# Heavy ion collisions from $\sqrt{s_{NN}}$ of 62.4 GeV down to 4 GeV in the
EPOS4 framework

K. Werner(1), J. Jahan (2), I. Karpenko (3), T. Pierog (4), M. Stefaniak
(5,6), D. Vintache (1)

###### Abstract

The EPOS4 project is an attempt to construct a realistic model for describing
relativistic collisions of different systems, from proton-proton ($pp$) to
nucleus-nucleus ($AA$), at energies from several TeV per nucleon down to
several GeV. We argue that a parallel scattering formalism (as in EPOS4) is
relevant for primary scatterings in AA collisions above 4 GeV, whereas
sequential scattering (cascade) is appropriate below. We present briefly the
basic elements of EPOS4, and then investigate heavy ion collisions from 62.4
GeV down to 4 GeV, to understand how physics changes with energy, studying in
particular the disappearance of the fluid component at low energies.

(1)SUBATECH, Nantes University – IN2P3/CNRS – IMT Atlantique, 44300 Nantes,
France

(2)Department of Physics, University of Houston, Houston, TX 77204, USA

(3)Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague,

Břehová 7, Prague, Czech Republic

(4)Institute for Astroparticle Physics, Karlsruhe Institute of Technology,
Karlsruhe, Germany

(5)Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

(6)GSI Helmholtz Centre for Heavy Ion Research, 64291 Darmstadt, Germany

##  1 Introduction

Scatterings of two nuclei are fundamentally different at high energies
(several TeV per nucleon) compared to low energies (several GeV). At high
energies, one can clearly separate “primary scatterings”, happening
instantaneously (at $t=0$) and “secondary scatterings” of the primary
particles (for $t>0$), which may lead to thermalized matter (quark gluon
plasma, QGP) and hadronic rescattering after hadronization. At low energies,
one cannot separate these two stages anymore, and at very low energies (< 4
GeV), heavy ion collisions can be treated by purely sequential hadronic
scatterings, also referred to as hadronic cascade [1, 2, 3, 4, 5, 6, 7, 8]. So
the physics picture changes drastically when going from several TeV down to a
few GeV, and we try to understand these changes. We will start at high
energies and we will see how (and when) the “high energy features” disappear
when lowering the collision energies.

Most important for the discussion of primary scatterings at very high energies
(several TeV per nucleon) is the observation that nucleon-nucleon scatterings
must happen in parallel, and not sequentially, based on very elementary
considerations concerning time scales. To take this "parallel scattering
scenario" into account, EPOS4 brings together ancient knowledge about S-matrix
theory (to deal with parallel scatterings) and modern concepts of perturbative
QCD and saturation, going much beyond the usual factorization approach; see
Refs. [9, 10, 11, 12].

Let us look more in detail at the relevant time scales (see [11] for details).
The particle (hadron) formation time $\tau_{\mathrm{form}}$ has to be compared
to the collision time $\tau_{\mathrm{collision}}$ (the duration of an $AA$
collision) and to the interaction time $\tau_{\mathrm{interaction}}$ (time
between two nucleon-nucleon interactions). We define the high energy threshold
$E_{\mathrm{HE}}$ (in the sense of energy $\sqrt{s_{NN}}$ per nucleon-nucleon
pair) by the identity

$\tau_{\mathrm{form}}=\tau_{\mathrm{collision}},$ (1)

and the low energy threshold $E_{\mathrm{LE}}$ by the identity

$\tau_{\mathrm{form}}=\tau_{\mathrm{interaction}}.s$ (2)

Considering central rapidity hadrons ($\gamma_{\mathrm{hadron}}=1$), a
formation time $\tau_{\mathrm{form}}=1\,\mathrm{fm}/c$, and a big nucleus with
$R=6.5\,\mathrm{fm}$, we get [11]

$E_{\mathrm{LE}}\approx 4\,\mathrm{GeV},\quad E_{\mathrm{HE}}\approx
24\,\mathrm{GeV}.$ (3)

Beyond $E_{\mathrm{HE}}$ particle production starts only after the two nuclei
have passed through each other, which means all the nucleon-nucleon collisions
should happen in parallel, instantaneously, there is no time sequence. Below
$E_{\mathrm{LE}}$ a hadronic cascade is appropriate. Between the two
thresholds, one needs some “partially parallel scattering scenario”, which is
not yet implemented in EPOS4. We will employ the full "parallel scattering
scenario" down to lowest energies, and we will investigate where precisely and
how it breaks down. Since PbPb at 5.02 ATeV and AuAu at 200 GeV have already
been discussed in [11], we will focus in this paper on energies below 200 GeV.

In the overview [9] and in detail in [10, 11] it is shown how such a “parallel
scattering scheme” for primary scatterings can be constructed, based on
S-matrix theory, which we will sketch very briefly in the following.

An early realization is the Gribov-Regge (GR) approach [13, 14, 15, 16] for
$pp$ and $AA$ scatterings. This S-matrix approach has a modular structure, it
is based on so-called “cut Pomerons”, representing elementary parton-parton
scatterings, with the associated mathematical object $G=\mathrm{cut}T$, with
$T$ being the Fourier transform (with respect to the momentum transfer) of the
corresponding T-matrix, divided by $2s$, with $s$ referring to the Mandelstam
variable. This so-called “impact parameter representation” with $G=G(b)$ with
an impact parameter $b$ makes formulas simpler. Although the GR approach is an
excellent tool to deal with parallel scatterings, a serious drawback is the
fact that the energy-momentum sharing between the multiple scatterings is not
taken care of. And obviously GR has no answer to the question of how to
connect the cut Pomeron expression $G$ and the corresponding QCD expression
$G_{\mathrm{QCD}}$ for parton-parton scattering.

In [17], a possible solution has been proposed, by taking into account energy-
momentum sharing (let us call this approach “GR+”) and based on the hypothesis
“$G$ is equal to $G_{\mathrm{QCD}}$”, where the latter is essentially a cut
parton ladder based on DGLAP parton evolutions [15, 18, 19]. A detailed
discussion about the calculation of $G_{\mathrm{QCD}}$ can be found in [10].
Unfortunately, it turned out that implementing energy-momentum sharing has a
very negative side effect: it ruins seriously elementary geometric properties
such as binary scaling in AA scattering. In EPOS4, the first step towards a
solution of the problem is a detailed understanding of what causes the problem
and that it is fundamental, and not just a wrong parameter choice. Using

$G=G_{\mathrm{QCD}}$ (4)

leads unavoidably to contradictions. In a second step, a solution could be
presented. Let us look at the arguments of $G$ and $G_{\mathrm{QCD}}$: Both
depend on $b$ (not written explicitly) and on the lightcone momenta of the
external legs $x^{+}$ and $x^{-}$. But most importantly, $G_{\mathrm{QCD}}$
also depends on some low virtuality cutoff fo the parton evolution, and this
cutoff is now considered to not anymore be simply a constant, but a dynamical
variable, named saturation scale $Q_{\mathrm{sat}}^{2}$. So we have
$G_{\mathrm{QCD}}=G_{\mathrm{QCD}}(Q_{\mathrm{sat}}^{2},x^{+}x^{-})$. The
fundamental relation between $G$ and $G_{\mathrm{QCD}}$ is now

$G(x^{+},x^{-})\\!=\frac{n}{R_{\mathrm{deform}}(N_{\mathrm{conn}},x^{+},x^{-})}G_{\mathrm{QCD}}(Q_{\mathrm{sat}}^{2},x^{+},x^{-}),$
(5)

with
$Q_{\mathrm{sat}}^{2}=Q_{\mathrm{sat}}^{2}(N_{\mathrm{conn}},x^{+},x^{-})$ and
with (being crucial)

$G\>\mathrm{independent\,of}\,N_{\mathrm{conn}},$ (6)

with the so-called connection number $N_{\mathrm{conn}}$ counting the number
of Pomerons being connected to the same projectile and target nucleon as the
given Pomeron. The quantity $R_{\mathrm{deform}}^{(N_{\mathrm{conn}})}$ is the
deformation of the distribution of $x^{+}x^{-}$ (the Pomeron’s energy squared)
in the case of $N_{\mathrm{conn}}>1$ compared to the case
$N_{\mathrm{conn}}=1$. This deformation (a consequence of several Pomerons
competing for energy sharing) destroys factorization and binary scaling unless
one uses eqs. (5,6). Finally, $n$ is a normalization constant.

The primary scattering will produce “parton-ladders”. The link between primary
and secondary scatterings is the “core-corona procedure”: The parton ladders
are treated as classical relativistic (kinky) strings. So in general, we have
a large number of (partly overlapping) strings. Based on the momenta and the
density of string segments (referred to as prehadrons in the following), one
separates at some early proper time $\tau_{0}$ the core (going to be treated
as fluid) from the corona (escaping hadrons, including jet hadrons).

We will first discuss in Sec. 2 about particle production and the role of
core, corona, and remnants for different energies, down to 4 GeV.

Then in Secs. 3, 4, and 5, we will show very detailed results from 62.4 GeV
down to 7.7 GeV, considering the transverse momentum dependencies of yields
and flow harmonics $v_{2}$

  * $\square$

for all energies,

  * $\square$

for all possible centrality choices, and

  * $\square$

for all hadron species where the corresponding data exist, namely pions,
kaons, (anti)protons, but also hyperons ($\Lambda$, $\Xi$, $\Omega$) and
$\Phi$ mesons.

This is the most complete collection of model/data comparisons concerning
gold-gold collisions in this energy domain, in particular concerning
differential yields ($p_{t}$ dependencies).

Over the last decade, a few other models involving a fluid dynamic picture
have been developed for the energy range $\sqrt{s_{\rm NN}}=7.7-62.4$ GeV,
where the UrQMD cascade [20], a decelerating string picture [21, 22] and a 3D
extension to Monte Carlo Glauber [23] have been employed for the initial
state. These models succeeded in reproducing the collision energy dependence
of the shape and the magnitude of the rapidity density of hadrons, and (except
[21, 22]) of $p_{t}$-integrated yields and elliptic flow coefficients. In Ref.
[20], transverse momentum spectra of several hadron species at selected
centralities and energies were shown.

##  2 The role of core, corona, and remnants for different energies

Multiple scattering diagrams (representing the primary scatterings) are the
origin of particle production, which means first the production of so-called
prehadrons, having the quantum numbers of hadrons, but not necessarily being
final. Prehadrons originate from Pomerons (via kinky strings) or from
remnants. A detailed description can be found in section 4 of Ref. [10], where
we also discuss the different types of Pomerons and their relation with pQCD,
including technical details for the pQCD computations.

The pQCD part (parton ladders) is a crucial element of a Pomeron at LHC
energies. But with decreasing energy, it becomes more and more likely that
these Pomerons are replaced by purely soft ones, see Sec. 3 of [12]. At $200$
GeV, the relative weight of “normal” compared to soft Pomerons is roughly 1:1,
at even lower energies, the soft dominates. Also the Pomerons get less
energetic, producing fewer particles.

Based on these abovementioned prehadrons, we employ a so-called core-corona
procedure (see [24], for a more recent discussion in the EPOS4 framework see
[12]), to distinguish core from corona particles, at some given (early)
proper-time $\tau_{0}$. The core prehadrons constitute “bulk matter” and will
be treated via hydrodynamics. The corona prehadrons become simply hadrons and
propagate with reduced energy (due to some energy loss). Corona particles are
either very energetic (then they move out even from the center), or they are
close to the surface, or we have a combination of both.

Figure 1: The prehadron yield as a function of space-time rapidity
($\eta_{s}=\frac{1}{2}\ln((t+z)/(t-z))$, with $t$ being the time and $z$ the
longitudinal coordinate), for central $AA$ collisions at different energies.
The curves refer to all prehadrons (red full), all core prehadrons (red
dotted), prehadrons from remnant decay (blue thin full), and core prehadrons
from remnant decay (blue thin dashed).

We will try to understand the relative importance of the core part and of the
fraction coming from remnant decay. In Fig. 1, we show results for central
(0-5%) PbPb collisions at 5.02 TeV, and central (0-5%) AuAu collisions from
200 down to 4 GeV (per nucleon). We plot four different curves: all prehadrons
(red full), all core prehadrons (red dotted), prehadrons from remnant decay
(blue thin full), and core prehadrons from remnant decay (blue thin dashed).
Looking at the results for PbPb at 5.02 TeV, we observe that almost all
prehadrons are core prehadrons, so the core dominates. Prehadrons from
remnants are preferentially produced at large rapidities, but also here almost
all are core prehadrons. Going down in energy, starting already at 200 GeV, we
see that the core still dominates around $\eta=0$, but at lage values of
$|\eta|$, the core contribution drops dramatically. We also observe that the
remnant contributions become more and more important, and very dominant below
20 GeV. Nevertheless, close to $\eta=0$, the core remains dominant - down to
7.7 GeV. There is actually little change from 39 down to 7.7 GeV, but things
change from 7.7 to 4 GeV: the core part decreases strongly.

In EPOS, $t=0$ is defined as the time corresponding to maximal overlap of the
colliding nuclei. Fluidization takes place at a fixed proper time
$\tau=\tau_{0}$, which we later call initial proper time. We set
$\tau_{0}=1.5$ fm/c for all collision energies except for 7.7 GeV, where
$\tau_{0}=2.0$ fm/c is used. Such setting ensures that the fluid stage starts
after all the primary nucleon-nucleon scatterings have taken place. Having
identified core pre-hadrons, we compute the corresponding energy-momentum
tensor $T^{\mu\nu}$ and the flavor flow vector at some position $x$ at initial
proper time $\tau=\tau_{0}$ as

$T^{\mu\nu}(x)=\sum_{i}\frac{p_{i}^{\mu}p_{i}^{\nu}}{p_{i}^{0}}g(x-x_{i})$ (7)

and

$N_{q}^{\mu}(x)=\sum_{i}\frac{p_{i}^{\mu}}{p_{i}^{0}}\,q_{i}\,g(x-x_{i}),$ (8)

with $q_{i}\in{u,d,s}$ being the net flavor content and $p_{i}$ the four-
momentum of prehadron $i$. The function $g$ is some Gaussian smoothing kernel
(see [12]). The Lorentz transformation into the comoving frame provides the
energy density $\varepsilon$ and the flow velocity components $v^{i}$, which
will be used as the initial condition for a hydrodynamical evolution [25].
This is based on the hypothesis that equilibration happens rapidly and affects
essentially the space components of the energy-momentum tensor. In Fig. 2, we
plot

Figure 2: Energy density at the initial proper-time $\tau_{0}$ as a function
of the transverse coordinate $r$ for central $AA$ collisions at different
energies. The blue dashed line is the freeze-out energy density.

the energy density at the initial proper-time $\tau_{0}$ as a function of the
transverse coordinate $r$ in AA collisions at different energies. We also
indicate as a blue dashed line the freeze-out energy density
$\varepsilon_{\mathrm{FO}}$. At the highest energies, the energy density is
way above the critical value $\varepsilon_{\mathrm{FO}}$, but with decreasing
beam energy, the energy density approaches this value. At 7.7 GeV, the energy
density is slightly above $\varepsilon_{\mathrm{FO}}$, and finally at 4 GeV,
there is no fluid created.

As a next step, a viscous hydrodynamic expansion follows. Starting from the
initial proper time $\tau_{0}$, the core part of the system evolves according
to the equations of relativistic viscous hydrodynamics [25, 26], where we use
presently $\eta/s=0.08$. The “core-matter” hadronizes on some hyper-surface
defined by a constant energy density $\epsilon_{\mathrm{FO}}$ (presently
$0.57\mathrm{\,GeV/fm^{3}}$). In earlier versions [27], we used a so-called
Cooper-Frye procedure. This is problematic in particular for small systems:
not only energy and flavor conservation become important, but we also
encounter problems due to the fact that we get small “droplets” with huge
baryon chemical potential, with strange results for heavy baryons. In EPOS4,
we will systematically use microcanonical hadronization, with major technical
improvements compared to earlier versions (see [12]).

In the following, we want to study core and corona contributions to hadron
production. We will distinguish:

(A)

The “core+corona” contribution: primary interactions (S-matrix approach for
parallel scatterings), plus core-corona separation, hydrodynamic evolution and
microcanonical hadronization of the core, but without hadronic rescattering.

(B)

The “core” contribution: as (A), but considering only core particles.

(C)

The “corona” contribution: as (A), but considering only corona particles.

(D)

The “full” EPOS4 scheme: as (A), but in addition hadronic rescattering.

In cases (A), (B), and (C) , we need to exclude the hadronic afterburner,
because the latter affects both core and corona particles, so in the full
approach, the core and corona contributions are not visible anymore. In the
following, we will focus on energies below 200 GeV, since the corresponding
plots for PbPb at 5.02 ATeV and AuAu at 200 GeV have already been discussed in
[11].

In Figs. 3, 4, and 5,

Figure 3: The X / core+coron ratio, with X being the corona contribution
(blue), the core (green), and the full contribution (red), for 4 centrality
classes and four different particle species, for AuAu at 62.4 GeV. Figure 4:
Same as Fig. 3, but for 39 GeV, and we consider $K^{+},$$K^{-}$, $p$, and
$\bar{p}$ hadrons. Figure 5: Same as Fig. 3, but for 19.6 GeV, and we consider
$K^{+},$$K^{-}$, $p$, and $\bar{p}$ hadrons.

we show ratios X / core+corona versus $p_{t}$, with X being the corona
contribution (blue), the core (green), and the full contribution (red), for
AuAu collisions at different energies. In Fig. 3, we show results at 62.4 GeV,
for (from top to bottom) pions ($\pi^{\pm}$), kaons ($K^{\pm}$), protons ($p$
and $\bar{p}$), and lambdas ($\Lambda$ and $\bar{\Lambda}$). The four columns
represent four different centrality classes, namely 0-5%, 20-40%, 60-80%,
80-100%. Looking at the green (core) and blue (corona) curves, we observe that
the core contribution increases from peripheral to central collisions.
Concerning the $p_{t}$ dependence, we observe a maximum of the green core
curves around 1-2$\,\mathrm{GeV/c}$, at very low $p_{t}$ the core contribution
goes down, so even at very small $p_{t}$ values the corona contributes. At
higher energies (see [11]), one observes a crossing of the green core and the
blue corona curves (core = corona) between around 2 GeV/c (mesons, peripheral)
and 5 GeV/c (baryons, central), and for larger values the corona contribution
dominates clearly. Here, the crossing is absent. There are two reasons. The
spectra are softer compared to higher energies, and around 3-6 GeV/c, the core
and corona curves are almost parallel. And since high $p_{t}$ becomes rare, we
cannot reach very high $p_{t}$ in the Monte Carlo simulations.

The red curve , full over core+corona, represents the effect of the hadronic
cascade in the full case. The pions are not much affected, but for kaons and
even more for protons and lambdas, rescattering makes the spectra harder. We
should keep in mind that rescattering involves particles from fluid
hadronization, but also corona particles from hard processes. Concerning the
baryons, rescattering reduces (considerably) low $p_{t}$ yields, due to
baryon-antibaryon annihilation. In Figs. 4 and 5, we show results for AuAu
collisions at 39 and 19.6 GeV. The high $p_{t}$ particles are getting rare, we
hardly get (for simulations with reasonable CPU times) beyond 5 GeV/c. Another
“low energy effect”: the difference between particles and antiparticles
becomes more and more important, this is why we consider separately $K^{+}$,
$K^{-}$, $p$, and $\bar{p}$. The biggest effect can be seen when comparing the
effect of hadronic rescattering (red curves) for protons and antiprotons: with
decreasing energy, the proton curve goes slightly up, and the antiproton curve
goes significantly down (they are rare, and most of them are annihilated).
Interesting observation: also at lower energies, the core ratios (green
curves) get down at low $p_{t}$, but stay close to unity at intermediate
$p_{t}$.

Corresponding plots for AuAu collisions at 27 GeV, 11.5 GeV, and 7.7 GeV can
be found in the appendix.

When discussing these results at low energies (7.7-19.6 GeV), we should keep
in mind our discussion in Sec. 1, where we estimated that our parallel
scattering scenario should be valid beyond 24 GeV per nucleon, and below we
have to take into account the fact that particle production starts before the
two nuclei have passed through each other (not yet done in EPOS4).

But anyway, in the following sections, we will explore to what extent the
model “works” (and can explain data), and where it fails, and what we can
learn from that.

##  3 Results concerning pt spectra

In this section, we show simulation results compared to data. We will not add
too many comments to each curve, the main purpose is to check if the concepts
discussed in the previous sections give a coherent picture (and reproduce the
data) or not.

The number of plots is huge, and it would be tempting to show only a selection
of results and provide a simple message. And it would be tempting to optimize
parameters to have “nice” plots. But the EPOS4 philosophy is different: It is
an attempt to

  * $\square$

have a single unique approach, with version number (here 4.0.0), and

  * $\square$

make a maximum of tests, as complete as possible, for all kinds of observables
and at various energies.

Modeling heavy ion collisions is complex, consisting of several stages, with
considerable uncertainties at each step. And there is nothing like the simple
“smoking gun” to prove or disprove certain concepts. Flow harmonics for
example are popular observables showing the effect of flow asymmetries, but
the flow affects also the $p_{t}$ spectra, more or less strongly depending on
the mass of the particle. So rather than looking at few selected curves and
expecting precision results, the idea is here more to get a global view and to
see to what extent a given theoretical scenario can provide an overall good
description of a very large set of data.

We will see later, that $p_{t}$ spectra and elliptical flow $v_{2}$, although
both are affected by flow, they are at a different degree sensitive to
particular details of the model, so we get complementary information, and
therefore both observables should be studied.

This “global view” strategy requires a huge amount of tests, and therefore
the“tuning” of parameters and options is far from being optimized, But we
expect reasonable agreement in the energy range (beyond 24 GeV) where the
parallel scattering approach is applicable.

In the following we discuss $p_{t}$ spectra. We will start with the highest
energy, and then move down till 7.7 GeV per nucleon. To go this way is
convenient in the sense that from the theory point of view, the high energy
case contains in principle everything, we do not need to add “features” at low
energies, simply certain phenomena “die out”.

We will focus on energies from 39 GeV down to 7.7 GeV, the corresponding plots
for PbPb at 5.02 ATeV and AuAu at 200 GeV have already been discussed in [11].

###  3.1 Results for 39 GeV

In Figs. 6 and 7, we show transverse momentum distributions of $\pi^{+}$,
$\pi^{-}$, $K^{+}$, $K^{-}$, $p$, $\bar{p}$ in AuAu collisions at 39 GeV for
different centrality classes. EPOS4 simulation (lines) are compared to data
from STAR [28]. From top to bottom, we multiply the curves by $3^{-i}$,
$i=0,1,2,3,...$.

In Fig. 8, we show transverse momentum distributions of $\phi$, $K_{0}$,
$\Lambda$, $\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$,
$\bar{\Omega}^{+}$ in AuAu collisions at 39 GeV at central rapidity for
different centralities. EPOS4 simulation (lines) are compared to data from
STAR [29].

In general, the simulation results are close to the data, concerning
identified particles as pions, kaons, and protons, and as well as hyperons,
although for kaons and antiprotons at low $p_{t}$ the simulations are slightly
above the data. Concerning the $\phi$ meson, the simulations are somewhat
above the data at low $p_{t}$

Figure 6: Transverse momentum distributions of $\pi^{+}$, $\pi^{-}$ in AuAu
collisions at 39 GeV for different centrality classes. EPOS4 simulation
(lines) are compared to data from STAR [28]. From top to bottom, we multiply
the curves by $3^{-i}$, $i=0,1,2,3,...$.

Figure 7: Same as Fig. 6, but for $K^{+}$, $K^{-}$, $p$, $\bar{p}$.

Figure 8: Transverse momentum distributions of $\phi$, $K_{0}$, $\Lambda$,
$\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$
in AuAu collisions at 39 GeV at central rapidity for different centralities.
EPOS4 simulation (lines) are compared to data from STAR [29].

###  3.2 Results for 27 GeV

In Fig. 9 and 10, we show transverse momentum distributions of $\pi^{+}$,
$\pi^{-}$, $K^{+}$, $K^{-}$, $p$, $\bar{p}$ in AuAu collisions at 27 GeV for
different centrality classes. EPOS4 simulation (lines) are compared to data
from STAR [28]. From top to bottom, we multiply the curves by $3^{-i}$,
$i=0,1,2,3,...$.

In Fig. 11, we show transverse momentum distributions of $\phi$, $K_{0}$,
$\Lambda$, $\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$,
$\bar{\Omega}^{+}$ in AuAu collisions at 27 GeV at central rapidity for
different centralities. EPOS4 simulation (lines) are compared to data from
STAR [29].

In general, the simulation results are relatively close to the data,
concerning identified particles as pions, kaons, and protons, and as well
hyperons. But compared to 39 GeV, for kaons, antiprotons, and the $\phi$
meson, the deviation (simulation compared to data) at low $p_{t}$ gets bigger.

Figure 9: Transverse momentum distributions of $\pi^{+}$, $\pi^{-}$ in AuAu
collisions at 27 GeV for different centrality classes. EPOS4 simulation
(lines) are compared to data from STAR [28]. From top to bottom, we multiply
the curves by $3^{-i}$, $i=0,1,2,3,...$.

Figure 10: Same as Fig. 9, but for $K^{+}$, $K^{-}$, $p$, $\bar{p}$.

Figure 11: Transverse momentum distributions of $\phi$, $K_{0}$, $\Lambda$,
$\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$
in AuAu collisions at 27 GeV at central rapidity for different centralities.
EPOS4 simulation (lines) are compared to data from STAR [29].

###  3.3 Results for 19.6 GeV

In Figs. 12 and 13, we show transverse momentum distributions of $\pi^{+}$,
$\pi^{-}$, $K^{+}$, $K^{-}$, $p$, $\bar{p}$ in AuAu collisions at 19.6 GeV for
different centrality classes. EPOS4 simulation (lines) are compared to data
from STAR [28]. From top to bottom, we multiply the curves by $3^{-i}$,
$i=0,1,2,3,...$.

In Fig. 14, we show transverse momentum distributions of $\phi$, $K_{0}$,
$\Lambda$, $\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$,
$\bar{\Omega}^{+}$ in AuAu collisions at 19.6 GeV at central rapidity for
different centralities. EPOS4 simulation (lines) are compared to data from
STAR [29].

In general, the simulation results are relatively close to the data,
concerning identified particles as pions, kaons, and protons, and as well
hyperons. Similar to what we have already seen at 27 GeV, for kaons,
antiprotons, and the $\phi$ meson, the simulation is somewhat above the data.

Figure 12: Transverse momentum distributions of $\pi^{+}$, $\pi^{-}$ in AuAu
collisions at 19.6 GeV for different centrality classes. EPOS4 simulation
(lines) are compared to data from STAR [28]. From top to bottom, we multiply
the curves by $3^{-i}$, $i=0,1,2,3,...$.

Figure 13: Same as Fig. 12, but for $K^{+}$, $K^{-}$, $p$, $\bar{p}$.

Figure 14: Transverse momentum distributions of $\phi$, $K_{0}$, $\Lambda$,
$\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$
in AuAu collisions at 19.6 GeV at central rapidity for different centralities.
EPOS4 simulation (lines) are compared to data from STAR [29].

###  3.4 Results for 11.5 GeV

In Figs. 15 and 16, we show transverse momentum distributions of $\pi^{+}$,
$\pi^{-}$, $K^{+}$, $K^{-}$, $p$, $\bar{p}$ in AuAu collisions at 11.5 GeV for
different centrality classes. EPOS4 simulation (lines) are compared to data
from STAR [28]. From top to bottom, we multiply the curves by $3^{-i}$,
$i=0,1,2,3,...$.

In Fig. 17, we show transverse momentum distributions of $\phi$, $K_{0}$,
$\Lambda$, $\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$,
$\bar{\Omega}^{+}$ in AuAu collisions at 11.5 GeV at central rapidity for
different centralities. EPOS4 simulation (lines) are compared to data from
STAR [29].

Here we see for the first time (compared to higher energies) significant
deviations between simulation and data. The most striking is a large proton
excess at low $p_{t}$. And (as already seen earlier) a $\phi$ excess.
Surprisingly, the hyperons are doing well.

Figure 15: Transverse momentum distributions of $\pi^{+}$, $\pi^{-}$ in AuAu
collisions at 11.5 GeV for different centrality classes. EPOS4 simulation
(lines) are compared to data from STAR [28]. From top to bottom, we multiply
the curves by $3^{-i}$, $i=0,1,2,3,...$.

Figure 16: Same as Fig. 15, but for $K^{+}$, $K^{-}$, $p$, $\bar{p}$.

Figure 17: Transverse momentum distributions of $\phi$, $K_{0}$, $\Lambda$,
$\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$
in AuAu collisions at 11.5 GeV at central rapidity for different centralities.
EPOS4 simulation (lines) are compared to data from STAR [29].

###  3.5 Results for 7.7 GeV

In Figs. 18 and 19, we show transverse momentum distributions of $\pi^{+}$,
$\pi^{-}$, $K^{+}$, $K^{-}$, $p$, $\bar{p}$ in AuAu collisions at 7.7 GeV for
different centrality classes. EPOS4 simulation (lines) are compared to data
from STAR [28]. From top to bottom, we multiply the curves by $3^{-i}$,
$i=0,1,2,3,...$.

In Fig. 20, we show transverse momentum distributions of $\phi$, $K_{0}$,
$\Lambda$, $\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$,
$\bar{\Omega}^{+}$ in AuAu collisions at 19.6 GeV at central rapidity for
different centralities. EPOS4 simulation (lines) are compared to data from
STAR [29].

Here, at 7.7 GeV, essentially all spectra from the simulation are too soft,
the yields at low $p_{t}$ too high, with the biggest excess observed for
protons and $K^{+}$ mesons. So here the model does not work.

Figure 18: Transverse momentum distributions of $\pi^{+}$, $\pi^{-}$ in AuAu
collisions at 7.7 GeV for different centrality classes. EPOS4 simulation
(lines) are compared to data from STAR [28]. From top to bottom, we multiply
the curves by $3^{-i}$, $i=0,1,2,3,...$.

Figure 19: Same as Fig. 18, but for $K^{+}$, $K^{-}$, $p$, $\bar{p}$.

Figure 20: Transverse momentum distributions of $\phi$, $K_{0}$, $\Lambda$,
$\bar{\Lambda}$ $\Xi^{-}$, $\bar{\Xi}^{+}$, $\Omega^{-}$, $\bar{\Omega}^{+}$
in AuAu collisions at 19.6 GeV at central rapidity for different centralities.
EPOS4 simulation (lines) are compared to data from STAR [29].

##  4 Results concerning integrated yields

In this section, we show the integrated yields $dN/dy$ for every particle
species as a function of $\sqrt{s_{NN}}$, at mid-rapidity in the most central
AuAu collisions. We summarise thus the discussion from the previous section,
by displaying how particle production changes with collision energy for every
hadronic species, compared to STAR data from [28, 29].

We compare in Fig. 21

Figure 21: Integrated yields of $\pi^{+}$, $\pi^{-}$, $K^{+}$, $K^{-}$, $p$,
and $\overline{p}$ at mid-rapidity ($|y|<0.1$) in the most central AuAu
collision class (0-5%), as a function of collision energy $\sqrt{s_{NN}}$.
EPOS4 simulation (lines) are compared to data from STAR [28].

the yields from EPOS4 (lines) of $\pi^{+}$, $\pi^{-}$ $K^{+}$, $K^{-}$, $p$,
and $\overline{p}$, measured within $|y|<0.1$, with STAR data from [28]. We do
the same in Fig. 22

Figure 22: Integrated yields of $\Lambda$, $\overline{\Lambda}$, $\Xi^{-}$,
$\overline{\Xi}^{+}$, $\phi$, $K^{0}_{s}$ at mid-rapidity ($|y|<0.1$) in the
most central AuAu collision class (0-5%), as a function of collision energy
$\sqrt{s_{NN}}$. EPOS4 simulation (lines) are compared to data from STAR [28].

for $\Lambda$, $\overline{\Lambda}$, $\Xi^{-}$, $\overline{\Xi}^{+}$, $\phi$,
and $K^{0}_{s}$, measured within $|y|<0.5$, compared with STAR data from [29].
For all hadronic species, the events considered are from the 0-5% centrality
class, except for the $\phi$ mesons for which we consider 0-10% centrality
events due to the available data. We only omit here $\Omega^{-}$ and
$\overline{\Omega}^{+}$, as the data published by STAR in [29] is not using
the same centrality classes for all energies, making it impossible to display
meaningful results about the evolution of integrated yields with collision
energy.

When looking at the whole ensemble of results, we see that particle production
in EPOS4 simulations above 19.6 GeV is, overall, in good qualitative and
quantitative agreement with STAR data. As expected, it reflects the
conclusions from the last section. At lower energies, we observe a clear
overproduction of protons, as well as $\Lambda$ and $\Xi^{-}$ baryons, unlike
their respective anti-baryon counterparts. The yield of $K^{+}$ mesons is also
largely overestimated at low energies, and displays notably almost no
dependence in energy.

Finally, the yield of $K^{0}_{s}$ shown in Fig. 22 is overestimated for the
whole range of collision energies displayed here, compatible with the $p_{t}$
spectra at low $p_{t}$. However, its energy dependence follows qualitatively
well the data.

On the contrary, the $\phi$ meson is the only species which is underproduced
in collision with energies 19.6 GeV and above. But this is in contradiction
with the $p_{t}$ spectra shown earlier, where we see an overproduction.

##  5 Results concerning v2

In Figs. 23, 24, 25, 26, 27, and 28, the transverse momentum dependence

Figure 23: Transverse momentum dependence of $v_{2}$ of identified particles
in AuAu collisions at 62.4 GeV at central rapidity for different centralities.
EPOS4 simulation, full simulations (thick red lines) and without hadronic
cascade (thin green lines), are compared to data from STAR [30] (dots).

Figure 24: Transverse momentum dependence of $v_{2}$ of identified particles
in AuAu collisions at 39.4 GeV at central rapidity for different centralities.
EPOS4 simulation, full simulations (thick red lines) and without hadronic
cascade (thin green lines), are compared to data from STAR [30] (dots).

Figure 25: Transverse momentum dependence of $v_{2}$ of identified particles
in AuAu collisions at 27 GeV at central rapidity for different centralities.
EPOS4 simulation, full simulations (thick red lines) and without hadronic
cascade (thin green lines), are compared to data from STAR [30] (dots).

Figure 26: Transverse momentum dependence of $v_{2}$ of identified particles
in AuAu collisions at 19.6 GeV at central rapidity for different centralities.
EPOS4 simulation, full simulations (thick red lines) and without hadronic
cascade (thin green lines), are compared to data from STAR [30] (dots).

Figure 27: Transverse momentum dependence of $v_{2}$ of identified particles
in AuAu collisions at 11.5 GeV at central rapidity for different centralities.
EPOS4 simulation, full simulations (thick red lines) and without hadronic
cascade (thin green lines), are compared to data from STAR [30] (dots).

Figure 28: Transverse momentum dependence of $v_{2}$ of identified particles
in AuAu collisions at 7.7 GeV at central rapidity for different centralities.
EPOS4 simulation, full simulations (thick red lines) and without hadronic
cascade (thin green lines), are compared to data from STAR [30] (dots).

of $v_{2}$ of identified particles within a pseudorapidity range of $\eta<1$
in AuAu collisions at energies from 62.4 GeV down to 7.7 GeV are shown.
Different centrality classes are considered: 0-10%, 10-40%, and 40-80%. We
show full simulations (thick red lines) and those without hadronic cascade
(thin green lines), and they are compared to data from STAR [30] in the Beam
Energy Scan at the Relativistic Heavy Ion Collider. We employ the same event
plane method as described in the experimental paper. For the upper plots, one
should note that the ranges on the ordinates change, depending on the
centrality, since $v_{2}$ increases with decreasing centrality. In general,
the simulation results describe the data reasonably well, also the centrality
dependence. The biggest deviation is observed for protons for 40-80%. Also for
the pions the agreement is not so great. But the hyperons ($\Xi^{-}$ and
$\Omega$ and their antiparticles $\bar{\Xi}^{+}$ and $\bar{\Omega}$) and as
well the $\phi$ and the $K_{s}$ are relatively close to the data.

##  6 Summary and Conclusions

We reviewed briefly the EPOS4 approach, a detailed discussion can be found in
[9, 10, 11, 12]. Most important is the concept of parallel scattering of
primary interactions, which is needed at high energies, to be more precise
above 24 GeV [11]. We also reviewed briefly “secondary interactions” in EPOS4,
composed of core-corona separation, hydrodynamic core evolution with
subsequent decay (microcanonically), and final state hadron cascade

In the EPOS4 formalism, there is a smooth transition from high to low energy,
certain features change (or disappear) gradually. The parton ladders become
less frequent, they are replaced by soft Pomerons, and most importantly, the
relative importance of particle production from remnant excitation and decay
increases. We discussed how this affects the core-corona procedure. Below 30
GeV, even central rapidities are dominated by particle production from
remnants, and essentially all prehadrons go into the core (for central
collisions). We computed the energy densities of the core (the fluid initial
condition) for the different systems, to observe that they drop from about
$40\mathrm{\,GeV/fm^{3}}$ at 5.02 ATeV to roughly $5\mathrm{\,GeV/fm^{3}}$ at
11.5 GeV, and then drop dramatically. At 4 GeV, there is no fluid anymore.

With all the model details already being published elsewhere [9, 10, 11, 12],
the main purpose of the paper is the presentation of a very detailed test,
considering a very large set of experimental data in the RHIC energy domain,
covering $p_{t}$ spectra and also the $p_{t}$ dependence of the elliptical
flow, for identified particles. Spectra and $v_{2}$ provide complementary
information about the fluid expansion. A particular aim was the investigation
of a possible breakdown of the model at low energies, expected at around 24
GeV from theoretical considerations.

We first showed comparisons of simulations with data concerning $p_{t}$
spectra of identified particles, from 39 GeV down to 7.7 GeV. For the higher
energies, down to 19.6 GeV, the simulation results are relatively close to the
data. At lower energies, we observe “problems”. At 11.5 GeV, we see for the
first time (compared to higher energies) significant deviations between
simulation and data. The most striking is a large proton excess at low
$p_{t}$. At 7.7 GeV, essentially all spectra from the simulation are too soft,
the yields at low $p_{t}$ too high, with the biggest excess observed for
protons and $K^{+}$ mesons. So here the model does not work. These results
concerning low $p_{t}$ yields are also summarized, more globally, through the
integrated yields of different hadronic species displayed as a function of
energy in the most central collisions.

The situation is quite different concerning $v_{2}.$ Here the simulation
results describe the data reasonably well, also the centrality dependence.
There is no “significant deterioration” at very low energies, as in the case
of transverse momentum spectra. Although the model is obviously wrong at low
energies, it works for $v_{2}$.

The success (at energies $\geq$ 19.6 GeV) and the failure (at energies below
19.6 GeV) of the model correspond (amazingly well) to our earlier estimate of
where the model should work and where not: it should work above 24 GeV
(roughly estimated). The main problem of the (full) parallel scenario at low
energies is the fact that a given nucleon hits all target nucleons on its way,
although in reality it does not exist anymore for the final scatterings. This
explains the large proton excess. One of the future projects will be to take
that into account.

### Acknowledgements

IK acknowledges support by the grant GA22-25026S of the Czech Science
Foundation. MS acknowledges support by the U.S. Department of Energy grant DE-
SC0020651. J.J. acknowledges support by the MUSES collaboration under National
Science Foundation (NSF) grant number OAC-2103680.

Appendix

##  Appendix A Core and corona contributions for 27 GeV, 11.5 GeV, and 7.7 GeV

In Figs. 29, 30, and 31, we show ratios X / core+corona versus $p_{t}$, with X
being the corona contribution (blue), the core (green), and the full
contribution (red), for different hadrons. The four columns represent four
different centrality classes, namely 0-5%, 20-40%, 60-80%, 80-100%. We show
results for AuAu collisions at 27 GeV, 11.5 GeV, and 7.7 GeV.

Figure 29: The X / core+coron ratio, with X being the corona contribution
(blue), the core (green), and the full contribution (red), for 4 centrality
classes and four different particle species, for 27 GeV. Figure 30: The X /
core+coron ratio, with X being the corona contribution (blue), the core
(green), and the full contribution (red), for 4 centrality classes and four
different particle species, for 11.5 GeV.  Figure 31: The X / core+coron
ratio, with X being the corona contribution (blue), the core (green), and the
full contribution (red), for 4 centrality classes and four different particle
species, for 7.7 GeV.

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