# Nuclear cluster structure effect in 16O+16O collisions at the top RHIC
energy

Xin-Li Zhao  College of Science, University of Shanghai for Science and
Technology, Shanghai $200093$, China  Key Laboratory of Nuclear Physics and
Ion-beam Application (MOE), Institute of Modern Physics, Fudan University,
Shanghai 200433, China  Shanghai Research Center for Theoretical Nuclear
Physics, NSFC and Fudan University, Shanghai $200438$, China  Guo-Liang Ma
<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan
University, Shanghai $200438$, China  You Zhou<EMAIL_ADDRESS>Niels Bohr
Institute, Jagtvej 155A, 2200 Copenhagen, Denmark  Zi-Wei Lin  Department of
Physics, East Carolina University, Greenville, NC 27858, USA  Chao Zhang
School of Science, Wuhan University of Technology, Wuhan, 430070, China

###### Abstract

The impact of nuclear structure has garnered considerable attention in the
high-energy nuclear physics community in recent years. This work focuses on
studying the potential nuclear cluster structure in ${}^{16}\text{O}$ nuclei
using anisotropic flow observables in $\rm O+O$ collisions at 200 GeV.
Employing an improved AMPT model with various cluster structure
configurations, we find that an extended effective parton formation time is
necessary to align with the recent STAR experimental data. In addition, we
reveal that the presented flow observables serve as sensitive probes for
differentiating configurations of $\alpha$-clustering of ${}^{16}\text{O}$
nuclei. The systematic AMPT calculations presented in this paper, along with
comprehensive comparisons to forthcoming experimental measurements at RHIC and
the LHC, pave the way for a novel approach to investigate the
$\alpha$-clustering structure of ${}^{16}\text{O}$ nuclei using $\rm O+O$
collisions at the ultra-relativistic energies.

##  I  Introduction

High-energy heavy ion collisions create an extremely dense and hot environment
in which the quarks and gluons confined inside nuclei can be released into a
deconfined state of matter known as the quark-gluon plasma (QGP), the hot and
dense matter that existed in the early universe. Extensive experimental
measurements from the Relativistic Heavy Ion Collider (RHIC) and the Large
Hadron Collider (LHC) show the recreation of QGP in the early stage of
relativistic nuclear collisions. A crucial experimental evidence of QGP is
anisotropic flow. This reflects the hydrodynamical evolution of the QGP, which
can transfer the asymmetries in the initial coordinate (geometry) space into
anisotropic expansion in the final momentum space Ollitrault (1992); Heinz and
Snellings (2013); Zhao et al. (2018). The measured different harmonics of the
anisotropic flow of the final particles in $\rm A+A$ collisions can be
successfully reproduced by hydrodynamic models Adams et al. (2005); Song et
al. (2017); Hirano et al. (2010), which reveal that QGP behaved nearly perfect
fluid. Moreover, recent observations show that anisotropic flow shows similar
patterns in small systems such as $p+p$ and $p+\rm A$ Acharya et al. (2019).
This challenges our current understanding of QGP, as there is controversy over
whether hydrodynamics applies to such small systems because of their small
size and short lifetime. To study how the system size affects the anisotropic
flow of the QGP, collisions of light nuclei at high energies, such as $\rm
O+O$ and $\rm Ne+Ne$ collisions, have been proposed Brewer et al. (2021);
Huang (2023).

A key aspect of the anisotropic flow is that it provides direct access to the
details of the initial conditions of nuclear collisions, thereby exploring the
geometric structure of colliding nuclei Acharya et al. (2018, 2022); Zhang and
Jia (2022); Zhang et al. (2022); Lu et al. (2023). The nuclear structure of
nuclei poses a longstanding challenge and offers an important research
opportunity in nuclear physics Ma (2023); Liu and Ye (2018). Over the years,
researchers have continuously attempted to understand the puzzles related to
the structure of nuclei using different approaches, especially in the field of
low and intermediate-energy nuclear physics Fang (2023); Ma et al. (2022).
Gamov et al. first proposed that clustered states exist in light nuclei such
as 12C, 16O, and 20Ne Gamow (1930). Determining the mechanism of the carbon-
generated fusion reaction led to the discovery of cluster states Hoyle (1954);
Cook et al. (1957). It has been proposed that $\alpha$-clustering
configurations can be detected by sequential decay Bishop et al. (2019); Yang
et al. (2023), giant dipole resonances He et al. (2014, 2016a); Wang et al.
(2023) or photonuclear reactions Huang et al. (2017a, b); Huang and Ma (2020);
Shi and Ma (2021). In recent years, many studies have revealed the significant
role of nuclear structures, such as neutron skin Li et al. (2020a); Giacalone
et al. (2023) or deformation nuclei Bally et al. (2022); Ryssens et al.
(2023); Nielsen et al. (2024), on some observables in high-energy nuclear
collisions. Thus, traditional topics related to nuclear structure can be
investigated from a new and unique perspective with high-energy nuclear
collisions. The anisotropic flow in high-energy nuclear collisions also serves
as an important probe to reveal the geometrical structure of cluster nuclei,
which has received significant attention Broniowski and Ruiz Arriola (2014);
Bozek et al. (2014); Zhang et al. (2017, 2018); He et al. (2021); Ma and Zhang
(2022); Dasgupta et al. (2021, 2023). On the other hand, theoretical studies
suggest that the $0_{6}^{+}$ state of 16O is a strong candidate for the
4-$\alpha$ condensed state Funaki et al. (2008); Ohkubo (2013); Suhara et al.
(2014); Funaki et al. (2013, 2010, 2009a). The condensed $\alpha$ particles
are weakly bound in 0$S$ orbits as quasi-bosons, leading to a diffuse and
gaseous spatial distribution Funaki et al. (2009b); Schuck et al. (2016).
Therefore, studying $\alpha$-clustering structure in $\rm O+O$ collisions at
high energies is crucial for understanding many important questions related to
nuclear structure, new states of matter, and nuclear astrophysics. Recent
theoretical studies also indicate the potential of using high-energy $\rm O+O$
collisions to study the existence of cluster structures within the nuclei of
16O Rybczyński and Broniowski (2019); Li et al. (2020b, 2021); Behera et al.
(2022); Summerfield et al. (2021); Ding et al. (2023); Behera et al. (2023,
2024); Wang et al. (2024); Giacalone et al. (2024). From the experimental
side, preliminary results from the STAR experiment demonstrate that
anisotropic flow and flow fluctuations can be a powerful tool to study
nucleon-nucleon correlations Gezerlis et al. (2013); Fomin et al. (2017); Ye
and Arrington (2018) in $\rm O+O$ collisions at $\sqrt{s_{{}_{\rm NN}}}=200$
GeV Huang (2023). Such efforts will be further strengthened after the
operation of $\rm O+O$ collisions at 6.37 TeV at the LHC Run 3 program in 2025
Acharya et al. (2021).

Various theoretical models, such as thermal, transport, and hydrodynamic
models, are used to simulate and study the phase-space evolution of QGP in
high-energy heavy ion collisions. Comparison of these model calculations with
experimental measurements extracts important information about QGP properties.
The transport models describe the dynamic evolution of matter in equilibrium
or non-equilibrium states at a more microscopic level and reasonably describe
systems with a sufficiently high average number of collisions per particle. A
multi-phase transport (AMPT) model successfully describes experimental
measurements of anisotropic flow coefficients in large and small systems Bzdak
and Ma (2014); Ma and Lin (2016). Recent studies show that the transport model
transforms asymmetries in the initial geometry space into anisotropies in the
final momentum space through the parton escape mechanism He et al. (2016b); Ma
and Bzdak (2016); Ma et al. (2021), in which the fraction of anisotropic flow
increases from small to large systems. By introducing the structure of the
initial colliding nuclei into the AMPT model, anisotropic flow in high-energy
heavy ion collisions serves as a new probe for exploring the cluster structure
of nuclei Zhang et al. (2017, 2018); He et al. (2021); Ma and Zhang (2022);
Zhang et al. (2024). We are motivated to study the effects of nuclear cluster
structure in $\rm O+O$ collisions using the AMPT model. Note that in this
work, we utilize the latest improved AMPT model. This model gives better
descriptions of more experimental results, especially for centrality
dependence of the mean transverse momentum that cannot be reproduced using the
public version of the AMPT model.

The paper is organized as follows. First, in Sec. II, we introduce four
different geometry configurations of oxygen into the improved AMPT model. In
Sec. III, the two-particle correlation method with non-flow subtraction is
introduced. In Sec. IV, we present our results on the anisotropic flow and its
related observables and compare them with the STAR data in $\rm O+O$
collisions at 200 GeV. Finally, we conclude Sec. V.

##  II  Geometry configurations and AMPT model

Figure 1: Illustration of the geometrical configurations of 16O with the
nucleon structures of (a) W-S distribution, (b) tetrahedron distribution, and
(c) square distribution.

Figure 2: The nucleon density distributions inside 16O in $\rm O+O$ collisions
with the geometrical configurations of W-S, tetrahedron, square, and
$ab~{}initio$ distributions.

Figure 3: The centrality dependence of $\left<p_{\rm T}\right>$ in $\rm O+O$
collisions at 200 GeV with the tetrahedron configuration of nuclear structure
for two versions of AMPT model.

Figure 4: The centrality dependence of (a) $\varepsilon_{2}$ and (b) $v_{2}$
in $\rm O+O$ collisions at 200 GeV with at different effective parton
formation times for the nuclear structure of tetrahedron configuration.

Figure 5: The centrality dependence of $\left<p_{\rm T}\right>$ in $\rm O+O$
collisions at 200 GeV with different nuclear structure configurations.

Figure 6: The centrality dependence of (a) $\varepsilon_{2}$ and (b)
$\varepsilon_{3}$ in $\rm O+O$ collisions at 200 GeV with different nuclear
structure configurations.

Figure 7: The $p_{\rm T}$ dependence of $v_{2}$ for (a) $0-10\%$, (b)
$10-20\%$, and (c) $20-40\%$ centrality bins in $\rm O+O$ collisions at 200
GeV with different nuclear structure configurations, compared to STAR data.

Figure 8: The $p_{\rm T}$ dependence of $v_{3}$ for (a) $0-10\%$, (b)
$10-20\%$, and (c) $20-40\%$ centrality bins in $\rm O+O$ collisions at 200
GeV with different nuclear structure configurations, compared to STAR data.
Figure 9: The centrality dependence of (a) $v_{2}$ and (b) $v_{3}$ in $\rm
O+O$ collisions at 200 GeV with different nuclear structure configurations,
compared to STAR data. Figure 10: The centrality dependence of (a)
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ and (b)
$v_{2}\\{4\\}/v_{2}\\{2\\}$ in $\rm O+O$ collisions at 200 GeV with different
nuclear structure configurations, compared to STAR data. Figure 11: The
$p_{\rm T}$ dependence of (a) $v_{2}$ and (b) $v_{3}$ for $0-10\%$ centrality
bins with different non-flow subtractions in $\rm O+O$ collisions at 200 GeV
with tetrahedron configuration, compared to STAR data.

Based on the experimental root-mean-square ($rms$) nuclear charge radius
$\left\langle r^{2}\right\rangle^{1/2}=2.6991$ fm for 16O Angeli and Marinova
(2013), we chose suitable parameters for the four nuclear structures of 16O as
follows:

(a) For the Woods-Saxon (W-S) distribution as shown in Fig. 1(a), i.e. the
three-parameter Fermi (3pF) model De Vries et al. (1987) is used for charged
density of 16O

$\displaystyle\rho(r)=\rho_{0}(1+wr^{2}/R_{0}^{2})/(1+{\rm
exp}(\frac{r-R_{0}}{a})),$ (1)

where $\rho_{0}$ is the nuclear saturation density, $R_{0}=2.608$ fm is the
radius of the nucleus, $a=0.513$ is the surface diffusion parameter, and
$w=-0.051$ is the weight parameter of the 3pF model.

(b) For the tetrahedron-shaped Bauhoff et al. (1984); and (2012); Girod and
Schuck (2013); Epelbaum et al. (2014); Bijker and Iachello (2014) and square-
shaped cluster structures, the nucleons are arranged as shown in Fig. 1(b) and
Fig. 1(c), respectively. The distance between the centers of the clusters is
$l$. The nucleons for each cluster are randomly distributed by the following
Gaussian probability distribution,

$\displaystyle f_{i}({\bf r})={\rm const~{}exp}(-\frac{3}{2}\frac{(\bf
r-c_{i})^{2}}{r_{c}^{2}}),$ (2)

where $\bf r$ is the coordinate of the nucleon, $\bf c_{i}$ is the center
position of cluster $i$, and $r_{c}$ is the width of the cluster, i.e., the
$rms$ radius of the cluster. For the tetrahedron configuration of the cluster
structure, $l=3.5$ fm and $r_{c}=1.23$ fm.

(c) For the square configuration Brink et al. (1970); Bauhoff et al. (1984) of
the cluster structure, $l=3.0$ fm, $r_{c}=1.23$ fm.

(d) For the $ab~{}initio$ case of cluster structure, the nucleon density
distribution for 16O is obtained by an $ab~{}initio$ method based on Nuclear
Lattice Effective Field Theory (NLEFT); please see the details in Refs.
Summerfield et al. (2021); Giacalone et al. (2024).

Figure 2 presents the nucleon density distributions for the four nuclear
structure configurations mentioned above. It can be observed that the nucleon
density distributions for the W-S and $ab~{}initio$ configurations are
significantly higher than that of the tetrahedron and square configurations at
the nucleus center. This feature can be inferred from Fig. 1. For the W-S
configuration, the nucleons are concentrated more toward the center of the
nucleus, while for the tetrahedron and square configurations, there are fewer
nucleons at the center of the nucleus.

Next, the above four configurations of 16O are introduced into the initial
conditions of the string-melting version of the AMPT (AMPT-SM) model Lin et
al. (2005) that could describe the evolution of the QGP and the hadrons in
high-energy nuclear collisions. This model includes the initial conditions
based on the HIJING model, parton interactions in the ZPC model, hadronization
with quark-coalescence, and hadron interactions with the ART model. In the
AMPT-SM model, the partons are generated by string melting after a formation
time of $\tau_{0}=E/m_{\rm T}^{2}$ where $E$ and $m_{\rm T}$ represent the
energy and transverse mass of the parent hadron. The positions of the partons
are determined based on the positions of their parent hadrons using straight-
line trajectories. In this work, we choose a longer effective parton formation
time of $\tau_{0}^{{}^{\prime}}=6.0\tau_{0}$ to mimic the diffuse density
spatial distribution of clustered 16O. The factor of 6.0 was chosen by fitting
the STAR measurements on the elliptic flow in $\rm O+O$ collisions at 200 GeV,
which will be discussed later. Unlike any publicly available AMPT-SM models,
including all the previous AMPT studies on the nuclear structure Li et al.
(2020b); Behera et al. (2023); Jia et al. (2023); Zhang and Jia (2022), here
we use the latest improved AMPT-SM model blue with parton cross-section 1.5 mb
Zhang et al. (2019); Zheng et al. (2020); Zhang et al. (2021, 2023a, 2023b).
This improved AMPT-SM model has implemented a new quark coalescence,
introduced modern parton distribution functions of the free proton, imparted
parameter-dependent nuclear shadowing, and improved heavy flavor productions.
The improved model can provide more reasonable descriptions of the particle
yield ratios, $p_{\rm T}$ spectra, and anisotropic flow in $\rm A+A$
collisions at RHIC and LHC energies. It is well-established that the average
transverse momentum $\left<p_{\rm T}\right>$ is highly sensitive to the
initial energy density distribution in collision systems, with $\left<p_{\rm
T}\right>$ being directly proportional to the temperature $T$ of the system.
The $\left<p_{\rm T}\right>$ is expected to decrease from central to
peripheral collisions, given that central collisions typically exhibit higher
temperatures than peripheral ones. As shown in Figure 3, the public AMPT-SM
model erroneously predicts an increase in $\left<p_{\rm T}\right>$ from
central to peripheral collisions in $\rm O+O$ collisions at 200 GeV, which is
contrary to the expected trend. In contrast, the improved AMPT-SM model
exhibits a more accurate centrality dependence, thanks to the recent
incorporation of local nuclear scaling in the initial condition parameters
Zhang et al. (2021). In this work, we will apply the improved AMPT-SM model
for the first time to investigate nuclear structure in high-energy nuclear
collisions.

The initial state anisotropy, such as eccentricity $\varepsilon_{2}$ and
triangularity $\varepsilon_{3}$, characterizes the initial state of nuclear
collisions through the event-by-event distribution of the participating
partons, which is essential for understanding the initial state geometry Alver
and Roland (2010); Ma et al. (2016). Figure 4 (a) shows the centrality
dependence of $\varepsilon_{2}$ for different effective formation times of
$\tau_{0}^{{}^{\prime}}=\tau_{0}\sim 10\tau_{0}$. We find that
$\tau_{0}^{{}^{\prime}}$ has a significant effect on $\varepsilon_{2}$, so
$\varepsilon_{2}$ decreases with increasing $\tau_{0}^{{}^{\prime}}$. Since
the linear response of $v_{2}\propto\varepsilon_{2}$ is expected, we present
the centrality dependence of $v_{2}$ for $\tau_{0}^{{}^{\prime}}=\tau_{0}\sim
10\tau_{0}$ in Fig. 4 (b). We can see that $v_{2}$ decreases with
$\tau_{0}^{{}^{\prime}}$, since $\varepsilon_{2}$ decreases with increasing
$\tau_{0}^{{}^{\prime}}$. Based on the fact that the $v_{2}$ result for
$\tau_{0}^{{}^{\prime}}=6.0\tau_{0}$ is the closest to the STAR data, we have
chosen a factor of 6.0 in our work. It should be noted that although we show
only the case for the nuclear structure of the tetrahedron, we have also
examined other nuclear structures and observed similar results.

Additionally, we investigated the impact of parton cross-section on $v_{2}$.
Our findings indicate that obtaining results similar to those of STAR data is
challenging, even with a very small cross-section. As a result, we ultimately
selected $\tau_{0}^{{}^{\prime}}=6.0\tau_{0}$ for further study.

There is a possible reason for the origin of this factor of 6.0. The ground
state of $\rm{}^{16}O$ has been predicted to be with a tetrahedron structure
of 4-$\alpha$ clusters, which is supported by theoretical calculations from
the chiral nuclear effective field theory Epelbaum et al. (2014), the
covariant density functional theory and (2012), and the algebraic model Bijker
and Iachello (2014). On the other hand, with the help of the electromagnetic
fields generated in high-energy nuclear collisions ($\sim$ 1 MeV per nucleon
in $\rm O+O$ collisions at 200 GeV Zhao et al. (2024)), the ground state of
$\rm{}^{16}O$ is likely to excite into its excited $0^{+}$ states which are
believed to be with a large occupation probability of 4-$\alpha$ condensate in
the 0$S$ state Funaki et al. (2009b); Schuck et al. (2016). Because the
clustered $\rm{}^{16}O$ is near the threshold energy and a large fraction of
the total binding energy is consumed in forming the $\alpha$ clusters, the
condensed $\alpha$ clusters are weakly coupled to each other. This weak
coupling results in a dilute and spatially extended density distribution of
the clustered $\rm{}^{16}O$ Ohkubo and Hirabayashi (2007); Tomita et al.
(2014); Zhou et al. (2023). We notice that the similar effects of so-called
compactness have also been studied in $\rm O+O$ collisions at 6.5 TeV, using
iEBE-VISHNU hydrodynamic simulation with different initial state
$\alpha$-clustering configurations Wang et al. (2024). We expect that
measuring the HBT radius is a good way to verify the presence of the diffuse
spatial distributions in clustered $\rm{}^{16}O$ He et al. (2020). A related
study is currently in progress.

##  III  Observables and methodology

In this paper, the study of anisotropic flow in $\rm O+O$ collisions is
performed using the two-particle correlation method. By Fourier expansion of
the two-particle correlations as

$\displaystyle\frac{dN^{\rm pairs}}{d\Delta\phi}\propto
1+2\sum_{n=1}^{\infty}v_{n}{\rm cos}(n\Delta\phi),$ (3)

the anisotropic flow can be extracted. Here the relative azimuthal angel of
the trigger particle $\phi^{\rm trig}$ and the associate particle $\phi^{\rm
assoc}$ is $\Delta\phi=\phi^{\rm trig}-\phi^{\rm assoc}$. And $v_{n}$ is the
coefficient of the $n$th-order flow. In order to obtain $v_{n}(p_{\rm T}^{\rm
trig})$, the Fourier fitting can be employed on the $Y(\Delta\phi)$
distributions:

$\displaystyle Y(\Delta\phi,p_{\rm T}^{\rm
trig})=c_{0}(1+2\sum_{n=1}^{n=4}c_{n}{\rm cos}(n\Delta\phi)).$ (4)

Then, the non-flow contaminations can be subtracted following

$\displaystyle c_{n}^{\rm sub}=c_{n}-c_{n}^{\rm non-flow}=c_{n}^{\rm
cent}-c_{n}^{\rm peri}\times f,$ (5)

where $f=c_{1}^{\rm cent}/c_{1}^{\rm peri}$, $c_{n}$ is the product of $v_{n}$
for the trigger and associate particles, i.e. $c_{n}=v_{n}^{\rm trig}\times
v_{n}^{\rm assoc}$. In this work, the anisotropic flow is extracted using the
Fourier fitting and the non-flow subtraction with $60-80\%$ peripheral
collisions, as the STAR experiment did Huang (2023).

##  IV  Results and discussions

Figure 5 shows the centrality dependence of $\left<p_{\rm T}\right>$ in $\rm
O+O$ collisions at 200 GeV using 16O structure from the W-S, tetrahedron,
square and $ab~{}initio$ configurations. The differences in $\left<p_{\rm
T}\right>$ between the different configurations are very striking, especially
for $ab~{}initio$ and tetrahedron. For central and mid-central collisions, the
$\left<p_{\rm T}\right>$ from $ab~{}initio$ is higher than the other
configurations, while the $\left<p_{\rm T}\right>$ from tetrahedral
configuration is the smallest. The $\left<p_{\rm T}\right>$ of the W-S and
square configurations are similar. Meanwhile, compared with Fig. 3,
$\left<p_{\rm T}\right>$ of tetrahedron for
$\tau_{0}^{{}^{\prime}}=6.0\tau_{0}$ is very close to
$\tau_{0}^{{}^{\prime}}=1.0\tau_{0}$. Given the considerable sensitivity of
$\left<p_{\rm T}\right>$ to $v_{2}$, it is reasonable to expect a similar
sensitivity of $v_{2}$ to different geometric configurations.

Figure 6 (a) and (b) present the AMPT-SM results on the centrality dependence
of $\varepsilon_{2}$ and $\varepsilon_{3}$ in $\rm O+O$ collisions at 200 GeV
for the W-S, tetrahedron, square, and $ab~{}initio$ configurations. We find
that both $\varepsilon_{2}$ and $\varepsilon_{3}$ increase from central to
peripheral collisions. In addition, Fig. 6(a) shows that $\varepsilon_{2}$ is
the largest for the square configuration and the smallest for the W-S
configuration in central and mid-central collisions. Meanwhile, Fig. 6(b)
shows that the $\varepsilon_{3}$ is the largest for the tetrahedron
configuration and the smallest for the square configuration at central and
mid-central collisions. Thus, the simultaneous studies on $v_{2}$ and $v_{3}$
could provide independent constraints on the nuclear structure.

Transverse momentum-dependent anisotropic flow, $v_{n}(p_{\rm T})$, has been a
popular tool in ultra-relativistic nuclear collisions. By measuring
$v_{n}(p_{\rm T})$ of charged and identified particles, one can extract unique
information on the initial state geometry and fluctuations, the transport
coefficients, the equation of state, and the hadronization mechanism of the
QGP. Considering its sensitivity to the initial geometry, it is an ideal probe
of nuclear structure in $\rm O+O$ collisions. Figures 7 (a)-(c) show the
$p_{\rm T}$ dependence of $v_{2}$ for (a) $0-10\%$, (b) $10-20\%$, and (c)
$20-40\%$ centrality bins in $\rm O+O$ collisions at 200 GeV from the AMPT-SM
model, respectively. The calculations from the W-S, tetrahedron, square, and
$ab~{}initio$ configurations, together with the comparisons to the STAR
measurements Huang (2023), are presented. We can see that $v_{2}(p_{\rm T})$
calculations first increase and then decrease with $p_{\rm T}$ for all the
three centrality bins for all configurations of the AMPT-SM model. Our AMPT-SM
calculations can describe STAR measurements for $p_{\rm T}$ below 1 GeV/c but
underestimate the experimental data for $p_{\rm T}>1$ GeV/c. We also observe
the apparent difference among the AMPT-SM results from the different
configurations of nuclear structure in $0-10\%$ central collisions. The square
configuration shows the largest $v_{2}(p_{\rm T})$, whereas the W-S
configuration shows the smallest $v_{2}(p_{\rm T})$. This can be understood
because the square configuration results in a maximum of $\varepsilon_{2}$ in
the transverse plane while the W-S configuration results in a minimum of
$\varepsilon_{2}$ in the central collisions, as shown in Fig. 6 (a).

Similarly, the $p_{\rm T}$ dependence of $v_{3}$ for (a) $0-10\%$, (b)
$10-20\%$, and (c) $20-40\%$ centrality intervals in $\rm O+O$ collisions at
200 GeV from the AMPT-SM model with different configurations are shown in Fig.
8. It can be seen that $v_{3}$ increases linearly as $p_{\rm T}$ increases.
The $p_{\rm T}$-dependent $v_{3}$ results from the AMPT-SM model with
different configurations show modest differences within the sizable
uncertainties, with the result from tetrahedron the largest and the one from
square configuration the smallest. All these calculations are compatible with
the STAR measurements, and the calculations work particularly well in central
collisions.

Figures 9(a) and (b) present the centrality dependence of $v_{2}$ and $v_{3}$
in $\rm O+O$ collisions at 200 GeV from the AMPT-SM model. Results from the
W-S, tetrahedron, square, and $ab~{}initio$ configurations are shown, together
with the comparisons to the recent STAR measurements Huang (2023). In Fig. 9
(a), the $v_{2}$ results from the AMPT-SM model are close to the $v_{2}$
measurement from STAR, while the AMPT-SM results present stronger centrality
dependence, especially for square configuration. When comparing the AMPT-SM
calculations from the four configurations, we find that the difference in
$v_{2}$ is most pronounced in central collisions, where the square
configuration shows the largest $v_{2}$ and the W-S configuration shows the
smallest $v_{2}$, which is consistent with the $p_{\rm T}$-dependent $v_{2}$
results in Fig. 7. In Fig. 9 (b), the $v_{3}$ results from the AMPT-SM model
overestimate the STAR data. The $v_{3}$ result for tetrahedron configuration
is the largest among all configurations, which is also consistent with the
$p_{\rm T}$-dependent $v_{3}$ results in Fig. 8. Considering that the
$v_{3}(p_{\rm T})$ result in Fig. 8 is in good agreement with the STAR
measurement, it indicates that the $p_{\rm T}$ distributions from the AMPT-SM
model may be different from those in actual data. However, since no
experimental measurements of the $p_{\rm T}$ distribution are available, we
leave a better tuning for future works.

Considering the linear response of the final state anisotropic flow to the
initial state geometry, i.e., $v_{n}\propto\varepsilon_{n}$ (for the $n=2,3$),
the study of the anisotropic flow and its event-by-event fluctuations can
provide direct access into the initial geometry and its fluctuations. It has
been reported recently that the sensitivity to the initial geometry reveals
additional information on the structure of the nuclei in the ultra-
relativistic nuclear collisions Jia (2022); Lu et al. (2023); Nielsen et al.
(2024). In this work, we will probe the nuclear structure via the study of the
relative flow fluctuation $v_{2}\\{4\\}/v_{2}\\{2\\}$ and the initial
eccentricity fluctuation carried by
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$. Here the two- and four-
particle cumulant of $\varepsilon_{2}$ are defined as

$\displaystyle\varepsilon_{2}\\{2\\}^{2}$ $\displaystyle=$
$\displaystyle\left\langle\varepsilon_{2}^{2}\right\rangle=\langle\varepsilon_{2}\rangle^{2}+\sigma_{\varepsilon_{2}}^{2},$
(6) $\displaystyle\varepsilon_{2}\\{4\\}^{2}$ $\displaystyle=$
$\displaystyle(-\left\langle\varepsilon_{2}^{4}\right\rangle+2\left\langle\varepsilon_{2}^{2}\right\rangle^{2})^{1/2}\approx\langle\varepsilon_{2}\rangle^{2}-\sigma_{\varepsilon_{2}}^{2}.$
(7)

where $\sigma_{\varepsilon_{2}}$ denotes the fluctuations of
$\varepsilon_{2}$. As indicated in Eqs. (6) and (7), a greater
$\sigma_{\varepsilon_{2}}$ leads to a larger deviation of the
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ ratio from unity. The
centrality dependence of $\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ ratio
is shown in Fig. 10(a). We find that for the square configuration, the
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ ratio decreases as centrality
increases. Conversely, the $\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$
ratios for the tetrahedron and $ab~{}initio$ configurations exhibit modest
increases with centrality, while the results from the W-S configuration show
no centrality dependence. Furthermore, the
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ ratio for the square
configuration is notably higher than those for the other configurations. This
difference among the four configurations is most pronounced in central
collisions and diminishes in more peripheral collisions.

Figure 10(b) shows the centrality dependence of $v_{2}\\{4\\}/v_{2}\\{2\\}$
for the four configurations. Compared to the
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ ratio shown in Fig. 10(a),
sizable statistical uncertainties of $v_{2}\\{4\\}/v_{2}\\{2\\}$ have been
seen. Nevertheless, the $v_{2}\\{4\\}/v_{2}\\{2\\}$ ratio for the square
configuration is the largest for the presented centrality ranges, and it
decreases with the centrality, whereas for other calculations, the
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ results seem to show an
increasing trend toward peripheral collisions. The centrality dependence of
$v_{2}\\{4\\}/v_{2}\\{2\\}$ seem to be consistent with
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ calculations. Furthermore,
compared to the STAR measurements on the $v_{2}\\{4\\}/v_{2}\\{2\\}$ ratio, we
found that the results from the tetrahedron and $ab~{}initio$ configurations
give better descriptions of the STAR measurements than the calculations using
W-S and square configurations, despite the sizable uncertainty. Considering
the fact that $\left<p_{\rm T}\right>$ and $v_{n}$ are sensitive to both the
initial conditions as well as the transport properties of QGP, the
$v_{2}\\{4\\}/v_{2}\\{2\\}$ ratio can serve as a better probe of the nuclear
structure as it is robust against the system’s dynamic evolution. The
presented $v_{2}\\{4\\}/v_{2}\\{2\\}$ results seem to suggest the cluster
structure for $\rm{}^{16}O$; they shed new light on using ultra-relativistic
nuclear collisions to complement the low-energy nuclear structure studies.

To eventually understand the existing discrepancy between the AMPT flow
calculations and STAR measurements, we perform careful investigations on the
non-flow effects on the anisotropic flows in this work. Figures 11 (a) shows
the $v_{2}(p_{\rm T})$ with four different non-flow subtractions, including
subtraction with $60-70\%$ collisions, subtraction with $70-80\%$ collisions,
subtraction with $60-80\%$ collisions, and template fit method, for
tetrahedron configuration at $0-10\%$ centrality. The results without nonflow
subtraction but directly from the Fourier fit are also presented for
comparison. Due to the largest non-flow effect, the $v_{2}(p_{\rm T})$ without
subtraction is the largest, especially at high $p_{\rm T}$. While
$v_{2}(p_{\rm T})$ for non-flow subtractions with $70-80\%$ collisions is the
smallest, possibly because non-flow is over-subtracted. The template fit
method gives the similar $v_{2}(p_{\rm T})$ results as non-flow subtractions
with $60-70\%$ collisions. Considering that the peripheral subtraction method
treats the entire correlation observed in peripheral collisions as nonflow, it
usually serves the lower limit of flow, whereas the template fit takes care of
the flow modulation in the peripheral collisions; it usually provides an upper
limit of flow. Thus, one would expect the true flow results to be somewhat
located between $v_{2}$ from the template and peripheral subtraction method,
while such a range could not cover the STAR measurement based on the template
fit method. Figure 11 (b) shows the $v_{3}(p_{\rm T})$ with the four different
non-flow subtractions and the results without non-flow subtraction. $v_{2}$
originates from the initial geometry of the collision region, while $v_{3}$
arises from the fluctuations in the initial geometry and subsequent
hydrodynamic evolution. In contrast to $v_{2}(p_{\rm T})$, the non-flow
subtraction approaches have the opposite effect on $v_{3}(p_{\rm T})$: the
$v_{3}(p_{\rm T})$ without subtraction is the smallest, while $v_{3}(p_{\rm
T})$ for non-flow subtractions with $70-80\%$ collisions is the largest. This
is because the fluctuation-induced $v_{3}(p_{\rm T})$ signal is more
pronounced after subtracting the more non-flow contribution. It is also
noticed that various non-flow subtraction methods do not yield a significant
difference in $v_{3}(p_{\rm T})$, which is not the case observed in the non-
flow study of $v_{2}(p_{\rm T})$. The reasonable agreement of AMPT
calculations and STAR measurement in $v_{3}(p_{\rm T})$ persists, independent
of what non-flow subtraction method is applied.

##  V  Summary

To summarize, we utilized the improved AMPT-SM model, which describes more
experimental results, to analyze the anisotropic flow and related observables
for the W-S, tetrahedron, square, and $ab~{}initio$ configurations in $\rm
O+O$ collisions at 200 GeV. We found that the AMPT-SM model, when adjusted for
a longer effective parton formation time $\tau_{0}^{{}^{\prime}}=6.0\tau_{0}$,
aligns better with the STAR data. Employing the same experimental method of
non-flow subtraction, we noted that the $p_{\rm T}$-dependent $v_{2}$ closely
matches the STAR $v_{2}(p_{\rm T})$ measurement at lower $p_{\rm T}$ values,
yet it underestimates the measurements at higher $p_{\rm T}$. The $p_{\rm
T}$-dependent $v_{3}$, however, is consistent with the STAR measurements
across the entire $p_{\rm T}$ range. Additionally, the integrated $v_{2}$
results are comparable in magnitude to the STAR measurements, while the
integrated $v_{3}$ values from the AMPT-SM model are marginally higher than
the observed $v_{3}$. In contrast to typical $v_{n}$ studies, the
$\varepsilon_{2}\\{4\\}/\varepsilon_{2}\\{2\\}$ ratio demonstrates reduced
sensitivity to dynamic evolution and robust sensitivity to fluctuations
originating from nuclear structures. Our findings indicate that the
implementation of $\alpha$-clustering structures yields a more accurate
description of the STAR $v_{2}\\{4\\}/v_{2}\\{2\\}$ measurements.

The presented studies, particularly using the latest developments of the AMPT-
SM model for the first time, provide insights into the presence of the cluster
structures in ${}^{16}\rm O$ nuclei. The findings suggest that the anisotropic
flow observables can serve as a sensitive probe for diagnosing the different
configurations of the ${}^{16}\rm O$ structure in high-energy nuclear
collisions. This opens up new possibilities for studying the nuclear structure
and exploring the cluster structure of nuclei using high-energy $\rm O+O$
collisions. Future studies can build upon these findings and further
investigate the nuclear structure in the $\rm O+O$ collisions using more
experimental measurements on the anisotropic flow phenomena at RHIC and the
ongoing LHC Run 3 program.

###### Acknowledgements.

We thank Prof. Bo Zhou and Dr. Chun-Jian Zhang for helpful discussions about
cluster structure for $\rm{}^{16}O$ and anisotropic flow for $\rm O+O$
collisions. In addition, we gratefully acknowledge Prof. Dean Lee and Prof.
Christopher Plumberg for sharing the nucleon density distribution of oxygen
nuclei. Finally, we acknowledge Dr. Chen Zhong for maintaining the high-
quality performance of the Fudan supercomputing platform for nuclear physics.
This work is supported by the National Natural Science Foundation of China
under Grant No. 12105054 (X.Z.), the European Union (ERC, InitialConditions),
the VILLUM FONDEN (grant number 00025462), and Independent Research Fund
Denmark (Y.Z.), the National Natural Science Foundation of China under Grants
No.12325507, No.12147101, No. 11961131011, No. 11890710, No. 11890714, No.
11835002, the Strategic Priority Research Program of Chinese Academy of
Sciences under Grant No. XDB34030000, and the Guangdong Major Project of Basic
and Applied Basic Research under Grant No. 2020B0301030008 (G.M.), the
National Science Foundation under Grant No. 2310021 (Z.W.L.).

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