# Proton correlations and apparent intermittency in the UrQMD model with
hadronic potentials

Pengcheng Li  School of Nuclear Science and Technology, Lanzhou University,
Lanzhou 730000, China  School of Science, Huzhou University, Huzhou 313000,
China  Yongjia Wang  School of Science, Huzhou University, Huzhou 313000,
China  Jan Steinheimer  Frankfurt Institute for Advanced Studies, Ruth-
Moufang-Str. 1, D-60438 Frankfurt am Main, Germany  Qingfeng
Li111Corresponding author<EMAIL_ADDRESS>School of Science, Huzhou
University, Huzhou 313000, China  Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  Hongfei Zhang  School of Nuclear Science
and Technology, Lanzhou University, Lanzhou 730000, China  Joint Department
for Nuclear Physics, Lanzhou University and Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China

###### Abstract

It is shown that the inclusion of hadronic interactions, and in particular
nuclear potentials, in simulations of heavy ion collisions at the SPS energy
range can lead to obvious correlations of protons. These correlations
contribute significantly to an intermittency analysis as performed at the NA61
experiment. The beam energy and system size dependence is studied by comparing
the resulting intermittency index for heavy ion collisions of different nuclei
at beam energies of $40A$, $80A$ and $150A$ GeV. The resulting intermittency
index from our simulations is similar to the reported values of the NA61
collaboration, if nuclear interactions are included. The observed apparent
intermittency signal is the result of the correlated proton pairs with small
relative transverse momentum $\Delta p_{t}$, which would be enhanced by
hadronic potentials, and this correlation between the protons is slightly
influenced by the coalescence parameters and the relative invariant four-
momentum $q_{inv}$ cut.

##  I INTRODUCTION

The exploration of the properties of the Quark Gluon Plasma (QGP) QGP1 ; QGP2
; QGP3  and the phase structure of quantum chromodynamics (QCD) QCD1973  are
the main objectives in relativistic heavy-ion collisions (HICs). Lattice QCD
calculations at vanishing baryon chemical potential ($\mu_{B}$) determined a
chiral crossover transition at $T\simeq 155$ MeV PRD71034504 ; Lattice ;
PRD90094503 . Various theoretical investigations suggest that at finite baryon
chemical potential and temperature, there could be a critical end point (CEP)
in the QCD phase diagram 1stptcep1 ; 1stptcep2 ; 1stptcep3 ; 1stptcep4 . In
order to explore the QCD phase diagram at finite net-baryon density, one tries
to vary the temperature and baryon chemical potential of the nuclear matter
created in HICs by changing the colliding energy and size of the colliding
nuclei as well as the centrality, leading to different freeze-out conditions
in $\mu_{B}$ and $T$ whitepaper ; SPSC-P-330-ADD-4 ; SCAN1 .

A non-monotonic behavior of fluctuation of observables is expected as a signal
for the CEP PRL835435 ; PRL852072 ; PRL95182301 ; PLB633275 ; PRL102032301 ;
sspma011012 . To investigate the phase transition and search for the CEP in
HICs, a lot of analyses on fluctuations were done. For example, the analysis
of event-by-event fluctuations of integrated quantities like the net-baryon
number, electric charge, and strangeness, in particular their higher order
cumulants NST281122017 ; prd96074510 ; prd101074502 ; prc101034915 ; pp583 ;
prl126092301 . In addition, the analysis of local power-law fluctuations of
the net-baryon density NA49SFM ; Local  has drawn much attention. It can be
detected through the measurement of the scaled factorial moments (SFMs) in
transverse momentum space within the framework of a proton intermittency
analysis SFM1 ; SFM2 ; Antoniou2005 ; Antoniou2006 . During the last decade,
NA49 and NA61 performed a systematic search for critical fluctuations
utilizing an intermittency analysis in central A+A collisions NA49prc81 ;
Davis2002 . An indication of power-law fluctuations, in the transverse
momentum phase space of protons at mid-rapidity in the central (0-12.5% )
Si+Si collisions at 158$A$ GeV/$c$, has recently been presented NA49SFM . In
addition, a non-trivial intermittency effect was shown in a preliminary
analysis of 40Ar+45Sc collisions at 150$A$ GeV/$c$ NA61 ; Davisapb .

A more indirect interpretation of recent STAR data also suggested a non-
trivial intermittency index which cannot be described by a pure cascade
simulation of the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model,
and therefore warrants further investigation plb801135186 ; appburqmd . In
this work, effects of hadronic potentials on the intermittency behaviour in
HICs will be evaluated by a systematic scan in collision energy and system
size within a modified version of the UrQMD model with and without hadronic
potentials.

This paper is organized as follows: in section II, we briefly present the
scaled factorial moments and the transport model (UrQMD model) used in this
work. In section III, the result for the intermittency effect is discussed.
Finally in section IV, the results and conclusions will be summarized.

##  II Methodology

In order to investigate the non-statistical fluctuations and quantitatively
understand the intermittent behavior in interactions, a technique based on the
scaled factorial moments (SFMs) measurement was first introduced by Bialas and
Peschanski SFM1 ; SFM2 . Here, momentum space is partitioned into equal-size
bins and the SFMs are defined as:

$F_{q}(M)=\frac{\left\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}(n_{i}-1)...(n_{i}-q+1)\right\rangle}{\left\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}\right\rangle^{q}},$
(1)

where $M^{D}$ is the number of equally sized bins in which the D-dimensional
space is partitioned, $n_{i}$ is the number of particles in the $i$-th bin,
the angular brackets denote an average over bins and events and $q$ is the
rank of the moment.

For a critical system, the fluctuations of the order parameter are self-
similar Vicsek , and the SFMs scale with the number of bins. In other words,
if the dynamical fluctuations are self-similar in nature in the limit of small
bin size, the SFMs are expected to increase with bin size following a power
law. This effect is called intermittency, and can be quantified by this
dependence:

$F_{q}(M)\sim(M^{D})^{\phi_{q}},~{}~{}~{}~{}M\rightarrow\infty.$ (2)

The exponent $\phi_{q}$ is the so called intermittency index, which not only
characterizes the strength of the intermittency, but also correlates with to
the anomalous fractal dimension $d_{q}$ of the system Antoniou2006 . Many
investigations suggest that one can explore the possible critical region of
the QCD matter by using an intermittency analysis NA49SFM ; Antoniou2006 ;
NA49prc81 ; plb801135186 ; CriticalReg . Universality class arguments
associate the power-law behaviour of the SFMs with a specific exponent
$\phi_{2}=\frac{2}{3}$ in case of the chiral condensate as order parameter
Antoniou2005  and $\phi_{2}=\frac{5}{6}$ for the net-baryon density as order
parameter Antoniou2006 ; NA49SFM .

Since such an analysis of higher order SFMs requires a large amount of data
one usually only considers the second scaled factorial moment (SSFMs) in
transverse momentum space, which is obtained by setting $q=2$ and $D=2$ in Eq.
1.

The above considerations are true for idealized systems which can be studied
in the thermal limit. In fact, heavy-ion collisions will never be such an
ideal system for a number of reasons NA49SFM . To reveal the predicted power-
law exponents, it is necessary to consider a large number of background
effects. Such effects include the finite lifetime of the system, global and
local conservation laws, finite experimental acceptance, freeze out dynamics,
resonance decays as well as other possible sources for correlations. In other
words it is important to understand the non-critical background. This can only
be done within well understood transport simulations.

In this study we will employ the UrQMD transport model ppnp41255 ; JPG251859
with (mean-field mode, UrQMD/M) and without the hadronic potentials (cascade
mode, UrQMD/C) to generate a large number of events for nuclear collisions at
different bombarding energies. From these events the second order SFMs will be
calculated. Similar to what was done in Ref.NA49SFM , a correlation-free
background can be generated from mixed events, where particles are taken from
different events randomly. Any critical behavior is expected to be encoded in
the approximate correlator, which can be estimated by the difference between
the SSFMs of the real events $F_{2}^{(d)}(M)$ and the associated mixed events
$F_{2}^{(m)}(M)$:

$\Delta F_{2}(M)=F_{2}^{(d)}(M)-F_{2}^{(m)}(M).$ (3)

The power law scaling of the correlator $\Delta F_{2}(M)$ can be extracted by
a fit to the $M^{2}$ dependence of the critical component ($\Delta
F_{2}(M)\sim(M^{2})^{\phi_{2}}$) for $M\gg 1$ NA49SFM , which will then be
investigated and compared to the expected critical exponents.

Using the mean-field mode of the UrQMD model plb659525 ; plb623395 ;
JPG36015111 ; mpla27 ; scpma59632001 ; mplalpc , density-dependent potentials
for both formed hadron and pre-formed hadron from string fragmentation are
treated in a similar way. The density dependent potential reads

$U=\alpha\left(\frac{\rho_{h}}{\rho_{0}}\right)+\beta\left(\frac{\rho_{h}}{\rho_{0}}\right)^{\gamma},$
(4)

where $\alpha$, $\beta$ and $\gamma$ are parameters. In this work, a soft
equation of state is adopted with $\alpha=$-110.49 MeV, $\beta$=182.014 MeV,
$\gamma$= 1.17. Here, $\rho_{0}=0.16$ fm-3 is the nuclear matter saturation
density, and $\rho_{h}=\sum_{i\neq j}c_{i}c_{j}\rho_{ij}$ is the hadronic
density, which has contributions from formed and pre-formed baryons with
$c_{i,j}=1$, and from pre-formed mesons with a reduction factor $c_{i,j}=2/3$
based on the light-quark counting rules.

The momentum-dependent term of the hadronic potentials reads as:

$U_{md}=\sum_{k=1,2}\frac{t_{md}^{k}}{\rho_{0}}\int
d\textbf{p}_{j}\frac{f(\textbf{r}_{i},\textbf{p}_{j})}{1+[(\textbf{p}_{i}-\textbf{p}_{j})/a_{md}^{k}]^{2}},$
(5)

where $t_{md}$ and $a_{md}$ are parameters. A detailed description on how the
model parameters be fixed can be found in Ref. prc72064908 . For simplicity,
the momentum-dependent term is only considered for formed baryons, and a two-
body Coulomb potential is included for formed charged particles only.

Furthermore, relativistic effects on the relative distance $\mathbf{r}_{ij}$
and relative momentum $\mathbf{p}_{ij}$ between two particles $i$ and $j$ are
taken into account:

$\displaystyle\tilde{\mathbf{r}}_{ij}^{2}$ $\displaystyle=$
$\displaystyle\mathbf{r}_{ij}^{2}+\gamma_{ij}^{2}(\mathbf{r}_{ij}\cdot\mathbf{\beta}_{ij})^{2},$
(6) $\displaystyle\tilde{\mathbf{p}}_{ij}^{2}$ $\displaystyle=$
$\displaystyle\mathbf{p}_{ij}^{2}-(E_{i}-E_{j})^{2}+\gamma_{ij}^{2}\left(\frac{m_{i}^{2}-m_{j}^{2}}{E_{i}+E_{j}}\right)^{2},$
(7)

where the velocity factor
$\beta_{ij}=(\mathbf{p}_{i}+\mathbf{p}_{j})/(E_{i}+E_{j}$) and the
$\gamma_{ij}=1/\sqrt{1-\mathbf{\beta}_{ij}^{2}}$. Additionally, $m_{j}/E_{j}$,
a covariance-related reduction factor for potentials in the Hamiltonian used
as in the simplified version of the relativistic quantum molecular dynamics
(RQMD/S) model prc101034915 ; prc72064908 ; PTP96263 ; EPJA5418 ; prc97064913
.

##  III results

In the following we will show the results of the proton intermittency analysis
for 40Ar+45Sc, 131Xe+139La collisions with centralities 0-5%, 5-10% and 10-15%
and central 197Au+197Au collisions (0-10%) (the centrality is defined by
impact parameter distribution $c=(b/b_{max})^{2}$ of the UrQMD model). The
analysis is done for protons (and neutrons) with transverse momenta of
$|p_{x,y}|\leqslant 1.5$ GeV/$c$, in the mid-rapidity region, i.e. with
rapidity $|y|\leq 0.75$. To accumulate enough statistics for the analysis,
more than 600K events are simulated for each scenario.

Figure 1: (Color online) Left plot: The second scaled factorial moment
($F_{2}(M)$) of the proton density in transverse momentum space
$|p_{x,y}|\leqslant 1.5$ GeV/$c$ at mid-rapidity ($|y|\leqslant 0.75$) for the
5-10% central 40Ar+45Sc collisions at 40$A$ GeV obtained from UrQMD model with
(UrQMD/M, solid symbols) and without (UrQMD/C, open symbols) hadronic
potentials. The squares and circles represent the $F_{2}(M)$ of the real and
mixed events, respectively. Right plot: The estimated correlator $\Delta
F_{2}M$ of protons for the corresponding system, shaded bands are 95%
confidence intervals around the fitted lines.

Fig.1 shows the resulting second scaled factorial moment $F_{2}(M)$ and the
estimator of the correlator $\Delta F_{2}(M)$ for 5-10% central 40Ar+45Sc
collisions at 40$A$ GeV calculated within UrQMD model with and without
hadronic potentials. In panel (a), the intermittency effect, i.e. the
difference in $F_{2}(M)$ between real events and mixed events, shows up for
$M^{2}\gtrsim 100$. Therefore, to determine the intermittency index, a fit of
$\Delta F_{2}(M)$ using Eq. 2 was used for $M^{2}>100$. The resulting curve is
shown in panel (b) of Fig.1. For the UrQMD cascade mode, the results for the
moments of the real and mixed events are almost identical, and the values of
$\Delta F_{2}(M^{2})$ fluctuate around zero, implying no effect. In the mean-
field mode, the $F_{2}(M)$ of real events rises significantly above those of
mixed events, the extracted intermittency index is $\phi_{2}=0.32\pm 0.03$.
This value of $\phi_{2}$ is similar to the one found in the NA61 preliminary
analysis NA61 .

Figure 2: (Color online) The correlations function of proton/neutron pairs
(ratio of pairs from real and mixed events) at mid-rapidity for the 5-10%
central 40Ar+45Sc collisions at 40$A$ GeV, left panel is for $\Delta p_{t}$
and right panel is for $\Delta R_{t}$. The errors are the standard deviation.

To understand what causes the non-vanishing intermittency index in the model
simulation, we further investigate the baryon pair correlations in momentum
and coordinate space. Fig.2 shows the distribution of $\Delta p_{t}$ and
$\Delta R_{t}$, over the mixed background, for proton and neutron pairs, in
the same simulations. Here, the relative transverse momentum $\Delta p_{t}$
and the relative radial distance $\Delta R_{t}$ are defined as:

$\displaystyle\Delta p_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{(p_{x1}-p_{x2})^{2}+(p_{y1}-p_{y2})^{2}},$ (8)
$\displaystyle\Delta R_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{(r_{x1}-r_{x2})^{2}+(r_{y1}-r_{y2})^{2}}.$ (9)

In Fig.2, the baseline scenario of the cascade simulation (magenta line with
cross symbols) does not show any structure in the relative transverse momentum
and coordinate difference distributions as compared to the mixed background.
Since in the cascade mode, the hadrons interact with each only through binary
scattering according to a geometrical interpretation of elastic and inelastic
cross sections. The ratio real/mixed is only slightly below unit due to
conservation law effects. On the other hand, the $\Delta p_{t}$ and $\Delta
R_{t}$ distributions in the simulation with potentials show a clear increase
for small relative transverse momentum $\Delta p_{t}\lesssim 60$ MeV/$c$ and
$\Delta R_{t}\lesssim 1.5$ fm. To further interpret the observed increase in
the $\Delta p_{t}$ and $\Delta R_{t}$ distributions, we found that with the
consideration of the hadronic potentials, the real collision case has 0.029
proton pairs per event more than that of mixed collisions case at $\Delta
p_{t}<$ 64 MeV/$c$. This is true for both, proton (blue line with squares) and
neutron pairs (olive line with up triangles), indicating that the Coulomb
interaction is less important for this distribution, and the effect of the
Coulomb interaction of formed charged particles on this distribution is weaker
than that of hadronic potentials. In addition, applying a cut (red line with
down triangles), similar to that of the NA61 analysis NA49SFM  in the relative
invariant four-momentum $q_{inv}\geqslant$ 25 MeV/$c$ only has a small effect
on the distributions ($\Delta p_{t}\lesssim$20 MeV/$c$).

Figure 3: (Color online) Similar as Fig.2, but for the proton-proton $\Delta
p_{t}$, $\Delta R_{t}$ distributions with (red line with circles and olive
line with triangles) and without (blue line with squares) the traditional
coalescence afterburner (Minimum Spanning Tree, MST). The errors are the
standard deviation.

Furthermore, to estimate the effect of the coalescence afterburner on these
distributions we employ a minimum spanning tree (MST) algorithm for phase
space coalescence. Fig.3 shows the $\Delta p_{t}$ and $\Delta R_{t}$
distributions of proton pairs from UrQMD/M mode with (UrQMD/M+MST) and without
(UrQMD/M, blue lines) the coalescence afterburner. The phase-space coalescence
afterburner (MST) used the parameter set of ($R_{0}$, $P_{0}$)=(2.8 fm, 0.25
GeV/$c$ and 3.8 fm, 0.3 GeV/$c$), indicated by red lines and olive lines
respectively, for the relative momenta and distances scpma2016 ) in the final
freeze-out stage as conditions for cluster formation. In the MST algorithm,
nonphysical clusters (such as di-neutrons or di-protons) are eliminated by
breaking them into free nucleons. The available data of deuteron and Helium-3
productions can be described fairly well by the UrQMD model with the above
empirical coalescence parametersscpma2016 ; Sombun:2018yqh . By adjusting the
parameter of MST, one can found that the coalescence afterburner has no effect
on the proton-proton correlations. This means that the observed correlations
in the low $\Delta p_{t}$ and $\Delta R_{t}$ can be readily understood as
result of the attractive potentials between the nucleons. This attractive
interaction eventually also is the reason for nuclear fragment formation, e.g.
for the formation of bound pronto-neutron systems, the deuteron.

###  III.1 Centrality dependence

Figure 4: (Color online) The second-order intermittency index as a function of
centrality in 40Ar+45Sc collisions at 150$A$ GeV, at 0-5% (left plot), 5-10%
(middle plot) and 10-15% (right plot) centrality range. The NA61/SHINE
preliminary data are taken from Ref. NA61 .

Next we study the centrality dependence of intermittent fluctuations in the
transverse plane. Fig.4 depicts the centrality dependence of the second-order
intermittency index in 40Ar+45Sc collisions. Panels (a), (b) and (c) are for
0-5%, 5-10% and 10-15% collisions, respectively. The calculated results from
UrQMD/C (open squares) and UrQMD/M (solid squares) are shown together with the
experimental data from NA61/SHINE (stars) with different proton purities NA61
for comparison. In experiment, the profile of SSFMs is affected by the proton
purity selection, thus a full scan in proton purity, at thresholds of 80%,
85%, and 90% are shown. The black dash line presents the expected $\phi_{2}$
for a second-order phase transition calculated based on the effective action
belonging to 3D Ising model Antoniou2006 . The intermittency index calculated
by the UrQMD/C model is essentially zero. Further, with the inclusion of
hadronic potentials, it is found that the intermittency index is increased and
comparable to the NA61/SHINE data in different centrality bins. In addition,
it can be seen that $\phi_{2}$ slightly increases with increasing impact
parameter, which is also consistent with the findings in the experiment
Davisapb . This result is due to the decreasing size of the system which goes
along with a shortened hadronic freeze out phase npa20201003 ; prc73044905 ;
prc90054907 .

###  III.2 Energy and system dependence of $\phi_{2}$

Figure 5: (Color online) Excitation functions of the second-order
intermittency index in the 0-5% (panels a and d), 5-10% (panels b and e),
10-15% (panels c and f) central 40Ar+45Sc (top plots) and 131Xe+139La (bottom
plots) collisions calculated within UrQMD model.

Finally, an energy scan of the intermittency in collisions of 40Ar+45Sc,
131Xe+139La and 197Au+197Au is presented. Fig.5 summaries the calculated
energy excitation function of the second-order intermittency index. Panels
(a), (b) and (c), respectively, are for 40Ar+45Sc at 0-5%, 5-10% and 10-15%.
And panels (d), (e) and (f) are for 131Xe+139La collisions. We observe that
the $\phi_{2}$ calculated with the UrQMD/C data (open squares) is essentially
zero no matter the centrality or energy, and no intermittency effect is
observed. In the mean-field mode, the second-order intermittency index (solid
squares) is slightly increasing with the energy and/or centrality.

Fig.6 shows the excitation function of $\phi_{2}$ from the UrQMD model for
0-10% central Au+Au collisions at several different energies. Here, we present
results for Au+Au systems as they are studied in the beam energy scan of the
STAR experiment. Again, we observe an increase of the intermittency index as
function of beam energy. The UrQMD results in Fig.6 are also compared to
conjectured values of $\phi_{2}$ which are based on a theoretical
interpretation of STAR data plb801135186 ; arxiv2002 . The data (red stars)
from Ref. plb801135186  are obtained by a mapping of the
$N_{\text{triton}}\cdot N_{\text{proton}}/N_{\text{deuteron}}^{2}$ ratio on
the neutron number fluctuations in most central (0-10%) Au + Au collisions
arxiv2002 . These indirectly reconstructed fluctuations are then again mapped
onto the obtained relation between the relative density fluctuation of baryons
$\Delta$n and $\phi_{2}$. The shadowed band corresponds to the systematic
errors from the experimental data. Even though the magnitude and energy
dependence of the indirectly calculated data is similar to the model results
plb801135186  one should be cautious with this comparison due to the indirect
nature of how $\phi_{2}$ is extracted. A future direct measurement would allow
a much more direct comparison with data.

Figure 6: (Color online) Energy dependence of the second-order intermittency
index in 197Au+197Au collisions. The data which represented by red stars are
taken from Ref. plb801135186  for Au+Au collisions.

##  IV summary

Investigations on the second order scaled factorial moments of proton number
fluctuations in transverse momentum space of HICs, within the UrQMD transport
model, have been presented. It was found that the inclusion of hadronic mean-
field potentials introduce a non-zero intermittency index $\phi_{2}$ which is
consistent with the reported data of the NA49/NA61 experiment.

The energy dependence ($40A$, $80A$, $150A$ GeV), system size
(${}^{40}Ar$+45Sc, 131Xe+139La, 197Au+197Au) and centrality (0-5%, 5-10%,
10-15%) dependence is studied in the UrQMD model with and without hadronic
potentials. A clear dependence of beam energy and centrality is observed. In
the present study the increase of the SFMs is due to an enhancement of proton
pairs (approximately 0.029 pairs per event for the case of 5-10% central Ar+Sc
collisions at 40$A$ GeV) with small relative momenta $\Delta p_{t}\lesssim 60$
MeV/$c$ due to attractive nuclear forces. With a further consideration of the
traditional coalescence afterburner, the observed correlations of protons were
found to not be influenced by the coalescence parameters. More theoretical and
experimental studies on the effect of both, hadronic interactions, the
stiffness of the equation of state, and the final-state interactions are
required to shed light on the intermittency in heavy ion collisions,
especially for lower collision energies.

###### Acknowledgements.

The authors acknowledge support by the computing server C3S2 in Huzhou
University. The work is supported in part by the National Natural Science
Foundation of China (Nos. 11875125, U2032145, and 12047568), the National Key
Research and Development Program of China under Grant No. 2020YFE0202002, and
the “Ten Thousand Talent Program” of Zhejiang province. JS thanks the Samson
AG and the BMBF through the ErUM-data project for funding.

## References

  * (1) J. C. Collins, M. J. Perry, Phys. Rev. Lett. 34 (1975) 1353.
  * (2) N. Cabibbo, G. Parisi, Phys. Lett. B 59 (1975) 67.
  * (3) E. V. Shuryak, Phys. Lett. B 78 (1978) 150.
  * (4) H. Fritzsch, M. Gell-Mann, H. Leutwyler, Phys. Lett. B 47 (1973) 365.
  * (5) C. Bernard, T. Burch, C. DeTar, et al., Phys. Rev. D 71 (2005) 034504.
  * (6) Y. Aoki, G. Endrodi, Z. Fodor, et al., Nature 443 (2006) 675.
  * (7) A. Bazavov, et al., (HotQCD Collaboration), Phys. Rev. D 90 (2014) 094503.
  * (8) P. Braun-Munzinger, V. Koch, T. Schäfer, J. Stachel, Phys. Rep. 621 (2016) 76.
  * (9) M. A. Stephanov, Int. J. Mod. Phys. A 20 (2005) 4387.
  * (10) K. Fukushima, C. Sasaki, Prog. Part. Nucl. Phys. 72 (2013) 99.
  * (11) G. Baym, T. Hatsuda, T. Kojo, et al., Rep. Progr. Phys. 81 (2018) 056902.
  * (12) STAR Collaboration, STAR Note 598, STAR white paper, “Studying the Phase Diagram of QCD Matter at RHIC”, 2014,
https://drupal.star.bnl.gov/STAR/starnotes/public/sn0598.

  * (13) NA61/SHINE Collaboration, CERN-SPSC-2009-001; SPSC-P-330-ADD-4, “Proposal for secondary ion beams and update of data taking schedule for 2009-2013”.
  * (14) A. Andronic, Int. J. Mod. Phys. A 29 (2014) 1430047.
  * (15) S. Jeon, V. Koch, Phys. Rev. Lett. 83 (1999) 5435.
  * (16) M. Asakawa, U. Heinz, B. Müller, Phys. Rev. Lett. 85 (2000) 2072.
  * (17) V. Koch, A. Majumder, J. Randrup, Phys. Rev. Lett. 95 (2005) 182301.
  * (18) S. Ejiri, F. Karsch, K. Redlich, Phys. Lett. B 633 (2006) 275.
  * (19) M. A. Stephanov, Phys. Rev. Lett. 102 (2009) 032301.
  * (20) X. H. Jin, J. H. Chen, Z. W. Lin, $et~{}al.$, Sci. Sin. Phys. Mech. Astron. 62 (2019) 011012.
  * (21) X. F. Luo, Nu Xu, Nucl. Sci. Tech. 28 (2017) 112.
  * (22) A. Bazavov, et al., (HotQCD Collaboration), Phys. Rev. D 96 (2017) 074510.
  * (23) A. Bazavov, et al., (HotQCD Collaboration), Phys. Rev. D 101 (2020) 074502.
  * (24) Y. Ye, Y. Wang, Q. Li, et al., Phys. Rev. C 101 (2020) 034915.
  * (25) A. Bzdak, S. Esumi, V. Koch, et al., Phys. Rep. 853 (2020) 1.
  * (26) J. Adam, et al., (STAR Collaboration), Phys. Rev. Lett. 126 (2021) 092301.
  * (27) T. Anticic, B. Baatar, J. Bartke, et al., Eur. Phys. J. C 75 (2015) 587.
  * (28) N. G. Antoniou, Y. F. Contoyiannis, F. K. Diakonos, et al., Nucl. Phys. A 693 (2001) 799.
  * (29) A. Bialas and R. Peschanski, Nucl. Phys. B 273 (1986) 703.
  * (30) A. Bialas and R. Peschanski, Nucl. Phys. B 308 (1988) 857.
  * (31) N. G. Antoniou, Y. F. Contoyiannis, F. K. Diakonos, et al., Nucl. Phys. A 761 (2005) 149.
  * (32) N. G. Antoniou, F. K. Diakonos, A. S. Kapoyannis, et al., Phys. Rev. Lett. 97, (2006) 032002.
  * (33) T. Anticic, et al., (NA49 Collaboration), Phys. Rev. C 81 (2010) 064907.
  * (34) N. Davis, (for the NA61/SHINE Collaboration), Acta Phys. Pol. B Proc. Suppl. 13 (2020) 637.
  * (35) N. Davis, et al., (for the NA61/SHINE Collaboration), in Proceedings of the CPOD 2018, Corfu Island, Greece, 24-28 September 2018.
  * (36) N. Davis, et al., (for the NA61/SHINE Collaboration), Acta Phys. Polon. B 50, 1029 (2019).
  * (37) J. Wu, Y. Lin, Y. Wu, Z. Li, Phys. Lett. B 801 (2020) 135186.
  * (38) P. Mali, A. Mukhopadhyay, G. Singh , Acta Phys. Pol. B 43 (2012) 463.
  * (39) Tamás Vicsek, Fractal growth phenomena, World Scientific, Singapore N. Jersey (1989), ISBN: 9971-50-830-3.
  * (40) N. G. Antoniou and F. K. Diakonos, J. Phys. G: Nucl. Part. Phys. 46 (2019) 035101.
  * (41) S. A. Bass, M. Belkacem, M. Bleicher, et al., Prog. Part. Nucl. Phys. 41 (1998) 255.
  * (42) M. Bleicher, E. Zabrodin, C. Spieles, et al., J. Phys. G 25 (1999) 1859.
  * (43) Q. Li, M. Bleicher, and H. Stöcker, Phys. Lett. B 659 (2008) 525.
  * (44) Q. Li, M. Bleicher, and H. Stöcker, Phys. Lett. B 663 (2008) 395.
  * (45) Q. Li, M. Bleicher, J. Phys. G 36 (2009) 015111.
  * (46) Q. Li, Z. Li, Mod. Phys. Lett. A 27 (2012) 1250004.
  * (47) Q. Li, Y. Wang, X. Wang $et~{}al.$, Sci. Sin. Phys. Mech. Astron. 59 (2016) 632001.
  * (48) P. Li, Y. Wang, J. Steinheimer $et~{}al.$, Mod. Phys. Lett. A 35 (2020) 2050289.
  * (49) T. Maruyama, K. Niita, T. Maruyama, et al., Prog. Theor. Phys. 96 (1996) 263.
  * (50) M. Isse, A. Ohnishi, N. Otuka, et al., Phys. Rev. C 72 (2005) 064908.
  * (51) Y. Nara, H. Niemi, A. Ohnishi, et al., Eur. Phys. J. A 54 (2018) 18.
  * (52) C. Zhang, J. Chen, X. Luo, et al., Phys. Rev. C 97 (2018) 064913.
  * (53) Q. F. Li, Y. J. Wang, X. B. Wang, et al., Sci. China-Phys. Mech. Astron. 59 (2016) 622001;59 (2016) 632002;59 (2016) 672013 (2016).
  * (54) S. Sombun, K. Tomuang, A. Limphirat, et al., Phys. Rev. C 99 (2019) 014901.
  * (55) N. G. Antoniou, N.Davis, F.K.Diakonos, et al., Nucl. Phys. A 1003 (2020) 122018.
  * (56) F. Becattini, J. Manninen, M. Gazdzicki, Phys. Rev. C 73 (2006) 044905 (2006).
  * (57) F. Becattini, M. Bleicher, E. Grossi, et al., Phys. Rev. C 90 (2014) 054907.
  * (58) D. W. Zhang, (for the STAR Collaboration), Nucl. Phys. A 1005 (2021) 121825.