# Impacts of the direct URCA and Superfluidity inside a Neutron Star on Type-I
X-Ray Bursts and X-Ray Superbursts

A. Dohi (土肥明) Department of Physics, Kyushu University, Fukuoka 819-0395,
Japan  Interdisciplinary Theoretical and Mathematical Sciences Program
(iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan  N. Nishimura (西村信哉)
Astrophysical Big Bang Laboratory, Cluster for Pioneering Research, RIKEN,
Wako, Saitama 351-0198, Japan  Nishina Center for Accelerator-Based Science,
Wako, Saitama 351-0198, Japan  Division of Science, National Astronomical
Observatory of Japan, Mitaka 181-8588, Japan  H. Sotani (祖谷元) Astrophysical
Big Bang Laboratory, Cluster for Pioneering Research, RIKEN, Wako, Saitama
351-0198, Japan  Interdisciplinary Theoretical and Mathematical Sciences
Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan  T. Noda (野田常雄)
Department of Education and Creation Engineering, Kurume Institute of
Technology, Kurume, Fukuoka 830-0052, Japan  He-Lei Liu  School of Physical
Science and Technology, Xinjiang University, Urumqi 830046, China  S. Nagataki
(長瀧重博) Astrophysical Big Bang Laboratory, Cluster for Pioneering Research,
RIKEN, Wako, Saitama 351-0198, Japan  Interdisciplinary Theoretical and
Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan
M. Hashimoto (橋本正章)  Department of Physics, Kyushu University, Fukuoka
819-0395, Japan

(Received —; Revised —; Accepted —)

###### Abstract

We investigate the impacts of neutrino cooling mechanism inside the neutron
star (NS) core on the light curves of type-I X-ray bursts and X-ray
superbursts. From several observations of NS thermal evolution, physical
processes of fast neutrino cooling, such as the direct Urca (DU) process, are
indicated. They significantly decrease the surface temperature of NSs, though
the cooling effect could be suppressed by nucleon superfluidity. In the
present study, focusing on the DU process and nucleon superfluidity, we
investigate the effects of NS cooling on the X-ray bursts using a general-
relativistic stellar-evolution code. We find that the DU process leads find
the longer recurrence time and the higher peak luminosity, which could be
obstructed by the neutrons superfluidity. We also apply our burst models to
the comparison with Clocked burster GS 1826$-$24, and to the recurrence time
of superburst triggered by carbon ignition. These effects are significant
within a certain range of binary parameters and uncertainty of the NS equation
of state.

X-ray bursts(1814) — Neutron stars(1108) — Neutron star core (1107) — Low-mass
x-ray binary stars(939)

##  1 Introduction

Thermonuclear explosions triggered by hydrogen or helium (H/He) burning at the
mass accreted layer on the neutron star (NS) have been observed as type-I
X-ray bursts. The type-I X-ray burst is initiated from the triple-$\alpha$
reaction, occurring around $T=0.2~{}{\rm GK}$. At low accretion rates of
$\dot{M}\lesssim 10^{-9}~{}M_{\odot}~{}{\rm yr^{-1}}$ (Fujimoto et al., 1981),
the most of hydrogen is consumed and transformed to helium. Thus, pure-helium
burning occurs and results in the rapid increase to near the Eddington
luminosity. Since there are few protons in this case, the next reaction after
3$\alpha$ reaction is not proton capture but $\alpha$ capture of carbons. Due
to a series of ($\alpha,\gamma$) reactions, even-even nuclei up to
${}^{56}{\rm Ni}$ are synthesized and finally the burst phase is terminated.

At moderate accretion rates of $\dot{M}\gtrsim 10^{-9}~{}M_{\odot}~{}{\rm
yr^{-1}}$ which we consider in this paper, on the other hand, the hot CNO
cycle occurs after 3$\alpha$ reaction. The rapid temperature increase by the
nuclear burning results in the linear increase of the luminosity. When the
temperature exceeds $0.5~{}{\rm GK}$, the breakthrough of the hot CNO cycle
occurs and the luminosity reaches the maximum, typically $10^{38}~{}{\rm
erg~{}s^{-1}}$. Additionally, nucleosynthesis by the $\alpha p$ and $rp$
processes begin, where proton-rich heavy nuclei up to $A\sim 106$ (Schatz et
al., 2001) are synthesized. Then, the luminosity decreases exponentially with
$t\sim 100~{}{\rm s}$ and finally the burst phase is terminated.

The physics of X-ray bursts is complicated as they involve in not only the
nuclear burning but also high dense matter of NSs and X-ray binary parameters.
Modeling of observed burst light curves, therefore, is subject to probe the
underlying properties of NS equations of state (EOSs), the accretion rate, and
the composition of accreting matter. In addition, the uncertainty of nuclear
reaction rates especially for $\alpha p$ and $rp$ processes affects the
modeling of burst light curves (Hu et al., 2021; Lam et al., 2021a, b; Meisel
et al., 2021). The dependence of the above physical parameters on X-ray burst
light curves has been investigated by many theoretical works (Woosley et al.,
2004; Heger et al., 2007; Cyburt et al., 2016; Meisel, 2018; Meisel et al.,
2019; Johnston et al., 2020; Lam et al., 2021b).

Currently, 115 X-ray bursters have been observed (see, e.g., Galloway et al.,
2020). Although there are various light-curve profiles, some bursters show the
almost same pattern. They are called Clocked bursters and can provide a
standard case to constrain theoretical model. In particular, bursts from GS
1826$-$24 are the most commonly used as such references. Based on numerical
models covering a wide range of burst parameters, Johnston et al. (2020)
constrain burst models by the observations of GS 1826$-$24, e.g., $Z_{\rm
CNO}=0.01^{+0.005}_{-0.004}$, and $\dot{M}_{-9}=1.5$–$3.0$, where $Z_{\rm
CNO}$ is the initial metallicity and $\dot{M}_{-9}$ is the accretion rate
normalized by $10^{-9}~{}M_{\odot}~{}{\rm yr^{-1}}$.

Previous theoretical modelings of X-ray bursters, however, have mostly focused
on the accreted layers, ignoring detailed physics inside the NS. For such
models, the effects of the NS cooling and heating are imposed on $Q_{b}$, the
energy change though the crust surface, as the boundary condition. Although we
can easily investigate parameter dependence of the heating strength with
$Q_{b}$, realistic physics in the NS core are not fully considered. To
overcome this, we have employed the original general relativistic (GR)
evolutionary code (see, Fujimoto et al., 1984; Dohi et al., 2020). Based on
X-ray burst models with realistic EOSs, we have shown that the neutrino
emission from the NS significantly affects the light curves (Dohi et al.,
2021): the recurrence time ($\Delta t$) and peak luminosity ($L_{\rm peak}$).

In our previous study (Dohi et al., 2021), in order to focus on the EOS and NS
mass dependence on the light curve, we took into account only the slow cooling
scenario, i.e., the effects of the fast cooling and the nucleon superfluidity
are ignored. However, recent NS thermal evolution observations in particular
Cassiopeia A (Page et al., 2011; Shternin et al., 2011), indicate minimal
cooling scenario (Page et al., 2004; Gusakov et al., 2004), where we must
consider the combination of slow neutrino cooling processes and the pair
breaking and formation (PBF) process due to nucleon superfluid state. In
addition, some cold NSs observations require even faster cooling by e.g. the
nucleon Direct Urca (DU) process (e.g., see Section 1 in Dohi et al. (2022)).

On the other hand, the cooling effects on the burst behavior have been also
widely discussed in the context of superbursts, which is longer duration burst
(1000 s) than mixed H/He burst (10–100 s) (in’t Zand et al., 2003; Serino et
al., 2017; Iwakiri et al., 2021). Superburst is thought to be a thermonuclear
runaway initially caused by carbon unstable burning (Cumming et al., 2006).
Generally, since the depth of the ignition of carbons is deeper than that of
the H/He ignition, superburst should be affected by the neutrino losses at the
relatively deeper layer inside NSs compared with usual X-ray bursts (Brown,
2004; Cumming et al., 2006; Gupta et al., 2007; Deibel et al., 2016). In
particular, the impacts of neutrino cooling on the recurrence time of
superburst strongly depend on the thermal neutrino losses (e.g., Gupta et al.
(2007) for nucleon-pair bremsstrahlung and crust PBF process). Thus, the DU
process may also affect the depth of carbon ignition and the observed
recurrence time of superburst, although no previous studies examine the effect
of the DU process. Not only the neutrino cooling effects but also the reaction
rate uncertainties of carbon burning on the superburst behavior have been
discussed (e.g, Cooper et al. (2009)).

In this study, we investigate the impacts of neutrino cooling on X-ray bursts
and superburst within our multi-zone models with the entire NSs considered
(Dohi et al., 2020). In particular, we discuss the impacts of the DU process
and nucleon superfluidity on both the mixed H/He bursts and on the carbon
ignition.

This paper is organized as follows. In Section 2, we describe our numerical
methods with an emphasis on the neutrino cooling processes and the initial
models for X-ray burst calculations. In Section 3, we show the effect of
neutrino cooling on the light curves of Type-I X-ray bursts, focusing on GS
1826$-$24\. In section 4, we discuss the impacts of NS cooling on the carbon
ignition, which results in the X-ray superburst, considering uncertainty of
the reaction rate. We finally give a concluding remarks in Section 5.

Fig. 1: Critical temperature with different neutron ${}^{3}P_{2}$ superfluid
models is shown as a function of normalized baryon density with
$\rho_{0}=2.57\times 10^{14}~{}{\rm g~{}cm^{-3}}$, which is the nuclear
saturation density for LS220 EOS. We also draw dotted lines at the threshold
density of the DU process and the central density of $2.0~{}M_{\odot}$ stars.

##  2 Numerical Setup

###  2.1 Input Physics

To calculate burst light curves, we use the numerical code developed in our
previous work (Dohi et al., 2020). Our formulation is based on one-dimensional
GR hydrostatic equilibrium condition with 88-nuclei approximated reaction
network for mixed H/He burning. In this study we especially adopt the EOS
proposed by Lattimer & Swesty (1991) with the incompressibility of
$K=220~{}{\rm MeV}$ (LS220).

With the LS220 EOS, the DU process works for the NS models with $M_{\rm
NS}\gtrsim 1.35~{}M_{\odot}$ (e.g., Dohi et al. (2019)), and the expected
maximum mass exceeds the $2M_{\odot}$ observations (Demorest et al., 2010;
Antoniadis et al., 2013; Cromartie et al., 2020). To examine the effect of DU
process, we focus only on the NS model with $M_{\rm NS}=2.0~{}M_{\odot}$
($R_{\rm NS}=11.3~{}{\rm km}$). We note that the stellar model considered in
this study agrees with the theoretical constraints obtained from best-fit
models of GS 1826$-$24 (Johnston et al., 2020), i.e., $M_{\rm NS}\gtrsim
1.7~{}M_{\odot}$ and $R_{\rm NS}=11.3\pm 1.3~{}{\rm km}$, (but see also Zamfir
et al. (2012) as a counterexample). For the heating mechanism inside NSs, the
effective process is the crustal heating, which is the energy release due to
the non-equilibrium nuclear reactions of accreted matter such as the electron
capture, neutron emission, and pycnonuclear reactions. For their reaction
rates, we simply adopt the conventional energy generation rates proposed by
Haensel & Zdunik (2008).

Regarding the neutrino cooling processes, we consider the DU process as a fast
cooling process, together with the minimal cooling processes. Once the DU
process works, the internal temperature of NS drastically decreases. On the
other hand, when the DU process is forbidden, the dominant cooling processes
are modified URCA, bremsstrahlung, and the PBF process of nucleon
superfluidity (Yakovlev et al., 2001). For isolated NSs, the PBF process
significantly decreases their surface temperature within $t=10^{1-3}~{}{\rm
yrs}$ (Page et al., 2004), but this process does not affect the temperature
observation of old accreting NSs. The PBF process works only when the
temperature of the layer declines through the critical temperature $T_{\rm
cr}$. This situation hardly occurs in the accreting NSs, and such an
additional cooling process is not so effective (Potekhin et al., 2019).

Focusing on the DU process, the neutron ${}^{3}P_{2}$ superfluidity is known
to decrease the neutrino emissivity in low-temperature regions (Takatsuka,
1972). Therefore, we adopt three kinds of neutron ${}^{3}P_{2}$ superfluid
models, which are different density dependence in $T_{\rm cr}$, i.e., TTav,
TToa (Takatsuka & Tamagaki, 2004; Ho et al., 2015), AO (Amundsen & Østgaard,
1985), and AO3 for which we multiply the density in AO model by a factor of
three. That is, the maximum critical temperature in AO3 is the same as in AO,
but the “peak” density at the maximum critical temperature is different from
each other. We show their superfluid models in Fig. 1. In all regions above
the saturation density, AO and AO3 are the models with the highest critical
temperature, while TTav is the model with the lowest critical temperature
among four neutron ${}^{3}P_{2}$ models considered in this study. Meanwhile,
for ${}^{1}S_{0}$-state neutrons/protons superfluidity, we simply fix them as
Chen-Clark-Davé-Khodel (CCDK) model (Chen et al., 1993; Ho et al., 2015),
where the maximum critical temperature is $T_{\rm cr}\simeq 6\times
10^{9}~{}{\rm K}$ at $\rho=0.56\rho_{\rm 0}$ for neutrons and $1.1\rho_{\rm
0}$ for protons, respectively, where $\rho_{\rm 0}=2.57\times 10^{14}~{}{\rm
g~{}cm^{-3}}$ is the saturation density for LS220 EOS. It is well known that
the ${}^{1}S_{0}$-state neutrons superfluidity does not contribute to the
surface luminosity at all for $t\gtrsim 10^{2}~{}{\rm yrs}$ (e.g., Page et al.
(2009)). For the protons superfluidity, the DU process could be suppressed,
whose effect is generally weaker than the neutron ${}^{3}P_{2}$ superfluidity
as we show later. In Table 1 we summarize the name of models and the effects
taken into account in this study.

Table 1: Correspondence between the name of models and effects taken into account. Model | direct URCA | ${}^{1}S_{0}(n/p)$ | ${}^{3}P_{2}(n)$
---|---|---|---
Normal | $\checkmark$ | – | –
CCDK | $\checkmark$ | $\checkmark$ | –
CCDK+TTav | $\checkmark$ | $\checkmark$ | TTav
CCDK+TToa | $\checkmark$ | $\checkmark$ | TToa
CCDK+AO | $\checkmark$ | $\checkmark$ | AO
CCDK+AO3 | $\checkmark$ | $\checkmark$ | AO($\rho_{\rm B}\times 3$)
No DU | – | – | –
Fig. 2: Redshifted temperature structure in steady state with
$\dot{M}_{-9}=2.0$. Normal(red) indicates the model without any superfluidity
and CCDK(magenta) without only neutron ${}^{3}P_{2}$ superfluidity. We note
that the result with Normal almost overlaps with that with CCDK. Other models
with all kinds of superfluidity are as follows: TTav(yellow), TToa(green), and
AO(blue) for neutron ${}^{3}P_{2}$ superfluid models, respectively. We draw
two vertical lines at the envelop-crust and crust-core boundaries, whose
densities respectively correspond to $10^{9}~{}{\rm g~{}cm^{-3}}$ and the
saturation density $\rho_{0}$. In addition, we also show the result without
the direct URCA labeled by No DU for reference.

###  2.2 Initial Thermal Profiles of Accreting NSs

To prepare an initial thermal profiles, we calculate thermal evolution of
accreting NSs without nuclear burning, adopting the same procedure as in Liu
et al. (2021a), but we also consider the homogeneous compressional heating,
not only the non-homogeneous one as explained by Matsuo et al. (2018). The
resultant temperature structure in steady state is considered as the initial
thermal profile for calculating the burst light curves. We show such a
temperature structure in steady state for various models in Fig. 2, where we
assume that $\dot{M}_{-9}=2.0$. Comparing to Fig. 1, one can recognize that
the steady-state NS maintains higher temperature, as the critical temperature
or the peak density are higher for superfluid models. That is, the neutron
${}^{3}P_{2}$ model with higher critical temperature corresponds to relatively
lower neutrino emissivity. This situation is understood as follows, i.e., once
the temperature decreases and reaches the critical temperature, the DU
process, which is fast cooling process, is effectively blocked by the pair
production due to neutrons superfluidity, while the PBF process as an effect
of superfulidity is less effective than the DU process.

Fig. 3: The bolometric luminosity in units of $10^{38}$ erg s-1 for the burst
sequence occurred during 40 hr for various superfluid models. The meaning of
symbol is the same as in Fig. 2. For AO(purple) and AO3(green), we show their
light curves in the same panel. Here, we adopt $\dot{M}_{-9}=2.0$.

Fig. 4: The averaged light curves of the burst phase within $1\sigma$ regions.
In the insets, only the most probable light curves are shown for the sake of
clarity. We adopt $\dot{M}_{-9}=2.0$ (left) and $2.5$ (right). Dashed curves
denote the averaged light curves while solid bands include the 1$\sigma$
errors in a numerous bursts.

##  3 Results for type-I X-ray bursts

In this section, first we discuss the overall of the model dependence of the
X-ray burst light curves. We also discuss the relation between the recurrence
time and ignition pressure. Then, we compare our results to the concrete
observational data in GS 1826$-$24, where we show the dependence of the
accretion rate.

###  3.1 Model dependence of the light curves

We calculate X-ray burst light curves by turning on the nuclear burning which
is triggered by mixed H/He burning, adopting the steady-state temperature
profile as the initial model for X-ray burst calculation. To eliminate the
initial model dependence of burst behavior, i.e., compositional inertia
(Woosley et al., 2004), we discard several dozens of burst profiles within
$t\lesssim 2\times 10^{5}~{}{\rm s}$ for all burst models. Then, for the
analysis of burst profiles, we select at least 15 successive burst profiles.

In Fig. 3, we show the superfluid model dependence in long-term light curves,
where the first bursts in the selected successive burst profiles for various
models are aligned as $t=0$. One observes that the interval between bursts,
i.e., the recurrence time $\Delta t$, becomes shorter, as the critical
temperature or the peak density in neutron ${{}^{3}P_{2}}$ superfluidity
becomes higher. We note that this result is significantly different from the
models without DU process (No DU). We also find that the behavior of
CCDK+TTav, CCDK and Normal are similar. The recurrence time of other models
with strong neutron ${}^{3}P_{2}$ superfluidity (TToa, AO, and AO3) are a
little shorter (see Table 2).

Table 2: Averaged recurrence time and ignition pressure with $\dot{M}_{-9}=2.0$ for various models. Model |  $\Delta t$ (hr) |  $P_{\rm ign}$ (${\rm dyn~{}cm}^{-2}$)
---|---|---
Normal | 3.90 | 4.60$\times 10^{22}$
CCDK | 3.90 | 4.60$\times 10^{22}$
CCDK+TTav | 3.89 | 4.59$\times 10^{22}$
CCDK+TToa | 3.83 | 4.53$\times 10^{22}$
CCDK+AO | 3.81 | 4.50$\times 10^{22}$
CCDK+AO3 | 3.76 | 4.43$\times 10^{22}$
No DU | 3.54 | 4.18$\times 10^{22}$

Next, to see the model dependence of the peak luminosity, the averaged burst
light curves are shown in Fig. 4, where the left and right panels correspond
to the results with $\dot{M}_{-9}=2.0$ and 2.5, respectively. From this
figure, we can classify the models considered in this study into three
families, i.e., (i) the lower luminosity (No DU), (ii) the higher luminosity
(Normal, CCDK, and CCDK+TTav), and (iii) the moderate luminosity (CCDK+TToa,
CCDK+AO and CCDK+AO3). Since Normal is quite similar to CCDK with respect to
the recurrence time and peak luminosity, we see that the ${}^{1}S_{0}$
superfluidity makes little impact on the burst behavior. We additionally see
that CCDK+TTav (TTav is a weak neutron ${}^{3}P_{2}$ superfluid model) is also
similar to (but a little smaller than) Normal in the burst peak luminosity,
while the peak luminosity with CCDK+TToa is similar to that with CCDK+AO model
and those are lower than that with Normal model. Furthermore, if the
appearance density of superfluidity is higher, the recurrence time and peak
luminosity become a little shorter as we compare with AO and AO3. Thus, we
find that the neutron ${}^{3}P_{2}$ superfluidity lowers the recurrence time
and peak luminosity through the suppression of the DU process, although the
range of reduction strongly depends on the neutron ${}^{3}P_{2}$ superfluidity
model.

###  3.2 Empirical relation between $\Delta t$ and the ignition pressure

For considering the burst behavior, it is important that the neutrino cooling
process inside the NS decreases the temperature near the typical ignition
pressure $P_{\rm ign}\simeq 10^{22-23}~{}{\rm dyn~{}cm^{-2}}$ (Dohi et al.,
2021) 111We note that the typical pressure for the H/He ignition corresponds
to the position, where the temperature becomes $\simeq 0.2$ GK
(log${}_{10}(T/{\rm[K]})\simeq 8.3$).. This tendency can be seen even in the
case with the DU process, as shown in Fig 5, but the temperature around
ignition pressure is changed in order of $\lesssim 0.2$, depending on the
superfluid models. As a result, the position where the temperature becomes
$\simeq 0.2$ GK slightly shifts to inner part. Due to the DU process, $P_{\rm
ign}$ is increased and indirectly enhance the burst behavior (or the peak
luminosity), because the column density $\sigma$ is estimated with the surface
gravity $g_{s}$ via $\sigma=P_{\rm ign}/g_{s}$, according to the shell-flash
model (Fujimoto et al., 1981). We note that the difference in the burst
behavior in our models comes from only the difference in $P_{\rm ign}$,
because the NS mass and radius are fixed in this study.

Fig. 5: Redshifted temperature structure without and with the DU process. The
dotted curves denote those just before the H/He ignition and the solid curves
denote just after the H/He ignition, respectively, where we adopt
$\dot{M}_{-9}=2.0$.

In Table 2, we list the values of ignition pressure at the time just after the
ignitions of hydrogen and helium. As we see, the ignition pressure is
maximally changed in order of $\lesssim 0.04$. According with the shell-flush
model and assuming the constant accretion rate, the column density $\sigma$ is
expressed in two ways (see also Gupta et al. (2007)):

$\displaystyle\sigma=\frac{\left(10^{-9}~{}M_{\odot}~{}{\rm
yr}^{-1}\right)\dot{M}_{-9}}{4\pi R_{\rm NS}^{2}}\Delta t=P_{\rm ign}/g_{\rm
s}~{},$ (1)

where $g_{\rm s}$ is the surface gravity expressed as

$\displaystyle g_{\rm s}=\frac{GM_{\rm NS}}{R_{\rm
NS}^{2}}\left(1-\frac{2GM_{\rm NS}}{R_{\rm NS}c^{2}}\right)^{-1/2}~{}.$ (2)

In this equation, $c$ is the speed of light and $G$ is the gravitational
constant. With the NS model considered in this study, $g_{\rm s}=3.01\times
10^{14}~{}{\rm cm~{}s^{-2}}$. Thus, if the NS mass, radius, and accretion rate
are fixed, $\Delta t$ should be proportional to $P_{\rm ign}$. In fact, one
can see that $P_{\rm ign}/\Delta t$ does not depend on the models considered
in this study, i.e., $P_{\rm ign}/\Delta t=1.2\times 10^{22}$ (dyn cm-2/hr),
using the values listed in Table 2. That is, one can discuss the recurrence
time of X-ray burst via Eq. (1) even with the effect of superfulidity.

Fig. 6: Recurrence time (top) and peak luminosity in units of $10^{38}~{}{\rm
erg~{}s^{-1}}$ (bottom) are shown as a function of accretion rate normalized
by $10^{-9}~{}M_{\odot}~{}{\rm yr}^{-1}$. The dotted line indicate the data of
GS 1826$-$24 observed in 2007, assuming the typical value of
$d\sqrt{\xi}_{b}=6.1~{}{\rm kpc}$ (Galloway et al., 2017), where $d$ is the
distance and $\xi_{b}$ is the anisotropy of burst flux.

###  3.3 Dependence on the accretion rate and implication for GS 1826$-$24

In order to see the dependence of the burst behavior on the accretion rate, in
Fig. 6, we show the recurrence time $\Delta t$ and peak luminosity $L_{\rm
pk}$ as a function of accretion rate. From the top panel of this figure, one
can observe that $\Delta t$ with the DU process is significantly different
from that without the DU process, where the deviation becomes at most $\sim
0.5$ hr. As we have already discussed with $\dot{M}_{-9}=2.0$, if the effect
of the superfluidity is taken into account, $\Delta t$ becomes slightly lower.
On the other hand, the peak luminosity seems to strongly depend on the effect
of the superfluidity, compared with the dependence in $\Delta t$.

Now, we compare these results to the data of GS 1826$-$24 observed in 2007. In
Fig. 6 we plot the observational data with the dotted lines. Unfortunately the
mass accretion rate in GS 1826$-$24 is still uncertain, but one can
distinguish the models if the mass accretion rate will be constrained somehow.
For example, if $\dot{M}_{-9}\simeq 2.0$, one can say that the DU process is
unfavorable for explaining the observed $\Delta t$ of GS 1826$-$24, at least
within the models we considered in this study. From this comparison, we find
that the DU process seems to be unfavorable for explaining the observed
$\Delta t$ of GS 1826$-$24, at least within the models we considered in this
study. We also compare the peak luminosities with the obsetvation of GS
1826$-$24 in 2007. For the observed peak luminosity, we take it from observed
peak flux in 2007 ($f_{\rm peak}=\left(2.76\pm 0.092\right)\times{\rm
10^{-8}~{}{\rm erg~{}cm^{-2}~{}s^{-1}}}$) and the typical observational value
of $d\sqrt{\xi_{b}}=6.1$ kpc, where $d$ is the distance and $\xi_{b}$ is the
burst anisotropy (Galloway et al., 2017). In this case, the models with DU
process can not explain the peak luminosity observed in GS 1826$-$24, which is
the same result as the constraints from $\Delta t$. Although the theoretical
results highly depends on the distance and other input parameters (e.g.,
Johnston et al. (2020)), we could thus extract the information of neutrino
cooling processes including nucleon superfluidity through the burst
observations through systematical examinations.

Light-curve fitting with the observations of Clocked bursters is also good
tool to constrain the NS core properties from X-ray burst observations but
this is very hard at present (e.g., Dohi et al. (2021)). The most uncertain
parameter is the accretion rate, which is in principle difficult to be
measured by observations. The uncertainties of reaction rate above all for the
$\alpha p$ and $rp$ processes also hasten the difficulty of constraints from
burst observations, while $\Delta t$ and $L_{{\rm pk},38}$ focused in this
study are not sensitive to heavy-element nucleosynthesis. As experiments to
measure the reaction rates of unstable $p-$nuclei are developed, the light-
curve fitting becomes more useful tool to constrain the NS properties and
accretion rate. Then, the light-curve fitting to probe the structure and
temperature of NS core will be presented elsewhere.

Fig. 7: Temperature structure just before the H/He ignition with (red) and
without (black) the DU process. Solid curves indicate $\dot{M}_{-9}=8$ while
dotted curves $\dot{M}_{-9}=4$. We ignore the effect of any kind of
superfluidity. We also plot three kinds of carbon ignition curves with the
carbon mass fraction $X({}^{12}{\rm C})=0.2$: CF88 (Caughlan & Fowler, 1988),
T18 (Tumino et al., 2018), and TK21 (Taniguchi & Kimura, 2021).

##  4 The ignition and recurrence time for superbursts

In this section, we investigate the effect of the DU process on the depth of
carbon ignition and the recurrence time of superburst based on Eq. (1). As the
temperature structure inside the NS just before the carbon ignition, we choose
that before the H/He ignition ($t\simeq 10^{6}~{}{\rm s}$), which is the same
as the temperature structure denoted with the dotted lines in Fig. 5. We note
that the temperature profile itself is not so changed regardless of the mass
fraction of carbons unless the nuclear burning works. We choose high accretion
rate of $\dot{M}_{-9}=4$ and 8, because the carbon burning should occur in hot
NS layers $\approx 0.6$ GK. We simply consider two extreme cases without and
with the DU process, where any kind of superfluidity is neglected. Burst
models are made in the same procedure as mentioned in Section 2.

The non-resonant experimental rate is often used as the standard reaction rate
of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ (Caughlan & Fowler (1988), hereafter
CF88), while, above the threshold energy of
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, many molecular resonances are
experimentally predicted from the indirect measurement of $\alpha$-inelastic
cross section (Kawabata et al., 2013). The effect of resonances tends to
enhance the reaction rates from theoretical models (e.g., Cooper et al. (2009)
for resonance assumed at $E_{R}=1.5$ MeV). Actually, recent measurements of
cross sections with use of the Trojan Horse Method showed some resonances in
low energies, which increases the reaction rate of
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ by $\gtrsim 25$ times at $T=$0.5 GK
compared with CF88 (Tumino et al. (2018), hereafter T18). The impact of the
new reaction rate on astrophysical phenomena has been investigated, such as
the type Ia supernovae (Mori et al., 2019) and evolution of massive stars
(Chieffi et al., 2021). Furthermore, the latest reaction rate has been
recently developed based on a full microscopic nuclear model to describe low-
energy resonances by handling the channel coupling and the rotation of nuclei
without any adjustable parameter (Taniguchi & Kimura (2021), hereafter TK21).
We adopt the three kinds of reaction rates of
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, i.e., CF88, T18, and TK21, in this
study.

To obtain the ignition curve for the reaction of
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, we employ the shell-flush model
(Fujimoto et al., 1981; Koike et al., 1999, 2004) with Helmholtz EOS (Timmes &
Swesty, 2000). For the compositions, we assume the carbon-iron plasma with the
carbon mass fraction of $X({}^{12}{\rm C})=0.2$, which is the typical mass
fraction to fit the observed superburst light curves with hydrogen accretion
(Cumming, 2003; Cumming et al., 2006) (but see also Cumming & Bildsten
(2001)).

Table 3: Ignition pressure of carbon burning $P_{\mathrm{C}}$ and the recurrence time of superburst $\Delta t_{\rm sb}$. Note that the model with $\dot{M}_{-9}=4$ and DU process does not cross the ignition curves as seen in Fig. 7 irrespective of the reaction rates of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. $\dot{M}_{-9}$ | Model | Rate |  $P_{\mathrm{C}}$ (${\rm dyn~{}cm}^{-2}$) | $\Delta t_{\rm sb}~{}({\rm yr})$
---|---|---|---|---
8 | Normal | CF88 | 3.49$\times 10^{24}$ | 1.17$\times 10^{-2}$
|  | T18 | 1.06$\times 10^{24}$ | 3.57$\times 10^{-3}$
|  | TK21 | 4.79$\times 10^{24}$ | 1.60$\times 10^{-2}$
8 | No DU | CF88 | 9.74$\times 10^{23}$ | 3.26$\times 10^{-3}$
|  | T18 | 4.87$\times 10^{23}$ | 1.63$\times 10^{-3}$
|  | TK21 | 1.04$\times 10^{24}$ | 3.48$\times 10^{-3}$
4 | No DU | CF88 | 6.88$\times 10^{26}$ | 4.61
|  | T18 | 1.71$\times 10^{25}$ | 0.115
|  | TK21 | 9.31$\times 10^{26}$ | 6.24

In Fig. 7, we show the temperature structure just before the H/He ignition,
together with the carbon ignition curves. With this figure, one can determine
the ignition pressure $P_{\mathrm{C}}$ as the intersection between the thermal
structure and the carbon ignition curves. The obtained values of
$P_{\mathrm{C}}$ and the recurrence time of superburst $\Delta t_{\rm sb}$ are
shown in Table 3, where $\Delta t_{\rm sb}$ is estimated from Eq. (1) together
with $P_{\mathrm{C}}$. Focusing on the superburst with $\dot{M}_{-9}=8$, we
can find that the DU process makes $P_{\mathrm{C}}$ (and also $\Delta t_{\rm
sb}$) larger by a factor of $\sim$2\. For the case with $\dot{M}_{-9}=4$,
since the DU process indirectly decreases the temperature for weaker crustal
heating than in $\dot{M}_{-9}=8$, there is no cross point between the thermal
structure and ignition curves with any type of reaction rate. This predicts
that the carbon burning does not occur for this model (and $X({}^{12}{\rm
C})=0.2$). Hence, if the DU process occurs in superbursters, the accretion
rate as the heat source inside NSs is preferred to be high because of the
necessity of hot regime enough to cause the carbon burning.

Next, we focus on the uncertainties of reaction rates of
${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. As we see Table 3, $\Delta t_{\rm sb}$
between CF88 and TK21 are not so changed even through the energy dependence of
astrophysical $S$ factor is highly different between them (see Fig. 2 in
Taniguchi & Kimura (2021)). Meanwhile, T18 is higher around twice than others
because the astrophysical $S$ factor of T18 is roughly higher than others by
an order of magnitude. Hence, the uncertainties of reaction rates are similar
with those of the DU process for $\dot{M}_{-9}=8$.

We should note that our estimation of $\Delta t_{\rm sb}$ for $\dot{M}_{-9}=8$
is three orders of magnitude smaller than the observations of superburst such
as the 4U 1820$-$30 ($\Delta t_{\rm sb}=5$–10 yr in Cumming (2003)). This is
due to the difference of ignition column density $P_{\mathrm{C}}/g_{s}$
(Cumming et al., 2006). For $\dot{M}_{-9}=4$ without DU process, CF88 and TK21
mostly match with the observations of 4U 1820$-$30, but for T18, $\Delta
t_{\rm sb}$ is lower by 1–2 orders of magnitude. Considering also the
necessity of hot regime implying high $\dot{M}_{-9}$, although the
uncertainties of crustal heating are crucial (e.g., Shchechilin et al. (2021,
2022)), T18 seems to be unpreferred.

Many previous studies of thermal evolution of accreting NSs consider the
shallow heating which is necessary for explaining some hot accreting NSs
(Brown & Cumming, 2009; Deibel et al., 2015; Waterhouse et al., 2016; Ootes et
al., 2016, 2018), though their physical mechanism has been still unknown and
some candidates have been investigated (e.g., Fattoyev et al. (2018); Liu et
al. (2021b) for neutrino heating scenario due to charged pion decay). We turn
off the shallow heating in this study, but depending on its depth and
strength, such a heating could have a significant impact on the carbon
ignition depth. In fact, Meisel (2022) investigates this and tries to
constrain the properties of shallow heating from recurrence time in some
superburst observations (see Figure 4 in his paper). Probing the origin of the
shallow heat source as well as the nature of dense matter in NSs are therefore
valuable and left for our future work.

After submitting the first manuscript, we became aware of similar work of
Meisel (2022) in terms of constraints on energy sources in accreting NSs from
the inferred depth of carbon ignition in X-ray superbursts. He also adopts T18
and TK21 as two of ${}^{12}{\rm C}+{}^{12}{\rm C}$ reaction rates as we do,
and his results are in good agreement with ours. He also examines the
uncertainties of neutrino Urca cooling in the crust (Schatz et al., 2014),
which does not impact the carbon ignition depth of superburst, unless the
densities where the heating and cooling sources greatly work are close each
other. We do not consider the neutrino Urca cooling in the crust, but this
seems not to be a problem at least in moderate accretion rates considered here
because the neutrino cooling above all for the DU process is clearly dominant
for the core temperature, while the crustal heating (and possibly shallow
heating) for the crust temperature.

##  5 Concluding Remarks

We investigate the neutrino cooling effect on the burst light curves with
emphasis on the DU process and nucleon superfluidity. We show that the DU
process makes the recurrence time $\Delta t$ longer and the peak luminosity
$L_{\rm pk}$ higher, but this is suppressed by neutrons superfluidity if the
critical temperature is relatively high. In our burst models, both of them are
changed by maximally $\lesssim 20\%$. In comparison with the observations of
GS 1826$-$24, we present the possibility to probe the occurrence of the DU
process and the strength of neutron superfluidty.

We also discuss the recurrence time of superburst estimated with the ignition
pressure of carbon burning, considering the DU process and various reaction
rates of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$. We show that if the DU process
occurs, it takes much time to cause the carbon burning or possibly carbon is
not ignited. Hence, the observation of superburst could probe the neutrino
cooling process because the carbon ignition occurs in deep layer, where the DU
process affects compared with the case of H/He burning. We also show that the
recent experimental reaction rate of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$
(T18) makes $\Delta t$ lower by 1–2 orders of magnitude and becomes
inconsistent with the observation in 4U 1820$-$30\. For conventional reaction
rate (CF88) and the latest theoretical one (TK21), we find that these
difference appears to be small in ignition curves.

Specifying the neutrino cooling processes including nucleon superfluidity from
burst observations is still difficult because there are many susceptible input
parameters. Hence, we might need to search other possible observations. One of
candidates is the gravitational wave with the gravity mode ($g$-mode), which
could be trapped in radiative layers in NSs. Even if the cooling (and heating)
processes inside NSs are not reflected on the observed luminosity, since they
highly change the temperature structure, we could specify the cooling and
heating processes through the signature of $g$-mode frequency. In fact, such
studies have been done in isolated NSs cooling (Krüger et al., 2015; Sotani &
Dohi, 2022). We leave an investigation of $g$-mode frequency in accreting NSs
for a future study.

Recently, a very long duration of bursts compared with the superburst
discovered in MAXI J0556$-$332 has been proposed to be triggered by heavier
elements than ${}^{\rm 12}{\rm C}$ (hyperburst, Page et al. (2022)). They
suggest that the hyperburst is triggered by unstable burning of neutron rich
isotopes of oxygen or neon around the high density of $\rho\approx
10^{11}~{}{\rm g~{}cm^{-3}}$. Thus, the heating and cooling processes inside
NSs including the DU process should effectively work and finally change the
ignition conditions of triggers for the hyperburst. As with the superburst
discussed in this paper, the observations of future very long duration bursts
would be useful for specifying heating and cooling processes, were the
scenario of hyperburst correct.

## Acknowledgments

The authors thank Y. Taniguchi and M. Kimura for discussion on the
${}^{12}{\rm C}+{}^{12}{\rm C}$ reaction rate and for providing the data. This
project was financially supported by JSPS KAKENHI (19H00693, 19KK0354,
20H05648, 20H04753, 21H01087, 21H01088) and a RIKEN pioneering project
“Evolution of Matter in the Universe (r-EMU)”. N.N. was supported by the
Incentive Research Project at RIKEN. H.L. was supported financially by the
National Natural Science Foundation of China under No. 11803026 and Xinjiang
Natural Science Foundation under No. 2020D01C063. T.N. acknowledges the
support from the Discretionary Budget of the President of Kurume Institute of
Technology. Parts of the computations in this study were carried out on
computer facilities at CfCA in National Astronomical Observatory of Japan.

##  Appendix A On the cooling effect on burst parameter $\alpha$

As with the recurrence time $\Delta t$ and peak luminosity $L_{\rm peak}$, the
burst parameter $\alpha$, which is the ratio of the accretion energy in a
cycle of burst to the total burst energy $E_{\rm burst}$, is also powerful for
constraining burst models. It can be calculated by

$\displaystyle\alpha=\frac{z_{g}}{1+z_{g}}\left(10^{-9}~{}M_{\odot}~{}{\rm
yr}^{-1}\right)\dot{M}_{-9}c^{2}\frac{\Delta t}{E_{\rm burst}},~{}$ (A1)

where $z_{g}$ is the gravitational redshift (Galloway et al., 2017). In Fig. 8
we show the neutrino cooling effect on $E_{\rm burst}$ and $\alpha$ as a
function of the accretion rate. As we see, $\alpha$ value is almost the same
regardless of neutrino cooling models. This is because the tendencies of
cooling effect on $E_{\rm burst}$ and $\Delta t$ are almost the same. Thus, we
cannot extract the information of neutrino cooling from $\alpha$. This feature
is consistent with our previous work (Dohi et al., 2021), in which $\alpha$
has a positive correlation with surface gravity with high accuracy and
therefore is more powerful for constraining NS structure than other output
values such as $\Delta t$ and $L_{\rm pk}$.

Fig. 8: Same as Fig. 6, but for total burst energy (left) and burst parameter
$\alpha$ (right).

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††thanks:
Presently at Department of Physics, Panchayat College, Sambalpur University,
Bargarh 768028, Odisha, India

# Energy dependence of particle production in Au + Au collisions
at $\sqrt{s_{\text{NN}}}$ = 7.7 - 200 GeV using a multiphase transport model

Krishan Gopal<EMAIL_ADDRESS>Chitrasen Jena
<EMAIL_ADDRESS>Kishora Nayak<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Science Education and Research (IISER) Tirupati,
Tirupati 517507, Andhra Pradesh, India

###### Abstract

In this study, we employ a multi-phase transport (AMPT) model to understand
the production of $\pi^{\pm}$, $K^{\pm}$, $p$, $\overline{p}$, $K^{0}_{s}$,
$\Lambda$, $\bar{\Lambda}$, and $\phi$ in Au + Au collisions at
$\sqrt{s_{NN}}=7.7$, $27$, $39$, $62.4$, and $200$ GeV. We have studied the
energy dependence of various bulk properties of the system such as transverse
momentum ($p_{T}$) spectra, particle yields ($dN/dy$), mean transverse mass
($\langle m_{T}\rangle$), and anti-particle to particle ratios. Model
calculations using both default and string melting versions of the AMPT with
three distinct sets of initial conditions are compared to the data from the
STAR experiment. In the case of $\pi^{\pm}$, $K^{\pm}$, $p$, and
$\overline{p}$, we observe that the string melting version shows better
agreement with data at higher energies, while the default version performs
better at lower collision energies. However, for $K^{0}_{s}$, $\Lambda$, and
$\phi$, it is observed that the default version is able to describe the data
better at all energies. In addition, we have used the blast-wave model to
extract the kinetic freeze-out properties, like the kinetic freeze-out
temperature and the radial flow velocity. We observe that these parameters are
comparable with the data.

keywords

AMPT Model, particle spectra, bulk properties, kinetic freeze-out properties.

††preprint: APS/123-QED

##  I Introduction

Heavy ion collision experiments provide an opportunity to probe the behaviour
of nuclear matter under extreme conditions of energy density and temperature.
Quantum Chromodynamics (QCD), the theory governing the strong nuclear force,
predicts that under the extreme energy densities achieved in heavy-ion
collisions, the hadronic matter transitions to a deconfined state of quarks
and gluons, known as Quark-Gluon Plasma (QGP) [1, 2, 3, 4, 5]. Experimental
facilities such as the Relativistic Heavy Ion Collider (RHIC) at the
Brookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC) at
the European Organization for Nuclear Research (CERN), aim to study various
properties of the QGP. One central goal of these experiments is to map out the
QCD phase diagram, typically plotted as temperature ($T$) versus baryon
chemical potential ($\mu_{B}$). In that regard, RHIC initiated the Beam Energy
Scan (BES) program in 2010, collecting the data in Au+Au collisions at
$\sqrt{s_{NN}}$ = 7.7 – 200 GeV. BES program spanned a wide range of $\mu_{B}$
from 20 to 420 MeV [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Lattice QCD
calculations predict that the phase transition from the hadronic matter to QGP
occurs somewhere in this range of $\mu_{B}$ [17].

In this study, the bulk properties of the system in Au+Au collisions at
$\sqrt{s_{NN}}$ = 7.7, 27, 39, 62.4, and 200 GeV are studied using a multi-
phase transport (AMPT) model. We employ three distinct sets of input
parameters for both the string melting (AMPT-SM) and the default (AMPT-Def)
versions of the AMPT [18]. Bulk properties, such as transverse momentum
($p_{T}$) spectra, integrated yields ($dN/dy$), mean transverse mass ($\langle
m_{T}\rangle$), and particle ratios at various collision energies are studied.
We also study the kinetic freeze-out (KFO) parameters obtained by fitting the
transverse momentum spectra using a hydrodynamically inspired blast-wave model
[19]. KFO refers to the surface of last scattering of hadrons produced in
heavy-ion collisions, effectively defining the surface of their final momentum
distribution. Freeze-out properties provide information on the evolution of
the medium produced in heavy-ion collisions.

The paper is organised as follows: Section II provides a brief overview of the
AMPT model. In Sec. III, we discuss the comparison of AMPT model results with
the experimental data. Section III.1 compares the $p_{T}$ spectra of various
particles obtained from the AMPT model with experimental data. Section III.2
presents the energy dependence of $dN/dy$, $\langle m_{T}\rangle$, and
particle ratios at various collision energies. In Sec. III.3, we show the
dependence of freeze-out parameters on the collision centrality and energy.
Finally, Section IV summarises our findings.

##  II AMPT Model

AMPT is a hybrid monte carlo event generator that allows the study of the
dynamics of relativistic heavy-ion collisions. It has been widely used to
analyse various collision systems at different centre-of-mass energies [20,
21, 22]. AMPT model generates fluctuating initial conditions using the Heavy
Ion Jet INteraction Generator (HIJING) model, which involves spatial and
momentum distributions of minijet partons and soft string excitations [23].
Evolution of partonic interactions is modelled using the Zhang’s Parton
Cascade (ZPC) [24]. The differential cross-section of parton-parton scattering
can be expressed as:

$\frac{d\sigma}{dt}\approx\frac{9\pi\alpha_{s}^{2}}{2(t-\mu^{2})^{2}},$ (1)

where $\sigma$ is cross section of partonic two body scattering, $t$ is the
standard Mandelstam variable for four-momentum transfer, $\alpha_{s}$ is the
strong coupling constant, and $\mu$ is the Debye screening mass in the
partonic matter. The $\mu$ is influenced by the medium effects, in turn
affecting $\sigma$. A medium dependent $\mu$ would produce results that are
closer to the experimental data. In addition, it has been observed that
particle multiplicity in a collision is not very sensitive to $\sigma$, but it
might affect the elliptic flow such that a larger $\sigma$ leads to larger
elliptic flow [25].

The two versions of AMPT, Default and String Melting, differ in how they treat
the formation of hadrons. In the AMPT-Def version, the formation of hadrons
from quarks and antiquarks is governed by Lund String model employing a
symmetric fragmentation function:

$f(z)\propto z^{-1}(1-z)^{a}\exp(-bm^{2}_{T}/z),$ (2)

where $z$ is the light-cone momentum fraction of the produced hadrons relative
to the fragmenting string, while $a$ and $b$ are called Lund string
fragmentation parameters [26]. The average squared transverse momentum,
$\langle p^{2}_{T}\rangle$, of the produced particles is proportional to the
string tension $\kappa$, which represents the energy stored per unit length of
a string:

$\langle p_{T}^{2}\rangle\propto\kappa=\frac{1}{b(2+a)}.$ (3)

According to Eq. (3), the parameters $a$ and $b$ determine the $p_{T}$
distribution of particles produced in heavy-ion collisions. A large value of
$a$ and/or $b$ will lead to a small $\langle p_{T}^{2}\rangle$, resulting into
a sharp $p_{T}$ spectrum. However, a smaller value of $a$ and $b$ will produce
a flat $p_{T}$ distribution [21].

The AMPT-SM version, on the other hand, is based on the concept that beyond a
certain critical energy density ($\sim$ 1 GeV/c), the coexistence of colour
strings and partons becomes unfavourable. Consequently, these strings “melt”
into low momentum partons at the start of ZPC. The transported partons finally
undergo hadronization by spatial quark coalescence mechanism [27].

|  $a$ |  $b$($GeV^{-2}$) |  $\alpha_{s}$ |  $\mu(fm^{-1})$ |  $\sigma$ (mb)
---|---|---|---|---|---
Set 1 |  0.55 |  0.15 |  0.33 |  2.265 |  3
Set 2 |  2.2 |  0.15 |  0.33 |  2.265 |  3
Set 3 |  0.5 |  0.9 |  0.33 |  3.2 |  1.5
Table 1: Different sets of input parameters for Lund string fragmentation and
parton scattering cross sections used in this study.

The subsequent interaction of hadronic matter is characterised by a hadronic
cascade, which is governed by a relativistic transport (ART) model [28].

Table 1 lists various choices of input parameters of the AMPT model we have
used in this work to study particle production in heavy-ion collisions. The
different choices of these parameters were made by insights derived from
previous studies [20, 21, 22]. We have performed a systematic study of
particle $p_{T}$ spectra using various set of input parameters as mentioned
below,

  * •

Set-1 and Set-2 differ in the $a$ parameter while keeping $b$ and $\sigma$ are
same. A larger value of $a$ is expected to yield a sharp $p_{T}$ spectrum.

  * •

In Set-1 and Set-3, all three parameters $a$, $b$, and $\sigma$ are different.
We have decreased all these three values to observe their effect on the bulk
properties.

We aim to find the best set of input parameters that properly describe the
experimentally measured $p_{T}$ spectra. In the following section experimental
results are compared with the results from different configurations of the
AMPT model.

##  III Results

We report the mid-rapidity $p_{T}$ spectra, $dN/dy$, $\langle m_{T}\rangle$ of
$\pi^{\pm}$, $K^{\pm}$, $p$, $\overline{p}$, $K^{0}_{S}$, $\Lambda$, and
$\phi$, along with particle ratios in most central Au+Au collisions at
$\sqrt{s_{NN}}=7.7$, $27$, 39, 62.4, and $200$ GeV. The results are obtained
for both AMPT-SM and AMPT-Def configurations. We compare the model results to
those obtained from the STAR experiment.

###  III.1  Transverse momentum spectra

Figure 1 shows the comparison of mid-rapidity $p_{T}$ spectra of $\pi^{+}$,
$K^{+}$, and $p$ in most central Au+Au collisions obtained using the AMPT-Def
with the STAR data at $\sqrt{s_{\text{NN}}}=7.7$, 27, 39, 62.4, and 200 GeV
[12]. The experimental data is fitted with the Levy-Tsallis function and the
lower panel in each plot shows the ratio of the invariant yield
($\frac{d^{2}N}{2\pi p_{T}dydp_{T}}$) obtained from the fit function to the
one obtained from AMPT-Def using different sets of input parameters. It is
observed that both Set-1 and Set-2 appear to adequately describe the data,
particularly at low energies, while Set-3 consistently underpredicts the data.

Figure 1: Mid-rapidity ($|y|<0.1$) invariant yield of $\pi^{+}$ (upper row),
$K^{+}$ (middle row) and proton (lower row) as a function of $p_{T}$ for
$0$-$5$% centrality in Au+Au collision at $\sqrt{s_{\text{NN}}}$ = 7.7, 27,
39, 62.4, and 200 GeV using AMPT-Def version. The AMPT results are compared
with the corresponding experimental data which is fitted with the Levy-Tsallis
function [29]. The ratio of data to fit function is shown in the lower panels
of each pad.

Similar to Fig. 1, Fig. 2 presents a comparison of mid-rapidity $p_{T}$
spectra of $\pi^{+}$, $K^{+}$, and $p$ in most central Au+Au collisions
obtained using the AMPT-SM with the STAR data at $\sqrt{s_{\text{NN}}}=7.7$,
27, 39, 62.4, and 200 GeV. STAR data is fitted with Levy-Tsallis function and
the lower panel in each plot shows the ratio of the invariant yield obtained
from the fit function to the one obtained with AMPT-SM using different sets of
input parameters. We observe that Set-2 provides an accurate description of
the data at higher energies among all the three sets, however, it fails to
describe the data at lower energies. We also observe that compared to
$\pi^{+}$ and $p$, $p_{T}$ spectra of $K^{+}$ deviates significantly from the
experimental data for all the different sets of AMPT-SM model considered in
this study, perhaps due to the difference of strangeness quantum number. We
have also explicitly checked that these result hold true for antiparticles as
well.

We also observe that the invariant yield shows a monotonically decreasing
trend with increasing $p_{T}$ across all particles and sets of parameters for
both AMPT-Def and AMPT-SM. In addition, the inverse slopes of particle spectra
display a consistent trend: $\pi<K<p$ similar to what has been observed in
experimental data.

Figure 2: Mid-rapidity ($|y|<0.1$) invariant yield of $\pi^{+}$ (upper row),
$K^{+}$ (middle row) and proton (lower row) as a function of $p_{T}$ for
$0$-$5$% centrality in Au+Au collision at $\sqrt{s_{\text{NN}}}$ = 7.7, 27,
39, 62.4, and 200 GeV using AMPT-SM version. The AMPT results are compared
with the corresponding experimental data which is fitted with the Levy-Tsallis
function [30]. The ratio of data to fit function is also shown in the lower
panels for each pad.

Figures 2 and 4 show a comparison of mid-rapidity $p_{T}$ spectra of
$K^{0}_{s}$, $\Lambda$, and $\phi$ obtained from the AMPT model with the STAR
data in most central Au+Au collisions at $\sqrt{s_{\text{NN}}}$ = 7.7, 27, 39,
62.4, and 200 GeV [29]. The experimental data is compared to predictions from
both the AMPT-Def and AMPT-SM model with distinct sets of input parameters.
STAR data is fitted with the Levy-Tsallis function and the lower panel in each
plot shows the ratio of the invariant yield obtained from the fit function to
the one obtained from the AMPT model. We observe that among the parameter
sets, Set-1 in AMPT-Def effectively describes the $p_{T}$ spectra of strange
hadrons. Whereas, in the case of AMPT-SM, no specific set is describing the
data consistently.

Figure 3: Invariant yield of $K^{0}_{S}$ (upper row), $\Lambda$ (middle row)
and $\phi$ (lower row) as a function of $p_{T}$ for $0$-$5$%
centrality222$\phi$-meson spectra is calculated in $0$-$10$% centrality of
Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 27, 29, and 200 GeV and $0$-$20$%
centrality at $\sqrt{s_{NN}}$ = 62.4 GeV as per the availability of the
experimental data. in Au+Au collision at $\sqrt{s_{\text{NN}}}$ = 7.7, 27, 39,
62.4, and 200 GeV at $|y|<0.5$ ($<1.0$, for $\Lambda$ in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV)) are shown in each column using AMPT-Def. The ratio
of data to fit function is also shown in the lower panels for each pad. Figure
4: Invariant yield of $K^{0}_{S}$ (upper row), $\Lambda$ (middle row) and
$\phi$ (lower row) as a function of $p_{T}$ for $0$-$5$%
centrality444$\phi$-meson spectra is calculated in $0$-$10$% centrality of
Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 27, 29, and 200 GeV and $0$-$20$%
centrality at $\sqrt{s_{NN}}$ = 62.4 GeV as per the availability of the
experimental data. in Au+Au collision at $\sqrt{s_{\text{NN}}}$ = 7.7, 27, 39,
62.4, and 200 GeV at $|y|<0.5$ ($<1.0$, for $\Lambda$ in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV) are shown in each column using AMPT-SM. The ratio
of data to fit function is also shown in the lower panels for each pad.

###  III.2  $p_{T}$ integrated yield, mean transverse mass, and particle
ratios

Figure 5 shows a comparison of the energy dependence of $dN/dy$ of
$\pi^{\pm}$, $K^{\pm}$, $p$, and $\bar{p}$, normalised by half of the average
number of participating nucleons ($\langle N_{\textrm{part}}\rangle/2$)
obtained using the AMPT model in the most central Au+Au collisions at
$\sqrt{s_{\text{NN}}}$ = 7.7, 27, 39, 62.4, and 200 GeV with the STAR data. We
observe that of all the choices of input parameters, Set-2 describes the
particle yields reasonably well for both AMPT-Def and AMPT-SM versions as
shown in Fig. 5(a) and Fig. 5(b), respectively. $dN/dy$ of $\pi^{\pm}$,
$K^{\pm}$, $p$ and $\bar{p}$ is observed to increase with increasing energy.
However, $dN/dy$ of $p$ decreases with increasing energy due to the baryon
stopping prominently observed at lower energies.

(a)

(b)

Figure 5: Energy dependence of (d$N$/dy)/($\langle
N_{\textrm{part}}\rangle/2$) for particle ($\pi^{+}$, $K^{+}$, $p$) and anti-
particle ($\pi^{-}$, $K^{-}$, $\bar{p}$) for central Au+Au collisions using
AMPT-Def (left panel) and AMPT-SM (right panel). The AMPT results shown in
open markers are compared with the corresponding experimental data (solid
marker).

Figure 6 shows a comparison of the energy dependence of $dN/dy$ of
$K_{s}^{0}$, $\Lambda$, $\bar{\Lambda}$, and $\phi$ normalised by $\langle
N_{\textrm{part}}\rangle/2$ in the most central Au+Au collisions at
$\sqrt{s_{\text{NN}}}$ = 7.7, 27, 39, 62.4, and 200 GeV calculated using AMPT
model with the STAR data. We observe that Set-1 and Set-2 of AMPT-Def better
describe the data for strange hadrons.

(a)

(b)

Figure 6: Energy dependence of (d$N$/dy)/($\langle
N_{\textrm{part}}\rangle/2$) for $K^{0}_{S}$, $\Lambda$, $\phi$, and
$\bar{\Lambda}$ for central Au+Au collisions using AMPT-Def (left panel) and
AMPT-SM (right panel). The AMPT results shown in open markers are compared
with the corresponding experimental data (solid marker).

(a)

(b)

Figure 7: $\langle m_{T}\rangle-m_{0}$ of $\pi^{\pm}$, $K^{\pm}$, $p$, and
$\bar{p}$ as a function of $\sqrt{s_{\text{NN}}}$ at midrapidity ($|y|<0.1$).
Results are presented for most central Au+Au collisions using AMPT-Def (left
panel) and AMPT-SM (right panel). The AMPT results shown in open markers are
compared with the corresponding experimental data (solid marker).

For a thermodynamic system, average transverse mass, ($\langle m_{T}\rangle-
m_{0}$), could be an indicative of the temperature of the system, where
$m_{0}$ is the rest mass of the particle. It has been suggested that the
energy dependence of $\langle m_{T}\rangle-m_{0}$ may be a possible signature
of first-order phase tranistion between the hadronic medium and the QGP [31].
Figure 7(a) and 7(b), show the energy dependence of $\langle m_{T}\rangle-
m_{0}$ for $\pi^{\pm}$, $K^{\pm}$, $p$, and $\bar{p}$ in most central Au+Au
collisions calculated using AMPT-Def and AMPT-SM, respectively. We observe
that all the three sets qualitatively capture the trend of the data.

Figure 8(a) and 8(b) show the energy dependence of $\langle m_{T}\rangle-
m_{0}$ for $K_{s}^{0}$, $\Lambda$, $\bar{\Lambda}$ and $\phi$ in most central
Au+Au collisions from AMPT-Def and AMPT-SM, respectively. We observe that
Set-1 of AMPT-Def version is describing the data well, however, all the sets
implemented in AMPT-SM tend to underpredict the data.

(a)

(b)

Figure 8: $\langle m_{T}\rangle-m_{0}$ of $K^{0}_{S}$, $\Lambda$, $\phi$, and
$\bar{\Lambda}$ as a function of $\sqrt{s_{\text{NN}}}$ at midrapidity.
Results are presented for most central Au+Au collisions using AMPT-Def (left
panel) and AMPT-SM (right panel). The AMPT results shown in open markers are
compared with the corresponding experimental data (solid marker).

The energy dependence of antiparticle-to-particle ratios helps in
understanding the particle production mechanism in heavy-ion collisions [32].
Figure 9 shows the collision energy dependence of the particle ratios
$\pi^{-}/\pi^{+}$, $K^{-}/K^{+}$, and $\bar{p}/p$ in most central Au+Au
collisions calculated using the AMPT model. The comparison of the calculations
of the AMPT-Def with the experimental data [33, 34, 35, 36, 37, 38, 39] are
shown in the upper panel, while the lower panel shows the comparison with the
AMPT-SM calculations. We observe that the $\pi^{-}/\pi^{+}$ ratio is greater
than unity at low energies, due to a significant contributions from resonance
decays, such as from $\Delta$ baryons. The $K^{-}/K^{+}$ ratio decreases with
decreasing energy due to the associate production of $K^{+}$. $\bar{p}/p$ is
also observed to decrease with decreasing energy due to the effects of baryon
stopping at lower energies. As the energy increases both $K^{-}/K^{+}$ and
$\bar{p}/p$ ratios approaches unity. The systematic effects resulting from
variations in model parameters cancel out in particle ratios, leading to an
absence of energy dependence on input parameters. We observe that all the sets
of both AMPT-Def and AMPT-SM are able to describe the data well.

Figure 9: $\pi^{-}/\pi^{+}$, $K^{-}/K^{+}$, and $\bar{p}/p$ ratios at
midrapidity ($|y|<0.1$) in central most Au+Au collisions at
$\sqrt{s_{NN}}=7.7,27,39,62.4$ and $200$ GeV using AMPT-Def (upper panel) and
AMPT-SM (lower panel). The AMPT results shown in open markers are compared
with the published experimental data (solid marker).

###  III.3  Kinetic freeze-out

In this section, we discuss kinetic freeze-out parameters obtained using a
blast-wave model in Au+Au collisions at various centre-of-mass energies. In
this approach, we perform a simultaneous fit of $p_{T}$ spectra of
$\pi^{\pm}$, $K^{\pm}$, $p$, and $\bar{p}$ with the blast-wave model similar
to how it has been done with experimental data [12] . The blast-wave model is
a hydrodynamically motivated model that provides an accurate description of
data at low $p_{T}$. However, it is not well-suited for describing hard
processes at high $p_{T}$. The blast-wave model assumes a common radial flow
velocity profile and a thermal freeze-out temperature for all particles [40].
Assuming a radially boosted thermal source with kinetic freeze-out temperature
($T_{kin}$) and transverse radial flow velocity ($\beta$), the $p_{T}$
distribution of the particles is given by:

$\begin{split}\frac{dN}{p_{T}dp_{T}}&\propto
m_{T}\int_{0}^{R}rdrI_{0}\left(\frac{p_{T}\sinh\rho(r)}{T_{kin}}\right)\\\
&\quad\times K_{1}\left(\frac{m_{T}\cosh\rho(r)}{T_{kin}}\right)\end{split},$
(4)

where $m_{T}=\sqrt{p_{T}^{2}+m_{0}^{2}}$ and $\rho(r)=\tanh^{-1}(\beta)$.
$I_{0}$ and $K_{1}$ are modified Bessel functions of the first and second
kind, respectively. $\beta=\beta_{S}\left(r/R\right)^{n}$ is the flow
velocity, where $\beta_{S}$ is the surface velocity, $r/R$ is the relative
radial position in the thermal source, and $n$ is the exponent of the flow
velocity profile.

Figure 10: Centality dependence of kinetic freeze-out temperature ($T_{kin}$
in MeV) in different collision energies for four different Sets in Au+Au
collisions using AMPT.

Figure 10 shows the energy and centrality dependence of $T_{kin}$ in Au+Au
collisions at $\sqrt{s_{NN}}$ = 7.7, 27, 39, 62.4, and 200 GeV. We observe
that predictions from all the sets of input parameters to the AMPT-SM model
are able to capture the decreasing trend of $T_{kin}$ with increasing
centrality, consistent with the experimental data.

Figure 11 shows the variation of $\langle\beta\rangle$ with centrality of
Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 27, 39, 62.4, and 200 GeV. We
observe that qualitatively all the different sets of input parameters of the
AMPT-SM model are able to describe the increasing trend of
$\langle\beta\rangle$ with increasing centrality.

An anticorrelation plot between $T_{kin}$ and $\langle\beta\rangle$ is shown
in Fig. 12. We observe that all the different configurations of the AMPT-SM
model capture this anti-correlation behavior, although Set-3 shows a
significant deviation from the data compared to the other two sets.

##  IV Summary

This study aims to understand the particle production mechanism in Au+Au
collisions at STAR energies in the framework of the AMPT model covering a wide
range of $\sqrt{s_{NN}}$ = 7.7, 27, 39, 62.4, and 200 GeV. We have studied the
production of $\pi^{\pm}$, $K^{\pm}$, $p$, $\overline{p}$, $K^{0}_{S}$,
$\Lambda$, $\bar{\Lambda}$, and $\phi$ in most central Au+Au collisions at
mid-rapidity. In addition, various bulk properties including the transverse
momentum spectra, particle yield, the mean transverse mass, and particle
ratios, as well as freeze-out properties have been studied. Both the Default
and String Melting versions of the AMPT model initialized with different sets
of Lund string fragmentation parameters (as listed in Table 1) are compared
with data from the STAR experiment.

Figure 11: Centality dependence of $\langle\beta\rangle$ in different
collision energies for four different Sets in Au+Au collisions using AMPT.
Figure 12: Variation of $T_{\text{kin}}$ with $\langle\beta\rangle$ for four
different Sets for different collision energies. The centrality increases from
left to right for a given energy.

We observe that the $p_{T}$ spectra of identified hadrons are well-described
by Set-2 of AMPT-SM at higher energies and by Set-2 of AMPT-Def at lower
energies. For strange hadrons we conclude that Set-1 of AMPT-Def version is
describing the data well at the studied collision energies. The comparison of
bulk properties with the results from the AMPT model suggests that:

  * •

Set-2 of input parameters describe the $dN/dy$ of $\pi^{\pm}$, $K^{\pm}$, $p$,
and $\overline{p}$ reasonably well for both AMPT-Def and AMPT-SM.

  * •

Set-1 and Set-2 of input parameters describe the $dN/dy$ of $K^{0}_{S}$,
$\Lambda$, $\bar{\Lambda}$, and $\phi$ reasonably well.

  * •

We observe that all three sets qualitatively capture the trend of $\langle
m_{T}\rangle-m_{0}$ for all studied particle as observed in the data.

  * •

All the three sets of input parameters are able to describe the particle
ratios reasonably well for both the versions of the AMPT model.

  * •

All three sets of input parameters of AMPT-SM qualitatively capture the
decreasing trend of $T_{kin}$ and the increasing trend of
$\langle\beta\rangle$ with increasing centrality. Additionally, they capture
the anti-correlation between $T_{kin}$ and $\langle\beta\rangle$, as observed
in the experimental data.

In conclusion, we observe that the bulk properties studied in the framework of
the AMPT model are sensitive to the input parameters. Comparing the AMPT
results with the experimental data helps in understanding the underlying
particle production mechanism. The results presented in this paper may help in
deciding a better set of input parameters for future studies.

###### Acknowledgements.

CJ acknowledges the financial support from DAE-DST Project No.
3015/I/2021/Gen/RD-I/13283 and KN is supported by OSHEC, Department of Higher.
Education, Government of Odisha, Index No. 23EM/PH/124 under MRIP-2023.

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