Multiplicity Dependencies of Midrapidity Transverse Momentum Distributions of Identified Charged Particles in Proton-Proton Collisions at (s)^1/2 = 7 TeV at the LHC

Abstract

Dependencies of midrapidity p_t distributions of the charged pions and kaons, protons and antiprotons on charged-particle multiplicity density (<dN_ch/dη>) in inelastic proton-proton collisions at (s)^1/2 = 7 TeV at the LHC, measured by ALICE Collaboration, are investigated. The simultaneous minimum χ² fits with the Tsallis function with thermodynamical consistence and the Hagedorn function with included transverse flow have well-described the p_t spectra of the particle species in the ten studied groups of charged-particle multiplicity density. The effective temperatures, T, of the Tsallis function with thermodynamical consistence have shown a steady rise with increasing the charged-particle multiplicity in proton-proton collisions at (s)^1/2 = 7 TeV, in agreement with the similar result obtained recently in proton-proton collisions at (s)^1/2 = 13 TeV at the LHC. The respective T versus <dN_ch/dη> dependence in proton-proton collisions at (s)^1/2 = 7 TeV is reproduced quite well by the simple power function with the same value (≈ 1/3) of the exponent parameter as that extracted in proton-proton collisions at (s)^1/2 = 13 TeV. The identical power dependence $T~{\epsilon }^{1/3}$ between the initial energy density and effective temperature of the system has been observed in proton-proton collisions at (s)^1/2 = 7 and 13 TeV. We have observed that the transverse radial flow emerges at <dN_ch/dη> ≈ 6 and then increases, becoming substantial at larger multiplicity events in proton-proton collisions at (s)^1/2 = 7 TeV. We have estimated, analyzing T₀ and $⟨{\beta }_t⟩$ versus <dN_ch/dη> dependencies, that the possible onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 7 TeV occurs at <dN_ch/dη> ≈ 6.1 ± 0.3, which is close to the corresponding recent estimate (<dN_ch/dη> ≈ 7.1 ± 0.2) in proton-proton collisions at (s)^1/2 = 13 TeV. The corresponding critical energy densities for probable onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 7 and 13 TeV at the LHC have been estimated to be 0.67 ± 0.03 and 0.76 ± 0.02 GeV/fm³, respectively.

1. Introduction

The widespread use [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25] of the Tsallis distribution function in high-energy proton-proton collisions is explained by its very good parameterization of the experimental p_t spectra of hadrons with just a few parameters: the first one—effective temperature (T), the second one—parameter of non-extensivity, q, which accounts for deviation of p_t distribution from the Boltzmann–Gibbs exponential distribution, and the third parameter—the fitting constant, assumed to be proportional to the system volume. It is important to note that the Tsallis function has the advantage of being connected through the entropy to thermodynamics, which is not true with other power law distributions [8]. The q and T parameters can also be employed for the identification of the system size scaling as well as initial conditions [25].

There are various modifications of the Tsallis function, which have equally well-described the p_t distributions of final hadrons in proton-proton collisions up to the largest available p_t values at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) experiments [3,4,5,6,7,8,9,10,15]. The non-extensivity parameter, q, of the Tsallis function has shown quite noticeable sensitivity to the large p_t region (p_t > 3 GeV/c) of the invariant p_t distributions of hadrons, suggesting the necessity of analyzing the longer p_t intervals for extracting the more correct q values [22,23,24].

The different transverse flow models have been incorporated into Tsallis statistics to describe the p_t distributions of hadrons in high-energy heavy-ion and proton-proton collisions at the RHIC and LHC. In most cases, the Blast-Wave model with Boltzmann–Gibbs statistics (the BGBW model) [26,27,28], the Blast-Wave model with Tsallis statistics (the TBW model) [29,30], the Tsallis distribution with transverse flow effect—improved Tsallis distribution [30,31,32], and the Hagedorn formula (function) with transverse flow [11,16,21,33] have been used to estimate the kinetic freeze-out temperature and transverse expansion velocity.

Most of the works analyzing high-energy collisions with various versions of the Tsallis function and its modified forms, incorporating other model functions, have used separate fits to p_t distribution of particular particle species in a studied collision. However, as indicated in [34], it is not feasible to assign the physical meaning to collective properties, such as kinetic freeze-out temperature or radial transverse flow velocity, extracted from separate model fits to each particle type. The simultaneous model fits to p_t distributions of various particle species in a studied collision system, performed by keeping the temperature and transverse flow velocity as the common (shared) fitted parameters for all particle species, can produce the physically meaningful collective parameters of a collision system [16,21,33,34]. The combined (global) fits have proved to be quite efficient for the extraction of collective properties and the comparison of different collision systems with the help of few parameters [16,21,33,34].

The modern LHC experiments have the goal to create the Quark-Gluon Plasma (QGP) in collisions of heavy ions at high energies to investigate the various properties of QGP matter, which is assumed to have been created a few microseconds after the so-called “Big Bang”, thought to be a starting point for the birth of our Universe. Nonetheless, analysis of tiny p + p collision systems at high energies is also important and interesting. Such analyses are needed not only because the results from proton-proton collisions are used as a baseline for the investigation of heavy-ion collisions, but also to study the collective properties of a system produced in proton-proton collisions at the largest energies achieved at modern collider experiments [15]. The observations [21,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50] of different QGP signals in high-multiplicity proton-proton collisions at the LHC, including strong similarity and resemblance of their collective properties to those of heavy ions, serve as a good motivation to further investigate the high-energy proton-proton collisions.

In the present article, we analyze the midrapidity (mid-y) p_t spectra of given particle species at ten groups of the average charged-particle (pseudo-rapidity) multiplicity density (<dN_ch/dη>) in proton-proton collisions at (s)^1/2 = 7 TeV at the LHC, measured by the ALICE collaboration and included in [45]. The goal of the present paper is to extract information on the evolution of collective parameters of the p + p system with changing <dN_ch/dη> through combined (simultaneous) fits of p_t distributions of the charged pions and kaons, protons and antiprotons in each group of <dN_ch/dη>, applying the Tsallis distribution with thermodynamical consistence and the Hagedorn function with included transverse flow implemented over the entire measured long p_t region. The present work follows and uses the same analysis methods as the recent paper [21], in which the mid-y p_t distributions of the same species of particles at ten different groups of <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 13 TeV at the LHC, measured by ALICE collaboration and presented in [51], have been investigated. Therefore, the results obtained in the present work for proton-proton collisions at (s)^1/2 = 7 TeV are compared systematically with the corresponding results in [21] extracted for proton-proton collisions at (s)^1/2 = 13 TeV.

2. The Data and Methods

ALICE collaboration has selected [45] inelastic proton-proton collision events at (s)^1/2 = 7 TeV with at least one produced charged particle in the pseudo-rapidity region $\left|\eta \right|$ < 1, which make up around 75% of the total inelastic cross-section. To study the dependence of light-flavor hadron production on multiplicity, the selected collision events have been divided into event classes (groups) according to the total charge accumulated in two V0 detectors (V0M amplitude). The V0M amplitude scales linearly with the total number of the charged particles produced in the pseudo-rapidity window corresponding to acceptance of V0 scintillators [45]. The respective average charged-particle pseudo-rapidity densities, <dN_ch/dη>, in each class of events have been measured at mid-pseudo-rapidity ($\left|\eta \right|$ < 0.5). The low p_t part of the particle spectra has been reconstructed in the rapidity interval $\left|y\right|$ < 0.5, while the high p_t part has been extracted in the pseudo-rapidity range $\left|\eta \right|$ < 0.8 in order to make use of the full statistics of inelastic proton-proton collisions [45]. The average charged-particle multiplicity density and corresponding fractions of inelastic cross-sections for different groups of event multiplicity are shown in

Table 1. The mean charged-particle multiplicity density and fraction of inelastic cross-section in ten multiplicity groups in inelastic proton-proton collisions at (s)^1/2 = 7 TeV.

V0M Mult. Class	<dNch/dη>	σ/σINEL>0 (%)
I	21.3 ± 0.6	0–0.95
II	16.5 ± 0.5	0.95–4.7
III	13.5 ± 0.4	4.7–9.5
IV	11.5 ± 0.3	9.5–14
V	10.1 ± 0.3	14–19
VI	8.45 ± 0.25	19–28
VII	6.72 ± 0.21	28–38
VIII	5.40 ± 0.17	38–48
IX	3.90 ± 0.14	48–68
X	2.26 ± 0.12	68–100

. The p_t ranges [45] of midrapidity p_t distributions in inelastic proton-proton collisions at (s)^1/2 = 7 TeV are as follows: [0.1–20.0] GeV/c for ${\pi }^{+}$ + ${\pi }^{-}$, [0.2–20.0] GeV/c for $K^{+}$ + $K^{-}$, and [0.3–20.0] GeV/c for p + $\overline{p}$. These p_t ranges coincide with those measured [21] by ALICE collaboration at midrapidity in ten groups of <dN_ch/dη> in inelastic proton-proton collisions at (s)^1/2 = 13 TeV.

It is well-known that the high p_t part of the p_t spectra of particles in high-energy nucleon–nucleon collisions can be reproduced well by the QCD-inspired Hagedorn function [52]:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{t}d{p}_{t}dy}=C{\left(1+\frac{m_t}{p_0}\right)}^{-n}$$

(1)

where C is the normalization constant, p₀ and n are free parameters, $m_t=\sqrt{p_t^2+m_0^2}$ is the transverse energy, and m₀ is the rest mass of a particle. As mentioned above, the various forms of the Tsallis function [1,2] can well-reproduce the p_t and m_t spectra of particles in proton-proton collisions at high energies [3,4,5,6,7,8,9,10,11,12,13,22,53]. In its simplest version [10,11,20], the Tsallis function is given at mid-y (y ≈ 0) by:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{t}d{p}_{t}dy}=C_x{\left(1+\left(q-1\right)\frac{m_t}{T}\right)}^{-1/\left(q-1\right)}$$

(2)

where $C_x$ is the normalization constant, T is the effective temperature, and q is the non-extensivity parameter, characterizing the deviation of p_t distribution from the exponential Boltzmann–Gibbs distribution. The function in Equation (2) is called a non-extensive generalization of the Boltzmann–Gibbs distribution ($~\exp (-\frac{E}{T}$)), with the q parameter added to the temperature parameter. The parameter q is assumed to also be a measure of non-thermalization [54]. When q approaches one (1), the Tsallis function tends to the usual Boltzmann–Gibbs distribution. The degree of system thermalization increases as q approaches one.

Compared to various versions of Tsallis distributions, the following one [3,9,10] at mid-y leads to consistent thermodynamics for the pressure, energy density, and particle number:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{t}d{p}_{t}dy}=C_q{m}_t{\left(1+\left(q-1\right)\frac{m_t}{T}\right)}^{-q/\left(q-1\right)}$$

(3)

which we call the Tsallis function with thermodynamical consistence in the present analysis. The values of T, extracted from Tsallis function (Equations (2) and (3)) fits, represent the effective temperatures, which include contributions of both the thermal motion and collective expansion effects. To separate these two factors, the transverse flow velocity is incorporated into the Tsallis distribution function. It has been demonstrated in [10,16] that the Tsallis distribution with thermodynamical consistence, provided in Equation (3), results in noticeably smaller values of T compared to T obtained using the simple non-consistent Tsallis function in Equation (2). This is due to the additional m_t term in Equation (3).

The functions represented by Equations (1) and (2) are mathematically equivalent when making substitutions, $n=1/\left(q-1\right)$ and $p_0=nT$. The larger n values correspond to lower q values. For the quark-quark point scattering, n ≈ 4, and the parameter n becomes larger in case of the multiple scattering centers involved [6,55,56]. To embed the effective temperature, T, the function in Equation (1) is modified, setting $p_0=n{T}_{}$:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{t}d{p}_{t}dy}=C{\left(1+\frac{m_t}{n T}\right)}^{-n}$$

(4)

In the present work, we include the transverse flow into Equation (4) by using the transformation $m_t\to \langle {\gamma }_t\rangle \left(m_t-p_t\langle {\beta }_t\rangle \right)$, as also performed in [11,15,16,21,33]. Then, Equation (4) is modified to:

$$\frac{d^{2}N}{2\pi N_{ev}{p}_{t}d{p}_{t}dy}=C{\left(1+\langle {\gamma }_t\rangle \frac{\left(m_t-p_t\langle {\beta }_t\rangle \right)}{n{T}_0}\right)}^{-n}$$

(5)

Here, $\langle {\gamma }_t\rangle =1/\sqrt{1-{\langle {\beta }_t^{}\rangle }^2}$, $<{\beta }_t>$ is the mean transverse flow velocity, and T₀ is an estimated kinetic freeze-out temperature. We call the function in Equation (5) the Hagedorn function with included transverse flow in the present work.

Successful analysis, using the function in Equation (5), of p_t distributions of particles in collisions at high energies has been performed in [11,15,16,21,33]. The model represented by the function in Equation (5) is a powerful tool, which probes the long p_t ranges of particles, allowing to compare different collision systems using few parameters [15,16,21,33]. The global parameters $<{\beta }_t>$ and T₀ obtained in [33] in central Cu + Cu, Au + Au, and Pb + Pb collisions at mid-y at (s_nn)^1/2 = 200–2760 GeV at the RHIC and LHC, applying the modified Hagedorn function in Equation (5) over the long p_t range, have reproduced all the observed dependencies of $<{\beta }_t>$ and T₀ on <N_part> and (s_nn)^1/2, extracted using three different transverse expansion (blast-wave) models in the low p_t range. The values of $<{\beta }_t>$ in the most central (0–5%) Pb + Pb collisions at (s_nn)^1/2 = 2.76 and 5.02 TeV, extracted using the function in Equation (5) over long p_t ranges in [16], have agreed within fit errors with the respective values of $\langle {\beta }_t\rangle$, obtained by ALICE collaboration in the most central (0–5%) Pb + Pb collisions at (s_nn)^1/2 = 2.76 and 5.02 TeV in [34] and [57], respectively, using the combined Boltzmann–Gibbs blast-wave fits applied over low p_t regions. Simultaneous fits with the modified Hagedorn function in Equation (5), applied over long p_t intervals in [16], have confirmed that $\langle {\beta }_t\rangle$ grows and T₀ decreases with increasing the collision centrality in Pb + Pb collisions at (s_nn)^1/2 = 2.76 and 5.02 TeV, which agrees well with the similar result of ALICE collaboration obtained from combined Boltzmann–Gibbs blast-wave fits in the low p_t ranges in the same collisions in [34,57].

In the present analysis, for the description of p_t distributions, d²N/(N_evdp_tdy), of the charged particles in ten groups of event multiplicity in inelastic proton-proton collisions at (s)^1/2 = 7 TeV, we used the Tsallis function with thermodynamical consistence (Equation (3)) in the following form:

$$\frac{d^{2}N}{N_{ev}d{p}_{t}dy}=2\pi C_q{p}_t{m}_t{\left(1+\left(q-1\right)\frac{m_t}{T}\right)}^{-\frac{q}{q-1}}$$

(6)

Similarly, to describe the p_t spectra, d²N/(N_evdp_tdy), of particles in the present work, we applied the Hagedorn function with included transverse flow (Equation (5)) in the following form:

$$\frac{d^{2}N}{N_{ev}d{p}_{t}dy}=2\pi C{p}_t{\left(1+\langle {\gamma }_t\rangle \frac{\left(m_t-p_t\langle {\beta }_t\rangle \right)}{n{T}_0 }\right)}^{-n}$$

(7)

In the present work, the simultaneous fits by the model functions of p_t distributions of all studied particle types in each group of <dN_ch/dη> have been conducted using the Nonlinear Curve Fitting of the Origin 9.1 Graphing and Data Analysis Software. The T/T₀ and <β_t> are extracted as global (common) parameters for all particle types in the fit procedures.

The combined statistical and systematic errors (added in quadrature) are shown in the figures for the experimental data points. The details for the calculation of systematic errors in the p_t spectra of the particles are provided in [45]. The minimum χ² fit processes have been conducted accounting for the combined statistical and systematic errors as the weights (1/(error)²) for data points. During fits of p_t distributions of the charged pions, the range p_t < 0.5 GeV/c, containing significant contribution from decays of baryon resonance, has been excluded, as is also the case in [15,16,21,33,34,57]. It is important to mention that in the present work, we used the same model functions, presented in Equations (6) and (7), and identical fitting p_t ranges for the studied particle species as used in recent work [21] for proton-proton collisions at (s)^1/2 = 13 TeV.

3. Analysis and Results

The results extracted from simultaneous minimum χ² fits by the Tsallis function with thermodynamical consistence (Equation (6)) of p_t distributions of the studied particle types in various groups of <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 7 TeV are shown in

Table 2. Parameters obtained using combined minimum χ² fits with the Tsallis function with thermodynamical consistence (Equation (6)) of p_t spectra of particles in various groups of <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 7 TeV. The fitted p_t ranges are [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 + $\overline{p}$. Here, n.d.f. denotes the number of degrees of freedom.

<dNch/dη>	$\mathit{q}({\mathit{\pi }}^{+}$$+{\mathit{\pi }}^{-})$	$\mathit{q}({\mathit{K}}^{+}$$+{\mathit{K}}^{-})$	$\mathit{q}(\mathit{p}+\overline{\mathit{p}})$	T (MeV)	${\mathit{\chi }}^2/\mathit{n}.\mathit{d}.\mathit{f}.(\mathit{n}.\mathit{d}.\mathit{f}.)$
21.3 ± 0.6	1.143 ± 0.001	1.153 ± 0.001	1.133 ± 0.001	131 ± 2	1.85 (114)
16.5 ± 0.5	1.145 ± 0.001	1.154 ± 0.001	1.131 ± 0.001	121 ± 2	1.54 (114)
13.5 ± 0.4	1.146 ± 0.001	1.155 ± 0.001	1.130 ± 0.001	114 ± 1	1.06 (114)
11.5 ± 0.3	1.147 ± 0.001	1.155 ± 0.001	1.128 ± 0.001	109 ± 1	0.79 (114)
10.1 ± 0.3	1.148 ± 0.001	1.155 ± 0.001	1.128 ± 0.001	104 ± 1	0.65 (114)
8.45 ± 0.25	1.149 ± 0.001	1.156 ± 0.001	1.128 ± 0.001	98 ± 1	0.61 (114)
6.72 ± 0.21	1.150 ± 0.001	1.156 ± 0.001	1.125 ± 0.001	91 ± 1	0.45 (114)
5.40 ± 0.17	1.151 ± 0.001	1.157 ± 0.001	1.123 ± 0.001	85 ± 1	0.39 (114)
3.90 ± 0.14	1.151 ± 0.001	1.156 ± 0.001	1.122 ± 0.001	75 ± 1	0.79 (114)
2.26 ± 0.12	1.149 ± 0.001	1.153 ± 0.001	1.114 ± 0.001	57 ± 1	1.00 (114)

. The parameter T is extracted as a shared (common) fitted parameter for all the particle species during the combined fitting procedures by Equation (6) in each group of <dN_ch/dη>. The results of this global fitting procedure are shown on Figure 1 for four different groups of <dN_ch/dη>.

Table 3. Parameters extracted using combined minimum χ² fits with the Hagedorn function with included transverse flow (Equation (7)) of p_t distributions of particles in various groups of <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 7 TeV. The fitted p_t ranges are [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 + $\overline{p}$.

<dNch/dη>	$\mathit{n}({\mathit{\pi }}^{+}$$+{\mathit{\pi }}^{-})$	$\mathit{n}({\mathit{K}}^{+}$$+{\mathit{K}}^{-})$	$\mathit{n}(\mathit{p}+\overline{\mathit{p}})$	$\langle {\mathit{\beta }}_{\mathit{t}}\rangle \left(\mathbf{in}\mathit{c}\mathbf{Units}\right)$	T0 (MeV)	${\mathit{\chi }}^2/\mathit{n}.\mathit{d}.\mathit{f}.(\mathit{n}.\mathit{d}.\mathit{f}.)$
21.3 ± 0.6	6.80 ± 0.05	6.61 ± 0.04	8.11 ± 0.06	0.29 ± 0.02	113 ± 3	0.60 (113)
16.5 ± 0.5	6.77 ± 0.05	6.57 ± 0.04	8.13 ± 0.06	0.23 ± 0.02	113 ± 3	0.72 (113)
13.5 ± 0.4	6.78 ± 0.04	6.55 ± 0.04	8.17 ± 0.06	0.18 ± 0.02	114 ± 3	0.63 (113)
11.5 ± 0.3	6.76 ± 0.04	6.52 ± 0.04	8.24 ± 0.06	0.15 ± 0.02	113 ± 3	0.54 (113)
10.1 ± 0.3	6.77 ± 0.04	6.52 ± 0.04	8.24 ± 0.06	0.12 ± 0.02	115 ± 3	0.58 (113)
8.45 ± 0.25	6.74 ± 0.04	6.48 ± 0.04	8.15 ± 0.06	0.07 ± 0.02	115 ± 4	0.74 (113)
6.72 ± 0.21	6.74 ± 0.04	6.47 ± 0.04	8.29 ± 0.06	0.04 ± 0.02	112 ± 3	0.60 (113)
5.40 ± 0.17	6.73 ± 0.04	6.46 ± 0.03	8.40 ± 0.06	0 ± 0.02	108 ± 3	0.56 (113)
3.90 ± 0.14	6.70 ± 0.05	6.49 ± 0.04	8.48 ± 0.09	0 ± 0.03	95 ± 4	0.97 (113)
2.26 ± 0.12	6.75 ± 0.06	6.59 ± 0.05	9.06 ± 0.13	0 ± 0.03	69 ± 4	1.19 (113)

shows the parameters extracted from combined minimum χ² fits with the Hagedorn function with included transverse flow (Equation (7)) of p_t distributions of particles in different groups of <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 7 TeV. The parameters $\langle {\beta }_t\rangle$ and T₀ are extracted as shared (common) fitted parameters for all the particle species during the combined fitting procedures by Equation (7) in each group of <dN_ch/dη>. Figure 2 illustrates the respective combined minimum χ² fit curves by the Hagedorn function with included transverse flow of mid-y p_t distributions of the studied charged particles in four different groups of <dN_ch/dη>. Figure 1 and the ${\chi }^2$/n.d.f. values in Table 2 show that the combined fits with the Tsallis function with thermodynamical consistence reproduce quite satisfactory p_t distributions of hadrons, except that for (anti)protons at high p_t values in the low-multiplicity events, in ten various groups of <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 7 TeV. Similarly, as observed from Figure 2 and the values of ${\chi }^2$/n.d.f. in Table 3, the simultaneous fits with the Hagedorn function with included transverse flow reproduce p_t distributions of particles quite well, except that for (anti)protons at high p_t values in the low-multiplicity events, in ten groups of <dN_ch/dη>.

As one can see from Figure 1 and Figure 2, the p_t spectra of (anti)protons could not be described well only in the low-multiplicity p + p collisions in region p_t > 6–8 GeV/c. Even in these low-multiplicity (with low <dN_ch/dη> < 6 values) p + p collision events at (s)^1/2 = 7 TeV, the quality of the combined fits is quite satisfactory, as one can see from the obtained χ²/n.d.f. values in Table 2 and Table 3. We attribute these deviations from model fits in the p_t spectra of (anti)protons in region p_t > 6–8 GeV/c in low-multiplicity p + p collisions to the fact that the protons are already available (as compared to the newly produced pions and kaons) as initial (colliding) particles, and it requires significantly larger energy to be deposited to the collision system (and hence, a significantly higher degree of thermalization) for the production of new (anti)protons, consisting of three (anti)quarks, as compared to that required for the production of pions and kaons, consisting of two (anti)quarks. Practically no thermalized collective system with collective transverse flow is expected or observed (see Table 3) in peripheral p + p interactions (with low <dN_ch/dη> < 6 values) with low energy transferred to the system.

Figure 3 illustrates the <dN_ch/dη> dependencies of the obtained q and T parameters of the Tsallis function with thermodynamical consistence for the studied charged particles in proton-proton collisions at (s)^1/2 = 7 TeV, extracted from simultaneous minimum χ² fits and presented in Table 2. Figure 4 summarizes the <dN_ch/dη> dependencies of the extracted parameters $\langle {\beta }_t\rangle$, T₀, and n of the Hagedorn function with included transverse flow for the studied particles in proton-proton collisions at (s)^1/2 = 7 TeV, presented in Table 3. The corresponding data obtained in [21] for the respective particle species in proton-proton collisions at (s)^1/2 = 13 TeV are also presented in Figure 3 and Figure 4 for comparison. Here, it is worth mentioning the importance of analyzing the behavior of $\langle {\beta }_t\rangle$ and T₀ and respective excitation functions [14] because of their relation to map the QCD phase diagram, even though the chemical freeze-out temperature (T_ch) is normally used in such phase diagrams.

As observed in Figure 3a, the overall values of q for mesons (pions and kaons) and baryons (protons and antiprotons) are clearly separated with a q(baryons) < q(mesons) relation holding in the entire studied <dN_ch/dη> region in proton-proton collisions at (s)^1/2 = 7 and 13 TeV. This is consistent with the q(baryons) < q(mesons) relation in minimum bias, high-energy proton-proton collisions obtained earlier in [15,16,22,25]. As observed in Figure 3a, the behavior of q versus <dN_ch/dη> dependence for all considered particle types in proton-proton collisions at (s)^1/2 = 7 TeV is similar to that obtained in proton-proton collisions at (s)^1/2 = 13 TeV in [21]. However, as observed in Figure 3a, on the whole, the values of q for the analyzed particle species in proton-proton collisions at (s)^1/2 = 7 TeV are found to be noticeably lower compared to those in proton-proton collisions at (s)^1/2 = 13 TeV in the whole analyzed <dN_ch/dη> range. This can suggest that the systems produced in proton-proton collisions at (s)^1/2 = 7 TeV are characterized by the noticeably larger degree of equilibrium and thermalization as compared to that in proton-proton interactions at (s)^1/2 = 13 TeV. This finding is consistent with that of the recent work [16], in which the non-extensivity parameter q has been shown to increase systematically for all studied particle species with increasing the energy (s)^1/2 of proton-proton collisions from 2.76 to 5.02 TeV. These results suggest that higher-energy proton-proton collisions are characterized by the lower degree of thermalization (larger degree of non-equilibrium).

In [8], the combined p_t spectra of the charged particles, consisting predominantly of pions, in minimum bias p + p interactions at wide energy regions (s)^1/2 = 0.54–7 TeV have been analyzed using the thermodynamically consistent Tsallis distribution. It has been obtained [8] that q increases weakly but consistently with beam energy, reaching the highest value ≈ 1.15 in minimum bias proton-proton interactions at (s)^1/2 = 7 TeV. This value agrees with q ≈ 1.15 for pions and kaons in Table 2 in the majority of multiplicity classes at (s)^1/2 = 7 TeV in the present analysis. The effective Tsallis temperature of 82 ± 1 MeV obtained in [8] in minimum bias proton-proton interactions at (s)^1/2 = 7 TeV agrees within uncertainties with the average effective Tsallis temperature <T> = 81 ± 1 MeV calculated in the present analysis. The average effective Tsallis temperature <T> for minimum bias proton-proton interactions at (s)^1/2 = 7 TeV in the present work has been calculated using the extracted temperature, T_i, from Table 2 and the corresponding fraction of events, f_i (with respect to the total inelastic cross-section), of the i-th group of <dN_ch/dη> from Table 1 as follows: $<T>={\displaystyle \sum }_{i=1}^{10}(T_i·f_i)$.

As seen from Figure 3b, the extracted T values of the Tsallis function with thermodynamical consistence show a similar consistent growth with increasing <dN_ch/dη> in proton-proton interactions at (s)^1/2 = 7 TeV as that observed in collisions of protons at (s)^1/2 = 13 TeV. This result can be explained by the fact that the larger multiplicity collisions are consistent with harder proton-proton collisions, in which a larger amount of energy-momentum is deposited to a system [21]. We have fitted the data points in Figure 3b, extracted from proton-proton collisions at (s)^1/2 = 7 TeV, employing the simple power function $T=A·{<\frac{d{N}_{ch}}{d\eta }>}^{\alpha }$, where A is the normalization constant and α is the exponent parameter, as was carried out in [21]. Similar to [21], we have conducted the minimum χ² fits by this simple power function in the entire studied <dN_ch/dη> region, and in the entire <dN_ch/dη> region excluding the first data point, i.e., excluding the softest (ultra-peripheral) proton-proton collisions. The respective results from minimum χ² fits are presented in

Table 4. Parameters extracted from minimum χ² fits using the simple power function $T=A·{<\frac{d{N}_{ch}}{d\eta }>}^{\alpha }$ of the T versus <dN_ch/dη> dependencies in Figure 3b in the whole <dN_ch/dη> region (I), and in the whole region excluding the first data point (II) with <dN_ch/dη> = 2.26 ± 0.12 and <dN_ch/dη> = 2.55 ± 0.04 in proton-proton collisions at (s)^1/2 = 7 and 13 TeV, respectively. The corresponding data extracted from [21] for proton-proton collisions at (s)^1/2 = 13 TeV are shown for comparison.

<dNch/dη> Fit Range	Collision Type, (s)1/2	A (MeV)	α	${\mathit{\chi }}^2/\mathit{n}.\mathit{d}.\mathit{f}.(\mathit{n}.\mathit{d}.\mathit{f}.)$
	p + p, 7 TeV	45.8 ± 1.4	0.353 ± 0.013	3.67 (8)
I	p + p, 13 TeV [21]	49.9 ± 1.7	0.33 ± 0.01	3.66 (8)
	p + p, 7 TeV	48.7 ± 0.6	0.327 ± 0.005	0.37 (7)
II	p + p, 13 TeV [21]	51.5 ± 0.9	0.31 ± 0.01	1.05 (7)

. The corresponding data extracted from [21] for proton-proton interactions at (s)^1/2 = 13 TeV are also shown in Table 4 for comparison. Figure 3b and ${\chi }^2$/n.d.f. values in Table 4 show that this power function is not able to well-reproduce the T versus <dN_ch/dη> dependence in the entire studied region. However, as seen from Figure 3b and values of ${\chi }^2$/n.d.f. from Table 4, the T versus <dN_ch/dη> dependence in proton-proton collisions at (s)^1/2 = 7 TeV is described quite satisfactorily, with the function $T=A·{<\frac{d{N}_{ch}}{d\eta }>}^{\alpha }$ having exponent parameter ≈ 1/3 in region <dN_ch/dη> > 3, when the first data point is excluded from the fit range. This agrees very well, as observed from Figure 3b and Table 4, with the similar result in [21], in which the T versus <dN_ch/dη> dependence in proton-proton interactions at (s)^1/2 = 13 TeV has been described quite well with the function $T=A·{<\frac{d{N}_{ch}}{d\eta }>}^{\alpha }$ with exponent parameter ≈ 1/3 in the whole <dN_ch/dη> region, excluding the first data point.

It was observed from Figure 4a that $\langle {\beta }_t\rangle$ values, obtained employing the Hagedorn function with included transverse flow, are essentially zero in the range <dN_ch/dη> < 6 (the first three data points) in proton-proton interactions at (s)^1/2 = 7 TeV. As observed in Figure 4a, beginning from <dN_ch/dη> ≈ 6, the transverse flow starts emerging and developing, with parameter $\langle {\beta }_t\rangle$ growing steadily in region <dN_ch/dη> > 6 until the largest <dN_ch/dη> values. The corresponding parameter T₀ grows systematically in the low-multiplicity region <dN_ch/dη> < 6 (the first three data points) and does not change within the fit uncertainties in the broad region <dN_ch/dη> > 6 in proton-proton interactions at (s)^1/2 = 7 TeV. The broad plateau range of T₀ starting at <dN_ch/dη> ≈ 6 in Figure 4b goes along with a simultaneous and systematic rise of $\langle {\beta }_t\rangle$ with increasing <dN_ch/dη> in the wide region <dN_ch/dη> > 6 in Figure 4a. The observation that the temperature, T₀, first rises in the range <dN_ch/dη> < 6, reaching a wide plateau beginning at <dN_ch/dη> ≈ 6, which matches the absence of transverse flow at <dN_ch/dη> < 6 and the emergence and systematic increase of $\langle {\beta }_t\rangle$ at the wide range <dN_ch/dη> > 6, suggests the probable onset of deconfinement phase transition at <dN_ch/dη> ≈ 6 in proton-proton interactions at (s)^1/2 = 7 TeV. The $\langle {\beta }_t\rangle$ and T₀ versus <dN_ch/dη> dependencies in collisions of protons at (s)^1/2 = 7 TeV, as observed from Figure 4a,b, qualitatively match those in collisions of protons at (s)^1/2 = 13 TeV obtained in [21], with some differences observed in the absolute values of parameters. It is seen in Figure 4a,b that the absolute values of $\langle {\beta }_t\rangle$ are noticeably smaller and the <dN_ch/dη> value for the probable onset of phase transition and the value of T₀ in the plateau region are larger in p + p collisions at (s)^1/2 = 13 TeV as compared to those at (s)^1/2 = 7 TeV. It has been estimated in [21] from the analysis of T₀ and $\langle {\beta }_t\rangle$ versus <dN_ch/dη> dependencies that the expected onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 13 TeV takes place at <dN_ch/dη> ≈ 7.1 ± 0.2. We have followed a similar procedure as that in [21] to better estimate the critical value of <dN_ch/dη> for the probable onset of deconfinement phase transition in collisions of protons at (s)^1/2 = 7 TeV: The critical <dN_ch/dη> is estimated to be the middle value on the <dN_ch/dη> axis between the third and fourth points in Figure 4b, which correspond to <dN_ch/dη> = 5.40 ± 0.17 and <dN_ch/dη> = 6.72 ± 0.21, respectively, in Table 1. The respective estimate for the probable onset of deconfinement phase transition in p + p collisions at (s)^1/2 = 7 TeV proved to be <dN_ch/dη> ≈ 6.1 ± 0.3, which is smaller than, but close to, the corresponding <dN_ch/dη> ≈ 7.1 ± 0.2 obtained in p + p collisions at (s)^1/2 = 13 TeV in [21].

As observed in Figure 4c, the n versus <dN_ch/dη> dependencies, obtained by employing the Hagedorn function with included transverse flow, in proton-proton collisions at (s)^1/2 = 7 and 13 TeV [21] approximately match the respective inverse dependencies of the q parameter for the studied particles in Figure 3a.

4. Further Analysis and Discussion

It has been observed that $\langle {\beta }_t\rangle$ becomes considerable at larger multiplicities in proton-proton collisions at (s)^1/2 = 7 TeV, attaining the maximal value (see Table 3) of $\langle {\beta }_t\rangle$ = 0.29 ± 0.02 at the highest studied value of <dN_ch/dη> = 21.3 ± 0.6. The transverse flow has been absent in proton-proton collisions in the low-multiplicity region <dN_ch/dη> < 6 (see Table 3 and Figure 4a). These results are consistent with the similar emergence and development of transverse radial flow in higher <dN_ch/dη> events in collisions of protons at (s)^1/2 = 13 TeV at the LHC recently deduced in [21]. As shown in Table 1, the $\langle {\beta }_t\rangle$ ≈ 0 events with <dN_ch/dη> < 6 make up around 62% of the total statistics of inelastic proton-proton collisions at (s)^1/2 = 7 TeV. In recent papers [15,16], almost zero transverse flow velocity has been extracted in minimum bias inelastic proton-proton collisions at (s)^1/2 = 2.76, 5.02, and 7 TeV, analyzing mid-y p_t distributions of particles with the Hagedorn function with embedded transverse flow. The practically zero $\langle {\beta }_t\rangle$ values obtained in [15,16] in minimum bias inelastic proton-proton interactions at (s)^1/2 = 2.76, 5.02, and 7 TeV are likely due to the dominant fraction of events with low-multiplicity with the absence of transverse flow in the total ensemble of inelastic proton-proton collisions.

The various features of QGP: the enhancement of strangeness [38], hardening of p_t distributions [39,40], and other QGP characteristics [35,36,37,41,42,43,44,45,46,47,48,49,50,58], obtained in proton-proton collisions at the LHC have been published. The onset of collective radial expansion in proton-proton collisions at (s)^1/2 = 900 and 7000 GeV at the LHC have been reported in [35]. The emergence and development of transverse radial flow in larger multiplicity proton-proton events at (s)^1/2 = 13 TeV at the LHC have been reported in [21]. The signatures of an equilibrated and collective system in large-multiplicity proton-proton events at (s)^1/2 = 7 TeV at the LHC were discovered in [45]. In [38], the integrated yields of strange and multi-strange particles, relative to pions, have been shown to increase significantly with increasing <dN_ch/dη> in proton-proton interactions at (s)^1/2 = 7 TeV at the LHC. This has been the first reported finding [38] of strangeness enhancement in large-multiplicity p + p events. These measurements [38] of ALICE collaboration have proven to be in remarkable agreement with those of p + Pb collisions at (s_nn)^1/2 = 5.02 TeV at LHC, implying that the observed phenomenon is due to the final system produced in a collision. In large-multiplicity p + p collisions [38], the strangeness production has reached values similar to those obtained in Pb + Pb collisions at (s_nn)^1/2 = 2.76 TeV, in which QGP is created. In a recent analysis [44], p_t distributions of hadrons as a function of charged-particle multiplicity and transverse sphericity have been investigated, using non-extensive Tsallis statistics and the Boltzmann–Gibbs blast-wave (BGBW) model in collisions of protons at (s)^1/2 = 13 TeV, with the help of the PYTHIA8 event generator. It has been deduced that the isotropic proton-proton collisions approach to thermal equilibrium, while the jetty events remain far from equilibrium [44].

The recent analyses in [46,47,48,49,50] present convincing explanations regarding the source of thermalization in proton-proton collisions at high energies. Thermal abundances of particle yield in proton-proton collisions at high energies with an exponential component in p_t distributions of hadrons are considered to be a clear signature of thermalization [46,50,58,59]. Since a few secondary interactions in high-energy proton-proton collisions do not support thermalization via interactions at the final state, the discovery of a thermal feature in proton-proton collisions has been quite surprising [46,50]. The thermalization in high-energy proton-proton interactions has been explained in [46,47,48,49] as the process occurring during the rapid quench caused by collisions due to the large extent of quantum entanglement inside the wave functions of partons of colliding protons. Therefore, the effective T extracted from the p_t spectra of hadrons depends on momentum transfer, constituting an ultraviolet cutoff of quantum modes resolved by collision [46,50]. In [50], p_t distributions at various multiplicities of Pb + Pb collisions at (s_nn)^1/2 = 2.76 TeV and proton-proton collisions at (s)^1/2 = 7 TeV have been fitted well by the sum of an exponential and power-like function, represented by thermal-like temperature, T_th, and hard temperature, T_h. The thermalization caused by quantum entanglement has been confirmed in [50], analyzing the multiplicity dependence of proton-proton and Pb + Pb collision data at the LHC. No thermal radiation is expected in diffractive proton-proton events with a large rapidity gap, because in such events the whole proton wave function becomes involved and entanglement entropy does not occur [50]. It has been indeed found in [46] that the thermal component vanishes in diffractive proton-proton interactions at (s)^1/2 = 13 TeV, in spite of many hadrons still being created. In the studied proton-proton and Pb + Pb collision data at LHC, the effective thermalization temperature, T_th, proved to be proportional [50] to the hard temperature, T_h, determined by the average p_t. The obtained coefficient of proportionality has been shown as being universal and independent from the type of collision [50]. The proportionality between T_th and T_h has been interpreted as that arising from color source clustering [50]. The T_th and T_h have been shown to rise with increasing multiplicity of both p + p and Pb + Pb collisions [50].

The systematic rise of parameter T (extracted employing the Tsallis function with thermodynamical consistence) with increasing particle multiplicity in proton-proton collisions at (s)^1/2 = 7 and 13 TeV [21], as seen from Figure 3b, is in agreement with the increase of T_th with multiplicity in collisions of protons at the LHC in [50]. We have confirmed an interesting result of [21], namely that T versus <dN_ch/dη> dependence in collisions of protons at (s)^1/2 = 13 TeV is described well with the function $T=A·{<\frac{d{N}_{ch}}{d\eta }>}^{\alpha }$ with exponent parameter ≈ 1/3. In the present paper, we have obtained that T versus <dN_ch/dη> dependence in p + p collisions at (s)^1/2 = 7 TeV is also described quite satisfactorily, with this power function having the exponent parameter ≈ 1/3 in the wide region <dN_ch/dη> > 3. Hence, assuming the proportionality of the charged-particle multiplicity density to the initial energy density (ε), it is deduced that $T~{\epsilon }^{1/3}$ in proton-proton collisions at (s)^1/2 = 7 and 13 TeV. The linear proportionality of the charged-particle multiplicity density to the initial energy density (ε) has been demonstrated in Figure 3 of [42], in which advanced calculations of the initial energy density of (s)^1/2 = 7 and 8 TeV p + p collisions have been performed, using the accelerating, exact, and explicit solutions of relativistic hydrodynamics [42]. It is interesting to mention that in a simple model of an ideal gas of massless pions [60], the energy density as a function of temperature is described by the Stefan–Boltzmann form ${\epsilon }_{\pi }=\left(\frac{{\pi }^2}{10}\right)T^4$, and hence $T~{\epsilon }_{\pi }{}^{1/4}$. Comparing $T~{\epsilon }^{1/3}$ extracted in the present analysis and in [21] with the relation $T~{\epsilon }_{\pi }{}^{1/4}$ for the model of an ideal gas of massless pions, one can see that the energy density dependencies of the effective temperatures of the systems, formed in proton-proton collisions at (s)^1/2 = 7 and 13 TeV [21], and of the temperature of an ideal gas of massless pions, are compatible with each other due to the closeness of the respective exponent parameters.

Employing the explicit and exact relativistic hydrodynamic solutions, the advanced calculations of initial energy density in minimum bias proton-proton interactions at mid-y at (s)^1/2 = 7 and 8 TeV at the LHC were performed in [42]. The improved advanced hydrodynamic evaluations [42] of initial energy density in minimum bias proton-proton collisions at mid-y at the LHC have yielded ε(7 TeV) = 0.645 GeV/fm³ and ε(8 TeV) = 0.641 GeV/fm³, respectively, which are below the critical energy density (1 GeV/fm³) from lattice QCD calculations. These results were extracted using the mean values of the charged-particle multiplicity density at mid-y in minimum bias proton-proton collisions at (s)^1/2 = 7 and 8 TeV at the LHC ALICE and CMS experiments. It has been concluded [42] that a large enough initial energy density to create a non-hadronic perfect fluid (QGP-like) is available in high-multiplicity proton-proton events at the LHC. The analogous advanced calculations, employing the relativistic hydrodynamic solutions in [43], have yielded ε(13 TeV) ≈ 0.69 GeV/fm³ using <dN_ch/dη> ≈ 6.5 in minimum bias proton-proton collisions at (s)^1/2 = 13 TeV at mid-y at the LHC. Assuming the proportionality of the charged-particle multiplicity density to the initial energy density (ε) and employing the results of [43], the critical value of <dN_ch/dη> at mid-y in proton-proton collisions at (s)^1/2 = 13 TeV, required to achieve the critical QCD energy density (1 GeV/fm³), has been calculated in [21] as follows:

$$<\frac{d{N}_{ch}}{d\eta }>\left(p+p\mathrm{at}13\mathrm{TeV}\right)=\frac{1 \mathrm{GeV}/{\mathrm{fm}}^3}{0.69 \mathrm{GeV}/{\mathrm{fm}}^3}·6.5\approx 9.4$$

(8)

Similarly, using the results of [42], we can evaluate the critical <dN_ch/dη> at mid-y in proton-proton collisions at (s)^1/2 = 7 TeV to achieve the critical QCD energy density (1 GeV/fm³):

$$<\frac{d{N}_{ch}}{d\eta }>\left(p+p\mathrm{at}7\mathrm{TeV}\right)=\frac{1 \mathrm{GeV}/{\mathrm{fm}}^3}{0.645 \mathrm{GeV}/{\mathrm{fm}}^3}·5.9\approx 9.1$$

(9)

Our estimate of <dN_ch/dη> ≈ 6.1 ± 0.3 for the probable onset of deconfinement phase transition was found to be lower than the critical value of <dN_ch/dη> for achieving the QCD critical energy density (1 GeV/fm³), evaluated above (Equation (9)) in collisions of protons at (s)^1/2 = 7 TeV at mid-y. It is an interesting result that <dN_ch/dη> ≈ 6.1 ± 0.3 for the probable deconfinement phase transition estimated in the present work in collisions of protons at (s)^1/2 = 7 TeV proved to be smaller than, but close to, the corresponding <dN_ch/dη> ≈ 7.1 ± 0.2 estimated in [21] in collisions of protons at (s)^1/2 = 13 TeV, which is also observed in Figure 4a,b. Using the results of Equations (8) and (9), in the present analysis we can estimate the respective critical energy densities for the probable onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 7 and 13 TeV corresponding to <dN_ch/dη> ≈ 6.1 ± 0.3 and <dN_ch/dη> ≈ 7.1 ± 0.2, respectively:

$${\epsilon }_{tr}\left(p+p\mathrm{at}7\mathrm{TeV}\right)=\frac{6.1}{9.1}\times 1\frac{\mathrm{GeV}}{{\mathrm{fm}}^3}\approx 0.67\pm 0.03 \frac{\mathrm{GeV}}{{\mathrm{fm}}^3}$$

(10)

and

$${\epsilon }_{tr}\left(p+p\mathrm{at}13\mathrm{TeV}\right)=\frac{7.1}{9.4}\times 1\frac{\mathrm{GeV}}{{\mathrm{fm}}^3}\approx 0.76\pm 0.02 \frac{\mathrm{GeV}}{{\mathrm{fm}}^3}$$

(11)

As seen from the above calculations, the estimated critical energy densities for the probable onset of deconfinement phase transitions in proton-proton collisions at (s)^1/2 = 7 and 13 TeV have proven to be significantly smaller than the critical value of 1 GeV/fm³ from lattice QCD calculations. From the comparison of non-extensivity parameter q versus <dN_ch/dη> dependencies for two collision energies in Figure 3a, it has been deduced that the systems produced in collisions of protons at (s)^1/2 = 7 TeV are characterized by a noticeably larger degree of equilibrium and thermalization as compared to that in proton-proton collisions at (s)^1/2 = 13 TeV, which could be due to the faster and more violent proton-proton collisions at (s)^1/2 = 13 TeV than those at (s)^1/2 = 7 TeV. The deduced larger degree of equilibrium and thermalization at (s)^1/2 = 7 TeV, compared to that at (s)^1/2 = 13 TeV, can probably explain our finding that the probable onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 7 TeV takes place at a lower estimated critical energy density (0.67 ± 0.03 GeV/fm³) than that (0.76 ± 0.02 GeV/fm³) in collisions of protons at (s)^1/2 = 13 TeV.

Here, it is worth mentioning the results of relevant works [61,62,63] of Campanini et al. In an early paper [61], the available experimental data on average p_t (<p_t>) versus charged-particle rapidity density, dn/dy, dependencies in proton-proton and proton–antiproton collisions at SPS collider and ISR have been examined. It has been suggested [61] that evidence may already be available of high-temperature phase transition to quark-gluon plasma in hadronic collision events at SPS and ISR with central rapidity density dn/dy within one or two units around 6. In [62], it has been shown that in various experiments in proton-proton and proton–antiproton collisions at (s)^1/2 energies ranging from 22 GeV to 7 TeV, there is a well-defined slope change in <p_t> versus charged-particle rapidity density, which may indicate a phase transition to a new production mechanism of particles at dn/dη = 5.5 ± 1.2. In [63], the approximate equations of state (EOS) have been extracted from the analysis of <p_t> versus dN_ch/dη dependencies in experimental data on proton-proton and proton–antiproton interactions at (s)^1/2 energies ranging from 31 GeV to 7 TeV. The obtained results [63] proved to be consistent with theory predictions in case of crossover from hadronic gas to quark gluon plasma, starting at (dN_ch/dη) ≈ 6 and ending at (dN_ch/dη) ≈ 24. Hence, the critical values <dN_ch/dη> ≈ 6.1 ± 0.3 and <dN_ch/dη> ≈ 7.1 ± 0.2 [21] for the probable onset of deconfinement phase transition in collisions of protons at (s)^1/2 = 7 and 13 TeV, respectively, estimated from the analysis of the dependency of T₀ and $\langle {\beta }_t\rangle$ on <dN_ch/dη>, are consistent and in good agreement with the respective estimates of Campanini, deduced from detailed analyses of the average p_t (<p_t>) versus charged-particle (pseudo)rapidity density dependencies in [61,62,63].

For more precise estimations of the critical <dN_ch/dη> and corresponding critical energy density for the probable onset of deconfinement phase transition in collisions of protons at high LHC energies, it is necessary to conduct a further comprehensive analysis of dependencies of particle production on <dN_ch/dη> in proton-proton collisions, preferably with smaller binning in <dN_ch/dη>.

5. Summary and Conclusions

We have analyzed experimental p_t distributions of particles at ten groups of <dN_ch/dη> in inelastic proton-proton collisions at (s)^1/2 = 7 TeV at mid-y at the LHC, measured by ALICE collaboration. We have studied the change of collective characteristics of the collision system with varying <dN_ch/dη> through combined minimum χ² model fits of p_t distributions of identified charged particles, using the Tsallis distribution function with thermodynamical consistence and the Hagedorn function with included transverse (radial) flow. The combined minimum χ² fits with the Tsallis function with thermodynamical consistence and the Hagedorn function with transverse (radial) flow quite well-describe the p_t distributions of the studied particle species in ten different groups of <dN_ch/dη> in collisions of protons at (s)^1/2 = 7 TeV. The findings of the present analysis for proton-proton collisions at (s)^1/2 = 7 TeV were systematically compared with the respective results of recent work [21] for proton-proton collisions at (s)^1/2 = 13 TeV.

The values of non-extensivity parameter q for pions and kaons, protons and antiprotons in proton-proton collisions at (s)^1/2 = 7 TeV were found to be noticeably smaller compared to those in collisions of protons at (s)^1/2 = 13 TeV in the whole analyzed <dN_ch/dη> range. This indicates that the systems produced in proton-proton collisions at (s)^1/2 = 7 TeV have a noticeably larger degree of equilibrium and thermalization than those at (s)^1/2 = 13 TeV.

The extracted effective temperatures, T, of the Tsallis function with thermodynamical consistence have shown a consistent increase with the increasing <dN_ch/dη> in proton-proton collisions at (s)^1/2 = 7 TeV, in agreement with the similar result [21] obtained in proton-proton collisions at (s)^1/2 = 13 TeV. The respective T versus <dN_ch/dη> dependence in proton-proton collisions at (s)^1/2 = 7 TeV has been well-described with the simple power function $T=A·{<\frac{d{N}_{ch}}{d\eta }>}^{\alpha }$ with the same value ≈ 1/3 of exponent parameter as that obtained recently [21] in proton-proton collisions at (s)^1/2 = 13 TeV. Comparing the relation $T~{\epsilon }^{1/3}$ extracted in the present analysis and in [21] with the relation $T~{\epsilon }_{\pi }{}^{1/4}$ for the simple model of an ideal gas of massless pions, it was found that dependencies on the initial energy density of the effective temperatures of the systems, created in collisions of protons at (s)^1/2 = 7 and 13 TeV, and that of an ideal pion gas, are compatible to each other due to the closeness of the corresponding exponent parameters.

It was obtained that the transverse (radial) flow emerged at <dN_ch/dη> ≈ 6 and then increased, becoming significant at higher multiplicity events and reaching the maximal value of $\langle {\beta }_t\rangle$ = 0.29 ± 0.02 at the highest studied value of <dN_ch/dη> = 21.3 ± 0.6 in proton-proton collisions at (s)^1/2 = 7 TeV. These results are consistent with the similar emergence and development of transverse radial flow in larger multiplicity proton-proton events at (s)^1/2 = 13 TeV at the LHC demonstrated in [21].

We have estimated, analyzing T₀ and $\langle {\beta }_t\rangle$ versus <dN_ch/dη> dependencies, extracted employing the Hagedorn function with included transverse flow, that the probable onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 7 TeV takes place at <dN_ch/dη> ≈ 6.1 ± 0.3, which is close to the corresponding estimate (<dN_ch/dη> ≈ 7.1 ± 0.2) obtained recently in collisions of protons at (s)^1/2 = 13 TeV in [21]. The critical values <dN_ch/dη> ≈ 6.1 ± 0.3 and <dN_ch/dη> ≈ 7.1 ± 0.2 [21] for the probable onset of deconfinement phase transition in collisions of protons at (s)^1/2 = 7 and 13 TeV, respectively, estimated by us from the analysis of T₀ and $\langle {\beta }_t\rangle$ versus <dN_ch/dη> dependencies, proved to be consistent and in good agreement with the respective estimates of Campanini et al. in [61,62,63] and theory predictions [63] in case of crossover from hadronic gas to quark gluon plasma starting at (dN_ch/dη) ≈ 6. We have also estimated the corresponding critical energy densities for the probable onset of deconfinement phase transitions in proton-proton collisions at (s)^1/2 = 7 and 13 TeV at the LHC to be 0.67 ± 0.03 and 0.76 ± 0.02 GeV/fm³, respectively, being significantly lower than the critical QCD energy density (1 GeV/fm³). The noticeably larger degree of equilibrium and thermalization deduced at (s)^1/2 = 7 TeV than that at (s)^1/2 = 13 TeV could probably explain our finding that the possible onset of deconfinement phase transition in proton-proton collisions at (s)^1/2 = 7 TeV takes place at the lower estimated critical energy density (0.67 ± 0.03 GeV/fm³), as compared to that (0.76 ± 0.02 GeV/fm³) in proton-proton collisions at (s)^1/2 = 13 TeV.