# Analysis of identified particle transverse momentum spectra produced in pp,
p–Pb and Pb–Pb collisions at the LHC using Tsallis–Pareto-type distribution

Pei-Pin Yang1,111E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>Mai-
Ying Duan1,222E-mail<EMAIL_ADDRESS>Fu-Hu Liu1,333Correspondence:
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>Raghunath
Sahoo2,3,444Correspondence: E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>1Institute of Theoretical Physics & College of Physics
and Electronic Engineering, State Key Laboratory of Quantum Optics and Quantum
Optics Devices & Collaborative Innovation Center of Extreme Optics, Shanxi
University, Taiyuan 030006, China
2Department of Physics, Indian Institute of Technology Indore, Indore 453552,
India
3Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland

###### Abstract

Abstract: In the framework of a multi-source thermal model at the quark level,
we have analyzed transverse momentum spectra of hadrons measured by the ALICE
Collaboration in proton–proton ($pp$ or $p$–$p$) collisions at the center-of-
mass energy of $\sqrt{s}=7$ and 13 TeV, proton–lead ($p$–Pb) collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV, and lead–lead (Pb–Pb) collisions at
$\sqrt{s_{\rm NN}}=2.76$ TeV. For meson(baryon), the contributions of
two(three) constituent quarks are considered, in which each quark contributes
to hadron transverse momentum to obey the revised Tsallis–Pareto-type
distribution (the TP-like function in short) with isotropic random azimuth.
Three main parameters, namely, the revised index $a_{0}$, effective
temperature $T$, and entropy-related index $n$ are obtained, which show the
same tendency for small and large systems with respect to the centrality (or
multiplicity) of events, rest mass of hadrons, and constituent mass of quarks.

Keywords: Transverse momentum spectra, Tsallis–Pareto distribution,
constituent quarks, proton–proton and heavy-ion Collisions

PACS numbers: 12.40.Ee, 13.85.Hd, 24.10.Pa

##  I Introduction

The deconfined high-temperature and high-density state of nuclear matter,
called quark-gluon plasma (QGP) is often studied by ultra-relativistic nuclear
collisions. The transverse momentum ($p_{T}$) spectra and ratios of identified
hadrons provide a means to study the properties of matter created in these
collisions and the mechanism by which quasi-free partons are transformed into
observable hadrons. The relative contributions of different hadronization
mechanisms rely on the change of hadron’s $p_{T}$. At low-$p_{T}$,
recombination may be dominant, but at high-$p_{T}$, hadrons may originate from
fragmentation processes. This mainly depends on the potential transverse
momentum distribution of quarks. Therefore, it is important to have the
$p_{T}$-spectra of identified mesons and baryons in a wide $p_{T}$-range.

Unlike up ($u$) and down ($d$) quarks those form ordinary matter, strange
($s$) quarks do not exist in the form of valence quarks in the colliding
species, but they are light enough to be produced in large quantities in the
process of ultra-relativistic collisions. In the early state of high-energy
collisions, strangeness is produced in hard (perturbative) $2\rightarrow 2$
partonic scattering processes by flavor creation ($gg\rightarrow s\bar{s}$,
$q\bar{q}\rightarrow s\bar{s}$) and flavor excitation ($gs\rightarrow gs$,
$qs\rightarrow qs$). Strangeness is also created during the subsequent
partonic evolution via gluon splitting process ($g\rightarrow s\bar{s}$).
These processes tend to dominate the production of strange hadrons with
high-$p_{T}$, and the production of strange hadrons with low-$p_{T}$ is mainly
dominated by non-perturbative processes.

The production of strange hadrons is suppressed compared to that of hadrons
containing only $u/d$ quarks, because $s$ quarks need higher threshold energy
to be excited. In basic particle-particle collisions, the degree of
suppression of strange hadrons is an important parameter in the model
analysis. Therefore, the measurement of strange hadron production imposes
restrictions on the models. The study of strange and multi-strange particles
in relativistic heavy-ion collisions is an important tool for exploring the
properties of the strong interaction system. The particle spectra provide
abundant information about the temperature and collective flow of the system,
which reflects the dynamic freezing condition, where the inelastic collision
stops. The enhancement of strangeness in heavy-ion collisions is one of the
most important signals of QGP 1 ; 2 ; 3 .

In the past decades, around the search for QGP, researchers have conducted
extensive study on hadrons containing one or more $s$ quarks 1 ; 4 ; 5 .
However, the origin of the strangeness enhancement is not yet clear, when
strangeness is also observed to be enhanced in proton–proton ($pp$ or $p$–$p$)
collisions 6 ; 7 . The azimuthal correlations and mass-dependent hardening of
the $p_{T}$ spectra observed in the high-multiplicity $pp$ and proton–nucleus
($pA$ or $p$–$A$) collisions are typically attributed to the formation of
strongly interacting quark-gluon medium 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ;
16 ; 17 ; 18 ; 19 . The abundance of strange particles at different center-of-
mass energies is in accordance with the calculation of the thermal statistical
model 20 ; 21 ; 22 . Strangeness, light flavor production, and heavy-ion
collision dynamics provide evidences for the properties of fluid-like and
collectiveness of the medium 23 ; 24 . Studying $pp$ collisions under high
multiplicity is of considerable significance, because it opens up the
possibility of understanding nuclear reaction phenomena from a microscopic
perspective.

Studies of high multiplicity charged particles in $pp$ and proton–lead
($p$–Pb) collisions at the Large Hadron Collider (LHC) have shown striking
similarities to lead–lead (Pb–Pb) collisions. The facts that the enhancements
of (multi-)strange hadrons 11 , azimuthal correlations and double-ridge
structure 12 ; 13 , nonzero elliptic flow ($v_{2}$) coefficients and other
anisotropic flow measurements 25 ; 26 ; 27 ; 28 ; 29 ; 30 , mass ordering in
hadron $p_{T}$ spectra, and characteristic modifications of baryon to meson
ratios 10 , have shown that collective phenomena also exist in small collision
systems. In addition, the continuous evolution of the ratio of light-flavor
hadrons to pions in $pp$, $p$–Pb, and Pb–Pb collisions have been found as a
function of charged particle multiplicity density 26 ; 31 ; 32 . The observed
similarities indicate that there is a common underlying mechanism that
determines the chemical composition that arises from these collisions. That
is, typically interactions among partons exit in these collisions.

Recently, we considered contributor quarks and based on the Tsallis statistics
33 ; 34 ; 35 ; 36 ; 37 , the available $p_{T}$ spectra of various particles
produced in collisions of small systems [$pp$, deuteron–gold ($d$–Au), $p$–Pb]
and large systems [gold–gold (Au–Au) and Pb–Pb] at high energies were studied
38 ; 39 , where we have used the convolution of two or three revised
Tsallis–Pareto-type distribution (the TP-like function in short) 40 ; 41 ; 42
; 43 . The application of convolution means that we have considered the
azimuths of contributor quarks to be the same or parallel to each other. A
detailed consideration shows that the azimuths of contributor quarks may be
isotropic and random which includes the parallel, vertical, and any other
cases. The analytical expressions for parallel and vertical cases are
available. For any other cases, we may use the Monte Carlo method, if it is
difficult to give the analytical expression.

In the current work, in the framework of multi-source thermal model at the
quark level, we use the Monte Carlo method to study the $p_{T}$ spectra of
identified hadrons produced in the collisions of different centralities (or
multiplicities) with small systems ($pp$ and $p$–Pb) and large system (Pb–Pb),
including non-strange hadrons ($\pi^{+}+\pi^{-}$ and $p+\overline{p}$),
strange hadrons ($K^{+}+K^{-}$, $K_{S}^{0}$, $K^{*}+\overline{K}{}^{*}$, and
$\Lambda+\overline{\Lambda}$), and multi-strange hadrons
($\Xi^{-}+\overline{\Xi}{}^{+}$ and $\Omega^{-}+\overline{\Omega}{}^{+}$). The
azimuths of contributor or constituent quarks are isotropic and random. The
size of transverse momentum of each quark contributed to hadron’s $p_{T}$ is
assumed to obey the TP-like function. Mathematically, we study the synthesis
of two or three vectors with changeable azimuths and sizes.

The $p_{T}$ spectra in a wide range can reflect more dynamical information of
the collision process. In order to verify the feasibility of the model and
extract relevant parameters, we collected the experimental data of $p_{T}$
spectra of identified hadrons produced in $pp$ collisions at $\sqrt{s}=7$ and
13 TeV 44 ; 45 ; 46 ; 47 , $p$–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
10 ; 48 ; 49 ; 50 , and Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV 51 ;
52 ; 53 ; 54 , measured by the ALICE Collaboration.

The remainder of this paper is structured as follows. The formalism and method
are described in Section II. Results and discussion are given in Section III.
In Section IV, we summarize our main observations with conclusions.

##  II Formalism and method

Inspired by the SU(3) super polymorphism theory, in 1964, M. Gell-Mann
proposed the quark model 55 . There are two commonly used masses of quarks.
One is the current mass of quark, which refers to the mass in the Lagrangian
of quantum field theory. The other is the constituent mass of quark that
refers to the equivalent mass after the interaction with the gluon being
included when considering the composition of hadrons. The quark model believes
that a meson is composed of a quark and antiquark pair, and a baryon is
composed of three quarks.

In our recent work 38 ; 39 , the TP-like function used for the $p_{T}$ spectra
of hadrons is given by

$\displaystyle
f(p_{T})=\frac{1}{N}\frac{dN}{dp_{T}}=C_{0}p_{T}^{a_{0}}\left(1+\frac{\sqrt{p_{T}^{2}+m_{0}^{2}}-m_{0}}{nT}\right)^{-n},$
(1)

where $N$ is the number of particles, $C_{0}$ is the normalization constant,
$a_{0}$ is the revised factor, $T$ is the effective temperature, $n$ is the
entropy-related index, and $m_{0}$ is the rest mass of given hadron. The three
parameters $a_{0}$, $T$, and $n$ can be determined from the $p_{T}$ spectra of
hadrons.

According to the multi-source thermal model 56 ; 57 , we think that the
$p_{T}$ spectra of hadrons are contributed by the contributor or constituent
quarks. The transverse momentum of each quark contributed to $p_{T}$ of given
hadron is assumed to follow the TP-like function, too. We have the
contribution, $p_{ti}$, of the $i$-th quark to be

$\displaystyle
f_{i}(p_{ti})=\frac{1}{N_{i}}\frac{dN_{i}}{dp_{T}}=C_{i}p_{ti}^{a_{0}}\left(1+\frac{\sqrt{p_{ti}^{2}+m_{0i}^{2}}-m_{0i}}{nT}\right)^{-n}.$
(2)

Here, $N_{i}$ is number of the $i$-th quark, $C_{i}$ is the normalization
constant, and $m_{0i}$ is the constituent mass of the $i$-th quark. A meson is
composed of a quark and antiquark pair, so $i$ is equal to 1 or 2, and a
baryon is composed of 3 quarks, so $i$ is equal to 1, 2 or 3. Regardless of
the value of $i$, we always have $N_{i}=N$.

The relations between the transverse momentum vectors $\bm{p_{t1}}$ and
$\bm{p_{t2}}$ of constituent quark and antiquark pair for meson, and those
among the transverse momentum vectors $\bm{p_{t1}}$, $\bm{p_{t2}}$, and
$\bm{p_{t3}}$ of constituent quarks for baryon, may be parallel, vertical, or
of any azimuth $\phi_{i}$. The analytical expressions for the parallel and
vertical cases are available in Refs. 38 ; 39 . For the cases of any azimuths,
we may use the Monte Carlo method 58 ; 59 to obtain $p_{T}$, if the analytical
expression is not available.

To obtain a discrete value of $p_{ti}$ that satisfies Eq. (2), we can perform
the solution of

$\displaystyle\int_{0}^{p_{ti}}f_{i}(p^{\prime}_{ti})dp^{\prime}_{ti}<r_{i}<\int_{0}^{p_{ti}+\delta
p_{ti}}f_{i}(p^{\prime}_{ti})dp^{\prime}_{ti}.$ (3)

Here, $r_{i}$ is a random number uniformly distributed in $[0,1]$, $\delta
p_{ti}$ denotes a small shift in $p_{ti}$. To obtain a discrete value of
$\phi_{i}$ that satisfies the isotropic or uniform distribution, we have

$\displaystyle\phi_{i}=2\pi R_{i},$ (4)

where $R_{i}$ denotes a random number uniformly distributed in $[0,1]$.

For a meson composed of a quark and antiquark pair, we have,

$\displaystyle p_{T}=$
$\displaystyle\sqrt{\bigg{(}\sum_{i=1}^{2}p_{ti}\cos\phi_{i}\bigg{)}^{2}+\bigg{(}\sum_{i=1}^{2}p_{ti}\sin\phi_{i}\bigg{)}^{2}}$
$\displaystyle=$
$\displaystyle\sqrt{p_{t1}^{2}+p_{t2}^{2}+2p_{t1}p_{t2}\cos|\phi_{1}-\phi_{2}|}.$
(5)

For a baryon composed of three quarks, we have,

$\displaystyle p_{T}=$
$\displaystyle\sqrt{\bigg{(}\sum_{i=1}^{3}p_{ti}\cos\phi_{i}\bigg{)}^{2}+\bigg{(}\sum_{i=1}^{3}p_{ti}\sin\phi_{i}\bigg{)}^{2}}$
$\displaystyle=$
$\displaystyle\big{(}p_{t1}^{2}+p_{t2}^{2}+p_{t3}^{2}+2p_{t1}p_{t2}\cos|\phi_{1}-\phi_{2}|$
$\displaystyle+2p_{t1}p_{t3}\cos|\phi_{1}-\phi_{3}|+2p_{t2}p_{t3}\cos|\phi_{2}-\phi_{3}|\big{)}^{1/2}.$
(6)

As the expression of $p_{T}$ for hadron with 2 or 3 quarks, Eq. (5) or (6) can
be easily extended to the hadron with multiple quarks, e.g. the hadron with 4
or 5 quarks. We have the expression of $p_{T}$ for hadron with $k$ quarks to
be

$\displaystyle
p_{T}=\sqrt{\bigg{(}\sum_{i=1}^{k}p_{ti}\cos\phi_{i}\bigg{)}^{2}+\bigg{(}\sum_{i=1}^{k}p_{ti}\sin\phi_{i}\bigg{)}^{2}}.$
(7)

In fact, Eq. (7) is a uniform expression for $p_{T}$ of any hadron. In a real
calculation, we need $k$ sets of $p_{ti}$ and $\phi_{i}$ for a given $p_{T}$.
After the calculation for many times, the distribution of $p_{T}$ can be
obtained in statistics.

It is worth emphasized that in the analysis performed in this paper, the mass
of the quark is the constituent mass, but not the current mass 60a ; 60 . Our
attempts show that the constituent mass is more suitable for the fit.
Meanwhile, we need to always distinguish the transverse momentum, $p_{T}$ of a
given hadron and the transverse momentum $p_{ti}$ of the $i$-th constituent
quark with the azimuth $\phi_{i}$.

Fig. 1. Multiplicity dependent $p_{T}$ spectra of $\pi^{+}+\pi^{-}$ (a),
$K^{+}+K^{-}$ (b), $p+\overline{p}$ (c), $K^{0}_{S}$ (d),
$K^{*}+\overline{K}{}^{*}$ (e), $\Lambda+\overline{\Lambda}$ (f),
$\Xi^{-}+\overline{\Xi}{}^{+}$ (g), and $\Omega^{-}+\overline{\Omega}{}^{+}$
(h) with $|y|<0.5$, produced in $pp$ collisions at $\sqrt{s}=7$ TeV. Different
symbols represent the experimental data for different multiplicity classes
measured by the ALICE Collaboration 44 , where in most cases the data are
scaled by constant multipliers marked in the panels for clarity. The curves
represent our fit results based on the Monte Carlo calculations.

Fig. 2. Same as Figure 1, but showing the multiplicity dependent $p_{T}$
spectra of $\pi^{+}+\pi^{-}$ (a), $K^{+}+K^{-}$ (b), $p+\overline{p}$ (c),
$K^{0}_{S}$ (d), $K^{*0}$ (e), $\Lambda+\overline{\Lambda}$ (f),
$\Xi^{-}+\overline{\Xi}{}^{+}$ (g), and $\Omega^{-}+\overline{\Omega}{}^{+}$
(h) with $|y|<0.5$ produced in $pp$ collisions at $\sqrt{s}=13$ TeV measured
by the ALICE Collaboration 45 ; 46 ; 47 .

Fig. 3. The centrality dependent $p_{T}$ spectra of $\pi^{+}+\pi^{-}$ (a),
$K^{+}+K^{-}$ (b), $p+\overline{p}$ (c), $K^{0}_{S}$ (d), $\Sigma(1385)^{+}$
(e), $\Lambda+\overline{\Lambda}$ (f), $(\Xi^{-}+\overline{\Xi}{}^{+})/2$ (g),
and $(\Omega^{-}+\overline{\Omega}{}^{+})/2$ (h) with $-0.5<y<0$ (a–c, e, g,
h) or $0<y<0.5$ (d, f) produced in $p$–Pb collisions at $\sqrt{s_{NN}}=5.02$
TeV measured by the ALICE Collaboration 10 ; 48 ; 49 ; 50 . Different symbols
represent the experimental data with different centrality classes, where
different constant multipliers are used to re-scale the data for clarity. The
curves are our fit results based on the Monte Carlo calculations.

Fig. 4. Same as Figure 3, but showing the centrality dependent $p_{T}$
spectra, of $\pi^{+}+\pi^{-}$ (a), $K^{+}+K^{-}$ (b), $p+\overline{p}$ (c),
$K^{0}_{S}$ (d), $(K^{*0}+\overline{K}{}^{*0})/2$ (e), $\Lambda$ (f),
$(\Xi^{-}+\overline{\Xi}{}^{+})/2$ (g), and
$(\Omega^{-}+\overline{\Omega}{}^{+})/2$ (h) with $|\eta|<0.8$ (a–c) or
$|y|<0.5$ (d–h) produced in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV
measured by the ALICE Collaboration 51 ; 52 ; 53 ; 54 .

Table 1. Values of the free parameters ($a_{0}$, $T$, and $n$), normalization
constant ($N_{0}$), $\chi^{2}$, and ndof corresponding to the curves in Figure
1 for $pp$ collisions at $\sqrt{s}=7$ TeV. The particle type (quark
structure), spectrum form, and multiplicity classes are given together. The
multiplicity classes I, II, III, IV, V, VI, VII, VIII, IX, and X correspond to
the centrality classes 0–0.95%, 0.95–4.7%, 4.7–9.5%, 9.5–14%, 14–19%, 19–28%,
28–38%, 38–48%, 48–68%, and 68–100%, respectively.

| Particle (quark structure)
---
and spectrum form
| Multiplicity
---
class
$a_{0}$ |  $T$ (GeV) | $n$ | $N_{0}$ |  $\chi^{2}$/ndof
| I | $-0.550\pm 0.004$ | $0.345\pm 0.002$ | $5.071\pm 0.054$ | $(2.130\pm 0.042)\times 10^{1}$ | $58/44$
| II | $-0.550\pm 0.003$ | $0.321\pm 0.003$ | $4.962\pm 0.035$ | $(1.700\pm 0.021)\times 10^{1}$ | $66/44$
| III | $-0.550\pm 0.004$ | $0.313\pm 0.002$ | $4.962\pm 0.031$ | $(1.350\pm 0.040)\times 10^{1}$ | $69/44$
$\pi^{+}+\pi^{-}$ | IV | $-0.550\pm 0.004$ | $0.299\pm 0.002$ | $4.934\pm 0.066$ | $(1.200\pm 0.030)\times 10^{1}$ | $66/44$
$(u\bar{d},d\bar{u})$ | V | $-0.550\pm 0.005$ | $0.291\pm 0.001$ | $4.910\pm 0.033$ | $(1.040\pm 0.101)\times 10^{1}$ | $65/44$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $-0.550\pm 0.005$ | $0.287\pm 0.001$ | $4.906\pm 0.029$ | $(8.580\pm 0.080)\times 10^{0}$ | $84/44$
$[({\rm GeV}/c)^{-1}]$ | VII | $-0.550\pm 0.004$ | $0.266\pm 0.002$ | $4.910\pm 0.086$ | $(7.230\pm 0.040)\times 10^{0}$ | $62/44$
| VIII | $-0.550\pm 0.003$ | $0.254\pm 0.002$ | $4.910\pm 0.115$ | $(5.860\pm 0.030)\times 10^{0}$ | $69/44$
| IX | $-0.550\pm 0.003$ | $0.233\pm 0.001$ | $4.916\pm 0.041$ | $(4.370\pm 0.101)\times 10^{0}$ | $104/44$
| X | $-0.550\pm 0.005$ | $0.187\pm 0.002$ | $4.910\pm 0.032$ | $(2.750\pm 0.080)\times 10^{0}$ | $194/44$
| I | $0.148\pm 0.010$ | $0.260\pm 0.002$ | $5.325\pm 0.028$ | $(2.810\pm 0.080)\times 10^{0}$ | $23/39$
| II | $0.148\pm 0.012$ | $0.248\pm 0.001$ | $5.321\pm 0.026$ | $(2.140\pm 0.040)\times 10^{0}$ | $11/39$
| III | $0.148\pm 0.024$ | $0.237\pm 0.002$ | $5.260\pm 0.018$ | $(1.740\pm 0.020)\times 10^{0}$ | $7/39$
$K^{+}+K^{-}$ | IV | $0.148\pm 0.016$ | $0.230\pm 0.001$ | $5.260\pm 0.014$ | $(1.490\pm 0.020)\times 10^{0}$ | $6/39$
$(u\bar{s},s\bar{u})$ | V | $0.148\pm 0.016$ | $0.224\pm 0.001$ | $5.260\pm 0.014$ | $(1.300\pm 0.010)\times 10^{0}$ | $5/39$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.148\pm 0.020$ | $0.216\pm 0.001$ | $5.261\pm 0.018$ | $(1.090\pm 0.020)\times 10^{0}$ | $10/39$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.148\pm 0.016$ | $0.207\pm 0.001$ | $5.252\pm 0.023$ | $(8.500\pm 0.080)\times 10^{-1}$ | $11/39$
| VIII | $0.148\pm 0.018$ | $0.194\pm 0.001$ | $5.240\pm 0.025$ | $(6.860\pm 0.100)\times 10^{-1}$ | $17/39$
| IX | $0.148\pm 0.016$ | $0.178\pm 0.001$ | $5.214\pm 0.029$ | $(5.000\pm 0.060)\times 10^{-1}$ | $22/39$
| X | $0.148\pm 0.021$ | $0.146\pm 0.002$ | $5.220\pm 0.030$ | $(2.900\pm 0.060)\times 10^{-1}$ | $34/39$
| I | $0.530\pm 0.008$ | $0.214\pm 0.002$ | $6.645\pm 0.035$ | $(1.109\pm 0.009)\times 10^{0}$ | $5/37$
| II | $0.530\pm 0.005$ | $0.205\pm 0.002$ | $6.646\pm 0.075$ | $(8.701\pm 0.150)\times 10^{-1}$ | $20/37$
| III | $0.530\pm 0.008$ | $0.194\pm 0.001$ | $6.579\pm 0.063$ | $(7.079\pm 0.040)\times 10^{-1}$ | $17/37$
$p+\overline{p}$ | IV | $0.530\pm 0.009$ | $0.185\pm 0.001$ | $6.603\pm 0.046$ | $(6.130\pm 0.080)\times 10^{-1}$ | $14/37$
$(uud,\bar{u}\bar{u}\bar{d})$ | V | $0.530\pm 0.008$ | $0.178\pm 0.003$ | $6.580\pm 0.086$ | $(5.450\pm 0.090)\times 10^{-1}$ | $23/37$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.530\pm 0.011$ | $0.170\pm 0.001$ | $6.480\pm 0.080$ | $(4.530\pm 0.080)\times 10^{-1}$ | $24/37$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.530\pm 0.004$ | $0.157\pm 0.002$ | $6.353\pm 0.066$ | $(3.650\pm 0.060)\times 10^{-1}$ | $16/37$
| VIII | $0.530\pm 0.009$ | $0.147\pm 0.001$ | $6.341\pm 0.077$ | $(2.900\pm 0.050)\times 10^{-1}$ | $22/37$
| IX | $0.530\pm 0.008$ | $0.134\pm 0.001$ | $6.181\pm 0.076$ | $(2.079\pm 0.020)\times 10^{-1}$ | $25/37$
| X | $0.530\pm 0.010$ | $0.110\pm 0.001$ | $6.180\pm 0.065$ | $(1.120\pm 0.020)\times 10^{-1}$ | $21/37$
| I | $0.148\pm 0.008$ | $0.281\pm 0.002$ | $5.722\pm 0.046$ | $(1.338\pm 0.015)\times 10^{0}$ | $10/34$
| II | $0.148\pm 0.007$ | $0.269\pm 0.002$ | $5.720\pm 0.057$ | $(1.030\pm 0.010)\times 10^{0}$ | $35/34$
| III | $0.148\pm 0.008$ | $0.257\pm 0.001$ | $5.716\pm 0.045$ | $(8.360\pm 0.010)\times 10^{-1}$ | $29/34$
$K^{0}_{S}$ | IV | $0.148\pm 0.009$ | $0.247\pm 0.002$ | $5.625\pm 0.057$ | $(7.080\pm 0.080)\times 10^{-1}$ | $35/34$
$(d\bar{s})$ | V | $0.148\pm 0.008$ | $0.239\pm 0.002$ | $5.560\pm 0.057$ | $(6.200\pm 0.080)\times 10^{-1}$ | $34/34$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.148\pm 0.011$ | $0.231\pm 0.001$ | $5.545\pm 0.066$ | $(5.150\pm 0.060)\times 10^{-1}$ | $46/34$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.148\pm 0.008$ | $0.222\pm 0.001$ | $5.538\pm 0.088$ | $(4.000\pm 0.040)\times 10^{-1}$ | $70/34$
| VIII | $0.148\pm 0.010$ | $0.210\pm 0.001$ | $5.480\pm 0.118$ | $(3.220\pm 0.040)\times 10^{-1}$ | $70/34$
| IX | $0.148\pm 0.007$ | $0.194\pm 0.001$ | $5.444\pm 0.096$ | $(2.350\pm 0.030)\times 10^{-1}$ | $63/34$
| X | $0.148\pm 0.011$ | $0.154\pm 0.001$ | $5.200\pm 0.057$ | $(1.270\pm 0.020)\times 10^{-1}$ | $35/34$

Table 1. Continued.

| Particle (quark structure)
---
and spectrum form
| Multiplicity
---
class
$a_{0}$ |  $T$ (GeV) | $n$ | $N_{0}$ |  $\chi^{2}$/ndof
| I | $1.450\pm 0.012$ | $0.196\pm 0.001$ | $7.073\pm 0.030$ | $(3.250\pm 0.040)\times 10^{-1}$ | $5/10$
| II | $1.450\pm 0.006$ | $0.186\pm 0.001$ | $7.062\pm 0.039$ | $(2.700\pm 0.029)\times 10^{-1}$ | $3/10$
$K^{*0}+\overline{K}{}^{*0}$ | III | $1.450\pm 0.013$ | $0.168\pm 0.002$ | $6.780\pm 0.035$ | $(2.300\pm 0.040)\times 10^{-1}$ | $5/10$
$(u\bar{s},s\bar{u})$ | IV+V | $1.450\pm 0.018$ | $0.154\pm 0.002$ | $6.569\pm 0.035$ | $(1.910\pm 0.030)\times 10^{-1}$ | $3/10$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $1.450\pm 0.020$ | $0.141\pm 0.002$ | $6.376\pm 0.086$ | $(4.519\pm 0.020)\times 10^{-1}$ | $9/10$
$[({\rm GeV}/c)^{-1}]$ | VII | $1.450\pm 0.012$ | $0.133\pm 0.002$ | $6.365\pm 0.057$ | $(1.249\pm 0.040)\times 10^{-1}$ | $9/10$
| VIII | $1.450\pm 0.016$ | $0.124\pm 0.003$ | $6.367\pm 0.066$ | $(1.039\pm 0.020)\times 10^{-1}$ | $6/10$
| IX | $1.450\pm 0.014$ | $0.114\pm 0.002$ | $6.346\pm 0.108$ | $(7.800\pm 0.040)\times 10^{-2}$ | $10/10$
| X | $1.450\pm 0.020$ | $0.093\pm 0.002$ | $6.315\pm 0.111$ | $(4.800\pm 0.030)\times 10^{-2}$ | $10/10$
| I | $1.100\pm 0.004$ | $0.177\pm 0.001$ | $7.149\pm 0.021$ | $(7.619\pm 0.089)\times 10^{-1}$ | $3/12$
| II | $1.100\pm 0.004$ | $0.166\pm 0.001$ | $7.138\pm 0.036$ | $(5.900\pm 0.070)\times 10^{-1}$ | $7/12$
| III | $1.100\pm 0.006$ | $0.158\pm 0.002$ | $7.078\pm 0.038$ | $(4.700\pm 0.060)\times 10^{-1}$ | $12/12$
$\Lambda+\overline{\Lambda}$ | IV | $1.100\pm 0.015$ | $0.151\pm 0.002$ | $6.996\pm 0.040$ | $(3.979\pm 0.060)\times 10^{-1}$ | $16/12$
$(uds,\bar{u}\bar{d}\bar{s})$ | V | $1.100\pm 0.013$ | $0.144\pm 0.002$ | $6.868\pm 0.042$ | $(3.450\pm 0.060)\times 10^{-1}$ | $16/12$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $1.100\pm 0.022$ | $0.136\pm 0.002$ | $6.780\pm 0.061$ | $(2.850\pm 0.040)\times 10^{-1}$ | $18/12$
$[({\rm GeV}/c)^{-1}]$ | VII | $1.100\pm 0.014$ | $0.126\pm 0.002$ | $6.526\pm 0.075$ | $(2.200\pm 0.039)\times 10^{-1}$ | $21/12$
| VIII | $1.100\pm 0.011$ | $0.118\pm 0.003$ | $6.470\pm 0.103$ | $(1.720\pm 0.040)\times 10^{-1}$ | $27/12$
| IX | $1.100\pm 0.012$ | $0.111\pm 0.002$ | $6.476\pm 0.085$ | $(1.130\pm 0.030)\times 10^{-1}$ | $23/12$
| X | $1.100\pm 0.023$ | $0.100\pm 0.002$ | $6.476\pm 0.129$ | $(4.500\pm 0.010)\times 10^{-2}$ | $29/12$
| I | $0.255\pm 0.004$ | $0.396\pm 0.003$ | $10.007\pm 0.060$ | $(9.700\pm 0.069)\times 10^{-2}$ | $7/9$
| II | $0.255\pm 0.003$ | $0.372\pm 0.001$ | $9.615\pm 0.043$ | $(7.000\pm 0.060)\times 10^{-2}$ | $10/9$
| III | $0.255\pm 0.006$ | $0.328\pm 0.001$ | $8.700\pm 0.026$ | $(6.000\pm 0.060)\times 10^{-2}$ | $4/9$
$\Xi^{-}+\overline{\Xi}{}^{+}$ | IV | $0.255\pm 0.005$ | $0.320\pm 0.001$ | $8.553\pm 0.034$ | $(4.800\pm 0.019)\times 10^{-2}$ | $2/9$
$(ssd,\bar{s}\bar{s}\bar{d})$ | V | $0.255\pm 0.005$ | $0.296\pm 0.001$ | $7.629\pm 0.027$ | $(3.999\pm 0.010)\times 10^{-2}$ | $4/9$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.255\pm 0.010$ | $0.276\pm 0.001$ | $7.366\pm 0.035$ | $(3.400\pm 0.020)\times 10^{-2}$ | $3/9$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.255\pm 0.005$ | $0.261\pm 0.001$ | $7.306\pm 0.031$ | $(2.579\pm 0.025)\times 10^{-2}$ | $4/9$
| VIII | $0.255\pm 0.007$ | $0.242\pm 0.001$ | $6.844\pm 0.035$ | $(1.899\pm 0.010)\times 10^{-2}$ | $4/9$
| IX | $0.255\pm 0.006$ | $0.210\pm 0.001$ | $6.255\pm 0.018$ | $(1.240\pm 0.012)\times 10^{-2}$ | $2/9$
| X | $0.255\pm 0.009$ | $0.175\pm 0.001$ | $6.060\pm 0.033$ | $(5.399\pm 0.199)\times 10^{-3}$ | $4/9$
$\Omega^{-}+\overline{\Omega}{}^{+}$ | I+II | $0.830\pm 0.004$ | $0.312\pm 0.002$ | $9.597\pm 0.031$ | $(8.099\pm 0.010)\times 10^{-3}$ | $1/2$
$(sss,\bar{s}\bar{s}\bar{s})$ | III+IV | $0.830\pm 0.004$ | $0.288\pm 0.002$ | $9.548\pm 0.033$ | $(4.899\pm 0.010)\times 10^{-3}$ | $2/2$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | V+VI | $0.830\pm 0.005$ | $0.270\pm 0.002$ | $9.572\pm 0.032$ | $(2.899\pm 0.010)\times 10^{-3}$ | $1/2$
$[({\rm GeV}/c)^{-1}]$ | VII+VIII | $0.830\pm 0.004$ | $0.190\pm 0.002$ | $6.477\pm 0.030$ | $(1.820\pm 0.009)\times 10^{-3}$ | $1/2$
| IX+X | $0.830\pm 0.004$ | $0.169\pm 0.002$ | $6.477\pm 0.031$ | $(5.200\pm 0.001)\times 10^{-4}$ | $3/2$

Table 2. Values of $a_{0}$, $T$, $n$, $N_{0}$, $\chi^{2}$, and ndof
corresponding to the curves in Figure 2 for $pp$ collisions at $\sqrt{s}=13$
TeV. The particle type (quark structure), spectrum form, and multiplicity
classes are given together. The multiplicity classes I, II, III, IV, V, VI,
VII, VIII, IX, and X for $\pi^{+}+\pi^{-}$, $K^{+}+K^{-}$, and
$p+\overline{p}$ correspond to the centrality classes 0–0.92%, 0.92–4.6%,
4.6–9.2%, 9.2–13.8%, 13.8–18.4%, 18.4–27.6%, 27.6–36.8%, 36.8–46%, 46–64.5%,
and 64.5–100%, respectively, and for $K_{S}^{0}$, $K^{*0}$,
$\Lambda+\overline{\Lambda}$, $\Xi^{-}+\overline{\Xi}{}^{+}$, and
$\Omega^{-}+\overline{\Omega}{}^{+}$ correspond to the centrality classes
0–0.9%, 0.9–4.5%, 4.5–8.9%, 8.9–13.5%, 13.5–18%, 18–27%, 27–36.1%, 36.1–45.3%,
45.3–64.5%, and 64.5–100%, respectively.

| Particle (quark structure)
---
and spectrum form
| Multiplicity
---
class
$a_{0}$ |  $T$ (GeV) | $n$ | $N_{0}$ |  $\chi^{2}$/ndof
| I | $-0.550\pm 0.003$ | $0.375\pm 0.002$ | $4.996\pm 0.045$ | $(2.599\pm 0.070)\times 10^{1}$ | $56/47$
| II | $-0.550\pm 0.002$ | $0.350\pm 0.002$ | $4.945\pm 0.022$ | $(2.030\pm 0.030)\times 10^{1}$ | $74/47$
| III | $-0.550\pm 0.004$ | $0.332\pm 0.004$ | $4.870\pm 0.021$ | $(1.670\pm 0.020)\times 10^{1}$ | $70/47$
$\pi^{+}+\pi^{-}$ | IV | $-0.550\pm 0.005$ | $0.326\pm 0.002$ | $4.874\pm 0.026$ | $(1.400\pm 0.019)\times 10^{1}$ | $60/47$
$(u\bar{d},d\bar{u})$ | V | $-0.550\pm 0.005$ | $0.315\pm 0.002$ | $4.874\pm 0.035$ | $(1.250\pm 0.002)\times 10^{1}$ | $55/47$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $-0.550\pm 0.005$ | $0.305\pm 0.001$ | $7.366\pm 0.020$ | $(1.030\pm 0.014)\times 10^{1}$ | $44/47$
$[({\rm GeV}/c)^{-1}]$ | VII | $-0.550\pm 0.002$ | $0.288\pm 0.001$ | $4.870\pm 0.016$ | $(8.229\pm 0.089)\times 10^{0}$ | $49/47$
| VIII | $-0.550\pm 0.004$ | $0.272\pm 0.001$ | $4.870\pm 0.029$ | $(6.730\pm 0.070)\times 10^{0}$ | $56/47$
| IX | $-0.550\pm 0.004$ | $0.249\pm 0.001$ | $4.865\pm 0.028$ | $(4.840\pm 0.050)\times 10^{0}$ | $92/47$
| X | $-0.550\pm 0.005$ | $0.203\pm 0.002$ | $4.861\pm 0.022$ | $(2.800\pm 0.050)\times 10^{0}$ | $324/47$
| I | $0.148\pm 0.003$ | $0.287\pm 0.003$ | $5.276\pm 0.082$ | $(3.179\pm 0.040)\times 10^{0}$ | $15/42$
| II | $0.148\pm 0.002$ | $0.270\pm 0.002$ | $5.273\pm 0.021$ | $(2.580\pm 0.040)\times 10^{0}$ | $9/42$
| III | $0.148\pm 0.003$ | $0.262\pm 0.001$ | $5.273\pm 0.018$ | $(2.060\pm 0.030)\times 10^{0}$ | $14/42$
$K^{+}+K^{-}$ | IV | $0.148\pm 0.005$ | $0.254\pm 0.002$ | $5.250\pm 0.021$ | $(1.740\pm 0.019)\times 10^{0}$ | $25/42$
$(u\bar{s},s\bar{u})$ | V | $0.148\pm 0.005$ | $0.243\pm 0.001$ | $5.180\pm 0.022$ | $(1.540\pm 0.020)\times 10^{0}$ | $24/42$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.148\pm 0.005$ | $0.237\pm 0.003$ | $5.242\pm 0.022$ | $(1.250\pm 0.019)\times 10^{0}$ | $43/42$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.148\pm 0.004$ | $0.224\pm 0.001$ | $5.217\pm 0.023$ | $(9.799\pm 0.100)\times 10^{-1}$ | $47/42$
| VIII | $0.148\pm 0.003$ | $0.213\pm 0.002$ | $5.223\pm 0.025$ | $(7.700\pm 0.080)\times 10^{-1}$ | $54/42$
| IX | $0.148\pm 0.003$ | $0.196\pm 0.002$ | $5.199\pm 0.027$ | $(5.370\pm 0.070)\times 10^{-1}$ | $67/42$
| X | $0.148\pm 0.005$ | $0.158\pm 0.002$ | $5.138\pm 0.024$ | $(2.970\pm 0.040)\times 10^{-1}$ | $464/41$
| I | $0.530\pm 0.011$ | $0.224\pm 0.002$ | $6.291\pm 0.012$ | $(1.379\pm 0.029)\times 10^{0}$ | $24/40$
| II | $0.530\pm 0.013$ | $0.208\pm 0.002$ | $6.278\pm 0.096$ | $(1.060\pm 0.020)\times 10^{0}$ | $24/40$
| III | $0.530\pm 0.012$ | $0.197\pm 0.002$ | $6.267\pm 0.044$ | $(8.799\pm 0.149)\times 10^{-1}$ | $22/40$
$p+\overline{p}$ | IV | $0.530\pm 0.011$ | $0.193\pm 0.001$ | $6.278\pm 0.012$ | $(7.299\pm 0.090)\times 10^{-1}$ | $26/40$
$(uud,\bar{u}\bar{u}\bar{d})$ | V | $0.530\pm 0.013$ | $0.184\pm 0.003$ | $6.262\pm 0.033$ | $(6.499\pm 0.079)\times 10^{-1}$ | $20/40$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.530\pm 0.023$ | $0.175\pm 0.001$ | $6.247\pm 0.055$ | $(5.600\pm 0.080)\times 10^{-1}$ | $27/40$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.530\pm 0.022$ | $0.165\pm 0.002$ | $6.240\pm 0.031$ | $(4.300\pm 0.050)\times 10^{-1}$ | $31/40$
| VIII | $0.530\pm 0.015$ | $0.155\pm 0.002$ | $6.228\pm 0.033$ | $(3.450\pm 0.060)\times 10^{-1}$ | $40/40$
| IX | $0.530\pm 0.009$ | $0.141\pm 0.001$ | $6.156\pm 0.036$ | $(2.360\pm 0.040)\times 10^{-1}$ | $48/40$
| X | $0.530\pm 0.005$ | $0.109\pm 0.001$ | $5.914\pm 0.028$ | $(1.249\pm 0.029)\times 10^{-1}$ | $36/40$
| I | $0.148\pm 0.003$ | $0.304\pm 0.002$ | $5.607\pm 0.033$ | $(1.610\pm 0.019)\times 10^{0}$ | $18/34$
| II | $0.148\pm 0.005$ | $0.288\pm 0.002$ | $5.515\pm 0.038$ | $(1.210\pm 0.080)\times 10^{0}$ | $33/34$
| III | $0.148\pm 0.004$ | $0.273\pm 0.002$ | $5.461\pm 0.063$ | $(9.899\pm 0.079)\times 10^{-1}$ | $37/34$
$K^{0}_{S}$ | IV | $0.148\pm 0.003$ | $0.257\pm 0.001$ | $5.332\pm 0.032$ | $(8.500\pm 0.040)\times 10^{-1}$ | $27/34$
$(d\bar{s})$ | V | $0.148\pm 0.005$ | $0.251\pm 0.002$ | $5.302\pm 0.024$ | $(7.500\pm 0.080)\times 10^{-1}$ | $35/34$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.148\pm 0.003$ | $0.243\pm 0.001$ | $5.303\pm 0.036$ | $(6.239\pm 0.069)\times 10^{-1}$ | $54/34$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.148\pm 0.004$ | $0.234\pm 0.002$ | $5.337\pm 0.048$ | $(4.850\pm 0.040)\times 10^{-1}$ | $65/34$
| VIII | $0.148\pm 0.004$ | $0.218\pm 0.001$ | $5.285\pm 0.043$ | $(3.930\pm 0.050)\times 10^{-1}$ | $64/34$
| IX | $0.148\pm 0.003$ | $0.200\pm 0.001$ | $5.229\pm 0.008$ | $(2.760\pm 0.030)\times 10^{-1}$ | $60/34$
| X | $0.148\pm 0.003$ | $0.158\pm 0.001$ | $5.000\pm 0.037$ | $(1.440\pm 0.029)\times 10^{-1}$ | $51/34$

Table 2. Continued.

| Particle (quark structure)
---
and spectrum form
| Multiplicity
---
class
$a_{0}$ |  $T$ (GeV) | $n$ | $N_{0}$ |  $\chi^{2}$/ndof
| I | $1.450\pm 0.022$ | $0.209\pm 0.002$ | $6.978\pm 0.013$ | $(4.569\pm 0.150)\times 10^{-1}$ | $2/8$
| II | $1.450\pm 0.028$ | $0.196\pm 0.002$ | $6.925\pm 0.008$ | $(3.619\pm 0.080)\times 10^{-1}$ | $8/10$
$K^{*0}$ | III | $1.450\pm 0.035$ | $0.190\pm 0.003$ | $6.805\pm 0.013$ | $(2.899\pm 0.070)\times 10^{-1}$ | $8/10$
$(d\bar{s})$ | IV+V | $1.450\pm 0.030$ | $0.166\pm 0.002$ | $6.318\pm 0.016$ | $(2.379\pm 0.010)\times 10^{-1}$ | $16/10$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $1.450\pm 0.028$ | $0.166\pm 0.004$ | $6.587\pm 0.006$ | $(1.829\pm 0.089)\times 10^{-1}$ | $12/10$
$[({\rm GeV}/c)^{-1}]$ | VII | $1.450\pm 0.018$ | $0.153\pm 0.005$ | $6.410\pm 0.008$ | $(1.519\pm 0.040)\times 10^{-1}$ | $18/10$
| VIII | $1.450\pm 0.025$ | $0.140\pm 0.003$ | $6.415\pm 0.013$ | $(1.219\pm 0.050)\times 10^{-1}$ | $17/10$
| IX | $1.450\pm 0.026$ | $0.127\pm 0.002$ | $6.224\pm 0.008$ | $(8.600\pm 0.026)\times 10^{-2}$ | $17/10$
| X | $1.450\pm 0.018$ | $0.103\pm 0.003$ | $6.218\pm 0.007$ | $(4.700\pm 0.029)\times 10^{-2}$ | $24/10$
| I | $0.300\pm 0.021$ | $0.357\pm 0.003$ | $8.885\pm 0.013$ | $(9.499\pm 0.130)\times 10^{-1}$ | $11/12$
| II | $0.300\pm 0.012$ | $0.328\pm 0.004$ | $8.478\pm 0.112$ | $(7.419\pm 0.180)\times 10^{-1}$ | $15/12$
| III | $0.300\pm 0.014$ | $0.306\pm 0.002$ | $8.301\pm 0.091$ | $(5.999\pm 0.100)\times 10^{-1}$ | $7/12$
$\Lambda+\overline{\Lambda}$ | IV | $0.300\pm 0.008$ | $0.287\pm 0.003$ | $7.782\pm 0.093$ | $(5.099\pm 0.150)\times 10^{-1}$ | $7/12$
$(uds,\bar{u}\bar{d}\bar{s})$ | V | $0.300\pm 0.011$ | $0.270\pm 0.002$ | $7.355\pm 0.043$ | $(4.500\pm 0.050)\times 10^{-1}$ | $5/12$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.300\pm 0.022$ | $0.257\pm 0.001$ | $7.351\pm 0.052$ | $(3.669\pm 0.080)\times 10^{-1}$ | $10/12$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.300\pm 0.019$ | $0.223\pm 0.002$ | $6.553\pm 0.063$ | $(3.060\pm 0.049)\times 10^{-1}$ | $8/12$
| VIII | $0.300\pm 0.014$ | $0.213\pm 0.003$ | $6.554\pm 0.081$ | $(2.339\pm 0.040)\times 10^{-1}$ | $7/12$
| IX | $0.300\pm 0.012$ | $0.192\pm 0.001$ | $6.456\pm 0.043$ | $(1.560\pm 0.040)\times 10^{-1}$ | $10/12$
| X | $0.300\pm 0.025$ | $0.152\pm 0.001$ | $6.000\pm 0.019$ | $(6.899\pm 0.026)\times 10^{-2}$ | $15/12$
| I | $0.255\pm 0.008$ | $0.431\pm 0.002$ | $10.059\pm 0.048$ | $(1.299\pm 0.013)\times 10^{-1}$ | $6/9$
| II | $0.255\pm 0.007$ | $0.405\pm 0.003$ | $10.013\pm 0.093$ | $(9.600\pm 0.140)\times 10^{-2}$ | $5/9$
| III | $0.255\pm 0.013$ | $0.381\pm 0.003$ | $9.603\pm 0.190$ | $(7.599\pm 0.219)\times 10^{-2}$ | $9/9$
$\Xi^{-}+\overline{\Xi}{}^{+}$ | IV | $0.255\pm 0.011$ | $0.371\pm 0.002$ | $9.590\pm 0.112$ | $(6.099\pm 0.120)\times 10^{-2}$ | $6/9$
$(ssd,\bar{s}\bar{s}\bar{d})$ | V | $0.255\pm 0.021$ | $0.353\pm 0.003$ | $9.558\pm 0.170$ | $(5.399\pm 0.140)\times 10^{-2}$ | $14/9$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | VI | $0.255\pm 0.012$ | $0.339\pm 0.002$ | $9.557\pm 0.087$ | $(4.299\pm 0.130)\times 10^{-2}$ | $14/9$
$[({\rm GeV}/c)^{-1}]$ | VII | $0.255\pm 0.013$ | $0.317\pm 0.002$ | $8.985\pm 0.152$ | $(3.199\pm 0.080)\times 10^{-2}$ | $12/9$
| VIII | $0.255\pm 0.007$ | $0.280\pm 0.002$ | $8.000\pm 0.122$ | $(2.600\pm 0.070)\times 10^{-2}$ | $13/9$
| IX | $0.255\pm 0.004$ | $0.242\pm 0.001$ | $6.593\pm 0.088$ | $(1.600\pm 0.010)\times 10^{-2}$ | $4/9$
| X | $0.255\pm 0.012$ | $0.201\pm 0.002$ | $6.587\pm 0.080$ | $(5.900\pm 0.160)\times 10^{-3}$ | $7/9$
$\Omega^{-}+\overline{\Omega}{}^{+}$ | I+II | $0.830\pm 0.021$ | $0.335\pm 0.001$ | $9.656\pm 0.013$ | $(1.020\pm 0.003)\times 10^{-2}$ | $1/2$
$(sss,\bar{s}\bar{s}\bar{s})$ | III+IV | $0.830\pm 0.018$ | $0.290\pm 0.001$ | $9.640\pm 0.013$ | $(4.100\pm 0.200)\times 10^{-3}$ | $5/2$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | V+VI | $0.830\pm 0.020$ | $0.288\pm 0.001$ | $9.640\pm 0.016$ | $(4.099\pm 0.150)\times 10^{-3}$ | $4/2$
$[({\rm GeV}/c)^{-1}]$ | VII+VIII | $0.830\pm 0.004$ | $0.276\pm 0.002$ | $9.640\pm 0.008$ | $(1.9000\pm 0.080)\times 10^{-3}$ | $8/2$
| IX+X | $0.830\pm 0.022$ | $0.233\pm 0.003$ | $9.622\pm 0.008$ | $(6.299\pm 0.150)\times 10^{-4}$ | $1/1$

Table 3. Values of $a_{0}$, $T$, $n$, $N_{0}$, $\chi^{2}$, and ndof
corresponding to the curves in Figure 3 for p–Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. The particle type (quark structure), spectrum form, and
centrality are given together.

| Particle (quark structure)
---
spectrum form
Centrality | $a^{0}$ |  $T$ (GeV) | $n$ | $N_{0}$ |  $\chi^{2}$/ndof
| 0–5% | $-0.550\pm 0.003$ | $0.382\pm 0.003$ | $6.244\pm 0.063$ | $(2.130\pm 0.025)\times 10^{1}$ | $148/54$
$\pi^{+}+\pi^{-}$ | 5–10% | $-0.550\pm 0.004$ | $0.370\pm 0.002$ | $6.004\pm 0.072$ | $(1.743\pm 0.028)\times 10^{1}$ | $135/54$
$(u\bar{d},d\bar{u})$ | 10–20% | $-0.550\pm 0.004$ | $0.358\pm 0.001$ | $5.619\pm 0.121$ | $(1.461\pm 0.025)\times 10^{1}$ | $148/54$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $-0.550\pm 0.005$ | $0.344\pm 0.001$ | $5.616\pm 0.030$ | $(1.153\pm 0.019)\times 10^{1}$ | $142/54$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $-0.550\pm 0.004$ | $0.317\pm 0.002$ | $5.330\pm 0.034$ | $(8.168\pm 0.251)\times 10^{0}$ | $130/54$
| 60–80% | $-0.550\pm 0.002$ | $0.300\pm 0.002$ | $5.332\pm 0.033$ | $(5.027\pm 0.094)\times 10^{0}$ | $93/54$
| 80–100% | $-0.550\pm 0.003$ | $0.255\pm 0.001$ | $5.032\pm 0.038$ | $(2.199\pm 0.072)\times 10^{0}$ | $173/54$
| 0–5% | $0.220\pm 0.007$ | $0.265\pm 0.002$ | $5.968\pm 0.021$ | $(2.859\pm 0.060)\times 10^{0}$ | $63/47$
$K^{+}+K^{-}$ | 5–10% | $0.220\pm 0.013$ | $0.260\pm 0.004$ | $5.948\pm 0.083$ | $(2.356\pm 0.063)\times 10^{0}$ | $36/47$
$(u\bar{s},s\bar{u})$ | 10–20% | $0.220\pm 0.006$ | $0.254\pm 0.002$ | $5.872\pm 0.061$ | $(1.948\pm 0.038)\times 10^{0}$ | $24/47$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $0.220\pm 0.006$ | $0.253\pm 0.002$ | $5.878\pm 0.046$ | $(1.477\pm 0.025)\times 10^{0}$ | $11/47$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $0.220\pm 0.003$ | $0.241\pm 0.001$ | $5.741\pm 0.053$ | $(9.990\pm 0.031)\times 10^{-1}$ | $28/47$
| 60–80% | $0.220\pm 0.006$ | $0.216\pm 0.003$ | $5.446\pm 0.048$ | $(5.969\pm 0.157)\times 10^{-1}$ | $32/47$
| 80–100% | $0.220\pm 0.007$ | $0.190\pm 0.002$ | $5.446\pm 0.056$ | $(2.419\pm 0.062)\times 10^{-1}$ | $88/47$
| 0–5% | $0.398\pm 0.021$ | $0.283\pm 0.003$ | $8.270\pm 0.063$ | $(2.262\pm 0.075)\times 10^{0}$ | $56/45$
$p+\overline{p}$ | 5–10% | $0.398\pm 0.012$ | $0.274\pm 0.002$ | $8.155\pm 0.120$ | $(1.847\pm 0.062)\times 10^{0}$ | $34/45$
$(uud,\bar{u}\bar{u}\bar{d})$ | 10–20% | $0.398\pm 0.024$ | $0.267\pm 0.004$ | $7.900\pm 0.111$ | $(1.558\pm 0.037)\times 10^{0}$ | $33/45$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $0.398\pm 0.021$ | $0.256\pm 0.004$ | $7.885\pm 0.112$ | $(1.206\pm 0.019)\times 10^{0}$ | $18/45$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $0.398\pm 0.024$ | $0.236\pm 0.004$ | $7.652\pm 0.052$ | $(8.357\pm 0.125)\times 10^{-1}$ | $36/45$
| 60–80% | $0.398\pm 0.029$ | $0.208\pm 0.003$ | $7.145\pm 0.066$ | $(5.152\pm 0.144)\times 10^{-1}$ | $45/45$
| 0–5% | $0.220\pm 0.005$ | $0.279\pm 0.002$ | $6.451\pm 0.052$ | $(1.445\pm 0.028)\times 10^{0}$ | $13/30$
$K^{0}_{S}$ | 5–10% | $0.220\pm 0.007$ | $0.279\pm 0.003$ | $6.436\pm 0.068$ | $(1.162\pm 0.028)\times 10^{0}$ | $14/30$
$(d\bar{s})$ | 10–20% | $0.220\pm 0.010$ | $0.278\pm 0.002$ | $6.426\pm 0.049$ | $(9.581\pm 0.157)\times 10^{-1}$ | $18/30$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $0.220\pm 0.008$ | $0.260\pm 0.002$ | $6.000\pm 0.062$ | $(7.383\pm 0.219)\times 10^{-1}$ | $20/30$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $0.220\pm 0.010$ | $0.244\pm 0.002$ | $5.762\pm 0.061$ | $(5.027\pm 0.126)\times 10^{-1}$ | $21/30$
| 60–80% | $0.220\pm 0.013$ | $0.217\pm 0.001$ | $5.527\pm 0.058$ | $(3.079\pm 0.072)\times 10^{-1}$ | $45/30$
| 80–100% | $0.220\pm 0.011$ | $0.181\pm 0.002$ | $5.125\pm 0.059$ | $(1.319\pm 0.053)\times 10^{-1}$ | $33/30$
$\Sigma(1385)^{+}$$(suu)$ | 0–20% | $1.300\pm 0.006$ | $0.201\pm 0.002$ | $7.682\pm 0.065$ | $(4.441\pm 0.050)\times 10^{-2}$ | $2/2$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–60% | $1.300\pm 0.004$ | $0.183\pm 0.001$ | $7.661\pm 0.056$ | $(2.396\pm 0.035)\times 10^{-2}$ | $4/2$
$[({\rm GeV}/c)^{-2}]$ | 60–100% | $1.300\pm 0.008$ | $0.157\pm 0.002$ | $7.523\pm 0.077$ | $(6.596\pm 0.035)\times 10^{-3}$ | $9/2$
| 0–5% | $1.300\pm 0.010$ | $0.187\pm 0.001$ | $8.512\pm 0.058$ | $(8.167\pm 0.220)\times 10^{-1}$ | $3/16$
$\Lambda+\overline{\Lambda}$ | 5–10% | $1.300\pm 0.010$ | $0.181\pm 0.001$ | $8.187\pm 0.055$ | $(6.439\pm 0.094)\times 10^{-1}$ | $4/16$
$(uds,\bar{u}\bar{d}\bar{s})$ | 10–20% | $1.300\pm 0.011$ | $0.174\pm 0.003$ | $7.949\pm 0.063$ | $(5.403\pm 0.063)\times 10^{-1}$ | $2/16$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $1.300\pm 0.005$ | $0.165\pm 0.001$ | $7.731\pm 0.052$ | $(4.083\pm 0.037)\times 10^{-1}$ | $4/16$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $1.300\pm 0.018$ | $0.154\pm 0.003$ | $7.631\pm 0.089$ | $(2.764\pm 0.069)\times 10^{-1}$ | $12/16$
| 60–80% | $1.300\pm 0.023$ | $0.131\pm 0.002$ | $6.958\pm 0.083$ | $(1.633\pm 0.041)\times 10^{-1}$ | $10/16$
| 80–100% | $1.300\pm 0.019$ | $0.111\pm 0.002$ | $6.692\pm 0.102$ | $(6.283\pm 0.251)\times 10^{-2}$ | $18/16$
| 0–5% | $1.000\pm 0.031$ | $0.259\pm 0.003$ | $9.399\pm 0.102$ | $(5.781\pm 0.079)\times 10^{-2}$ | $13/13$
$(\Xi^{-}+\overline{\Xi}^{+})/2$ | 5–10% | $1.000\pm 0.019$ | $0.249\pm 0.004$ | $8.964\pm 0.103$ | $(4.618\pm 0.110)\times 10^{-2}$ | $9/13$
$(ssd,\bar{s}\bar{s}\bar{d})$ | 10–20% | $1.000\pm 0.018$ | $0.244\pm 0.003$ | $8.882\pm 0.089$ | $(3.770\pm 0.085)\times 10^{-2}$ | $7/13$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $1.000\pm 0.007$ | $0.235\pm 0.001$ | $8.824\pm 0.086$ | $(2.733\pm 0.050)\times 10^{-2}$ | $6/13$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $1.000\pm 0.023$ | $0.222\pm 0.002$ | $8.819\pm 0.079$ | $(1.791\pm 0.053)\times 10^{-2}$ | $11/13$
| 60–80% | $1.000\pm 0.002$ | $0.203\pm 0.001$ | $8.030\pm 0.001$ | $(9.204\pm 0.003)\times 10^{-3}$ | $40/12$
| 80–100% | $1.000\pm 0.000$ | $0.153\pm 0.000$ | $6.324\pm 0.001$ | $(3.299\pm 0.003)\times 10^{-3}$ | $11/11$
| 0–5% | $0.830\pm 0.008$ | $0.319\pm 0.002$ | $8.346\pm 0.062$ | $(6.597\pm 0.087)\times 10^{-3}$ | $4/4$
$(\Omega^{-}+\overline{\Omega}{}^{+})/2$ | 5–10% | $0.830\pm 0.006$ | $0.319\pm 0.002$ | $8.139\pm 0.088$ | $(5.655\pm 0.094)\times 10^{-3}$ | $3/4$
$(sss,\bar{s}\bar{s}\bar{s})$ | 10–20% | $0.830\pm 0.021$ | $0.294\pm 0.004$ | $7.722\pm 0.103$ | $(4.398\pm 0.069)\times 10^{-3}$ | $7/4$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $0.830\pm 0.021$ | $0.278\pm 0.002$ | $7.233\pm 0.111$ | $(3.047\pm 0.038)\times 10^{-3}$ | $2/4$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $0.830\pm 0.008$ | $0.243\pm 0.003$ | $6.832\pm 0.193$ | $(1.885\pm 0.063)\times 10^{-3}$ | $7/4$
| 60–80% | $0.830\pm 0.011$ | $0.234\pm 0.002$ | $6.901\pm 0.098$ | $(1.005\pm 0.019)\times 10^{-2}$ | $5/4$
| 80–100% | $0.830\pm 0.018$ | $0.182\pm 0.003$ | $6.003\pm 0.105$ | $(2.984\pm 0.016)\times 10^{-4}$ | $2/4$

Table 4. Values of $a^{0}$, $T$, $n$, $N_{0}$, $\chi^{2}$, and ndof
corresponding to the curves in Figure 4 for Pb–Pb collisions at $\sqrt{s_{\rm
NN}}=2.76$ TeV. The particle type (quark structure), spectrum form, and
centrality are given together.

| Particle (quark structure)
---
spectrum form
Centrality | $a_{0}$ |  $T$ (GeV) | $n$ | $N_{0}$ |  $\chi^{2}$/ndof
| 0–5% | $-0.300\pm 0.004$ | $0.236\pm 0.001$ | $6.599\pm 0.012$ | $(2.325\pm 0.042)\times 10^{3}$ | $562/59$
$\pi^{+}+\pi^{-}$ | 5–10% | $-0.300\pm 0.002$ | $0.236\pm 0.002$ | $6.506\pm 0.014$ | $(1.930\pm 0.021)\times 10^{3}$ | $469/59$
$(u\bar{d},d\bar{u})$ | 10–20% | $-0.300\pm 0.004$ | $0.236\pm 0.002$ | $6.226\pm 0.023$ | $(1.414\pm 0.040)\times 10^{3}$ | $471/59$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $-0.300\pm 0.005$ | $0.237\pm 0.002$ | $6.206\pm 0.022$ | $(8.043\pm 0.302)\times 10^{2}$ | $331/59$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $-0.300\pm 0.004$ | $0.225\pm 0.002$ | $5.850\pm 0.025$ | $(3.086\pm 0.101)\times 10^{2}$ | $158/59$
| 60–80% | $-0.300\pm 0.009$ | $0.218\pm 0.004$ | $5.668\pm 0.026$ | $(8.193\pm 0.367)\times 10^{1}$ | $99/59$
| 0–5% | $0.300\pm 0.008$ | $0.222\pm 0.002$ | $7.198\pm 0.031$ | $(3.177\pm 0.093)\times 10^{2}$ | $607/54$
$K^{+}+K^{-}$ | 5–10% | $0.300\pm 0.009$ | $0.221\pm 0.001$ | $7.178\pm 0.025$ | $(2.724\pm 0.070)\times 10^{2}$ | $524/54$
$(u\bar{s},s\bar{u})$ | 10–20% | $0.300\pm 0.014$ | $0.216\pm 0.003$ | $6.926\pm 0.028$ | $(2.161\pm 0.058)\times 10^{2}$ | $415/54$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $0.300\pm 0.022$ | $0.211\pm 0.003$ | $6.526\pm 0.032$ | $(1.190\pm 0.035)\times 10^{2}$ | $291/54$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $0.300\pm 0.012$ | $0.202\pm 0.005$ | $6.157\pm 0.040$ | $(4.363\pm 0.151)\times 10^{1}$ | $95/54$
| 60–80% | $0.300\pm 0.009$ | $0.201\pm 0.003$ | $6.102\pm 0.026$ | $(1.116\pm 0.035)\times 10^{1}$ | $25/54$
| 0–5% | $1.000\pm 0.005$ | $0.209\pm 0.003$ | $10.357\pm 0.023$ | $(1.089\pm 0.050)\times 10^{2}$ | $490/45$
$p+\overline{p}$ | 5–10% | $1.000\pm 0.020$ | $0.207\pm 0.002$ | $10.337\pm 0.035$ | $(9.178\pm 0.402)\times 10^{1}$ | $444/45$
$(uud,\bar{u}\bar{u}\bar{d})$ | 10–20% | $1.000\pm 0.015$ | $0.205\pm 0.002$ | $10.077\pm 0.080$ | $(6.977\pm 0.312)\times 10^{1}$ | $374/45$
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ | 20–40% | $1.000\pm 0.019$ | $0.193\pm 0.004$ | $9.391\pm 0.100$ | $(3.921\pm 0.090)\times 10^{1}$ | $259/45$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $1.000\pm 0.003$ | $0.170\pm 0.003$ | $8.183\pm 0.082$ | $(1.553\pm 0.060)\times 10^{1}$ | $104/45$
| 60–80% | $1.000\pm 0.002$ | $0.155\pm 0.002$ | $8.005\pm 0.007$ | $(4.524\pm 0.090)\times 10^{0}$ | $14/45$
| 0–5% | $0.220\pm 0.033$ | $0.251\pm 0.005$ | $8.022\pm 0.012$ | $(1.003\pm 0.050)\times 10^{2}$ | $545/29$
$K^{0}_{S}$ | 5–10% | $0.220\pm 0.022$ | $0.251\pm 0.005$ | $8.018\pm 0.017$ | $(8.660\pm 0.340)\times 10^{1}$ | $406/29$
$(d\bar{s})$ | 10–20% | $0.220\pm 0.018$ | $0.251\pm 0.003$ | $8.018\pm 0.068$ | $(6.930\pm 0.300)\times 10^{1}$ | $317/29$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | 20–40% | $0.220\pm 0.012$ | $0.239\pm 0.004$ | $7.238\pm 0.053$ | $(3.889\pm 0.179)\times 10^{1}$ | $229/29$
$[({\rm GeV}/c)^{-2}]$ | 40–60% | $0.220\pm 0.013$ | $0.222\pm 0.002$ | $6.490\pm 0.049$ | $(1.469\pm 0.056)\times 10^{1}$ | $61/29$
| 60–80% | $0.220\pm 0.008$ | $0.214\pm 0.002$ | $6.083\pm 0.040$ | $(3.579\pm 0.039)\times 10^{0}$ | $19/29$
| 80–90% | $0.220\pm 0.009$ | $0.195\pm 0.002$ | $5.719\pm 0.067$ | $(7.959\pm 0.219)\times 10^{-1}$ | $22/28$
| 0–5% | $1.950\pm 0.009$ | $0.156\pm 0.004$ | $8.468\pm 0.064$ | $(1.728\pm 0.094)\times 10^{1}$ | $29/9$
$(K^{*0}+\overline{K}{}^{*0})/2$ | 5–10% | $1.950\pm 0.031$ | $0.148\pm 0.003$ | $8.276\pm 0.061$ | $(1.521\pm 0.125)\times 10^{1}$ | $23/9$
$(u\bar{s},s\bar{u})$ | 10–20% | $1.950\pm 0.040$ | $0.148\pm 0.003$ | $7.978\pm 0.105$ | $(1.175\pm 0.157)\times 10^{1}$ | $50/9$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | 20–30% | $1.950\pm 0.026$ | $0.144\pm 0.005$ | $7.646\pm 0.203$ | $(8.734\pm 0.817)\times 10^{0}$ | $32/9$
$[({\rm GeV}/c)^{-1}]$ | 30–40% | $1.950\pm 0.019$ | $0.140\pm 0.002$ | $7.646\pm 0.104$ | $(6.723\pm 0.565)\times 10^{0}$ | $13/9$
| 40–50% | $1.950\pm 0.018$ | $0.138\pm 0.002$ | $7.613\pm 0.066$ | $(4.398\pm 0.163)\times 10^{0}$ | $7/9$
| 0–5% | $2.600\pm 0.014$ | $0.125\pm 0.002$ | $11.456\pm 0.015$ | $(2.403\pm 0.060)\times 10^{1}$ | $84/27$
$\Lambda$ | 5–10% | $2.600\pm 0.012$ | $0.124\pm 0.001$ | $11.406\pm 0.013$ | $(2.170\pm 0.060)\times 10^{1}$ | $43/27$
$(uds)$ | 10–20% | $2.600\pm 0.013$ | $0.127\pm 0.002$ | $11.406\pm 0.021$ | $(1.740\pm 0.090)\times 10^{1}$ | $30/27$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | 20–40% | $2.600\pm 0.008$ | $0.110\pm 0.001$ | $9.993\pm 0.033$ | $(9.899\pm 0.299)\times 10^{0}$ | $25/27$
$[({\rm GeV}/c)^{-1}]$ | 40–60% | $2.600\pm 0.006$ | $0.100\pm 0.001$ | $9.367\pm 0.045$ | $(3.700\pm 0.079)\times 10^{0}$ | $5/27$
| 60–80% | $2.600\pm 0.014$ | $0.085\pm 0.001$ | $8.383\pm 0.057$ | $(8.800\pm 0.099)\times 10^{-1}$ | $21/27$
| 80–90% | $2.600\pm 0.023$ | $0.078\pm 0.001$ | $8.169\pm 0.062$ | $(1.720\pm 0.069)\times 10^{-1}$ | $37/25$
$(\Xi^{-}+\overline{\Xi}{}^{+})/2$ | 0–10% | $1.955\pm 0.008$ | $0.194\pm 0.002$ | $16.514\pm 0.018$ | $(3.390\pm 0.099)\times 10^{0}$ | $68/23$
$(ssd,\bar{s}\bar{s}\bar{d})$ | 10–20% | $1.955\pm 0.010$ | $0.181\pm 0.001$ | $14.343\pm 0.089$ | $(2.579\pm 0.059)\times 10^{0}$ | $18/23$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | 20–40% | $1.955\pm 0.014$ | $0.168\pm 0.001$ | $12.385\pm 0.103$ | $(1.460\pm 0.039)\times 10^{0}$ | $21/23$
$[({\rm GeV}/c)^{-1}]$ | 40–60% | $1.955\pm 0.014$ | $0.164\pm 0.002$ | $12.152\pm 0.100$ | $(4.850\pm 0.159)\times 10^{-1}$ | $27/21$
| 60–80% | $1.955\pm 0.025$ | $0.130\pm 0.002$ | $9.000\pm 0.013$ | $(1.099\pm 0.019)\times 10^{-1}$ | $18/21$
$(\Omega^{-}+\overline{\Omega}{}^{+})/2$ | 0–10% | $2.150\pm 0.009$ | $0.186\pm 0.002$ | $12.707\pm 0.048$ | $(5.500\pm 0.099)\times 10^{-1}$ | $6/9$
$(sss,\bar{s}\bar{s}\bar{s})$ | 10–20% | $2.150\pm 0.008$ | $0.179\pm 0.002$ | $12.492\pm 0.056$ | $(4.099\pm 0.109)\times 10^{-1}$ | $8/9$
$(1/N_{evt})d^{2}N/dp_{T}dy$ | 20–40% | $2.150\pm 0.008$ | $0.176\pm 0.002$ | $12.400\pm 0.088$ | $(2.280\pm 0.140)\times 10^{-1}$ | $10/9$
$[({\rm GeV}/c)^{-1}]$ | 40–60% | $2.150\pm 0.011$ | $0.138\pm 0.001$ | $9.008\pm 0.066$ | $(7.599\pm 0.129)\times 10^{-2}$ | $3/8$
| 60–80% | $2.150\pm 0.014$ | $0.136\pm 0.002$ | $9.032\pm 0.101$ | $(1.299\pm 0.100)\times 10^{-2}$ | $7/6$

##  III Results and discussion

###  III.1 Comparison with data

Figure 1 shows the multiplicity dependent $p_{T}$ spectra, the double-
differential yield $(1/N_{evt})d^{2}N/dp_{T}dy$, of $\pi^{+}+\pi^{-}$ (a),
$K^{+}+K^{-}$ (b), $p+\overline{p}$ (c), $K^{0}_{S}$ (d),
$K^{*}+\overline{K}{}^{*}$ (e), $\Lambda+\overline{\Lambda}$ (f),
$\Xi^{-}+\overline{\Xi}{}^{+}$ (g), and $\Omega^{-}+\overline{\Omega}{}^{+}$
(h) with rapidity $|y|<0.5$, produced in $pp$ collisions at the center-of-mass
energy, $\sqrt{s}=7$ TeV, where $N_{evt}$ denotes the number of events which
can be omitted in the vertical axis according to the format in the cited
reference. Different symbols represent the experimental data for different
multiplicity classes determined by the values of multiplicity (VOM) measured
by the ALICE Collaboration 44 , where in most cases the data are scaled by
constant multipliers marked in the panels for clarity. The curves represent
our fit results from the Monte Carlo calculations. The values of free
parameters ($a_{0}$, $T$, and $n$), normalization constant ($N_{0}$),
$\chi^{2}$, and number of degree of freedom (ndof) are listed in Table 1 in
which the particle type, quark structure that makes up hadrons, and form of
spectra are mentioned. The corresponding serial numbers of VOM classes marked
with the Roman numerals in the figure are also listed in Table 1 in terms of
the percentage classes which can be regarded as the centrality classes. One
can see that our fit results are in well agreement with the experiment data
measured by the ALICE Collaboration at midrapidity in $pp$ collisions at
$\sqrt{s}=7$ TeV.

Similar to Figure 1, Figure 2 shows the multiplicity dependent $p_{T}$
spectra, the double-differential yield $(1/N_{evt})d^{2}N/dp_{T}dy$, of
$\pi^{+}+\pi^{-}$ (a), $K^{+}+K^{-}$ (b), $p+\overline{p}$ (c), $K^{0}_{S}$
(d), $K^{*0}$ (e), $\Lambda+\overline{\Lambda}$ (f),
$\Xi^{-}+\overline{\Xi}{}^{+}$ (g), and $\Omega^{-}+\overline{\Omega}{}^{+}$
(h) with $|y|<0.5$, produced in $pp$ collisions at $\sqrt{s}=13$ TeV measured
by the ALICE Collaboration 45 ; 46 ; 47 . Different symbols represent the data
for different serial numbers of VOM classes marked by the Roman numerals, and
the corresponding centrality classes are listed in Table 2. The curves are our
results of the Monte Carlo calculations which are used to fit the data. Some
of the data are scaled by multiplying different amounts marked in the panels
for clarity. The values of $a_{0}$, $T$, $n$, $N$, $\chi^{2}$/ndof, and other
related information are listed in Table 2. From the figure and
$\chi^{2}$/ndof, one can see that our fit results are in good agreement with
the experimental data measured by the ALICE Collaboration at midrapidity in
$pp$ collisions at $\sqrt{s}=13$ TeV.

Similar to Figures 1 and 2, Figure 3 displays the centrality dependent $p_{T}$
spectra, the invariant yield $(1/2\pi p_{T})d^{2}N/dp_{T}dy$ (a–c) or
$(1/N_{evt}2\pi p_{T})d^{2}N/dp_{T}dy$ (d, f–h) or the double-differential
yield $(1/N_{evt})d^{2}N/dp_{T}dy$ (e), of $\pi^{+}+\pi^{-}$ (a),
$K^{+}+K^{-}$ (b), $p+\overline{p}$ (c), $K^{0}_{S}$ (d), $\Sigma(1385)^{+}$
(e), $\Lambda+\overline{\Lambda}$ (f), $(\Xi^{-}+\overline{\Xi}{}^{+})/2$ (g),
and $(\Omega^{-}+\overline{\Omega}{}^{+})/2$ (h) with $-0.5<y<0$ (a–c, e, g,
h) or $0<y<0.5$ (d, f), produced in $p$–Pb collisions at $\sqrt{s_{NN}}=5.02$
TeV measured by the ALICE Collaboration 10 ; 48 ; 49 ; 50 . Different symbols
represent the experimental data with different centrality classes, where
different constant multipliers are used to re-scale the data for clarity. The
curves are our fit results based on the Monte Carlo calculations. The values
of $a_{0}$, $T$, $n$, $N_{0}$, and $\chi^{2}$/ndof are listed in Table 3 with
other information. One can see that our fit results are in good agreement with
the experimental data measured by the ALICE Collaboration at mid-$y$ in $p$–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV.

Similar to Figures 1–3, Figure 4 gives the centrality dependent $p_{T}$
spectra, the invariant yield $(1/2\pi p_{T})d^{2}N/dp_{T}dy$ (a–c, e) or the
double-differential yield $(1/N_{evt})d^{2}N/dp_{T}dy$ (d, f–h), of
$\pi^{+}+\pi^{-}$ (a), $K^{+}+K^{-}$ (b), $p+\overline{p}$ (c), $K^{0}_{S}$
(d), $(K^{*0}+\overline{K}{}^{*0})/2$ (e), $\Lambda$ (f),
$(\Xi^{-}+\overline{\Xi}{}^{+})/2$ (g), and
$(\Omega^{-}+\overline{\Omega}{}^{+})/2$ (h) with $|\eta|<0.8$ (a–c) or
$|y|<0.5$ (d–h), produced in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV
measured by the ALICE Collaboration 51 ; 52 ; 53 ; 54 . The symbols represent
the experimental data and the curves are our fit results. The value of
$a_{0}$, $T$, $n$, $N_{0}$, and $\chi^{2}$/ndof are listed in Table 4. One can
see that our fit results are approximately in agreement with the experimental
data measured by the ALICE Collaborations at mid-$\eta$ or mid-$y$ in Pb–Pb
collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV.

From the above comparisons one can see that the multi-source thermal model at
the quark level can fit the $p_{T}$ spectra of identified hadrons produced at
midrapidity in $pp$, $p$–Pb, and Pb–Pb collisions at the LHC. We note that the
degree of fit for $pp$ collisions is better than that for $p$–Pb collisions,
and the degree of fit for $p$–Pb collisions is better than that for Pb–Pb
collisions. The phenomenon that the degree of fit for small system is better
than that for large system is caused by the multiple scattering in large
system. Due to the multiple scattering, there are more factors affecting
particle production. In particular, in central Pb–Pb collisions, for light
particles ($\pi$, $K$, and $p$) with high-$p_{T}$, the departure of the fit
from data is more obvious. This is indeed caused by multiple scattering and
other medium effects.

In the model, the contribution of each quark to $p_{T}$ of given hadron is
assumed to obey the TP-like function with isotropic azimuth. Although the
analytical expression of $p_{T}$ distribution is not available, the Monte
Carlo method is performed in the calculations. As we know, for a wide $p_{T}$
range, there are at least two components namely the soft and hard components
in the structure of $p_{T}$ spectrum. The present work shows that we do not
need to distinguish the two components. Instead, we may use a set of
parameters to fit a wide $p_{T}$ spectrum, though the parameters are multi-
factor dependent.

###  III.2 Tendencies of parameters

Fig. 5. Dependence of $T$ on $C$ for $pp$ collisions at $\sqrt{s}=7$ TeV (a),
$pp$ collisions at $\sqrt{s}$ = 13 TeV (b), $p$–Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV (c), and Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV (d).
Different symbols represent the parameters from the $p_{T}$ spectra of
different particles marked in the panels.

Fig. 6. Same as Figure 5, but showing the dependence of $n$ on $C$.

Fig. 7. Same as Figure 5, but showing the dependence of $a_{0}$ on $m_{0}$.

Fig. 8. Same as Figure 5, but showing the dependence of $T$ on $m_{0}$.

Fig. 9. Same as Figure 5, but showing the dependence of $n$ on $m_{0}$.

Fig. 10. Same as Figure 5, but showing the dependence of $a_{0}$ on $m_{q}$.

Fig. 11. Same as Figure 5, but showing the dependence of $T$ on $m_{q}$.

Fig. 12. Same as Figure 5, but showing the dependence of $n$ on $m_{q}$

In order to see more intuitively the dependence of parameters on the
centrality $C$ (the multiplicity is converted to centrality in $pp$
collisions), rest mass $m_{0}$ of the hadron, and constituent mass $m_{q}$ of
the quark, we show the multi-factor dependent parameters in Figures 5–12,
where the values of parameters are cited in Tables 1–4. Because of the revised
index $a_{0}$ being independent of the centrality, we have not shown the plot
for the centrality dependent $a_{0}$.

Figures 5 and 6 show the dependence of effective temperature $T$ and entropy-
related index $n$ on centrality $C$ respectively. Panels (a–d) are for $pp$
collisions at $\sqrt{s}=7$ TeV, $pp$ collisions at $\sqrt{s}=13$ TeV, $p$–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and Pb–Pb collisions at
$\sqrt{s_{\rm NN}}=2.76$ TeV, respectively. Different symbols represent the
parameters from the $p_{T}$ spectra of different particles marked in the
panels. One can see that with the decrease of centrality from central to
peripheral collisions, $T$ and $n$ decrease in most of the cases. Central
collisions create higher concentration of energies because of larger number of
participants and hence a higher system temperature, $T$ is expected. As a
consequence, this is expected to create a more thermalized system leading to
larger value of $n$ or smaller entropy index $q$, as $n=1/(q-1)$. These
results are consistent with the fact that more central is the collision, the
system has a higher tendency of creating a high temperature thermalized
system.

This result is consistent with our previous work 39 which also shows that $T$
in central collisions is larger. In Ref. 39 we have studied the light (also
including strangeness) and heavy particles produced in small system ($pp$,
$d$–Au, and $p$–Pb collisions) and large system (Au–Au and Pb–Pb collisions)
in the case of $\bm{p_{ti}}$ being parallel. This result is also consistent
with our another work published recently 61 , which analyzes the kinetic
freeze-out temperature $T_{0}$ extracted from narrow $p_{T}$ spectra
($p_{T}<4.5$ GeV/$c$) of identified particles ($\pi^{+}$, $K^{+}$, $p$,
$K^{0}_{S}$, $\Lambda$, $\Xi$, $\Omega+\overline{\Omega}$) produced in
copper–copper (Cu–Cu), Au–Au, and Pb–Pb collisions by the blast-wave model.
That result shows larger $T_{0}$ in central collisions.

Figures 7–9 show the dependences of parameters $a_{0}$, $T$, and $n$ on the
rest mass $m_{0}$ of particle, respectively. Same as Figures 5 and 6, panels
(a–d) are for $pp$ collisions at $\sqrt{s}=7$ TeV, $pp$ collisions at
$\sqrt{s}$ =13 TeV, $p$–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and
Pb–Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV, respectively. Different
symbols represent the parameters from the $p_{T}$ spectra of different
particles marked in the panels. With the increase of $m_{0}$, one can see that
$a_{0}$ and $n$ increases significantly for the four cases, and the tendency
of $T$ is strange: $T$ decreases firstly and then increases, the boundary is
at around $m_{0}=1$ GeV/$c^{2}$.

The results of the $m_{0}$ dependent parameters are not contradictory to our
previous work 39 if we examine minutely the parameter plots around $m_{0}=1$
GeV/$c^{2}$ there, though in which the analysis was done for a special case.
The amplitudes of the $m_{0}$ dependent parameters in Figures 7–9 are
different for the four cases. These differences are explained by different
collision energies and system sizes. In particular, in Pb–Pb collisions,
multiple scattering of particles in hot and dense matter affects the
parameters. This multiple scattering also reflects the shadowing effect in
which the subsequent medium has no chance to be collided due to the change of
direction of motion of the particles in the multiple scattering.

The dependences of parameters $a_{0}$, $T$, and $n$ on the constituent mass
$m_{q}$ of quark are given in Figures 10–12, respectively. Same as Figures
5–9, panels (a–d) are for $pp$ collisions at $\sqrt{s}=7$ and 13 TeV, $p$–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and Pb–Pb collisions at
$\sqrt{s_{\rm NN}}=2.76$ TeV, respectively. Different symbols represent the
parameters from the $p_{T}$ spectra of different particles marked in the
panels. One can see nearly the same parameters for $m_{u}$ ($m_{d}$) and
$m_{s}$, and some differences for the four collisions.

The present work on $m_{q}$ dependent parameters is not contradictory to our
previous work 39 , though two more quarks (charm and bottom) are included
there. The nearly the same parameters are observed due to very small
difference between $m_{u}$ ($m_{d}$) and $m_{s}$ ($m_{u}=m_{d}=0.31$
GeV/$c^{2}$ and $m_{s}=0.5$ GeV/$c^{2}$ 60a ). The differences in amplitudes
for the four collisions are explained by different collision energies and
system sizes, in particular by different system sizes, where the influence of
multiple scattering in large system is considerable, as the explanations for
Figures 7–9.

###  III.3 Further discussion

The values of revised index $a_{0}$ extracted from the $p_{T}$ spectra of
$\pi^{+}+\pi^{-}$ are negative, which means the upward tendency of the spectra
in low-$p_{T}$ region 38 . This is contributed by the resonance decays. The
values of $a_{0}$ extracted from the $p_{T}$ spectra of other particles are
positive or even larger than 1, which means the downward tendency of the
spectra in low-$p_{T}$ region. This is caused by the constraints in the
production of other particles. Larger constraint seems to be observed for
strange particles/quark. Due to the limitation of normalization, any upward or
downward tendency in low-$p_{T}$ region will cause the shape change of the
curve in intermediate- and high-$p_{T}$ regions.

The values of effective temperature $T$ extracted from the present work
contain both the contributions of thermal motion and collective flow, which
shows that $T$ is larger than 0.2 GeV even 0.3 GeV. To dissociate the two
contributions and obtain the kinetic freeze-out temperature $T_{0}$ and
transverse flow velocity $\beta_{T}$, one has to apply different methods which
shows some level of inconsistencies in some cases. These inconsistencies
mainly show in different centrality and size dependent behaviour of $T_{0}$
and $\beta_{T}$. In our opinion, $T$ displays the coincident results and
should be paid more attention. To obtain $T_{0}$ and $\beta_{T}$, a uniform
method is needed at the first place. This issue is beyond the focus of the
present work. In all cases, the entropy-related index $n>4$ which means that
$q<1.25$, as expected for high-energy collisions. This is a $q$ close to 1 and
implies that the system at the quark level is in approximate equilibrium or
local equilibrium.

As an extension of the special (parallel) case to general (arbitrary) one, the
results from our previous work 38 ; 39 are confirmed more broadly here.
Combining with our previous work, we may infer that the method used in the
present work is suitable for wide ranges of collision energy, system size,
event centrality, and particle species. Although the analytical expression of
$p_{T}$ distribution for the special case can be obtained, we may use the
Monte Carlo method to calculate the $p_{T}$ distribution for the general case.

We note that the fit for the $p_{T}$ spectra in Pb–Pb collisions is not good,
in particular for the spectra in high-$p_{T}$ region in central collisions.
This is explained by the influence of multiple scattering of produced
particles in hot and dense matter. To include the influence of multiple
scattering, except for the superposition of the contributions of quarks with
the TP-like form of transverse momenta and isotropic azimuths, we may need
another function in the present framework. Or, we may use a two-component
function to fit the $p_{T}$ spectra in Pb–Pb collisions, where the second
component can describe the contribution of multiple scattering.

The present work includes the contributions of soft excitations and hard
scattering processes together, without having a scope to separate them in the
ambient of the present analysis of identified particle transverse momentum
spectra using TP-like distribution function. The possible departure of the fit
from the data in high-$p_{T}$ region in central Pb–Pb collisions could be due
to multiple scatterings and medium effects and in addition, a thermalized
medium being formed in these nuclear collisions.

##  IV Summary and Conclusions

The present analysis is summarized below with important observations and
conclusions.

(a) The transverse momentum, $p_{T}$ spectra of various light hadrons
(including some strange particles) produced in $pp$ collisions at $\sqrt{s}=7$
and 13 TeV, $p$–Pb collisions at $\sqrt{s_{NN}}=5.02$ GeV, and Pb–Pb
collisions at $\sqrt{s_{NN}}=2.76$ TeV for different multiplicity or
centrality classes have been studied in the framework of a multi-source
thermal model at the quark level. The contribution of each constituent quark
to hadron’s $p_{T}$ is assumed to obey the TP-like function with isotropic
azimuth. The calculations are performed by Monte Carlo method and compared
with the experimental data measured by the ALICE Collaboration. A reasonably
good description of the spectra across various collision species and energies
available at the LHC is observed.

(b) With the decrease of final state multiplicity from central to peripheral
collisions, the revised index, $a_{0}$ being nearly invariant, the effective
temperature, $T$ and the entropy-related index, $n$ decrease in most of the
cases. With the increase of the rest mass $m_{0}$ of the particles, $a_{0}$
and $n$ increases significantly, and $T$ decreases first and then increases
with the boundary being at around 1 GeV/$c^{2}$. Higher collision energy is
deposited in central collisions, which results in larger $T$. The system in
central collisions is closer to the equilibrium, which results in larger $n$
or smaller entropy index, $q$. Both the soft excitation and hard scattering
processes are considered uniformly. The influence of multiple scattering in
large system is considerable.

Acknowledgments

The work of Pei-Pin Yang was supported by the China Scholarship Council
(Chinese Government Scholarship) under Grant No. 202008140170, the Shanxi
Provincial Innovative Foundation for Graduate Education under Grant No.
2019SY053, and the Innovative Foundation for Graduate Education of Shanxi
University. The work of Shanxi Group was supported by the National Natural
Science Foundation of China under Grant Nos. 12047571, 11575103, and 11947418,
the Scientific and Technological Innovation Programs of Higher Education
Institutions in Shanxi (STIP) under Grant No. 201802017, the Shanxi Provincial
Natural Science Foundation under Grant No. 201901D111043, and the Fund for
Shanxi “1331 Project” Key Subjects Construction. Raghunath Sahoo acknowledges
the financial supports under the CERN Scientific Associateship, CERN, Geneva,
Switzerland and the financial grants under DAE-BRNS Project No.
58/14/29/2019-BRNS of Government of India.

Author Contributions All authors listed have made a substantial, direct, and
intellectual contribution to the work and approved it for publication.

Data Availability Statement This manuscript has no associated data or the data
will not be deposited. [Authors’ comment: The data used to support the
findings of this study are included within the article and are cited at
relevant places within the text as references.]

Ethical Approval The authors declare that they are in compliance with ethical
standards regarding the content of this paper.

Disclosure The funding agencies have no role in the design of the study; in
the collection, analysis, or interpretation of the data; in the writing of the
manuscript, or in the decision to publish the results.

Conflict of Interest The authors declare that there are no conflicts of
interest regarding the publication of this paper.

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