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# What Can We Learn from Directed Flow at STAR-FXT Energies?

Yu. B. Ivanov  Bogoliubov Laboratory of Theoretical Physics, JINR Dubna,
141980 Dubna, Russia  National Research Center ”Kurchatov Institute”, 123182
Moscow, Russia  M. Kozhevnikova  Veksler and Baldin Laboratory of High Energy
Physics, JINR Dubna, 141980 Dubna, Russia

###### Abstract

We present results of simulations of directed flow of various hadrons in Au+Au
collisions at collision energies of $\sqrt{s_{NN}}=$ 3 and 4.5 GeV.
Simulations are performed within the model three-fluid dynamics (3FD) and the
event simulator based on it (THESEUS). The results are compared with recent
STAR data. The directed flows of various particles provide information on
dynamics in various parts and at various stages of the colliding system
depending on the particle. However, the information on the equation of state
is not always directly accessible because of strong influence of the
afterburner stage or insufficient equilibration of the matter. It is found
that the crossover scenario gives the best overall description of the data.
This crossover EoS is soft in the hadronic phase. The transition into QGP in
Au+Au collisions occurs at collision energies between 3 and 4.5 GeV, at baryon
densities $n_{B}\mathrel{\hbox to0.0pt{ \lower
4.0pt\hbox{\hskip-3.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}4n_{0}$ and
temperatures $\approx 150$ MeV. In-medium effects in the directed flow of
(anti)kaons are discussed.

##  I Introduction

The directed flow is one of the most sensitive quantities to the dynamics of
nucleus-nucleus collisions and properties of the matter produced in these
collisions. It provides information about the stopping power of the nuclear
matter, its equation of state (EoS), transition to quark-gluon phase (QGP) and
more. All these issues were addressed in the analysis of the STAR data
STAR:2014clz  obtained within Beam Energy Scan (BES) program at the
Relativistic Heavy-Ion Collider (RHIC). The analysis was performed within
various approaches Konchakovski:2014gda ; Ivanov:2014ioa ; Ivanov:2016sqy ;
Steinheimer:2014pfa ; Nara:2016phs ; Shen:2020jwv ; Ryu:2021lnx ; Du:2022yok ;
Nara:2016hbg ; Nara:2020ztb ; Nara:2019qfd ; Nara:2021fuu ; Nara:2022kbb ,
which include both hydrodynamic and kinetic models. An important conclusion of
these studies is that the transition to the quark-gluon phase is most probably
of the crossover or weak-first-order type and it stars at collision energies
of $\sqrt{s_{NN}}<$ 8 GeV in Au+Au collisions. A promising recent development
is prediction of correlation between the directed flow and the angular
momentum accumulated in the participant region of colliding nuclei Ryu:2021lnx
; Ivanov:2020wak ; Tsegelnik:2022eoz ; Jiang:2023fad ; Jiang:2023vxp ;
Karpenko:2023bok , which allows a deeper insight into collision dynamics.

The STAR-FXT (fixed-target) data on the directed flow of identified particles
at energies $\sqrt{s_{NN}}=$ 3 and 4.5 GeV were recently published in Refs.
STAR:2020dav ; STAR:2021yiu . These data were also analyzed within various,
mostly kinetic models Nara:2020ztb ; Nara:2022kbb ; Oliinychenko:2022uvy ;
Steinheimer:2022gqb ; OmanaKuttan:2022aml ; Li:2022cfd ; Wu:2023rui ;
Parfenov:2022brq ; Mamaev:2023yhz ; Yao:2023yda ; Kozhevnikova:2023mnw ;
Kozhevnikova:2024itb ; Yong:2023uct ; Wei:2024oca  in relation to various
problems: the hyperon production Nara:2022kbb ; Wu:2023rui ; Wei:2024oca , the
production of light (hyper)nuclei Kozhevnikova:2023mnw ; Kozhevnikova:2024itb
, etc. The EoS of the matter produced in the nucleus-nucleus collisions was
the prime topic of the above theoretical considerations. It was discussed
mostly in terms of softness and stiffness of the EoS Nara:2020ztb ;
Oliinychenko:2022uvy ; Steinheimer:2022gqb ; OmanaKuttan:2022aml ; Wu:2023rui
; Yao:2023yda . These studies were performed within different transport
models: The relativistic version of the quantum molecular dynamics implemented
into the transport code JAM Nara:2020ztb , the hadronic transport code SMASH
Oliinychenko:2022uvy ; Yao:2023yda , the Ultrarelativistic Quantum Molecular
Dynamics (UrQMD) Steinheimer:2022gqb ; OmanaKuttan:2022aml , and a multi-phase
transport model Wu:2023rui .

All the aforementioned papers Nara:2020ztb ; Oliinychenko:2022uvy ;
Steinheimer:2022gqb ; OmanaKuttan:2022aml ; Wu:2023rui ; Yao:2023yda  reported
that stiff (to a different extent) EoS’s are preferable for the reproduction
of the directed flow ($v_{1}$) at $\sqrt{s_{NN}}=$ 3 GeV, while the $v_{1}$
data at 4.5 GeV require a softer EoS. The latter was interpreted as indication
of onset of the phase transition into QGP. This conclusion about preference of
the stiff EoS at the energy of 3 GeV appears to contradict the earlier
findings. The analysis of KaoS KAOS:2000ekm  and FOPI FOPI:2011aa  data at
collision energies $E_{lab}\leq 2A$ GeV ($\sqrt{s_{NN}}\leq$ 2.7 GeV) within
the Isospin Quantum Molecular Dynamics model led to the conclusion that the
soft EoS with the incompressibility $K=$ 210 MeV is strongly preferable
FOPI:2011aa ; Fuchs:2000kp ; Hartnack:2005tr ; Hartnack:2011cn . Although,
this energy range is somewhat below of the STAR-FXT one.

The energy range of the BNL Alternating Gradient Synchrontron (AGS),
$E_{lab}=2-10.7$ $A\cdot$GeV ($\sqrt{s_{NN}}=$ 2.7-4.9 GeV), practically
coincide with the currently explored STAR-FXT range. The results of the
analysis of the AGS data E895:2000maf ; E895:2000sor  are more controversial.
Strong preference of the soft EoS was reported in Refs. Ivanov:2014ioa ;
Ivanov:2016sqy ; Pal:2000yc ; Danielewicz:2002pu ; Russkikh:2006ae . In Refs.
Ivanov:2014ioa ; Ivanov:2016sqy , the EoS additionally softens at
$\sqrt{s_{NN}}>$ 4 GeV because of onset of the deconfinement transition.
However, in Ref. Sahu:2002ku  it was found that the best description of the
data on the transverse flow is provided by a rather stiff EoS at 2$A\cdot$GeV
(NL3) while at higher bombarding energies (4–8 $A\cdot$GeV) a medium EoS ($K=$
300 MeV) leads to better agreement with the data, while the differences in the
soft-EoS and stiff-EoS transverse flows become of minor significance at 4–8
$A\cdot$GeV. In Ref. Isse:2005nk , the proton flow was found to be also
independent of stiffness of the EoS, however provided the momentum dependence
in the nuclear mean fields is taken into account.

As recent studies Nara:2020ztb ; Oliinychenko:2022uvy ; Steinheimer:2022gqb ;
OmanaKuttan:2022aml ; Li:2022cfd ; Wu:2023rui ; Yao:2023yda ; Yong:2023uct  of
the STAR-FXT $v_{1}$ data deduced comparatively stiff EoS’s at
$\sqrt{s_{NN}}=$ 3 GeV, some of them predicted comparatively low baryon
densities ($n_{B}$) for onset of the denfinement transition. This transition
was associated with the softening of the EoS required for $v_{1}$ reproduction
at the energy of 4.5 GeV. In terms of the normal nuclear density $n_{0}$, the
deduced transition densities are 3-4$n_{0}$ Oliinychenko:2022uvy , 4$n_{0}$
OmanaKuttan:2022aml , 2.5$n_{0}$ Li:2022cfd , 3-5$n_{0}$ Wu:2023rui , $n_{B}>$
2-3$n_{0}$ Yao:2023yda , and 5$n_{0}$ Yong:2023uct . The model of three-fluid
dynamics (3FD) Ivanov:2005yw ; Ivanov:2013wha  predicts that the denfinement
transition starts at approximately $n_{B}>$ 4-5$n_{0}$ at temperatures 100–150
MeV for the crossover EoS. However, the STAR-FXT data on the directed flow of
identified particles at energies $\sqrt{s_{NN}}=$ 3 and 4.5 GeV have not yet
been fully considered within the 3FD model, with the exception of the proton
and $\Lambda$-hyperon data at 3 GeV, which were analyzed with respect to the
light (hyper)nuclei production Kozhevnikova:2023mnw ; Kozhevnikova:2024itb .

In view of the above reviewed developments, the debate about the EoS stiffness
and onset of the QGP transition is far from being completed. A more extended
discussion of the EoS constraints deduced from the directed-flow analysis can
be found in recent review Sorensen:2023zkk .

In the present paper, we present results of calculations of the directed flow
of various hadrons at energies $\sqrt{s_{NN}}=$ 3 and 4.5 GeV and compare them
with recent STAR-FXT data STAR:2020dav ; STAR:2021yiu . The calculations are
performed within the 3FD model Ivanov:2005yw ; Ivanov:2013wha  and also within
the Three-fluid Hydrodynamics-based Event Simulator Extended by UrQMD final
State interactions (THESEUS) Batyuk:2016qmb ; Batyuk:2017sku ;
Kozhevnikova:2020bdb . The THESEUS simulations are intended to study the
effect of the UrQMD afterburner stage on the directed flow. We present some
conclusions that can be drawn from agreement or disagreement of the calculated
results with the data.

##  II 3FD model and THESEUS generator

The 3FD model Ivanov:2005yw ; Ivanov:2013wha  simulates nonequilibrium at the
early stage of nuclear collisions by means of two counterstreaming baryon-rich
fluids. The third (fireball) fluid accumulates newly produced particles,
dominantly populating the midrapidity region. These fluids are governed by
conventional hydrodynamic equations coupled by friction terms in the right-
hand sides of the Euler equations. The friction terms describe the
energy–momentum exchange between the fluids. The hydrodynamic evolution ends
with the freeze-out procedure described in Refs. Russkikh:2006aa ;
Ivanov:2008zi . The 3FD model does not include any kinetic afterburner stage.

The THESEUS event generator Batyuk:2016qmb ; Batyuk:2017sku ;
Kozhevnikova:2020bdb  does include the afterburner stage that is described by
the UrQMD model. The THESEUS generator transforms the 3FD output, which is
recorded in terms of local flow velocities and thermodynamic quantities on the
freeze-out hypersurface, into a set of observed particles, i.e. performs a
particlization. The particlization is followed by the afterburner stage.

The afterburner stage is of prime importance for collisions at lower energies,
where there is no clear rapidity separation between participant and spectator
nucleons at the freeze-out. When the time for the nuclei to pass each other
becomes long relative to the characteristic time scale for the participant
evolution, the interaction between participants and spectators (so-called
shadowing) becomes important Bass:1993ce ; Heiselberg:1998es ; Liu:1998yc . In
particular, the squeeze-out effect Sorge:1996pc ; Danielewicz:1998vz ;
Ivanov:2014zqa , is the consequence of this shadowing, i.e. results from
blocking of the expanding central blob by the spectator matter. This shadowing
only partially is taken into account within the 3FD evolution because the
central fireball remains to be shadowed even after the freeze-out while
particles escape from this fireball without interacting with spectators in the
3FD model.

The afterburner stage should, in principle, correct this deficiency. However,
it does not do it completely. The reason is that the THESEUS assigns the same
time instant to all produced particles during the particlization procedure,
while different parts of the system are frozen-out at different time instants
in 3FD. A time-extended transition from hydrodynamic evolution to afterburner
dynamics would need treatment of the interaction of the kinetic afterburner
phase with still hydrodynamically evolved matter. This is a difficult task
both technically and conceptually. The lack of this interaction is the prime
reason of shortcoming of the THESEUS particlization. At lower collision
energies, participants are frozen out earlier than spectators. The spectators
evolve slower because of the lower excitation energy and hence require longer
time before the freeze-out. If the particlization is isochronous like in
THESEUS, the evolution of the frozen-out participants stops untill the
spectators also become frozen-out. Therefore, we skip the stage of shadowing
the afterburner expansion of the central fireball by spectators still being in
the hydrodynamic phase. The afterburner evolution is switched on only when the
spectators also become frozen out, when they have already partially passed the
expanding central fireball. Thus, the shadowing by spectators turns out to be
reduced compared to what it would be if the entire collision process were
kinetically treated, like in UrQMD or JAM.

The 3FD model has been extensively used to simulations of Au+Au collisions at
AGS energies, which almost coincide with the STAR-FXT ones. Quantities, which
are low sensitive to the afterburner stage, were well reproduced by the 3FD
simulations. These are various bulk observables Ivanov:2013wha ;
Ivanov:2013yqa ; Ivanov:2013yla , proton directed Ivanov:2014ioa ;
Ivanov:2016sqy  and elliptic (at higher AGS energies) Ivanov:2014zqa  flow,
bulk properties and directed flow of light (hyper)nuclei at $\sqrt{s_{NN}}=$ 3
GeV Kozhevnikova:2023mnw ; Kozhevnikova:2024itb . Problems with reproduction
of the elliptic flow of protons and light nuclei at $\sqrt{s_{NN}}=$ 3 GeV in
Ref. Kozhevnikova:2023mnw  are related to the aforementioned deficiency of the
isochronous particlization in THESEUS. Precisely the same parameters of the
3FD model as those in Refs. Ivanov:2014ioa ; Ivanov:2016sqy ;
Kozhevnikova:2023mnw ; Kozhevnikova:2024itb ; Ivanov:2013wha ; Ivanov:2013yqa
; Ivanov:2013yla  are used in the present simulations.

##  III Equations of State

The 3FD model is designed to work with different EoS’s. Three different EoS’s
are traditionally used in the 3FD simulations: a purely hadronic EoS gasEOS
and two EoS’s with deconfinement transitions Toneev06 , i.e. an EoS with a
first-order phase transition (1PT EoS) and one with a smooth crossover
transition. While the hadronic EoS is quite flexible, i.e., it allows for
changes of incompressibility, the EoS’s with deconfinement transitions are
strictly tabulated. These EoS’s are illustrated in Fig. 1.

Figure 1: Pressure (scaled by product of the normal nuclear density, $n_{0}=$
0.15 1/fm3, and the nucleon mass, $m_{N}$) at three temperatures, $T=$ 10, 100
and 150 MeV (from bottom upwards for corresponding curves), as function of the
net baryon density (scaled by $n_{0}$) for the hadronic, crossover and 1PT
EoS’s.

As seen, all three EoS’s are similar in the hadronic phase. Note that the
displayed version of the hadronic EoS is characterized by incompressibility
$K=$ 190 MeV. The simulations below are performed with this version of the
hadronic EoS. The crossover pressure starts to deviate from the hadronic one
at $n_{B}>$ 4-5$n_{0}$ at temperatures 100–150 MeV that are typical for the
collisions at STAR-FXT energies, see Fig. 2.

Dynamical trajectories of the matter in the central cell of the colliding
Au+Au nuclei in semicentral collisions ($b=$ 6 fm) at energies
$\sqrt{s_{NN}}=$ 3 and 4.5 GeV are presented in Fig. 2 in terms of the baryon
density and temperature.

Figure 2: Dynamical trajectories of the matter in the central cell of the
colliding Au+Au nuclei in semicentral collisions (impact parameter is $b=$ 6
fm) at energies $\sqrt{s_{NN}}=$ 3 and 4.5 GeV. The trajectories are plotted
in terms of the baryon density ($n_{B}$, scaled by the normal nuclear density
$n_{0}$) and temperature $T$. The trajectories are presented for the three
EoS’s. The mixed phase of the 1PT EoS is displayed by the shadowed region
marked as “mixed phase”. The wide shadowed area displays the region of the
crossover EoS between the QGP fractions $W_{QGP}=$ 0.1 and 0.5.

Evolution starts from the normal nuclear density and zero temperature and then
follows an almost universal trajectory for some time. Shortly before reaching
the turning point, at which density and temperature are maximal, the matter in
this central cell becomes equilibrated, as it was demonstrated in Ref.
Ivanov:2019gxm , and therefore the temperature takes its conventional meaning.
The turning points at the energy of 3 GeV only touches the QGP region
according to the crossover EoS, see Fig. 1. The the same time the 4.5-GeV
trajectories fall well into the crossover QGP region and even enter the the
1PT mixed phase. The trajectories for different EoS’s move away from each
other at higher densities and temperatures. In particular, it demonstrates
that the hadronic EoS and the 1PT one are not identical in the whole hadronic
phase.

Figure 3: Directed flow of protons, pions, $\Lambda$ hyperons, and kaons as
function of rapidity in semicentral ($b=$ 6 fm) Au+Au collisions at collision
energy of $\sqrt{s_{NN}}=$ 3 GeV. Results are calculated within the 3FD model
(upper raw of panels) and the THESEUS (lower raw of panels) with hadronic,
1PT, and crossover EoS’s. STAR data are from Ref. STAR:2021yiu .  Figure 4:
Directed flow of protons, pions, $\Lambda$ hyperons, and kaons ($K^{0}$ short)
as function of rapidity in semicentral ($b=$ 5 fm) Au+Au collisions at
collision energy of $\sqrt{s_{NN}}=$ 4.5 GeV. Results are calculated within
the 3FD model (left block of panels) and the THESEUS (right block of panels)
with hadronic, 1PT, and crossover EoS’s. STAR data are from Ref. STAR:2020dav
.

The crossover and 1PT phase diagrams require some comments. The QCD lattice
calculations demonstrated that the transition into QGP at zero baryon chemical
potential is a smooth crossover Aoki:2006we . Due to that, the transition
temperature is ambiguous because different definitions can lead to different
values for it. Observables related to chiral symmetry result in the transition
temperature around 155–160 MeV Borsanyi:2020fev . As seen from Fig. 2, the
transition regions at zero baryon density (i.e. chemical potential) in EoS’s
of Ref. Toneev06  are located at noticeably higher temperatures than 155–160
MeV. This happens because the EoS’s of Ref. Toneev06  were fitted to the old,
still imperfect lattice data Fodor:2002sd ; Csikor:2004ik ; Karsch:2000kv .
Moreover, the crossover transition constructed in Ref. Toneev06  is very
smooth. The hadronic fraction survives up to very high temperatures. In
particular, this is seen from Fig. 2: the fraction of the quark-gluon phase
($W_{QGP}$) reaches value of 0.5 only at very high temperatures. Such a smooth
crossover is also used in the PHSD model (Parton-Hadron-String Dynamics)
Cassing:2009vt . However, this version of the crossover Toneev06  certainly
contradicts results of the lattice QCD calculations at zero chemical
potential, where a fast crossover was found Aoki:2006we . However, the
aforementioned shortcomings are not severe for the present simulations at
relatively low collision energies, because the system evolution takes place in
the region of high baryon densities, where the EoS is not known from the first
principles.

##  IV Directed Flow

The calculated directed flow of protons, pions, $\Lambda$ hyperons, and
(anti)kaons as function of rapidity in semicentral Au+Au collisions at
collision energies of $\sqrt{s_{NN}}=$ 3 and 4.5 GeV are presented in Figs. 3
and 4, respectively. These calculations were performed in the 3FD model
without any afterburner and within THESEUS (i.e. with the UrQMD afterburner).
The results are compared with STAR data STAR:2021yiu ; STAR:2020dav .

As seen, the proton $v_{1}$ flow is well reproduced with and without
afterburner. The afterburner slightly improves the description at 4.5 GeV,
while worsens it at forward/backward rapidities at 3 GeV without changing the
midrapidity slope. The midrapidity proton flow turns out to be almost
independent of the used EoS even at 4.5 GeV, where the QGP transition already
takes place, see Fig. 2. This is because the proton flow is formed at the
early stage of the collision Baumgardt:1975qv ; Stoecker:2004qu ;
Poskanzer:1998yz . At considered collision energies, this stage is developed
in the hadronic phase for all considered EoS’s, see Fig. 2, where all
considered EoS’s are very similar, see Fig. 1. Moreover, the stopping power of
the matter, i.e. friction forces of the 3FD model Ivanov:2013wha , are
identical in the hadronic phase for all considered scenarios. Consequently,
the flow appears to be quite independent of the used EoS even at 4.5 GeV. Note
that the proton directed flow does depend on the EoS at the BES RHIC energies
Konchakovski:2014gda ; Ivanov:2014ioa ; Ivanov:2016sqy , where the transition
to QGP occurs already the early stage of the collision.

Figure 5: Transverse flow of protons as function of rapidity in semicentral
($b=$ 6 fm) Au+Au collisions at collision energies of $\sqrt{s_{NN}}=$ 2.7–4.3
GeV ($E_{\rm lab}=$ 2$A$, 4$A$, 6$A$, and 8$A$ GeV). Results are calculated
within the 3FD model with hadronic, 1PT, and crossover EoS’s. E895 data are
from Ref. E895:2000maf .

Figure 5 illustrates the description of the old E895 data E895:2000maf  in the
same collision energy range. These data are presented in terms of transverse
flow defined as Danielewicz:1985hn

$\displaystyle\langle P_{x}\rangle(y)=\frac{\displaystyle\int d^{2}p_{T}\
p_{x}\ E\ dN/d^{3}p}{\displaystyle\int d^{2}p_{T}\ E\ dN/d^{3}p},$ (1)

where $p_{x}$ is the transverse momentum of the proton in the reaction plane,
$E\ dN/d^{3}p$ is the invariant momentum distribution of protons with $E$
being the proton energy, and integration runs over the transverse momentum
$p_{T}$. This is done because the E895 data in terms of $v_{1}(y)$ raised many
questions, as it was discussed in Ref. Ivanov:2014ioa  in detail. In addition,
they contradict the new STAR-FXT data. As seen from Fig. 5, the crossover EoS
gives almost perfect description of the proton transverse flow in the
midrapidity regions. This gives hope that the future STAR-FXT proton data at
the energies between 3 and 4.5 GeV will be also well reproduced with the
crossover scenario. The dependence on the EoS is quite moderate, similar to
that at 3 and 4.5 GeV in Figs. 3 and 4.

The $\Lambda$ flow turns out to be more sensitive to the EoS, see Figs. 3 and
4, because $\Lambda$’s are produced in highly excited but still baryon-rich
regions of the colliding system. These regions are formed later, when the
temperature reaches high values, see Fig. 2. The afterburner stronger affects
evolution in these regions. It reduces the midrapidity slope of the $\Lambda$
flow, making the crossover EoS definitely preferable at 3 GeV, while the
hadronic EoS turns out to be preferable at 4.5 GeV. In view of this
sensitivity to the afterburner, definite conclusions on the EoS relevance can
hardly be made based on the $\Lambda$ flow.

The meson flow probes dynamics in highly excited baryon-rich and baryon-
depleted regions of the system. The highly excited baryon-depleted regions are
formed even later than the excited baryon-rich regions, when the transverse
expansion already dominates. Nevertheless, the mesonic flow is very similar to
the baryon one after the 3FD stage at 3 GeV, see Fig. 3, while substantially
differ from the baryon flow at 4.5 GeV, see Fig. 4. Note that this difference
at 4.5 GeV concerns only the EoS’s involving the transition to the QGP,
whereas the hadronic EoS results in the mesonic flow being similar to the
baryon one. The reason is that the EoS becomes softer in the QGP and hence the
pressure causing the directed flow is reduced. Therefore, the mesonic flow may
indicate the transition to the QGP. However, there are other circumstances
that may prevent us from drawing definite conclusions from the mesonic flow.
One of them is the afterburner.

The pion flow is strongly affected by the afterburner. If the 3FD-calculated
pion flow hardly resembles the corresponding data, after the afterburner
stage, it almost perfectly describes these data at 4.5 GeV within the
crossover and 1PT scenarios. The hadronic scenario evidently fails to
reproduce the pion flow at 4.5 GeV. The afterburner even changes the sign of
the midrapidity slope of the crossover and 1PT flows at 4.5 GeV. Note that the
3FD model and hence THESEUS do not distinguish positive, neutral, and negative
pions. Therefore, the calculated pion flow refers to the flow of all pions.
This strong dependence on the afterburner is a consequence of the shadowing
discussed in Sec. II. At 3 GeV, the afterburner shifts the 3FD-calculated flow
closer to the data but still not enough to reproduce them. This insufficient
effect of the afterburner is a result of the shortcoming of the THESEUS
particlization discussed in Sec. II: The afterburner skips the stage of
shadowing the afterburner expansion of the central fireball by spectators
still being in the hydrodynamic phase. At 4.5 GeV, this skipped stage is
already of miner importance because the time for the nuclei to pass each other
becomes shorter relative to the time scale of the participant evolution.

The $K^{+}$ directed flow is not changed by the afterburner stage because of
small cross sections of their interactions with other hadrons. Indeed, the
kaon-nucleon cross section is about 10 mb, while the nucleon-nucleon one is
about 40 mb Hartnack:2011cn ; Bleicher:1999xi ; Song:2020clw . Indeed, at
somewhat lower energies (2$A$ GeV) it was concluded that after their
production the K-mesons suffer not more than one rescattering before escaping
Zwermann:1984pq ; Russkikh:1992tg . At the energy of 3 GeV ($\approx 3A$ GeV),
higher densities are achieved in the collisions and therefore rescatterings
become more frequent, however not frequent enough to thermalize the kaon. This
explains why the calculated flow (on the assumption of the kaon
thermalization) is so different from the data, see Fig. 3.

At slightly higher collision energy of 3.85 GeV (6$A$ GeV), the rescatterings
of the kaons with the nucleons in the dense matter already cause them to flow
in the direction of the nucleons, as it was reported in Ref. Pal:2000yc .
Thus, the transition from rare-collisional to collisional regime occurs in the
considered energy range. At the energy of 4.5 GeV, the kaons can be considered
well thermalized at the late stage of nuclear collision, however the
afterburner still does not affect the flow, as seen from Fig. 4.

At the same time, the $K^{-}$ directed flow at 3 GeV is reduced by the
afterburner because the $NK^{-}$ cross section is of the order of 40 mb or
even higher at low realative $NK^{-}$ energies Hartnack:2011cn ;
Bleicher:1999xi ; Song:2020clw . However, the calculated $K^{-}$ flow is still
essentially stronger than the experimental one. Apparently, this is a result
of the aforementioned shortcoming of the THESEUS particlization, i.e. a lack
of shadowing of the central fireball by still hydrodynamically evolving
spectators.

Figure 6: Directed flow of kaons, antikaons and $K^{0}_{s}$ mesons as function
of rapidity in semicentral ($b=$ 5 fm) Au+Au collisions at collision energy of
$\sqrt{s_{NN}}=$ 4.5 GeV. Results are calculated within THESEUS (the middle
raw of panels) and also within the 3FD model with (the lower raw of panels)
and without (the upper raw of panels) in-medium modifications of (anti)kaons.
STAR data are from Ref. STAR:2020dav .

Within the 3FD model, we calculate $v_{1}(y)$ for $K^{0}_{s}$ mesons in terms
of those for $K^{0}$ and $\bar{K}^{0}$ as follows

$\displaystyle v_{1}^{K0s}(y)$ $\displaystyle=$
$\displaystyle\left(v_{1}^{K0}(y)\frac{dN^{K0}}{dy}+v_{1}^{\overline{K}0}(y)\frac{dN^{\overline{K}0}}{dy}\right)$
(2) $\displaystyle/$
$\displaystyle\left(\frac{dN^{K0}}{dy}+\frac{dN^{\overline{K}0}}{dy}\right),$
(3)

where $dN^{K0}/dy$ and $dN^{\overline{K}0}/dy$ are rapidity distributions of
the $K^{0}$ and $\bar{K}^{0}$ mesons. Eq. (2) does not imply that $K^{0}_{s}$
consists of $K^{0}$ and $\bar{K}^{0}$ in this proportion. It only means that
$K^{0}_{s}$ mesons originate from $K^{0}$ and $\bar{K}^{0}$ mesons that are
emitted from the interaction region. These $K^{0}$ and $\bar{K}^{0}$ mesons
keep their momenta and thus their flow pattern after escaping from the
interaction region. Therefore, the corresponding fractions of produced
$K^{0}_{s}$ mesons carry these $K^{0}$ and $\bar{K}^{0}$ flow patterns. The
$\bar{K}^{0}$ number is about 20% of that of $K^{0}$ at 4.5 GeV.

The directed flow of $K^{0}_{s}$ mesons at 4.5 GeV strongly depends on the EoS
and moderately depends on the afterburner. This moderate dependence on the
afterburner is a consequence of the large fraction of $K^{0}$ mesons in
produced $K^{0}_{s}$. The $K^{0}$ mesons are practically unaffected by the
afterburner. Therefore, the $K^{0}_{s}$ directed flow is a good probe of the
hot and dense stage of the collision. As seen from Fig. 4, the crossover EoS
is certainly preferable for reproduction of the data.

Thus, the directed flows of various particles provide information on dynamics
in various parts and at various stages of the colliding system depending on
the particle. However, the information on the EoS is not always directly
accessible because of strong influence of the afterburner stage or
insufficient thermalization of kaons. The crossover scenario gives the best
overall description of the data, of course, with all reservations regarding
the above-mentioned difficulties in applying the model.

##  V Directed Flow of Kaons

The kaons deserve a separate discussion. As has been mentioned above, the
afterburner does not affect the flow of kaons because of small cross sections
of their interactions with other hadrons but noticeably changes the antikaon
flow at 3 GeV, see Fig. 3. It is instructive to consider the kaon and antikaon
flows at 4.5 GeV, in spite of absence of the corresponding data.

The directed flow of kaons, antikaons and $K^{0}_{s}$ mesons as function of
rapidity in semicentral ($b=$ 5 fm) Au+Au collisions at collision energy of
$\sqrt{s_{NN}}=$ 4.5 GeV is presented in Fig. 6. The flows of kaons and
antikaons are marked as $(K^{0},K^{+})$ and $(\bar{K}^{0},K^{-})$,
respectively, because the 3FD model does not distinguish the corresponding
mesons. The kaon flow again turns out to be insensitive to the afterburner.

The flow of antikaons is enhanced by the afterburner, contrary to the
reduction of the antikaon flow at 3 GeV. Notably, midrapidity slopes of the
kaon and antikaon flow are of the opposite sign for the crossover and 1PT
EoS’s while they are both non-negative within the hadronic scenario.
Apparently, the antiflow of the antikaons is again related to the
aforementioned shadowing of the decay of central blob by the spectator matter.
This shadowing is present already in the 3FD stage of the evolution, as seen
from the upper raw of panels in Fig. 6. The afterburner additionally enhances
this shadowing and hence the antiflow, see the middle raw of panels in Fig. 6.
These opposite signs of the midrapidity slopes of the kaon and antikaon flows
can be considered as a prediction for the flow at 4.5 GeV.

In-medium modifications of kaons are discussed in connection with chiral
symmetry restoration and neutron star properties, see review Tolos:2020aln .
In Refs. Pal:2000yc ; Hartnack:2011cn ; Song:2020clw ; Cassing:2014xca , it
was found that in-medium modifications of kaons are very important for
description of the kaon observables, in particular, the kaon directed flow. It
was reported that these in-medium effects can even change the midrapidity
slope of the kaon flow at the energy of 3.85 GeV (6$A$ GeV) Pal:2000yc , i.e.
in the energy range we consider here.

In the relativistic mean-field approximation for the baryon degrees of freedom
Li:1994cu ; Schaffner:1994bx , the in-medium (anti)kaon energy reads

$\displaystyle E({\bf p})=\left[m_{K}^{2}+{\bf
p}^{2}-\frac{\Sigma_{KN}}{f^{2}_{K}}\rho+\left(\frac{3}{8}\frac{n}{f^{2}_{K}}\right)^{2}\right]^{1/2}\\!\\!\\!\\!\pm\frac{3}{8}\frac{n}{f^{2}_{K}},$
(4)

where ${\bf p}$ is the three-momentum of the (anti)kaon, the upper(lower) sign
refers to $K$($\bar{K}$).

$\displaystyle n$ $\displaystyle=$
$\displaystyle\sum_{B}\langle{\bar{B}}\gamma^{0}B\rangle,$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle\sum_{B}\langle{\bar{B}}B\rangle$

are the proper baryon density and scalar baryon density, respectively, which
are sums over various baryons $B$. Numerical values of the kaon decay
constant, $f_{K}=106$ MeV, and the kaon-nucleon sigma term, $\Sigma_{KN}=350$
MeV, are taken from Ref. Hartnack:2011cn . The term proportional to
$\Sigma_{KN}$ results from the attractive scalar interaction due to explicit
chiral symmetry breaking.

The above expression was derived for the so-called $s$-wave interaction.
Importance of $p$-wave kaon-baryon interactions was indicated in Refs.
Kolomeitsev:1995xz ; Kolomeitsev:2002pg . The treatment of the kaon-baryon
interaction beyond the mean-field approximation, i.e. with the G-matrix
approach Song:2020clw ; Tolos:2020aln , also turned out to be important.
Therefore, Eq. (4) can only serve as a basis for the estimation of the in-
medium effects in (anti)kaon production. Below we use Eq. (4) for this
purpose. The same form of the in-medium kaon energy was used in Ref.
Pal:2000yc .

The version of the UrQMD that is implemented in THESEUS is not suitable for
treatment of the medium modified kaons. Therefore, we study the in-medium
effects within the 3FD model. Results of the 3FD calculation of the directed
flow taking into account the in-medium kaon modification at collision energy
of $\sqrt{s_{NN}}=$ 4.5 GeV is shown in the lower raw of panels of Fig. 6. As
seen, the effect of this in-medium modification is quite moderate. However, it
slightly improves the agreement with the $K^{0}_{s}$ data within the 1PT EoS
and especially the crossover scenario. This improvement is practically the
same as that resulted from the afterburner. It is remarkable that the change
of the antikaon flow due to the in-medium effects is opposite to that caused
by the afterburner.

Figure 7: Directed flow of kaons (left panel) and antikaons (right panel) as
function of rapidity in semicentral ($b=$ 6 fm) Au+Au collisions at collision
energy of $\sqrt{s_{NN}}=$ 3 GeV. Results are calculated with and without in-
medium modifications of (anti)kaons for the crossover EoS’s. STAR data are
from Ref. STAR:2020dav .

As seen from Fig. 7, the effect of the in-medium modifications of (anti)kaons
is also small contrary to that found in Refs. Pal:2000yc ; Hartnack:2011cn ;
Song:2020clw ; Cassing:2014xca . Apparently, this is because the kaons were
incompletely equilibrated in the matter in kinetic simulations of Refs.
Pal:2000yc ; Hartnack:2011cn ; Song:2020clw ; Cassing:2014xca  and hence the
in-medium effect was accumulated throughout the evolution of the system. In
the 3FD simulations they are completely equilibrated and the in-medium
modifications appear only at the freeze-out, leaving them insufficient time to
manifest themselves. Again the in-medium modifications and the afterburner
result in opposite changes in the antikaon directed flow. Only the afterburner
decreases and the in-medium modification increases the flow, contrary to that
at 4.5 GeV. This is because there is the normal flow at 3 GeV instead of
antiflow at 4.5 GeV.

We can conclude that the directed flow of kaons or $K^{0}_{s}$ is a promising
probe of the EoS at hot and dense stage of the collision at 4.5 GeV because it
is not affected by the afterburner stage. At 3 GeV, the kaons do not appear to
be fully equilibrated in matter and therefore do not reflect the EoS of the
matter. The antikaon flow is also a good EoS probe, which is however strongly
modified during the afterburner evolution.

##  VI Summary

The directed flow of various hadrons at energies $\sqrt{s_{NN}}=$ 3 and 4.5
GeV were calculated and compared with recent STAR-FXT data STAR:2020dav ;
STAR:2021yiu . The calculations were performed within the 3FD model
Ivanov:2005yw ; Ivanov:2013wha  and also within the THESEUS generator
Batyuk:2016qmb ; Batyuk:2017sku ; Kozhevnikova:2020bdb  in order to to study
the effect of the UrQMD afterburner stage on the directed flow. Three
different EoS’s are used in the simulations: a purely hadronic EoS gasEOS  and
two EoS’s with deconfinement transitions Toneev06 , i.e. an EoS with a strong
first-order phase transition and one with a smooth crossover transition.

At these collision energies, the time for the nuclei to pass each other is
long relative to the time scale of the participant evolution and therefore the
interaction between participants and spectators (shadowing) is important. In
particular, the squeeze-out effect is a consequence of this shadowing. This
shadowing only partially is taken into account within the 3FD evolution
because the central fireball remains to be shadowed even after the freeze-out.
Therefore, the afterburner stage becomes of prime importance.

The afterburner shifts the 3FD-calculated flow closer to the data but still
not enough to reproduce the pion and antikaon flow at 3 GeV. This insufficient
effect of the afterburner results from shortcoming of the THESEUS isochronous
particlization: The afterburner skips the stage of shadowing of the post-
freeze-out expansion of the central fireball by spectators still
hydrodynamically evolving.

The directed flows of various particles provide information on dynamics in
various parts and at various stages of the colliding system depending on the
particle. However, the information on the EoS is not always directly
accessible because of strong influence of the afterburner stage or
insufficient equilibration, as it happens with kaons at 3 GeV. Based on these
simulations, the following conclusions were drawn:

  * •

The proton flow is formed at the early stage of the collision, where the
matter is not yet equilibrated. Therefore, it probes the properties of this
nonequilibrium matter rather than its EoS that implies the equilibrated
matter. The proton flow is well reproduced within all three considered
scenarios and is practically independent of the afterburner.

  * •

The $\Lambda$ flow turns out to be more sensitive to the EoS because
$\Lambda$’s are produced in highly excited but still baryon-rich regions of
the colliding system. These regions are formed later, when the temperature
reaches high values. The afterburner stronger affects evolution in these
regions.

  * •

The meson flow probes dynamics of highly excited baryon-rich and baryon-
depleted regions of the system. The highly excited baryon-depleted regions are
formed even later than the excited baryon-rich regions, when the transverse
expansion already dominates.

  * •

The pion flow is strongly affected by the afterburner. This strong dependence
on the afterburner is a consequence of the shadowing.

  * •

The directed flow of kaons or $K^{0}_{s}$ is a promising probe of the EoS at
hot and dense stage of the collision at 4.5 GeV because it is not affected by
the afterburner stage. At 3 GeV, the kaons do not appear to be fully
equilibrated in matter and therefore do not reflect the EoS of the matter. The
antikaon flow is also a good EoS probe, which is however strongly modified
during the afterburner evolution.

In conclusion, the crossover scenario gives the best overall description of
the data, of course, with all reservations regarding the above-mentioned
difficulties in applying the model. This crossover EoS is soft in the hadronic
phase. This result agrees with that in Refs. Ivanov:2014ioa ; Ivanov:2016sqy ;
Pal:2000yc ; Danielewicz:2002pu  but is in contrast to Refs. Nara:2020ztb ;
Oliinychenko:2022uvy ; Steinheimer:2022gqb ; OmanaKuttan:2022aml ; Wu:2023rui
; Yao:2023yda , where stiff EoS’s were found being preferable for the
reproduction of the directed flow at 3 GeV. The conclusion about the
preference of the stiff EoS Nara:2020ztb ; Oliinychenko:2022uvy ;
Steinheimer:2022gqb ; OmanaKuttan:2022aml ; Wu:2023rui ; Yao:2023yda , was
mostly based on the proton flow, which is formed at the early nonequilibrium
stage of the collision. Therefore, the proton flow is a combined result of the
EoS and the stopping power of the matter. Different combinations of the EoS
and the stopping power can properly describe the proton flow. Directed flows
of different hadrons, as well as other bulk observables should be considered
together to decouple effects of the EoS and the stopping power.

Within the preferred crossover scenario, the transition into QGP in Au+Au
collisions occurs at collision energies between 3 and 4.5 GeV, at baryon
densities $n_{B}\mathrel{\hbox to0.0pt{ \lower
4.0pt\hbox{\hskip-3.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}4n_{0}$ and
temperatures $\approx 150$ MeV. This implies that the EoS additionally softens
at 4.5 GeV. This softening and hence transition into QGP at 4.5 GeV well
agrees with conclusions made in Refs. Nara:2020ztb ; Oliinychenko:2022uvy ;
Steinheimer:2022gqb ; OmanaKuttan:2022aml ; Wu:2023rui ; Yao:2023yda .

###### Acknowledgements.

Fruitful discussions with D.N. Voskresensky are gratefully acknowledged. This
work was carried out using computing resources of the federal collective usage
center “Complex for simulation and data processing for mega-science
facilities” at NRC ”Kurchatov Institute” ckp.nrcki.ru . and computing
resources of the supercomputer ”Govorun” at JINR govorun .

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