††thanks: Email<EMAIL_ADDRESS>Email<EMAIL_ADDRESS>
# Signatures of an $\alpha$ \+ core structure in 44Ti + 44Ti collisions at
$\sqrt{s_{NN}}=5.02$ TeV by a multiphase transport model

Yu-Xuan Zhang(张宇轩)  Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
Song Zhang(张松) ID 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  Yu-Gang Ma(马余刚) ID 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

###### Abstract

It is important to understand whether $\alpha$-clustering structures can leave
traces in ultra-relativistic heavy ion collisions. Using the modified AMPT
model, we simulate three $\alpha$ \+ core configurations of 44Ti in 44Ti+44Ti
collisions at $\sqrt{s_{NN}}=5.02$ TeV as well as other systems with Woods-
Saxon structures. One of these configurations has no additional constraint,
but the other two have the Mott density edge $r_{\mathrm{Mott}}$ set as either
a lower or upper bound on the cluster position $r_{\alpha}$ to check the
influence of $\alpha$ dissolution. This is the first time that the initial
stage of the geometric properties in heavy-ion collisions has been configured
using the traditional treatment of the nuclear structure. We compare the
radial nucleon density, multiplicity distribution, transverse momentum
spectra, eccentricity, triangularity, elliptic flow and triangular flow of
these six systems. $\alpha$ \+ core structures can alter all these
observations especially in the most-central collisions, among which elliptic
flow is the most hopeful as a probe of such structures.

##  I Introduction

Predicted by quantum chromodynamics (QCD) [1], the confinement of hadronic
matter may be broken under extreme conditions at high temperature or density,
resulting in a new matter state called quark-gluon plasma (QGP) [2]. QGP
contains deconfined quarks and gluons, providing us a chance to understand the
essence of strong interaction on a new scale, which can help develop QCD
further in return. Besides, the extreme conditions required by QGP are
expected to exist in the early universe right after the Big Bang [3], so the
investigation of QGP is also of great importance for revealing the universe
origin. One of the important tools to gain such conditions is the ultra-
relativistic heavy-ion collision, which is currently performed by the CERN
(European Organization for Nuclear Research) Large Hadron Collider (LHC) [4,
5, 6] and the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven
National Laboratory [7, 8, 9]. Extensive studies are helpful in exploring the
QCD phase structure and diagram [10, 11, 12, 13, 14]. Among the results
presented by different QGP probes, particle production [15] and collectivity
[16] play instrumental roles in describing the evolution of collision systems.
It is widely recognized that they reflect the initial state of system and
efforts to explain them usually involve modeling of the basic collision
mechanism.

Recently, the influences of nuclear structure are taken into account in this
community by transport or hydrodynamics models configured with initial nuclear
structure [17, 18, 19, 20, 21, 22, 23]. And these works suggest that there is
a potential window on investigating nuclear structure by relativistic heavy-
ion collisions [24, 25]. Among various nuclear structures, the
$\alpha$-clustering structure inside nuclei is of particular interest. The
$\alpha$ cluster model, first proposed by Gamow [26], has been demonstrated to
be a powerful tool in describing nuclear structure [27, 28, 29], $\alpha$
decay [30, 31], ground state bands [32] and so on. In this model, light nuclei
could be thought to be made of $\alpha$ clusters as well as some nucleons or
other smaller clusters [27, 33, 34, 35], while for heavy nuclei, only part of
nucleons may cluster and a core could be formed by the remained nucleons [30,
32, 36]. The clustering effect is important to nuclear equations of state,
nucleosynthesis and many other problems [37, 38, 39, 40]. Various observables
have therefore been proposed to study the clustering of nuclei in the heavy-
ion reaction, such as collective flow [41, 42, 43, 44], multiplicity
correlation [45, 46] as well as giant resonance [38, 47, 48, 49] etc. It is
still an interesting question whether such cluster structures will form
signatures in ultra-relativistic heavy-ion collisions. Some review on
$\alpha$-clustering effects can be found in [50, 51, 24, 52]. For light
nuclei, there is prediction [41, 53, 54] that clustering may lead to the
variance of harmonic flow measures, implying a granular geometry preserved in
ultra-relativistic heavy-ion collisions. Some positive simulation results [54,
42, 55, 56] have been reported under a multiphase transport (AMPT) model [57].
However, for heavy nuclei, behaviors of $\alpha$-cluster structures remain
unclear in ultra-relativistic collisions, and further investigation is
required.

Among heavy nuclei, those with potential $\alpha$ \+ doubly magic core
structures draw extra concentration since they match well with the binary
$\alpha$ cluster model [32, 58] and avoid a complex many-body problem [59].
Through a modified AMPT model, we test this cluster model by simulating 44Ti +
44Ti collisions at $\sqrt{s_{NN}}=5.02$ TeV with $\alpha$ \+ core ($\alpha$ \+
c) or Woods-Saxon (W-S) structures. We analyze the influence of this mode on
yields and harmonic flow of major charged particles ($\pi^{\pm}$, $K^{\pm}$,
$p$, $\bar{p}$) and find possible signatures related to the $\alpha$ \+ 40Ca
structure. Sec. II presents our approaches to acquiring the $\alpha$ \+ core
simulation and analyzing the results. Results and discussion are shown in Sec.
III, and the last section is the conclusion.

##  II METHODOLOGY

Figure 1: The normalized radial component $f_{L}(r_{\alpha})$ of the $\alpha$
cluster wave function for 44Ti at the ground state (a) and the cumulative
$r_{\alpha}$ distribution function $F_{L}(r_{\alpha})$ (b).

The AMPT model is a hybrid transport model aimed at simulating heavy ion
collisions at RHIC and LHC energy [57, 60]. It contains several sub-models,
such as the heavy-ion jet interaction generator (HIJING) model [61], Zhang’s
parton cascade (ZPC) model [62], the Lund JETSET fragmentation model [63, 64],
a quark coalescence model and a relativistic transport (ART) model [65]. In
our work, heavy-ion collisions at the center of mass energy
$\sqrt{s_{NN}}=5.02$ TeV will reach ultra-high temperature, leading to high
QGP formation possibility. Hence, a string-melting version of the AMPT model
is chosen for our simulation. In this model version, all excited strings are
fragmented into partons and hadronized together with minijet partons, which is
closer to the QGP case.

In the AMPT model, the initial conditions of the collisions are provided by
the HIJING model, where projectile and target nuclei are shaped into Woods-
Saxon distributions and then the positions of constituent nucleons are set
event by event. A 3-parameter Fermi function of the position $\mathbf{r}$ is
used to describe Woods-Saxon distributions:

$\displaystyle\begin{aligned} f(r)=&A\frac{1+\omega
r^{2}/c^{2}}{1+\exp[(r-c)/z]}.\end{aligned}$ (1)

Here $A$ is the normalization factor, $c$ is the radius parameter, $z$ is the
diffusion parameter, and $\omega$ is the newly added third parameter. If
$\omega<0$, this function is cut off at $1+\omega r^{2}/c^{2}=0$. For 40Ca,
there are optimized parameters preset in the HIJING model, while for 44Ti,
only automatically fitted data is available. 50Ti, a natural nuclei with a
neutron magic number $N=28$, is considered as a singly closed shell core in
the binary $\alpha$ cluster model [66] and also not preset in the HIJING
model. We additionally include it in our simulation to evaluate this fitting
and test the system size effects.

Instead of fixed cluster configurations common in light nucleus research [54,
42], a non-localized $\alpha$ \+ 40Ca structure is introduced to modify the
initial conditions by using the local potential model (LPM) [67, 30]. This
structure features global motion of clusters and has well described the alpha
condensate state of light nuclei like 12C [68] and the $\alpha$ \+ core
structure of heavy nuclei such as 20Ne [69] and 212Po [59]. In LPM, we utilize
the cluster position $r_{\alpha}$ [67, 30] to describe the $\alpha$ cluster
potential $V(\mathbf{r_{\alpha}})$, which can be divided into the Coulomb
potential $V_{C}(r_{\alpha})$, the centrifugal potential $V_{L}(r_{\alpha})$,
and the nuclear potential $V_{N}(\mathbf{r_{\alpha}})$. With a uniformly
charged spherical core assumption, $V_{C}(r_{\alpha})$ can be written as

$\displaystyle V_{C}(r_{\alpha})=\left\\{\begin{aligned}
&\frac{Z_{\alpha}Z_{c}e^{2}}{2R}\left(3-\frac{r_{\alpha}^{2}}{R^{2}}\right),&r_{\alpha}<R\\\
&\frac{Z_{\alpha}Z_{c}e^{2}}{r_{\alpha}},&r_{\alpha}\geq
R\end{aligned}\right.$ (2)

where $Z_{\alpha}$ and $Z_{c}$ are the charge numbers of the $\alpha$ cluster
and the core, and $R$ is the core radius decided later by fitting
$V_{N}(\bm{r_{\alpha}})$. The form of $V_{L}(r_{\alpha})$ is

$\displaystyle
V_{L}(r_{\alpha})=\frac{\hbar^{2}}{2\mu_{\alpha}r_{\alpha}^{2}}L(L+1),$ (3)

with $L$ the azimuthal quantum number and $\mu_{\alpha}$ is the reduced mass
in $\alpha$ two-body systems. Here we suppose that projectile and target
nuclei are at their ground states, so $L$ is equal to zero and
$V_{L}(r_{\alpha})$ can be neglected. As for $V_{N}(\bm{r_{\alpha}})$, we
select a (1 + Gaussian) $\times$ (W.S. + W.S.3) potential model [58], with
which $V_{N}(\mathbf{r_{\alpha}})$ can be simply decided by the length
$r_{\alpha}$. In this model there is

$\displaystyle\begin{aligned} V_{N}(r_{\alpha})=&-V_{0}\left(1+\lambda
e^{-r_{\alpha}^{2}/\sigma^{2}}\right)\left\\{\frac{b}{1+e^{(r_{\alpha}-R)/a}}+\right.\\\
&\left.\frac{1-b}{[1+e^{(r_{\alpha}-R)/3a}]^{3}}\right\\},\end{aligned}$ (4)

where $V_{0}$, $\lambda$, $a$ and $b$ are fixed parameters, $R$ and $\sigma$
are free parameters for fitting. Here these parameters are the same as in Ref.
[58], which well describes the ground state bands of 44Ti. Specifically, we
set $V_{0}=220~{}\mathrm{MeV}$, $a=0.65~{}\mathrm{fm}$, $b=0.3$,
$\lambda=0.14$, $R=4.551~{}\mathrm{fm}$, and $\sigma=0.425~{}\mathrm{fm}$.
Then with $\alpha$ decay energy $E_{\alpha}=-5.1271~{}\mathrm{MeV}$ [70], we
can gain the cluster’s radial wave function $\varphi_{L}(\mathbf{r_{\alpha}})$
by numerically solve its stationary Schr$\ddot{\mathrm{o}}$dinger equation.
Figure 1 (a) shows the normalized radial component of the cluster wave
function
$f_{L}(r_{\alpha})=\varphi_{L}(\mathbf{r_{\alpha}})r_{\alpha}/Y_{LM}(\theta,\phi)$.
The probability density of $r_{\alpha}$ can be gained from
$\rho(r_{\alpha})=|f_{L}(r_{\alpha})|^{2}$ and the corresponding cumulative
distribution function $F_{L}(r_{\alpha})$ is presented in Fig. 1 (b).

With $F_{L}(r_{\alpha})$, we are able to insert an $\alpha$ cluster before
core nucleons are placed. In the binary $\alpha$ cluster model, external
influence on cluster or core nucleons is usually neglected, so here both
$\alpha$ clusters and 40Ca cores have Woods-Saxon inner structures, and their
parameters are simply decided by the HIJING preset data. For $\alpha$, it
means $c=0.964~{}\mathrm{fm}$, $z=0.322~{}\mathrm{fm}$, and $\omega=0.517$ in
Eq. (1). For 40Ca, there is $c=3.766~{}\mathrm{fm}$, $z=0.586~{}\mathrm{fm}$,
and $\omega=-0.161$. According to Ref. [59], the Pauli blocking may lead to
the dissolution of $\alpha$ clusters at the Mott density
$\rho_{\mathrm{Mott}}=0.02917~{}\mathrm{fm^{-3}}$, which is
$r_{\mathrm{Mott}}=4.498~{}\mathrm{fm}$ for 40Ca. This Mott density of a
nucleus is given by the maximum density at which a nucleus of zero momentum
can still be bound [71, 72]. That is to say, $\alpha$ clusters may not always
exist in the $\alpha$ \+ core nuclei even if the model fully matches the
reality. As pointed out in Refs. [30, 73, 74], the $\alpha$ cluster
preformation probability $P_{\alpha}$ can be estimated by calculating its
distribution outside $r_{\mathrm{Mott}}$. And according to these references,
$P_{\alpha}$ usually has a magnitude of a few tenths, which can be high enough
to form observable alteration. Shown in Fig. 1 (b), about 55% of clusters are
in $r_{\alpha}>r_{\mathrm{Mott}}$ for the classical binary cluster model,
which is acceptable considering the existing results. Since the clustering
state is not completely dominant, we also test the influence of cluster
dissolution by cutting off $r_{\alpha}$ at $r_{\mathrm{Mott}}$ like in
$P_{\alpha}$ estimation in addition to the widely adopted no dissolution case.
Though $r_{\alpha}<r_{\mathrm{Mott}}$ case results in the breaking of the
nuclear matter saturation as discussed following, the simulation can be a
comparison for $\alpha$ cluster effect with other cases.

Figure 2: The nucleon density $\rho$ of projectile and target nuclei shown
linearly in the main graph and logarithmically in the top right as a function
of the distance $r$. Figure 3: The multiplicity distribution of major charged
particles within $|\eta|<0.5$ (a) and ratios to the Woods-Saxon structure in
44Ti + 44Ti systems (b).

To characterize the initial geometry, we utilize the eccentricity
$\varepsilon_{n}$ of participant partons and the anisotropic flow $v_{n}$ of
final charged particles. $\varepsilon_{n}$ is a direct description of spatial
anisotropy. In specific heavy-ion collision, it can be defined as [55]

$\displaystyle\varepsilon_{n}=\frac{\sqrt{\left\langle
r^{n}\cos(n\varphi_{part})\right\rangle^{2}+\left\langle
r^{n}\sin(n\varphi_{part})\right\rangle^{2}}}{\left\langle
r^{n}\right\rangle},$ (5)

where $\varphi_{part}$ is the azimuthal angle and $r$ the position.
$\varepsilon_{2}$ is usually called eccentricity and $\varepsilon_{3}$ is
called triangularity. $\left<\cdots\right>$ denotes average over participant
nucleons here, but later $\left<\varepsilon_{n}\right>$ means average over
events. Hydrodynamics demonstrates a picture that the initial asymmetry in
coordinate space will transfer to the final momentum space [75]. And we should
point out the initial geometry asymmetry in the collisions includes the
intrinsic structures and the overlapped region of the colliding nuclei, as
well as fluctuation. In actual collision experiments, the participant parton
positions are undetectable due to quark confinement, so description from
collective flow reflecting anisotropy of particles at the final state is also
necessary. On the other hand, the investigation of the final momentum space
will disclose the properties of the initial stage and in further the intrinsic
structure of nuclei.

The anisotropic components of collective flow can be characterized by the
Fourier expansion of particle momentum distribution [76, 77, 78, 79, 80, 81]:

$\displaystyle
E\frac{\mathrm{d}^{3}N}{\mathrm{d}^{3}p}=\frac{1}{2\pi}\frac{\mathrm{d}^{2}N}{p_{\mathrm{T}}\mathrm{d}p_{\mathrm{T}}dy}\left\\{1+\sum_{n=1}^{\infty}2v_{n}\cos[n(\varphi-\Psi_{n})]\right\\}.$
(6)

Here $E$ is the particle energy, $p_{\mathrm{T}}$ is the transverse momentum,
and $y$ is the rapidity. The Fourier expansion coefficient $v_{n}$ is the
$n$-th order of the anisotropic flow and $\Psi_{n}$ is the corresponding event
plane angle. Among all orders of flow, the elliptic flow $v_{2}$ and the
triangular flow $v_{3}$ draw extra attention since they represent the initial
collision geometry and its fluctuations, respectively [82]. A common way to
extract anisotropic flow is the cumulant method [83], which allows us to build
multi-particle azimuthal correlations without looping over all particle
multiplets. In a two sub-event case, with
$\bm{Q}_{n}=\sum_{i=1}^{M}e^{in\varphi_{i}}$ storing the azimuthal angle
information of $M$ particles, the two-particle correlation and its average
over all events in this method can be written as [84]

$\displaystyle\begin{aligned} \left\langle
2\right\rangle_{a|b}&=\frac{\bm{Q}_{n,a}\bm{Q}^{*}_{n,b}}{M_{a}M_{b}},\\\
\left\langle\left\langle
2\right\rangle\right\rangle_{a|b}&=\frac{\Sigma_{events}M_{a}M_{b}\left\langle
2\right\rangle_{a|b}}{\Sigma_{events}M_{a}M_{b}},\end{aligned}$ (7)

where $a$ and $b$ are symbols for these two sub-events. The corresponding
$n$-th order flow can be written as

$\displaystyle
v_{n}^{a|b}\\{2\\}=\sqrt{c_{n}^{a|b}\\{2\\}}=\sqrt{\left\langle\left\langle
2\right\rangle\right\rangle_{a|b}},$ (8)

where $c_{n}^{a|b}\\{2\\}$ is the two-particle cumulant. Here we assume the
flow in two sub-events is the same, which is reasonable in a symmetric system
with the same sub-event kinetic windows.

We can also generate two-particle correlation directly to extract collective
flow [85, 82, 86]. For trigger-associated particle correlation, we can go over
all particle pairs and gain a per-trigger-particle associated yield,

$\displaystyle\begin{aligned}
\frac{1}{N_{trig}}\frac{\mathrm{d}^{2}N^{pair}}{\mathrm{d}\Delta\eta\mathrm{d}\Delta\varphi}&=B(0,0)\frac{S(\Delta\varphi,\Delta\eta)}{B(\Delta\varphi,\Delta\eta)},\\\
S(\Delta\varphi,\Delta\eta)&=\frac{1}{N_{trig}}\frac{\mathrm{d}^{2}N^{same}}{\mathrm{d}\Delta\varphi\mathrm{d}\Delta\eta},\\\
B(\Delta\varphi,\Delta\eta)&=\frac{1}{N_{trig}}\frac{\mathrm{d}^{2}N^{mix}}{\mathrm{d}\Delta\varphi\mathrm{d}\Delta\eta},\end{aligned}$
(9)

where $\Delta\varphi$ and $\Delta\eta$ are the differences in $\varphi$ and
pseudorapidity $\eta$ of the pair, and $N_{trig}$ is the trigger particle
yield. $S(\Delta\phi,\Delta\eta)$ is the per-trigger-particle pair yield in
the same event. $B(\Delta\phi,\Delta\eta)$ is generated by pairing trigger
particles with associated particles from other events. The Fourier expansion
of associated yields is

$\displaystyle\frac{1}{N_{trig}}\frac{\mathrm{d}N^{pair}}{\mathrm{d}\Delta\varphi}=\frac{N_{asso}}{2\pi}\left[1+\sum_{n=1}^{\infty}2V_{n\Delta}\cos(n\Delta\varphi)\right],$
(10)

where $V_{n\Delta}$ is the coefficient and $N_{asso}$ is the associated
particle yield. With non-flow correlation reduced, there is

$\displaystyle V_{n\Delta}=v_{n}^{asso}v_{n}^{trig},$ (11)

and

$\displaystyle
v_{n}^{corr}=\sqrt{V_{n\Delta}}~{}~{}(v_{n}^{asso}=v_{n}^{trig}),$ (12)

where $v_{n}^{asso}$, $v_{n}^{trig}$ and $v_{n}^{corr}$ are the flow of
associated particles, triggers, and the entire correlation system
respectively. In practice a $\Delta\eta$ gap is applied to Eq.(9) while
projected into one-dimensional correlation function of Eq.(10), which is
considered to reduce non-flow effectively.

Figure 4: The transverse momentum spectra of major charged particles within
$|\eta|<0.5$ for integrating all centralities. The red line connected with red
squares is plotted for the case of W-S as a baseline for comparing other 44Ti
+ 44Ti systems. The same situation is presented in following figures.

##  III Result and discussion

In this work, three collision systems of Woods-Saxon nuclei and three of
$\alpha$ \+ core 44Ti are simulated by the string-melting AMPT model version
2.26t7b. The first three systems are 50Ti + 50Ti, 40Ca + 40Ca, and 44Ti +
44Ti. In other three systems, one has no $r_{\alpha}$ constraint, one follows
$r_{\alpha}>r_{\mathrm{Mott}}$, and the last one follows
$r_{\alpha}<r_{\mathrm{Mott}}$. Each system has $8\times 10^{5}$ events, which
are uniformly divided into ten centrality classes according to the
multiplicity of major charged particles within
$p_{\mathrm{T}}>0.2~{}\mathrm{GeV/c}$ and $|\eta|<0.5$ ($N_{\mathrm{ch}}$).

To assess the initialization configuration, we first extract the radial
nucleon density of projectile and target nuclei, which is presented in Fig. 2.
The central nucleon density is almost the same for Woods-Saxon 40Ca, 50Ti and
44Ti, showing typical nuclear matter saturation. However, this feature is
broken for nuclei with a classical $\alpha$ \+ core structure. Further density
increase appears for $r_{\alpha}<r_{\mathrm{Mott}}$, while for
$r_{\alpha}>r_{\mathrm{Mott}}$, the saturation is restored, indicating that
this breaking is due to clusters neglecting Pauli blocking. This negligence
widely exists in energy level calculation, where the result is mainly drawn
from stationary Schr$\ddot{\mathrm{o}}$dinger equations regardless of the
cluster dissolution [58, 67]. Some progress has been made in Ref. [69], where
cluster motion avoids mutual overlap due to the Pauli blocking effect.
However, this just falls in the $r_{\alpha}>r_{\mathrm{Mott}}$ case and the
cluster dissolution is not directly handled. So future optimization to
introduce such transition is still necessary in relative areas. Another
difference is in the peripheral area. For Woods-Saxon nuclei, the nucleon
density decreases exponentially in peripheral areas, but for $\alpha$ \+ core
structures, the clusters introduce another density enhancement. This time the
Mott density constraint concentrates more clusters outside and hence leads to
higher peripheral density than the classical cluster model’s.

Seeing such density distribution differences, we expect alteration related to
nuclear structures in particle production. Fig. 3 (a) reports the probability
distribution of $N_{\mathrm{ch}}$ as $\mathrm{P}(N_{\mathrm{ch}})$.
Distributions of all systems look alike at low $N_{\mathrm{ch}}$, reflecting
the low possibility of clusters appearing in collision zones and similar
surface behaviors of Woods-Saxon nuclei or cores. But when $N_{\mathrm{ch}}$
is high, the influence of nuclear structures starts to play an important role
in the multiplicity distribution. A $\mathrm{P}(N_{\mathrm{ch}})$ ratio
comparison is presented for 44Ti + 44Ti systems in Fig. 3 (b), where clear
depression is seen in $r_{\alpha}>r_{\mathrm{Mott}}$ at high
$N_{\mathrm{ch}}$, while enhancement is shown instead when
$r_{\alpha}<r_{\mathrm{Mott}}$. The mixed result of free $\alpha$ tends to
decrease slightly since there are a bit more clusters outside the cores as
discussed in Sec. II. We also present the transverse momentum spectra of major
charged particles within $|\eta|<0.5$ for integrating all centralities in Fig.
4, in which the red line connected with red squares is plotted for the case of
W-S as a baseline (the same in the following figures). The distinction due to
nuclear structures is rather small but still exists this time, which is
consistent with Fig. 3 (b) where the deviation only appears in the ultra-high
$N_{\mathrm{ch}}$ events and is lower than one order of magnitude. This
behaviour is similar to the multiplicity distribution and ratio in isobar
collisions [87, 22] while considering the neutron skin effect.

Figure 5: The nuclear modification factor $R_{\mathrm{CP}}$ of the centrality
class 0-10% as a function of $p_{\mathrm{T}}$. The centrality class 80-90% is
the denominator. Figure 6: Eccentricity (a) and triangularity (b) of
participant partons as a function of $N_{\mathrm{ch}}$.

To further look into this phenomenon, we introduce the nuclear modification
factor $R_{\mathrm{CP}}$, which can be defined as [88]

$\displaystyle
R_{\mathrm{CP}}=\frac{\mathrm{d}^{2}N^{\mathrm{cent}}_{\mathrm{ch}}/(\mathrm{d}p_{\mathrm{T}}\mathrm{d}y)/\left<N^{\mathrm{cent}}_{\mathrm{coll}}\right>}{\mathrm{d}^{2}N^{\mathrm{peri}}_{\mathrm{ch}}/(\mathrm{d}p_{\mathrm{T}}\mathrm{d}y)/\left<N^{\mathrm{peri}}_{\mathrm{coll}}\right>}.$
(13)

Figure 7: The elliptic flow and triangular flow by the two sub-event cumulant
method or by the direct two-particle correlation method as a function of
$N_{\mathrm{ch}}$. (a) and (b) are $v_{2}^{a|b}\\{2\\}$ and
$v_{3}^{a|b}\\{2\\}$ by the two sub-event cumulant method, and (c) and (d) are
$v_{2}^{corr}$ and $v_{3}^{corr}$ by the direct two-particle correlation
method.  Figure 8: Ratios of $v_{2}^{a|b}\\{2\\}$ to
$\left<\varepsilon_{2}\right>$ (a) and $v_{3}^{a|b}\\{2\\}$ to
$\left<\varepsilon_{3}\right>$ (b) as a function of $N_{\mathrm{ch}}$. Figure
9: $v_{2}^{a|b}\\{2\\}$ (a) and $v_{3}^{a|b}\\{2\\}$ (b) as a function of
$p_{\mathrm{T}}$ in the centrality class 0-10%.

Here $\left<N_{\mathrm{coll}}\right>$ is the average inelastic binary
collision number over events. The superscript “cent” represents central
collisions and “peri” means peripheral events. $R_{\mathrm{CP}}$ is usually
used to estimate the particle production suppression, and is widely considered
to be negatively correlated with multiplicity. Figure 5 shows the
$R_{\mathrm{CP}}$ results of the centrality class 0-10% with the class 80-90%
as the denominator. The 40Ca + 40Ca system shows the highest
$R_{\mathrm{CP}}$, while 50Ti + 50Ti has the lowest, reflecting typical
multiplicity correlation. Among 44Ti + 44Ti systems, in the low
$p_{\mathrm{T}}$ range the difference caused by nuclear structures is more
significant than in Fig. 4, which is also negatively correlated with the
multiplicity. At other $p_{\mathrm{T}}$, $R_{\mathrm{CP}}$ is almost the same
for all 44Ti + 44Ti systems. Unfortunately, such difference is still far from
distinguishing itself compared with the system size influence.

Another issue about the initialization configuration is the collision zone
shape, which can be characterized by $\varepsilon_{n}$ defined in Eq. (5).
Fig. 6 presents $\left<\varepsilon_{2}\right>$ and
$\left<\varepsilon_{3}\right>$ of participant partons in all centrality
classes as a function of $N_{\mathrm{ch}}$. $\left<\varepsilon_{2}\right>$ and
$\left<\varepsilon_{3}\right>$ decrease with the increasing of $N_{ch}$ in
these collision systems, which presents a system size dependence like that in
our previous work [89]. For Woods-Saxon nuclei, their eccentricity follows an
order of multiplicity, implying that for similar systems $\varepsilon_{n}$ is
mainly affected by the system size. For cluster + core nuclei, at low
$N_{\mathrm{ch}}$ the eccentricity is almost the same as for the Woods-Saxon
structure, while at high $N_{\mathrm{ch}}$ an eccentricity enhancement appears
when $\alpha$ clusters are in $r_{\alpha}>r_{\mathrm{Mott}}$ or have no
$r_{\alpha}$ constraint. The enhancement is stronger in
$r_{\alpha}>r_{\mathrm{Mott}}$, suggesting that it is $\alpha$ clusters
outside the cores that cause this $\varepsilon_{n}$ enhancement. These
behaviors of $\varepsilon_{n}$ also match the analysis of particle production
above, where the influence of cluster + core structures plays an important
role in high multiplicity events but diminishes in peripheral collisions.
Interestingly, the $\alpha$-cluster structure effects on the eccentricity are
consistent with our previous investigations [54, 42] for
${}^{12}\mathrm{C}+^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}+^{16}\mathrm{O}$
collisions where the chain structure enhances $\varepsilon_{2}$ and reduces
$\varepsilon_{3}$ and the triangle or tetrahedron structure do the opposite.
This implies that the nuclear intrinsic structure plays an important role in
the geometry shape formation at the collision initial stage.

Figure 7 shows the elliptic flow and the triangular flow of major charged
particles gained with two sub-event cumulants or direct two-particle
correlation. For the two sub-event cumulant method, the kinetic window is
$-2.4<\eta_{a}<0$, $0<\eta_{b}<2.4$ and
$0.3<p_{\mathrm{T}}<4.0~{}\mathrm{GeV/c}$, where $\eta_{a}$ and $\eta_{b}$ are
the pseudorapidity of particles in sub-events a and b, respectively. For
direct two-particle correlation, the kinetic window of the trigger and
associated particles is $0.3<p_{\mathrm{T}}<4.0~{}\mathrm{GeV/c}$ and
$-2.4<\eta<2.4$, so that the measurement involves the same particles as in the
two sub-event cumulant method. A pseudorapidity gap $\Delta\eta>2.0$ is
required for effective correlation to avoid non-flow effects. The result of
peripheral events in centrality $60-100\%$ is omitted since jet-like
correlation is the main source of $v_{2}$, and $v_{3}$ here. In Fig. 7 (a)
$v_{2}^{a|b}\\{2\\}$ shows obvious enhancement for cluster + core structures
except in $r_{\alpha}<r_{\mathrm{Mott}}$, while in Fig. 7 (b)
$v_{3}^{a|b}\\{2\\}$ of all systems follows similar multiplicity order except
in the highest multiplicity class. In Fig. 7 (c) and (d), the results of
$v_{2}^{corr}$ and $v_{3}^{corr}$ show great similarity with
$v_{2}^{a|b}\\{2\\}$ and $v_{3}^{a|b}\\{2\\}$, showing that the difference
between $v_{2}$ and $v_{3}$ is independent of the exact tool to extract the
flow.

To carry out further investigation, we try a new perspective of the flow-to-
eccentricity ratio. Hydrodynamic calculations [75, 90] point out that there is
a proportional relationship between $\varepsilon_{n}$ and $v_{n}$ for $n=2,3$,
so their ratio might be able to reflect the efficiency to transform initial
geometry asymmetry to final momentum space asymmetry [42]. The ratios of
$v_{n}^{a|b}\\{2\\}$ to $\left<\varepsilon_{n}\right>$ are shown in Fig. 8.
Although the behaviors of $v_{2}^{a|b}\\{2\\}$ and $v_{3}^{a|b}\\{2\\}$ are
quite different, their ratios to corresponding eccentricity do share the same
pattern. Like the case of $\left<\varepsilon_{n}\right>$,
$v_{n}^{a|b}\\{2\\}/\left<\varepsilon_{n}\right>$ of Woods-Saxon systems
follows the order of multiplicity, while for core + cluster nuclei,
$v_{n}^{a|b}\\{2\\}/\left<\varepsilon_{n}\right>$ shows influence of nuclear
structures at high $N_{\mathrm{ch}}$ except in $r_{\alpha}<r_{\mathrm{Mott}}$.
This time binary structures lead to a slight depression, indicating that the
cluster effects may be weakened by particle interaction during collisions to
some extent. Combining $\left<\varepsilon_{n}\right>$, $v_{n}^{a|b}\\{2\\}$
and $v_{n}^{a|b}\\{2\\}/\left<\varepsilon_{n}\right>$, we can say that the
different behaviors between $v_{2}$ and $v_{3}$ are likely to be the natural
result of system evolution.

To investigate the kinetic window dependence of the influence of
$\alpha$-cluster nuclear structure, the $p_{\mathrm{T}}$-differential
$v_{n}^{a|b}\\{2\\}$ of the centrality class 0-10% is calculated and shown in
Figure 9. $v_{2}^{a|b}\\{2\\}$ and $v_{3}^{a|b}\\{2\\}$ increase with the
increasing of $p_{T}$ and reach the maximum around $p_{T}\sim$ 2.5 GeV/$c$,
and then present a decreasing tend. The core + cluster enhancement is still
obvious this time for $v_{2}^{a|b}\\{2\\}$, while $v_{3}^{a|b}\\{2\\}$ remains
similar for all six systems. So the $\alpha$-cluster nuclear structure effect
on collective flow dose not depend on the selected $p_{T}$ window and
$p_{\mathrm{T}}$-differential collective flow may also be a good probe for
nuclear structures.

From the above analysis of the differential flow in the most-central
collisions and the $N_{ch}$ dependence of flow, it is shown that elliptic flow
coefficients in the natural nucleus collision systems, 50Ti + 50Ti, 44Ti(W-S)
+ 44Ti(W-S), 40Ca + 40Ca, follow the order of $v_{2}(^{40}\mathrm{Ca})$ $>$
$v_{2}(^{44}\mathrm{Ti(W-S)})$ $>$ $v_{2}(^{50}\mathrm{Ti})$ which is
consistent with our previous study of the system size dependence of collective
flow [89]. And it is obvious that 44Ti + 44Ti configured with
$\alpha$-clusters is significantly biased against the order of $v_{2}$ in the
natural nucleus collision systems, up to 10%, no matter whether the Mott
density is taken into account for 44Ti. In the experiment, a system scan of
relativistic heavy-ion collisions near 44Ti could be considered as a probe to
explore the signature of the $\alpha$-cluster structure in 44Ti.

##  IV Conclusion

In this work, we test the influence of $\alpha$ cluster + core structures on
particle production and collectivity in 44Ti + 44Ti collisions at
$\sqrt{s_{NN}}=5.02$ TeV with the AMPT model where the initial geometry
properties is configured by using the traditional treatment method of the
nuclear structure. Due to $\alpha$ dissolution, there might be the so-called
Mott density restricting the positions of $\alpha$ clusters. So we try three
modes of binary structures, which are $r_{\alpha}<r_{\mathrm{Mott}}$,
$r_{\alpha}>r_{\mathrm{Mott}}$ and free $r_{\alpha}$. For real nuclei, only
the last two are possible. The absence of $\alpha$ dissolution greatly
increases central nucleon density and leads to the breaking of nuclear matter
saturation, implying the necessity of including Mott density in future
$\alpha$ \+ core structure research. Clusters outside cores also raise the
peripheral nucleon density but to a much smaller degree.

In particle production, the structure influence mainly contributes to high
multiplicity events. Systems with clusters inside cores show a higher fraction
of high $N_{\mathrm{ch}}$ events, and clusters outside cores lead to the
contrary. The mixed mode is similar to the Woods-Saxon situation, with a much
weaker enhancement at high $N_{\mathrm{ch}}$. The alteration is less than one
order of magnitude, so it is hard to detect binary structures in transverse
momentum spectra. We also calculate the nuclear modification factor
$R_{\mathrm{CP}}$ in the centrality class 0-10% and found that the signal is
stronger, but still not enough to decide the structure. The initial geometry
is also affected by cluster + core structures. Both $\varepsilon_{2}$ and
$\varepsilon_{3}$ are increased by clusters outside cores, while clusters
inside seem to have no effect. The difference in $v_{2}$ is strengthened in
comparison with $\varepsilon_{2}$, so there may be possibility to distinguish
the binary structure with a cross-check of elliptic flow in similar systems.
However, in $v_{3}$ the difference almost vanishes due to system evolution,
leaving traces only in central events.

Although it is still under debate whether $\alpha$ cluster structures can
generate strong signatures in ultra-relativistic heavy-ion collisions, there
have been many predictions about the all-cluster structures. Our result
provides a new insight of $\alpha$ \+ core structures into this issue, showing
that elliptic flow might be a good probe of binary structures, while
triangular flow and particle production are less sensitive to these
structures. And a system scan of relativistic heavy-ion collisions is proposed
to be performed in the experiment to investigate $\alpha$ \+ core structures
through the system dependence of elliptic flow.

###### Acknowledgements.

This work was supported in part by National Key R&D Program of China under
Grant No. 2018YFE0104600 and 2016YFE0100900, the National Natural Science
Foundation of China under contract Nos. 12275054, 12147101, 11925502,
11890710, 11890714, 12061141008 and 11875066, the Strategic Priority Research
Program of CAS under Grant No. XDB34000000, the Guangdong Major Project of
Basic and Applied Basic Research No. 2020B0301030008, Shanghai Special Project
for Basic Research No. 22TQ006 and the STCSM under Grant No. 23590780100.

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Our design idea can be extended to $MPC_v^{5+}$ with some effort. However, new ideas are needed for the general $k \ge 6$.
It is possible that the local operations we designed inside our algorithm can be developed into local search algorithms with better performance ratios.
Currently we do not have any inapproximability result for $MPC_v^{k+}$ for any fixed constant $k \ge 4$.
Such negative results can be interesting to pursue.

§ ACKNOWLEDGMENTS

ZZC is supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan, under Grant No. 18K11183.
GL is supported by the NSERC Canada.
ZZ is supported by National Science Foundation of China (NSFC: 61972329),
GRF grants for Hong Kong Special Administrative Region, China (CityU 11210119, CityU 11206120, CityU11218821),
and a grant from City University of Hong Kong (CityU 11214522).

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