Centrality Evolution of Transverse Momentum Spectra of Identified Light-Flavor Hadrons in p+Pb Collisions at √s_nn = 5.02 TeV at the LHC Within a Two-Component Model

1. Introduction

High-energy Xe+Xe and Pb+Pb collisions at the LHC (Large Hadron Collider, Switzerland) are primarily aimed to produce QGP (Quark-Gluon Plasma), very dense state of nearly free quarks and gluons, and study its various properties and evolution along with understanding the strong interaction at extreme matter densities. Our Universe is believed to be in QGP state a few microseconds after the so-called Big Bang, which is assumed to be a starting moment of Universe creation. Production of QGP was firmly established [1,2,3,4,5,6,7,8,9] in high-energy Au+Au and Pb+Pb collisions at the RHIC (Relativistic Heavy-Ion Collider, USA) and LHC/CERN experiments. QGP with very short lifetime of the order of 10⁻²³ seconds behaves as nearly perfect quark-gluon liquid with very small viscosity [8,9]. Produced QGP decays very quickly into large number of particles during hadronization stage. Afterwards, very dense and hot system of produced hadrons, called fireball, expands and cools down very swiftly, going through the chemical and kinetic freeze-out stages. At the chemical freeze-out, the particles of a fireball stop interacting inelastically, which fixes the numbers of produced hadron species. The temperature of the system (T_ch) and chemical potential (μ) at the chemical freeze-out stage are extracted from yield ratios of various particle species using thermal or statistical hadronization models [10,11,12,13,14,15]. At the kinetic freeze-out, the particles of the fireball stop interacting elastically, and the momenta and energies of the final particles do not change anymore until they reach detectors. Therefore, it is important to analyze the transverse momentum ($p_T$) distributions of the final particles with the help of various thermodynamic and phenomenological models to extract the thermodynamic and hydrodynamic parameters of the fireball at the moment of kinetic freeze-out.

Traditionally, analysis of high-energy proton-proton and proton-nucleus collisions was considered as a reference study to extract the collective parameters of the collision system, including those of QGP, in high-energy nucleus-nucleus collisions at the same center-of-mass collision energy per nucleon pair. However, many signatures of QGP production, like significant enhancement of yield of multi-strange particles with respect to non-strange hadrons and development of a strong radial flow, were also observed [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] in high-energy proton-proton and proton-nucleus collisions. Proton to pion yields ratio did not change with multiplicity, while the ratios of a strange-hadron to pion yields increased considerably with an increase in multiplicity in proton-proton collisions at $\sqrt{s}$ = 7 TeV at midrapidity at the LHC [19]. Larger enhancement was observed for hadrons with a higher strangeness content [19]. The thermal abundances in the hadron yields in high-energy proton-proton collisions along with exponential component found in their p_T spectra were considered as the clear signature of thermalization [27,31]. Measurements revealed [41,42] strong azimuthal, long-range particle correlations (the "ridge" effect) in p+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV at the LHC. The development of a significant collective radial flow in high-energy proton-proton collisions with large enough multiplicities was demonstrated in Refs. [20,24,25,33,34]. Strong flow-like effects were also reported [39,40,41,42,43] in high-multiplicity p+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV at the LHC. The evidences of significant radial flow developed with increasing centrality of p+Pb collisions at the LHC were given in Refs. [44,45]. Investigation of various properties of proton-lead (p+Pb) collisions at the LHC is especially important because it bridges the gap between elementary proton-proton collisions and heavy-ion (Pb+Pb) collisions.

The present work is an important extension of our recent work [45], which was devoted to analysis of evolution of the kinetic freeze-out parameters in p+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV at the LHC with changing collision centrality, expressed by $\langle N_{part}\rangle$, via simultaneous minimum χ² fits in p_T region up to 5 GeV/c of the midrapidity p_T spectra of the charged pions and kaons, protons and antiprotons, measured by ALICE Collaboration [43], using Tsallis distribution function with embedded transverse flow. In Ref. [43] the $\langle N_{part}\rangle$ dependencies of the global transverse flow velocity parameter,$\langle {\beta }_T\rangle$, common kinetic freeze-out temperature, T₀, and non-extensivity parameter, q, for particle species of Tsallis function with incorporated transverse flow in p+Pb collisions at $\sqrt{s_{nn}}$ = 5.02 TeV were analyzed. The transverse flow velocity ($\langle {\beta }_T\rangle$) was absent at the first $\langle N_{part}\rangle$ point, corresponding to 80-100% centrality of p+Pb collisions, and then $\langle {\beta }_T\rangle$ started emerging and increased further with an increase in $\langle N_{part}\rangle$. The border (threshold) $\langle N_{part}\rangle$ value, where the collective transverse flow starts emerging simultaneously with T₀ transitioning to almost constant (plateau) region in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV, was estimated to be $\langle N_{part}\rangle$ ≈ 4.0 ± 0.5. Non-extensivity parameter q decreased systematically for all studied particle species with an increase in $\langle N_{part}\rangle$, suggesting an increase in degree of the system thermalization and equilibrium with centrality of p+Pb collisions at $\sqrt{s_{nn}}$ =5.02 TeV, in agreement with the similar result obtained in high-energy Xe+Xe and Pb+Pb collisions at the LHC [46,47,48]. Inequality q (kaons) > q (pions) > q ((anti)protons) was observed for non-extensivity parameter q in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV in agreement with the relation q(baryons) < q(mesons), established previously [33,34,46,47,48] in high-energy proton-proton and nucleus-nucleus collisions.

Transverse momentum distributions of particles generated in high-energy collisions are characterized by exponential behavior in the low-p_T part and power-like distribution, typical for perturbative Quantum-Chromo-Dynamics (pQCD), in the high-p_T part. The low-p_T part (p_T < 2 GeV/c) of hadron spectra contains valuable information about the collective radial expansion and thermal parameters of the system at kinetic freeze-out along with resonance decays [9]. Quark recombination processes, which lead to an enhancement of proton-to-pion yield ratio in nuclear collisions compared to proton-proton collisions, contribute mainly to intermediate p_T range (2 GeV/c <p_T < 4−5 GeV/c) of transverse momentum spectra. Quark and gluon fragmentation, QCD hard scatterings, production of quark-gluon jets and their energy loss in the QGP medium, contribute mostly to the high-p_T part (p_T > 5 GeV/c) of hadron spectra [9]. While high-p_T hadrons originate mostly from parton (quark or gluon) fragmentation, the dense QGP matter formed in nuclear collisions allows for partons to recombine, which favors baryon over meson production mainly in intermediate p_T range [9]. There are no exact p_T borders between various processes and mechanisms, contributing to the measured p_T spectra of hadrons and overlapping significantly in different p_T regions. Unfortunately, these various processes and mechanisms, contributing differently to the final p_T spectra of hadrons, still lack unambiguous theoretical description and can be reproduced by various phenomenological models only.

To reproduce accurately the measured long $p_T$ ranges of the midrapidity $p_T$ distributions of charged particles at various centralities of high-energy heavy-ion collisions at the LHC, the two-component model function, which is the sum of thermodynamically consistent Tsallis function with embedded transverse flow and inverse power-law function (which is asymptotic of Hagedorn function at high p_T values), was proposed in recent work [49]. The first and second terms of this two-component model function are assumed to describe the soft (together with intermediate region) and hard components, respectively, of midrapidity transverse momentum distributions of hadrons in high-energy collisions. This new two-component model function, called Tsallis-Hagedorn model with transverse flow (THMTF), described [49] very well the experimental midrapidity transverse momentum spectra of primary charged particles in the whole $p_T$ range up to 50 GeV/c, measured by ALICE Collaboration in minimum bias p+p collisions at $\sqrt{s}$=2.76 and 5.02 TeV [50], minimum bias p+Pb collisions at $\sqrt{s_{nn}}$= 5.02 TeV [50], and nine collision centrality intervals of Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [51] and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV [50]. The dependencies of the extracted parameters of THMTF function on Xe+Xe and Pb+Pb collision centrality at the LHC, expressed by $\langle N_{part}\rangle$ as well as $\langle d{N}_{ch}/d\rangle$, and c.m. collision energy (per nucleon pair) were analyzed and interpreted successfully in Ref. [49].

The main aim of the present work is to analyze the centrality ($\langle N_{part}\rangle$) dependencies of midrapidity p_T spectra of the charged pions and kaons, protons and antiprotons in the whole measured p_T range at seven centrality classes, expressed by $\langle N_{part}\rangle$, in p+Pb collisions at $\sqrt{s_{nn}}$ =5.02 TeV at the LHC, measured [43] by the ALICE collaboration, using the new two-component THMTF model function, introduced and probed successfully in Ref. [49]. The trends of $\langle N_{part}\rangle$ dependencies of the parameters of the two-component THMTF model function, obtained in present work in p+Pb collisions at $\sqrt{s_{nn}}$ =5.02 TeV at the LHC, will be compared systematically with the $\langle N_{part}\rangle$ tendencies of the corresponding parameters of the same model function, established recently [49] in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV at the LHC.

2. The Experimental Data and Two-Component Model

In present work, we analyze the midrapidity ($\left|y\right|$<0.5) transverse momentum distributions of the charged pions and kaons, protons and antiprotons measured [43] at seven centrality intervals in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV by ALICE Collaboration. The average numbers of participant nucleons ($\langle N_{part}\rangle$), calculated in Ref. [52] in seven centrality classes of p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV, are shown in Table 1.

The following equation [53,54,55,56] of non-extensive Tsallis distribution [57,58,59] function, derived for a zero chemical potential (μ ≈ 0 at the TeV energy range) at midrapidity (<y>=0), satisfies the thermodynamic relations for pressure, temperature, particle number, energy, and entropy densities [53]:

$${\displaystyle \frac{d^{2}N}{2\pi N_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_q{m_{\mathrm{T}}\left(1+(q-1){\displaystyle \frac{m_{\mathrm{T}}}{T}}\right)}^{-q/(q-1)},$$

(1)

where C_q is the fitting constant, N_ev – the total number of inelastic collision events,$m_T=\sqrt{p_T^2+m_0^2}$ – the transverse energy (mass), and m₀ is the hadron's rest mass. Parameter T is the effective temperature, and q is the non-extensivity parameter. The parameter, q, is the fundamental for Tsallis distribution, and it describes the fluctuations in system’s temperature through the relationship $q-1={\displaystyle \frac{Var\left(T\right)}{{<T>}^2}}$. The difference from unity of this parameter, q−1, is assumed to quantify the degree of non-thermalization and non-equilibrium within the system. When q approaches 1, the Tsallis distribution transitions to the exponential Boltzmann-Gibbs distribution, which describes systems in equilibrium. The closer the q parameter is to one (1), the larger the degree of thermalization (equilibrium) of the system. We call the function in Eq. (1) the thermodynamically consistent Tsallis function. The thermodynamic consistency of the function in Eq. (1) was proved in Ref. [53]. The normalization constant is related to the system’s freeze-out volume (V) [53,54] via expression C_q=gV/(2π)³, where g is the degeneracy factor of hadron species.

The effective temperature (T) in Eq. (1) comprises contributions from both chaotic thermal motion and transverse radial flow. To separate these two factors, the velocity of transverse flow should be incorporated into the Tsallis distribution function. The transverse is incorporated into thermodynamically consistent Tsallis distribution in Eq. (1) by substitution $m_{\mathrm{T}}\to \langle {\gamma }_{\mathrm{T}}\rangle (m_{\mathrm{T}}-p_{\mathrm{T}}\langle {\beta }_{\mathrm{T}}\rangle )$, as made for the first time in Ref. [46], to obtain the function:

$${\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_q{\langle {\gamma }_{\mathrm{T}}\rangle \left(m_{\mathrm{T}}-p_{\mathrm{T}}\langle {\beta }_{\mathrm{T}}\rangle \right)\left(1+\langle {\gamma }_{\mathrm{T}}\rangle {\displaystyle \frac{\left(q-1\right)\left(m_{\mathrm{T}}-p_{\mathrm{T}}\langle {\beta }_{\mathrm{T}}\rangle \right)}{T_0}}\right)}^{-q/(q-1)},$$

(2)

The substitution $m_T\to \langle {\gamma }_T\rangle (m_T-p_T\langle {\beta }_T\rangle )$ means [46] the Lorentz transformation to a system co-moving with an average transverse flow velocity, $\langle {\beta }_T\rangle$, of the system of particles in the transverse plane assuming that such a flow of particles exists in this system. The thermodynamically consistent Tsallis function with embedded transverse flow in Eq. (2) reproduced [45,46,47,48] very well the midrapidity p_T spectra of identified charged hadrons in the low and intermediate p_T ranges up to 5 GeV/c in high-energy p+Pb, Xe+Xe, and Pb+Pb collisions at the LHC, allowing for the physically meaningful interpretation of the obtained freeze-out parameters and their dependencies on collision centrality ($\langle N_{part}\rangle$) and system’s size.

The following inverse power law − QCD inspired Hagedorn function [60,61,62] reproduced very well the high transverse momentum part of p_T spectra of hadrons in nucleon-nucleon collisions at high energies:

$${\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_n{\left(1+{\displaystyle \frac{p_T}{P_0}}\right)}^{-n},$$

(3)

where $C_n$ is the fitting constant, proportional to the freeze-out volume of the system, n and P₀ are free parameters, which are related to underlying mechanisms of hadron production. Parameter $P_0$, is the characteristic momentum scale, and it represents the typical momentum scale, where the production mechanism transitions from soft (thermal/exponential) to hard (power-law/perturbative) processes. In modified Hagedorn function, $P_0$can also be expressed through temperature of the system (T) and the power-index n via relation $P_0=nT$. This parameter could also feature the average transverse momentum, contributed by partons before they fragment into hadrons. The power-index parameter n characterizes the hardness of the interaction. A low value of n reflects a hard spectrum dominated by perturbative QCD (pQCD) scattering and jet production. The QCD cross sections at high energies are demonstrated [60,63] to scale as ${\left(p_T\right)}^{-n}$, in which the power-index n can be phenomenologically expressed as $n=2n_{\alpha }-4$, where $n_{\alpha }$ is the number of active flavors or the number of partons (quarks or gluons) participating in a scattering. For the leading twist, simple scattering, the dominant process is a simple point-like scattering between two quarks, $q+qq+q$, with the number of participant partons, $n_{\alpha }$=4. This leads to the power-index parameter $n=2n_{\alpha }-4=4$. For the higher twist, multiple scattering, multiple quark-quark or quark-hadron scatterings occur, resulting in an increase in n value. This shows involvement of more constituent participants in particle production. The n value varies significantly as a function of the constituent quark content, differing between mesons and baryons. These variations in the n-value stem from the distinct quark structures and production mechanisms of mesons and baryons, explicitly depending on the number of constituent partons required to coalesce into the final hadron. Within a statistical framework, the Hagedorn parameter n scales inversely with the Tsallis non-extensivity parameter q according to the relation $n=1/(q-1)$.

In the limit of high $p_T$, the Hagedorn function in Eq. (3) reduces to the asymptotic form of a simple inverse power law:

$${\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=C_n{\left({\displaystyle \frac{p_T}{P_0}}\right)}^{-n}.$$

(4)

In order to accurately describe the measured long $p_T$ ranges of midrapidity transverse momentum spectra for charged particles at different centralities in high-energy heavy-ion and proton–proton collisions, a new two-component model function composed of the sum of a thermodynamically consistent Tsallis function with embedded transverse flow (Eq. (2)) and a simple inverse power-law function (Eq. (4)) was proposed recently in our work [49]:

$$\begin{array}{}{\displaystyle \frac{d^{2}N}{{2\pi N}_{ev}{p}_{\mathrm{T}}d{p}_{\mathrm{T}}dy}}=\\ =\theta \left(x_{thr}-p_T\right)C_q{\langle {\gamma }_{\mathrm{T}}\rangle \left(m_{\mathrm{T}}-p_{\mathrm{T}}\langle {\beta }_{\mathrm{T}}\rangle \right)\left(1+\langle {\gamma }_{\mathrm{T}}\rangle {\displaystyle \frac{\left(q-1\right)\left(m_{\mathrm{T}}-p_{\mathrm{T}}\langle {\beta }_{\mathrm{T}}\rangle \right)}{T_0}}\right)}^{-q/(q-1)}+{\theta \left(p_T-x_{thr}\right)C}_n{\left({\displaystyle \frac{p_T}{P_0}}\right)}^{-n},\end{array}$$

(5)

where $\theta \left(x\right)$ is the Heaviside function with the following properties: $\theta \left(x\right)=0$ if $x<0,$and $\theta \left(x\right)=1$ if $x>0$. This two-component model function in Eq. (5) is called Tsallis-Hagedorn model with transverse flow (THMTF). Parameter $x_{thr}$ is the optimal value of $p_T$border between the corresponding $p_T$ fit ranges of the first and second term in Eq. (5). The first term in Eq. (5) is assumed to describe the low (soft) and intermediate $p_T$ range, while the second term should describe the high (hard) $p_T$ region. It is well known that at the TeV collision energy range at the LHC, soft bulk particle production dominates below $p_T$ ≈ 2 GeV/c, while above $p_T$ ≈ 4−5 GeV/c hard fragmentation becomes the primary mechanism for particle production. In our work, we use fixed $x_{thr}=5$GeV/c to make sure that $p_T$ > 5 GeV/c region is practically entirely governed by the hard scattering perturbative QCD processes, described by the second power-law term in Eq. (5). Moreover, the first term in Eq. (5), expressed by thermodynamically consistent Tsallis function with incorporated transverse flow, given in Eq. (2), reproduced very well the midrapidity transverse momentum distributions of identified charged particles at various centralities of high-energy heavy-ion collisions in the low and intermediate $p_T$ range up to $p_T$ = 5 GeV/c [45,46,47,48]. Using $x_{thr}=5$GeV/c in Eq. (5) denotes that the hadron $p_T$ spectrum is reproduced by thermodynamically consistent Tsallis function with embedded transverse flow (Eq. (2)) in $p_T$ ≤ 5 GeV/c region and by inverse power-law function (Eq. (4)) in $p_T$ > 5 GeV/c range.

To study the correlation between two sets of parameters x and y of the two-component model function in Eq. (5) in present work, we calculate the Pearson coefficient of linear correlation as follows:

$$R(x,y)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}(x_i-<x>)\bullet (y_i-<y>)}{\sqrt{{\displaystyle {\sum }_{i=1}^n}{(x_i-<x>)}^2\bullet {\displaystyle {\sum }_{i=1}^n}{(y_i-<y>)}^2}}},$$

(6)

where $<x>={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{x}_i}{n}}$and $<y>={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{y}_i}{n}}$ are the mean values of the parameters x and y. The Pearson correlation coefficient, $R(x,y)$, changes from −1 to +1, being a statistical measure of a linear correlation between two sets of data. The $R(x,y)$ = ±1 values denotes a perfect linear dependence between parameters x and y with all data points (x_i, y_i) lying on a line in XY plane. The $R(x,y)$ = 0 value implies an absence of a linear correlation between parameters x and y. The positive and negative values of $R(x,y)$ show the positive and negative (linear) correlation, respectively, between parameters x and y.

To evaluate the uncertainty in the $R(x,y)$ values, the standard error of Pearson correlation coefficient is calculated as follows:

$$s_r=\sqrt{{\displaystyle \frac{1-R^2(x,y)}{n-2}}}.$$

(7)

The formula in Eq. (7) is obtained under an assumption that the data are normally distributed and with the null hypothesis that there is a zero correlation between x and y.

In present work, we analyze the midrapidity p_T spectra of the charged pions and kaons, protons and antiprotons at seven multiplicity classes (see Table 1) in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV at the LHC, measured [43] by the ALICE collaboration, using the THMTF function presented in Eq. (5). References [64,65] state that analyzing a single particle species via model fits fails to yield physically meaningful collective properties, including kinetic freeze-out temperature and radial transverse flow velocity. In contrast, extracting collective parameters like global freeze-out temperature and average transverse flow velocity becomes physically meaningful when performing simultaneous fits to various particle species produced in a collision [64,65]. Simultaneously fitting pion, kaon, proton, and antiproton spectra allows for an efficient comparison of different collision systems using only a few parameters [33,34,45,46,47,48,64,65]. In the present analysis, simultaneous minimum χ² fits using the THMTF function (Eq. 5) were applied to the transverse momentum spectra of charged pions and kaons, protons and antiprotons across each centrality class in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV. This procedure was executed using the Nonlinear Curve Fitting package within Origin 9.1 Data Analysis and Graphing Software. During the simultaneous fitting procedure, the parameters T₀, <β_T>, and P₀ were treated as global (shared) variables across all analyzed particle species. Error bars depicted in the figures of present work show the combined (added) statistical and systematic uncertainties. This total uncertainty is mostly dominated by systematic components, which are significantly larger than the statistical ones. Minimum χ² fits were executed by weighting each experimental data point by the inverse square of its combined uncertainty, (1/(combined error)²). Following the approach in Refs. [45,46,47,48,64,65], the low-transverse-momentum region of p_T < 0.5 GeV/c for charged pions was excluded from the fitting procedure. This exclusion is a standard, commonly accepted procedure [64,65] because the low p_T < 0.5 GeV/c region contains a significant fraction of pions originating from baryon resonance decays. The simultaneously fitted p_T ranges have been as follows: [0.5-20.0] GeV/c for ${\pi }^{+}$+${\pi }^{-}$, [0.2-20.0] GeV/c for $K^{+}$+$K^{-}$, and [0.3-20.0] GeV/c for p+$\overline{p}$.

3. Analysis and Results

The parameters obtained from simultaneous fits with two-component model (THMTF) function (Eq. (5)) of experimental midrapidity $p_T$ spectra of the charged pions and kaons, protons and antiprotons in seven multiplicity classes in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV are presented in Table 2. The corresponding minimum ${\chi }^2$ simultaneous fit curves of experimental midrapidity $p_T$ spectra of the studied particle species in various centrality classes of p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV are shown in Figure 1. As seen from Figure 1 and ${\chi }^2$/n.d.f. values in Table 2, the THMTF function in Eq. (5) describes quite well the experimental midrapidity $p_T$ spectra of the charged pions and kaons, protons and antiprotons in the whole measured $p_T$ range up to 20 GeV/c at all seven multiplicity classes in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV.

The dependencies of the obtained parameters T₀, ${<\beta }_{\mathrm{T}}>$, and q of two-component model (THMTF) function (Eq. (5)), given in Table 2, on the average number of participant nucleons ($\langle N_{part}\rangle$) in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV are illustrated in Figure 2. The dependencies of the obtained parameters P₀ and n of two-component THMTF function, given in Table 2, on $\langle N_{part}\rangle$ in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV are demonstrated in Figure 3. The dependencies of the parameters q, T₀, P₀, and n on $\langle N_{part}\rangle$, obtained in Ref. [49] from minimum χ² fits with two-component THMTF function of the experimental $p_T$ spectra of the primary charged particles at nine centrality classes in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV, are presented in Figure 4 for a comparison. Pearson correlation coefficients, $R(x,y)$, between various parameters of two-component THMTF function, extracted in present work from simultaneous minimum χ² fits of experimental midrapidity $p_T$ spectra of the charged pions and kaons, protons and antiprotons in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV and given in Table 2, are presented in Table 3. The corresponding Pearson correlation coefficients, $R(x,y)$, between various parameters of two-component THMTF function, obtained in Ref. [49] from minimum χ² fits of experimental midrapidity $p_T$ spectra of primary charged particles in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV, are shown in Table 4 for a comparison.

Figure 2 (b) shows that the transverse flow velocity is initially absent at the first data point, which corresponds to the 80–100% centrality interval of p+Pb collisions. Beyond this point, the average transverse velocity, ${<\beta }_{\mathrm{T}}>$, emerges and increases monotonically with growing collision centrality, represented by $\langle N_{part}\rangle$. At the same time, Figure 2 (a) indicates that the kinetic freeze-out temperature T₀ rises with $\langle N_{part}\rangle$ up to approximately $\langle N_{part}\rangle$ ≈ 4, after which it remains constant within the experimental uncertainties in region $\langle N_{part}\rangle$ > 4. A closer inspection of Figure 2 (b) reveals that while the transverse flow velocity is zero at the first point $\langle N_{part}\rangle$ = 2.99, it becomes statistically noticeable ${<\beta }_{\mathrm{T}}>$ ≈ 0.1 by the second point $\langle N_{part}\rangle$ = 4.9. Consequently, the threshold value for the onset of collective transverse radial flow in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV is estimated to be $\langle N_{part}\rangle$ ≈ 4.0 ± 0.5. This boundary is determined as the midpoint in $\langle N_{part}\rangle$ between the first and second data points in Figure 2 (b), which correspond to the 80–100% and 60–80% centrality classes listed in Table 1, respectively. This boundary value,$\langle N_{part}\rangle$ ≈ 4.0 ± 0.5, estimated in present work from analysis of $\langle N_{part}\rangle$ dependencies of the parameters ${<\beta }_{\mathrm{T}}>$ and T₀ of the two-component model (THMTF) function (Eq. (5)), has coincided with the corresponding boundary $\langle N_{part}\rangle$ value, evaluated in Ref. [45] from simultaneous analysis of midrapidity $p_T$ distributions of the charged pions and kaons, protons and antiprotons in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV in region $p_T$ ≤ 0.5 GeV/c using thermodynamically consistent Tsallis function with embedded transverse flow (Eq. (2)). Simultaneous analyses of ${<\beta }_{\mathrm{T}}>$, T₀ and q as functions of $\langle N_{part}\rangle$ in Refs. [46,47,48] have suggested that these boundary $\langle N_{part}\rangle$ values reflect not only the onset of transverse radial flow, but potentially also the threshold for a possible crossover phase transition from a dense hadronic state to a QGP. Indications of this potential transition threshold have been observed across several collision systems and energies at the LHC, including Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [46], as well as Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 TeV [48] and $\sqrt{s_{nn}}$=5.02 TeV [47]. The corresponding threshold values have been estimated to be $\langle N_{part}\rangle$ ≈ 44 ± 5, 71±5, and 71±7 in Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [46], Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 TeV [48], and Pb+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV [47], respectively.

The threshold $\langle N_{part}\rangle$ values for Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 TeV [48] and 5.02 TeV [47] are significantly higher than those observed in Xe+Xe collisions [46] at $\sqrt{s_{nn}}$=5.44 TeV. As noted in Refs. [47,48], this difference is likely due to the larger mass number (A) of Pb relative to Xe, implying that the greater nuclear size naturally yields a higher number of participant nucleons within the overlap zone, even though the threshold in both systems corresponds to a similar centrality range (between the 40–50% and 50–60% classes). Conversely, the consistency of the threshold $\langle N_{part}\rangle$ values between the two Pb+Pb energies can be attributed to the identical collision system and geometry [48].

In contrast, the threshold value of $\langle N_{part}\rangle$ ≈ 4.0 ± 0.5 estimated for p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV in present work is substantially smaller than those reported earlier for the heavier Xe+Xe [46] and Pb+Pb [47,48] collision systems at the LHC. This difference is expected due to the considerably smaller overlap volume and the resulting lower number of participant nucleons in asymmetric p+Pb collisions compared to symmetric nucleus-nucleus (Pb+Pb) systems.

Interestingly, the behavior of the freeze-out temperature T₀ as a function of $\langle N_{part}\rangle$ (or collision centrality) in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV (Figure 2 (a)) closely resembles the T₀ dependence on midrapidity charged-particle multiplicity density ($\langle d{N}_{ch}/d\rangle$) previously observed [33,34] in p+p collisions at $\sqrt{s}$ = 7 and 13 TeV. In both small systems, T₀ initially rises up to the threshold $\langle N_{part}\rangle$ value and subsequently stabilizes within uncertainties, exhibiting a distinct plateau-like behavior at higher multiplicities [33,34]. This initial increase of T₀ in small systems can be attributed to higher initial energy density and intense partonic hard scattering in higher-multiplicity events, which creates a hotter system that decouples rapidly before undergoing significant expansion and cooling.

In contrast, nucleus-nucleus (A+A) collisions display a fundamentally different trend, as recently reported for Xe+Xe collisions at $\sqrt{s_{nn}}$=5.44 TeV [46], Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 TeV [48], and Pb+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV [47]. In these heavy-ion systems, T₀ initially decreases up to the threshold $\langle N_{part}\rangle$ value before flattening into a plateau in the more central region [46,47,48]. This behavior can be seen also in Figure 4 (b), where the $\langle N_{part}\rangle$ dependence of T₀, obtained in Ref. [49] from minimum χ² fits with two-component THMTF function of the experimental $p_T$ spectra of the primary charged particles in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV, is presented. This contrasting behavior can be understood through a geometric and expansion-driven mechanism: in large nuclear systems, more central collisions lead to a larger initial volume and stronger collective radial flow. The resulting prolonged hydrodynamic expansion phase allows the system to cool down more extensively before reaching the low-density decoupling limit, effectively lowering the kinetic freeze-out temperature T₀ before it stabilizes in more central events.

Figure 2 (c) shows that non-extensivity parameter q consistently decreases across all analyzed particle species as the centrality (or $\langle N_{part}\rangle$) of p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV increases. This decreasing trend of q can be seen also in Figure 4 (a), where the $\langle N_{part}\rangle$ dependence of parameter q, obtained in Ref. [49] from minimum χ² fits with two-component THMTF function of the experimental $p_T$ spectra of the primary charged particles in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV, is shown. This decline of q implies that the degree of system thermalization and statistical equilibrium increases in more central collisions, as evidenced by q approaching unity (+1). Furthermore, the data reveal a clear hierarchy among particle species: q (mesons) > q ((anti)protons). While the q values for kaons and pions remain close to each other, they are significantly higher than those for (anti)protons. This mass- or quark number-dependent ordering agrees well with the relation q (baryons) < q (mesons) previously observed [45,46,47,48,49,66] in high-energy collisions.

Physically, this hierarchy can be understood in terms of quark coalescence and flavor dynamics. Baryons, composed of three constituent quarks, require a higher number of local partonic interactions to form compared to two-quark mesons. Consequently, baryons preferentially sample the denser, more frequently interacting, and highly thermalized core of the collision volume. This intensive interaction history drives baryons closer to a classical Boltzmann-Gibbs thermal distribution, resulting in a lower q value. In contrast, lighter mesons are more affected by surface production and hard scattering. These processes limit thermalization, which preserves stronger non-extensive correlations and increases their q parameters.

As illustrated in Figure 3 (a), the momentum scale parameter P₀ of the inverse power-law component in Eq. (5) rises steadily with $\langle N_{part}\rangle$ for p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV. This rise in P₀ stems from increased energy-momentum transfer and more frequent hard scatterings in more central collisions, which ultimately elevates the average $p_T$. Furthermore, comparing Figure 2 (b) and Figure 3 (a) reveals that this continuous growth of P₀ closely tracks the onset and evolution of the average transverse flow velocity (${<\beta }_{\mathrm{T}}>$). This is supported by the strong positive correlation between parameters ${<\beta }_{\mathrm{T}}>$ and P₀ with corresponding Pearson correlation coefficient, R(<β_T>, P₀) = +0.98±0.08, given in Table 3. This alignment confirms that more central p+Pb collisions generate stronger pressure gradients in the interaction zone, inducing a stronger transverse flow that pushes particles toward higher $p_T$ and consequently increases P₀.

As seen from Figure 3 (b), the power-index n for protons and antiprotons demonstrates a significant increase with $\langle N_{part}\rangle$ in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV, whereas parameter n for pions and kaons, on the whole, shows a very weak increase with an increase in collision centrality, staying nearly constant within uncertainties. In contrast to p+Pb collisions, the power-index n for primary charged particles in Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV in Figure 4 (d) demonstrates a significant decrease with $\langle N_{part}\rangle$ in region up to $\langle N_{part}\rangle$ ≈ 100, and then it stabilizes in region $\langle N_{part}\rangle$ > 100, showing a very weak decrease or almost plateau-like behavior.

While the momentum scale parameter P₀ reflects the initial local energy density and collective expansion across these systems, the structural evolution of the high-$p_T$ spectral tail, governed by the power index n, reveals a strong contrast in the underlying physics between p+Pb and Pb+Pb systems. Formally, the power index n quantifies the rate of fall-off of the high-$p_T$ tail, where a decreasing n denotes spectral hardening due to hard partonic processes or collective dynamics, and an increasing n indicates a softer spectral shape. In large-scale Pb+Pb collisions, central events generate a dense, strongly-coupled Quark-Gluon Plasma (QGP) that induces immense hydrodynamic pressure gradients. This drives a powerful collective radial flow (${<\beta }_{\mathrm{T}}>$) that hydrodynamically shifts the bulk particle distribution toward the high-$p_T$ region, resulting in a decrease in n, observed in Figure 4 (d), due to this spectral flattening. In contrast, the p+Pb system is too small for long-lasting fluid expansion. Light mesons in p+Pb collisions form primarily via vacuum-like jet fragmentation. Consequently, their spectral slope remains stable, yielding a negligible increase in n with $\langle N_{part}\rangle$, as seen in Figure 3 (b). Meanwhile, protons and antiprotons are strongly affected by non-perturbative processes like parton recombination or coalescence at intermediate $p_T$ values. This baryon enhancement heavily populates the mid-$p_T$ region, leaving the high-$p_T$ tail looking isolated and steeper, which mathematically forces a sharp increase in the fitted power index n with an increase in $\langle N_{part}\rangle$, observed for protons and antiprotons in p+Pb collisions in Figure 3 (b).

The similarities and differences in the $\langle N_{part}\rangle$ dependencies of the parameters of two-component THMTF function, observed in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV collisions in Figs. 2 and 3 and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV in Figure 4, are reflected in the similarities and differences in the Pearson linear correlation coefficients between corresponding parameters, presented in Table 3 and Table 4, respectively. As seen from Table 3 and Table 4, the correlation between parameters ${<\beta }_{\mathrm{T}}>$ and P₀, as well as between parameters P₀ and $\langle N_{part}\rangle$, is strongly positive in both p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV and Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV. This is because both ${<\beta }_{\mathrm{T}}>$ and P₀ rise with an increase in $\langle N_{part}\rangle$ in both p+Pb and Pb+Pb collisions, as seen in Figs. 2-4. On the other hand, the correlation between parameters q and P₀, between parameters q and ${<\beta }_{\mathrm{T}}>$ and between parameters q and $\langle N_{part}\rangle$, is strongly negative in both p+Pb and Pb+Pb collisions. It is due to the similar $\langle N_{part}\rangle$ dependencies of the parameters q, P₀, and ${<\beta }_{\mathrm{T}}>$ observed in p+Pb collisions and Pb+Pb collisions in Figs. 2-4. As seen from Table 3 and Table 4, the correlation between parameters ${<\beta }_{\mathrm{T}}>$ and n is significantly positive in p+Pb collisions, whereas it is strongly negative with R(<β_T>, n) ≈ −1 in Pb+Pb collisions. This is due to opposite trends in $\langle N_{part}\rangle$ dependencies of power-index n in p+Pb and Pb+Pb collisions, observed in Figs. 3 (b) and 4 (d): parameter n on the whole increases with $\langle N_{part}\rangle$ in p+Pb collisions in Figure 3 (b), whereas n generally decreases with an increase in $\langle N_{part}\rangle$ in Pb+Pb collisions in Figure 4 (d). Table 3 shows quite weak correlation between parameters q and T₀ in p+Pb collisions, whereas correlation between these parameters is strongly positive with R(q, T₀) ≈ +0.9 in Pb+Pb collisions in Table 4. This is, as discussed earlier, due to significant differences in the $\langle N_{part}\rangle$ dependencies of T₀ in p+Pb and Pb+Pb collisions, as seen in Figs. 2 (a) and 4 (b). The differences in the $R(x,y)$ values of other parameters in p+Pb collisions in present work and Pb+Pb collisions in Ref. [49] (Table 3 and Table 4) can be explained similarly. These discrepancies come from the corresponding differences observed in the $\langle N_{part}\rangle$ dependencies of either the kinetic freeze-out temperature parameter (T₀) or the power-index (n) in p+Pb and Pb+Pb collision systems.

4. Summary and Conclusions

In this work, we have presented a comprehensive systematic analysis of the centrality ($\langle N_{part}\rangle$) dependencies of midrapidity transverse momentum ($p_T$) spectra for identified light-flavor hadrons − specifically charged pions and kaons, and protons/antiprotons in asymmetric p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV. We have employed the new two-component Tsallis-Hagedorn Model with Transverse Flow (THMTF) to reproduce the experimental midrapidity $p_T$ spectra for identified light-flavor hadrons, measured by ALICE collaboration, across seven multiplicity classes, quantified by the average number of participant nucleons ($\langle N_{part}\rangle$), in p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV. This two-component THMTF function is the sum of thermodynamically consistent Tsallis function with embedded transverse flow and inverse power-law function (which is asymptotic of Hagedorn function at high p_T values), which are assumed to describe the soft (together with intermediate region) and hard $p_T$ components, respectively. The extracted THMTF function parameters and their physical trends with an increase in $\langle N_{part}\rangle$ in asymmetric p+Pb collisions at $\sqrt{s_{nn}}$=5.02 TeV were systematically compared with those of heavy-ion symmetric Pb+Pb collisions at $\sqrt{s_{nn}}$=2.76 and 5.02 TeV.

The two-component THMTF model has successfully described the experimental transverse momentum distributions across the entire measured $p_T$ range up to 20 GeV/c. The high quality of the simultaneous fits of midrapidity $p_T$ spectra of identified light-flavor hadrons, evidenced by the ${\chi }^2$/n.d.f. values, establishes this new two-component model as an efficient phenomenological tool capable of bridging the physics of small asymmetric and large symmetric collision systems using just few parameters.

The systematic mapping of the fit parameters as a function of $\langle N_{part}\rangle$ has revealed key insights into the thermodynamic and collective properties of the media produced in high-energy collisions at the LHC. While collective radial flow is absent in peripheral p+Pb collisions at $\langle N_{part}\rangle$ = 2.99, it emerges at higher multiplicities. The threshold for the onset of collective transverse radial flow has been estimated to be $\langle N_{part}\rangle$ ≈ 4.0 ± 0.5. This boundary is further supported by the $\langle N_{part}\rangle$ dependence of the kinetic freeze-out temperature T₀, which rises steadily with increasing centrality up to $\langle N_{part}\rangle$ ≈ 4.0 before transitioning into a stable, constant plateau region. This threshold, representing the onset of collective expansion in p+Pb collisions and potentially the critical boundary for a crossover phase transition to a QGP, is substantially smaller than those reported earlier for symmetric systems ($\langle N_{part}\rangle$ ≈ 44 ±5 and ≈ 71 ±7 in Xe+Xe and Pb+Pb collisions at the LHC, respectively). This scaling reflects the fundamental role of initial collision geometry, where smaller overlap volumes naturally require fewer participant nucleons to initiate collective behavior.

The kinetic freeze-out temperature parameter (T₀) exhibits fundamentally divergent trends between small and large systems. In p+Pb collisions, T₀ initially rises up to the threshold $\langle N_{part}\rangle$ value before stabilizing into a plateau, closely resembling the similar behavior observed in p+p collisions at $\sqrt{s}$ = 7 and 13 TeV at the LHC. This is driven by high initial energy densities and intense partonic hard scattering in high-multiplicity proton-proton events, causing the hot tiny-volume system to decouple rapidly before undergoing significant expansion. Conversely, heavy-ion systems (Pb+Pb and Xe+Xe) exhibit an initial decrease in T₀ due to a larger initial volume and a prolonged hydrodynamic expansion phase, which allows the medium to cool down extensively before reaching the low-density decoupling limit.

In both p+Pb and Pb+Pb collision systems, the non-extensivity parameter q consistently decreases toward unity (+1) as centrality increases, indicating enhanced thermalization and a transition toward classical Boltzmann-Gibbs statistical equilibrium in central collisions. The observed hierarchy, q(mesons) > q(baryons), can be interpreted by whether these hadrons are produced in the highly thermalized core or the peripheral regions of the expanding medium. Three-quark baryons need more interactions to assemble, so they are likely produced in the highly thermalized center of the collision zone (lower q). In contrast, lighter two-quark mesons are often produced near the outer edges, meaning they stay further from thermal equilibrium (higher q).

The momentum scale parameter (P₀) rises steadily with centrality in both p+Pb and Pb+Pb collisions, strongly correlating with the average flow velocity (${<\beta }_{\mathrm{T}}>$) (R(<β_T>, P₀) ≈ +1), suggesting that larger pressure gradients drive both collective flow and high-$p_T$ particle production. However, the power-index n, which governs the high-$p_T$ spectral tail, reveals a strong contrast in underlying physics of p+Pb and Pb+Pb systems. In large-scale Pb+Pb collisions, n decreases with centrality (spectral flattening) likely due to massive hydrodynamic pressure shifting the bulk distribution toward higher $p_T$. In small p+Pb collisions, n remains nearly constant for light mesons (consistent with their production via vacuum-like jet fragmentation), but increases sharply for protons and antiprotons with an increase in $\langle N_{part}\rangle$. This can be a clear signature of intermediate-$p_T$ baryon enhancement driven by non-perturbative parton recombination/coalescence, which steepens the remaining high-$p_T$ tail with an increase in centrality of p+Pb collisions.

The physical differences between these two p+Pb and Pb+Pb collision environments are further validated by Pearson linear correlation analyses. The correlation between the average transverse flow velocity (${<\beta }_{\mathrm{T}}>$) and the power-index n is significantly positive in p+Pb collisions, but strongly negative (R(<β_T>, n) ≈ −1) in Pb+Pb collisions. This is due to nearly opposite trends observed in $\langle N_{part}\rangle$ dependencies of power-index n in p+Pb and Pb+Pb collisions: parameter n on the whole increases with $\langle N_{part}\rangle$ in p+Pb collisions, whereas n generally decreases with an increase in $\langle N_{part}\rangle$ in Pb+Pb collisions. Similarly, the correlation between non-extensivity parameter q and freeze-out temperature, T₀, is quite weak in p+Pb, but strongly positive (R(q, T₀) ≈ +0.9) in Pb+Pb collisions. This is due to significant differences in the $\langle N_{part}\rangle$ dependencies of T₀ in p+Pb and Pb+Pb systems. Similarly, the remaining differences in parameter correlations between the p+Pb and Pb+Pb systems in Table 3 and Table 4 can be explained simply by distinct centrality ($\langle N_{part}\rangle$) dependencies of parameters T₀ and n in p+Pb and Pb+Pb collisions.

Although both the small asymmetric (p+Pb) and large symmetric (Pb+Pb) collision systems show signs of partial thermalization and collective radial expansion in more central collisions, they appear to evolve through different dynamics. While high-energy heavy-ion (Pb+Pb) collisions are characterized by strong fluid-like expansion and cooling, the smaller p+Pb systems seem to be more influenced by localized partonic interactions and rapid decoupling. These results establish critical benchmarks for theoretical models describing the transition from microscopic interactions to collective behavior in high-energy collisions.