# Anisotropic flow fluctuation as a possible signature of clustered nuclear
geometry
in O–O collisions at the Large Hadron Collider

Suraj Prasad<EMAIL_ADDRESS>Neelkamal Mallick
<EMAIL_ADDRESS>Raghunath Sahoo<EMAIL_ADDRESS>(Corresponding Author) Department of Physics, Indian Institute of Technology
Indore, Simrol, Indore 453552, India  Gergely Gábor Barnaföldi
<EMAIL_ADDRESS>Wigner Research Center for Physics, 29-33
Konkoly-Thege Miklós Str., H-1121 Budapest, Hungary

###### Abstract

Nuclei having $4n$ number of nucleons are theorized to possess clusters of
$\alpha$ particles (4He nucleus). The Oxygen nucleus (16O) is a doubly magic
nucleus, where the presence of an $\alpha$-clustered nuclear structure grants
additional nuclear stability. In this study, we exploit the anisotropic flow
coefficients to discern the effects of an $\alpha$-clustered nuclear geometry
w.r.t. a Woods-Saxon nuclear distribution in O–O collisions at $\sqrt{s_{\rm
NN}}=7$ TeV using a hybrid of IP-Glasma + MUSIC + iSS + UrQMD models. In
addition, we use the multi-particle cumulants method to measure anisotropic
flow coefficients, such as elliptic flow ($v_{2}$) and triangular flow
($v_{3}$), as a function of collision centrality. Anisotropic flow
fluctuations, which are expected to be larger in small collision systems, are
also studied for the first time in O–O collisions. It is found that an
$\alpha$-clustered nuclear distribution gives rise to an enhanced value of
$v_{2}$ and $v_{3}$ towards the highest multiplicity classes. Consequently, a
rise in $v_{3}/v_{2}$ is also observed for the (0-10)% centrality class.
Further, for $\alpha$-clustered O–O collisions, fluctuations of $v_{2}$ are
larger for the most central collisions, which decrease towards the mid-central
collisions. In contrast, for a Woods-Saxon 16O nucleus, $v_{2}$ fluctuations
show an opposite behavior with centrality. This study, when confronted to
experimental data may reveal the importance of nuclear density profile on the
discussed observables.

##  I Introduction

The primary goal of relativistic heavy-ion collisions at the RHIC and the LHC
is to create a deconfined thermalised medium of partons, also known as Quark
Gluon Plasma (QGP), which is, by our understanding, have existed a few
microseconds after the Big Bang. However, due to the very low lifespan of QGP,
it can not be observed directly in experiments. Instead, several indirect
signatures from experimentally measurable observables can signify the presence
of such a deconfined strongly interacting matter in heavy-ion collisions. One
such observable is azimuthal anisotropy, which characterises the collective
behaviour of the medium formed in heavy-ion collisions. It is quantified in
terms of the Fourier expansion coefficients of azimuthal distribution of final
state charged hadrons, as follows [1]:

$\frac{dN}{d\phi}\propto\Bigg{(}1+2\sum_{n=1}^{\infty}v_{n}\cos\left[n(\phi-\psi_{n})\right]\Bigg{)}\
.$ (1)

Here, $\phi$ is the azimuthal angle, $\psi_{n}$ stands for the $n$th order
symmetry plane angle according to the give $v_{n}$ coefficients of the Fourier
expansion. Here, $v_{2}$ quantifies the elliptic flow, $v_{3}$ gives the
estimates for triangular flow, and so on. Both $v_{2}$ and $v_{3}$ depend upon
the equation of state and transport properties of the QCD medium formed during
heavy-ion collisions. In addition, studies based on hydrodynamical
calculations show that the flow coefficients are affected strongly by a change
in the ratio of shear viscosity to entropy density ($\eta/s$) [2, 3].
Moreover, the sensitivity of flow coefficients to $\eta/s$ is large for
higher-order harmonics, i.e., $v_{3}$ is more sensitive to $\eta/s$ as
compared to $v_{2}$ [3]. In several MC studies, $v_{2}$ and $v_{3}$ are found
to be associated with the initial eccentricities of the participating
nucleons, where the values of anisotropic flow coefficients can change by
modifying the nuclear distribution [6, 4, 5]. In addition, in Xe–Xe
collisions, an observation of a large value of $v_{2}$ in most central
collisions, as compared to that observed in Pb–Pb collisions, is anticipated
due to a deformed nuclear structure of Xe nucleus [7, 8, 10, 9]. In a similar
line, the nuclear deformation of Uranium, being a prolate shape nucleus is
reflected in a higher $v_{2}$, when compared to that is observed in the most
central collisions of Au nuclei, which is close to a spherical nucleus [11].
Similarly, one notices that the values of $v_{2}$ and $v_{3}$ vary with the
change in the deformation parameters of Uranium in U–U collisions, and the
effect is enhanced for $v_{2}$, as compared to $v_{3}$ [4, 5, 12].

Figure 1: Pictorial representation of an $\alpha$-clustered (left) and non-
clustered (right) distribution of nucleons inside the oxygen nucleus.

Elliptic flow is one of the observables that can probe the initial state of
heavy-ion collisions [13]. However, the value of elliptic flow fluctuates from
one event to another, which reflects the fluctuation in the initial energy
density and event-by-event participant nucleon distribution before QGP is
formed [14]. These fluctuations in the initial density profile of the
participating nucleons can lead to fluctuations in the even harmonic flow
coefficients and generate odd flow harmonics [14]. Further, the fluctuations
in the initial energy density can develop symmetry planes of different orders
in various kinematic regions [15, 16, 17]. In other words, the initial density
fluctuations can develop transverse momentum ($p_{\rm T}$) and pseudorapidity
($\eta$) dependence of $\psi_{n}$. Therefore, to comprehend the event-by-event
initial density fluctuations and their impact on medium evolution, the study
of anisotropic flow fluctuations becomes crucial. In both experimental and
theoretical frontiers, a few studies on anisotropic flow fluctuations have
been performed in heavy-ion collisions [18, 19, 20, 21]. The Pb–Pb collisions
at the LHC energies hint at non-Gaussian initial density fluctuations, and
consequently, it led the heavy-ion physicists to constrain their probability
distribution functions [22, 23]. However, the studies of anisotropic flow
fluctuations in small collision systems are not yet well understood. Moreover,
one may conceive that for a small number of participants, the contribution to
fluctuations can increase. Finally, if the evolution of anisotropic flow
fluctuations has more than one contributor, then the results can vary
significantly from the expectations.

In addition to the collective behaviour of the QGP, several other signatures
for the presence of a QGP medium have been observed in heavy-ion collisions.
For the measurement of such signatures, the baseline measurements are obtained
from proton-proton ($pp$) collisions, where the formation of a QCD medium is
not anticipated. However, recent observations of strangeness enhancement [24],
ridge-like structure [26, 25], radial flow effects [27, 28, 29], etc., in high
multiplicity $pp$ collisions have highlighted the significance of studying the
small collision systems from different aspects. Recently, LHC and RHIC have a
plan to inject Oxygen nuclei (16O) to perform p-O and O–O collisions [30, 31],
which can bridge the multiplicity gap between $pp$, p-Pb and Pb–Pb collisions
and may provide insight into the possible formation of QGP droplets in small
systems. Furthermore, Oxygen is a stable nucleus having a double magic number
and is theorized to possess clusters of $\alpha$ (4He) particles, where the
$\alpha$-particles arrange themselves in the corners of a tetrahedron [32, 33,
34, 35, 36, 37, 38]. Figure 1 shows a pictorial representation of the nucleons
inside an oxygen (16O) nucleus for an $\alpha$-clustered structure (left) and
non-clustered distribution of nucleons (right). In recent years, there have
been several studies performed to understand the O–O collisions both at RHIC
and LHC energies [44, 45, 43, 39, 40, 41, 6, 42, 46]. A few of them
investigate the use of hydrodynamical models [47, 48, 49], and Glauber model
calculations [39, 40, 41]. Some intend to study the parton energy loss and jet
quenching effects in O–O collisions [50, 51]. Interestingly, there have been a
few phenomenological studies that aim to establish the effect of the presence
of an $\alpha$-cluster nucleus in the final state particle production [44, 45,
43, 53, 54, 55, 56, 6, 52, 57, 58, 59]. A few of these studies employ O–O
collisions, while others employ O-Au, C-Au, O-Pb, and Ne-Pb collisions.
Nevertheless, the presence of a clustered nuclear structure of 16O is
reflected in final state azimuthal anisotropy when compared to a Woods-Saxon
nuclear density profile. In Ref. [52], using a multi-phase transport model,
authors have shown that an away side broadening is observed in the two-
particle correlation function for an $\alpha$-clustered particle. In addition,
an increased value of $v_{3}$, and $v_{3}/v_{2}$ for most central O–O
collisions are also reported for an $\alpha$-clustered structure. A similar
observation was reported in Ref. [44], where authors study the anisotropic
flow in C+Au collisions with a clustered structure of 12C nucleus. However, so
far, no studies have shown the effect of change in nuclear density profile on
event-by-event flow fluctuations. In this paper, for the first time, we report
the behaviour of azimuthal anisotropy and flow fluctuations in O–O collisions
at $\sqrt{s_{\rm NN}}=7$ TeV using a hydrodynamically expanding system. For
this study, we use a hybrid of IPGlasma initial condition, MUSIC
hydrodynamics, iSS as a particlisation and UrQMD as an afterburner for the
hadronic cascade. Here, we compared results from an $\alpha$-clustered nuclear
distribution with a Woods-Saxon nuclear distribution.

This paper is organized as follows. We start with a brief introduction and
motivation to this study in Section I. We briefly discuss the event generation
and methodology in Section II. Then, we present our findings and discuss the
results in Section III. Finally, we summarise our findings with a brief
outlook in Section IV.

##  II Event Generation and Methodology

In this section, we briefly introduce the hybrid simulation framework used to
generate events, including a discussion on the multi-particle cumulant method
for the anisotropic flow estimation.

###  II.1 Simulation framework

In this work, the simulation framework implements the impact-parameter-
dependent Glasma (IP-Glasma) initial conditions, MUSIC relativistic viscous
hydrodynamics, particle sampling using the iSS package, and performs hadronic
cascade through the UrQMD transport model. This model provides a good
description of the particle production and flow in heavy-ion collisions [60].
The key aspects of these models are also discussed briefly below, including
the parameters and settings necessary to reproduce the results reported in
this study.

####  II.1.1 IP-Glasma: Initial condition

The IP-Glasma model is used to describe the dynamics of the gluon fields
during the collisions of nuclear matter at relativistic speeds [61, 62]. It
provides the initial energy-momentum tensor of the classical Yang-Mills (CYM)
color Glasma fields ($T_{\rm YM}^{\mu\nu}$) which can be further evolved using
relativistic hydrodynamics. This model is based on the color glass condensate
effective theory [63, 64] and uses a classical description of gluon
production. This model includes fluctuations in the distributions of nucleons
in the nuclear wave functions and also fluctuations in the color charge
distributions inside the nucleons. The color charge fluctuations are modeled
via the impact-parameter-dependent dipole saturation model (IP-Sat) [65, 66].
For the sub-nucleonic fluctuations, three hot spots are introduced in the
proton thickness function, and each hot spot is parametrized using a Gaussian
distribution in the transverse plane. The spatial positions of the nucleons
inside the colliding nuclei are obtained from the Woods-Saxon distribution
function as given below,

$\rho(r)=\rho_{0}\frac{1+w\left(\frac{r}{r_{0}}\right)^{2}}{1+\exp\left(\frac{r-r_{0}}{a}\right)}\
.$ (2)

Here, $\rho(r)$ is the nuclear charge density at a radial distance $r$ from
the center of the nucleus. $r_{0}$, $w$, and $a$ are the mean radius of the
nucleus, the nuclear deformation parameter, and nuclear skin depth,
respectively. For 16O nucleus, $r_{0}=2.608$ fm, $a=0.513$ fm, $w=-0.051$
[67]. In addition, the exotic $\alpha$–cluster geometry for the 16O nucleus
has also been implemented in this study. Four such $\alpha$–particles
($2p,~{}2n$) are placed at the vertices of a regular tetrahedron with side
length 3.42 fm, which makes the root mean square (RMS) radius of 16O nucleus
to 2.699 fm [6, 42, 43]. The spatial positions of the nucleons inside an
$\alpha$–particle are sampled using the Woods-Saxon distribution given in Eq.
(2), with the parameters chosen for the 4He nucleus, which are $r_{0}=0.964$
fm, $w=0.517$ and $a=0.322$ fm. This gives the RMS radius of 1.676 fm for the
$\alpha$–particle. A minimum separation distance of 0.4 fm between the
nucleons is imposed. Finally, to randomize each nucleus, the entire
tetrahedral configuration is rotated through its center randomly in the polar
and azimuthal coordinates before its initialization in IP-Glasma. Figure 2
shows the nucleon distribution of 16O nuclei in the x-y plane following
$\alpha$-cluster (left) and Woods-Saxon (right) nuclear density profiles. In
Fig. 2, for the $\alpha$-cluster nuclear density profile of 16O nuclei, the
tetrahedral configuration is not rotated for better visualization of the
clustered geometry.

Figure 2: The nucleon distribution of 16O nuclei in the x-y plane following
$\alpha$-cluster (left) and Woods-Saxon (right) nuclear density profiles
plotted with one million nuclei in each case.

Once the nucleon centers are obtained, the color charges are sampled using the
IP-Sat/McLerran-Venugopalan model [64, 68]. Now, color currents arise due to
the motion of color charges present inside the colliding nuclei along the beam
direction. These currents act as the sources in the Yang-Mills equation,
$[D_{\mu},F^{\mu\nu}]=J^{\nu}$. The initial condition at $\tau=0$ is given by
the solution of CYM equations in the Schwinger gauge. For the subsequent
evolution of the equations of motion for Glasma fields, a lattice formulation
has been used, which was first introduced in Ref. [69]. These equations of
motion are solved using a leapfrog algorithm on a 2D lattice. For our case, we
use the lattice size, $L=14$, and lattice spacing $a=0.02$ fm [49]. Finally,
all the component of the energy-momentum tensor ($T_{\rm YM}^{\mu\nu}$) for
the Yang-Mills system is computed at $\tau_{\rm switch}=0.4$ fm, and the
initial-state configuration is transferred to the hydrodynamic simulation.

####  II.1.2 MUSIC: Hydrodynamics

Under the assumption of system under local thermal equilibrium, the initial
energy-momentum tensor obtained from the IP-Glasma model at thermalization
time $\tau_{0}=0.4$ fm is evolved using the MUSIC framework of relativistic
hydrodynamics following the conservation law $\partial_{\mu}T^{\mu\nu}=0$ [70,
71, 72]. The energy-momentum tensor is composed of both ideal and dissipative
parts, which are combined to give,
$T^{\mu\nu}=eu^{\mu}u^{\nu}-(P+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu}$. Here, $e$ is
the energy density, $u^{\mu}$ is the fluid flow four-velocity, $P$ is the
thermodynamic pressure, $\Pi$ is the bulk viscous pressure, and $\pi^{\mu\nu}$
is the shear viscous tensor. Again, the projection operator is defined as
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$. In this study, we use the boost
invariant picture of hydrodynamics and use the equation-of-state
parametrization "s95p-v1.2", which is obtained from the interface of lattice
data and hadron resonance gas [73]. We use a fixed $\eta/s=0.12$, and a
temperature dependent $\zeta/s(T)$ as described in Ref. [49]. In MUSIC, the
hydrodynamic equations are solved using the Kurganov-Tadmor (KT) algorithm
[74, 75].

####  II.1.3 iSS: Particle sampler

The fluid-dynamic description is finally switched to the particle description
when the local energy density drops to the switching energy density, $e_{\rm
sw}=0.18$ GeV/fm3 [49]. At this point, the freeze-out hypersurface from MUSIC
is fed to the iSS particle sampler [76, 77], which follows the usual Cooper-
Frye formalism for computing the particle spectra using equilibrium
distribution with both shear and bulk viscous correction terms [78, 79, 49].
Each MUSIC freeze-out hypersurface can be over-sampled using the iSS particle
sampler to increase event statistics and save computation time. So, we have
sampled 200 events from each MUSIC output hypersurface.

####  II.1.4 UrQMD: Hadronic cascade

Finally, the produced particles are evolved using the UrQMD microscopic
transport model [80, 81]. This model handles both elastic and inelastic
processes of hadronic scatterings and resonance decay with particles of masses
up to 2.25 GeV. UrQMD simulates a more realistic description of the final-
stage hadron evolution and performs a dynamical-freezeout for different
particle species. All parameters in the UrQMD model are set to their default
values. The output of UrQMD includes final-state particle four momenta and
particle identification (PID) code which can be written to disk for further
analysis.

###  II.2 Multiparticle cumulant method

Anisotropic flow can be expressed as the coefficients of Fourier expansion, as
shown in Eq. (1). One can estimate the anisotropic flow coefficients using the
following equation [82, 83, 84]:

$v_{n}=\langle\cos[n(\phi-\psi_{n})]\rangle\ .$ (3)

Here $\langle\dots\rangle$ stands for the average over all the charged
particles in a single event. Obtaining the anisotropic flow coefficients using
Eq. (3) requires $\psi_{n}$, whose estimation is not trivial in experiments.
In addition, Eq. (3) is prone to nonflow effects caused by resonance decays,
jet fragmentations, and Bose-Einstein correlations. To avoid the above-
mentioned uncertainties in the estimations of anisotropic flow coefficients,
we use a multi-particle Q-cumulant method [85, 86], which was used in the
experiments at RHIC [87] and LHC [89, 90, 88, 91, 92], as well. Since we aim
to estimate the anisotropic flow fluctuations, thus for this study, we limit
our measurements to two- and four-particle cumulants, which can be estimated
using the flow vector ($Q_{n}$) defined as follows.

$Q_{n}=\sum_{j=1}^{M}e^{in\phi_{j}},$ (4)

where $M$ is the multiplicity of the event, and $\phi$ is the azimuthal angle
of the charged hadrons. One can obtain a single event average two- and four-
particle correlation function using the following expressions.

$\displaystyle\langle 2\rangle$ $\displaystyle=\frac{|Q_{n}|^{2}-M}{M(M-1)},$
(5) $\displaystyle\langle 4\rangle$
$\displaystyle=\frac{|Q_{n}|^{4}+|Q_{2n}|^{2}-2\cdot{\rm{Re}}[Q_{2n}Q_{n}^{*}Q_{n}^{*}]}{M(M-1)(M-2)(M-3)}$
$\displaystyle~{}~{}-2\frac{2(M-2)\cdot|Q_{n}|^{2}-M(M-3)}{M(M-1)(M-2)},$

One can obtain the event-average correlation function using the following
expression.

$\displaystyle\langle\langle 2\rangle\rangle$
$\displaystyle=\frac{\sum_{i=1}^{N_{\rm ev}}(W_{\langle 2\rangle})_{i}\langle
2\rangle_{i}}{\sum_{i=1}^{N_{\rm ev}}(W_{\langle 2\rangle})_{i}},$ (6)
$\displaystyle\langle\langle 4\rangle\rangle$
$\displaystyle=\frac{\sum_{i=1}^{N_{\rm ev}}(W_{\langle 4\rangle})_{i}\langle
4\rangle_{i}}{\sum_{i=1}^{N_{\rm ev}}(W_{\langle 4\rangle})_{i}}.$

Here, $\langle\langle\dots\rangle\rangle$ denotes the average taken over all
the charged hadrons over all events. The ‘*’ stands for the complex conjugate.
$N_{\rm ev}$ is the total number of events used for the measurements. Finally,
$(W_{\langle 2\rangle})_{i}$ and $(W_{\langle 4\rangle})_{i}$ are the weight
factors for the $i$th event which takes into account the number of different
two- and four-particle combinations in the event of multiplicity $M$. Weight
factors, $W_{\langle 2\rangle}$ and $W_{\langle 4\rangle}$ can be calculated
using the following equations.

$\displaystyle W_{\langle 2\rangle}$ $\displaystyle=M(M-1),$ (7)
$\displaystyle W_{\langle 4\rangle}$ $\displaystyle=M(M-1)(M-2)(M-3).$

Consequently, the two- and four-particle cumulants can be obtained as:

$\displaystyle c_{n}\\{2\\}$ $\displaystyle=\langle\langle 2\rangle\rangle,$
(8) $\displaystyle c_{n}\\{4\\}$ $\displaystyle=\langle\langle
4\rangle\rangle-2\cdot\langle\langle 2\rangle\rangle^{2},$

Now, the reference flow of the particles can be estimated using the following
expression.

$\displaystyle v_{n}\\{2\\}$ $\displaystyle=\sqrt{c_{n}\\{2\\}},$ (9)
$\displaystyle v_{n}\\{4\\}$ $\displaystyle=\sqrt[4]{-c_{n}\\{4\\}}.$

To estimate the differential flow of the particles of interest (POIs), the
$p_{n}$ and $q_{n}$ vectors for specific kinematic range and/or for specific
hadron species are defined as follows:

$\displaystyle p_{n}$ $\displaystyle=\sum_{j=1}^{m_{p}}e^{in\phi_{j}},$ (10)
$\displaystyle q_{n}$ $\displaystyle=\sum_{j=1}^{m_{q}}e^{in\phi_{j}},$

where $m_{p}$ is the total number of particles labeled as POIs, and $m_{q}$ is
the total number of particles tagged as both reference flow particles (RFPs)
and POIs. RFPs serve as a reference frame for POIs to quantify the collective
motion of the system and help establish the orientation of the event plane,
which is crucial for determining the anisotropic flow coefficients. The
single-event average differential two- and four-particle azimuthal correlation
functions are calculated as:

$\displaystyle\langle 2^{{}^{\prime}}\rangle$
$\displaystyle=\frac{p_{n}Q_{n}^{*}-m_{q}}{m_{p}M-m_{q}},$ (11)
$\displaystyle\langle 4^{{}^{\prime}}\rangle$
$\displaystyle=\left[p_{n}Q_{n}Q_{n}^{*}Q_{n}^{*}-q_{2n}Q_{n}^{*}Q_{n}^{*}-p_{n}Q_{n}Q_{2n}^{*}\right.$
$\displaystyle-2\cdot Mp_{n}Q_{n}^{*}-2\cdot m_{q}|Q_{n}|^{2}+7\cdot
q_{n}Q_{n}^{*}-Q_{n}q_{n}^{*}$ $\displaystyle\left.+q_{2n}Q_{2n}^{*}+2\cdot
p_{n}Q_{n}^{*}+2\cdot m_{q}M-6\cdot m_{q}\right]$
$\displaystyle/\left[(m_{p}M-3m_{q})(M-1)(M-2)(M-3)\right]$

Similarly, one obtains the event-average differential azimuthal correlations
using the following expressions.

$\displaystyle\langle\langle 2^{{}^{\prime}}\rangle\rangle$
$\displaystyle=\frac{\sum_{i=1}^{N_{\rm ev}}(w_{\langle
2^{{}^{\prime}}\rangle})_{i}\langle
2^{{}^{\prime}}\rangle_{i}}{\sum_{i=1}^{N_{\rm ev}}(w_{\langle
2^{{}^{\prime}}\rangle})_{i}},$ (12) $\displaystyle\langle\langle
4^{{}^{\prime}}\rangle\rangle$ $\displaystyle=\frac{\sum_{i=1}^{N_{\rm
ev}}(w_{\langle 4^{{}^{\prime}}\rangle})_{i}\langle
4^{{}^{\prime}}\rangle_{i}}{\sum_{i=1}^{N_{\rm ev}}(w_{\langle
4^{{}^{\prime}}\rangle})_{i}}.$

Here, $w_{\langle 2^{{}^{\prime}}\rangle}$ and $w_{\langle
4^{{}^{\prime}}\rangle}$ are the weights corresponding to two- and four-
particle cumulants, given by:

$\displaystyle w_{\langle 2^{{}^{\prime}}\rangle}$
$\displaystyle=m_{p}M-m_{q},$ (13) $\displaystyle w_{\langle
4^{{}^{\prime}}\rangle}$ $\displaystyle=(m_{p}M-3m_{q})(M-1)(M-2)(M-3).$

Thus, one finds the two- and four-particle differential cumulants using the
following equations:

$\displaystyle d_{n}\\{2\\}$ $\displaystyle=\langle\langle
2^{{}^{\prime}}\rangle\rangle,$ (14) $\displaystyle d_{n}\\{4\\}$
$\displaystyle=\langle\langle 4^{{}^{\prime}}\rangle\rangle-2\langle\langle
2^{{}^{\prime}}\rangle\rangle\langle\langle 2\rangle\rangle.$

Finally, one can calculate the differential flow $v_{2}(p_{\rm T})$ using two-
and four-particle correlations as follows:

$\displaystyle v_{n}\\{2\\}(p_{\rm T})$
$\displaystyle=\frac{d_{n}\\{2\\}}{\sqrt{c_{n}\\{2\\}}},$ (15) $\displaystyle
v_{n}\\{4\\}(p_{\rm T})$
$\displaystyle=-\frac{d_{n}\\{4\\}}{\sqrt[4]{(-c_{n}\\{4\\})^{3}}}.$

Figure 3: Pictorial representation of the pseudorapidity regions $A$ and $B$
separated with a pseudorapidity gap ($\Delta\eta$).

Unfortunately, the $v_{n}$ obtained from the two-particle Q-Cumulant method
may contain contributions from non-flow effects, which can be suppressed by
appropriate kinematic cuts. One can introduce a pseudorapidity gap between the
particles in the two-particle Q-cumulant method [91, 94, 93] to suppress the
non-flow contributions. Accordingly, the whole event is divided into two sub-
events [95, 96, 97], $A$ and $B$, which are separated by a $|\Delta\eta|$ gap,
as depicted in Fig. 3. This modifies Eq. (5) to:

$\langle 2\rangle_{\Delta\eta}=\frac{Q_{n}^{A}\cdot Q_{n}^{B*}}{M_{A}\cdot
M_{B}},$ (16)

where $Q_{n}^{A}$ and $Q_{n}^{B}$ are the flow vectors from the sub-event $A$
and $B$, respectively. $M_{A}$ and $M_{B}$ are the corresponding
multiplicities. Thus, the two-particle Q-cumulant with a $|\Delta\eta|$ gap is
given by:

$c_{n}\\{2,|\Delta\eta|\\}=\langle\langle 2\rangle\rangle_{\Delta\eta}.$ (17)

In calculations of differential flow with a pseudorapidity gap, there is no
overlap of POIs and RFPs if we select RPs from one subevent and POIs from the
other. Thus Eq. (11) can be modified to:

$\langle
2^{{}^{\prime}}\rangle_{\Delta\eta}=\frac{p_{n,A}Q_{n,B}^{*}}{m_{p,A}M_{B}},$
(18)

and we obtain the differential two-particle cumulant as:

$d_{n}\\{2,|\Delta\eta|\\}=\langle\langle
2^{{}^{\prime}}\rangle\rangle_{\Delta\eta}.$ (19)

Finally, the differential flow from the two-particle cumulant can be obtained
by inserting the two-particle reference flow (with $\eta$ gap) into the
differential two-particle cumulant:

$v_{n}\\{2,|\Delta\eta|\\}(p_{\rm
T})=\frac{d_{n}\\{2,|\Delta\eta|\\}}{\sqrt{c_{n}\\{2,|\Delta\eta|\\}}}.$ (20)

In this paper, the elliptic and triangular flow coefficients are calculated
using the above equations by setting $n=$ 2 and 3, respectively. Using Eqs.
(9) and (20), one can obtain the mean and fluctuations of anisotropic flow
using the following expressions.

$\langle v_{n}\rangle=\sqrt{\frac{v_{n}^{2}\\{2\\}+v_{n}^{2}\\{4\\}}{2}}$ (21)
$\sigma_{v_{n}}=\sqrt{\frac{v_{n}^{2}\\{2\\}-v_{n}^{2}\\{4\\}}{2}}$ (22)

Using Eqs. (21), and (22), one can obtain relative anisotropic flow
fluctuation as follows.

$F(v_{n})=\frac{\sigma_{v_{n}}}{\langle v_{n}\rangle}$ (23)

In this study, we use all the charged hadrons within the pseudorapidity
region, $|\eta|<2.5$, for the estimation of anisotropic flow coefficients
using the two- and four-particle Q-cumulants method. We tag the charged
hadrons within $|\eta|<2.5$ and $0.2<p_{\rm T}<4.0$ GeV/c as RFPs. We apply a
pseudorapidity gap, $|\Delta\eta|>1.0$ in the two-subevent method, as shown in
Fig. 3, to subtract the non-flow effects from two-particle Q-cumulants method.
The statistical uncertainties are calculated using the relations shown in Ref.
[98].

##  III Results and discussions

In this section, we start with a brief discussion of the multiplicity
selection, followed by discussions on the results for flow coefficients.
Finally, we discuss the centrality dependence of flow fluctuations.

Figure 4: Distribution of charged particle produced in the pseudorapidity
acceptance of $-3.7<\eta<-1.7$ and $2.8<\eta<5.1$ (V0 Multiplicity or V0M) in
O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS +
UrQMD model for Woods-Saxon (upper) and $\alpha$-cluster (middle) nuclear
density profiles. The lower panel shows the ratio of V0M distributions from
Woods-Saxon to that of in $\alpha$-cluster nuclear profile.

Figure 4 shows the distribution of charged particles produced in the
pseudorapidity region $-3.7<\eta<-1.7$ and $2.8<\eta<5.1$ (V0M) in O–O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD
model for a Woods-Saxon (upper) and $\alpha$-cluster (lower) nuclear density
profiles. The lower panel shows the ratio of normalized yields of V0M for
Woods-Saxon and $\alpha$-cluster nuclear distributions. Also, different
centrality classes are defined based on the percentile slices on the V0M
distribution for the Woods-Saxon nuclear density profile. Here, for a fair
comparison, we use the same cuts of V0M distribution for all the centrality
classes for both the nuclear density profiles. The lowest central class, e.g.,
(0-10)%, refers to the most central collisions, while the highest centrality
class implies a peripheral collision. One can refer to Fig. 8 in the Appendix
.1 for the average value of the impact parameter corresponding to each
centrality class for both Woods-Saxon and $\alpha$-cluster nuclear density
profiles. In Fig. 4, one finds a higher yield for V0M for the O–O collisions
having an $\alpha$-clustered nucleus. In addition, for a similar value of V0M,
one finds a higher yield for the $\alpha$-clustered distribution on nucleons
inside the oxygen nuclei for V0M $>300$, which can be seen in the bottom ratio
plot. This signifies the compact nature of an $\alpha$-clustered oxygen
nucleus, which is reflected in the most central O–O collisions.

Figure 5: Upper panel shows the elliptic flow and triangular flow using two-
particle Q-cumulant method as a function of centrality class (%) in O–O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD
model for Woods-Saxon and $\alpha$-cluster nuclear density profiles. The lower
panel shows the ratio of $v_{n}\\{2\\}$ for Woods-Saxon to a $\alpha$-cluster
nuclear density profile. Figure 6: Upper panel shows the ratio
$v_{3}\\{2\\}/v_{2}\\{2\\}$ as a function of centrality class (%) in O–O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD
model for Woods-Saxon and $\alpha$-cluster nuclear density profiles. Lower
panel shows the ratio of $v_{3}\\{2\\}/v_{2}\\{2\\}$ from Woods-Saxon to a
$\alpha$-cluster nuclear density profile.

Figure 5 shows $v_{2}\\{2\\}$ and $v_{3}\\{2\\}$ with a pseudorapidity gap,
$|\Delta\eta|>1.0$, as a function of centrality for Woods-Saxon and
$\alpha$-cluster nuclear density profiles of 16O in O–O collisions at
$\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. One
notices a fair centrality dependence on both elliptic and triangular flow in
both Woods-Saxon and $\alpha$-cluster nuclear density profiles. The values of
$v_{2}\\{2\\}$ for the $\alpha$-cluster nuclear density profile are higher
than the Woods-Saxon nuclear density profiles throughout the centrality
classes except for the peripheral collisions, where the trend is reversed.
Here, the value of elliptic flow is highest for the central collisions and
decreases towards the peripheral collisions. In most central collisions, the
observed high values of elliptic flow can be attributed to the initial
eccentricity fluctuations. In contrast, towards the mid-central collisions,
although the geometry of the overlap region plays a role, the presence of a
lower number of participants makes the initial eccentricity completely
transform into the final state elliptic flow of hadrons, thus lowering the
value of elliptic flow. However, The observed behavior of elliptic flow as a
function of collision centrality and nuclear density profiles assuming a
hydrodynamical expansion vary significantly from that observed in a kinetic
theory-based model, such as AMPT [6]. In contrast, we observe an interesting
behavior for $v_{3}\\{2\\}$ as a function of collision centrality for both the
nuclear density profiles. One finds that $v_{3}\\{2\\}$ decreases from most
central to peripheral collisions, having similar slopes for the Woods-Saxon
and $\alpha$-clustered nuclear density profiles within (20-90)% centrality
classes. This consistency in the triangular flow between two nuclear density
profiles for the mentioned centrality class is consistent with the AMPT
predictions, as shown in Ref. [6]. Interestingly, it can be observed from the
lower panel of Figure 5, a sudden rise of $v_{3}\\{2\\}$ in the more central
collisions for the $\alpha$-clustered in contrast to a Woods-Saxon density
profile. These results are similar to the AMPT model calculations in Ref. [6].
We believe that the sudden rise in $v_{3}\\{2\\}$ for most central collisions
of oxygen nuclei having an $\alpha$-clustered nuclear density profile can be
attributed to the presence of an initial tetrahedral arrangement of four
$\alpha$-particles inside the oxygen nucleus.

Figure 7: Collision centrality dependence of $v_{2}$, $\langle v_{2}\rangle$,
$\sigma_{v_{2}}$, and $F(v_{2})$ in O–O collisions, from (a) to (d),
respectively, at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS +
UrQMD model for Woods-Saxon and $\alpha$-cluster nuclear density profiles.

The upper panel of Fig. 6 shows the ratio $v_{3}\\{2\\}/v_{2}\\{2\\}$ as a
function of collision centrality in O–O collisions at $\sqrt{s_{\rm NN}}=7$
TeV using IPGlasma + MUSIC + iSS + UrQMD model for Woods-Saxon and
$\alpha$-cluster nuclear density profiles. The bottom panel shows the ratio of
$v_{3}\\{2\\}/v_{2}\\{2\\}$ for Woods-Saxon to a $\alpha$-cluster nuclear
density profile. Interestingly, when the oxygen nuclei having the Woods-Saxon
nuclear distributions collide, their elliptic and triangular flow values
follow a smooth dependence on collision centrality, as one can see in Fig. 5.
Thus, the ratio $v_{3}\\{2\\}/v_{2}\\{2\\}$ also exhibits a smooth decrease
from most central-to-peripheral collisions. However, one finds a notably sharp
increase in $v_{3}\\{2\\}/v_{2}\\{2\\}$ for (0-5)% centrality case for the
$\alpha$-cluster case, followed by a smooth decrease in
$v_{3}\\{2\\}/v_{2}\\{2\\}$ as a function of collision centrality. This
reflects the presence of a tetrahedral arrangement of $\alpha$-clusters inside
the oxygen nuclei giving rise to a large intrinsic triangular flow in the most
central collisions. Although one finds $v_{2}\\{2\\}$ for $\alpha$-cluster
density profile is larger compared to that of Woods-Saxon nuclear density
profile, this tetrahedral structure raises $v_{3}\\{2\\}$ even higher for the
most central collisions, which is also consistent with transport models such
as AMPT, shown in Ref. [6]. Thus the observation of a sudden rise in
tetrahedral $\alpha$-clustered case can lead to an observation of a spike in
$v_{3}\\{2\\}/v_{2}\\{2\\}$ for most central collisions and consequently, the
observation of which in O–O collisions may signify the presence of a
$\alpha$-clustered nuclear geometry of 16O nuclei.

Figure 7 (a) shows the evolution of elliptic flow using two- and four-particle
Q-cumulant method as a function of centrality for Woods-Saxon and
$\alpha$-cluster nuclear density profiles in O–O collisions at $\sqrt{s_{\rm
NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. One finds a small
centrality dependence for $v_{2}\\{2,|\Delta\eta|>1.0\\}$ for both the nuclear
density profiles within (0-30)% centrality. In contrast, using the four-
particle cumulants method, we observe a significant centrality dependence of
elliptic flow. The value of $v_{2}\\{4\\}$ for the $\alpha$-cluster case
increases from (0-10)% to (10-20)%, and again decreases for (20-30)%
centrality. However, for the Woods-Saxon case, $v_{2}\\{4\\}$ maintains a
consistent decreasing feature with an increase in the centrality. Another
important point to note here is that
$v_{2}\\{2,|\Delta\eta|>1.0\\}>v_{2}\\{4\\}$ for both the nuclear density
profiles, which can consequently lead to real values of flow fluctuations.
Figure 7 (b) shows the mean value of elliptic flow ($\langle v_{2}\rangle$),
evaluated using Eq. (21), for Woods-Saxon and $\alpha$-cluster nuclear density
profiles in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC
+ iSS + UrQMD model. $\langle v_{2}\rangle$ is found to be larger for O–O
collisions having a $\alpha$-cluster nuclear density profile as compared to a
Woods-Saxon distribution. The centrality dependence of $\langle v_{2}\rangle$
for both the nuclear density profiles is similar to $v_{2}\\{4\\}$, as
$v_{2}\\{2,|\Delta\eta|>1.0\\}$ is nearly independent of collision centrality
in (0-30)% centrality region. Figure 7 (c) shows a centrality dependence of
elliptic flow fluctuation ($\sigma_{v_{2}}$) for Woods-Saxon and
$\alpha$-cluster nuclear density profiles in O–O collisions at $\sqrt{s_{\rm
NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. One finds that
$\sigma_{v_{2}}$ for Woods-Saxon and $\alpha$-cluster nuclear density profiles
have complementing centrality dependence, i.e., when going from most central
to mid-central collisions, $\sigma_{v_{2}}$ for Woods-Saxon increases while
for an $\alpha$-cluster nuclei it shows a decreasing trend. This makes one of
the interesting distinctions between the two nuclear density profiles with the
study of flow fluctuations. Although, in Pb–Pb collisions, due to the absence
of elliptic geometry in the initial state, the elliptic flow in the most-
central collisions has a large contribution from elliptic flow fluctuations;
however, towards the mid-central collisions, the contribution of elliptic
flow-fluctuations to elliptic flow decreases, as the elliptic geometry of the
nuclear participants leads the elliptic flow contribution [93, 96, 97].
Interestingly, the $\alpha$-cluster case of O–O collisions shows a similarity
in the evolution of elliptic flow fluctuations with the Pb–Pb collisions,
which is clearly different in O–O collisions with a Woods-Saxon nuclear
density profile. The source of this opposite behavior of $\sigma_{v_{2}}$ for
both the nuclear density profiles can thus intrinsically be attributed to the
difference in the nuclear geometry of the 16O nucleus. Interestingly, the
centrality dependence of $\langle v_{2}\rangle$ and $\sigma_{v_{2}}$ are
complementary to each other for the Woods-Saxon case in O–O collisions.
However, this correlation between $\langle v_{2}\rangle$ and $\sigma_{v_{2}}$
is not completely applicable for O–O collisions with an $\alpha$-cluster
nuclear density profile. Figure 7 (d) shows the evolution of relative elliptic
flow fluctuation ($F(v_{2})$) as a function of collision centrality in O–O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV. Interestingly, we find a similar
centrality dependence of $F(v_{2})$ as found for $\sigma_{v_{2}}$. The
centrality dependence of $F(v_{2})$ in O–O collisions with Woods-Saxon nuclear
distribution resembles a p–Pb collision [93]. In contrast, for
$\alpha$-cluster nuclear density profiles, $F(v_{2})$ is higher for most
central collisions and decreases towards the mid-central collisions. This is
an important observation where we find that a change in the initial nucleon
distribution (Woods-Saxon versus $\alpha$-cluster) can cause a difference in
the collision centrality dependence of relative elliptic flow fluctuations for
a hydro-dynamically evolving system.

##  IV Summary

In this paper, we study the anisotropic flow coefficients in O–O collisions at
$\sqrt{s_{\rm NN}}=7$ TeV using a hybrid of IPGlasma + MUSIC + iSS + UrQMD
model and try to probe the final state effects of the presence of an exotic
$\alpha$-cluster nuclear geometry in contrast to a uniform Woods-Saxon nuclear
distribution inside a 16O nucleus. Here, we study the centrality dependence of
anisotropic flow coefficients. In addition, for the first time, we report
studies of elliptic flow fluctuations as a motivation to distinguish O–O
collisions having $\alpha$-cluster geometry from a regular Woods-Saxon
distribution. The observation of large triangular flow in the most central O–O
collisions having an $\alpha$-clustered nuclei compared to a Woods-Saxon
nuclear distribution is consistent with transport model predictions. In
addition, we observe a significant centrality dependence of relative elliptic
flow fluctuations in both the nuclear distributions. With this study, it is
clear that the presence of an $\alpha$-cluster nuclear geometry has a
significant effect on both elliptic and triangular flow. However, we note a
distinctive feature appears when observables related to fluctuations are
studied, i.e., triangular flow and elliptic flow fluctuations. While probing a
nuclear density profile is a subject matter of low-energy nuclear physics, it
will be very interesting to find an appropriate observable that can be
measured in TeV nuclear collisions to do the job. This makes the present study
more interesting.

## Acknowledgements

S. P. acknowledges the doctoral fellowship from UGC, Government of India. R.
S. sincerely acknowledges the DAE-DST, Government of India funding under the
mega-science project – “Indian participation in the ALICE experiment at CERN”
bearing Project No. SR/MF/PS-02/2021-IITI (E-37123). G. G. B. gratefully
acknowledges the Hungarian National Research, Development and Innovation
Office (NKFIH) under Contracts No. OTKA K135515, No. NKFIH NEMZ_KI-2022-00031
and Wigner Scientific Computing Laboratory (WSCLAB, the former Wigner GPU
Laboratory). The authors gratefully acknowledge the MoU between IIT Indore and
Wigner Research Centre for Physics (WRCP), Hungary, for the techno-scientific
international cooperation.

## Appendix

###  .1 Centrality versus impact parameter

Figure 8: Average value of impact parameter as a function of centrality class
in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS +
UrQMD for a Woods-Saxon and $\alpha$-cluster nuclear density profile.

Figure 8 shows the average value of the impact parameter ($\langle b\rangle$)
as a function of centrality class in O–O collisions at $\sqrt{s_{\rm NN}}=7$
TeV using IPGlasma + MUSIC + iSS + UrQMD for a Woods-Saxon and
$\alpha$-cluster nuclear density profiles. As expected, one finds a
significant dependence of $\langle b\rangle$ on collision centrality selection
based on the final state-charged particle multiplicity in the V0 acceptance
region, i.e., the collisions having a lower impact parameter tend to produce a
higher charged particle yield at the final state. In addition, one finds a
significant dependence of $\langle b\rangle$ on the nuclear density profile of
the colliding oxygen nucleus to produce a similar charge particle yield in the
final state. For the $\alpha$-cluster case, one finds that $\langle b\rangle$
is significantly smaller compared to a Woods-Saxon density profile because of
the compact nature of $\alpha$-cluster nuclear structure, the effects of which
are enhanced in the peripheral collisions. In addition, the compact nature of
$\alpha$-cluster nuclear density profile is also reflected in the higher yield
in most central collisions, as shown in Fig. 4.

###  .2 Four particle cumulants

Figure 9: Upper panel shows the second and third-order four-particle
Q-cumulants ($c_{n}\\{4\\}$) as a function of multiplicity class (%) in O–O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD
model for Woods-Saxon and $\alpha$-cluster nuclear density profiles. Lower
panel shows the ratio of $c_{n}\\{4\\}$ from Woods-Saxon to a $\alpha$-cluster
nuclear density profile.

The upper panel of Fig. 9 shows the centrality dependence of four-particle
cumulants ($c_{n}\\{4\\}$) as a function of collision centrality in O–O
collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD
model for Woods-Saxon and $\alpha$-cluster nuclear density profiles. The lower
panel depicts the ratio of $c_{n}\\{4\\}$ from Woods-Saxon to a
$\alpha$-cluster nuclear density profile. One can note that, $v_{n}\\{4\\}$ is
related with $c_{n}\\{4\\}$ using Eq. (15), thus a negative value of
$c_{n}\\{4\\}$ results in a physically acceptable value of $v_{n}\\{4\\}$.
However, a positive value of $c_{n}\\{4\\}$ leads to an imaginary value of
$v_{n}\\{4\\}$, which sometimes occur when the system is dominated with non-
flow contributions [93]. In the upper panel of Fig. 9, one finds a significant
difference of $c_{n}\\{4\\}$ between two nuclear density profiles. For an
$\alpha$-cluster case, $c_{2}\\{4\\}$ starts with a small negative value and
starts to decrease further with an increase in collision centrality, i.e.,
towards (10-20)% centrality, attains a minimum and goes on increasing, until
attains a positive value beyond (30-40)% centrality class. In contrast,
$c_{2}\\{4\\}$ for a Woods-Saxon nuclear density profile starts a minimum
value for the most central case, i.e., (0-10)% centrality class, and starts to
rise, becomes positive beyond (30-40)% centrality. Interestingly, for both the
nuclear density profiles, $c_{2}\\{4\\}$ attains a positive value beyond a
similar (30-40)% centrality class, which naively implies the dominance of non-
flow contribution for events with low final state multiplicity. The positive
values of $c_{2}\\{4\\}$ on and beyond (30-40)% centrality class restricts us
from the calculation of four-particle elliptic flow and its fluctuations below
(30-40)% centrality class. On the other hand, $c_{3}\\{4\\}$ for both the
nuclear density profiles fluctuate around 0, with a much smaller value as
compared to corresponding $c_{2}\\{4\\}$. The positive values of
$c_{3}\\{4\\}$ correspond to large non-flow azimuthal correlations.

###  .3 $p_{\rm T}$-dependence of anisotropic flow

Figure 10: Upper panel shows the elliptic flow and triangular flow using two-
particle Q-cumulant method as a function of transverse momentum for different
multiplicity classes in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using
IPGlasma + MUSIC + iSS + UrQMD model for Woods-Saxon and $\alpha$-cluster
nuclear density profiles. The lower panel shows the ratio of
$v_{n}\\{2\\}(p_{\rm T})$ for the Woods-Saxon density profile to
$\alpha$-cluster density profile.

Figure 10 shows the transverse momentum dependence of elliptic and triangular
flow calculated using the two-particle Q-cumulants method in O–O collisions at
$\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model for
Woods-Saxon and $\alpha$-cluster nuclear density profiles in the upper panel.
The bottom panel shows the ratio of $v_{n}\\{2\\}(p_{\rm T})$ the Woods-Saxon
density profile to $\alpha$-cluster density profile. One finds both
$v_{2}\\{2\\}(p_{\rm T})$ and $v_{3}\\{2\\}(p_{\rm T})$ rise linearly with an
increase in $p_{\rm T}$, which is a exclusive feature of hydro-based models.
In addition, one observes that for the most central (0-10)% class,
$v_{2}\\{2\\}(p_{\rm T})$ and $v_{3}\\{2\\}(p_{\rm T})$ are closer to one
another, where the separation further increases as one moves towards the mid-
central collisions. This is because the contributions for $v_{2}\\{2\\}(p_{\rm
T})$ and $v_{3}\\{2\\}(p_{\rm T})$ are the consequences of the interplay
between the geometry of the overlap region and the eccentricity fluctuations
during the collisions, etc., where the eccentricity fluctuations dominate the
contributions to the anisotropic flow in the central collisions. Another
interesting observation is that, in the most central collisions,
$v_{n}\\{2\\}(p_{\rm T})$ for the $\alpha$-cluster case is larger compared to
corresponding values from a Woods-Saxon distribution. This observation is
consistent with the Fig 5. However, towards the mid-central collisions,
$v_{n}\\{2\\}(p_{\rm T})$ from Woods-Saxon nuclear distribution is larger
compared to an $\alpha$-cluster nuclear density profile. For the central
collisions, the lower ratio plot of Woods-Saxon to an $\alpha$-cluster nuclear
density profile for both elliptic and triangular flow shows a flat ratio,
almost independent of $p_{\rm T}$. However, it no longer holds towards the
mid-central collisions, indicating a $p_{\rm T}$ dependence of medium effects
for different nuclear profiles to the anisotropic flow coefficients.

## Software

In this work, we used publicly available software for data generation. The
software for IPGlasma, MUSIC, iSS and UrQMD can be accessed using the web
links provided in Refs. [99, 100, 101, 102].

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