# Measurement of the flow harmonic correlations via multi-particle symmetric
and asymmetric cumulants in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV

Kaiser Shafi<EMAIL_ADDRESS>Prabhupada Dixit
<EMAIL_ADDRESS>Sandeep Chatterjee<EMAIL_ADDRESS>Md.
Nasim<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Berhampur

###### Abstract

We study multi-particle azimuthal correlations in Au+Au collisions at
$\sqrt{s_{NN}}=200$ GeV. We use initial conditions obtained from a Monte-Carlo
Glauber model and evolve them within a viscous relativistic hydrodynamics
framework that eventually gives way to a transport model in the late hadronic
stage of the evolution. We compute the multi-particle symmetric and asymmetric
cumulants and present the results for their sensitivity to the shear and bulk
viscosities during the hydrodynamic evolution. We show that these observables
are more sensitive to the transport coefficients than the traditional flow
observables.

††preprint: APS/123-QED

##  I Introduction

In ultra-relativistic heavy-ion collisions, a thermalized, strongly
interacting, deconfined medium, known as Quark-Gluon Plasma (QGP), is expected
to form [1, 2, 3, 4, 5, 6]. The information about such an event is carried
both by anisotropies in the initial state collision geometry and by the
transport properties of QGP [7, 8, 3]. After hadronization, particles are
emitted anisotropically in the transverse plane to the beam direction, as
detected by a detector. Traditionally, Fourier series in flow amplitudes
$v_{n}$ and symmetry planes $\psi_{n}$ are used to quantify this azimuthal
distribution in the momentum space [9],

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

Collective flow is crucial for studying the properties of the QGP medium
formed in these collisions [10, 7, 11]. The impact parameter driven spatial
geometry of the fireball, event-by-event nucleon position as well as sub-
nucleonic fluctuations within the overlap region of the two colliding nuclei
lead to the development of various orders of anisotropic flow harmonics. Due
to the almond shape of the overlap region, the primary source of the second-
order flow harmonic, known as elliptic flow ($v_{2}$), is the initial spatial
anisotropy [12, 13, 14, 15, 16, 17, 18]. The third-order flow harmonic, known
as triangular flow ($v_{3}$), arises from event-by-event fluctuations in the
positions of the colliding nucleons as well as their sub-nucleonic
fluctuations [19, 20, 21, 22, 23]. These fluctuations also give rise to
higher-order flow harmonics like $v_{4}$, $v_{5}$, etc. Previous studies have
demonstrated that these flow harmonics are sensitive to the equation of state
(EoS) and the transport properties of the fireball created in the collision,
such as shear and bulk viscosity [24, 25, 26, 27, 28].

Anisotropic flow analysis involves measuring $v_{n}$, $\psi_{n}$ and their
event-by-event correlations and fluctuations. Due to the random fluctuations
of the impact parameter vector, however, the estimation of $v_{n}$ and
$\psi_{n}$ involves the complications associated with reaction plane
dispersion in conventional flow analyses. As a result, they are indirectly
estimated using correlation techniques, in which there is no need for an
event-by-event determination of the reaction plane [29, 30]. Therefore, there
is no need to correct for dispersion in an estimated reaction plane. The
foundation of this alternative approach is based on the following outcome,

$\langle
e^{i\left(n_{1}\phi_{1}+n_{2}\phi_{2}+...+n_{k}\phi_{k}\right)}\rangle=v_{1}v_{2}...v_{k}e^{i\left(n_{1}\psi_{1}+n_{2}\psi_{2}+...+n_{k}\psi_{k}\right)}$
(2)

which analytically relates multiparticle azimuthal correlators and flow
degrees of freedom [31]. The average is calculated over all unique sets of $k$
different particles in a single event. This expression can be used to
determine the properties of flow amplitudes $v_{n}$ and symmetry planes
$\psi_{n}$ on an event-by-event basis. Apart from collective flow, other
sources of correlations called non-flow are also present that typically
involve only a subset of particles.

Multivariate cumulants were introduced in anisotropic flow analyses in the
early 2000s in Refs. [32, 33], which tackled long-standing issues in the field
and transformed the approach to anisotropic flow analysis in high-energy
physics. For example, the four-variate cumulant that is used to estimate the
flow amplitude, $v_{n}$, from four-particle correlation is defined as [33]

$\displaystyle c_{n}\\{4\\}=\langle\langle
e^{in(\phi_{1}+\phi_{2}-\phi_{3}-\phi_{4})}\rangle\rangle-2\langle\langle
e^{in(\phi_{1}-\phi_{3})}\rangle\rangle^{2},$ (3)

The double angular brackets indicate that the averaging procedure is performed
in two steps—first, averaging over all distinct particle multiplets in an
event and then averaging these single-event averages with appropriately chosen
event weights in the second step. By generalizing this concept for non-
identical harmonics, new observables, which strictly satisfy all defining
mathematical properties of cumulants, were constructed to quantify the
correlations among different flow harmonics [34]. Ref. [35] further
generalized these observables to probe the genuine correlation between
different moments of flow harmonics.

###  I.1 Symmetric Cumulants of Flow Amplitudes

Reference [34] proposed a general algorithm to measure multi-particle
correlation where the harmonics inside the correlator bracket of Eq. (3) can
be different. This introduces a new set of observables known as symmetric
cumulants $(SCs)$ which can be used to measure the correlation between event-
by-event fluctuation of flow harmonics $v_{m}$ and $v_{n}$. These four-
particle symmetric cumulants, $SC(m,n)$, are defined as [36]

$\displaystyle SC(m,n)$ $\displaystyle\equiv$
$\displaystyle\left<\left<\cos(m\varphi_{1}\\!+\\!n\varphi_{2}\\!-\\!m\varphi_{3}-\\!n\varphi_{4})\right>\right>_{c}$
(4) $\displaystyle=$
$\displaystyle\left<\left<\cos(m\varphi_{1}\\!+\\!n\varphi_{2}\\!-\\!m\varphi_{3}-\\!n\varphi_{4})\right>\right>$
$\displaystyle-$
$\displaystyle\left<\left<\cos[m(\varphi_{1}\\!-\\!\varphi_{2})]\right>\right>\left<\left<\cos[n(\varphi_{1}\\!-\\!\varphi_{2})]\right>\right>$
$\displaystyle=$
$\displaystyle\left<v_{m}^{2}v_{n}^{2}\right>-\left<v_{m}^{2}\right>\left<v_{n}^{2}\right>.$

The subscript $c$ indicates the cumulant. Positive (Negative) values of
$SC(m,n)$ suggest the correlation (anti-correlation) between $v_{m}^{2}$ and
$v_{n}^{2}$, which means that if $v_{m}^{2}$ is larger than $\langle
v_{m}^{2}\rangle$ in an event then the probability of $v_{n}^{2}$ being larger
than $\langle v_{n}^{2}\rangle$ in that same event is enhanced (suppressed).
The $SC(m,n)$ observables focus on the correlations among different orders of
flow harmonics and enable the quantitative comparison between experimental
data and model calculations. To eliminate the effect of the magnitudes of
$v_{m}$ and $v_{n}$ on the value of the symmetric cumulant, we divide
$SC(m,n)$ by their average values, $\left<v_{m}^{2}\right>$ and
$\left<v_{n}^{2}\right>$, and define the normalized symmetric cumulant. This
enables us to compare data and model calculations in a quantitative way, and
to compare the fluctuations of the initial and final states. The normalized
symmetric cumulant, denoted by $NSC(m,n)$, is achieved following the standard
method from Ref. [37]:

$NSC(m,n)=\frac{SC(m,n)}{\langle v_{m}^{2}\rangle\langle v_{n}^{2}\rangle}.$
(5)

These correlations have been measured by both STAR experiment at RHIC [38] and
ALICE experiment at LHC [39] with observation of a positive correlation
between $v_{2}$ and $v_{4}$ while a negative correlation between $v_{2}$ and
$v_{3}$. The sensitivity of these observables to the shear viscosity to
entropy density ($\eta/s$) has been observed in both hydrodynamics and
transport model studies [40, 41].

###  I.2 Asymmetric Cumulants of Flow Amplitudes

Recent studies indicate that insightful information can be obtained about the
properties of Quark-Gluon Plasma by using higher-order observables [42]. These
observables can probe the genuine correlations between the different moments
of different flow harmonics. They are robust against non-flow correlations
which can be verified using the HIJING Monte Carlo generator [43]. The
asymmetric cumulant ${\rm AC}_{2,1}(m,n)$ is defined as [35]

$\begin{split}{\rm
AC}_{2,1}(m,n)&\quad\equiv\langle(v_{m}^{2})^{2}v_{n}^{2}\rangle_{c}\quad\equiv\langle
v_{m}^{4}v_{n}^{2}\rangle_{c}\\\ &\quad=\langle
v_{m}^{4}v_{n}^{2}\rangle-\langle v_{m}^{4}\rangle\langle
v_{n}^{2}\rangle-2\langle v_{m}^{2}v_{n}^{2}\rangle\langle v_{m}^{2}\rangle\\\
&\quad+2\langle v_{m}^{2}\rangle^{2}\langle v_{n}^{2}\rangle\end{split}$ (6)

In Eq. (6), the subscript (2,1) on the left-hand side indicates the exponents
of $v_{m}^{2}$ and $v_{n}^{2}$ on the right-hand side. The final combinations
of azimuthal correlators used to estimate the $ACs$ experimentally are [35]:

$\begin{split}{\rm AC}_{2,1}(m,n)&=\langle\langle{\rm
e}^{i(m\varphi_{1}+m\varphi_{2}+n\varphi_{3}-m\varphi_{4}-m\varphi_{5}-n\varphi_{6})}\rangle\rangle\\\
&\quad-\langle\langle{\rm
e}^{i(m\varphi_{1}+m\varphi_{2}-m\varphi_{3}-m\varphi_{4})}\rangle\rangle\langle\langle{\rm
e}^{i(n\varphi_{1}-n\varphi_{2})}\rangle\rangle\\\ &\quad-2\langle\langle{\rm
e}^{i(m\varphi_{1}+n\varphi_{2}-m\varphi_{3}-n\varphi_{4})}\rangle\rangle\langle\langle{\rm
e}^{i(m\varphi_{1}-m\varphi_{2})}\rangle\rangle\\\ &\quad+2\langle\langle{\rm
e}^{i(m\varphi_{1}-m\varphi_{2})}\rangle\rangle^{2}\langle\langle{\rm
e}^{i(n\varphi_{1}-n\varphi_{2})}\rangle\rangle\end{split}$ (7)

These expressions are genuine multivariate cumulants. ${\rm AC}_{1,1}(m,n)$
corresponds to SC$(m,n)$, illustrating the generalisation aspect of the $ACs$.
We can also normalise the $ACs$. This procedure has two benefits. Normalizing
the results allows proper comparisons and determination of the initial state
effects and the changes brought by the hydrodynamic evolution since the
predictions for $ACs$ in the initial and final state do not have the same
scale. Flow amplitudes depend on the transverse momentum, $p_{T}$, which leads
to a similar dependence in any linear combinations of them, such as the $SCs$
and the $ACs$. The normalization eliminates this dependence and enables
comparisons between models and data with different $p_{T}$ ranges [43]. Again,
the normalization of the $ACs$ is done following the standard method from Ref.
[37],

$\displaystyle{\rm NAC}_{2,1}(m,n)$ $\displaystyle=$ $\displaystyle\frac{{{\rm
AC}}_{2,1}(m,n)}{\langle v_{m}^{2}\rangle^{2}\langle v_{n}^{2}\rangle}.$ (8)

In this paper, we have presented the measurement of $SC(m,n)$ and
$AC_{2,1}(m,n)$ in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV using a hybrid
hydrodynamic model. $SC(m,n)$ have been measured in previous studies from
hydrodynamic and transport models [44, 43, 37, 45, 46, 47, 48], while
$AC_{2,1}(m,n)$ have been measured in transport models [46, 47]. ALICE
collaboration has measured $AC_{2,1}(m,n)$ for Pb+Pb collisions at
$\sqrt{s_{NN}}=5.02\ TeV$ [49]. In addition to symmetric cumulants, various
other correlators have also been studied [50, 51, 52, 53]. We show the
sensitivity of these measurements to the transport properties, such as the
shear viscosity to entropy density ratio, $\eta/s$, and the bulk viscosity to
entropy density ratio, $\zeta/s$, of the medium.

##  II Framework

We use a framework with multiple components to simulate various stages of
heavy ion collisions. The hydrodynamic evolution has been initialized using a
Glauber-based model. To generate the incoming nuclei, we use a Woods-Saxon
distribution to sample nucleons.

$\rho=\frac{\rho_{0}}{1+exp[(r-R)/a]}.$ (9)

Here, $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $a$ characterizes the diffuseness of the
nuclear surface, $R$ is is the radius parameter, and $\rho_{0}$ is the
saturation density determined by $\int\rho\ dr^{3}=A$ (mass number of the
nucleus). We have taken $R=6.38$ fm, and $a=0.535$ fm [54].

The evolution of the energy-momentum tensor starts at $\tau_{0}=0.6$ fm. The
hydrodynamic equations are solved using the Music simulation [55, 56, 57, 58],
which utilizes a Kurganov-Tadmor algorithm. A constant effective $\eta/s$ =
0.08 was fixed by reproducing the measured anisotropic flow coefficients of
charged hadrons. We use a temperature-dependent specific bulk viscosity
parametrized in the following way [59]:

$\frac{\zeta}{s}=\left\\{\begin{array}[]{ll}\lambda_{1}e^{-(x-1)/\sigma_{1}}+\lambda_{2}e^{-(x-1)/\sigma_{2}}+0.001&(T>1.05T_{C})\\\
\lambda_{3}e^{(x-1)/\sigma_{3}}+\lambda_{4}e^{(x-1)/\sigma_{4}}+0.03&(T<0.995T_{C})\end{array}\right.$
(10)

where $x=T/T_{C}$. The fitted parameters are $\lambda_{1}=\lambda_{3}=0.9$ ,
$\lambda_{2}=0.25$, $\lambda_{4}=0.22$, $\sigma_{1}=0.025$, $\sigma_{2}=0.13,$
$\sigma_{3}=0.0025$, $\sigma_{4}=0.022,$ $A_{1}=-13.77$, $A_{2}=27.55$ and
$A_{3}=13.45$.

We use a QCD equation of state (EoS), NEoS-B [60], based on continuum
extrapolated lattice calculations at zero net baryon chemical potential
published by the HotQCD Collaboration [61, 62, 63]. It is smoothly matched to
a hadron resonance gas EoS in the temperature region between 110 and 130 MeV
[64].

A hyper-surface is generated from the hydrodynamic space-time evolution of the
fluid. To describe the dilute hadronic phase, we utilize the iSS code [65, 66]
to sample the primordial hadrons from the hypersurface with a constant energy
density, $e_{sw}=0.26\ GeV/fm^{3}$, equivalent to a local temperature of
approximately 151 MeV. Then we use the UrQMD code [67, 68] to simulate
scatterings and decays of hadrons during the late stage of heavy ion
collisions. To increase statistics, each hydrodynamic switching hyper-surface
is sampled multiple times. The number of oversampling events for every
hydrodynamic event is determined to ensure sufficient statistics for every
centrality class. We generated ensembles of 2000 hydrodynamic events for each
centrality class, each with a suitable number of repeated samplings. Table 1
lists the number of events generated for each of the centrality classes with
the three sets of parameters for the hydrodynamic stage shown in Table 2:

Centrality class | 0-10% | 10-20% | 20-30% | 30-40% | 40-50%
---|---|---|---|---|---
# Events (x$10^{6}$) | 0.8 | 1.6 | 2.4 | 3.2 | 4
Table 1: Ensemble size (in millions) of the centrality classes. | $\eta/s$ | $\zeta/s$
---|---|---
Parameter(Par.) Set I | 0 | 0
Parameter(Par.) Set II | 0.08 | 0
Parameter(Par.) Set III | 0.08 | $\zeta/s(T)$
Table 2: The transport coefficients’ values for the parameter sets used in the
ensembles generated for this study.

In the following results, some observables will be presented by scaling with
the average number of participants, $\langle N_{part}\rangle$. Table 3 lists
the values of $\langle N_{part}\rangle$ in different centrality bins, which
are determined using the Monte-Carlo Glauber model.

Centrality class |  $\langle N_{part}\rangle$
---|---
0-10% |  327.7
10-20% |  240.3
20-30% |  172.7
30-40% |  120.5
40-50% |  81.5
Table 3: $\langle N_{part}\rangle$ in the centrality classes from a Monte-
Carlo Glauber model.

##  III Results

We calibrated our model by using the produced particle yields, transverse
momentum spectra and elliptic flow. We then compared these results to
experimental data from the PHENIX and STAR Collaborations. The normalization
factor for the system’s energy density was determined by matching the model
and STAR data [69] charged hadron multiplicity $dN_{Ch}/d\eta$ in the 0-5%
centrality bin. We also compared the $p_{T}$ spectra of identified particles
in 0-5% centrality Au+Au collisions to STAR results [69]. Additionally, we
compared the single-particle anisotropic flow coefficient $v_{2}$ of charged
particles in 20-30% centrality Au+Au collisions with the PHENIX measurement
[70]. The hydrodynamic evolution, of the MCGM initial conditions, coupled to
UrQMD can reproduce the elliptic flow upto the mid-central collisions.

###  III.1 Sensitivity to transport coefficients

####  III.1.1 Four-particle symmetric cumulants

In this section, we present results for $SC(m,n)$ from our simulations. To
check their sensitivity to the hydro transport coefficients, we compute these
observables for the three sets of ensembles mentioned in Table 2. We observe
that both $SC(2,3)$ and $SC(2,4)$ are suppressed by shear and bulk
viscosities. In Fig.[1], we show the effect of shear viscosity on
$v_{2}\\{4\\}$ and $SC(2,3)$. In $v_{2}\\{4\\}$, we see a maximum change of
around 10% in the mid-central collisions. While in $SC(2,3)$, we see a change
of around 40-60% throughout the centrality range. We observe a similar
dependence on the bulk viscosity of the medium. Thus, these observables are
more sensitive to the transport coefficients. Similar to $SC(2,3)$, $SC(2,4)$
is also affected by the transport coefficients in the same manner. The
comparison of the symmetric cumulants with $v_{3}\\{4\\}$ also reveals a
similar difference in their sensitivity to shear and bulk viscosity.

Figure 1:  Ratios of $v_{2}\\{4\\}$ and $SC(2,3)$, with different hydro
parameter sets, vs centrality in Au+Au collisions at $\sqrt{s_{NN}}$ = 200
GeV. The figure compares the effects of $\eta/s$ and $\zeta/s$ on
$v_{2}\\{4\\}$ and $SC(2,3)$. The plotted results show ratios of
$v_{2}\\{4\\}$ using bands and $SC(2,3)$ using markers. Certain data points
with high uncertainty were excluded for clarity.

####  III.1.2 Six-particle asymmetric cumulants

Now, we present the results for $AC_{2,1}(m,n)$. In Fig.[2], we show the
effect of shear viscosity on elliptic flow from the four-particle correlations
and $AC_{2,1}(2,3)$. For $AC_{2,1}(2,3)$, we see a change of around 50%
throughout the centrality range. We observe that the medium’s bulk viscosity
also has a similar effect on these observables. Compared to the traditional
flow observables, we see that these are more sensitive to the transport
coefficients. Hence, these will put better constraints on the transport
properties of the medium created in heavy-ion collisions.

Figure 2:  Ratios of $v_{2}\\{4\\}$ and $AC_{2,1}(2,3)$, with different hydro
parameter sets, vs centrality in Au+Au collisions at $\sqrt{s_{NN}}$ = 200
GeV. The figure compares the effects of $\eta/s$ and $\zeta/s$ on
$v_{2}\\{4\\}$ and $AC_{2,1}(2,3)$. The plotted results show ratios of
$v_{2}\\{4\\}$ using bands and $AC_{2,1}(2,3)$ using markers. Certain data
points with high uncertainty were excluded for clarity.

###  III.2 Centrality dependence of $SC(m,n)$ and $AC_{2,1}(m,n)$

####  III.2.1 Four-particle symmetric cumulants

Next, we compare the results from our simulations with the experimental
measurements from RHIC. $SC(m,n)$ have been measured in previous studies from
hydrodynamic and transport models [44, 43, 37, 45, 46, 47, 48], while
$AC_{2,1}(m,n)$ have been measured in some transport models [46, 47]. Fig. [3]
shows comparisons between the theory calculations for the symmetric cumulants
$SC(2,3)$ and $SC(2,4)$ multiplied by $\langle N_{part}\rangle$ and the same
measured by the STAR Collaboration [71]. The factor of $\langle
N_{part}\rangle$ is multiplied to scale out the trivial dilution of
correlation with the increase in the number of triplets. We are able to
reproduce the qualitative variations with centrality. Here, we don’t aim to
reproduce the experimental data because our aim is to check the sensitivity of
these observables to the hydro transport coefficients. The comparison in
Fig.[3] reflects that the correlation in initial spatial eccentricities and
the following hydrodynamic evolution can capture the correlation among flow
harmonics of different orders. Negative values of $SC(2,3)$ throughout the
centrality range reveal the anti-correlation between $v_{2}$ and $v_{3}$.
While as $SC(2,4)$ is positive for all centralities, indicating positive
correlations between $v_{2}$ and $v_{4}$. Symmetric cumulants do not have
contributions from non-flow effects, where non-flow refers to azimuthal
correlations not related to the reaction plane orientation, like those from
resonances, jets, quantum statistics, etc. This is verified by computing these
observables for HIJING model, which includes only non-flow physics, for which
these are consistently zero [35]. The model captures the variation of
$SC(m,n)$ with the centrality. In central collisions, the model underestimates
$SC(2,4)$, possibly due to inadequate fluctuations in the initial conditions
or some shortcomings in the transport model since the nonlinear response in
$v_{4}$ and $\varepsilon_{4}$ is sensitive to these effects. However,
$SC(2,3)$ is better aligned with the data because the $v_{2}-v_{3}$
correlations mainly arise from the initial state
$\varepsilon_{2}-\varepsilon_{3}$ correlations due to linear response in
$v_{2}$ and $v_{3}$.

Figure 3:  Symmetric cumulant vs centrality from four-particle correlations
compared with STAR data [71] in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV.
$SC(2,3)$ is consistently negative while as $SC(2,4)$ is consistently
positive.

The normalized symmetric cumulants in Au+Au collisions at $\sqrt{s_{NN}}$ =
200 GeV, evaluated using the Monte Carlo Glauber model in coordinate space
(using Eq.[11]), are presented in Fig.[4] and compared with $NSC(m,n)$
measured in momentum space. If only eccentricity drives $v_{n}$, then we can
expect that the $NSC(m,n)$ in the final state would be equal to the $NSC(m,n)$
initial state. Fig.[4] shows that the initial anti-correlation between
$\varepsilon_{2}$ and $\varepsilon_{3}$ is mainly responsible for the observed
anti-correlation between $v_{2}$ and $v_{3}$. However, the correlation between
$\varepsilon_{2}$ and $\varepsilon_{4}$ is smaller than the observed
correlation between $v_{2}$ and $v_{4}$. The contribution to anisotropic flow
$v_{4}$ not only comes from the linear response of the system to
$\varepsilon_{4}$ but also has a contribution proportional to
$\varepsilon_{2}^{2}$. As collisions become more peripheral, the difference
between $NSC(2,4)$ in the final state and the initial state increases, likely
due to $\varepsilon_{2}$ playing a more significant role in $v_{4}$. This has
also been observed in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV by the
ATLAS [72] and ALICE [39] experiments. The properties of the medium were
suggested to affect the relative contribution of $\varepsilon_{2}$ in $v_{4}$
compared to $\varepsilon_{4}$ [73]. Therefore, $NSC(2,4)$ provides a probe
into the medium properties.

$\displaystyle\varepsilon_{n}e^{in\psi_{n}}$ $\displaystyle=$
$\displaystyle-\frac{\int r^{n}e^{in\phi}\rho\left(r,\phi\right)rdrd\phi}{\int
r^{n}\rho\left(r,\phi\right)rdrd\phi},\hskip 2.84526ptn>2$ (11)

Figure 4:  Normalized symmetric cumulant vs centrality from four-particle
correlations compared with STAR data [71] in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV. $NSC(2,3)$ is consistently negative while as
$NSC(2,4)$ is consistently positive.

####  III.2.2 Six-particle asymmetric cumulants

$AC_{2,1}(m,n)$ have measured by the ALICE collaboration for Pb+Pb collisions
at $\sqrt{s_{NN}}=5.02\ TeV$ [49]. In Fig.[5], we should the centrality
dependence of $AC_{2,1}(m,n)$ for Au+Au at $\sqrt{s_{NN}}=200\ GeV$. The
trends are similar to the results by the ALICE collaboration. As for symmetric
cumulants, negative values of $AC(2,3)$ throughout the centrality range reveal
the anti-correlation between $v_{2}$ and $v_{3}$. While as $AC(2,4)$ is
positive for all centralities, indicating positive correlations between
$v_{2}$ and $v_{4}$. To check their sensitivity to the hydro transport
coefficients, we compute these observables for the three sets of ensembles
mentioned in Table 2. We observe that both $AC_{2,1}\\{2,3\\}$ and
$AC_{2,1}\\{2,4\\}$ are suppressed by shear and bulk viscosities. For a
clearer picture, the same results are plotted with a log scale for the y-axis
in Fig.[6].

Figure 5:  Asymmetric cumulant, $AC_{2,1}\\{2,3\\}$/$AC_{2,1}\\{2,4\\}$, vs
centrality from six-particle correlations in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV.

Figure 6:  Asymmetric cumulants, -$AC_{2,1}\\{2,3\\}$ and $AC_{2,1}\\{2,4\\}$,
vs centrality from six-particle correlations in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV. The upper panel shows $AC_{2,1}\\{2,3\\}$
multliplied by -1 to show the y-axis in log scale while as the lower panel
shows $AC_{2,1}\\{2,4\\}$.

###  III.3 Conclusions

We have presented results for multi-particle correlation functions in heavy-
ion collisions at top RHIC energy using a hybrid framework based on the Monte-
Carlo Glauber model, MUSIC viscous hydrodynamics simulations, and the UrQMD
hadronic cascade.

First, we adjusted the free parameters, such as shear and bulk viscosities, to
accurately describe particle multiplicities, mean transverse momentum, and
anisotropic flow. After that, we have discussed the impact of shear and bulk
viscosities on traditional and novel observables.

We have studied correlators that measure the correlations between flow
harmonics of varying orders, specifically four and six-particle correlations,
including both symmetric and asymmetric cumulants as functions of centrality
at midrapidity in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV. These new
observables manifestly satisfy all mathematical properties of cumulants in the
basis in which they are expressed. This new approach allows for the separation
of non-flow and flow contributions and provides initial insights into how the
combinatorial background contributes to flow measurements using correlation
techniques.

The hybrid framework that we have used yields good agreement with the majority
of the latest multi-particle observables from RHIC. In addition, the hadronic
afterburner plays an essential role in describing the data collected at RHIC
[44]. Study based on HIJING demonstrated that these new observables are robust
against systematic biases due to non-flow correlations [43].

Compared to traditional flow observables, the novel observables are more
sensitive to transport coefficients, resulting in better constraints on the
transport properties of the medium created in heavy-ion collisions. We
observed anti-correlation between event-by-event fluctuations of $v_{2}$ and
$v_{3}$, while the event-by-event fluctuations of $v_{2}$ and $v_{4}$ are
found to be positively correlated. The observed anti-correlation between
$v_{2}$ and $v_{3}$ appears to be described by the initial-stage anti-
correlation between $\varepsilon_{2}$ and $\varepsilon_{3}$, which supports
the idea of linearity between $\varepsilon_{n}$ and $v_{n}$ [23]. The
hydrodynamic response suggests that the final state fluctuation originates
from the initial state. However, including the nonlinear hydrodynamic response
of the medium is necessary to reproduce the measured correlation between
$v_{2}$ and $v_{4}$, as the initial-stage correlation alone is insufficient.
Therefore, we can gain new knowledge regarding the initial conditions and
properties of QGP in high-energy nuclear collisions from these higher-order
observables. Symmetric and asymmetric cumulants of flow amplitudes can be used
directly to constrain the multivariate probability density function of flow
fluctuations since its functional form can be reconstructed only from its true
moments or cumulants.

###  III.4 Acknowledgements

We would like to acknowledge Tribhuban Parida for fruitful discussions and
providing us the computational setup.

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