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# Collective flows of protons and deuterons in Au + Au collisions at
$E_{beam}=1.23$$A$ GeV by the IQMD model

Ling-Meng Fang(房灵猛)  Shanghai Institute of Applied Physics, Chinese Academy of
Sciences, Shanghai 201800, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Yu-Gang Ma(马余刚) ID 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

###### Abstract

Collective flows of protons and deuterons for Au + Au collisions at beam
energy $E_{beam}$ = 1.23$A$ GeV were simulated by an Isospin dependent Quantum
Molecular Dynamics (IQMD) model. Two coalescence models, namely naive
coalescence and dynamical coalescence models, for the formation of deuterons
are compared. After reasonable match of rapidity spectra of protons and
deuterons to the High Acceptance DiElectron Spectrometer (HADES) data is
reached， we apply an event-plane method to calculate the first four-order
collective flow coefficients as well as the ratios of $\left\langle
v_{4}\right\rangle/\left\langle v_{2}\right\rangle^{2}$ and $\left\langle
v_{3}\right\rangle/(\left\langle v_{1}\right\rangle\left\langle
v_{2}\right\rangle)$, and observe the number of constituent nucleon scaling
among protons and deuterons. In addition, the dependence of $\varepsilon_{n}$
and $\left\langle v_{n}\right\rangle$ as well as the ratio $\left\langle
v_{n}\right\rangle/\varepsilon_{n}$ on the centrality is obtained. Lastly, we
further investigate the Pearson coefficients $corr(v_{n},v_{m})$ between the
first four harmonic flows for protons and deuterons as a function of rapidity
and centrality.

##  I Introduction

In heavy-ion collisions, a highly excited nuclear medium is created, and its
collective expansion produces the associated particle emission. In a perfectly
central collision, the expansion should be isotropic in the transverse plane,
as observed in the transverse mass spectra of the ejected particles. The shape
of the overlapping regions becomes more anisotropic in more off-central
collisions. In heavy-ion collisions, the collective motion of final-state
particles can be described by the collective flows, which can be divided into
longitudinal flow and transverse flow according to the motion direction of the
final-state particles. The anisotropic flow is essentially originated from the
asymmetrical azimuthal distribution of participant nucleons, which can be
classified into directed flow, elliptic flow, triangular flow, quadruple flow
and so on according to different terms of the Fourier expansion of the
azimuthal distribution.

The Fourier expansion of the azimuthal distribution of the final-state
emission particles in momentum space can be expressed as follows [1, 2, 3]:

$E\frac{d^{3}N}{d^{3}p}=\frac{1}{2\pi}\frac{d^{2}N}{p_{t}dp_{t}dy}\left(1+\sum_{n=1}^{\infty}2v_{n}\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right)$
(1)

where $E$ is the energy of the final particle, $p_{t}$ is the transverse
momentum of the particle, $y$ is the rapidity, $\phi$ is the azimuthal angle
of the transverse momentum relative to the fixed plane $XZ$, $\Psi_{r}$ is the
azimuthal angle of the reaction plane relative to the fixed plane $XZ$.
$v_{n}$ at $n$ = $1,2,3,4$ are defined as directed flow, elliptic flow,
triangular flow, and quadruple flow, respectively, as mentioned before.

In general we know that, the elliptic shape of the transverse momentum
distribution of the final particles is located in-plane in lower energy below
a hundred MeV per nucleon due to the collective rotation dominated by the
attractive mean-field [4, 5, 6, 7, 8]. With the increasing of beam energy to a
few hundred MeV energy, the elliptic shape could be perpendicular to the
reaction plane in the mid-rapidity region which is mainly because the
spectators have not moved away from the reaction area timely at such energy
range [9, 10, 11]. The spectators have a shadowing effect on participants,
making particles tend to eject perpendicular to the direction of the reaction
plane. This phenomenon is called ”squeeze-out effect”, i.e. an elliptical flow
outside of the reaction plane. While in the high energy region, because of the
Lorentz contraction in two nuclei collisions, the transverse size of the
nucleons is negligible relative to the longitudinal alignment. The time for
the two nuclei to cross is extremely short in comparison with the
characteristic time of elliptic flow formation, thus the bystander leaves the
reaction area quickly and almost has no shadowing effect on the reaction zone,
so the final particles tend to exude in the reaction plane, and the elliptic
flow is in the reaction plane [12].

In 1992, Ollitrault et al. found that the spatial energy density distribution
at the early stage of the collision was related to the spatial angular
distribution of the freeze-out particles at the later stage of the reaction
[13]. In 1996, Voloshin et al. carried out the Fourier expansion of the
particle spectrum of the final state particles, and proposed a method to
express the size of the collective flows of the final state particles by the
coefficients of the expansion terms [3]. After that, with the continuous in-
depth theoretical researches, people studied the collective flows of each
order in details, and put forward different calculation methods of collective
flows, for example, event-plane method [14, 15], energy momentum tensor method
[16] and two particle correlation method [16, 17, 2, 18], etc. With the
development of the accelerators, high-energy heavy ion collision experiments
can be carried out under different conditions to study the collective flows of
final state particles at different energies. In 1999, Heiselberg and Levy
studied the azimuthal asymmetry of the system reflected by elliptical flow in
noncentral collisions [19]. Stachel concluded that the energy dependence of
elliptical flow in high-energy heavy ion collisions is related to QGP phase
transition after analyzing the experimental data of several accelerators [20].
Voloshin and Poskanzer analyzed Pb + Pb collisions on SPS and found that the
elliptic flow has the centrality and rapidity dependence [21]. In 2000, Heinz
et al. investigated anisotropic flows and established a deeper connection with
QGP phase transitions [22].

Recently, the HADES Collaboration made systematic measurements on properties
of baryon-rich matter formed in Au + Au collisions at $\sqrt{s_{NN}}$ = 2.4
GeV. Different probes, including dilepton and virtual photons [23], identical
pion intensity interferometry [24, 25] as well high-order harmonic flows of
light nuclei [26], which provide an opportunity to investigate the nuclear
fireball properties as well as light nuclei production mechanism [27, 28, 29,
30], and then constrains theoretical model in this reaction energy region and
contributes the understanding of the ‘ice in the fire’ puzzle [31].

The paper is organized as follows. First, a concise introduction to the IQMD
model and coalescence model as well as the event-plane method for flow
analysis are given in Sec. II. Next, the results of first to fourth order
coefficients of collective flow of protons and deuterons are presented in Sec.
III. The results about the linear correlations between different-order flows
and eccentricity are also given in this section. Finally, a brief summary is
presented in Sec. IV.

##  II Models and methods

In the study of heavy-ion collisions, various models have been established to
simulate the collision processes. At present, the commonly used heavy-ion
reaction models can be divided into statistical models and transport models.

In this study, an Isospin dependent Quantum Molecular Dynamics (IQMD) model, a
kind of transport models, is employed to study the reaction system from
initial state to final stage in medium-high energy heavy-ion collisions. The
coalescence model is used to simulate the generation of light nuclei by using
the nucleon phase-space from IQMD model. And the collective flow of light
nuclei are calculated from the phase-space information at the freeze-out stage
simulated by the IQMD model with help of the event-plane method. In the
following, the IQMD model, coalescence model and event plane calculation
methods will be introduced, separately.

###  II.1 The IQMD model

Quantum Molecular Dynamics (QMD) model can provide the information on both the
collision dynamics and the phase space information [32, 33, 34, 35, 36]. The
IQMD model is based on the traditional QMD model, by including the isospin
degree of freedom of nucleons [37].

In the IQMD model, the normalized wave function of each nucleon is expressed
in the form of a Gaussian wave packet,

$\phi_{i}(\vec{r},t)=\frac{1}{(2\pi
L)^{3/4}}\exp(\frac{-(\vec{r}-\vec{r_{i}}(t))^{2}}{4L})\exp(\frac{i\vec{r}\cdot\vec{p_{i}}(t)}{\hbar}),$
(2)

here $\vec{r_{i}}(t)$ and $\vec{p_{i}}(t)$ are time-dependent variables
describing the center of the wave packet in coordinate space and momentum
space, respectively. Given the direction of $\vec{r_{i}}$ and $\vec{p_{i}}$,
$\phi_{i}(\vec{r},t$) is a four-dimensional function. The parameter $L$ is the
width of the wave packet, which is related to the size of the reaction system
and usually fixed in the simulations. Here the width $L$ is fixed as
$2.16fm^{2}$ for Au + Au reactions [38, 39].

All the nucleons interact with each other through an effective mean field and
two-body scatterings. The interaction potential can be expressed as

$U=U_{Sky}+U_{Coul}+U_{Yuk}+U_{Sym}+U_{MDI},$ (3)

where $U_{Sky}$, $U_{Coul}$, $U_{Yuk}$, $U_{Sym}$, and $U_{MDI}$ represent the
density-dependent Skyrme potential, Coulomb potential, Yukawa potential,
isospin asymmetric potential, and the momentum-dependent interaction
potential, respectively. The nucleon-nucleon collision cross section in the
medium ($\sigma_{NN}^{med}$) can be expressed as taken in Refs. [40, 41, 42]

$\sigma_{NN}^{med}=(1-\eta\frac{\rho}{\rho_{0}})\sigma_{NN}^{free},$ (4)

where $\rho_{0}$ is the density of normal nuclear matter, $\rho$ is the local
density, $\eta$ is the in-medium correction factor, which is chosen as 0.2 in
this paper to better reproduce the flow data [43], and $\sigma_{NN}^{free}$ is
the free nucleon-nucleon cross section.

###  II.2 Coalescence model

There are two types of coalescence models, naive coalescence model and
dynamical coalescence model. In this article, we use both of two coalescence
models, and compare the difference between them.

The naive coalescence model uses the following criteria to judge the formation
of deuterons:

$\Delta p<p_{0},~{}~{}~{}~{}~{}\Delta r<r_{0},$ (5)

where $\Delta p=\left|\vec{p_{1}}-\vec{p_{2}}\right|$, $\Delta
r=\left|\vec{r_{1}}-\vec{r_{2}}\right|$, and $p_{0}=0.35$ GeV/$c$, $r_{0}=3.5$
$fm$ are selected in this paper. It is emphasized that here the momentum and
coordinate should be at the rest frame of the pair, such as proton and
neutron.

The dynamical coalescence model can give the probability of light nuclei by
the overlap of the cluster Wigner phase-space density with the nucleon phase
space distributions at an equal time in the $M$-nucleon rest frame at the
freeze-out stage [44]. The momentum distribution of a cluster in a system
containing $A$ nucleons can be expressed by:

$\displaystyle\begin{aligned}
&\frac{d^{3}N_{M}}{d^{3}K}=G\binom{A}{M}\binom{M}{Z}\frac{1}{A^{M}}\int\left[\prod_{i=1}^{Z}f_{p}\left(\vec{r}_{i},\vec{k}_{i}\right)\right]\\\
&\qquad\left[\prod_{i=Z+1}^{M}f_{n}\left(\vec{r}_{i},\vec{k}_{i}\right)\right]\times\rho^{W}\left(\vec{r}_{i_{1}},\vec{k}_{i_{1}},\cdots,\vec{r}_{i_{M-1}},\vec{k}_{i_{M-1}}\right)\\\
&\qquad\times\delta\left(\vec{K}-\left(\vec{k}_{1}+\cdots+\vec{k}_{M}\right)\right)d\vec{r}_{1}d\vec{k}_{1}\cdots
d\vec{r}_{M}d\vec{k}_{M},\end{aligned}$ (6)

where $M$ and $Z$ are the number of the nucleon and proton of the cluster,
respectively; $f_{n}$ and $f_{p}$ are the neutron and proton phase-space
distribution functions at freeze-out, respectively; $\rho_{W}$ is the Wigner
density function; $\vec{r}_{i_{1}}\cdots\vec{r}_{i_{M-1}}$ and
$\vec{k}_{i_{1}}\cdots\vec{k}_{i_{M-1}}$ are the relative coordinate and
momentum in the $M$-nucleon rest frame; the spin-isospin statistical factor
$G$ is $3/4$ for deuteron in this paper [44, 45, 46]. While the neutron and
proton phase-space distribution comes from the transport model simulations,
the multiplicity of a $M$-nucleon cluster is then given by:

$\displaystyle\begin{aligned}
&N_{M}=G\int\sum_{i_{1}>i_{2}>\cdots>i_{M}}d\vec{r}_{i_{1}}d\vec{k}_{i_{1}}\cdots
d\vec{r}_{i_{M-1}}d\vec{k}_{i_{M-1}}\\\
&\qquad\left\langle\rho_{i}^{W}\left(\vec{r}_{i_{1}},\vec{k}_{i_{1}},\cdots,\vec{r}_{i_{M-1}},\vec{k}_{i_{M-1}}\right)\right\rangle,\end{aligned}$
(7)

where the $\left\langle\cdots\right\rangle$ denotes the event averaging.

###  II.3 The event-plane method for flow analysis

A common method for calculating collective flow is the event-plane method. The
$n$-th order event-plane angle $\Psi_{EP}^{(n)}$ can be defined by the event
flow vector $Q_{n,x}$ and $Q_{n,y}$ as [2, 47, 1, 2, 48, 49]:

$\displaystyle\begin{aligned}
&\Psi_{EP}^{(n)}=\frac{1}{n}\tan^{-1}\left(\frac{Q_{n,y}}{Q_{n,x}}\right),\\\
&Q_{n,x}=\sum_{i}\omega_{i}\cos(n\Phi_{i}),\quad
Q_{n,y}=\sum_{i}\omega_{i}\sin(n\Phi_{i}),\end{aligned}$ (8)

where $\Phi_{i}$ and $\omega_{i}$ are the azimuthal angle of the momentum and
the weight for the $i$-th particle, respectively. $\omega_{i}$ is usually set
to unit for theoretical simulation but set as charges $|Z|$ in this paper,
which is suggested in Ref. [26]. The sums extend over all particles used in
the event plane reconstruction. For systems with finite multiplicity, the
harmonic flow coefficients can be calculated by:

$\displaystyle\begin{aligned} &\left\langle
v_{n}\right\rangle=\frac{\left\langle
v_{n}^{obs}\right\rangle}{Res\left\\{\Psi_{n}\left\\{EP\right\\}\right\\}},\\\
&\left\langle
v_{n}^{obs}\right\rangle=\left\langle\cos\left(km\left(\phi-\Psi_{n}\left\\{EP\right\\}\right)\right)\right\rangle,\\\
&Res\left\\{\Psi_{n}\left\\{EP\right\\}\right\\}=\left\langle\cos\left(km\left(\Psi_{n}\left\\{EP\right\\}-\Psi_{RP}\right)\right)\right\rangle.\end{aligned}$
(9)

The angular brackets indicate an average over all particles in all events and
$km$ = $n$ in this work. The resolution of event plane angle
$Res\left\\{\Psi_{n}\left\\{EP\right\\}\right\\}$ owing to the finite number
of particles can be calculated by:

$\displaystyle\begin{aligned}
&Res\left\\{\Psi_{n}\left\\{EP\right\\}\right\\}=\left\langle\cos\left(km\left(\Psi_{n}\left\\{EP\right\\}-\Psi_{RP}\right)\right)\right\rangle\\\
&=\frac{\sqrt{\pi}}{2\sqrt{2}}\chi_{m}\exp\left(-\chi_{m}^{2}/4\right)\\\
&\times\left[I_{\left(k-1\right)/2}\left(\chi_{m}^{2}/4\right)+I_{\left(k+1\right)/2}\left(\chi_{m}^{2}/4\right)\right],\end{aligned}$
(10)

where the $\chi_{m}$ can be estimated by the sub-event method. The event used
to calculate the event plane angle would randomly be split into two sub-
events, event $A$ and $B$, with maximum difference of particle number equal to
1. $\chi_{m}$ from sub-event resolution $\cos(km(\Psi_{m}^{A}-\Psi_{m}^{B}))$
multiplying $\sqrt{2}$ would be the $\chi_{m}$ for full event resolution
$Res\left\\{\Psi_{n}\left\\{EP\right\\}\right\\}$. The details for this
analysis can be found in Refs. [47, 2, 48, 49, 50]

##  III Analysis and discussion

In this paper, we use an IQMD model to simulate Au + Au collisions at beam
energy $E_{beam}$ = 1.23$A$ GeV, which corresponds to a center of mass energy
$\sqrt{s_{NN}}$ = 2.4 GeV. The total number of events included in the
simulation is 1,600,000. The centrality is characterized as $c=(\pi
b^{2})/(\pi b_{max}^{2})\times 100\%$, where $b$ is the impact parameter, and
$b_{max}=1.15(A_{P}^{1/3}+A_{T}^{1/3})$ is the sum of effective shape radius
of projectile and target. With this definition of centrality, the smaller the
$c$ value, the more central the collisions.

Figure 1: Rapidity distribution of protons (a) and deuterons (b) in Au + Au
collisions at $E_{beam}$ = 1.23$A$ GeV for 4 centrality classes with 10% bin
width. The solid and hollow points in (a) represent the HADES experimental
data [51] and the IQMD simulation results, respectively. The points in (b) are
from the HADES experimental data (black) [52], naive coalescence model (red),
and dynamical coalescence model (blue), respectively. Figure 2: The variation
of first to fourth order resolutions with the centrality in Au + Au collisions
at $E_{beam}$ = 1.23$A$ GeV. The solid and hollow points represent the HADES
experimental data and the IQMD simulation results, respectively.

Figure 3: (a) Different harmonic flows as a function of rapidity in Au + Au
collisions at $E_{beam}$ = 1.23$A$ GeV for 20-30% centrality. The protons and
deuterons are selected within transverse momentum of 1 - 1.5 GeV/$c$. (b)
Different harmonic flows as a function of transverse momentum in Au + Au
collisions at $E_{beam}$ = 1.23$A$ GeV for 20-30% centrality. For the odd-
order collective flows, the protons and deuterons are selected within rapidity
of -0.25 -0.15, while for the even-order collective flows, the rapidity is
selected within -0.05 0.05. In the figure, the solid symbols with error bars
represent the HADES experimental data, the dotted lines with error bands
represent the IQMD simulation results, respectively.

###  III.1 Yield of protons and deuterons

In this paper, we use naive coalescence model and dynamical coalescence model
to estimate deuterons formations. Figure 1 shows the rapidity distributions of
protons and deuterons for the 0-10% centralities as well as the comparison
with the HADES results.

It is seen from Fig. 1(b) that the yields of deuterons from two coalescence
models are consistent with each other and are all in good agreement with the
HADES experimental data. We notice that the yield of protons from the IQMD
model is higher than experimental data in the most central collisions from
Fig. 1(a), but there is an overtaking in more off-centralities. This behavior
reproduces previous IQMD results or other models as given in Ref. [52].

###  III.2 Collective flows of protons and deuterons

To be consistent with the method used by the HADES experiment in Ref. [26], we
use the charges $|Z|$ as the weight in this paper, and the flow coefficients
of all orders discussed here are defined relative to $\Psi_{EP,1}$ as:

$\displaystyle\begin{aligned} &\left\langle
v_{n}^{obs}\right\rangle=\left\langle\cos\left[n\left(\phi-\Psi_{EP,1}\right)\right]\right\rangle\\\
&\mathfrak{R}_{n}=\left\langle\cos\left[n\left(\Psi_{EP,1}-\Psi_{RP}\right)\right]\right\rangle\\\
&\left\langle v_{n}\right\rangle=\left\langle
v_{n}^{obs}\right\rangle/\mathfrak{R}_{n}.\end{aligned}$ (11)

Via this method, we obtain the variation of first to fourth order resolutions
versus centrality as shown in Fig. 2. As we can see from Fig. 2, the value of
resolution decreases significantly as the order increasing, and the resolution
obtained by the event-plane method has basically the same trend as the HADES
experimental data. In the most central collisions, the emission particles tend
to be more isotropic, so the values of all order resolutions are the smallest.
With the increase of the centrality value (i.e. more off-central collision),
the anisotropy of the emission particles is gradually obvious, so the value of
resolution tends to increase gradually. As we can see from Fig. 2, in the most
central collisions, the resolution of IQMD model is higher than that from the
HADES, which corresponds to the higher proton yield from IQMD model in the
most central collision as shown in Fig. 1. With the increase of centrality
value, the proton yield from the IQMD is gradually lower than that from the
HADES, which explains the overtaking phenomenon of resolution in more off-
central collisions in Fig. 2.

Using the event-plane method as in Eq. (11), we  can calculate the
distribution of the collective flow as a function of rapidity for light
nuclei, as shown in Fig. 3(a). As we can see from Fig. 3(a), $\left\langle
v_{1}\right\rangle$ and $\left\langle v_{3}\right\rangle$ are anti-symmetric
with rapidity, while $\left\langle v_{2}\right\rangle$ and $\left\langle
v_{4}\right\rangle$ are axis-symmetric. As for $\left\langle
v_{2}\right\rangle$, a negative value in middle rapidity region indicates an
out-of-plane emission, which is caused by the so-called squeeze-out effect,
where particles are blocked from being emitted in the reaction plane by the
spectator nucleons and are therefore emitted mainly in the out-of-plane-
direction. As rapidity increases, the value of $\left\langle
v_{2}\right\rangle$ becomes positive due to the reduced shadowing effect of
bystanders on the reaction zone. And in the middle rapidity region, the
$\left\langle v_{2}\right\rangle$ of the protons is lower than that of
deuterons, which indicates that after the collision, the protons are more
likely to eject out of the plane, while deuterons prefer an in-plane emission.
Also, $\left\langle v_{n}\right\rangle$ has a larger magnitude for lower
harmonics than higher harmonics. Moreover, we can see that the result of
collective flow obtained by the IQMD model is lower than that from the HADES
experiment especially for the elliptic flow, this phenomenon is consistent
with the results from the UrQMD model in Ref. [53, 54].

The distributions of different order collective flows as a function of light
nuclei transverse momentum are shown in Fig. 3(b). The collective flow
coefficients of deuterons follow that of the free protons, and show a similar
strong dependence on the transverse momentum. And the absolute values of
collective flow of each order increase with transverse momentum, which
indicates that light nuclei with higher $p_{t}$ tend to emit more out of
plane, as they are from earlier emission. $\left\langle v_{1}\right\rangle$ of
the deuterons have larger negative values than the protons which can be
inferred from the coalescence mechanism. We can find that the IQMD model can
well describe the experimental results of $\left\langle v_{1}\right\rangle$,
$\left\langle v_{3}\right\rangle$ and $\left\langle v_{4}\right\rangle$, but
$\left\langle v_{2}\right\rangle$ obtained by the IQMD model is slightly lower
than that from the HADES experiment.

The scaling of elliptic flow of hadrons with the number of constituents has
been established for more than a decade with quark recombination [55] or quark
coalescence model [56] at RHIC energies, and an empirical function can also
fits the experimental elliptic flow data [57]. For the coalescence of nucleons
into deuterons the same scaling should be there in terms of the baryon number.
It has been first claimed that nucleon-number scaled flows should be observed
if the coalescence mechanism is satisfied for light nuclei production in Ref.
[6, 58] and later on the experimental confirmation has been achieved by the
STAR Collaboration [59]. The nucleon-number scaling of elliptic flow results
in the expectation that $\left\langle
v_{2}^{d}\right\rangle\left(p_{T}^{d}\right)=2\left\langle
v_{2}^{p}\right\rangle\left(\frac{1}{2}p_{T}^{d}\right)$. Thus $\left\langle
v_{2}\right\rangle/A$ as a function of $p_{T}/A$, with $A$ being the baryon
number, should yield the same curves for protons and light nuclei in the
coalescence picture. Moreover, instead scaling by the baryon number $A$ for
$\left\langle v_{2}\right\rangle$, the measured data $\left\langle
v_{4}\right\rangle$ seems to be scaled by $A^{2}$ in previous studies [53, 58,
26, 60]. Taking the data of Fig. 3 we show that the flow of protons and the
scaled deuterons for Au + Au collisions in 20-30% centrality at a beam energy
of 1.23 AGeV in Fig. 4. From Fig. 4(a) we observe that the simulation predicts
a good scaling among protons and deuterons. Fig. 4(b) display $\left\langle
v_{4}\right\rangle/A^{2}$ as a function of $(p_{t}/A)^{2}$, from which we can
see that the $\left\langle v_{4}\right\rangle$ can still be roughly scaled by
$A^{2}$ for protons and deuterons. However, the scaling behavior is not
perfect within the present statistics. For example, the $\left\langle
v_{2}\right\rangle/A$ from the IQMD model has a lower magnitude than that from
the HADES, which is probably due to the underestimate of $\left\langle
v_{2}\right\rangle$ as shown in Fig. 3.

Figure 4: Mass number $A$ scaled $\left\langle v_{2}\right\rangle$(a) and
$\left\langle v_{4}\right\rangle$(b) of protons and deuterons for 1.23$A$ GeV
Au + Au collisions in 20-30% centrality as a function of transverse momentum
per nucleon for $|y|<0.05$.

The initial fluctuation can affect the initial geometric asymmetry of the
overlapping region which could be transferred into the momentum space
partially, and then significantly contribute to higher-order harmonic flows
[61]. In earlier studies in intermediate energy [6, 58] as well as at ultra-
relativistic energies [61], it was found that triangular and quadrangular
flows also roughly present a constituent nucleon number scaling in the
intermediate-$p_{T}$ region, similar to the behaviors of elliptic flow. From
those results, a nucleon-number scaling of $\left\langle
v_{n}\right\rangle/n^{n/2}$ for different light nuclei holds for harmonic flow
($\left\langle v_{n}\right\rangle$, $n$ = 2, 3, and 4), which can be related
to $\left\langle v_{n}\right\rangle$ scaling. In ultra-relativistic energies,
such extended flow scaling for high-order harmonic flows has been demonstrated
by the PHENIX Collaboration [62] and STAR Collaboration [63, 64]. Fig. 5 show
the ratio $\left\langle v_{4}\right\rangle/\left\langle
v_{2}\right\rangle^{2}$ and $\left\langle v_{3}\right\rangle/(\left\langle
v_{1}\right\rangle\left\langle v_{2}\right\rangle)$ distributions on rapidity
and transverse momentum [65]. As we can see from Fig. 5(a), for protons and
deuterons, the $\left\langle v_{4}\right\rangle/\left\langle
v_{2}\right\rangle^{2}$ value approaches to the experimental data of 0.5
within the larger error in mid-rapidity region. However, in the off-middle
rapidity interval, the $\left\langle v_{4}\right\rangle/\left\langle
v_{2}\right\rangle^{2}$ of protons and deuterons decreases. Fig. 5(b)
demonstrates that the asymptotic values of $\left\langle
v_{4}\right\rangle/\left\langle v_{2}\right\rangle^{2}$ of protons and
deuterons (naive or dynamical) approach 0.42 and 0.41 or 0.78, respectively,
which is overall in agreement with the experimental values. As for
$\left\langle v_{3}\right\rangle/(\left\langle v_{1}\right\rangle\left\langle
v_{2}\right\rangle)$, the results obtained by the IQMD model are higher than
those from the HADES experiment, and all of them do not show a significant
rapidity correlation.

Figure 5: The ratio $\left\langle v_{4}\right\rangle/\left\langle
v_{2}\right\rangle^{2}$ (upper row) and $\left\langle
v_{3}\right\rangle/(\left\langle v_{1}\right\rangle\left\langle
v_{2}\right\rangle)$ (bottom row) distributions on rapidity (left column) and
transverse momentum (right column) for protons and deuterons of 1.23A GeV
Au+Au collisions at 20-30% centrality.

To quantify initial geometric asymmetry and fluctuation, an eccentricity is
introduced to describe the geometric anisotropy of the overlapping region at
the initial state. After considering the initial fluctuation, the eccentricity
under the center-of-mass frame is defined as [66, 67, 68]:

$\displaystyle\begin{aligned} \varepsilon_{n}=\frac{\sqrt{\left\langle
r^{n}\cos\left(n\varphi\right)\right\rangle^{2}+\left\langle
r^{n}\sin\left(n\varphi\right)\right\rangle^{2}}}{\left\langle
r^{n}\right\rangle},\end{aligned}$ (12)

where $r=\sqrt{x^{2}+y^{2}}$ and $\varphi$ are the coordinate position and
azimuthal angle of participants in the reaction zone with $\langle x\rangle$ =
0 and $\langle y\rangle$ = 0.

Fig. 6 shows the dependence of the $\varepsilon_{n}$ and $\left\langle
v_{n}\right\rangle$ as well as the ratio $\left\langle
v_{n}\right\rangle/\varepsilon_{n}$ on the centrality. It is obvious that the
collision eccentricity increases as centrality (here larger centrality
corresponds to more peripheral collision), indicating a more elliptical
structure under more off-central collisions. The $\left\langle
v_{2}\right\rangle$ and $\left\langle v_{2}\right\rangle/\varepsilon_{2}$ is
negative, and decrease with the increase of centrality. The large positive
elliptic flow in RHIC energy region (above 3 GeV) in semi-central collisions
are always understood by the hydrodynamic picture. With the dynamic evolution
of the fireball [69, 66, 70], the geometric anisotropy of the initial state
will be transformed into the anisotropy of the momentum space at final state
which is characterized by the collective flow, $\left\langle
v_{n}\right\rangle$. However at lower energy the large negative elliptic flow
in non-central collisions was explained by the squeeze-out mechanism [9, 10,
11]. From Fig. 6, it is seen that the negative elliptic flow presents larger
absolute value in non-central collisions where the $\varepsilon_{2}$ also
takes larger value than that in central collisions. This indicates that more
spectators in non-central collisions enhance the particle emission out of
plane by the squeeze-out mechanism.

The initial fluctuation results in the triangular and higher-order asymmetry
of the geometry shape in the reaction zone. The third ($\varepsilon_{3}$) and
fourth ($\varepsilon_{4}$) order eccentricity coefficients are calculated for
the reaction system by using the participants, as shown in Fig. 6 (b) and (c).
As $\varepsilon_{2}$, both $\varepsilon_{3}$ and $\varepsilon_{4}$ increase
with centrality, which implies that the initial fluctuation is more obvious in
peripheral collisions than in central collisions. The initial geometry
fluctuation determines the high-order harmonic flows in momentum space and the
centrality dependent trend, which is shown in Fig. 6(e) and (f). $\left\langle
v_{3}\right\rangle$ present the similar centrality dependence as
$\varepsilon_{3}$ which is consistent with the phenomena in ultra-relativistic
heavy-ion collisions [66]. $v_{4}$ weakly depends on centrality and the
nonlinear-mode is not separated [71], which is beyond the scope for this work.
The ratios of $\left\langle v_{n}\right\rangle/\varepsilon_{n}$ ($n=3,4$) as
shown in Fig. 6(h) and (i) also indicates that there is more significant
initial fluctuation effects in peripheral collisions than in central
collisions.

Figure 6: The dependence of the $\varepsilon_{n}$ (top row) and $\left\langle
v_{n}\right\rangle$ (middle row) as well as the ratio $\left\langle
v_{n}\right\rangle/\varepsilon_{n}$ (bottom row) on the centrality for $n$ = 2
(left column), 3 (middle column), 4 (right column) in Au + Au collisions at
$E_{beam}=1.23AGeV$ for 0-40% centralities from IQMD. For the odd-order
collective flow, the protons and deuterons are selected within rapidity of
-0.5 - 0, while for the even-order collective flow, the rapidity is selected
within -0.1 - 0. The transverse momentum is selected within 1-1.5 GeV/c.

###  III.3 Linear-correlation between collective flows

Figure 7: The Pearson correlation function $corr(v_{n},v_{m})$ of protons and
deuterons as a function of rapidity in Au + Au collisions at 1.23A GeV from
IQMD. The transverse momentum of protons and deuterons are selected as 1-1.5
GeV/c.  Figure 8: The Pearson correlation function $corr(v_{n},v_{m})$ of
protons and deuterons as a function of centrality in Au + Au collisions at
1.23A GeV from IQMD. The transverse momentum of protons and deuterons are
selected as 1-1.5 GeV/c. The open symbols represent for $0<y_{cm}<0.5$ and the
solid symbols represent for $-0.5<y_{cm}<0$.

To further understand the coalescence mechanism of deuteron in the collisions,
the linear correlation functions $corr({v_{n},v_{m}})$ known as the Pearson
coefficient for protons and deuterons are calculated as [72]:

$\displaystyle\begin{aligned} corr\left(v_{n},v_{m}\right)=\frac{\left\langle
v_{n}v_{m}\right\rangle-\left\langle v_{n}\right\rangle\left\langle
v_{m}\right\rangle}{\sigma_{v_{n}}\sigma_{v_{m}}}.\end{aligned}$ (13)

Here, the standard deviation $\sigma_{v_{i}}=\sqrt{\left\langle
v_{i}^{2}\right\rangle-\left\langle v_{i}\right\rangle^{2}}$ is used to
normalize the covariance. We know that the Pearson coefficient provides a
measure for linear dependence of two random variables, which equals to 1
implies a perfect linear dependence, but a vanishing Pearson coefficient does
not rule out any nonlinear correlation.

We show the Pearson correlation function $corr(v_{n},v_{m})$ between the first
four flow coefficients of protons and deuterons in Au+Au collisions at 1.23
AGeV from IQMD modle as a function of rapidity in Fig. 7, and as a function of
centrality in Fig. 8.

In Fig. 7, we can see that the correlation between the even and odd flow
harmonic, for example, $corr(v_{1},v_{2})$ (Fig. 7(a)), $corr(v_{2},v_{3})$
(Fig. 7(b)) and $corr(v_{3},v_{4})$ (Fig. 7(c)) as well as $corr(v_{1},v_{4})$
(Fig. 7(f)), are antisymmetric, and while the correlation between the even/odd
flow harmonic, for example, $corr(v_{1},v_{3})$ (Fig. 7(d)) and
$corr(v_{2},v_{4})$ (Fig. 7(e)), are symmetric around $y_{cm}=0$. This
phenomenon is consistent with the conclusion given in Ref. [72]. Moreover, we
can see that the correlation between adjacent-order $v_{n}$, for example,
$corr(v_{1},v_{2})$ (Fig. 7(a)), $corr(v_{2},v_{3})$ (Fig. 7(b)) and
$corr(v_{3},v_{4})$ (Fig. 7(c)), is stronger, and the results are basically
the same. Furthermore, we can find an interesting phenomenon that the
correlations between different order $v_{n}$ is closely related to the
differentials between the order numbers. As differential between orders equals
to 1, such as $corr(v_{1},v_{2})$ (Fig. 7(a)), $corr(v_{2},v_{3})$ (Fig. 7(b))
and $corr(v_{3},v_{4})$ (Fig. 7(c)), has the same result. As differential
between orders equals to 2, such as $corr(v_{1},v_{3})$ (Fig. 7(d) and
$corr(v_{2},v_{4})$ (Fig. 7(e)), has the similar result. Moreover, the smaller
the differential between the order numbers, the larger the correlation between
$v_{n}$, which indicates that the correlation between $v_{n}$ of the more
adjacent order is stronger.

Considering the (anti-)symmetry behavior of flow correlation coefficients as a
function of rapidity, we extract the average $corr(v_{n},v_{m})$ in positive
and negative rapidity intervals, which is shown in Fig. 8. We observe that as
the centrality increasing, the Pearson coefficient between different order
$v_{n}$ increases gradually, but the increasing trend is somewhat different.
Compared with the obvious increase of the correlation between the first and
second (Fig. 8(a)), the second and third (Fig. 8(b)), and the third and fourth
(Fig. 8(c)) flow harmonic, we notice the correlation between the first and
third (Fig. 8(d)), the second and fourth flow (Fig. 8(e)), and the first and
fourth (Fig. 8(f)) harmonic increases slightly which can be ignored overall.
Overall, the above correlation phenomenon is interesting, but the deeper
understanding needs to be further studied in the future.

##  IV Summary

In summary, the yields of protons and deuterons were calculated by a
simulation of Au + Au collision at 1.23A GeV with the IQMD model and
coalescence models. Then by an event-plane method, we calculate the first four
order collective flows of protons and deuterons. The results show that a good
nucleon-number scaling of elliptic flow among proton and deuteron holds. The
ratio $\left\langle v_{4}\right\rangle/\left\langle v_{2}\right\rangle^{2}$
approaches to the experimental value of 1/2 with a large error between
$\pm$0.3 rapidity but decreases beyond mid-rapidity interval, and
$\left\langle v_{3}\right\rangle/(\left\langle v_{1}\right\rangle\left\langle
v_{2}\right\rangle)$ is higher than those from the HADES experiment. In
addition, we give the dependence of $\varepsilon_{n}$, $\left\langle
v_{n}\right\rangle$ as well as $\left\langle
v_{n}\right\rangle/\varepsilon_{n}$ ratio on the centrality, indicating a more
elliptical structure under more off-central collisions, and more spectators in
non-central collisions enhance particle emission out of plane by the squeeze-
out mechanism. At last, we show the rapidity and centrality dependence of the
linear correlation coefficients $corr(v_{n},v_{m})$ between the first four
flow coefficients, and notice that the correlation between the even/odd flow
harmonic are symmetric around $y_{cm}=0$, while the correlation between the
even and odd flow harmonic are antisymmetric. The correlations between
different order $v_{n}$ is closely related to the differentials between the
order numbers. For the Pearson correlation functions $corr(v_{n},v_{m})$ with
the same differentials have the same result. And the smaller the differential
between the order numbers, the larger the correlation between $v_{n}$, which
indicates that the correlation between $v_{n}$ of the more adjacent order is
stronger. From the centrality dependence, the Pearson coefficient between
different order $v_{n}$ increases gradually. Further understanding of physics
insight to different harmonic flow correlation is expected.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under contract Nos. 11890714, 11875066, 11421505, 11775288, and
12147101, the National Key R&D Program of China under Grant Nos.
2016YFE0100900 and 2018YFE0104600, and by Guangdong Major Project of Basic and
Applied Basic Research No. 2020B0301030008.

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