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# Light Nuclei Production in Au+Au Collisions at $\sqrt{s_{NN}}=$ 3 GeV within
Thermodynamical Approach: Bulk Properties and Collective Flow

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

###### Abstract

We present results of simulations of light-nuclei production in Au+Au
collisions at collision energy of $\sqrt{s_{NN}}=$ 3 GeV within updated Three-
fluid Hydrodynamics-based Event Simulator Extended by UrQMD (Ultra-
relativistic Quantum Molecular Dynamics) final State interactions (THESEUS).
The results are compared with recent STAR data. The light-nuclei production is
treated within the thermodynamical approach on equal basis with hadrons. The
only additional parameter related to the light nuclei is the energy density of
late freeze-out that imitates afterburner stage of the collision because the
light nuclei do not participate in the UrQMD evolution. It is found that the
late freeze-out is preferable for deuterons, tritons, and 3He. Remarkably, the
4He observables are better reproduced with the standard freeze-out. This
suggests that the 4He nuclei better survive in the afterburner stage because
they are more spatially compact and tightly bound objects. This is an argument
in favor of dynamical treatment of light nuclei. The simulations indicate that
the collision dynamics is determined by the hadronic phase. The calculated
results reveal not perfect but a good reproduction of the data on bulk
observables and directed flow. The elliptic flow turns out to be more
intricate.

relativistic heavy-ion collisions, hydrodynamics, light nuclei

###### pacs:

25.75.-q, 25.75.Nq, 24.10.Nz

##  I Introduction

During recent years the light-nuclei production has become again one of the
central topics in studies of relativistic heavy-ion collisions. As predicted,
an enhanced production of light nuclei is a promising signal of the critical
endpoint (CEP) Shuryak:2019ikv ; Shuryak:2020yrs ; Sun:2020zxy . This
prediction is based on enhancement of the nucleon attraction near the CEP due
to a rapid increase of correlation length Stephanov:1998dy  and slowing down
the equilibration of the density fluctuations Berdnikov:1999ph  in the
critical region. Abundant production of light nuclei also results from
formation of baryon clusters due to spinodal decomposition associated with
mechanically unstable region in the first-order phase transition Skokov:2008zp
; Skokov:2009yu ; Randrup:2009gp ; Steinheimer:2012gc ; Steinheimer:2019iso ;
Sun:2022cxp . These expectations revived interest to study of the light-nuclei
production at high collision energies. At lower energies, a noticeable part of
the baryon charge is emitted in the form of light nuclei. Therefore, even the
proton data cannot be described without proper reproduction of the light-
nuclei yield. There are different approaches to the light-nuclei production,
which are still actively debated Mrowczynski:2020ugu ; Motornenko:2021nds ;
Kittiratpattana:2022ety ; Kireyeu:2023bye .

Coalescence is the most popular approach, see, e.g., Aichelin:1991xy ;
Russkikh:1993ct ; Ivanov:2005yw ; Liu:2019nii ; Zhu:2015voa ;
Steinheimer:2012tb ; Dong:2018cye ; Sombun:2018yqh ; Zhao:2020irc ;
Hillmann:2021zgj ; Zhao:2021dka , which however needs additional parameters
for description of the light-nuclei yield. The recently developed transport
models Oliinychenko:2018ugs ; Staudenmaier:2021lrg ; Aichelin:2019tnk ;
Glassel:2021rod ; Bratkovskaya:2022vqi ; Sun:2021dlz ; Sun:2022xjr ;
Wang:2023gta  treat light nuclei microscopically on equal basis with other
hadrons. However, these transport models also require an extensive additional
input for the light-nuclei description.

The thermodynamical approach does not need any additional parameters for the
light-nuclei treatment. It describes the light nuclei on equal basis with
hadrons, i.e. in terms of temperatures and chemical potentials. This approach
was first realized within the statistical model, which fairly well described
deuteron midrapidity yields STAR:2019sjh ; STAR:2022hbp  at the energies from
7.7 to 200 GeV Andronic:2010qu ; Vovchenko:2020dmv  while overestimated the
tritium yield by roughly a factor of two Vovchenko:2020dmv ; Zhang:2020ewj .
The statistical model gave a good description of even hypernuclei and
antinuclei Andronic:2017pug .

Inspired by relative success of the statistical model Andronic:2010qu ;
Vovchenko:2020dmv ; Zhang:2020ewj ; Andronic:2017pug , we implemented the
thermodynamic approach to the light-nuclei production into the updated THESEUS
event generator Kozhevnikova:2020bdb . In general, this approach involves no
additional parameters inherent in light nuclei, which makes its predictive
power the same for light nuclei and hadrons. However, because of the lack of
the (post-hydrodynamical) afterburner stage for the light nuclei we had to
introduce a parameter of the late freeze-out for them Kozhevnikova:2022wms .
This late freeze-out imitates the afterburner evolution. In Ref.
Kozhevnikova:2022wms  we considered Pb+Pb and Au+Au collisions in the
collision energy range of $\sqrt{s_{NN}}=$ 6.4–19.6 GeV. The updated THESEUS
resulted in an imperfect but reasonable reproduction of data on bulk
observables of the light nuclei, especially their functional dependence on the
collision energy and light-nucleus mass.

Data on light-nuclei production in Au+Au collisions at $\sqrt{s_{NN}}=$ 3 GeV
were recently published Liu:2021xfy ; Liu:2022ump ; STAR:2021ozh , some of
them Liu:2021xfy ; Liu:2022ump  preliminary. Apart from several Blast-Wave
fits, these data on bulk properties of light nuclei were analyzed in
coalescence-based 3D simulations within JAM (Jet AA microscopic transport
Model Nara:1999dz ; Isse:2005nk ), SMASH (Simulating Many Accelerated
Strongly-interacting Hadrons SMASH:2016zqf ), and UrQMD Bass:1998ca , which
were presented in the STAR paper Liu:2022ump , as well as in the JAM-based
calculation Xu:2023xul . Simulations were also performed within the PHQMD
(Parton-Hadron-Quantum-Molecular-Dynamics) approach Bratkovskaya:2022vqi , in
which the cluster formation occurs dynamically due to interactions, and the
hybrid dynamical-statistical approach Buyukcizmeci:2023azb . These models
reproduce, albeit to varying degrees, experimental trends of the data. The
data on the collective flow of light nuclei at 3 GeV STAR:2021ozh  were
analyzed within the coalescence-based JAM generator, presented in Ref.
STAR:2021ozh . These JAM simulations well describe the data.

In the present paper, the treatment of our previous paper Kozhevnikova:2022wms
is extended to Au+Au collisions at $\sqrt{s_{NN}}=$ 3 GeV. We calculate bulk
properties, directed and elliptic flows of protons and light nuclei ($d$, $t$,
3He and 4He) within the updated THESEUS approach Kozhevnikova:2020bdb  based
on thermodynamic treatment of light nuclei. In contrast to the collision
energy range considered in Ref. Kozhevnikova:2022wms , the yield of the light
nuclei at 3 GeV plays noticeable role in the total balance of the baryon
charge.

##  II updated THESEUS

The THESEUS event generator Batyuk:2016qmb ; Batyuk:2017sku  is based on the
model of the three-fluid dynamics (3FD) Ivanov:2005yw ; Ivanov:2013wha
complemented by the UrQMD Bass:1998ca  for the afterburner stage. The 3FD
takes into account counterstreaming of the leading baryon-rich matter at the
early stage of nuclear collisions. This nonequilibrium stage is modeled by
means of two counterstreaming baryon-rich fluids. Newly produced particles,
which dominantly populate the midrapidity region, are assigned to a so-called
fireball fluid. These fluids are governed by set of hydrodynamic equations
coupled by friction terms, which describe the energy–momentum exchange between
the fluids.

The output of the 3FD model, i.e. the freeze-out hypersurface, is recorded in
terms of local flow velocities and thermodynamic quantities. The THESEUS
generator transforms the 3FD output into a set of observed particles, i.e.
performs the particlization. After the particlization the afterburner stage is
described by the UrQMD. First applications of the THESEUS generator to
description of heavy-ion collisions were demonstrated in Refs. Batyuk:2016qmb
; Batyuk:2017sku .

In the initial version of the THESEUS Batyuk:2016qmb ; Batyuk:2017sku ,
spectra of the so-called primordial nucleons, i.e. both observable nucleons
and those bound in the light nuclei, were calculated. These spectra were
intended for the subsequent use in the coalescence model Ivanov:2005yw ;
Ivanov:2017nae  for the light-nuclei production. Therefore, the nucleons bound
in the light nuclei should be subtracted from the primordial ones in order to
obtain the observable nucleons. Such subtraction is performed in the 3FD,
where production of the light nuclei is calculated within the coalescence
approach Ivanov:2005yw ; Ivanov:2017nae . The initial version of the THESEUS
took temperature and chemical potential fields for hadron sampling from the
hydrodynamic output of the 3FD (where the clusters are not included in the
EoS), and produced both hadrons and clusters within the thermodynamical
approach. This led to an overestimation of the total baryon charge in the
final state containing both baryons and clusters. Therefore, a compensating
correction was required. Such correction was made in the updated version of
THESEUS Kozhevnikova:2020bdb  by means of the recalculation of the baryon
chemical potentials, proceeding from the local baryon number conservation in
the system of hadrons extended by the light-nuclei species listed in Tab. 1.
The list of the light nuclei includes the stable nuclei [deuterons ($d$),
tritons ($t$), helium isotopes 3He and 4He], and low-lying 4He resonances
decaying into stable species Shuryak:2019ikv . The corresponding anti-nuclei
are also included.

Nucleus($E$[MeV]) | $J$ | decay modes, in %
---|---|---
$d$ | $1$ | Stable
$t$ | $1/2$ | Stable
3He | $1/2$ | Stable
4He | $0$ | Stable
4He(20.21) | $0$ |  $p$ = 100
4He(21.01) | $0$ |  $n$ = 24, $p$ = 76
4He(21.84) | $2$ |  $n$ = 37, $p$ = 63
4He(23.33) | $2$ |  $n$ = 47, $p$ = 53
4He(23.64) | $1$ |  $n$ = 45, $p$ = 55
4He(24.25) | $1$ |  $n$ = 47, $p$ = 50, $d$ = 3
4He(25.28) | $0$ |  $n$ = 48, $p$ = 52
4He(25.95) | $1$ |  $n$ = 48, $p$ = 52
4He(27.42) | $2$ |  $n$ = 3, $p$ = 3, $d$ = 94
4He(28.31) | $1$ |  $n$ = 47, $p$ = 48, $d$ = 5
4He(28.37) | $1$ |  $n$ = 2, $p$ = 2, $d$ = 96
4He(28.39) | $2$ |  $n$ = 0.2, $p$ = 0.2, $d$ = 99.6
4He(28.64) | $0$ |  $d$ = 100
4He(28.67) | $2$ |  $d$ = 100
4He(29.89) | $2$ |  $n$ = 0.4, $p$ = 0.4, $d$ = 99.2
Table 1: Stable light nuclei and low-lying resonances of the 4He system (from
BNL properties of nuclides www.nndc.bnl ). $J$ denotes the total angular
momentum. The last column represents branching ratios of the decay channels,
in per cents. The $p,n,d$ correspond to the emission of protons, neutrons, or
deuterons, respectively.  Figure 1: Midrapidity ($|y|<$ 0.1) transverse-
momentum spectra of protons and light nuclei (deuterons, tritons, 3He and 4He)
in Au+Au collisions at collision energy of $\sqrt{s_{NN}}=$ 3 GeV and
different centralities (impact parameters $b$). Results are calculated with
hadronic, 1PT and crossover EoS’s. Results of the THESEUS simulations with the
late freeze-out ($\varepsilon_{\rm frz}=$ 0.2 GeV/fm3) for three EoS’s and the
conventional 3FD freeze-out ($\varepsilon_{\rm frz}=$ 0.4 GeV/fm3) for the
crossover EoS are displayed for light nuclei. Protons are calculated within
the conventional 3FD freeze-out with the subsequent UrQMD afterburner.
Preliminary STAR data are from Refs. Liu:2021xfy ; Liu:2022ump .

In the updated version of the THESEUS Kozhevnikova:2020bdb , the light nuclei
were included on equal basis with other hadrons. These nuclei are sampled
similarly to other hadrons, i.e. accordingly to their phase-space distribution
functions. However, there is an important difference. While the hadrons pass
through the UrQMD afterburner stage after the particlization, the light nuclei
do not, because the UrQMD is not able to treat them. This is a definite
shortcoming because the light nuclei are destroyed and again re-produced
during the afterburner stage Oliinychenko:2018ugs ; Staudenmaier:2021lrg ;
Glassel:2021rod ; Sun:2021dlz ; Sun:2022xjr . Following the recipe of Ref.
Kozhevnikova:2020bdb , we imitate the afterburner for light nuclei by late
freeze-out in the 3FD.

Three different equations of state (EoS’s) are used in the 3FD simulations: a
purely hadronic EoS gasEOS  (hadr. EoS) and two EoS’s with deconfinement
Toneev06 , i.e. an EoS with a first-order phase transition (1PT EoS) and one
with a smooth crossover transition (crossover EoS). Consequently, the 3FD
output for these EoS’s is used in the THESEUS generator.

##  III Bulk Observables

To partially overcome the aforementioned problem of the afterburner stage for
the light nuclei, we imitate the afterburner effect by late freeze-out for
light nuclei. Similarly to Ref. Kozhevnikova:2020bdb , we take the freeze-out
energy density $\varepsilon_{\rm frz}=$ 0.2 GeV/fm3 for this late freeze-out,
which looks quite suitable for all considered quantities, as seen below. No
other additional tuning of the parameters was carried out for the light
nuclei. Note that the conventional 3FD freeze-out energy density for all other
hadrons, subjected to the UrQMD afterburner, is 0.4 GeV/fm3. Details of the
freeze-out procedure in the 3FD are described in Refs. Russkikh:2006aa ;
Ivanov:2008zi . The $\varepsilon_{\mbox{\scriptsize frz}}$ quantity has
meaning of a “trigger” that indicates possibility of the freeze-out. The
freeze-out procedure begins when the local (i.e. in a cell) energy density
drops below the freeze-out value $\varepsilon_{\mbox{\scriptsize frz}}$, then
testing for additional freeze-out conditions starts. If all the freeze-out
conditions are met, the cell is declared frozen-out. Thus, the actual energy
density of a frozen-out cell turns out to be lower than
$\varepsilon_{\mbox{\scriptsize frz}}$, as it is demonstrated, e.g., in Ref.
Ivanov:2020wak .

###  III.1 Transverse-momentum spectra

Midrapidity ($|y|<$ 0.1) transverse-momentum spectra of protons and light
nuclei (deuterons, tritons, 3He and 4He) in Au+Au collisions at collision
energy of $\sqrt{s_{NN}}=$ 3 GeV and different centralities (impact parameters
$b$) are displayed in Fig. 1. The proton spectra are calculated within full
THESEUS, i.e. with the standard 3FD freeze-out and the UrQMD afterburner. The
spectra of light nuclei are evaluated for the late 3FD freeze-out without the
afterburner stage. As seen, the results for different EoS’s are almost
identical, which means that the dynamics is dominated by the hadronic phase.
The difference between the late freeze-out and the conventional 3FD freeze-out
for the crossover EoS is mostly seen at low $p_{T}$ for light nuclei. In the
$p_{T}$ spectra, this difference does not look dramatic. However, in rapidity
distributions, Fig. 2, which mostly determined by the low-$p_{T}$ spectra, the
difference is quite noticeable.

Figure 2: Rapidity distributions of protons and light nuclei (deuterons,
tritons, 3He and 4He) in Au+Au collisions at collision energy of
$\sqrt{s_{NN}}=$ 3 GeV and different centralities (impact parameters $b$).
Results are calculated with hadronic, 1PT and crossover EoS’s. Results of the
THESEUS simulations with the late freeze-out ($\varepsilon_{\rm frz}=$ 0.2
GeV/fm3) for three EoS’s and the conventional 3FD freeze-out
($\varepsilon_{\rm frz}=$ 0.4 GeV/fm3) for the crossover EoS are displayed for
light nuclei. Protons are calculated within the conventional 3FD freeze-out
with the subsequent UrQMD afterburner. Preliminary STAR data are from Refs.
Liu:2021xfy ; Liu:2022ump . Full symbols display measured experimental points,
whereas the open ones are those reflected with respect to the midrapidity.

Similarly to that found previously Kozhevnikova:2022wms , the experimental
$p_{T}$ spectra Liu:2021xfy ; Liu:2022ump  turn out to be steeper than the
calculated ones. This is not only the problem of light-niclei description. The
proton spectra are also more flat than the experimental ones. This is a
shortcoming of the 3FD model. The slight extra flatness of the proton spectra
transforms into larger extra flatness of light-nuclei spectra.

As has been already noted in Ref. Kozhevnikova:2022wms , the 3FD predictions
overestimate the high-$p_{T}$ ends of the spectra because of finiteness of the
considered system. Even abundant hadronic probes become rare at high momenta.
For the rare probes, treatment on the basis of the canonical ensemble is
needed rather than within the grand canonical ensemble. The grand canonical
ensemble results in overestimation of their yields. Of course, it is difficult
to indicate how much of this overestimation is due to the grand canonical
treatment, and not to the shortcomings of the model.

###  III.2 Rapidity distributions

Rapidity distributions of protons and light nuclei in Au+Au collisions at
collision energy of $\sqrt{s_{NN}}=$ 3 GeV and different centralities are
presented in Fig. 2. Again the proton distributions are calculated within full
THESEUS, i.e. with the standard 3FD freeze-out and the UrQMD afterburner,
while the light-nuclei distributions, within the late 3FD freeze-out and
without the afterburner stage. The light-nuclei distributions, calculated at
the conventional 3FD freeze-out, are also displayed for comparison. The same
value of the late-freeze-out energy density ($\varepsilon_{\rm frz}=$ 0.2
GeV/fm3) as that at higher collision energies Kozhevnikova:2022wms  turned out
to be the most suitable at 3 GeV. The reproduction of the experimental
distributions turns out to be even better than that at higher collision
energies Kozhevnikova:2022wms . The THESEUS simulations well describe
difference in the form of proton and light-nuclei distributions and its
dependence on the centrality.

For the experimental centrality of 20-40% we present comparison with results
for two impact parameters ($b=$ 7 and 8 fm) in order to illustrate the
sensitivity of the results to the choice of $b$. As seen, the proton rapidity
density is underestimated in midrapidity at $b=$ 8 fm in spite of perfect
reproduction of the low-$p_{T}$ experimental spectrum, see Fig. 1. The reason
is that the extrapolation of the experimental spectrum to even lower $p_{T}$
exceeds the THESEUS predictions. Similar situation takes place for the light
nuclei. Thus, the results for two impact parameters ($b=$ 7 and 8 fm)
illustrate uncertainty of the THESEUS predictions.

The 4He distributions deserve a separate discussion. The late-freeze-out
calculation strongly underestimates these distributions. Expanding the list of
light-nuclei resonances by those of 5H, 5He, and 5Li Vovchenko:2020dmv , which
decay into 4He, makes an additional contribution to the 4He yield. This
additional contribution is large, i.e. of the order of 60%, in central
collisions at the energy of 3 GeV, accordingly to Ref. Vovchenko:2020dmv .
However, it is not large enough to compensate the obtained underestimation. At
the same time, the standard-freeze-out calculation results in much better
(almost perfect in midrapidity at 10-20% and 20–40% centralities) reproduction
of the data. The $p_{T}$ spectra are also much better described by the
standard freeze-out, see Fig. 1. This suggests that the 4He nuclei better
survive in the afterburner stage because they are more spatially compact and
tightly bound objects. Then the standard freeze-out is more relevant for their
description. Note that the feed-down contribution to the 4He yield,
$\approx$60% Vovchenko:2020dmv , is quite enough to drastically improve the
reproduction midrapidity data in central collisions by the standard-freeze-out
calculation.

At lower collision energies, the enhancement of the 4He production is even
more spectacular FOPI:2010xrt . In central Au+Au collisions, the 4He and 3He
yields are approximately equal at $E_{lab}=$ 0.4$A$ GeV and the 4He yield even
exceeds that of 3He at 0.15$A$ GeV. It seemingly contradicts to the
thermodynamic picture of the light-nuclei production. However, this
contradiction is removed, if the chemical freeze-out occurs earlier for 4He
nuclei than for $d$, $t$ and 3He because of larger binding energy of the 4He
nuclei. In Ref. Wang:2023gta  it is formulated in terms of the Mott transition
Typel:2009sy : The observed enhancement of the 4He yield can be attributed to
the weaker Mott effect on 4He nuclei than that on deuteron, triton and 3He, as
a result of its much larger binding energy. The cutoff value for the average
nucleon phase-space density, $f_{A}^{\rm cut}$, that is used in Ref.
Wang:2023gta , see Eq. (5) in Ref. Wang:2023gta , in fact plays the role of
the effective freeze-out for different light nuclei. As found in Ref.
Wang:2023gta , the $f_{A=4}^{\rm cut}$ value is approximately twice as large
as that for lighter nuclei. This is consistent with our conclusion about the
earlier freeze-out of 4He. In particular, the results of the kinetic approach
of Ref. Wang:2023gta  imply that the freeze-out parameters for each light-
nuclei specie are individual and depend on the binding energy of considered
nucleus.

###  III.3 Medium Effects

Study of the in-medium effects in light-nuclei production has started long ago
Danielewicz:1992mi . Later, coupled quantum kinetic equations were derived
which describe the time evolution of the Wigner distribution functions for
nucleons and light clusters Kuhrts:2000zs . An alternative approach has also
been proposed within the antisymmetrzed molecular dynamics (AMD) approach
Ono:2016mqk , in which nucleons are represented in terms of quantal wave
packets, which are antisymmeztrized with each other. These approaches have
been successfully applied to analysis of results of the GANIL experiment at
50$A$ MeV.

Figure 3: Rapidity distributions of protons and light nuclei (top down:
protons, deuterons, tritons, 3He and 4He) in Au+Au collisions at collision
energy of $\sqrt{s_{NN}}=$ 3 GeV and $b=$ 5 fm. Results are calculated in the
crossover scenario with (with SE) and without (w/o SE) the self-energy
contributions, i.e. the $\Delta E^{\rm SE}_{A}({\bf p})$ and $\Delta E^{\rm
Pauli}_{A}({\bf p})$ terms in Eq. (1). Protons are calculated within the
conventional 3FD freeze-out with the subsequent UrQMD afterburner. Deuterons,
tritons, and 3He are calculated with the late freeze-out ($\varepsilon_{\rm
frz}=$ 0.2 GeV/fm3), while 4He, the conventional 3FD freeze-out
($\varepsilon_{\rm frz}=$ 0.4 GeV/fm3). Preliminary STAR data are from Refs.
Liu:2021xfy ; Liu:2022ump . Full symbols display measured experimental points,
whereas the open ones are those reflected with respect to the midrapidity.

In Refs. Bastian:2016xna ; Ropke:2017dur , it was proposed to extend the study
of these in-medium effects to nuclear collisions at NICA (Nuclotron-based Ion
Collider Facility) and FAIR (Facility for Antiproton and Ion Research)
energies. A suitable frame for such a study is the thermodynamic approach to
light-nuclei production complemented by quantum statistical approach that
includes medium effects due to Pauli blocking and self-energies Typel:2009sy ;
Ropke:2012qv ; Ropke:2014fia . This quantum statistical approach is based on
the relativistic mean-field model of Ref. Ropke:2012qv . The energies of light
nuclei are given by the formula

$\displaystyle E_{A}({\bf p})=E^{0}_{A}({\bf p})+\Delta E^{\rm SE}_{A}({\bf
p})+\Delta E^{\rm Pauli}_{A}({\bf p})$ (1)

where $E^{0}_{A}(p)$ is the vacuum energy of $A$-nucleus with momentum ${\bf
p}$, $\Delta E^{\rm SE}_{A}(p)$ is in-medium self-energy shift, $\Delta E^{\rm
Pauli}_{A}(p)$ is the energy correction due to the Pauli blocking. The last
two quantities also depend on the baryon density, temperature and
proton/neutron asymmetry of the matter. More details can be found in Ref.
Ropke:2014fia . This quantum statistical approach was incorporated into
THESEUS at the particlization stage.

Preliminary results showed Blaschke:2020gqr ; Blaschke:2020jmv  that the best
description of the light-nuclei yields is obtained when the self-energy
effects are discarded at few-GeV collision energies ($\sqrt{s_{NN}}$).
However, those results were not conclusive because the proper conservation of
the baryon charge was not achieved in that version of the THESEUS, see sect.
II. The Pauli-blocking effects were disregarded in those calculations.

The present calculation are performed within the updated version of the
THESEUS, where the conservation of the baryon charge is strictly fulfilled. An
upper estimate of the Pauli-blocking effect is also made by means of its
approximation by that at zero light-nucleus momentum in the rest frame the
medium. The Pauli-blocking is strongest at this zero momentum. The Pauli-
blocking effect turned out to be negligibly small because of high freeze-out
temperatures as compared with the corresponding Fermi energies. The results of
this calculation with the crossover EoS are displayed in Fig. 3. As seen, the
medium effect accordingly to Ref. Ropke:2014fia  turns out to be too strong.
It results in strong disagreement with data, as has been already seen from the
preliminary simulations Blaschke:2020gqr ; Blaschke:2020jmv . Even the proton
yield is overestimated. Apparently this is because the relativistic mean-field
model Ropke:2012qv , underlying these medium corrections, has been
parametrized to reproduce low-energy nuclear phenomena. This parametrization
is simply inapplicable to highly excited nuclear matter. Of course, we could
introduce a temperature-dependent attenuation factor to reduce the strength of
these medium corrections. However, it would be a purely phenomenological
tuning parameter. Therefore, we avoid doing this.

##  IV Collective flow

Collective flow is a more subtle observable of the heavy-ion collisions. Its
calculation with the thermodynamic approach is straightforward because light
nuclei are treated on the equal basis with other hadrons. Results for the
collective flow, which are presented below, are calculated with respect to the
reaction plane that is exactly defined in the simulations. We have also done
calculations with the event plane determined in the afterburner. The reaction-
plane and event-plane results turned out to be practically identical.

###  IV.1 Directed flow

The calculated directed flow of protons and light nuclei (deuterons, 3He and
4He) as function of rapidity in semicentral ($b=$ 6 fm) Au+Au collisions at
collision energy of $\sqrt{s_{NN}}=$ 3 GeV is presented in Fig. 4. The results
are compared with STAR data STAR:2021ozh ; STAR:2021yiu . We do not display
results for the tritons because they are very similar to those for 3He,
including the degree of agreement with the data. The THESEUS simulations for
light nuclei are performed for the late freeze-out ($\varepsilon_{\rm frz}=$
0.2 GeV/fm3) for three EoS’s. Protons are calculated within the conventional
3FD freeze-out with the subsequent UrQMD afterburner.

Figure 4: Directed flow of protons and light nuclei (deuterons, 3He and 4He)
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
hadronic, 1PT and crossover EoS’s. Results of the THESEUS simulations with the
late freeze-out ($\varepsilon_{\rm frz}=$ 0.2 GeV/fm3) are displayed for light
nuclei. Protons are calculated within the conventional 3FD freeze-out with the
subsequent UrQMD afterburner. STAR data are from Ref. STAR:2021ozh ;
STAR:2021yiu . Full symbols display measured experimental points, whereas the
open ones are those reflected with respect to the midrapidity. Figure 5: The
same as in Fig. 4 but for various versions of hadronic EoS: the standard
hadronic EoS ($K=$ 190 MeV) and very soft hadronic EoS ($K=$ 130 MeV). Results
of the THESEUS simulations with the late freeze-out ($\varepsilon_{\rm frz}=$
0.2 GeV/fm3) and conventional 3FD freeze-out ($\varepsilon_{\rm frz}=$ 0.4
GeV/fm3) are displayed for light nuclei. Protons are calculated within the
conventional 3FD freeze-out with the subsequent UrQMD afterburner. The proton
$v_{1}$ within the 3FD model, i.e. before the UrQMD afterburner, is also
presented for the standard EoS ($K=$ 190 MeV, thin solid line) and stiff EoS
($K=$ 380 MeV, thin short-dashed line).

The directed flow turns out to be independent of the used EoS, which again
suggests that the dynamics is dominated by the hadronic phase. The calculated
results almost perfectly (except for very forward and backward rapidities)
reproduce the experimental proton directed flow STAR:2021yiu . Agreement with
the data STAR:2021ozh  is getting worse with increase of atomic number of
light nucleus. If the calculated midrapidity slope of the directed flow is
only slightly steeper than the experimental one for deuterons, for 4He it is
already noticeably steeper.

To check if this disagreement is related to the above observed preference of
the conventional freeze-out for the 4He, see Fig. 2, we present the results
with conventional 3FD freeze-out in Fig. 5. As seen, the 4He flow is
independent of the type of the freeze-out while the flow slopes of lighter
nuclei become only slightly steeper at the conventional freeze-out.

Stiffness of the hadronic EoS with the 3FD model can be easily changed. The
stiffness is characterized by incompressibility of nuclear matter that is
conventionally defined as

$\displaystyle
K=9n_{0}^{2}\frac{d^{2}}{dn^{2}}\left(\frac{\varepsilon(n,T=0)}{n}\right)_{n=n_{0}},$
(2)

where $\varepsilon(n,T=0)$ is the energy density of the nuclear matter at zero
temperature ($T=0$) as a function of the baryon density ($n$), $n_{0}$ is the
normal nuclear density. The conventionally used hadronic EoS is characterized
by $K=$ 190 MeV. This is a quite soft EoS. It is very similar (but not
identical) to the EoS of the hadronic phase in the 1PT and crossover EoS’s
Toneev06 . To study the effect of the EoS stiffness on the directed flow, we
present results for a very soft hadronic EoS ($K=$ 130 MeV) in Fig. 5. As
seen, the 4He flow again turns out to be independent of the EoS stiffness. The
very soft EoS gives slightly better agreement with the data for lighter nuclei
than the conventionally used EoS but results in disagreement with the
experimental proton flow. In Ref. STAR:2021yiu  it is reported that the stiff
hadronic EoS ($K=$ 380 MeV) well reproduces the proton directed flow within
the UrQMD and JAM models. In contrast, our calculation shows that the stiff
EoS ($K=$ 380 MeV) results in too steep slope of the proton flow (see the thin
short-dashed line in Fig. 5), which leads to even stronger disagreement with
flow data for light nuclei. Therefore, the conventionally used soft hadronic
EoS with $K=$ 190 MeV seems to be the optimal choice. This conclusion agrees
with that in famous paper Danielewicz:2002pu  made more than twenty years ago.

The proton $v_{1}$ within the 3FD model, i.e. before the UrQMD afterburner, is
also presented in Fig. 5. The afterburner does not change the midrapidity
slope of the flow but worsens agreement with the data at very forward and
backward rapidities. The proton flow at the late freeze-out and without
afterburner (not displayed in Fig. 5) is very similar to that at the
conventional 3FD freeze-out and the subsequent afterburner, which once again
confirms the correctness of the choice of energy density for the late freeze-
out.

In contrast to the proton flow, the afterburner imitation (i.e. the late
freeze-out) does change the midrapidity slope of deuterons and 3He albeit
slightly. However, the 4He flow is not affected by the late freeze-out.

###  IV.2 Elliptic flow

The calculated elliptic flow of protons and light nuclei (deuterons, 3He and
4He) as function of rapidity in semicentral ($b=$ 6 fm) Au+Au collisions at
collision energy of $\sqrt{s_{NN}}=$ 3 GeV is presented in Fig. 6. The results
are compared with STAR data STAR:2021ozh ; STAR:2021yiu . Results for tritons
are again omitted because they are very similar to those for 3He. The THESEUS
simulations for deuterons and 3He are performed for the late freeze-out
($\varepsilon_{\rm frz}=$ 0.2 GeV/fm3), while for 4He, the conventional 3FD
freeze-out ($\varepsilon_{\rm frz}=$ 0.4 GeV/fm3) in view of its preference
for $p_{T}$ spectra and $y$ distributions, see sect. III. Protons are
calculated within the conventional 3FD freeze-out with the subsequent UrQMD
afterburner. For comparison, the light-nuclei flow with the conventional 3FD
freeze-out for the 1PT EoS and the proton $v_{2}$ flow before the UrQMD
afterburner are also demonstrated.

Figure 6: Elliptic flow of protons and light nuclei (deuterons, 3He and 4He)
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
hadronic, 1PT and crossover EoS’s. Results of the THESEUS simulations with
three EoS’s are displayed for deuterons and 3He at the late freeze-out
($\varepsilon_{\rm frz}=$ 0.2 GeV/fm3) and for the 4He nuclei at the
conventional 3FD freeze-out ($\varepsilon_{\rm frz}=$ 0.4 GeV/fm3). The
conventional 3FD freeze-out with the 1PT EoS is also presented for for
deuterons and 3He. Protons are calculated within the conventional 3FD freeze-
out with the subsequent UrQMD afterburner. The proton $v_{2}$ flows before the
UrQMD afterburner are also presented. STAR data are from Refs. STAR:2021ozh ;
STAR:2021yiu . Full symbols display measured experimental points, whereas the
open ones are those reflected with respect to the midrapidity. Figure 7: The
same as in Fig. 6 but for various versions of the hadronic EoS. Results with
the standard hadronic EoS ($K=$ 190 MeV) are displayed by bold dash-dotted
lines for the late freeze-out ($\varepsilon_{\rm frz}=$ 0.2 GeV/fm3) for all
light nuclei and by thin dash-dotted line for the standard 3FD freeze-out
($\varepsilon_{\rm frz}=$ 0.4 GeV/fm3) for 4He nuclei. Results of calculations
with the stiff hadronic EoS ($K=$ 380 MeV) are also presented: for deuterons
and 3He at the late freeze-out and for all nuclei at the standard 3FD freeze-
out. Protons are calculated within the conventional 3FD freeze-out with the
subsequent UrQMD afterburner. The proton $v_{2}$ before the UrQMD afterburner
is also presented for the stiff EoS ($K=$ 380 MeV) by solid line.

As seen from Fig. 6, the calculated elliptic flow considerably overestimates
the data STAR:2021ozh ; STAR:2021yiu . Even the $v_{2}$ sign is different in
the midrapidity region. The afterburner (for protons) and the late freeze-out
(for light nuclei) even worsen agreement with the data. The large disagreement
of light-nuclei elliptic flow with the data STAR:2021ozh  does not mean that
the thermodynamic approach fails to describe this flow. This only means that
the 3FD model has troubles describing the elliptic flow of protons, which
transform into even biger troubles for light nuclei.

As stated in Refs. STAR:2021ozh ; STAR:2021yiu ; Danielewicz:2002pu , a stiff
EoS is needed for describing the elliptic flow. Therefore, we performed
simulations with stiff hadronic EoS ($K=$ 380 MeV), see Fig. 7. The results
became closer to the data. The proton elliptic flow is even reproduced in the
midrapidity. However, the overall disagreement with data remains. The
afterburner (late freeze-out) still worsens the agreement with data.

Note that dips and even negative values of $v_{2}$ in the midrapidity are
consequences of the squeeze-out effect Sorge:1996pc ; Danielewicz:1998vz ;
Ivanov:2014zqa , which result from blocking of the expanding central blob by
the spectator matter. The squeeze-out is a characteristic feature of
moderately relativistic collisions, in which the expanding central fireball is
shadowed by spectators. This shadowing only partially is taken into account
within the 3FD evolution because the frozen-out matter of the central fireball
remains to be shadowed even after the freeze-out while in the 3FD model it
escapes without interacting with spectators. The afterburning stage should, in
principle, correct this deficiency. However, it does not, as we see from Figs.
7 and 6. 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. Participants
are frozen out earlier than spectators. If the particlization is isochronous,
the evolution of the frozen-out participants stops untill the spectators also
become frozen-out. Therefore, we skip the stage of shading 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. The spectators are frozen out when they
have already passed the expanding central fireball. Thus, the shadowing by
spectators turns out to be strongly reduced after such isochronous
particlization compared to what it would be if the entire collision process
were kinetically treated, as in UrQMD or JAM. Apparently, this is the prime
reason of failure of the elliptic-flow description. A time-extended transition
from hydrodynamic evolution to afterburner dynamics would need to take into
account the interaction of the kinetic afterburner phase with still
hydrodynamically evolved matter. This is a difficult task both technically and
conceptually.

##  V Feed-Down from Unstable ${}^{4}He$

Figure 8: Rapidity distributions of light nuclei (deuterons and 3He) in
central Au+Au collisions at collision energy of $\sqrt{s_{NN}}=$ 3 GeV.
Results are calculated in the crossover scenario with and without (w/o 4He∗
decays) the feed-down contributions. Preliminary STAR data are from Refs.
Liu:2021xfy ; Liu:2022ump . Full symbols display measured experimental points,
whereas the open ones are those reflected with respect to the midrapidity.

As found in Ref. Kozhevnikova:2022wms , the feed-down contributions from
unstable 4He to deuterons are negligibly small, while to tritons and 3He are
less than 20% at $\sqrt{s_{NN}}>$ 6 GeV in the midrapidity. At the
forward/backward rapidities, these contributions are essential even at
$\sqrt{s_{NN}}>$ 6 GeV. It was predicted Vovchenko:2020dmv , that feed-down
contributions reach values of the order of 60% for tritons and 3He even in
midrapidity at $\sqrt{s_{NN}}=$ 3 GeV.

Results of our calculations are presented in Fig. 8 at the example of the
crossover EoS. In agreement with Ref. Vovchenko:2020dmv , the feed-down
contribution amounts $\sim$20% for deuterons and 50–100% (depending on the
rapidity) for 3He. While the feed-down contribution into the deuteron yield is
inessential for the data reproduction, it plays important role for 3He.
Without this feed-down the 3He yield is noticably underestimated.

The $v_{1}$ flow of deuterons, tritons and 3He turns out to be insensitive to
the feed-down contributions from unstable 4He. Without these contributions,
the corresponding $v_{2}$ flows are reduced by $\sim$20%, which, however, does
not essentially change the degree of their agreement with the data. The effect
on the proton yield and flow is negligible.

##  VI Summary

Simulations of the proton and light-nuclei production in Au+Au collisions at
$\sqrt{s_{NN}}=$ 3 GeV were performed within the updated THESEUS event
generator Kozhevnikova:2020bdb . The results were compared with recent STAR
data Liu:2021xfy ; Liu:2022ump ; STAR:2021ozh , some of them Liu:2021xfy ;
Liu:2022ump  are preliminary. The updated THESEUS treats the light-nuclei
production within the thermodynamical approach on the equal basis with
hadrons. The protons (as well as other hadron) are calculated with the
standard 3FD freeze-out, characterized by the energy density of
$\varepsilon_{\rm frz}=$ 0.4 GeV/fm3, followed by the UrQMD afterburner. The
only additional parameter related to the light nuclei is the energy density of
the late freeze-out that imitates the afterburner stage because the light
nuclei do not participate in the UrQMD evolution.

It was found that the late freeze-out characterized by $\varepsilon_{\rm
late\>frz}=$ 0.2 GeV/fm3 is preferable for deuterons, tritons, and 3He. This
is precisely the same value of $\varepsilon_{\rm late\>frz}$ that was found in
Ref. Kozhevnikova:2022wms  at higher collision energies. Remarkably, the 4He
yield and $p_{T}$ spectra are better reproduced with the standard 3FD freeze-
out. This suggests that the 4He nuclei better survive in the afterburner stage
because they are more spatially compact and tightly bound objects. This is an
argument in favor of dynamical treatment of light nuclei.

Results of simulations with different EoS’s (with and without transition to
the quark-gluon phase) indicated that the dynamics is determined by the
hadronic phase. The calculated results revealed not perfect, but a good
reproduction of the data on bulk observables of the light nuclei. The
calculated proton directed flow almost perfectly (except for very forward and
backward rapidities) reproduces the experimental flow STAR:2021yiu . Agreement
with the data on the directed flow STAR:2021ozh  becomes worse with increase
of atomic number of light nucleus. If the calculated midrapidity slope of the
directed flow is only slightly steeper than the experimental one for
deuterons, for 4He it is already noticeably steeper.

The model failed to properly describe the data on the elliptic flow of both
protons and light nuclei. We attribute this to shortcomings of the transition
from the 3FD evolution to the UrQMD afterburner, which prevents us from proper
description of the squeeze-out effect. The squeeze-out results from shadowing
of the expanding central fireball by spectators. This shadowing only partially
taken into account within the 3FD evolution because the frozen-out matter of
the central fireball remains to be shadowed even after the freeze-out while in
the 3FD model it escapes without interacting with spectators. The afterburning
stage should, in principle, correct this deficiency but it does not. 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 get frozen-out at different time instants in 3FD. The shadowing by
spectators is strongly reduced after such isochronous particlization because
the participants and spectators turn out to be well separated in thus
constructed pre-afterburner configuration.

We also studied the feed-down contributions from unstable 4He and possible in-
medium effects. As found, the feed-down contribution amounts $\sim$20% for
deuterons and 50–100% (depending on the rapidity) for tritium and 3He. While
the feed-down contribution into the deuteron yield is inessential for the data
reproduction, it plays important role for tritium and 3He. Without this feed-
down the tritium and 3He yields are noticably underestimated. The medium
effect accordingly to Ref. Ropke:2014fia  turned out to be too strong and
resulted in strong disagreement with data. Apparently this is because the
relativistic mean-field model Ropke:2012qv , underlying these medium
corrections, has been parametrized to reproduce low-energy nuclear phenomena.
This parametrization is simply inapplicable to highly excited nuclear matter.

Development of new hybrid model called MUFFIN (MUlti Fluid simulation for Fast
IoN collisions) was recently announced in Ref. Cimerman:2023hjw . This is a
next-generation hybrid three-fluid model for simulating heavy-ion collisions
at energies from few to few tens of GeV. Several methodical and conceptual
improvements, as compared with THESEUS, are introduced in MUFFIN. An important
conceptual improvement is the inclusion of initial state fluctuations, which
are important when considering the collective flow and allow study of
fluctuations associated with the CEP. The afterburner based on SMASH will make
it possible to describe the afterburner evolution of at least deuterons. This
could resolve some of the aforementioned problems in THESEUS.

###### Acknowledgements.

We are sincerely grateful to Iurii Karpenko and David Blaschke who made
enormous contributions early on in the implementation of this project.
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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