#  Measurement methods of radial flow in relativistic heavy-ion collisions

Peng Yang  Key Laboratory of Quark and Lepton Physics (MOE), Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China  Lin Li
School of Science, Wuhan University of Technology, Wuhan 430070, China  Yu
Zhou  Department of Mathematics, University of California, Los Angeles,
California 90095, USA  Zhiming Li  Key Laboratory of Quark and Lepton Physics
(MOE), Institute of Particle Physics, Central China Normal University, Wuhan
430079, China  Mingmei Xu  Key Laboratory of Quark and Lepton Physics (MOE),
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China  Yeyin Zhao  Key Laboratory of Quark and Lepton Physics (MOE), Institute
of Particle Physics, Central China Normal University, Wuhan 430079, China
Yuanfang Wu<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics
(MOE), Institute of Particle Physics, Central China Normal University, Wuhan
430079, China

###### Abstract

Radial flow can be directly extracted from the azimuthal distribution of mean
transverse rapidity. We apply the event-plane method and the two-particle
correlation method to estimate the anisotropic Fourier coefficient of the
azimuthal distribution of mean transverse rapidity. Using the event sample
generated by a multiphase transport model with string melting, we show that
both methods are effective. For the two-particle correlation method to be
reliable, the mean number of particles in an azimuthal bin must be above a
certain threshold. Using these two methods, anisotropic radial flow can be
estimated in a model-independent way in relativistic heavy-ion collisions.

###### pacs:

25.75.Nq, 25.75.Dw, 25.75.Ld

##  I Introduction

Collectivity, or flow, is one of the main characteristics of the newly formed
quark-gluon plasma in relativistic heavy-ion collisions lab1 ; lab2 ; lab3 ;
lab4 . Traditionally, we have observed radial flow, direct flow, elliptic
flow, triangular flow, and so on. Radial flow and elliptic flow are the two
prominent classes among them.

Elliptic flow is defined as the second Fourier component of the azimuthal
multiplicity distribution. It is generated by the initial geometric asymmetry
of noncentral collisions where the overlap of two incident nuclei is of an
almond shape in the transverse coordinate plane. The minor axis of the overlap
is on the reaction plane, which is spanned by the vector of impact parameter
and beam direction lab1 ; lab4 ; lab5 . This initial geometric asymmetry leads
to a larger density gradient along the in-plane direction, i.e., the
anisotropic distribution of final-state particles in momentum space.
Therefore, elliptic flow provides information about initial conditions and
system properties lab2 ; lab6 .

Radial flow is originally deduced from an analysis of transverse momentum
spectra in central collisions lab1 ; lab4 ; lab7 . Later, it is generalized
into two parameters: isotropic radial velocity and anisotropic radial velocity
lab8 ; lab9 ; lab10 ; lab11 ; lab12 . The isotropic radial velocity features
the isotropic transverse expansion of the source at kinetic freeze-out. The
anisotropic radial velocity presents the difference of the radial flow between
the in-plane direction and the out-of-plane direction and arises in noncentral
collisions.

The interplay of radial expansion and elliptic flow results in what we observe
as the particle mass splitting of the differential elliptic flow lab3 ; lab4 ;
lab6 . Namely, the heavier particles show smaller elliptic flow values. This
mass ordering of elliptic flow has been well understood by hydrodynamics with
a set of kinetic freeze-out constraints, i.e., radial flow, temperature, and
source deformation lab13 ; lab14 .

In addition, momentum transfer due to viscosity is proportional to the first
derivative of the velocity in hydrodynamics lab15 ; lab16 . Bulk viscosity is
associated with isotropic transverse velocity, whereas shear viscosity is
associated with the anisotropic transverse velocity. The proportion constants
are equal to the bulk and shear viscosities, respectively. Therefore, the
determination of anisotropic radial flow is essential for hydrodynamic
calculations and for measuring the shear viscosity in relativistic heavy-ion
collisions lab17 ; lab18 ; lab19 .

Conventionally, anisotropic radial flow is extracted by fitting the transverse
momentum spectrum of particles with the blast-wave parametrization lab20 ;
lab21 . To minimize $\chi^{2}$ of the fitting, the spectra of all particle
species are fitted simultaneously. For the relativistic heavy-ion collisions
at the Brookhaven National Laboratory (BNL) Alternating Gradient Synchrotron,
the BNL Relativistic Heavy Ion Collider (RHIC), and the CERN Large Hadron
Collider energies, the fitting results have all been presented lab22 ; lab23 ;
lab24 ; lab25 . However, these results are obviously model dependent.
Therefore, a model-independent measurement is called for.

In order to measure radial flow, we had introduced the mean-transverse
rapidity (MTR) and its azimuthal distribution, replacing velocity with
rapidity in the original definition of radial flow lab12 ; lab15 ; lab26 . The
MTR is averaged over the number of particles in an azimuthal angle bin. It
presents the raw kinetic expansion in a specified azimuthal direction.

The azimuthal distribution of the MTR presents the transverse expansion of the
source at kinetic freeze-out lab15 ; lab27 ; lab28 . The second Fourier
coefficient of this distribution features the anisotropic kinetic expansion
and is consistent with the anisotropic radial flow extracted from the blast-
wave parametrization lab12 .

The aim of this paper is to estimate the anisotropic radial flow in
relativistic heavy-ion collisions. This estimation is similar to how we
estimate elliptic flow where the reaction plane cannot be measured directly
due to random fluctuations of the impact parameter vector and is instead
estimated from the reconstructed particles. This estimation method is the so-
called event-plane method (EPM) lab29 ; lab30 ; lab31 . Later, it is found
that the two-particle correlation method (TPCM) can also be used in place of
the EPM to estimate the elliptic flow lab32 ; lab33 ; lab34 ; lab35 ; lab36 .

In this paper, we first describe the EPM and the TPCM that are designed to
estimate the anisotropic coefficients of azimuthal distributions for both
total-transverse rapidity (TTR) and MTR in Sec. II. Then, we demonstrate the
effectivity of both methods using the event sample generated by the AMPT model
with string melting in Sec. III. The limitation of the TPCM is discussed in
Sec. IV. Finally, summary and conclusions are presented in Sec. V.

##  II Measurement methods

Transverse rapidity is defined as lab2 ; lab12 ; lab15 ; lab26

$y_{T}=\ln\left(\frac{m_{T}+p_{T}}{m_{0}}\right),$ (1)

where $m_{0}$ is the particle mass in the rest frame, $p_{T}$ is transverse
momentum, and $m_{T}=\sqrt{m_{0}^{2}+p_{T}^{2}}$ is the transverse mass.

The TTR of the _m_ th azimuthal bin is the summation of all particles’
transverse rapidities in an event, and its average is lab12 ; lab15 ; lab26
as follows:

$\langle Y_{T}(\phi_{m}-\Psi_{r})\rangle=\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm
event}}\sum_{i=1}^{N_{m}^{k}}y_{T,i}^{k}(\phi_{m}-\Psi_{r}).$ (2)

It measures total transverse expansion in a specified azimuthal bin
$(\phi_{m}-\Psi_{r})$, where $\Psi_{r}$ is the reaction plane angle.
$\phi_{m}$ is the _m_ th azimuthal angle bin. The bin width is $2\pi/N_{\rm
bin}$. $N_{\rm bin}$ is the total number of bins.
$y_{T,i}^{k}(\phi_{m}-\Psi_{r})$ is the transverse rapidity of the _i_ th
particle in the _k_ th event and _m_ th azimuthal angle bin. $N_{m}^{k}$ is
the total number of particles in the _k_ th event and the _m_ th azimuthal
angle bin. $N_{\rm event}$ is the total number of events.

The MTR of the _m_ th azimuthal bin is defined as

$\langle\langle y_{{\rm T}}(\phi_{m}-\Psi_{r})\rangle\rangle=\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm
event}}\frac{1}{N_{m}^{k}}\sum_{i=1}^{N_{m}^{k}}y_{T,i}^{k}(\phi_{m}-\Psi_{r}),$
(3)

where $\langle\langle\cdots\rangle\rangle$ is first averaged over the number
of particles in the _k_ th event and the _m_ th azimuthal angle bin
$N_{m}^{k}$, and then over the total number of events. It measures the raw
kinetic expansion in a specified azimuthal direction. The contribution of the
number of particles is removed by the first average.

The Fourier expansions of azimuthal TTR and MTR distributions are as follows:

$\begin{split}&\frac{d\langle
Y_{T}(\phi-\Psi_{r})\rangle}{d(\phi-\Psi_{r})}\\\ &\
=v_{0}(Y_{T})\left(1+\sum_{n=1}^{\infty}2v_{n}(Y_{{\rm
T}})\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right),\end{split}$ (4)

and

$\begin{split}&\frac{d\langle\langle
y_{T}(\phi-\Psi_{r})\rangle\rangle}{d(\phi-\Psi_{r})}\\\ &\
=v_{0}(y_{T})\left(1+\sum_{n=1}^{\infty}2v_{n}(y_{T})\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right),\end{split}$
(5)

where azimuthal-angle-independent $v_{0}(Y_{T})$ and $v_{0}(y_{T})$ are
isotropic TTR flow and radial flow, respectively. The second Fourier
coefficients $v_{2}(Y_{T})$ and $v_{2}(y_{T})$ are defined as anisotropic TTR
flow and radial flow, respectively lab12 ; lab15 ; lab26 .

They are similar to the mean multiplicity in the _m_ th azimuthal angle bin,
i.e.,

$\langle N(\phi_{m}-\Psi_{r})\rangle=\frac{1}{N_{\rm event}}\sum_{k=1}^{N_{\rm
event}}N^{k}(\phi_{m}-\Psi_{r}),$ (6)

where $N^{k}(\phi_{m}-\Psi_{r})$ is the number of particles in the _k_ th
event and the $(\phi_{m}-\Psi_{r})$ direction. Its azimuthal distribution in
Fourier expansion is as follows:

$\begin{split}&\frac{d\langle N(\phi-\Psi_{r})\rangle}{d(\phi-\Psi_{r})}\\\ &\
=v_{0}(N)\left(1+\sum_{n=1}^{\infty}2v_{n}(N)\cos\left[n\left(\phi-\Psi_{r}\right)\right]\right),\end{split}$
(7)

where $v_{0}(N)$ is azimuthal-angle independent. $v_{2}(N)$ is the elliptic
flow, which is estimated, in practice, by the EPM and the TPCM. In the
following, we apply these two methods to estimate $v_{2}(Y_{T})$ and
$v_{2}(y_{T})$.

###  II.1 The event-plane method

In nuclear-nuclear collisions, the azimuthal angle of the final-state particle
is measured with respect to the reaction plane. However, the angle of reaction
plane is unknown in experiments and fluctuates from event to event. Usually, a
reconstructed event plane is considered as a substitute lab29 , i.e.,
$\Psi_{n}$ of the _n_ th coefficient is given by

$\Psi_{n}=\left.\left(\tan^{-1}\frac{Q_{n,y}}{Q_{n,x}}\right)\middle/n\right.,$
(8)

where $Q_{n}$ is the flow vector and defined as,

$Q_{n,x}=\sum_{i=1}^{M}\omega_{i}\cos(n\phi_{i})=Q_{n}\cos(n\Psi_{n}),$ (9)

$Q_{n,y}=\sum_{i=1}^{M}\omega_{i}\sin(n\phi_{i})=Q_{n}\sin(n\Psi_{n}).$ (10)

_M_ is the number of particles used in the event-plane determination.
$\phi_{i}$ and $\omega_{i}$ are the azimuthal angle and the weight of the _i_
th particle.

A general Fourier expansion with respect to the event-plane ($\Psi_{n}$) can
be expressed as

$\begin{split}&\frac{d\langle\omega
N(\phi-\Psi_{n})\rangle}{d(\phi-\Psi_{n})}\\\ &\ =v_{0}(\omega
N)\left(1+\sum_{n=1}^{\infty}2v_{n}^{\rm obs}(\omega
N)\cos\left[n\left(\phi-\Psi_{n}\right)\right]\right),\end{split}$ (11)

where weight $\omega$ changes with the observable lab29 , such as different
kinds of flow defined in Eqs. (4), (5), and (7).

For multiplicity, the weight is unity, and Eq. (11) corresponds to Eq. (7).
Its $v_{2}^{\rm obs}(N)$ is as follows:

$v_{2}^{\rm obs}\left(N\right)=\frac{1}{N_{\rm event}}\sum_{k=1}^{N_{\rm
event}}\left(\frac{1}{N^{k}}\sum_{i=1}^{N^{k}}\cos\left[2\left(\phi_{i}^{k}-\Psi_{2}\right)\right]\right),$
(12)

where $\Psi_{2}$ is the event-plane angle of the second harmonic and $N^{k}$
is the total number of particles in the _k_ th event.

For the TTR and MTR, their weights are $\omega_{i}=y_{T,i}$ and
$\omega_{i}=\left.y_{T,i,m}\middle/N_{m}\right.$, respectively. Their
$v_{2}^{\rm obs}(Y_{T})$ and $v_{2}^{\rm obs}(y_{T})$ are as follows:

$\begin{split}v_{2}^{\rm obs}\left(Y_{T}\right)=&\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm event}}\\\
&\left(\frac{1}{\sum_{i=1}^{N^{k}}y_{T,i}^{k}}\sum_{i=1}^{N^{k}}y_{T,i}^{k}\cos\left[2\left(\phi_{i}^{k}-\Psi_{2}\right)\right]\right),\end{split}$
(13)

and

$\begin{split}v_{2}^{\rm obs}\left(y_{T}\right)=&\frac{1}{N_{\rm
event}}\sum_{k=1}^{N_{\rm event}}\left(\frac{1}{\sum_{m=1}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}}\sum_{k=1}^{N_{m}^{k}}y_{T,i,m}^{k}}\right.\\\
&\left.\sum_{m=1}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}}\sum_{k=1}^{N_{m}^{k}}y_{T,i,m}^{k}\cos\left[2\left(\phi_{i,m}^{k}-\Psi_{2}\right)\right]\right).\end{split}$
(14)

With finite multiplicity, $v_{2}^{\rm obs}(N)$ has to be corrected for the
event-plane resolution lab4 ; lab29 , i.e.,

$\rm Re_{2}=\left\langle\cos\left[2(\Psi_{2}-\Psi_{r})\right]\right\rangle,$
(15)

which is obtained by the iteration of sub-event-plane resolution. The subevent
is usually constructed in a longitudinal rapidity window that is different
from the particles of interest lab23 ; lab37 . Therefore, the elliptic flow is
as follows:

$v_{2}(N)=\left.v_{2}^{\rm obs}(N)\middle/\rm Re_{2}\right..$ (16)

After the same corrections for both $v_{2}^{\rm obs}(Y_{T})$ and $v_{2}^{\rm
obs}(y_{T})$, $v_{2}(Y_{T})$ and $v_{2}(y_{T})$ are as follows:

$v_{2}(Y_{T})=\left.v_{2}^{\rm obs}(Y_{T})\middle/\rm Re_{2}\right.,$ (17)

and

$v_{2}(y_{T})=\left.v_{2}^{\rm obs}(y_{T})\middle/\rm Re_{2}\right..$ (18)

###  II.2 The two-particle correlation method

The coefficients of Fourier expansion Eq. (7) can also be estimated by the
TPCM lab30 , i.e.,

$\left\langle\cos\left[n\left(\phi_{1}-\phi_{2}\right)\right]\right\rangle=\left\langle
e^{in\left(\phi_{1}-\phi_{2}\right)}\right\rangle=\left\langle
v_{n}^{2}\right\rangle+\delta_{n}.$ (19)

Here a cross term that depends both on $v_{n}$ and on $\delta_{n}$ is
neglected as it is very small in most cases of interest. $\delta_{2}$
represents the so-called nonflow contribution and is irrelevant to initial
geometry. All correlations are first averaged over all particle pairs in a
given event and then over all events. The latter average involves weight
depending on event. For convenience, single-event average two-particle
azimuthal correlations is defined as

$\left\langle 2\right\rangle^{k}\equiv\frac{1}{\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}}\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}.$
(20)

Its average over all events is as follows:

$\begin{split}\left\langle\left\langle
2\right\rangle\right\rangle&\equiv\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j=1,i\neq j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}\left\langle
2\right\rangle^{k}}{\sum_{k=1}^{N_{\rm event}}\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}}\\\ &=\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j=1,i\neq
j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}}{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j=1,i\neq j}^{N^{k}}\omega_{i}^{k}\omega_{j}^{k}}.\end{split}$
(21)

For multiplicity and TTR, their weights are $\omega_{i}=1$ and
$\omega_{i}=y_{T,i}$, respectively. So their $\left\langle\left\langle
2\right\rangle\right\rangle$ are

$\left\langle\left\langle
2\right\rangle\right\rangle_{N}=\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j,i\neq
j}^{N^{k}}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}}{\sum_{k=1}^{N_{\rm
event}}N^{k}\left(N^{k}-1\right)},$ (22)

and

$\left\langle\left\langle
2\right\rangle\right\rangle_{Y_{T}}=\frac{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j,i\neq
j}^{N^{k}}y_{T,i}^{k}y_{T,j}^{k}e^{in\left(\phi_{i}^{k}-\phi_{j}^{k}\right)}}{\sum_{k=1}^{N_{\rm
event}}\sum_{i,j,i\neq j}^{N^{k}}y_{T,i}^{k}y_{T,j}^{k}}.$ (23)

Elliptic flow can be estimated from two-particle correlations by lab30 ; lab31

$v_{2}(N)=\sqrt{\left\langle\left\langle 2\right\rangle\right\rangle_{N}}.$
(24)

Similarly, corresponding anisotropic coefficient of TTR can be estimated by,

$v_{2}(Y_{T})=\sqrt{\left\langle\left\langle
2\right\rangle\right\rangle_{Y_{T}}}.$ (25)

Equation (25) is obtained in the same way as Eq. (24) except for the nonunit
weight factor.

For MTR, its weight is $\omega_{i}=\left.y_{T,i,m}\middle/N_{m}\right.$. The
two correlated particles could come from the same azimuthal angle bin or from
different bins. If they come from different bins, the single-event average
two-particle azimuthal correlations is as follows:

$\begin{split}\langle 2\rangle_{\rm d}^{k}=&\sum_{m,n=1,m\neq n}^{N_{\rm
bin}}\left(\frac{1}{N_{m}^{k}N_{n}^{k}}\right.\\\
&\left.\sum_{i=1}^{N_{m}^{k}}\sum_{j=1}^{N_{n}^{k}}y_{T,i,m}^{k}y_{T,j,n}^{k}e^{in(\phi_{i,m}^{k}-\phi_{j,n}^{k})}\right).\end{split}$
(26)

Here $N_{m}^{k}$ and $N_{n}^{k}$ are the numbers of particles in the _m_ th
and _n_ th bin, respectively. If they come from the same bin, the single-event
average two-particle azimuthal correlations are as follows:

$\begin{split}\langle 2\rangle_{\rm s}^{k}=&\sum_{m=1}^{N_{\rm
bin}}\left(\frac{1}{N_{m}^{k}(N_{m}^{k}-1)}\right.\\\ &\left.\sum_{i,j=1,i\neq
j}^{N_{m}^{k}}y_{T,i,m}^{k}y_{T,j,m}^{k}e^{in\left(\phi_{i,m}^{k}-\phi_{j,m}^{k}\right)}\right).\end{split}$
(27)

Therefore, $\left\langle\left\langle 2\right\rangle\right\rangle$ of the
azimuthal distribution of MTR is as follows:

$\langle\left\langle 2\rangle\right\rangle_{y_{T}}=\frac{\sum_{k=1}^{N_{\rm
event}}\langle 2\rangle^{k}_{\rm d}+\langle 2\rangle^{k}_{\rm
s}}{\sum_{k=1}^{N_{\rm event}}W^{k}},$ (28)

where $W^{k}$ is the event weight, i.e.,

$\begin{split}W^{k}=&\sum_{m,n=1,m\neq n}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}N_{n}^{k}}\sum_{i=1}^{N_{m}^{k}}\sum_{j=1}^{N_{n}^{k}}y_{T,i,m}^{k}y_{T,j,n}^{k}\\\
\ &+\sum_{m=1}^{N_{\rm
bin}}\frac{1}{N_{m}^{k}\left(N_{m}^{k}-1\right)}\sum_{i,j=1,i\neq
j}^{N_{m}^{k}}y_{T,i,m}^{k}y_{T,j,m}^{k}.\end{split}$ (29)

The corresponding anisotropic radial flow is as follows:

$v_{2}(y_{T})=\sqrt{\left\langle\left\langle
2\right\rangle\right\rangle_{y_{T}}}.$ (30)

Up to now, we derive the anisotropic coefficients of azimuthal multiplicity,
TTR and MTR distributions by both the EPM and the TPCM. In order to check how
these two methods work in practice, we apply them to the event sample
generated by Monte Carlo simulation in the following section.

##  III Application

Figure 1:  Centrality dependence of elliptic flow (solid black dots),
anisotropic radial flow (solid red stars), anisotropic TTR flow (solid blue
triangles), and the summation of elliptic flow and anisotropic radial flow
(solid purple squares). (a) is the true results. (b) and (c) are the results
estimated by the EPM and the TPCM, respectively. The error of each point is
smaller than the symbol size.

The AMPT model with string melting lab38 ; lab39  can reproduce the observed
elliptic flow at the RHIC lab32 ; lab33 ; lab34 ; lab35 . The event sample
generated by this model has the basic characteristics of flows, in general,
and can be used to test the effectivity of the two aforementioned methods.

With this model, we generate $4\times 10^{6}$ events for Au + Au collisions at
200 GeV. The analysis is presented in the kinetic ranges $|\eta|\leq 2.5$, and
$p_{T}\in\left[0.15,2\right]\rm GeV/\mathit{c}$ which is the same as the
RHIC/STAR Collaboration lab33 ; lab35 . Nine centrality bins are defined by
the $N_{\rm part}$ corresponding to the nine multiplicity-ranges, consistent
with those at the RHIC/STAR Collaboration lab35 .

For the EPM, two subevents [(a) and (b)] for resolution corrections are
constructed in two longitudinal symmetry windows
$|\eta|\in\left[3.3,4.5\right]$, similar to those at the RHIC/STAR
Collaboration lab23 . We also use a random method to make the number of
charged particles in each subevent even lab29 ; lab33 .

Since the azimuthal angle of the true reaction plane $\Psi_{r}$ is known and
set to zero in this model, $v_{2}(Y_{T}),v_{2}(N)$, and $v_{2}(y_{T})$ can be
directly obtained by the corresponding azimuthal distributions. These directly
obtained $v_{2}(Y_{T}),v_{2}(N)$, and $v_{2}(y_{T})$ can be regarded as their
true values. Their centrality dependence are represented in Fig. 1(a) by solid
blue triangles, black dots, and red stars, respectively.

Figure 2:  Centrality dependence of (a) $v_{2}(N)$, (b) $v_{2}(Y_{T})$, and
(c) $v_{2}(y_{T})$ where solid black dots are true values, solid red stars,
and solid blue triangles are estimated by the EPM and the TPCM, respectively.
The error of each point is smaller than the symbol size.

In Fig. 1(a), three kinds of flows have similar centrality dependence,
consistent with those obtained from UrQMD lab27 ; lab28 ; lab40 . In each of
the centrality interval, $v_{2}(Y_{T})$ is the largest, $v_{2}(N)$ is medial,
and $v_{2}(y_{T})$ is the smallest. It is no surprise that $v_{2}(N)$ and
$v_{2}(y_{T})$ are both smaller than $v_{2}(Y_{T})$. From the definitions of
Eqs. (2), (3), and (6), TTR contains the contributions from both the number of
particles (multiplicity) and the transverse movement. MTR is raw kinetics, and
the influence of multiplicity is excluded. Therefore, $v_{2}(y_{T})$ and
$v_{2}(N)$ are both smaller than $v_{2}(Y_{T})$.

It is interesting that $v_{2}(y_{T})+v_{2}(N)$ (purple squares) overlaps with
$v_{2}(Y_{T})$ (blue triangles) at each of the nine centralities. This
relationship shows that the anisotropic radial flow and elliptic flow are
additive, although MTR and multiplicity are not simply related to TTR.

In order to compare the anisotropic coefficients estimated by the two methods
with their true values of $v_{2}(N),v_{2}(Y_{T})$, and $v_{2}(y_{T})$ are
presented in Figs. 2(a)-2(c), respectively, where solid black dots are the
true values, solid red stars, and blue triangles are estimated by the EPM and
the TPCM.

For $v_{2}(N)$ as shown in Fig. 2(a), in each centrality interval, the black
dot is the lowest, the red star is in the middle, and the blue triangle is the
highest. This shows that the estimation given by TPCM is higher that given by
the EPM, and the estimations given these two methods are both higher than the
true value. The excess parts are the contributions of nonflow and flow
fluctuations lab41 ; lab42 . Nevertheless, the difference is roughly an
overall shift. The general trends of three data sets are consistent with each
other and with the expectation that the larger elliptic flow appears in the
midcentral collisions and small elliptic flow in peripheral and central
collisions. Therefore, both methods can be considered as a valid estimation of
true elliptic flow.

Similarly, for $v_{2}(Y_{T})$ as shown in Fig. 2(b), in each centrality
interval, three kinds of points have the same position ordering as shown in
Fig. 2(a), i.e., the lowest black dot, the middle red star, and the highest
blue triangle. By the same reasoning, these results demonstrate that the both
methods are effective in estimating $v_{2}(Y_{T})$.

For $v_{2}(y_{T})$ as shown in Fig. 2(c), three kinds of points also display
the same ordering as shown in Figs. 2(a) and 2(b), i.e., the lowest black
dots, the middle red stars, and the highest blue triangles for the seven
middle and central collisions. These results are not really surprising. As
even in some toy Monte Carlo studies where there is only flow fluctuations and
no systematic biases due to nonflow, the elliptic flows given by the EPM and
the TPCM are still different lab42 .

The centrality dependence of elliptic flow, anisotropic TTR flow, and radial
flow estimated by the EPM and the TPCM are presented in Figs. 1(b) and 1(c),
respectively, in comparison to their true values in Fig. 1(a). The summations
of elliptic flow $v_{2}(N)$ and anisotropic radial flow $v_{2}(y_{T})$ (purple
squares) estimated by these two methods also overlap with corresponding
anisotropic TTR flow $v_{2}(Y_{T})$ (blue triangles), consistent with the case
in Fig. 1(a). Therefore, the elliptic flow and anisotropic radial flow
estimated by these two methods are also additive, just like their true values.
This demonstrates that the two methods are equally effective.

However, the TPCM is ineffective in estimating $v_{2}(y_{T})$ for the two
peripheral-collision bins. We will show in the following section why and how
the TPCM fails.

##  IV The limitation of the TPCM

$v_{2}(y_{T})$ given by the TPCM in Eq. (30) is the square root of
$\langle\langle 2\rangle\rangle_{y_{T}}$. For the two peripheral-collision
bins, $\langle\langle 2\rangle\rangle_{y_{T}}$ becomes negative. This is why
the TPCM fails.

We can understand the cause of this problem from the original definition of
MTR. MTR is defined to be averaged over $N_{m}$ (the number of particles in
the _m_ th bin). If all particles are uniformly distributed in the whole
azimuthal region $N_{m}\sim N/N_{\rm bin}$ ($N$ is multiplicity, and $N_{\rm
bin}$ is the total number of bins). Now, assuming $\langle\langle
2\rangle\rangle_{y_{T}}$ is well defined, we should have selected a proper
$N_{\rm bin}$ for which $N_{m}$ is large enough.

To examine the proper range of $N_{\rm bin},\langle\langle
2\rangle\rangle_{y_{T}}$ versus $N_{\rm bin}$ at the nine centralities are
presented in Fig. 3. For midcentral collisions and central collisions,
$\langle\langle 2\rangle\rangle_{y_{T}}$ decreases with $N_{\rm bin}$ when
$N_{\rm bin}\leq 20$, and then becomes independent of $N_{\rm bin}$ when
$N_{\rm bin}\geq 20$. There is a wide range of $N_{\rm bin}$, where
$\langle\langle 2\rangle\rangle_{y_{T}}$ stays flat. This range guarantees
that $\langle\langle 2\rangle\rangle_{y_{T}}$ is independent of $N_{\rm bin}$.
Therefore, we choose $N_{\rm bin}=25$ for all the aforementioned analysis.

Figure 3:  $\langle\langle 2\rangle\rangle_{y_{T}}$ versus $N_{\rm bin}$ at
the nine centralities where the error of each point is smaller than the symbol
size.

However, for the two peripheral-collision bins, $\langle\langle
2\rangle\rangle_{y_{T}}$ rapidly decreases to negative as $N_{\rm bin}$
increases. Obviously, this is not caused by the change in $N_{\rm bin}$ but
instead by the decrease of multiplicity $N$ from central to peripheral
collisions. When $N_{\rm bin}$ is fixed, $N_{m}$ is solely proportional to
$N$. Therefore, $N_{m}$ also decreases from central to peripheral collisions.

To find the lower threshold of $\langle N_{m}\rangle$ in general, we randomly
drop some particles in each event. Using this method, we can show how
$\langle\langle 2\rangle\rangle_{y_{T}}$ changes with $\langle N_{m}\rangle$,
and where in each centrality interval the TPCM becomes invalid.

$\langle\langle 2\rangle\rangle_{y_{T}}$ versus $\langle N_{m}\rangle$ for
four cases are presented in Figs. 4(a)-4(d). For the two upper subfigures, we
choose the pseudorapidity range $|\eta|\leq 2.5$ and $N_{\rm bin}=25$ for Fig.
4(a) and $N_{\rm bin}=35$ for Fig. 4(b). In Fig. 4(a), with a decrease in
$\langle N_{m}\rangle,\langle\langle 2\rangle\rangle_{y_{T}}$ stays flat and
positive. This implies that $\langle\langle 2\rangle\rangle_{y_{T}}$ can still
be estimated correctly, even though some particles in each of the events are
dropped. When $\langle N_{m}\rangle$ decreases even lower, $\langle\langle
2\rangle\rangle_{y_{T}}$ rapidly drops to negative. This implies that the
correlations between the two particles decrease significantly, and the method
becomes invalid. Meanwhile, Fig. 4(b) also shows the same $\langle
N_{m}\rangle$ which makes $\langle\langle 2\rangle\rangle_{y_{T}}$ rapidly
drop to negative. Therefore, independent of $N_{\rm bin}$, as long as the mean
number of particles in a bin is less than 10, the TPCM will fail.

Figure 4:  $\langle\langle 2\rangle\rangle_{y_{T}}$ versus $\langle
N_{m}\rangle$ at the nine centralities where the error of each point is
smaller than the symbol size.

For the two lower subfigures, we choose the pseudorapidity range $|\eta|\leq
1$ and $N_{\rm bin}=25$ for Fig. 4(c) and $N_{\rm bin}=35$ for Fig. 4(d). In
contrast with the two upper row subfigures, narrow pseudorapidity range makes
the total multiplicity smaller. The general trends in Figs. 4(c) and 4(d) are
consistent with each other and with the two upper subfigures. This means that
$\langle\langle 2\rangle\rangle_{y_{T}}$ is independent of the total mean
multiplicity $\langle N\rangle$ and $N_{\rm bin}$ but is dependent on $\langle
N_{m}\rangle$. Both Figs. 4(c) and 4(d) show negative $\langle\langle
2\rangle\rangle_{y_{T}}$ when $\langle N_{m}\rangle<10$. Therefore, the lower
threshold of $\langle N_{m}\rangle$ is ten for the TPCM.

In addition, one may ask why there is no such lower threshold in the EPM. This
is because $v_{2}(y_{T})$ given by Eq. (30) is of higher order than that given
by Eq. (18). The right side of Eq. (30) is the square root of two-particle
correlations. Therefore, higher precision is required for the TPCM.

Moreover, as shown in Fig. 1, anisotropic radial flow is the smallest one
among three. Its measurement is more difficult than anisotropic TTR flow, or
elliptic flow. Therefore, a larger number of two-particle pairs is required.

##  V Summary and conclusions

Anisotropic radial flow can be directly extracted from the azimuthal
distribution of the MTR. First, we apply the EPM and the TPCM to estimate the
anisotropic coefficients of the azimuthal distributions of TTR and MTR in
relativistic heavy-ion collisions.

Then, using the event sample of Au + Au collisions at 200 GeV generated by the
AMPT model with string melting, we show that the EPM and the TPCM are both
effective in estimating the anisotropic TTR flow. The EPM is also effective in
estimating anisotropic radial flow.

For central collisions and midcentral collisions, the TPCM is effective in
estimating anisotropic radial flow. However, for the two peripheral-collision
bins, this method fails when the mean number of particles in a bin becomes
less than ten.

Therefore, anisotropic radial flow can be estimated in a model-independent way
by the EPM and the TPCM in relativistic heavy-ion collisions. For the TPCM to
be reliable, the mean number of particles in an azimuthal bin must be above a
certain threshold.

We find that the summation of the true elliptic flow and true anisotropic
radial flow is consistent with the true anisotropic TTR flow. This
relationship also holds for the corresponding coefficients estimated by the
EPM and the TPCM. Therefore, elliptic flow and anisotropic radial flow are
additive.

##  VI Acknowledgement

We are very grateful to Dr. G. Wang for his valuable comments and suggestions.
We thank Dr. A. Tang, F. Wang, Y. Zhou, and Z. Lin for helpful discussions.
This work was supported, in part, by the Ministry of Science and Technology
(MoST) under Grant No. 2016YFE0104800 and the Fundamental Research Funds for
the Central Universities under Grant No. CCNU19ZN019.

## References

  * (1) J. Y. Ollitrault, Nucl. Phys. A 638, 195c (1998).
  * (2) P. Braun-Munzinger, J. Stachel, Nature (London) 448, 302 (2007).
  * (3) S. A. Voloshin, Phys. Rev. C 55, R1630(R) (1997).
  * (4) S. A. Voloshin, A. M. Poskanzer, and R. Snellings, in _Landolt-Boernstein, Relativistic Heavy Ion Physics_ , Vol. 1/23 (Springer-Verlag, 2010), pp. 5–54.
  * (5) T. Hirano, arXiv:nucl-th/9904082.
  * (6) P. Danielewicz, Phys. Rev. C 51, 716 (1995).
  * (7) C. M. Hung and E. Shuryak, Phys. Rev. C 57, 1891 (1998).
  * (8) P. Huovinen and P. V. Ruuskanen, Annu. Rev. Nucl. Part. Sci. 56, 163 (2006).
  * (9) M. Shao, L. Yi, Z. B. Tang, H. F. Chen _et al._ , J. Phys. G: Nucl. Part. Phys. 37, 085104 (2010).
  * (10) Z. B. Tang, L. Yi, L. J. Ruan _et al._ , Chin. Phys. Lett. 30, 031201 (2013).
  * (11) J. Chen, J. Deng, Z. Tang, Z. Xu, and L. Yi, arXiv: 2012.02986.
  * (12) L. Li, N. Li, and Y. F. Wu, J. Phys. G: Nucl. Part. Phys. 40, 075104 (2013).
  * (13) P. Huovinen, P. F. Koblb, U. W. Heinz, P. V. Ruuskanen, and S. A. Voloshin, Phys. Lett. B 503, 58 (2001).
  * (14) N. Borghini and J. Y. Ollitrault, Phys. Lett. B 642, 227 (2006).
  * (15) L. Li, N. Li, and Y. F. Wu, Chin. Phys. C 36, 423 (2012).
  * (16) L. D. Landau and E. M. Lifshitz, _Fluid Mechanics_ , 2nd. ed., Vol. 6, Course of Theoretical Physics (Pergamon, Oxford, 1987), p.44.
  * (17) C. Shen and U. Heinz, Phys. Rev. C 85, 054902 (2012).
  * (18) Iu. A. Karpenko, P. Huovinen, H. Petersen, and M. Bleicher, Phys. Rev. C 91, 064901 (2015).
  * (19) P. Liu and R. A. Lacey, Phys. Rev. C 98, 021902 (2018).
  * (20) E. Schnedermann, J. Sollfrank, and U. W. Heinz, Phys. Rev. C 48, 2462 (1993).
  * (21) Y. Oh, Z. W. Lin, and C. M. Ko, Phys. Rev. C 80, 064902 (2009).
  * (22) J. Barrette _et al._ (E877 Collaboration), Phys. Rev. C 62, 024901 (2000).
  * (23) L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 93, 014907 (2016).
  * (24) L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 96, 044904 (2017).
  * (25) B. Abelev _et al._ (ALICE Collaboration), Phys. Rev. C 88, 044910 (2013).
  * (26) P. Yang, L. Li, and Y. F. Wu, Beijing: Sciencepaper Online, 201404-454 (2014) [arXiv:1405.0686].
  * (27) S. Sarkar, P. Mali, and A. Mukhopadhyay, Phys. Rev. C 95, 014908 (2017).
  * (28) S. Sarkar, P. Mali, S. Ghosh, and A. Mukhopadhyay, Adv. High Energy Phys. 2018, 7453752 (2018).
  * (29) A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58, 1671 (1998).
  * (30) A. Bilandzic, R. Snellings, and S. A. Voloshin, Phys. Rev. C 83, 044913 (2011).
  * (31) N. Borghini, P. M. Dinh, and J. Y. Ollitrault, Phys. Rev. C 64, 054901 (2001).
  * (32) C. Adler _et al._ (STAR Collaboration), Phys. Rev. C 66, 034904 (2002).
  * (33) B. I. Abelev _et al._ (STAR Collaboration), Phys. Rev. C 77, 054901 (2008).
  * (34) L. Adamczyk _et al._ (STAR Collaboration), Phys. Rev. C 86, 054908 (2012).
  * (35) J. Adams _et al._ (STAR Collaboration), Phys. Rev. C 72, 014904 (2005).
  * (36) S. Wang, Y. Z. Jiang, Y. M. Liu, D. Keane, D. Beavis, S. Y. Chu, S. Y. Fung, M. Vient, C. Hartnack, and H. Stöcker, Phys. Rev. C 44, 1091 (1991).
  * (37) J. Y. Ollitrault, Phys. Rev. D 48, 1132 (1993).
  * (38) Z. W. Lin and C. M. Ko, Phys. Rev. C 65, 034904 (2002).
  * (39) Z. W. Lin, C. M. Ko, B. A. Li, B. Zhang and S. Pal, Phys. Rev. C 72, 064901 (2005).
  * (40) S. Sarkar and A. Mukhopadhyay, Proc. DAE-BRNS Symp. Nucl. Phys. 60, 752 (2015).
  * (41) J. Y. Ollitrault, A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 80, 014904 (2009).
  * (42) M. Luzum and J. Y. Ollitrault, Phys. Rev. C 87, 044907 (2013).