quark-level effects, which needs more sophisticated theoretical modeling than a simple thermal description. In Figure 4c, the $p_T$ distributions of ${\Xi }^{-}$ show that the production yield of ${\Xi }^{-}$ at 7 TeV is greater than at 0.9 TeV. Being a heavier particle than $\Lambda$, the ${\Xi }^{-}$ spectrum is steeper and the overall production yield is lower. Although this Single-Slope SDF fit describes the low $p_T$ data relatively well, at the highest measured $p_T$ (e.g., $p_T\ge 4$ GeV/c), the data points clearly fall above the fit, once again highlighting the limitation of a single exponential model to describe the full range of particle production, which extends into the hard-scattering regime. In Figure 4d, the yield of d is several orders of magnitude lower (down to ${10}^{-7}$ GeV/c) than for the elementary hadrons, which is a characteristic of coalescence. The 7 TeV data is slightly higher than 0.9 TeV, demonstrating an energy dependence for this secondary production mechanism. The sharp steepening of the spectrum indicates that the probability for two nucleons to coalesce decreases rapidly with the deuteron’s $p_T$. Therefore, this Single-Slope SDF fit (Equation (4)) provides a fair, though not perfect, representation of this steep fall-off. In Figure 4e, the distribution of $\overline{d}$ is nearly identical in shape and yield to the d distribution, consistent with the expectation of charge-conjugation (C) symmetry in p–p collisions, where the production of matter and antimatter nuclei should be equal. Like the deuteron, the antideuteron production yield is extremely low and the spectrum is very steep. The failure of the Single-Slope fit demonstrates that the system cannot be described by a simple static thermal source, and therefore, more complex models are presented later in this paper. The fit parameters, such as volume V, effective temperature T, chemical potential $\mu$, and ${\chi }^2$, are extracted from each subfigure, as given in

Table 2. Thermodynamic parameters ($V,T,\mu ,{\chi }^2$) for different hadrons in p–p collisions at various center-of-mass energies, extracted using the Single-Slope Standard Distribution Function (Equation (4)). ${\sigma }_V,{\sigma }_T,{\sigma }_{\mu }$ denote the uncertainties of the respective fit parameters ($V,T,\mu$), respectively.

Particle	$\sqrt{{\mathit{s}}_{\mathit{NN}}}$ (TeV)	V (GeV −3)	${\mathit{\sigma }}_{\mathit{V}}$	T (MeV)	${\mathit{\sigma }}_{\mathit{T}}$	$\mathit{\mu }$ (MeV)	${\mathit{\sigma }}_{\mathit{\mu }}$	${\mathit{\chi }}^2$
$K_S^0$	0.9	8873	61	183	10	118	7	544.146
	7	11,348	74	223	11	115	6	567.126
$\Lambda$	0.9	8376	56	212	8	114	8	346.407
	7	10,231	69	259	11	111	8	348.247
${\Xi }^{-}$	0.9	7542	48	248	11	105	7	67.2332
	7	9121	59	276	10	101	6	107.212
d	0.9	2322	34	293	9	91	5	8.0141
	7	3967	41	307	9	84	5	0.6900
$\overline{d}$	0.9	2251	33	299	11	90	5	10.1237
	7	4101	38	311	10	82	6	0.2367

. Both the volume and effective temperature increase proportionally to the rise of center-of-mass energy, while the chemical potential $\mu$ decreases. The gradual increase in T and decrease in $\mu$ are consistent with higher kinetic freeze-out temperatures and lower net baryon density at high center-of-mass energies. In addition, the increase in volume V confirms that the expected system is expanded at higher center-of-mass energies. The fit of the extracted temperature and volume reveals a marginal increase in thermalization and system size, which corroborates the onset of a near-equilibrated hadronic phase.

In parallel to the previous two fit functions, a third approach was employed to characterize the transverse momentum $p_T$ distributions of $K_s^0$, $\Lambda$, ${\Xi }^{-}$, d, and $\overline{d}$, which were then compared to the experimental data. Figure 5 presents the results obtained by fitting the distributions with the Double-Slope Standard Distribution Function (Equation (6)), which generally parameterizes the low $p_T$ soft physics component and the high $p_T$ hard scattering component with separate inverse slope parameters. In Figure 5a, the $p_T$ distribution of $K_s^0$ at 7 TeV shows a higher yield than at 0.9 TeV across the measured $p_T$ range, reflecting the expected increase in the production cross-section with energy. A stronger representation of the data is obtained with Double-Slope SDF, which is eventually designed to better capture the spectral shape compared to the Single-Slope SDF. The fit follows the steep initial drop (soft component) and successfully incorporates the shallower slope, or spectral hardening, observed at $p_T\gtrsim 2$ GeV/c (hard component). The agreement suggests that the two primary production mechanisms (soft thermal/statistical processes and hard pQCD scattering) are well-modeled by the two exponential or power-law terms of the Double-Slope SDF. Likewise, in Figure 5b, the $p_T$ distributions of $\Lambda$ baryon also show the expected increase in yield from 0.9 TeV to 7 TeV. The Double-Slope SDF presents a much better parametrization than the Single-Slope SDF fit. This improved fit is crucial because $\Lambda$ production, particularly at intermediate $p_T$, is sensitive to hadronization mechanisms like recombination/coalescence, which could enhance the baryon yield relative to mesons. The success of the Double-Slope SDF fit function suggests that this framework is robust enough to model the $p_T$ distributions of both mesons and baryons by accommodating the shift in dynamics between the low and high $p_T$ regions. In a parallel manner, for ${\Xi }^{-}$ in Figure 5, the $p_T$ distributions continue to show a higher ${\Xi }^{-}$ production of at 7 TeV as compared to 0.9 TeV. The ${\Xi }^{-}$ yield is lower and the distribution is steeper than for $\Lambda$ due to its larger mass and higher strangeness content. The Double-Slope SDF provides a good fit across the measured $p_T$ range. This consistency across various hadron masses and strangeness (from $K_s^0$ to ${\Xi }^{-}$ ) suggests that the underlying dynamics, which necessitate a two-component description, are universal for primary hadron production in p–p collisions. In Figure 5d,e, $d/\overline{d}$ have significantly lower yields and much steeper $p_T$ distributions than the primary hadrons. The distributions of $d/\overline{d}$ are dominated by a very steep drop. The Double-Slope SDF provides a very good fit over the narrow measured $p_T$ range. These distributions are primarily determined by the momentum-space overlap of the constituent nucleons, and the sharp fall-off essentially acts as the soft component, even up to the highest measured $p_T$. The spectra for $d/\overline{d}$ (matter and antimatter) are almost perfectly superimposed, confirming Charge-conjugation (C) symmetry in p–p collisions for light nuclei production, where the net baryon number is zero at mid-rapidity. The fit parameters are given in

Table 3. Values of extracted parameters (V, $T_1$, $T_2$, ${\chi }^2$) using the Double-Slope Standard Distribution Function Equation (6). ${\sigma }_V,{\sigma }_{T_1},{\sigma }_{T_2}$ denote the uncertainties of the respective fit parameters ($V,T_1,T_2$), respectively.

Particle	$\sqrt{{\mathit{s}}_{\mathit{NN}}}$ (TeV)	V (${\mathrm{GeV}}^{-3}$)	${\mathit{\sigma }}_{\mathit{V}}$	${\mathit{T}}_1$ (MeV)	${\mathit{\sigma }}_{{\mathit{T}}_1}$	${\mathit{T}}_2$ (MeV)	${\mathit{\sigma }}_{{\mathit{T}}_2}$	${\mathit{\chi }}^2$
$K_S^0$	0.9	2310	32	109	9	253	10	29.2362
	7	3123	41	126	6	353	14	165.179
$\Lambda$	0.9	2101	26	119	9	300	17	11.3671
	7	2709	37	177	11	368	16	17.5391
${\Xi }^{-}$	0.9	1977	21	184	14	376	17	0.8619
	7	2609	33	317	18	385	15	2.4188
d	0.9	1758	19	191	11	383	16	0.0996
	7	2407	27	209	15	398	14	0.1256
$\overline{d}$	0.9	1716	17	189	10	382	15	0.1086
	7	2391	33	206	13	401	14	0.1352

. The two extracted temperatures $T_1$ and $T_2$ represent the kinetic freeze-out features, with $T_1$ characterizing the bulk medium and $T_2$ hinting at a higher-temperature tail due to semi-hard partonic scatterings. The extracted volume V grows with $\sqrt{s_{NN}}$, which is consistent with an expanding fireball at higher center-of-mass energies. The low ${\chi }^2$ values (fit quality) show the best fitting of the Double-Slope SDF.

Figure 6a presents the dependence of average transverse momentum $<p_T>$ on center-of-mass energy for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. Here, the $<p_T>$ spectra are presented as a function of $\sqrt{s_{NN}}$. The most apparent feature is the general increase in $<p_T>$ with increasing $\sqrt{s_{NN}}$ for all types of particles. This trend is expected in QCD, as higher collision energy leads to more hard scattering processes, resulting in the production of particles with higher momentum components, including the transverse momentum. For a given $\sqrt{s_{NN}}$, heavier particles generally exhibit higher $<p_T>$ values than lighter ones (i.e., $<p_T{>}_d><p_T{>}_{{\Xi }^{-}}><p_T{>}_{\Lambda }><p_T{>}_{K_s^0}$). This mass ordering is a characteristic feature of hadronization in a vacuum and can be partially explained by kinematic effects and the different quark content of the particles. Additionally, for comparison purposes, the $<p_T>$ spectra of $\Lambda$ and ${\Xi }^{-}$ obtained from $Au-Au$ collisions (heavy-ion collisions [24,36,37,38]) are shown in Figure 6b. $Au-Au$ collisions create a high-energy density medium, often referred to as a quark gluon plasma (QGP), especially at higher $\sqrt{s_{NN}}$ values. In p–p collisions, the collisions serve as a baseline to understand particle production without the effects of the extended, thermalized medium present in heavy-ion collisions. For both systems, $<p_T>$ generally increases with increasing $\sqrt{s_{NN}}$. This is expected, as higher collision energy translates to higher available energy for particle production and greater momentum transfer. In both $Au-Au$ and p–p collision systems, the $<p_T>$ distribution of ${\Xi }^{-}$ (heavy and has two strange quarks) is consistently higher than $\Lambda$ (lighter and has one strange quark); $<p_T{>}_{{\Xi }^{-}}><p_T{>}_{\Lambda }$. This trend is a well-known feature of particle production, often related to the mass of the produced particle. At lower energies ($\sqrt{s_{NN}}$ below ∼100 GeV), the $<p_T>$ values for $Au-Au$ and p–p are comparable. At higher energies ($\sqrt{s_{NN}}$∼ at ${10}^3$ GeV), the $<p_T>$ values for particles in $Au-Au$ collisions tend to be significantly enhanced compared to the p–p baseline. This enhancement of $<p_T>$ in heavy-ion collisions, particularly for heavier and multi-strange hadrons, is a classic signature of the collective radial flow (hydrodynamic expansion) of the medium created in $Au-Au$ collisions. The pressure gradients in the QGP push the particles outward, boosting their transverse momentum. Thus, the data clearly illustrate the transition from a regime where particle production is dominated by elementary interactions to a hydrodynamically expanding medium (QGP) at higher collision energies in heavy-ion collisions, indicated by the collective $<p_T>$ enhancement. The average transverse momentum $<p_T>$ values at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV are given in

Table 4. The average transverse momentum $\langle p_T\rangle$ values of different particles ($K_s^0$, $\Lambda$, ${\Xi }^{-}$, d and $\overline{d}$ at various center-of-mass energies. ${\sigma }_{p_T}$ is the uncertainty in $<p_T>$.

Particle	$\sqrt{{\mathit{s}}_{\mathit{NN}}}$ (TeV)	$\langle {\mathit{p}}_{\mathit{T}}\rangle$ (GeV/c)	${\mathit{\sigma }}_{\langle {\mathit{p}}_{\mathit{T}}\rangle }$
${\mathrm{K}}_S^0$	0.9	0.645875	0.080721
	7	0.797204	0.077651
$\Lambda$	0.9	0.830980	0.099871
	7	1.007138	0.099461
${\Xi }^{-}$	0.9	0.957280	0.098914
	7	1.186169	0.165432
d	0.9	1.075948	0.122132
	7	1.248989	0.187654
$\overline{d}$	0.9	1.059312	0.141031
	7	1.249180	0.165781

.

Figure 7a presents the dependence of the effective temperature T on center-of-mass energy $\sqrt{s_{NN}}$ for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. It shows T for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ as a function of $\sqrt{s_{NN}}$. For all types of particles, there is a direct dependence of T on energy ($\sqrt{s_{NN}}$ = [0.9–7] TeV), with the former exhibiting a clear increase in response to the latter. This suggests that at higher collision energies, the system formed in p–p interactions becomes harder, implying greater kinetic energy imparted to the produced particles. It also indicates that, even in p–p collisions, a degree of collectivity or increasing phase-space density influences the particle spectra. Additionally, for comparison purposes, the effective temperature spectra of $\Lambda$ and ${\Xi }^{-}$ obtained from $Au-Au$ collisions [24,36,37,38] are shown in Figure 7b. The plot superimposes the effective temperature data for $\Lambda$ and ${\Xi }^{-}$ from both heavy-ion $Au-Au$ and p–p collisions. A striking observation is the significantly higher effective temperatures obtained in $Au-Au$ collisions compared to p–p collisions for both particles, particularly for the heavier particle. Thus, the plots collectively highlight the fundamental difference between p–p collisions and $Au-Au$ collision systems.

Figure 8a presents the dependence of the initial temperature $T_i$ on center-of-mass energy $\sqrt{s_{NN}}$ for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. For all types of particles, a direct dependence is evident between $\sqrt{s_{NN}}$ (0.9 and 7 TeV) and $T_i$, with the latter exhibiting a clear increase in response to the former. This shows that as $\sqrt{s_{NN}}$ increases, the initial conditions of the p–p collisions become hotter, i.e., higher-energy collisions lead to harder scattering processes and a higher density of produced partons. Figure 8b provides a comparison of $T_i$ for $\Lambda$ and ${\Xi }^{-}$ between $Au-Au$ collisions [24,36,37,38] and p–p collisions. A prominent feature is the significantly higher initial temperatures observed in $Au-Au$ collisions compared to p–p collisions, especially for heavier particles. This substantial enhancement of $T_i$ in $Au-Au$ collisions is a crucial piece of evidence for the formation of a de-confined state of matter (QGP). In $Au-Au$ collisions, $T_i$ reflects the energy density that is achieved in the first stages of the collisions, which is much higher due to the greater number of participating nucleons. The mass ordering of $T_i$ is also important in $Au-Au$ collisions, supporting the idea of a collectively expanding medium where particles of various masses acquire different kinetic energies from the radial flow, which influences the extracted temperature. The initial temperature in $Au-Au$ collisions generally shows a plateau or a weaker dependence on $\sqrt{s_{NN}}$ at higher energies, suggesting that the system reaches a maximum temperature or that the extraction method becomes less sensitive to further increases in collision energy. The large disparity in $T_i$ between $Au-Au$ and p–p collisions underscores the qualitative difference in the systems formed: a hot, dense, and collectively expanding medium in heavy-ion collisions versus more localized, string-like, or jet-like fragmentation in p–p collisions.

In Figure 9a, the dependence of the system freeze-out volume V on the center-of-mass energy $\sqrt{s_{NN}}$ is displayed for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. A direct correlation is clearly observed between V and $\sqrt{s_{NN}}$ across all types of particles. As $\sqrt{s_{NN}}$ increases to a few TeV, a larger energy deposition produces denser and more extended systems, and the freeze-out volume becomes large, indicating a system transitioning toward a thermally equilibrated medium with extensive spatial dimensions. This rise in volume is also consistent with the hydrodynamic expansion of the system. Caution must be taken when comparing this extracted volume to heavy-ion geometric volumes, as it is derived from spectral normalization. Moreover, in Figure 9b, the dependence of the chemical potential $\mu$ on the center-of-mass energy $\sqrt{s_{NN}}$ is shown for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. An inverse correlation is clearly observed between $\mu$ and $\sqrt{s_{NN}}$ across all types of particles. As the energy increases, collisions become more transparent, resulting in a dilute baryon environment with $\mu$ approaching zero. It is important to note that at LHC energies, the physical baryon chemical potential is expected to be near zero. The non-zero values extracted here should be interpreted as effective parameters required by the specific shape of the chosen fit function, rather than the global thermodynamic potential of the system.

Figure 10 presents the dependence of $<p_T>$, the effective temperature T, the chemical potential $\mu$, and the volume of the system V, on the rest mass $m_0$ for $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. Figure 10a shows an increase in $<p_T>$ with increasing $m_0$ for each value of $\sqrt{s_{NN}}$. It suggests that heavier mass particles tend to gain more $<p_T>$ than lighter mass particles. Because of the large inertia in heavy mass particles, they retain more of the $<p_T>$ from the hard scattering explained partially by kinematic effects. Furthermore, in p–p collisions, if there is even a small degree of collective expansion or mini-flow; heavier particles would gain more $p_T$ from this collective velocity boost. Similar to $<p_T>$, Figure 10b shows a general increase in T with increasing $m_0$ for each value of $\sqrt{s_{NN}}$. This mass dependence of the effective temperature is another strong indicator of the underlying dynamics. If the system undergoes a collective expansion, heavier particles, having more inertia, gain a larger kinetic energy component from the collective velocity, thus appearing hotter when their spectra are fitted with a thermal distribution. On the other hand, Figure 10c shows that the values of $\mu$ are generally small and exhibit little to no significant dependence on the particle mass. Slight variations are observed in chemical potentials with different energies of collisions, reflecting the different initial energy densities and subsequent hadronization conditions, but the overall trend across masses remains flat. This suggests that the chemical freeze-out temperature, which dictates the particle productions, is relatively uniform across different masses of hadron for these types of collisions. Finally, Figure 10d, presents the volume of the system V as a function of $m_0$ for all types of particles. The lack of a strong mass dependence or a consistent trend might indicate that the emission volume is not primarily determined by the particle’s mass in p–p collisions or that the extraction methods are sensitive to other factors. The values for V are relatively small, as expected for p–p collisions, where a large, extended medium like in heavy-ion collisions is not formed. The different collision energies show some variations in the extracted volume, possibly reflecting the expansion dynamics at different initial energy densities. However, without more context on the specific model used to extract these volume parameters, a definitive interpretation is challenging beyond stating the observed lack of a strong mass dependency.

Furthermore, the modified Standard Distribution Function (Equation (15)) is also applied to fit $p_T$ distributions of $K_s^0$, $\Lambda$, ${\Xi }^{-}$, d, and $\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV, as shown in Figure 11. The general shape of all spectra is characterized by a rapid, exponential-like decrease in the differential yield as $p_T$ increases, which is a common feature of particle production in high-energy hadronic collisions. In Figure 11a,b, the plots show the distributions for the $K_s^0$ meson and the $\Lambda$ baryon, respectively. A notable observation is the increase in the yield (the vertical shift) at the higher collision energy $\sqrt{s_{NN}}$ = 7 TeV. This behavior is expected, as higher collision energy leads to greater energy available for particle production. The spectra for $\Lambda$ appear slightly harder (slower fall-off with $p_T$) than those for $K_s^0$ at higher $p_T$. Figure 11c–e show the distributions for the ${\Xi }^{-}$ baryon, light deuteron (d), and anti-deuteron ($\overline{d}$), respectively. The ${\Xi }^{-}$ and d distributions also exhibit the expected increase in yield with collision energy. The remarkable similarity between the d and $\overline{d}$ distributions in Figure 11d,e, particularly in the 7 TeV data, highlights the principle of charge conjugation symmetry in particle production, where the production of a nucleus and its anti-nucleus are similar. For all types of particles, the differential yield is consistently higher at 7 TeV than at 0.9 TeV, reflecting the increased kinetic energy available to the produced particles. The spectral shapes for a given particle and energy are well-described by the modified SDF, and they provide an excellent fit to the experimental data points across all analyzed particle species and energies. In the context of particle and heavy-ion physics, such functions are often rooted in the hydrodynamical or thermal models (like the Blast-Wave model or Hagedorn’s thermal model) that describe particle production from a collectively expanding, thermalized medium (fireball). The core concept is that particles are emitted from a thermal source that has an overall transverse collective flow. The high-quality fit (low ${\chi }^2$) supports the fitness of the function, which is extracted and given in

Table 5. The values of extracted V, $T_0$, ${\beta }_T$, and $\mu$ using the modified Standard Distribution Function with Chemical Potential ($\mu$). ${\sigma }_V,{\sigma }_{T_0},{\sigma }_{{\beta }_T}$, and ${\sigma }_{\mu }$ are the uncertainties in the values of extracted V, $T_0$, ${\beta }_T$, and $\mu$, respectively.

Coll.	Particle	$\sqrt{{\mathit{s}}_{\mathit{NN}}}$ (TeV)	V (${\mathrm{GeV}}^{-3}$)	${\mathit{\sigma }}_{\mathit{V}}$	${\mathit{T}}_0$ (MeV)	${\mathit{\sigma }}_{{\mathit{T}}_0}$	${\mathit{\beta }}_{\mathit{T}}$ (c)	${\mathit{\sigma }}_{{\mathit{\beta }}_{\mathit{T}}}$	$\mathit{\mu }$ (MeV)	${\mathit{\sigma }}_{\mathit{\mu }}$	${\mathit{\chi }}^2$
p–p	${\mathrm{K}}_S^0$	0.9	4851	54	116	8	133	8	116	7	31.8532
		7	6881	67	151	7	143	8	103	8	166.211
p–p	$\Lambda$	0.9	2597	33	133	7	72	5	109	7	9.2317
		7	4393	47	159	7	91	6	96	5	14.3528
p–p	${\Xi }^{-}$	0.9	2451	31	156	8	69	5	106	6	0.9562
		7	4271	42	164	9	86	6	94	5	2.3498
p–p	d	0.9	2194	28	178	8	61	5	101	7	0.1893
		7	4045	39	185	9	66	5	91	5	0.1248
p–p	$\overline{d}$	0.9	2158	29	180	9	63	4	102	7	0.1191
		7	4073	36	186	8	67	4	92	7	0.1360

.

To further investigate the thermodynamic properties of these particles, we extract the $T_i$, T, and $T_0$ at both energies. The variation in these parameters is presented in Figure 12. $T_i$ shows a gradual increase with the increase in $\sqrt{s_{NN}}$, signifying higher energy densities at early stages of p–p collisions. Meanwhile, T, often linked to both kinetic motion and radial flow, also increases but more gently, indicating growing transverse collective expansion. On the other hand, $T_0$, representing the point where particles decouple, remains significantly lower and more stable, indicating that even as collisions grow more energetic, the system cools to a similar temperature before freeze-out. The clear separation between these temperatures ($T_i>T>T_0$) indicates a three-stage freeze-out scenario, representing small systems with strong collective effects. For $K_s^0$ and $\Lambda$, all parameters increase weakly with $\sqrt{s_{NN}}$, suggesting a slight increase in the energy density or lifetime of the system.

It is also important to investigate the dependence of the transverse flow velocity and center-of-mass energy, as this can provide key information for understanding the evolution of the system from initial to final states. This is shown in Figure 13a for all types of particles ($K_s^0$$\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$) and for both energies. As $\sqrt{s_{NN}}$ increases, ${\beta }_T$ also steadily increases, indicating that higher $\sqrt{s_{NN}}$ values generate systems with stronger collective transverse expansion. This behavior is a signature of increased initial pressure gradients and partonic activity at higher $\sqrt{s_{NN}}$ values, contributing to stronger radial flow velocity. The trend is evident across all types of particles and emphasizes the role of energy density in driving collective phenomena, even in small systems of collisions like p–p collisions. Figure 13b shows the inverse relationship between ${\beta }_T$ and the rest mass $m_0$ for all types of particles ($K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$) and both energies. It is clear from the plot that light mass particles ($K_s^0$) have higher ${\beta }_T$ values, and heavier mass particles ($d/\overline{d}$) receive a lower ${\beta }_T$ value. The inverse relationship between ${\beta }_T$ and $m_0$ signifies that lighter particles ($K_s^0$) exhibit higher transverse flow velocities compared to heavier particles ($\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$). This mass-dependent flow is a hallmark of hydrodynamic expansion, where the lighter particles are more effectively pushed by the collective pressure gradient of the expanding medium.

Another classic study in relativistic heavy-ion physics is to investigate the relationship between $T_0$ and ${\beta }_T$ for all types of particles under study and at both energies, as this provides insight into the thermodynamic and collective properties of the system at the point where particle interactions cease (freeze-out). This relationship is displayed in Figure 14a, which shows that the studied particles do not all freeze-out at a single, universal ($T_0$, ${\beta }_T$) point, suggesting a non-equilibrium or multi-stage freeze-out scenario. At high ${\beta }_T$ and low $T_0$, $K_s^0$ and $\Lambda$ data points show a clear increase in ${\beta }_T$ with a corresponding decrease in $T_0$. In addition, higher ${\beta }_T$ values signify stronger collective expansion or hydrodynamic flow in the system. The inverse correlation suggests that particles that experience more collective push (higher ${\beta }_T$) decouple from the system when it is cooler (lower $T_0$). At low ${\beta }_T$ and high $T_0$, $K_s^0$, $\Lambda$, and ${\Xi }^{-}$ points clustered around ${\beta }_T\sim$70–100, possibly indicating a chemical freeze-out or a thermal freeze-out that happens earlier, when the system is hotter and less expanded (lower ${\beta }_T$). In the meantime, $K_s^0$ and $\Lambda$ points in the high ${\beta }_T$ region may represent the kinetic freeze-out, where elastic interactions cease. On the other hand, $d/\overline{d}$ data points are notably distinct, suggesting that they might have different freeze-out dynamics compared to the mesons and baryons, possibly forming via coalescence at a later, cooler stage. Furthermore, Figure 14b presents the dependence of $\mu$ on ${\beta }_T$ for all particles under study and at both energies $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. We observe that as ${\beta }_T$ increases, $\mu$ decreases, which is consistent with the expectations from statistical hadronization models and the phase diagram of QCD. At lower collision energies, the system is characterized by a higher net-baryon density (hence a larger ${\beta }_T$). As $\sqrt{s_{NN}}$ increases, the system becomes more symmetric in terms of matter–antimatter production, leading to a decrease in the net-baryon density (hence a lower ${\beta }_T$). The increase in ${\beta }_T$ with decreasing $\mu$ further supports the concept that higher $\sqrt{s_{NN}}$ values lead to a larger and longer-lived system that undergoes more significant collective expansion before hadronization and kinetic freeze-out. This also provides insights into the evolution of the QCD matter from a baryon-rich to a more meson-dominated environment as $\sqrt{s_{NN}}$ increases. Finally, Figure 14c displays a direct relationship between V and ${\beta }_T$ for all types of particles under study and at both energies $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. It is clear from the data that V and ${\beta }_T$ are directly proportional to each other, which signifies that stronger flow is associated with larger particle-emitting sources. This is acceptable, as higher $\sqrt{s_{NN}}$ produce more particles, leading to an expanded freeze-out volume. This direct relationship supports hydrodynamic expansion models, where larger systems tend to exhibit more pronounced collective behavior, even in small collision systems like p–p collisions.

Finally, in Figure 15, we investigated the variation in $T_i$, T, and $T_0$ with rest mass $m_0$ for all studied particles at the two energies, $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV. It is generally observed that $T_i>T>T_0$ at both energy levels. This order is physically very important; the system cools as it expands, leading to a hierarchy of temperatures from the initial hot and dense state to the final freeze-out state where interactions cease. It is also usually observed from both the subfigures that $T_i$, T, and $T_0$ increase with the increase in rest mass of the given particles. $T_0$ is consistently the lowest, reflecting the point at which particles cease to interact strongly and their yields become fixed. The values of $T_0$ are generally in the range of typical hadronic freeze-out temperatures, further supporting the interpretation of a hadron gas phase. These systematic studies are vital for understanding the thermodynamic properties of the strongly interacting matter created in proton–proton collisions and for establishing a baseline for comparisons with heavy-ion collisions, where the formation of a quark-gluon plasma is expected. The values of $T_i$, T, and $T_0$ are given in

Table 6. Extracted $T_i$, T, and $T_0$ for various particles in p–p collisions at different center-of-mass energies.

Particle	$\sqrt{{\mathit{s}}_{\mathit{NN}}}$ (TeV)	${\mathit{T}}_{\mathit{i}}$ (MeV)	T (MeV)	${\mathit{T}}_0$ (MeV)
${\mathrm{K}}_S^0$	0.9	437	183	116
	7	489	223	151
$\Lambda$	0.9	705	212	133
	7	736	259	159
${\Xi }^{-}$	0.9	739	248	156
	7	773	276	164
d	0.9	792	293	178
	7	862	307	185
$\overline{d}$	0.9	789	299	180
	7	867	311	186

.

4. Conclusions

The $p_T$ distributions of $K_s^0$, $\Lambda$, ${\Xi }^{-}$, and $d/\overline{d}$ produced in p–p collisions at $\sqrt{s_{NN}}$ = 0.9 TeV and 7 TeV were effectively investigated using the Single-Slope (with and without $\mu$) and Double-Slope, and the modified Standard Distribution Function (SDF). The Double-Slope SDF provided an excellent fit across all the particles at both center-of-mass energies, particularly by accounting for separate contributions from thermal motion ($T_1$) and hard QCD processes ($T_2$). Additionally, the modified SDF gave an excellent fit through the $p_T$ range and good results of freeze-out temperature $T_0$ and effective flow velocity ${\beta }_T$. Furthermore, all thermodynamic parameters are dependent on the center-of-mass energy $\sqrt{s_{NN}}$ and rest mass $m_0$ of the particles, i.e., $<p_T>$, $T_i$, T, $T_0$, V, and ${\beta }_T$ increase with $\sqrt{s_{NN}}$, while only $\mu$ shows a decrease with $\sqrt{s_{NN}}$. In addition, $<p_T>$, T, $T_i$, and $T_0$ showed an increase with $m_0$, while $\mu$, V, and ${\beta }_T$ decreased with $m_0$.

The comparison of p–p collisions with $Au-Au$ collisions shows a lower degree of collectivity but still indicates some thermodynamic features such as flow-like effects and freeze-out patterns. These results suggest that collective behavior may not be exclusive to large systems. The initial temperature $T_i$ shows an increase with $\sqrt{s_{NN}}$, reflecting a hotter and denser medium formation at the initial stage in high-energy collisions. The behavior of $T_i$ is suppressed relative to the case of heavy-ion collisions ($Au-Au$ collisions), but it still suggests significant partonic activity even in p–p collisions. Furthermore, the observed trends in T, $\mu$, V etc., with different center-of-mass energies provide empirical input relevant to mapping the QCD phase diagram. While p–p collisions are simpler than heavy-ion collisions, their thermodynamic signatures could suggest closeness to the QCD crossover region, justifying further investigation.

Finally, this work contributes to the discussion regarding collective-like behavior in small systems p–p collisions at LHC energies. In addition, it supports the hypothesis that the dynamics of p–p collisions at LHC energies are not purely perturbative but involve a significant degree of collective and thermalized behavior, marking an essential step toward understanding the continuum from small to large systems in high-energy nuclear or particle physics.