#  $\mathrm{K}^{+}/\pi^{+}$ horn: appropriate statistics for nonextensive
particle production and nonequilibrium phase transition

Abdel Nasser Tawfik<EMAIL_ADDRESS>Future University in Egypt (FUE), 5th
Settlement, 11835 New Cairo, Egypt  Eman R. Abou Elyazeed  Physics Department,
Faculty of Women for Arts, Science and Education, Ain Shams University, 11577
Cairo, Egypt  Hayam Yassin  Physics Department, Faculty of Women for Arts,
Science and Education, Ain Shams University, 11577 Cairo, Egypt

###### Abstract

We suggest that the long–standing problem with incapability of the thermal
models in reproducing the horn–like structure of the kaon–to–pion ratio
measured at AGS, SPS and low RHIC energies and confirmed in the beam energy
scan program, was raised due to applying inappropriate statistics, mainly,
extensive additive Boltzmann–Gibbs (BG). The assumption that the particle
production, the dynamical nonequilibrium process, is priori to be analyzed by
extensive BG or nonextensive Tsallis statistics, seems being a failed path
followed over the past decades. With generic (non)extensive statistics, two
equivalence classes $(c,d)$ belittle such an ad hoc assumption. Accordingly,
the statistical ensemble itself determines its degree of (non)extensivity,
including extensive BG and nonextensive Tsallis statistics, which are
characterized by $(1,1)$ and $(0,q)$, respectively. The energy dependence of
light–, $\gamma_{q}$, and strange–quark occupation factor, $\gamma_{s}$,
indicates that the produced particles are best described as a nonequilibrium
ensemble, where a remarkable nonmonotonic behavior associates the
$\mathrm{K}^{+}/\pi^{+}$ horn, for example. On other hand, the resulting
equivalence classes $(c,d)$ refer to a generic nonextensivity associated with
extended exponential and Lambert–$W_{0}$ exponentially generating distribution
function, which obviously emerge from free, short– and long–range
correlations. With this generic statistical approach, the hadron resonance gas
model excellently reproduces the $\mathrm{K}^{+}/\pi^{+}$ ratio, including its
nonmonotonic horn–like structure.

Nonequilibrium and irreversible thermodynamics, Self–organized systems,
Relativistic heavy-ion collisions, Other topics in statistical physics,
thermodynamics, and nonlinear dynamical systems

###### pacs:

05.70.Ln, 05.65.+b, 25.75.-q, 05.90.+m

††preprint: ECTP-2021-08††preprint: WLCAPP-2021-08

##  I Introduction

A nonmonotonic horn–like structure of the $\mathrm{K}^{+}/\pi^{+}$ ratio was
confirmed in various experiments, at AGS, SPS and low RHIC energies
Gazdzicki:1998vd ; E866:1999ktz ; NA44:2002wha ; NA49:2002pzu ; NA49:2004iqm .
At $\sqrt{s_{NN}}\lesssim 20~{}$GeV, there is a monotonic rapid increase
followed by a monotonic rapid decrease, i.e., a maximum appears, while at
energies $\gtrsim 20~{}$GeV, the RHIC and LHC energies, the
$\mathrm{K}^{+}/\pi^{+}$ ratio seems to be nearly energy–independent, i.e., a
plateau is formed. Various interpretations have been given in literature
Gazdzicki:1998vd ; Cleymans:2005zx ; Andronic:2005yp ; Tawfik:2005gk ;
Tomasik:2006qs ; Chatterjee:2009km ; Tawfik:2010uh ; Nayak:2010uq . The
thermal models which are either based on extensive additive Boltzmann–Gibbs
(BG) and/or on Tsallis–type nonextensive statistics Tawfik:2019oct  aren’t
well succeeded in reproducing the nonmonotonic horn–like structure of the
$\mathrm{K}^{+}/\pi^{+}$ ratio, Fig. 4 and Fig. 3. Despite of this, various
extensions to the thermal models are allegedly assumed to deliver explanations
for the underlying processes in the particle production Shen:2019kil ;
Shen:2019zgi ; Shen:2017pyo ; Biro:2016dgd ; Parvan:2010vn ; Wilk:2014zka ;
Rybczynski:2012vj !

To improve the capability of the thermal models; the ideal gas of identified
hadronic resonances, one of the authors (AT) proposed i) a substantial change
in the single–particle distribution function due to a dynamical change in the
phase–space volume, especially at the phase transition, whose imprints seem to
survive till the chemical freezeout, where the produced particles are detected
Tawfik:2010uh , ii) integrating nonequilibrium light– and strange–quark
occupation factors, $\gamma_{q,s}$, in the phase–space volume Tawfik:2005gk ;
Tawfik:2017ehz ; Tawfik:2017oyn , iii) applying Tsallis–type nonextensive
statistics Tawfik:2019oct , and iv) deriving the ideal gas of hadron
resonances away from equilibrium by assigning eigen–volume to each of its
constituents Tawfik:2014eba ; Tawfik:2013eua . Castorina et al. assumed a
suppression in the strange–quark occupation factor, $\gamma_{s}$, that
accounts for reducing the production rates of the strange particles in $pp$,
$pA$, $AA$ collisions, at different energies and found that the strangeness
suppression disappears with the onset of the color deconfinement and
full–chemical nonequilibrium is subsequently attained Castorina:2017dvt ;
Castorina:2018zxi . The occupation factors, $\gamma_{q,s}$, introduce
controllers over the numbers of light– and strange–quarks fitting within the
phase–space element, i.e., a similar role as that of the chemical potentials.
None of these ingratiates was able to satisfactorily reproduce the various
particle ratios, including $\mathrm{K}^{+}/\pi^{+}$ Tawfik:2018ahq ;
Tawfik:2017ehz ; Tawfik:2017oyn .

Hence, there is a conceptional reason. Imposing nonequilibrium (either as
partial– or full–chemical nonequilibrium $\gamma_{q,s}$ or as finite sizes of
the constituents of the ideal gas, etc.) in an equilibrium ensemble of BG–type
extensive additive statistics seems conceptionally unsuccessful. Even if
Tsallis–type nonextensivity is assumed, the matching of nonequilibrium with
nonextensivity would be improved, on one hand. On the other hand, the price to
be paid in return is huge. This isn’t less than lose the meaning of the
thermodynamic quantities, such as the freezeout temperature Deppman:2012qt ;
Biro:2016myk ; NasserTawfik:2016sqs . With this regard, it should be
highlighted that the Tsallis–type nonextensivity is limited to a
one–dimensional line within the entire space of (non)extensivity utilized in
present calculations, in which both BG and Tsallis statistics are merely
special cases.

The relativistic high–energy collisions are assumed to go through several
critical processes, such as deconfinement and hadronization. A radical change
in properties, symmetries, and degrees of freedom (dof) of the QCD matter is
obviously depending on the collision energy. The horn–like structure of the
kaon–to–pion ratio, which normalizes strange to light quark flavors, likely
signals onset of nonextensive particle production and nonequilibrium phase
transition that might dynamically enhance the strange quark flavor or supress
the light quark flavor or vice versa. The present script suggests that the
incapability of the extensive additiove thermal models in reproducing the
horn–like structure measured at AGS, SPS and low RHIC energies, Fig. 4 and
Fig. 3, is due to inappropriate statistics. Generic (non)extensive statistics,
section II, autonomously describes the equivalence classes of the generalized
entropies in the entire $(c,d)$–plane Thurner1 . At the chemical freezeout,
where detection of the produced particles takes place, the statistical
ensemble is therefore dynamic and probably violates the fourth
Shannon–Khinchin axiom NasserTawfik:2016sqs . Even if this wouldn’t be the
case and the produced particles would be in full–chemical nonequilibrium,
generic (non)extensive statistics is also well applicable. Extensive or
nonextensive ensemble and equilibrium or out–of–equilibrium system are
properly encoded by the equivalence classes $(c,d)$ including extensive BG and
Tsallis–type nonextensive statistics, which are characterized by $(c,d)=(1,1)$
and $(c,d)=(0,q)$, respectively.

We first assume partial–chemical nonequilibrium in generic (non)extensive
statistics, i.e., $\gamma_{q}=1$, while $\gamma_{s}$ is to be determined from
the statistical fit of various particle ratios, in a wide range of energies,
to the available experimental measurements. We also assume full–chemical
nonequilibrium, i.e., both $\gamma_{q}\neq 1$ and $\gamma_{s}\neq 1$ are
determined from the statistical fits, as well. In the same manner, the
energy–dependence of the equivalence classes $(c,d)$ is also determined, Fig.
1. In the present script, the occupation factors $\gamma_{q,s}$, the
equivalence classes $(c,d)$, and the freezeout parameters; the temperature $T$
and the baryon chemical potential $\mu_{B}$, aren’t fitting variables. They
are inputs varying with the collision energies. With these inputs, the
$\mathrm{K}^{+}/\pi^{+}$ ratio calculated in generic (non)extensive statistics
excellently reproduces its measured result including the nonmonotonic
horn–like structure, at $\sqrt{s_{NN}}\lesssim 20~{}$GeV; the AGS, SPS and low
RHIC energies, Fig. 3.

The present paper is organized as follows. The generic (non)extensive approach
is reviewed in section II. The energy–dependence of the equivalence classes
$(c,d)$, the light– and strange–quark occupation factors $\gamma_{s,q}$, and
the $\mathrm{K}^{+}/\pi^{+}$ ratio is presented in section III.1, III.2, and
III.3, respectively. Section IV is devoted to the conclusions. Appendices A, B
and C respectively elaborate details on probability distribution, entropy, and
exponential and logarithm functions in Boltzmann, Tsallis, and generic
(non)extensive statistics, The impacts of the inclusion of very high–mass
resonances and missing states in the hadron resonance gas model, and the
contour plots of the quality of fit for $\gamma_{q,s}$. The quality of the fit
for the equivalence classes $(c,d)$ is depicted in Fig. 1.

##  II Generic (non)extensive statistical approach

In generic (non)extensive statistics, the partition function can be given as
Tawfik:2018ahq ; Tawfik:2017bsy

$\displaystyle\ln\,Z(T,\mu)$ $\displaystyle=$ $\displaystyle\pm
V\,\sum_{i}\,\frac{g_{i}}{(2\,\pi)^{3}}\,\int_{0}^{\infty}\,\ln\left[1\pm\varepsilon_{c,d,r}(x_{i})\right]\;d^{3}\,{\bf
p},$ (1)

where $x_{i}=[\mu_{i}-E_{i}({\bf p})]/T$ with $E_{i}(p)=\sqrt{{\bf
p}^{2}+m_{i}^{2}}$; the dispersion relation of $i$–th particle, $\mu_{i}$ is
the chemical potential of that particle;
$\mu_{i}=B_{i}\mu_{B}+S_{i}\mu_{S}+...$, with $S_{i}$ and $B_{i}$ are the
corresponding strangeness and baryon quantum numbers and $\mu_{S}$ and
$\mu_{B}$ are the strangeness and baryon chemical potential, respectively. $g$
is the degeneracy factor, $V$ is the volume of the fireball, ${\bf p}$ is the
momentum, and $\pm$ represent fermions and bosons, respectively. The extended
exponential function $\varepsilon_{c,d,r}(x_{i})$ is given as Thurner1 ;
Thurner2

$\displaystyle\varepsilon_{c,d,r}(x_{i})=\exp\left[\frac{-d}{1-c}\left(W_{k}\left[B\left(1-\frac{x_{i}}{r}\right)^{\frac{1}{d}}\right]-W_{k}[B]\right)\right],$
(2)

where $W_{k}$ is the Lambert–$W_{k}$ function which has real solutions, at
$k=0$ with $d\geq 0$, and at $k=1$ with $d<0$.

$B=\frac{(1-c)r}{1-(1-c)r}\exp\left[\frac{(1-c)r}{1-(1-c)r}\right],$ (3)

where $r=(1-c+cd)^{-1}$ with $c$, $d$ are two critical exponents defining the
equivalence classes for both types of extensive and nonoextensive statistical
ensembles. The latter are statistical systems violating the fourth Shannon-
Khinchin axiom; the generalized Shannon additivity 111If $p_{ij}\geq 0$,
$p_{i}=\sum_{j=1}^{m_{i}}p_{ij}$ and $\sum_{i=1}^{n}p_{i}=1$, where
$i=1,\cdots,n$ and $j=1,\cdots,m_{i}$, then
$S(p_{11},\cdots,p_{nm_{n}})=S_{1}(p_{1},\cdots,p_{n})+\sum_{i=1}^{n}p_{i}S_{1}\left(\frac{p_{i1}}{p_{i}}\cdots,\frac{p_{im_{i}}}{p_{i}}\right)$.
Accordingly, $S_{n}$ is the Shannon entropy given by
$S_{n}(P)=\tau\cdot\sum_{k=1}^{n}p_{k}\log_{2}p_{k}$ with $\tau<0$, i.e., the
entropy of a system, which could be divided into sub–systems $A$ and $B$ is
given as $S_{A}$ and the expectation value of $S_{B}$ but conditional on $A$;
$S_{nm}(AB)=S_{n}(A)+S_{m}(B|A)$, where $S_{m}(B|A)=\sum_{k}p_{k}\cdot
S_{m}(B_{k})$ Tawfik:2017bsy . The generalized entropy reads
$S_{c,d}[p]\propto\sum_{i=1}^{\Omega}\Gamma(d+1,1-c\log(p_{i}))$, where the
constants $(c,d)$ characterize the universality class of entropy of the system
completely in the thermodynamic limit and also specifies its distribution
functions. The fourth Shannon–Khinchin axiom is corresponding to Markovian
processes, which are random processes, whose future is independent of the
past, and the natural stochastic analogy of the deterministic processes
described by differential equations., especially in their large size limit, in
which two asymptotic properties of the generalized entropies are associated
with one scaling function each. Each of the scaling functions is characterized
by one exponent defining equivalence relations of the entropic forms, i.e.,
two entropic forms are equivalent if their exponents are the same.

The equivalence classes $(c,d)$ which are two exponents determining two
scaling functions with two asymptotic properties, are intrinsic properties of
all types of the underlying statistics. For instance, extensive BG and
Tsallis–type nonextensive statistics assume $(c,d)=(1,1)$ and $(0,q)$,
respectively Thurner2 . The remaining $c$–$d$ space is inhabited by various
types of nonextensivity or superstatistics enabling us to reveal
nonequilibrium phenomenon such as critical endpoint, correlations,
fluctuations, higher–order cumulants, etc. The present script suggests that
the horn of the kaon–to–pion ratio likely signals onset of nonextensive
particle production and impact of nonequilibrium phase transition. The
incapability of the equilibrium thermal models in reproducing the enhancement
in both particle yields, the kaons and the pions, Fig. 4 and Fig. 3, could be
interpretted due to inappropriate statistics Tawfik:2014eba ; Tawfik:2010uh ;
Tawfik:2017bsy ; NasserTawfik:2016sqs ; Tawfik:2018ahq , i.e., either
extensive BG or Tsallis–type nonextensive statistics.

With the occupation factors of light– and strange–quarks, $\gamma_{q}$ and
$\gamma_{s}$, respectively, the generic (non)extensive partition function
reads

$\displaystyle\ln\,Z(T,\mu)$ $\displaystyle=$ $\displaystyle\pm
V\,\sum_{i}\,\frac{g_{i}}{(2\,\pi)^{3}}\,\int_{0}^{\infty}\,\ln\left[1\pm(\gamma_{q}^{n_{q}})_{i}\;(\gamma_{s}^{n_{s}})_{i}\;\varepsilon_{c,d,r}(x_{i})\right]\;d^{3}\,{\bf
p},$ (4)

where $n_{q}$ and $n_{s}$ are the number of light and strange quarks,
respectively 222Since we are interested on the QCD phase diagram, the
contributions of the heavier quarks could be neglected.. As BG statistics
assumes extensivity, additivity, and equilibrium particle production,
$\gamma_{q}$ and $\gamma_{s}$ could plainly be seen as additional variables
imposed on BG statistics. But they are essential ingredients to (non)extensive
statistics 333This should not necessarily be limited to the Tsallis–type. The
entire $c$–$d$ space characterizes various types of superstatistics including
extensive and nonextensive ensembles.. In generic (non)extensive statistics,
the degree of nonextensivity shall subjectively be determined by the
changeable statistical nature of the system of interest, at various energies.

With this regard, we recall that the statistical properties of an ensemble
consisting of individual objects is given by a single–particle distribution
function. The probability of finding an object in infinitesimal region of the
phase–space is proportional to the volume of the phase–space element and the
single–particle distribution function. The latter is assumed to fulfill the
Boltzmann-Vlasov (BV) master equation, which is the semi–classical limit of a
time–dependent Hartee–Fock theory through the Wigner transform of the one–body
density matrix and can be used to study the dynamics of the constituents. At
high energies, the phase–space volume and its configurations are apparently
modified. So does the single–particle distribution function Tawfik:2010uh .

The occupation factors $\gamma_{q,s}$ act as controllers over the number of
the quarks fitting within the phase–space element, i.e., similar to the
chemical potentials $a=\ln E-\ln T-\ln N$, where $\lambda=\exp(-a)$
Tawfik:2010aq . At $\gamma_{s}=\gamma_{q}=1$, the statistical ensemble refers
to a system in chemical and thermal equilibrium, but if $\gamma_{s}$ or
$\gamma_{q}$ varies from unity, while its counterpart remains equilibrium, the
statistical system is modified to partial–chemical nonequilibrium. At
$\gamma_{s}\neq 1$ and $\gamma_{q}\neq 1$, the statistical ensemble is
apparently out–of–equilibrium.

In the thermodynamic limit, the number density can then be deduced from the
partition function, Eq. (4) RAFELSKI ; Tawfik:2018ahq ,

$\displaystyle\textit{n}(T,\mu)$ $\displaystyle=$
$\displaystyle\pm\sum_{i}\,\frac{g_{i}}{2\pi^{2}}\int_{0}^{\infty}\frac{(1-c)^{-1}\,(\gamma_{q}^{n_{q}})_{i}\;(\gamma_{s}^{n_{s}})_{i}\;\varepsilon_{c,d,r}(x_{i})\;W_{0}\left[B(1-\frac{x_{i}}{r})^{\frac{1}{d}}\right]}{\left[1\pm(\gamma_{q}^{n_{q}})_{i}\;(\gamma_{s}^{n_{s}})_{i}\;\varepsilon_{c,d,r}(x_{i})\right]\left(r-x_{i}\right)\left(1+W_{0}\left[B(1-\frac{x_{i}}{r})^{\frac{1}{d}}\right]\right)}\;\textbf{p}^{2}d\textbf{p}.\hskip
14.22636pt$ (5)

At the chemical freezeout, which in both extensive BG Tawfik:2016jzk ;
Tawfik:2012si ; Cleymans:2005zx ; Andronic:2005yp ; Tawfik:2004ss  and
genertic (non)extensive statistics Tawfik:2018ahq  is thermodynamically
conditioned along the entire QCD phase diagram, the various hadron resonances
decay either to stable particles, $n_{i}^{direct}$, or to other resonances
$n_{j}$. In the final state, we take this into account so that a specific
particle yield, such as pion and kaon, is determined as

$\displaystyle\langle n_{i}^{final}\rangle$ $\displaystyle=$
$\displaystyle\langle n_{i}^{direct}\rangle+\sum_{j\neq i}b_{j\rightarrow
i}\langle n_{j}\rangle,$ (6)

where $b_{j\rightarrow i}$ is the branching ratio for the decay of $j$–th
resonance to $i$–th particle.

In the present work, we first assume partial–chemical nonequilibrium, i.e.,
$\gamma_{q}=1$, while $\gamma_{s}$ is determined from the statistical fits of
the generic (non)extensive calculations of the $\mathrm{K}^{+}/\pi^{+}$ ratio,
in a wide range of center–of–mass energies $\sqrt{s_{NN}}$ to available
experimental measurements, Fig. 2 panel (a). We also assume full–chemical
nonequilibrium, i.e., both $\gamma_{q}\neq 1$ and $\gamma_{s}\neq 1$ are
determined from the same statistical fits of the $\mathrm{K}^{+}/\pi^{+}$
ratio, Fig. 2 panels (b) and (c). The degree of (non)extensivity is
autonomously determined by the statistical ensemble. The results shall be
discussed in the next section.

##  III Results

By statistical fits of various particle ratios, at the chemical freezeout
444At which the various quantum numbers, including baryon, strangeness, and
electric charge, are conserved., whose parameters $T$ and $\mu_{B}$ are well
expressed as functions of the collision energies and thermodynamically
conditioned, for example constant entropy density $s/T^{3}=7$, in either
extensive BG Tawfik:2004ss ; Tawfik:2012si ; Tawfik:2013eua ; Tawfik:2016jzk
or genertic (non)extensive statistics Tawfik:2018ahq , the energy dependence
of the equivalence classes $(c,d)$, section III.1, and that of the light– and
strange–quark occupation factors $\gamma_{q}$ and $\gamma_{s}$, section III.2,
have been determined in generic (non)extensive statistics, at various
center–of–mass energies $\sqrt{s_{NN}}$. The three sets of parameters, the
freezeout parameters, the equivalence classes and the occupation factors,
enter our calculations for $\mathrm{K}^{+}/\pi^{+}$ ratio as inputs. At a
given $\sqrt{s_{NN}}$, which is related to $\mu_{B}$ and a freezeout
temperature $T$, the corresponding equivalence classes and occupation factors
are substituted in Eq. (5), which together with Eq. (6) determine the number
density of a specific particle.

###  III.1 Equivalence classes ($c,d$) and nonextensivity

In Fig. 1, the resulting equivalence classes $(c,d)$, at various energies
Tawfik:2018ahq , are respectively depicted as squares and circles. These
results could be well parameterized by the following expressions:

$\displaystyle c(\sqrt{s_{NN}})$ $\displaystyle=$
$\displaystyle\left[a_{1}+\left(\sqrt{s_{NN}}\right)^{-b_{1}}\right]^{-c_{1}}\;d_{1}\sqrt{s_{NN}},$
(7) $\displaystyle d(\sqrt{s_{NN}})$ $\displaystyle=$
$\displaystyle\left[a_{2}-\left(\sqrt{s_{NN}}\right)^{-b_{2}}\right]^{c_{2}}\;\exp\left(\frac{d_{2}}{\sqrt{s_{NN}}}\right),$
(8)

where $a_{1}=1.005\pm 0.01$, $b_{1}=3.242\pm 0.106$, $c_{1}=3.247\pm 0.24$,
$d_{1}=6.526\times 10^{-06}\pm 1.003\times 10^{-06}$, $a_{2}=1.947\pm 0.0006$,
$b_{2}=0.008\pm 0.00015$, $c_{2}=3.493\pm 0.0739$, and $d_{2}=0.506\pm 0.019$.
Both expressions are respectively depicted as dashed and solid curves in Fig.
1.

Fig. 1: Contour plot showing the equivalence classes $(c,d)$ as functions of
$\sqrt{s_{NN}}$. The dashed curve stands for $c$, Eq. (7), while the solid
curve represents $d$, Eq. (8). The horizontal line refers to extensive BG
statistics. The thin dashed line marks the well–defined equivalence classes of
extensive BG statistics, $(c,d)=(1,1)$. The colored band to the left
illustrates the corresponding statistical errors.

At AGS energies, we find that $d$ seems to start from a large value (nearly
unity) and rapidly decreases, while $c$ starts from $\sim 0.94$ and
exponentially increases with increasing energy. At $\sqrt{s_{NN}}\simeq 4\pm
1~{}$GeV, the values of both equivalence classes become equal,
$(c,d)\simeq(0.97,0.97)$. At SPS and low RHIC energies, another remarkable
conclusion can be drawn. While $c$ reaches a maximum value, at
$\sqrt{s_{NN}}\simeq 10\pm 1~{}$GeV, $d$ continues its decrease and approaches
its minimum $d\sim 0.93\pm 0.02$, at $\sqrt{s_{NN}}\simeq 20\pm 2~{}$GeV. At
RHIC and LHC energies, $c$ sets on a wide plateau, at $c\simeq 0.985$. At
$\sqrt{s_{NN}}\gtrsim 20$GeV, $d$ switches to a nice exponential increase with
increasing energy. $c$ turns to set on such an exponential increase, at LHC
energies, where both equivalence classes exceed the unity, i.e., the second
Shannon–Khinchin axiom also becomes violated.

The degree of (non)extensivity is to be determined by the equivalence classes
$(c,d)$, whose energy–dependence was best parameterized by Eqs. (7) and (8),
respectively, and depicted in Fig. 1. We notice that the resulting equivalence
classes are positive and close to unity. This means that:

  1. 1.

the underlying statistics isn’t extensive BG, especially at AGS and SPS
energies, but certainly not Tsallis–type nonextesnive statistics,

  2. 2.

at AGS, SPS and RHIC energies, the statistical systems holds first 555The
Shannon entrpy $S_{n}(p)=\tau\cdot\sum_{k=1}^{n}p_{k}\log_{2}p_{k}$, where
$\tau<0$, is continuous in $\Delta_{n}$. i.e., the entropy continuously
depends on $p$., second 666For the uniform distribution
$U_{n}=(1/n,\cdots,1/n)\in\Delta_{n}$, the Shannon entropy is maximum, i.e.,
$S_{n}(p)\leq S_{n}(U_{n})$ for any $p\in\Delta_{n}$ Tawfik:2017bsy . and
third 777The Shannon entropy is expandable;
$S_{n+1}(p_{1},p_{2},\cdots,p_{n},0)=S_{n}(p_{1},p_{2},\cdots,p_{n})$ for all
$(p_{1},p_{2},\cdots,p_{n})\in\Delta_{n}$, i.e., the requirement that adding a
zero–probability state to a system does not change the entropy.
Shannon–Khinchin axioms but violates the fourth Shannon–Khinchin axiom,

  3. 3.

at LHC energies, both second and fourth Shannon–Khinchin axioms are violated,
and d) the corresponding entropy is Lambert–$W_{0}$ exponential; exponentially
generating function $W_{0}(x)=\sum_{n=1}^{\infty}(-n)^{n-1}x^{n}/n!$.

Even the slight difference from unity depicted in Fig. 1, the resulting
$(c,d)$ play an essential role by deriving the entire statistical system from
extensivity, additivity, and equilibrium. Boltzmann statistics is sharply
located at $(c,d)=(1,1)$. Any deviation from $(c,d)=(1,1)$ apparently replaces
Boltzmann distribution, entropy, exponential, and logarithm functions by their
counterparts characterizing generic (non)extensive statistics, Appendix A. The
extended exponential and Lambert–$W_{o}$ exponentially generating distribution
function, are substantially different from the ordinary exponential and
distribution functions.

###  III.2 Nonequilibrium quark occupation factors

In partial– and full–chemical nonequilibrium, various particles ratios
measured, at various $\sqrt{s_{NN}}$, are fitted to their calculations in
generic (non)extensive statistics. At a given $\sqrt{s_{NN}}$, the
corresponding equivalence classes $(c,d)$, section III.1, and freezeout
parameters $\mu_{B}$ and $T$ are substituted in the expression for the number
density of a specific particle yield. With the fitting parameters either
$\gamma_{s}$, at $\gamma_{q}=1$ for partial–chemical nonequilibrium or
$\gamma_{s}$ and $\gamma_{q}$ for full–chemical nonequilibrium various
particles ratios calculated in generic (non)extensive statistics are fitted to
the measured ones. The overall energy dependence of both $\gamma_{q}$ and
$\gamma_{s}$, Fig. 2, is given as

$\displaystyle\gamma_{s,q}(\sqrt{s_{NN}})$ $\displaystyle=$
$\displaystyle\alpha\,\exp\left(-\beta\,\sqrt{s_{NN}}\right)\,\sin\left(\nu\,\sqrt{s_{NN}}+\eta\right)+\zeta,$
(9)

where the values of the parameters $\alpha,\;\beta,\;\nu,\;\eta,\;\zeta$
distinguish between partial– ($\gamma_{q}=1$ while $\gamma_{s}$ is the fitting
variable) and full–chemical nonequilibrium, Tab. 1.

nonequilibrium |  | $\alpha$ | $\beta$ | $\nu$ | $\eta$ | $\zeta$
---|---|---|---|---|---|---
partial at $\gamma_{q}=1$ | $\gamma_{s}$ | $10.373\pm 0.625$ | $0.361\pm 0.009$ | $0.233\pm 0.003$ | $5.362\pm 0.014$ | $0.751\pm 0.018$
full | $\gamma_{s}$ | $2.913\pm 0.259$ | $0.262\pm 0.062$ | $-0.257\pm 0.018$ | $4.353\pm 0.081$ | $0.804\pm 0.025$
$\gamma_{q}$ | $-5.264\pm 0.256$ | $0.381\pm 0.018$ | $0.373\pm 0.017$ | $4.88\pm 0.071$ | $1.037\pm 0.041$
Tab. 1: Fitting parameters of the dependence of $\gamma_{s,q}$ on
$\sqrt{s_{NN}}$, Eq. (9).

Fig. 2: Top panel (partial–chemical nonequilibrium): $\gamma_{s}$ is given as
a function of $\sqrt{s_{NN}}$, where $\gamma_{q}=1$. Bottom panel
(full–chemical nonequilibrium): both $\gamma_{s}$ (left) and $\gamma_{s}$
(right) are the fitting variables. Solid symbols with errors depict our
results from the statistical fits of the measured $\mathrm{K}^{+}/\pi^{+}$
with the generic (non)extensive calculations. Open symbols with errors
represent the statistical fits of the measured $\mathrm{K}^{+}/\pi^{+}$ to the
extensive BG statistics Tawfik:2017oyn . The solid and double–dotted curves
show Eq. (9) of the present study and refs. Tawfik:2017ehz ; Tawfik:2017oyn ,
respectively.

The dependence of $\gamma_{s}$ and $\gamma_{q}$ on $\sqrt{s_{NN}}$ is
presented in Fig. 2. The solid symbols refer to the present results from the
statistical fits of the measured $\mathrm{K}^{+}/\pi^{+}$ ratio to the generic
(non)extensive calculations, Eq. (5), while the open symbols represent the
calculations from the extensive hadron resonance gas (HRG) model based on
extensive BG statistics Tawfik:2017ehz ; Tawfik:2017oyn . Top panel of Fig. 2
shows $\gamma_{s}$ in dependence on $\sqrt{s_{NN}}$ as calculated in generic
(non)extensive statistics, where $\gamma_{q}=1$, i.e., partial–chemical
nonequilibrium. Bottom panel of Fig. 2 depicts the dependence of $\gamma_{s}$
(left panel) and $\gamma_{q}$ (right panel) on $\sqrt{s_{NN}}$, i.e.,
full–chemical nonequilibrium. The proposed expression, Eq. (9), based on
generic (non)extensive statistics for the dependence of $\gamma_{s}$ and/or
$\gamma_{q}$ on $\sqrt{s_{NN}}$ is depicted by solid curves. The double–dotted
curves are the corresponding expressions deduced from the extensive HRG model,
which is based on exensive BG statistics Tawfik:2017ehz ; Tawfik:2017oyn .

In top panel of Fig. 2, at $\gamma_{q}=1$ and partial–chemical nonequilibrium,
we notice that $\gamma_{s}$ exponentially increases, at $\sqrt{s_{NN}}\lesssim
7$ GeV. The same behaviour was quantitatively observed in the extensive BG
calculations Tawfik:2017ehz . The rate of increase in the present calculations
(generic (non)extensive statistics) is larger than in extensive BG statistics.
At higher energies ($7\lesssim\sqrt{s_{NN}}\lesssim 20$ GeV), $\gamma_{s}$
rapidly decreases. Also here, the rate of decreasing $\gamma_{s}$ in generic
(non)extensive statistics is larger than that in extensive BG statistics. At
$\sqrt{s_{NN}}>20$ GeV, $\gamma_{s}$ becomes energy independent. At this
plateau, the resulting $\gamma_{s}$ from both types of statistics is in good
agreement with each other, $\gamma_{s}\simeq 0.78\pm 0.01$. We conclude that
with the global assumption that $\gamma_{q}=1$, the resulting $\gamma_{s}$
equally encodes the partial–chemical nonequilibrium in both types of
statistics, at $\sqrt{s_{NN}}>20$ GeV. At lower energies, both types of
statistics significantly disagree with each other, especially at
$\sqrt{s_{NN}}\simeq 7$ GeV. Here, the generic (non)extensive $\gamma_{s}$ is
$\sim 40\%$ larger than the extensive BG $\gamma_{s}$, i.e., the
partial–chemical nonequilibrium in strange–quark occupation factor is $\sim
40\%$ larger. In other words, the signals for the strangeness enhancement, for
instance, at the peak (horn) of the kaon–to–pion ratio, are more than $40\%$
overcast shrouded when measurements, such as multiplicity of the produced
particles, are restricted to extensive BG statistics. Hence, a recommendation
to the fellow experimentalists can be now addressed. Applying generic
(non)extensive statistics, especially in Run II of the STAR beam energy scan
program Nonaka:2019fad ; STAR:2015vvs , and the future facilities, FAIR and
NICA, likely exposes essential experimental signals.

Bottom panel of Fig. 2 shows nonequilibrium $\gamma_{s}$ (left) and
$\gamma_{q}$ (right). In bottom left panel, we notice that at
$\sqrt{s_{NN}}\lesssim 7$ GeV and varying $\gamma_{q}$, $\gamma_{s}$
calculated from generic (non)extensive statistics exponentially increases with
increasing $\sqrt{s_{NN}}$, while $\gamma_{s}$ deduced from the extensive HRG
calculations exponentially decreases Tawfik:2017oyn . It is apparent that the
rate of the increase in the earlier is more rapid than that of the decrease in
the latter. Also here, we find that both maximum [generic (non)extensive
statistics] and minimum (BG statistics) are likely positioned, at
$\sqrt{s_{NN}}\simeq 7$ GeV. At $7\lesssim\sqrt{s_{NN}}\lesssim 20$ GeV, the
generic (non)extensive $\gamma_{s}$ rapidly decreases, while the extensive BG
$\gamma_{s}$ rapidly increases.

At $\sqrt{s_{NN}}>20$ GeV, $\gamma_{s}$ from both types of statistics becomes
energy independent. Opposite to the partial–chemical nonequilibrium (top
panel), the corresponding plateaus are not coincident. While the extensive BG
$\gamma_{s}\simeq 0.98\pm 0.1$, i.e., very close to equilibrium, the generic
(non)extensive $\gamma_{s}\simeq 0.81\pm 0.01$ referring to large
nonequilibrium. To draw a conclusion about the degree of (non)equilibrium, the
resulting $\gamma_{q}$ which is depicted in the bottom left panel should also
be taken into consideration. Relative to the plateaus formed, at high
energies, we notice that the height of the $\gamma_{s}$ crest [generic
(non)extensive] is larger than the depth of the $\gamma_{s}$ trough [extensive
BG], at $\sqrt{s_{NN}}\simeq 7$ GeV.

The bottom right panel of Fig. 2 depicts the energy dependence of
$\gamma_{q}$, at varying $\gamma_{s}$. Here, we find that at
$\sqrt{s_{NN}}\lesssim 7$ GeV, both generic (non)extensive $\gamma_{q}$ and
extensive BG $\gamma_{q}$ (HRG model) exponentially decrease with increasing
energy. The rate of the earlier is larger than that of the latter. While both
minima are nearly positioned, at $\sqrt{s_{NN}}\simeq 7$ GeV, their depths
aren’t the same. The depth of the generic (non)extensive $\gamma_{q}$ is
larger than that of the extensive BG $\gamma_{q}$. Also, $\gamma_{q}$ becomes
energy independent, at $\sqrt{s_{NN}}>20$ GeV. The heights of both plateaus
aren’t coincident. The generic (non)extensive are lower than the extensive BG
results. We find that, both generic (non)extensive extensive BG
$\gamma_{q}>1$.

Both bottom panels illustrate that a) the resulting values of $\gamma_{q}$ and
$\gamma_{s}$ obtained from generic (non)extensive statistics are lower than
their values in extensive BG statistics, b) at $\sqrt{s_{NN}}\lesssim 7$ GeV,
there is an exception that the generic (non)extensive $\gamma_{s}$ reaches
maximum, c) the span of the values that $\gamma_{q}$ and $\gamma_{s}$ take in
extensive BG statistics is narrower than in generic (non)extensive statistics,
and d) at $\sqrt{s_{NN}}\gtrsim 20$ GeV, both generic (non)extensive extensive
BG $\gamma_{q,s}$ are nearly energy–independent, where $\gamma_{q}>1$ and
$\gamma_{s}<1$.

###  III.3 $\mathrm{K}^{+}/\pi^{+}$ ratio

Figure 3 depicts the $\mathrm{K}^{+}/\pi^{+}$ ratio as a function of
$\sqrt{s_{NN}}$. Symbols refer to the experimental results, at energies
ranging from $5$ GeV to $5.44$ TeV E866:1999ktz ; NA44:2002wha ; NA49:2002pzu
; NA49:2004iqm ; STAR:2005gfr ; ALICE:2012ovd ; STAR:2019vcp ; ALICE:2021lsv ;
ALICE:2019hno . The dotted curve shows the generic (non)extensive
calculations, at partial–chemical nonequilibrium, i.e., varying $\gamma_{s}$,
while $\gamma_{q}=1$. The solid curve illustrates the results, at
full–chemical nonequilibrium, i.e., varying $\gamma_{s}$ and $\gamma_{q}$. The
double–dashed curve gives the extensive BG calculations, at partial–chemical
nonequilibrium Tawfik:2017ehz , while the dashed curve presents the extensive
BG calculations, at full–chemical nonequilibrium Tawfik:2017oyn . The
incapability of the equilibrium HRG model, i.e., $\gamma_{q}=\gamma_{s}=1$, in
reproducing both the horn–structure of the $\mathrm{K}^{+}/\pi^{+}$ ratio, at
AGS, SPS, and low RHIC energies, and their measurements, at RHIC and LHC
energies, is shown by the long–dashed curve.

In the HRG model; an ideal gas of point–like hadrons characterized by the
Hagedorn mass spectrum which is consisting of a discrete (so–far measured) and
a continuous (missing hadron states), various higher–order thermodynamic
quantities have been analyzed ManLo:2016pgd . In a previous study
Tawfik:2017ggu , one of the authors (AT) concluded that the missing states
aren’t considerably contributing to the lower–order thermodynamic quantities,
especially the first–order ones, the particle number or particle multiplicity.
Thus, the inclusion of missing states wouldn’t improve the thermal model
incapability to reproduce the horn–structure of the $\mathrm{K}^{+}/\pi^{+}$
ratio, Appendix B.

Fig. 3: The experimental results of the $\mathrm{K}^{+}/\pi^{+}$ ratio are
given in dependence on $\sqrt{s_{NN}}$ E866:1999ktz ; NA44:2002wha ;
NA49:2002pzu ; NA49:2004iqm ; STAR:2005gfr ; ALICE:2012ovd ; STAR:2019vcp ;
ALICE:2021lsv ; ALICE:2019hno . The solid and dotted curves depict the generic
(non)extensive calculations, at full– ($\gamma_{q}$ and $\gamma_{s}\neq 1$)
and partial–chemical nonequilibrium ($\gamma_{s}\neq 1$), respectively.
Double–dashed and dashed curves represent the extensive BG results, at full–
Tawfik:2017ehz  and partial–chemical nonequilibrium Tawfik:2017oyn . The
long–dashed curve shows the HRG calculations, at equilibrium, i.e.,
$\gamma_{q}=\gamma_{s}=1$.

We find that the generic (non)extensive calculations in the entire energy
spectrum excellently agree with the experimental results. The horn–like
structure of the kaon–to–pion ratio is perfectly reproduced. Comparing to the
other types of statistics, the generic (non)extensive approach is well capable
in describing the rapid increase, at $\sqrt{s_{NN}}\lesssim 7$ GeV and also
the fast decrease, at $7\lesssim\sqrt{s_{NN}}\lesssim 20$ GeV. At
$\sqrt{s_{NN}}\gtrsim 20$ GeV, a fine structure of the plateau is predicted.
Because of the wide gab between top RHIC and LHC energies, this is an
essential estimation. Another finding is that both partial– and full–chemical
nonequilibrium results significantly vary, especially at
$\sqrt{s_{NN}}\lesssim 7\pm 1$ GeV. This emphasizes that the light quark
occupation factor, $\gamma_{q}$ becomes significant, at $\sqrt{s_{NN}}\lesssim
7\pm 1$ GeV. As depicted in bottom right panel of Fig. 2, at this energy,
$\gamma_{q}$ becomes minimum, while $\gamma_{s}$ reaches a maximum. The span
between both values increases when moving from partial– to full–chemical
nonequilibrium. So does the horn of the kaon–to–pion ratio. In light of this,
we conclude that the horn–structure of the kaon–to–pion ratio is likely due to
a) Lambert–$W_{0}$ exponentially generating distribution function [genenric
(non)extensive statistics] and b) full–chemical nonequilibrium quark
occupation factors.

The full–chemical nonequilibrium predicts a further enhancement in the
$\mathrm{K}^{+}/\pi^{+}$ ratio, especially at $3\lesssim\sqrt{s_{NN}}\lesssim
10$ GeV. At these energies, a detailed beam energy scan by the STAR and the
CBM experiments, for instance, is recommended to refine the conclusion whether
partial– or full–chemical nonequilibrium excellently reproduces the
nonmonotonic horn–structure of the $\mathrm{K}^{+}/\pi^{+}$ ratio, at AGS, SPS
and low RHIC energies.

##  IV Conclusion

The present paper introduces generic (non)extensive statistics. The physical
motivation isn’t just to propose a approximate solution to a long–standing
problem, the failure of thermal models in reproducing the nonmonotonic
horn–like structure of the $\mathrm{K}^{+}/\pi^{+}$ ratio. The assumption that
the particle production could be analyzed by extensive additive equilibrium
statistics is badly contradicts the nature of the high–energy particle
production. On one hand, this could be tolerated as long as no other
alternative existed. The nonextensive Tsallis approach is a very special case
and seems to fail, as well. The present paper introduces a first–principle
solution based on proper statistical approach, namely statistics whose
extensivity and nonextensivity match well with the statistical nature of the
particle production.

Various ingrediants, such as modifying single–particle distribution function
Tawfik:2010uh , assuming nonequilibrium light– and strange–quark occupation in
the phase–space volume, i.e., $\gamma_{q,s}\neq 1$ Tawfik:2005gk ;
Tawfik:2017ehz ; Tawfik:2017oyn , and integrating an excluded volume in the
pointlike ideal gas of hadron resonances Tawfik:2014eba ; Tawfik:2013eua ,
have been introduced to improve the capability of the thermal approaches,
which are either based on extensive additive BG statistics Gazdzicki:1998vd ;
Cleymans:2005zx ; Andronic:2005yp ; Tawfik:2005gk ; Tomasik:2006qs ;
Chatterjee:2009km ; Tawfik:2010uh ; Nayak:2010uq  or on Tsallis–type
nonextensive statistics Tawfik:2019oct . These weren’t fully succeeded in
characterizing the particle ratios including the nonmonotonic horn–like
structure of the $\mathrm{K}^{+}/\pi^{+}$ ratio, which likely signals onset of
nonextensive particle production and impacts of nonequilibrium phase
transition.

The present script assumes that the incapability of the thermal models in
reproducing the horn–like structure (long–dashed curve in Fig. 3) is due to
inappropriate statistics; either extensive additive BG or Tsallis–type
nonextensive statistics or their extensions. Inserting nonequilibrium
ingredients such as modified single–particle distribution function or
nonequilibrium occupation factors $\gamma_{q,s}$ or finite volume of the
constituents of the ideal gas to extensive additive BG statistics is
conceptionally fruitless. But these are eligible in nonextensivity statistics.
Relative to the entire $c$–$d$ space of (non)extensivity, the Tsallis–type
nonextensivity is limited to a line, where $c=0$ and $d=q$. On the other hand,
generic (non)extensive statistics covers the entire space of the two
equivalence classes $(c,d)$. With this statistical approach, the ensemble
itself determines its statistical extensivity and nonextensivity. Even both
extensive BG and Tsallis–type nonextensive statistics are included in. They
are sharply characterized by $(1,1)$ and $(0,q)$, respectively.

First, the partial– and full-chemical nonequilibrium, $\gamma_{q}$ or/and
$\gamma_{s}$ have been determined, at different energies. Second, in the same
manner, the energy–dependence of $(c,d)$ are estimated. In calculating the
$\mathrm{K}^{+}/\pi^{+}$ ratio, the occupation factors $\gamma_{q,s}$, the
equivalence classes $(c,d)$, and the freezeout parameters ($T$, $\mu_{B}$),
are no longer fitting variables. They are inputs, at various energies. We
found that the $\mathrm{K}^{+}/\pi^{+}$ ratio calculated in generic
(non)extensive statistics excellently reproduces the experimental results,
including the horn–like structure. The nonmonotonic energy–dependence of
$\gamma_{q,s}$ and that of $(c,d)$ together are essential. We conclude that
the excellent reproduction of the $\mathrm{K}^{+}/\pi^{+}$ ratio requires:

  1. 1.

Lambert–$W_{0}$ exponentially generated distribution rather than BG or Tsallis
and

  2. 2.

partial– or full–chemical nonequilibrium occupation factors.

With this regard, we assure you that other ratios such proton–to–pion are also
perfectly reproduced, Appendix C.

While both partial– and full–chemical nonequilibrium in generic (non)extensive
statistics seem to excellently reproduce the measured $\mathrm{K}^{+}/\pi^{+}$
ratio, at all energies, the full–chemical nonequilibrium is likely
significant, at $3\lesssim\sqrt{s_{NN}}\lesssim 10$ GeV. To refine the
conclusion whether partial– or full–chemical nonequilibrium is essential for
the nonmonotonic horn–structure of the $\mathrm{K}^{+}/\pi^{+}$ ratio, a
detailed beam energy scan by the STAR and CBM experiments, for instance, is
highly recommended.

##  Appendix A Probability distribution, entropy, exponential and logarithm
functions in Boltzmann, Tsallis and generic (non)extensive statistics

For a reliable comparison between Boltzmann, Tsallis, and generic
(non)extensive statistics, we first recall the corresponding probability
distributions

$\displaystyle\mathtt{Extensive\;BG:}$ $\displaystyle
p_{i}(x)=\frac{\exp(-x)}{{\cal Z}(x)}=\frac{1}{{\cal
Z}(x)}\sum_{n=0}^{\infty}\frac{x^{n}}{n!},$ (10)
$\displaystyle\mathtt{Nonextensive\;Tsallis:}$ $\displaystyle
p_{i}^{(q)}(x)=\frac{\exp^{(q)}(-x)}{{\cal Z}^{(q)}(x)}=\frac{1}{{\cal
Z}^{(q)}(x)}[1-x+\frac{q}{2}x^{2}-\frac{q^{2}}{3!}x^{3}+\cdots]$ (11)
$\displaystyle\hskip 34.1433pt=\frac{1}{{\cal
Z}^{(q)}(x)}\left[1+\sum_{n=1}^{\infty}\frac{x^{n}}{n!}Q_{n-1}(q)\right]=\frac{1}{{\cal
Z}^{(q)}(x)}\left[1+(1-q)x\right]^{1/(1-q)},\hskip 22.76219pt$
$\displaystyle\mathtt{(Non)extensive:}$ $\displaystyle
p_{i}^{(c,d,r)}(x)=\frac{\exp^{(c,d,r)}(-x)}{{\cal Z}^{(c,d,r)}(x)}$ (12)
$\displaystyle\hskip 48.36967pt=\frac{1}{{\cal
Z}^{(c,d,r)}(x)}\exp\left[\frac{-d}{1-c}\left\\{W_{k}\left[B\left(1-\frac{x}{r}\right)^{1/d}\right]\right\\}-W_{k}[B]\right],$

where ${\cal Z}(x)$ is the corresponding canonical partition function,
$Q_{n-1}(q):=\Pi_{i=0}^{n}[iq-(i-1)]$ Borges:1998xr , and the asymptotic
expansion of $0$–th branch Lambert–$W_{k}$ function is given as

$\displaystyle
W_{k=0}(x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{n^{n-1}}{n!}\;x^{n}.$

The probability distribution function can be obtained from the integral of the
probability density function determining the probability of an individual
event to take place in a given interval. Also, The probability distribution
function can be derived from the maximization of the corresponding entropy
under appropriate constraints

$\displaystyle\mathtt{Extensive\;BG:}$ $\displaystyle
s[p]=-\kappa\sum_{i=1}^{\Omega}p_{i}\;\ln\;p_{i},$ (13)
$\displaystyle\mathtt{Nonextensive\;Tsallis:}$ $\displaystyle
s^{(q)}[p]=\frac{\kappa}{1-q}\sum_{i=1}^{\Omega}\left(p_{i}^{(q)}-p_{i}\right),$
$\displaystyle\mathtt{(Non)extensive:}$ $\displaystyle
s^{(c,d,r)}[p]=\kappa\sum_{i=1}^{\Omega}\left[{\cal A}\,\Gamma(d+1,1-c\log
p_{i})-{\cal B}\,p_{i}\right],$ (14)

where $\Omega$ is the number of micro–states or processes. $\kappa$, ${\cal
A}$, and ${\cal B}$ are arbitrary parameters. The incomplete gamma–function is
given as Pereira2012AnalyticalSO

$\displaystyle\Gamma(a,x)=\int_{x}^{\infty}\,dt\,t^{a-1}\exp(-t)$
$\displaystyle=$
$\displaystyle\Gamma(a)-\frac{x^{a}}{a\,\exp(x)}\sum^{\infty}_{i=0}\frac{x^{i}(a)!}{(a-i)!}.$
(15)

The differences between Boltzmann, Tsallis and generic (non)extensive
statistical statistics can be characterized by the mathematical meaning of the
corresponding exponential and logarithm functions; namely for Boltzmann
$\exp(x)$ and $\ln(x)$, for Tsallis $\exp^{(q)}(x)$ and $\ln^{(q)}(x)$, and
for generic (non)exrensive functions $\exp^{(c,d,r)}(x)$ and
$\ln^{(c,d,r)}(x)$. For example, the Tsallis exponential and logarithm
functions, respectively read Umarov2008OnAQ ,

$\displaystyle\exp^{(q)}(-x)$ $\displaystyle=$ $\displaystyle
1+\sum_{n=1}^{\infty}(-1)^{n}\frac{q^{n-1}}{n!}x^{n}=1-x+\frac{q}{2!}x^{2}-\frac{q^{2}}{3!}x^{3}+\cdots,$
$\displaystyle\ln^{(q)}(x)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{q^{n-1}}{n}(x-1)^{n}=(x-1)-\frac{q}{2}(x-1)^{2}+\frac{q^{2}}{3}(x-1)^{3}-\cdots.$
(16)

For generic (non)exrensive exponential and logarithm functions,
generaltization of BG $\exp(x)$ and $\ln(x)$ gives Tawfik:2016jol

$\displaystyle\exp^{(c,d,r)}(-x)$ $\displaystyle=$
$\displaystyle\exp\left\\{\frac{-d}{1-c}\left[\mathtt{W}_{k}\left(B\left(1-x/r\right)^{1/d}\right)-\mathtt{W}_{k}(B)\right]\right\\},\hskip
8.53581pt$ (17) $\displaystyle\ln^{(c,d,r)}(x)$ $\displaystyle=$
$\displaystyle
r\left\\{1-x^{c-1}\left[1-\frac{1-r(1-c)}{d\;r}\log(x)\right]^{d}\right\\}.$
(18)

It is now in order to comment on the interpretations of the Tsallis
nonextensitivity parameter $q$. First, the assumption that $q>1$ manifests
fluctuations in $1/\lambda$ of the so–called Levy exponential distribution,
$\exp(-x/\lambda)$, is apparently superficial Wilk:1999dr . The nature of such
fluctuations wasn’t determined, yet. On the other hand, such fluctuations
should be also accompanied by a remarkable change in the phase space, itself
Tawfik:2010uh . Likewise, we assess the connection between the departure of
$q$ from its extensive value, the unity, from above and the large fluctuations
of temperature. Second, the claim on the existence of fundamental differences
between BG– and Tsallis–statistics888There is no fundamental difference
between BG– and Tsallis–statistics, unless the ealier is extensive and the
latter is nonextensive statistical framework! leads to an ad hoc prerequisite
that the thermodynamic temperature, the one deduced from BG statistrics,
differs from the one determined in Tsallis statistics,
$T_{Tsallis}=T_{BG}+(q-1)K$, where the parameter $K$ should be modeled as a
function of the energy transfer between the source and the surrounding
Deppman:2012qt . This phenomenological interpretation is closely related to
blind improper implementation of Tsallis–algebra999Replacing $\exp(x)$ with
$\exp_{q}(x)$, Eq. (A) and $\ln(x)$ with $\ln_{q}(x)$, Eq. (16). It is worthy
highlighting that the reproduction of the transverse momentum distributions
was misinterpreted and accordingly the resulting temperature Bialas:2015pla .

The present script suggests that Boltzmann and Tsallis statistics are
subdomains of generic (non)extensive statistics. The Tsallis nonextensive
parameter $q$ is also included in BG; $q=1$, as well as in generic
(non)extensive statistics; $c=0$ and $d=q$. The equivalence classes $(c,d)$
are intrinsic properties of the entire statistical space, where $(c,d)=(1,1)$
characterize Boltzmann statistics and $(c,d)=(0,q)$ represents the Tsallis
type. The thermodynamic quantities either intensive or extensive, such as
temperature, entropy, pressure, and energy density, should remain identical in
the entire statistical space, i.e., they shouldn’t be changing when moving
from BG to Tsallis or to any other type of nonextensive statistics. The
interpretations of $q$ and of $(c,d)$ aren’t just dummy fluctuations but
rather nonextensive statistical characterizations. Their nature is related to
the properties of the relevant statistical ensemble.

##  Appendix B Inclusion of very high–mass resonances and missing states in
the hadron resonance gas model

Fig. 4: Energy dependence of the relative production ratio
$\mathrm{K}^{+}/\pi^{+}$ measured at various energies. With the dotted curve,
we show the $\mathrm{K}^{+}/\pi^{+}$ ratio estimated in the hadron resonance
gas model with extensive additive equilibrium BG statistics and dof determined
by mass–truncated PDG compilation of hadrons and resonances. The solid curve
shows the same but dof are the entire PDG compilation of hadron and resonances
and up to $56$ missing states.

The authors of ref. Andronic:2008gu  introduced two ingredients to the
extensive additive equilibrium hadron resonance gas model. The first one is
very high–mass resonances ($>2~{}$GeV). The second one is the $\sigma$ meson
or $f_{0}(600)$. The various decay channels are also taken into consideration.

The $\sigma$ meson is characterized by energy–depending mass and varying decay
constants. Its dominant decay channel produces pion pair. This might be the
motivation of ref. Andronic:2008gu . Accordingly, it has been estimated that
$\sigma$ meson comes up with about $3.5\%$ enhancement in the pion yields. The
inclusion of very high–mass resonances also increase the pion yields, this
time by about $13\%$. Enhancing pion yields against Kaon yields seems to
reproduce the nonmonotonic $\mathrm{K}^{+}/\pi^{+}$ ratio, on one hand. On the
other, the standard of evidence prerequisites that the Kaon yields should be
enhanced, similarly! Alternatively, the ensemble of the hadron resonances is
left biased unbalanced.

Fig. 4 presents unbaised balanced reproduction of $\mathrm{K}^{+}/\pi^{+}$
ratio (solid curve), where all hadrons and resonances reported in recent PDG,
including $\sigma$ meson and its heavier resonances Zyla:2020zbs , as well as
$56$ messing states are included in. Most if not all decay channels are also
taken into consideration, so that the final number of any particle is summed
over that particle and all other decay channels producing it weighted with the
corresponding decay width. The masses of the resonances can be as heavier as
$12~{}$GeV. We conclude that the inclusion of the very heavy–mass resonances
would slightly enhance the respective particle yields, on one hand. On the
other hand, such an enhancement is faded away in the relative production
ratio. The $\mathrm{K}^{+}/\pi^{+}$ ratio isn’t well reproduced either with or
without the inclusion of missing states.

Fig. 5: Temperature dependence of number density, energy density and pressure
calculated in the extensive additive equilibrium hadron resonance gas model,
at $7.7$, $19.6$, $39$ and $200~{}$GeV. The curves show the results with
truncated hadron mass spectrum, $<2~{}$GeV. With the symbols, we show the
results, where $56$ missing states are included in and the hadron mass can be
as heavy as $12~{}$GeV.

The contributions of the missing states to the various thermodynamic
quantities ManLo:2016pgd  are depicted in Fig. 5. At $7.7$, $19.6$, $39$ and
$200~{}$GeV, the number density, energy density, and pressure calculated in
the extensive additive equilibrium hadron resonance gas model, are compared
with (symbols) and without (curves) missing stated and when including
(symbols) and excluding (curves) very–heavy mass resonances. We conclude that
the inclusion of the missing states and very–heavy mass resonances minimally
contributes to the particle yields. These contributions are apparently
balanced in the relative particle production ratios, Fig. 4. With this regard,
we recall that the authors of ref. ManLo:2016pgd  also have drawn the same
conclusion that ”susceptibilities of conserved charges, […] can be attributed
to the missing resonances in the strange baryon sector.” In other words, the
missing states become first dominant in the higher–order moments of conserved
charges, such as susceptibilities and cumulants. But they slightly contribute
to lower–order moments, such as the number density. Even this slight
contribution is faded away in the particle ratios.

##  Appendix C Contour plots of the quality of fits for $\gamma_{q,s}$

Fig. 6: Top left: contour plot of the quality of fit for $\gamma_{q}$, where
$\gamma_{s}=1$, over different collision energies. Top left: the same as in
top left but for $\gamma_{s}$, where $\gamma_{q}=1$. Bottom panel shows
contour plot for nonequilibrium $\gamma_{q}$ and $\gamma_{s}$, at different
collision energies.

A suppression in $\gamma_{s}$ was assumed as a signature of onset of the color
deconfinement Castorina:2017dvt ; Castorina:2018zxi . The nonequilibrium quark
occupation factors, $\gamma_{q,s}\neq 1$, are conjectured to dictate the
numbers of light– and strange–quarks, respectively, which can be accomodated
in the phase–space volume and accordingly the production rates of light and
strange particle yields Tawfik:2005gk ; Tawfik:2017ehz ; Tawfik:2017oyn . In
all these studies, extensive additive equilibrium BG statistics was utilized.

When $\gamma_{q}$ is determined from the statistical fit of one particle
ratio, the other particle ratios aren’t well reproduced. Although various
other particle ratios can simutaneously be reproduced for nonequilibrium
$\gamma_{q,s}$, the resulting values of $\gamma_{q,s}$ are scattered, Table 1
of ref. Tawfik:2005gk . Values of $\gamma_{q,s}$ much greater than unity are
tough to incorporate with a nonequilibrium approach.

Figure 6 shows the contour plots of the quality of fits for $\gamma_{q}$ (top
left panel), $\gamma_{s}$ (top right panel), and $\gamma_{q,s}$ (bottom
panel). The resulting of $\gamma_{q,s}$ are parameterized in Eq. (9). When
comparing with Table 1 of ref. Tawfik:2005gk , we conclude that the present
estimations can be well incorporated with anonequilibrium approach.

With the same sets of equivalence classes given by Eqs. (7)-(8) and
nonequilibrium quark–occupation factors $\gamma_{q,s}$ given by Eq. (9), not
only $\mathrm{K}^{+}/\pi^{+}$ ratio is well reproduced, but other particle
ratios, as well, for example, $\mathrm{P}/\pi^{+}$ and $\Lambda/\pi^{-}$, Fig.
7. We like to highlight that at $>4~{}$TeV, the ensemble of the experimental
results available so–far isn’t large enough to deduce the equivalent classes.
Thus, we have assumed that their results, at lower LHC energies, remain
unchanged. There is a little overestimation, which can be improved with the
release of LHC results in the near future.

Fig. 7: Energy dependence of $\mathrm{P}/\pi^{+}$ (left panel) and
$\Lambda/\pi^{-}$ (right panel), where the same sets of the equivalence
classes are given by Eqs. (7)-(8) and $\gamma_{q,s}$ by Eq. (9), as the ones
for $\mathrm{K}^{+}/\pi^{+}$ are utilized.

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