$\|W\|_{{\boldsymbol{S}}_{p}}\leq\|\Psi\|_{L^{\infty}\otimes^{\rm
h}\\!L^{\infty}\otimes_{\rm
h}\\!L^{\infty}}\|T\|\cdot\|R\|_{{\boldsymbol{S}}_{p}}.$

These definitions of triple operator integrals were given in the paper [4]; we
also refer the reader to the paper [13], where triple operator integrals were
defined in a more general situation.

Now we are ready to state the theorem on Lipschitz type estimates for
functions of noncommuting self-adjoint operators under perturbation.

######  Theorem 7.18.

Let $f$ be a function in the Besov class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$
and let $1\leq p\leq 2$. Suppose that $(A_{1},B_{1})$ are $(A_{2},B_{2})$
pairs of bounded not necessarily commuting self-adjoint operators such that
$A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$ and $B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$.
Then the divided differences $\big{(}\mathfrak{D}^{[1]}f\big{)}$ and
$\big{(}\mathfrak{D}^{[2]}f\big{)}$ defined by (7.32) and (7.33) belong to the
Haagerup like tensor products $L^{\infty}(E_{1})\\!\otimes_{\rm
h}\\!L^{\infty}(E_{2})\\!\otimes^{\rm h}\\!L^{\infty}(E_{3})$ and
$L^{\infty}(E_{1})\\!\otimes^{\rm h}\\!L^{\infty}(E_{2})\\!\otimes_{\rm
h}\\!L^{\infty}(E_{3})$ for arbitrary Borel spectral measures $E_{1}$, $E_{2}$
and $E_{3}$; moreover, representation (7.8) holds, in which the first triple
operator integral should be understood in the sense of Definition 2 while the
second triple operator integral should be understood in the sense of
Definition 4.

Moreover, the following Lipschitz type estimate holds

$\displaystyle\|f(A_{1},B_{1})-f(A_{2},B_{2})\|_{{\boldsymbol{S}}_{p}}\leq\operatorname{const}\|f\|_{B_{\infty,1}^{1}}\max\big{\\{}\|A_{1}-B_{1}\|_{{\boldsymbol{S}}_{p}},\|A_{2}-B_{2}\|_{{\boldsymbol{S}}_{p}}\big{\\}}.$
(7.39)

Theorem 7.18 was established in [4]. In the same paper it was shown that for
$p>2$ as well as in the operator norm, such a Lipschitz estimate does not
hold.

Note also that it was shown in [86] that such Lipschitz type estimates cannot
be generalized to the case of functions of three noncommuting self-adjoint
operators.

###  7.9. Functions of pairs of unbounded noncommuting operators

We proceed now to the case of pairs of unbounded not necessarily commuting
self-adjoint operators.

Let $1\leq p\leq 2$ and let $f$ a function in the Besov class
$B_{\infty,1}^{1}({\mathbb{R}})$. Suppose now that $(A_{1},B_{1})$ and
$(A_{2},B_{2})$ are pairs of unbounded not necessarily commuting self-adjoint
operators such that $A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$ and
$B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$.

Note, first of all, that both triple operator integrals on the right-hand side
of representation (7.8) are still well defined as in the case of pairs of
bounded operators and determine operators of class ${\boldsymbol{S}}_{p}$
while their norms in ${\boldsymbol{S}}_{p}$ can still be estimated in terms of
the right-hand side of inequality (7.39).

The problem is how to define functions of operators $f(A_{1},B_{1})$ and
$f(A_{2},B_{2})$ as unbounded densely defined operators. It is to this problem
the papers [16] and [17] are devoted.

In Subsection 7.8 we have defined the functions $f(A,B)$ of not necessarily
commuting self-adjoint operators in the case when $f\in
L^{\infty}_{E_{A}}\otimes_{\rm h}L^{\infty}_{E_{B}}$. Let now $f$ be a
function of two variables and let $f_{\sharp}$ be the function defined by
$f_{\sharp}(s,t)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(1-{\rm
i}t)^{-1}f(s,t)$. Assume that $f_{\sharp}\in L^{\infty}_{E_{A}}\otimes_{\rm
h}L^{\infty}_{E_{B}}$. We define the operator $f(A,B)$ by

$f(A,B)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}f_{\sharp}(A,B)(I-{\rm
i}B)=\left(\,\,\,\iint\limits_{{\mathbb{R}}\times{\mathbb{R}}}f_{\sharp}(s,t)\,{\rm
d}E_{A}(s)\,{\rm d}E_{B}(t)\right)(I-{\rm i}B).$

Then $f(A,B)$ is a densely defined operator whose domain coincides with the
domain ${\rm D}(B)$ of $B$. It is not necessarily bounded but the operator
$f(A,B)(I-{\rm i}B)^{-1}$ is bounded.

It was established in [15] and [17] that if
$f\in{\mathscr{E}}_{\sigma}^{\infty}({\mathbb{R}}^{2})$, $\sigma>0$, then
$f_{\sharp}\in L^{\infty}_{E_{1}}\\!\otimes_{\rm h}\\!L^{\infty}_{E_{2}}$ for
arbitrary Borel spectral measure $E_{1}$ and $E_{2}$. Therefore, if
$f\in{\mathscr{E}}_{\sigma}^{\infty}({\mathbb{R}}^{2})$, then the operator
$f_{\sharp}(A,B)$ is bounded while $f(A,B)$ is a not necessarily bounded
densely defined operator with domain ${\rm D}(B)$. Herewith the following
inequality holds

$\|f_{\sharp}\|_{L^{\infty}_{E_{1}}\otimes_{\rm
h}L^{\infty}_{E_{2}}}\leq\operatorname{const}(1+\sigma)\|f\|_{L^{\infty}({\mathbb{R}}^{2})}.$

This easily implies that if $f$ is a function in the inhomogeneous Besov class
$\Be_{\infty,1}^{1}({\mathbb{R}}^{2})$, then $f_{\sharp}\in
L^{\infty}_{E_{1}}\\!\otimes_{\rm h}\\!L^{\infty}_{E_{2}}$ and

$\|f_{\sharp}\|_{L^{\infty}_{E_{1}}\otimes_{\rm
h}L^{\infty}_{E_{2}}}\leq\operatorname{const}\|f\|_{\Be_{\infty,1}^{1}}$

for arbitrary spectral Borel measures $E_{1}$ and $E_{2}$ (see [16] and [17]).

Thus, for an arbitrary pair $(A,B)$ of not necessarily bounded self-adjoint
operators and for a function $f$ in the inhomogeneous Besov class
$\Be_{\infty,1}^{1}({\mathbb{R}}^{2})$, we can define the operator $f(A,B)$ as
a densely defined operator. Herewith the following result holds (see [16] and
[17]):

######  Theorem 7.19.

Let $f\in\Be_{\infty,1}^{1}({\mathbb{R}}^{2})$ and let $p\in[1,2]$. Suppose
that $(A_{1},B_{1})$ and $(A_{2},B_{2})$ are pairs of not necessarily bounded
and not necessarily commuting self-adjoint operators such that
$A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$ and $B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$.
Then

$\|f(A_{1},B_{1})-f(A_{2},B_{2})\|_{{\boldsymbol{S}}_{p}}\leq\operatorname{const}\|f\|_{\Be_{\infty,1}^{1}}\max\big{\\{}\|A_{1}-A_{2}\|_{{\boldsymbol{S}}_{p}},\|B_{1}-B_{2}\|_{{\boldsymbol{S}}_{p}}\big{\\}}.$

Note here that an analogue of this result for functions of pairs of
noncommuting dissipative operators was obtained in [15].

###  7.10. Functions of pairs of almost commuting operators

Let $A$ and $B$ be bounded self-adjoint operators. They are said to be almost
commuting if their commutator
$[A,B]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}AB-BA$ is a trace class
operator.

A polynomial calculus for a pair $(A,B)$ of almost commuting self-adjoint
operators is defined in the following way. If ${\varphi}$ is a polynomial of
the form ${\varphi}(s,t)=\sum_{j,k}a_{jk}s^{j}t^{k}$, then the operator
${\varphi}(A,B)$ is defined by

${\varphi}(A,B)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum_{j,k}a_{jk}A^{j}B^{k}.$

It is easy to verify that if ${\varphi}$ are $\psi$ polynomials of two real
variables, then $[{\varphi}(A,B),\psi(A,B)]\linebreak\in{\boldsymbol{S}}_{1}$.

In the paper [38] by Helton and Howe the authors obtained a trace formula for
commutators of polynomials of almost commuting self-adjoint operators. Let $A$
and $B$ be almost commuting self-adjoint operators and let ${\varphi}$ and
$\psi$ be polynomials of two real variables. Then

$\displaystyle\operatorname{trace}[{\varphi}(A,B),\psi(A,B)]=-{\rm
i}\int_{{\mathbb{R}}^{2}}\left(\frac{\partial{\varphi}}{\partial
x}\frac{\partial\psi}{\partial y}-\frac{\partial{\varphi}}{\partial
y}\frac{\partial\psi}{\partial x}\right){\rm d}P,$ (7.40)

where $P$ is the real-valued measure with compact support, which is determined
by the pair $(A,B)$. It was shown in [90] that the measure $P$ is absolutely
continuous with respect to planar Lebesgue measure and

${\rm d}P(x,y)=\frac{1}{2\pi}g(x,y)\,{\rm d}{\boldsymbol{m}}_{2}(x,y),$

where $g$ is the so-called Pincus principal function introduced in [89].

The polynomial calculus was extended in [30] to the calculus on the class of
functions ${\varphi}={\mathscr{F}}{\omega}$ that are Fourier transforms of
complex Borel measure ${\omega}$ on ${\mathbb{R}}^{2}$ such that

$\int_{{\mathbb{R}}^{2}}(1+|t|)(1+|s|)\,{\rm d}|{\omega}|(s,t)<\infty;$

moreover the Helton–Howe trace formula (7.40) was extended to the case of such
functions.

Next, in the author’s paper [78] such results were obtained for a yet broader
class of functions. Moreover, in the same paper [78] it was shown that it is
impossible to extend such a functional calculus to the class of all
continuously differentiable functions on ${\mathbb{R}}^{2}$ that would satisfy
certain natural requirements and
$[{\varphi}(A,B),\psi(A,B)]\in{\boldsymbol{S}}_{1}$ for arbitrary continuously
differentiable functions ${\varphi}$ and $\psi$.

Most advanced results in this direction were obtained in the paper [10], in
which the desired functional calculus is constructed on the Besov class
$B_{\infty,1}^{1}({\mathbb{R}}^{2})$ and it is shown that this functional
calculus is almost multiplicative.

Let us proceed now to the main results of the paper [10]. Note that these
results use triple operator integrals for Haagerup-like tensor products of the
$L^{\infty}$ spaces that were introduced in [4] and the estimates of the trace
norms of such operator integrals that were found in [4], see Subsection 7.8.

######  Theorem 7.20.

Let $A$ and $B$ be self-adjoit operators and let $Q$ be a bounded linear
operator such $[A,Q]\in{\boldsymbol{S}}_{1}$ and
$[B,Q]\in{\boldsymbol{S}}_{1}$. Suppose that ${\varphi}$ is a function in the
Besov class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$. Then
$[{\varphi}(A,B),Q\big{]}\in{\boldsymbol{S}}_{1}$,

$\displaystyle\big{[}{\varphi}(A,B),Q\big{]}$
$\displaystyle=\iiint\frac{{\varphi}(x,y_{1})-{\varphi}(x,y_{2})}{y_{1}-y_{2}}\,{\rm
d}E_{A}(x)\,{\rm d}E_{B}(y_{1})[B,Q]\,{\rm d}E_{B}(y_{2})$
$\displaystyle+\iiint\frac{{\varphi}(x_{1},y)-{\varphi}(x_{2},y)}{x_{1}-x_{2}}\,{\rm
d}E_{A}(x_{1})[A,Q]\,{\rm d}E_{A}(x_{2})\,{\rm d}E_{B}(y)$

and

$\big{\|}[{\varphi}(A,B),Q\big{]}\big{\|}_{{\boldsymbol{S}}_{1}}\leq\operatorname{const}\|{\varphi}\|_{B_{\infty,1}^{1}({\mathbb{R}}^{2})}\big{(}\big{\|}[A,Q]\big{\|}_{{\boldsymbol{S}}_{1}}+\big{\|}[B,Q]\big{\|}_{{\boldsymbol{S}}_{1}}\big{)}.$

######  Theorem 7.21.

Let $A$ and $B$ be almost commuting self-adjoint operators and let ${\varphi}$
and $\psi$ be functions in the Besov $B_{\infty,1}^{1}({\mathbb{R}}^{2})$.
Then

$\displaystyle\\!\\!\\!\\!\\!\big{[}{\varphi}(A,B),\psi(A,B)\big{]}\\!$
$\displaystyle=\iiint\frac{{\varphi}(x,y_{1})-{\varphi}(x,y_{2})}{y_{1}-y_{2}}\,{\rm
d}E_{A}(x)\,{\rm d}E_{B}(y_{1})[B,\psi(A,B)]\,{\rm d}E_{B}(y_{2})$
$\displaystyle+\\!\\!\iiint\frac{{\varphi}(x_{1},y)-{\varphi}(x_{2},y)}{x_{1}-x_{2}}{\rm
d}E_{A}(x_{1})[A,\psi(A,B)]{\rm d}E_{A}(x_{2}){\rm d}E_{B}(y)$

and

$\big{\|}[{\varphi}(A,B),\psi(A,B)\big{]}\big{\|}_{{\boldsymbol{S}}_{1}}\leq\operatorname{const}\|{\varphi}\|_{B_{\infty,1}^{1}({\mathbb{R}}^{2})}\|\psi\|_{B_{\infty,1}^{1}({\mathbb{R}}^{2})}\big{\|}[A,B]\big{\|}_{{\boldsymbol{S}}_{1}}.$

Now we can extend the Helton–Howe trace formula to the case of functions in
the Besov class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$.

######  Theorem 7.22.

Let $A$ and $B$ be almost commuting self-adjoint operators and let ${\varphi}$
and $\psi$ be functions in the Besov class
$B_{\infty,1}^{1}({\mathbb{R}}^{2})$. Then

$\displaystyle\operatorname{trace}\big{(}{\rm
i}\big{(}{\varphi}(A,B)\psi(A,B)$
$\displaystyle-\psi(A,B){\varphi}(A,B)\big{)}\big{)}$
$\displaystyle=\frac{1}{2\pi}\iint_{{\mathbb{R}}^{2}}\left(\frac{\partial{\varphi}}{\partial
x}\frac{\partial\psi}{\partial y}-\frac{\partial{\varphi}}{\partial
y}\frac{\partial\psi}{\partial x}\right)g(x,y)\,{\rm d}x\,{\rm d}y,$

where $g$ is the Pincus principal function for the pair $(A,B)$.

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St.Petersburg State University
---
Universitetskaya nab., 7/9
199034 St.Petersburg
Russia
St.Petersburg Department
Steklov Institute of Mathematics
Russian Academy of Sciences
Fontanka 27, 191023 St.Petersburg
Russia
People’s Friendship University
of Russia (RUDN University)
6 Miklukho-Maklaya St, Moscow,
6117198, Russian Federation


# Transverse momentum decorrelation of the flow vector in Pb–Pb collisions at
$\sqrt{s_{{}_{\rm NN}}}$ = 5.02 TeV

Emil Gorm Nielsen<EMAIL_ADDRESS>You Zhou<EMAIL_ADDRESS>Niels
Bohr Institute, University of Copenhagen

###### Abstract

The individual studies of the anisotropic flow vector, flow angle and flow
magnitude fluctuations with multi-particle correlations provide insight into
the initial conditions and properties of the quark-gluon plasma (QGP) created
in high-energy heavy-ion collisions. Recent measurements of these fluctuations
have been available and the comparison to hydrodynamic models shows unique
sensitivities to the initial conditions of the system, but also a puzzling
dependence on the specific shear viscosity $\eta/s$. In this paper, a
systematic study with A Multi-Phase Transport (AMPT) model using different
tunings of the initial conditions, partonic cross section and hadronic
interactions investigates the $p_{\rm T}$-dependent flow vector, flow angle
and flow magnitude fluctuations. It is found that the transport model
reasonably describes the flow vector, flow angle and flow magnitude
fluctuations observed in data and that the fluctuations are driven by
fluctuations in the initial state. The comparison of data and model presented
in this paper enables further constraints on the initial conditions of the
heavy-ion collisions.

††preprint: APS/123-QED

##  I Introduction

The primary purpose of ultra-relativistic heavy-ion collisions, such as those
at the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider
(LHC), is to create the strongly-coupled Quark-Gluon Plasma (QGP) and measure
its properties [1, 2, 3, 4, 5]. The azimuthal anisotropy of produced particles
is a key observable in probing the QGP [6, 7], and it is typically
characterized by a Fourier expansion of the azimuthal angle distribution [8]

$\displaystyle
P(\varphi)=\sum_{n=-\infty}^{\infty}\overline{V}_{n}e^{-in\varphi},$ (1)

where $\varphi$ is the azimuthal angle of the produced particles and
$\overline{V}_{n}=v_{n}e¨^{in\Psi_{n}}$ is the n-th order flow vector with
magnitude $v_{n}$ and orientation $\Psi_{n}$. The magnitude $v_{n}$ is the
n-th order flow coefficient, and the orientation $\Psi_{n}$ is the symmetry
plane angle. The anisotropic flow has been studied extensively at RHIC [9, 10,
11, 12] and the LHC [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Comparison
to theoretical model calculations have shown that the QGP behaves like an
almost perfect fluid [24, 25, 26, 27] with very low specific shear viscosity,
$\eta/s$. A good theoretical model description on the experimental data have a
significant impact on the understanding of the QGP. The current state-of-the-
art understanding of the QGP comes from models based on Bayesian inference
[28, 29, 30], which use multiple experimental measurements to simultaneously
constrain the parameters of the initial conditions and the hydrodynamical
model. The extracted information can be better constrained by including
higher-order flow and correlations of different order flow harmonics in the
experimental observables [31]. Additional constraints can be obtained by
including the $p_{\rm T}$-differential studies of the flow coefficient
$v_{n}(p_{\rm T})$ and its fluctuations, as they contain additional
information about the dynamics of the expansion and the initial conditions of
the heavy-ion collisions [30]. Fluctuations of the flow vector with the
transverse momentum, $p_{\rm T}$, are predicted in hydrodynamical simulations
[32, 33] and have been measured at the LHC [34, 35, 36, 37]. The event-by-
event fluctuations of the flow coefficient $v_{n}$ are well known [7, 24, 38];
however, the decorrelation of the flow coefficient in $p_{\rm T}$ has not been
studied separately from the flow angle fluctuations [34]. Additionally, the
fluctuations of the flow angle $\Psi_{n}$ has proved hard to measure
experimentally [39, 40, 41, 42, 43, 33, 44], and the dependence of the flow
angle on the transverse momentum [33, 45, 46, 44, 47] results in a
decorrelation between the integrated symmetry plane, $\Psi_{n}$, and the
$p_{\rm T}$-differential plane, $\Psi_{n}(p_{\rm T})$. Such fluctuations
potentially affect measurements that rely on the assumption of a common
symmetry plane. The overall flow vector fluctuation gives insight into the
event-by-event fluctuating initial conditions and separating the fluctuations
into the flow angle and flow magnitude fluctuations enables the individual
study of the sources of the fluctuations in heavy-ion collisions. Measurements
of $p_{\rm T}$-dependent flow vector fluctuations have traditionally been
measured with two-particle correlations [32, 33, 34, 35, 36]. Recently, we
proposed new four-particle correlations to measure the flow angle and flow
magnitude fluctuations separately [48], which cannot be done with two-particle
correlations. A follow-up study confirmed that the flow angle and magnitude
fluctuation phenomena were present in hydrodynamical models [44, 49].
Additional hydrodynamical model calculations are compared to the ALICE data in
[50], which were successful in capturing the trend of the data. However, the
model calculations of the flow magnitude fluctuations showed a puzzling trend
when using large values of $\eta/s$. At low $p_{\rm T}$, where the expectation
is to see little to no fluctuations, these calculations show values of the
flow magnitude fluctuations greatly above 1. Meanwhile, systematic studies
with a transport model are completely missing. Thus, we present the first
transport model study on this topic.
In this paper, the $p_{\rm T}$-dependent flow vector, flow angle and flow
magnitude fluctuations are studied with A Multi-Phase Transport model (AMPT)
[51, 52]. The AMPT model is well suited to the study of anisotropic flow
observables. It reproduces measurements of anisotropic flow at RHIC [53, 54]
and the LHC [55, 56]. The analysis in this work is performed with different
tunings of the model in order to explore the effects of the initial
conditions, transport properties of the QGP and the hadronic rescatterings. In
section II, details of the model and the different configurations are
presented. Section III describes the observables and the method used to
extract information about the decorrelation of the flow vector. The results
are presented and discussed in section IV. Finally, the work is summarized in
section V. Ratio plots between the different AMPT configurations for all
observables can be found in appendix A.

##  II The model

In this paper, AMPT [51, 52] is used to study the transverse momentum
decorrelation of the flow vector, flow angle and flow magnitude in Pb–Pb
collisions at 5.02 TeV. The model consists of four stages: initial conditions,
partonic scattering, hadronization, and hadronic rescattering. The initial
conditions are generated with HIJING [57], which includes the spatial and
momentum distributions of mini-jet partons and soft string excitations.
Hadrons produced from string fragmentation are then converted with string
melting into their valence quarks and anti-quarks. The interaction of partons
is treated with ZDC [58], which only includes two-body scatterings. The cross
sections are obtained from pQCD with screening masses

$\displaystyle\sigma=\frac{9\pi\alpha_{s}^{2}}{2\mu^{2}},$ (2)

where $\alpha_{s}$ is the strong coupling and $\mu$ is the Debye screening
mass. At LHC energies, a strong coupling constant of $\alpha_{s}=0.33$ is used
[59]. Partons are converted into hadrons with a quark coalescence model,
combining two quarks into mesons and three quarks into baryons [60]. Finally,
the hadronic rescattering is handled by ART [61, 62], where the dynamics of
the hadronic matter are described by a hadron cascade. In order to study the
effects of the initial state on the flow vector fluctuations, the Lund string
parameters a and b, which are related to the string tension, are varied. The
default setting is $a=0.3$ and $b=0.15$ [59, 56], and the variation is $a=0.5$
and $b=0.9$, which are the default HIJING values [63, 64]. The effect of the
QGP properties is studied by varying the partonic cross section using
different values of the screening mass $\mu$. The values
$\mu=3.2042~{}\mathrm{fm}^{-1}$ and $\mu=2.2814~{}\mathrm{fm}^{-1}$ correspond
to cross sections of $\sigma\approx 1.5$ mb and $\sigma\approx 3$ mb,
respectively. A smaller cross section in AMPT corresponds to a larger
viscosity in viscous hydrodynamics [65, 66]. The specific shear viscosity can
be approximated by [55]

$\displaystyle\eta/s\approx\frac{3\pi}{40\alpha_{s}^{2}}\frac{1}{\left(9+\frac{\mu^{2}}{T^{2}}\right)ln\left(\frac{18+\mu^{2}/T^{2}}{\mu^{2}/T^{2}}\right)-81},$
(3)

where $T$ is the initial temperature. At the LHC, the initial temperature is
approximately $468$ MeV [55], which leads to values of specific shear
viscosity around $0.18$ for 3 mb and $0.28$ for 1.5 mb. Additionally, the
effects of the hadronic rescatterings are explored by changing the
rescattering time in ART between 0.6 fm/c (ART effectively OFF) and 30 fm/c
(ART ON). In total, five different sets of parameters are used for AMPT in
this study, as seen in table 1. The results in this paper are presented in
different centrality intervals. The centrality is determined by the impact
parameter b as in [55]. Around 320K events are generated in the 0-10% central
regions, and around 300K events are generated for each 10% interval of the
other centralities.

| Partonic cross section | a | b | ART
---|---|---|---|---
Par1 | 1.5 mb | 0.3 | 0.15 | ON
Par2 | 1.5 mb | 0.3 | 0.15 | OFF
Par3 | 1\. 5mb | 0.5 | 0.9 | OFF
Par4 | 3.0 mb | 0.3 | 0.15 | ON
Par5 | 3.0 mb | 0.3 | 0.15 | OFF
Table 1: Table of the different sets of parameters used for AMPT in this
study. The cross section is determined from the screening mass $\mu$ with Eq.
(2).

##  III Observables

The $p_{\rm T}$-dependent flow vector fluctuations are studied with $p_{\rm
T}$-differential flow coefficients. The regular $p_{\rm T}$-differential flow
coefficient is defined as

$\displaystyle v_{n}\\{2\\}$ $\displaystyle=\frac{\langle\langle\cos
n(\varphi_{1}^{\mathrm{POI}}-\varphi_{2})\rangle\rangle}{\sqrt{\langle\langle\cos
n(\varphi_{1}-\varphi_{2})\rangle\rangle}}$ $\displaystyle=\frac{\langle
v_{n}(p_{\rm T})v_{n}\cos n[\Psi_{n}(p_{\rm
T})-\Psi_{n}]\rangle}{\sqrt{\langle v_{n}^{2}\rangle}},$ (4)

where $\varphi^{\mathrm{POI}}$ is the azimuthal angle of a particle of
interest (POI) chosen from a narrow $p_{\rm T}$ range and $\varphi$ is the
azimuthal angle of a reference particle (RP) chosen from a wide $p_{\rm T}$
range. The single bracket denotes an average over all events, and double
brackets indicate an average over all particles and all events. The
$v_{2}\\{2\\}$ probes the $p_{\rm T}$-differential flow and is sensitive to
fluctuations of both the flow magnitude and flow angle. Another $p_{\rm
T}$-differential flow observable was proposed in [32], which is unaffected by
the flow vector fluctuations

$\displaystyle v_{n}[2]$ $\displaystyle=\sqrt{\langle\langle\cos
n(\varphi_{1}^{\mathrm{POI}}-\varphi_{2}^{\mathrm{POI}})\rangle\rangle}$
$\displaystyle=\sqrt{\langle v_{n}^{2}(p_{\rm T})\rangle}.$ (5)

Since $v_{n}[2]$ is not affected by the flow angle and flow magnitude
fluctuations, the $p_{\rm T}$-dependent flow vector fluctuations can be probed
by taking the ratio of $v_{n}\\{2\\}$ and $v_{n}[2]$:

$\displaystyle\frac{v_{n}\\{2\\}}{v_{n}[2]}=\frac{\langle v_{n}(p_{\rm
T})v_{n}\cos n[\Psi_{n}(p_{\rm T})-\Psi_{n}]\rangle}{\sqrt{\langle
v_{n}^{2}(p_{\rm T})\rangle}\sqrt{\langle v_{n}^{2}\rangle}}.$ (6)

A value of $v_{n}\\{2\\}/v_{n}[2]<1$ indicates the presence of $p_{\rm
T}$-dependent flow vector fluctuations. Similarly, the double-differential
factorization ratio [33], which tests the factorization of the two-particle
correlation into the product of single-particle flow coefficients, is a
measure of the flow vector fluctuations. It provides more detailed information
about the correlation structure

$\displaystyle r_{n}$ $\displaystyle=\frac{\langle\langle\cos
n(\varphi_{1}^{a}-\varphi_{2}^{t})\rangle\rangle}{\sqrt{\langle\langle\cos
n(\varphi_{1}^{a}-\varphi_{2}^{a})\rangle\rangle\langle\langle\cos
n(\varphi_{1}^{t}-\varphi_{2}^{t})\rangle\rangle}}$
$\displaystyle=\frac{\langle v_{n}(p_{\rm T}^{\rm a})v_{n}(p_{\rm T}^{\rm
t})\cos n[\Psi_{n}(p_{\rm T}^{\rm a})-\Psi_{n}(p_{\rm T}^{\rm
t})]\rangle}{\sqrt{\langle v_{n}^{2}(p_{\rm T}^{\rm a})\rangle}\sqrt{\langle
v_{n}^{2}(p_{\rm T}^{\rm t})\rangle}}.$ (7)

Here, $\varphi_{2}^{a}$ is the azimuthal angle of the associate particle, and
$\varphi_{2}^{t}$ is the azimuthal angle of the trigger particle with
transverse momentum $p_{\rm T}^{\rm a}$ and $p_{\rm T}^{\rm t}$, respectively.
The $p_{\rm T}^{\rm t}$-range is fixed for all values of $p_{\rm T}^{\rm a}$.
Again, a value of $r_{n}<1$ indicates that factorization is broken and that
$p_{\rm T}$-dependent flow vector fluctuations are present. Both Eqs. (6) and
(7) are based on two-particle correlations and may be affected by fluctuations
in both the flow magnitude and flow angle. Previously we proposed to use four-
particle correlations to separate the two types of fluctuations [48] so that
they can be measured experimentally. The flow angle fluctuations are given by

$\displaystyle A_{n}^{\mathrm{f}}$ $\displaystyle=\frac{\langle\cos
n(\varphi_{1}^{a}+\varphi_{2}^{a}-\varphi_{3}-\varphi_{4})\rangle}{\langle\cos
n(\varphi_{1}^{a}+\varphi_{2}-\varphi_{3}^{a}-\varphi_{4})\rangle}$
$\displaystyle=\frac{\langle v_{n}^{2}(p_{\rm T})v_{n}^{2}\cos
2n[\Psi_{n}(p_{\rm T})-\Psi_{n}]\rangle}{\langle v_{n}^{2}(p_{\rm
T})v_{n}^{2}\rangle}$ (8) $\displaystyle\approx\langle\cos 2n[\Psi_{n}(p_{\rm
T})-\Psi_{n}]\rangle,$

where the last equality holds if the non-flow is approximately the same in
numerator and denominator. Then $A^{\mathrm{f}}_{n}<1$ suggests $p_{\rm
T}$-dependent flow angle fluctuations. The flow magnitude is given by the
double ratio of the $p_{\rm T}$-differential flow fluctuations against the
reference flow baseline fluctuations

$\displaystyle M_{n}^{\mathrm{f}}$ $\displaystyle=[\langle\langle\cos
n[\varphi_{1}^{\mathrm{a}}+\varphi_{2}-\varphi_{3}^{\mathrm{a}}-\varphi_{4}]\rangle\rangle/$
$\displaystyle\left(\langle\langle\cos
n[\varphi_{1}^{\mathrm{a}}-\varphi_{3}^{\mathrm{a}}]\rangle\rangle\langle\langle\cos
n[\varphi_{2}-\varphi_{4}]\rangle\rangle\right)]/$
$\displaystyle[\langle\langle\cos
n[\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4}]\rangle\rangle/\langle\langle\cos
n[\varphi_{1}-\varphi_{2}]\rangle\rangle^{2}]$ $\displaystyle=\frac{\langle
v_{n}^{2}(p_{\rm T})v_{n}^{2}\rangle/(\langle v_{n}^{2}(p_{\rm
T})\rangle\langle v_{n}^{2}\rangle}{\langle v_{n}^{4}\rangle/\langle
v_{n}^{2}\rangle^{2}}$ (9)

Any deviation from unity indicates the presence of $p_{\rm T}$-dependent flow
magnitude fluctuations.

##  IV Method

The anisotropic flow observables are calculated using the two- and multi-
particle correlation method [67, 68]. This method has been widely used at RHIC
[69] and the LHC [13, 14] in the study of anisotropic flow. In this study, the
Generic Framework [70, 71] is used, which enables a fast and exact calculation
of the multi-particle correlations. However, additional correlators had to be
implemented in the Generic Framework to separate the flow angle and flow
magnitude fluctuations as in Eqs. (8) and (9). While the method has been used
for the measurements in [50], a detailed explanation of the method has not
been available before this paper.

###  IV.1 Standard multi-particle correlations

The anisotropic flow correlations are expressed in terms of the Q-vector

$\displaystyle Q_{n,p}=\sum_{k=1}^{M}w_{k}^{p}e^{in\varphi_{k}},$ (10)

where $M$ is the multiplicity of the selected particles in a given kinematic
range, $\varphi$ is the azimuthal angle of the particles, and $w$ is a
particle weight that corrects for detector inefficiencies in experimental
measurements. In general, the single-event m-particle correlation can be
calculated with

$\displaystyle\langle
m\rangle_{n_{1},\ldots,n_{m}}=\frac{\displaystyle\sum_{k_{1}\neq\ldots\neq
k_{m}}^{M}w_{k_{1}}\ldots
w_{k_{m}}e^{i(n_{1}\varphi_{1}+\ldots+n_{m}\varphi_{m})}}{\displaystyle\sum_{k_{1}\neq\ldots\neq
k_{m}}^{M}w_{k_{1}}\ldots w_{k_{m}}},$ (11)

The numerator and denominator in Eq. (11) are trivially related

$\displaystyle\textrm{N}\langle m\rangle_{n_{1},\ldots,n_{m}}$
$\displaystyle=\displaystyle\sum_{k_{1}\neq\ldots\neq
k_{m}}^{M}w_{k_{1}}\ldots
w_{k_{m}}e^{i(n_{1}\varphi_{1}+\ldots+n_{m}\varphi_{m})}$ (12)
$\displaystyle\textrm{D}\langle m\rangle_{n_{1},\ldots,n_{m}}$
$\displaystyle=\displaystyle\sum_{k_{1}\neq\ldots\neq
k_{m}}^{M}w_{k_{1}}\ldots w_{k_{m}}$ (13) $\displaystyle=\textrm{N}\langle
m\rangle_{0,\ldots,0}$

Due to the trivial nature of the denominators, they will be omitted from the
following part unless necessary. The exact equations for the two- and four-
particle azimuthal single-event correlations $\mathrm{N}\langle
2\rangle_{n_{1},n_{2}}$ and $\mathrm{N}\langle
4\rangle_{n_{1},n_{2},n_{3},n_{4}}$ calculated with the Generic Framework are
given by Eqs. (19)-(22) in [70]. The event-averaged two- and four-particle
correlations can then be obtained with

$\displaystyle\langle\langle 2\rangle\rangle$
$\displaystyle=\frac{\displaystyle\sum_{\mathrm{events}}\mathrm{N}\langle
2\rangle_{n_{1},n_{2}}}{\displaystyle\sum_{\mathrm{events}}\mathrm{D}\langle
2\rangle_{n_{1},n2}},$ (14) $\displaystyle\langle\langle 4\rangle\rangle$
$\displaystyle=\frac{\displaystyle\sum_{\mathrm{events}}\mathrm{N}\langle
4\rangle_{n_{1},n_{2},n_{3},n_{4}}}{\displaystyle\sum_{\mathrm{events}}\mathrm{D}\langle
4\rangle_{n_{1},n_{2},n_{3},n_{4}}},$ (15)

where the double brackets refer to an average over all particles and events.
$\mathrm{D}\langle m\rangle_{n_{1},n_{2},\ldots,n_{m}}$ is used as an event
weight in order to reduce the effect of the varying multiplicity [70].

For $p_{\rm T}$-differential anisotropic flow, one or more of the particles
are selected from a narrow $p_{\rm T}$ range by constructing a p-vector
analogous to the Q-vector consisting only of particles of interest.

$\displaystyle p_{n,p}=\sum_{k=1}^{M_{\mathrm{POI}}}w_{k}^{p}e^{in\varphi},$
(16)

where $M_{\mathrm{POI}}$ is the multiplicity of the POIs in the selected
kinematic region. The p-vector enables the calculation of various $p_{\rm
T}$-differential anisotropic flow coefficients, such as $v_{n}\\{2\\}$ by
selecting one POI and one RP or $v_{n}[2]$ by selecting two POIs, in the
calculation of the two-particle correlations. Additionally, the double-
differential two-particle correlations are calculated with one associate
particle and one trigger particle from different narrow $p_{\rm T}$-ranges.
The $p_{\rm T}$-differential two-particle correlations are then

$\displaystyle\mathrm{N}\langle 2^{\prime}\rangle_{n_{1},n_{2}}$
$\displaystyle=Q_{n_{1},1}p_{n_{2},1}-q_{n_{1}+n_{2},2},$ (17)
$\displaystyle\mathrm{N}\langle 2^{\prime\prime}\rangle_{n_{1},n_{2}}$
$\displaystyle=p_{n_{1},1}p_{n_{2},1}-p_{n_{1}+n_{2},2},$ (18)
$\displaystyle\mathrm{N}\langle 2^{\prime\prime}\rangle_{n_{1},n_{2}}^{a,t}$
$\displaystyle=p_{n_{1},1}^{a}p_{n_{2},1}^{t},$ (19)

where $q_{n,p}$ is the overlap-vector of particles that are both POIs and RPs,
and the superscript $a$ refers to associate particles, and $t$ refers to
trigger particles. Eq. (17) takes one POI and correlates it with the reference
particles, whereas Eq. (18) correlates two POIs from the same $p_{\rm
T}$-range. Eq. (19) is a special case of Eq. (18) when the two POIs are taken
from different $p_{\rm T}$-ranges. The event-averaged $p_{\rm T}$-differential
correlations then follow similarly to Eqs. (14) and (15)

$\displaystyle\langle\langle 2^{\prime}\rangle\rangle$
$\displaystyle=\frac{\displaystyle\sum_{\mathrm{events}}\mathrm{N}\langle
2^{\prime}\rangle_{n_{1},n_{2}}}{\displaystyle\sum_{\mathrm{events}}\mathrm{D}\langle
2^{\prime}\rangle_{n_{1},n2}},$ (20) $\displaystyle\langle\langle
2^{\prime\prime}\rangle\rangle$
$\displaystyle=\frac{\displaystyle\sum_{\mathrm{events}}\mathrm{N}\langle
2^{\prime\prime}\rangle_{n_{1},n_{2}}}{\displaystyle\sum_{\mathrm{events}}\mathrm{D}\langle
2^{\prime\prime}\rangle_{n_{1},n2}},$ (21) $\displaystyle\langle\langle
2^{\prime}\rangle\rangle$
$\displaystyle=\frac{\displaystyle\sum_{\mathrm{events}}\mathrm{N}\langle
2\rangle_{n_{1},n_{2}}^{a,t}}{\displaystyle\sum_{\mathrm{events}}\mathrm{D}\langle
2\rangle_{n_{1},n2}^{a,t}},$ (22)

###  IV.2 Subevent method

Non-flow effects are correlations not due to collective effects, such as
resonance decays and jets. They can be suppressed by applying a gap in
pseudorapidity, $|\Delta\eta|$, between two subevents, A and B, from which the
particles are selected.

$\displaystyle\mathrm{N}\langle 2\rangle_{n1,n2}^{|\Delta\eta|}$
$\displaystyle=Q_{n_{1},1}^{A}\cdot Q_{n_{2},1}^{B},$ (23)
$\displaystyle\mathrm{N}\langle
4\rangle_{n_{1},n_{2},n_{3},n_{4}}^{|\Delta\eta|}$
$\displaystyle=\mathrm{N}\langle
2\rangle_{n_{1},n_{2}}^{A}\cdot\mathrm{N}\langle 2\rangle_{n_{1},n_{2}}^{B}.$
(24)

For $p_{\rm T}$-differential correlators the subevents are implemented
analogous to Eq. (23)

$\displaystyle\mathrm{N}\langle
2^{\prime}\rangle_{n_{1},n_{2}}^{|\Delta\eta|}=p_{n_{1},1}^{A}Q_{n_{2},1}^{B},$
(25) $\displaystyle\mathrm{N}\langle
2^{\prime\prime}\rangle_{n_{1},n_{2}}^{|\Delta\eta|}=p_{n_{1},1}^{A}p_{n_{2},1}^{B},$
(26) $\displaystyle\mathrm{N}\langle
2^{\prime\prime}\rangle_{n_{1},n_{2}}^{a,t,|\Delta\eta|}=p_{n_{1},1}^{a,A}p_{n_{2},1}^{t,B}.$
(27)

Figure 1: The event divided into two subevents separated by a gap in
pseudorapidity with two POIs selected from the same subevent (top) and one POI
selected from each subevent (bottom).

The four-particle correlations, which are needed to calculate Eq. (8) and (9),
are obtained with the subevent method to reduce the possible non-flow
contamination. For four-particle correlations with a pseudorapidity gap, there
is a choice of how to group the POIs and RPs. Either the POIs are chosen from
the same subevents (SS), or one POI is chosen from opposite subevents (OS), as
shown in fig. 1.

$\displaystyle\mathrm{N}\langle 4^{\prime}\rangle_{\mathrm{SS}}=$
$\displaystyle(p_{n_{1},1}p_{n_{2},1}-p_{n_{1}+n_{2},2})^{A}$
$\displaystyle\cdot(Q_{n_{3},1}Q_{n_{4},1}-Q_{n_{3}+n_{4},2})^{B},$ (28)
$\displaystyle\mathrm{N}\langle 4^{\prime}\rangle_{\mathrm{OS}}=$
$\displaystyle(p_{n_{1},1}Q_{n_{2},1}-q_{n_{1}+n_{2},2})^{A}$
$\displaystyle\cdot(p_{n_{3},1}Q_{n_{4},1}-q_{n_{3}+n_{4},2})^{B}.$ (29)

The event average of the correlators described in Eqs. (17)-(29) are obtained
by replacing the correlators in Eqs. (14) and (15). The four-particle
correlations are, by construction, less sensitive to non-flow correlations, as
the latter typically involves fewer particles. By also applying a
pseudorapidity gap, it ensures that the non-flow contamination is minimal. The
non-flow can also be tested by performing the analysis with only particles
with the same sign charge, also known as like-sign method. It was found in
[50] that the method is robust against non-flow within 1% from the like-sign
method as well as from model studies with HIJING, which only contains non-flow
correlations. The four-particle correlations presented here are identical to
the method used in the ALICE measurements. This is, however, different to the
four-particle correlations used in [72], where short-range correlations
between particles in each subevents are present. These can be significantly
biased by non-flow and thus lead to a different conclusion.

With the above definitions, the observables to probe the $p_{\rm T}$-dependent
flow vector fluctuations can easily be calculated. The $p_{\rm
T}$-differential flow coefficient $v_{n}\\{2\\}$ is

$\displaystyle v_{n}\\{2\\}$ $\displaystyle=\frac{\langle\langle
2^{\prime}\rangle\rangle}{\sqrt{\langle\langle 2\rangle\rangle}},$ (30)

and the alternative $p_{\rm T}$-differential flow coefficient $v_{n}[2]$
becomes

$\displaystyle v_{n}[2]$ $\displaystyle=\sqrt{\langle\langle
2^{\prime\prime}\rangle\rangle}.$ (31)

The factorization ratio $r_{n}$ is given by

$\displaystyle r_{n}$ $\displaystyle=\frac{\langle\langle
2^{\prime\prime}\rangle\rangle_{a,t}}{\sqrt{\langle\langle
2^{\prime\prime}\rangle\rangle_{a}\langle\langle
2^{\prime\prime}\rangle\rangle_{t}}}.$ (32)

Finally, the observables based on four-particle correlations,
$A_{n}^{\mathrm{f}}$ and $M_{n}^{\mathrm{f}}$, which describe the flow angle
and flow magnitude fluctuations, respectively, are given by

$\displaystyle A_{n}^{\mathrm{f}}$ $\displaystyle=\frac{\langle\langle
4^{\prime}\rangle\rangle_{\mathrm{SS}}}{\langle\langle
4^{\prime}\rangle\rangle_{\mathrm{OS}}},$ (33) $\displaystyle
M_{n}^{\mathrm{f}}$ $\displaystyle=\frac{\langle\langle
4^{\prime}\rangle\rangle_{\mathrm{OS}}/(\langle\langle
2^{\prime\prime}\rangle\rangle\langle\langle 2\rangle\rangle)}{\langle\langle
4\rangle\rangle/\langle\langle 2\rangle\rangle^{2}}.$ (34)

Figure 2: The $p_{\rm T}$-dependence of $v_{2}\\{2\\}/v_{2}[2]$ in different
centrality classes from the AMPT model with different configurations:
$\sigma=$ 1.5 mb and ART ON (Par1), $\sigma=$ 1.5 mb and ART OFF (Par2),
$\sigma=$1.5 mb, ART ON, $a=0.5$ and $b=0.9$ (Par3), $\sigma=$ 3.0 mb and ART
ON (Par4), and $\sigma=$ 3.0 mb and ART OFF (Par5). ALICE data points are
shown as black circles for comparison.

As the non-flow contribution is small for multi-particle cumulants [73], a
smaller $\eta$-gap is used for the four-particle correlations compared to the
two-particle correlations to increase the event sample size. No particle
weights are used to correct detector inefficiencies and acceptance for the
generated AMPT events.

##  V Results and discussion

Figure 3: The $p_{\rm T}$-dependence of the factorization ratio $r_{2}$ in
different centrality and trigger particle transverse momentum ranges from the
AMPT model with different configurations: $\sigma=$1.5 mb and ART ON (Par1),
$\sigma=$1.5 mb and ART OFF (Par2), $\sigma=$1.5 mb, ART ON, $a=0.5$ and
$b=0.9$ (Par3), $\sigma=$3.0 mb and ART ON (Par4), and $\sigma=$3.0 mb and ART
OFF (Par5). ALICE data points are shown as black circles for comparison.
Figure 4: The $p_{\rm T}$-dependence of the flow angle fluctuations,
$A_{2}^{\mathrm{f}}\approx\langle\cos 2n[\Psi_{n}(p_{\rm
T})-\Psi_{n}]\rangle$, in different centrality ranges from the AMPT model with
different configurations: $\sigma=$1.5 mb and ART ON (Par1), $\sigma=$1.5 mb
and ART OFF (Par2), $\sigma=$1.5 mb, ART ON, $a=0.5$ and $b=0.9$ (Par3),
$\sigma=$3.0 mb and ART ON (Par4), and $\sigma=$3.0 mb and ART OFF (Par5).
ALICE data points are shown as black circles for comparison [50].

The $p_{\rm T}$-dependence of the $v_{2}\\{2\\}/v_{2}[2]$ ratio calculated
from the different configurations of the AMPT model is shown in fig. 2 for
centrality ranges 0-5% to 40-50%. The deviation of $v_{2}\\{2\\}/v_{2}[2]$
from unity is largest in the 0-5% central collisions and then becomes
progressively smaller as the centrality increases. The effect of changing the
initial conditions via the Lund string parameters, a and b, can be seen by
comparing the results from Par3 and Par2. The effect depends on the transverse
momentum with an increasing effect up to 4 GeV/c, after which it decreases
slightly. The string parameters affect the ratio $v_{2}\\{2\\}/v_{2}[2]$ by up
to 35% in central collisions and up to 30% in non-central collisions. The
$v_{2}\\{2\\}/v_{2}[2]$ shows a weak dependence on the partonic cross section:
comparing the different cross sections with ART ON shows at most a 10%
difference (Par1 vs Par4). Without ART, the effect of changing the cross
section is negligible except in the 0-5% central collisions (Par2 vs Par5).
The comparison with ART OFF may better reflect the true sensitivity to the QGP
properties as the final state hadronic interactions may differ significantly
when ART is enabled (Par1 vs Par2, Par4 vs Par5). A smaller partonic cross
section in AMPT corresponds to a larger specific viscosity in hydrodynamics.
The expectation is therefore that the calculation of $v_{2}\\{2\\}/v_{2}[2]$
with a smaller partonic cross section should show less deviation from unity as
a larger specific shear viscosity tends to dampen fluctuations from the
initial state [74]. This expectation is consistent with what is observed in
fig. 2: the Par1 calculation with smaller partonic cross section deviates less
from unity than the Par4 calculations. However, the difference is small
compared to the sensitivity to the initial state. The effect of the hadronic
rescattering on the ratio $v_{2}\\{2\\}/v_{2}[2]$ increases with $p_{\rm T}$
and is at most 10% in the presented $p_{\rm T}$-range across all centralities
for the Par5 vs Par4 calculations. The same is true for the Par2 vs Par1
results except in the 0-5% most central collisions, where a 20% difference is
seen in the highest $p_{\rm T}$ bin. The above comparisons suggest that the
flow vector fluctuations are highly sensitive to the initial state of heavy-
ion collision and, unlike the $v_{2}$ flow coefficient itself, are less
sensitive to the transport properties of the created matter. The effect of the
hadronic rescattering (ART ON vs ART OFF) is constant across the non-central
collisions. Measurements of $v_{2}\\{2\\}/v_{2}[2]$ from the ALICE experiment
in Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}$ = 5.02 TeV are shown for
comparison. All the configurations of the AMPT model overestimate the
deviation from unity, especially in non-central collisions, suggesting that
the model needs further tuning to the data, particularly of the initial
conditions. The calculation with 1.5 mb cross section, a = 0.3, b = 0.15 and
ART ON (Par1) is closest in describing the data and only overestimates the
data in the highest $p_{\rm T}$ data point.

The factorization ratio of the second-order flow harmonic, $r_{2}$, is shown
in fig. 3 as a function of associate particle transverse momentum, $p_{\rm
T}^{\rm a}$, in centrality ranges 0-5%, 10-20% and 30-40% and different
trigger particle transverse momentum, $p_{\rm T}^{\rm t}$, ranges. The
factorization ratio in centrality ranges 5-10%, 20-30% and 40-50% is shown in
Appendix B. The deviation of the factorization ratio from unity is largest in
the 0-5% central collisions and increases with the difference $|p_{\rm T}^{\rm
a}-p_{\rm T}^{\rm t}|$, which has also been observed in both hydrodynamics
[33] and data [75, 36]. The different configurations of AMPT all predict
factorization breaking, with the largest deviation from unity observed for the
Par3 model, which has different parameters for the initial conditions. The
effect of the initial stages on the factorization ratio by changing the Lund
string parameters from $a=0.3$, $b=0.15$ to $a=0.5$, $b=0.9$ is up to 40% in
central collisions and increases with $|p_{\rm T}^{\rm a}-p_{\rm T}^{\rm t}|$.
However, for the lower $p_{\rm T}^{\rm t}$-ranges, $p_{\rm T}^{\rm t}<1.5$
GeV/c, the effect does not increase for $p_{\rm T}^{\rm a}>3$ GeV/c for
central collisions and $p_{\rm T}^{\rm a}>4$ GeV/c for non-central collisions.
In non-central collisions, the effect is up to 30% for large values of
$|p_{\rm T}^{\rm a}-p_{\rm T}^{\rm t}|$. Changing the cross section from 1.5
mb to 3.0 mb affects $r_{2}$ by up to 10% at high $|p_{\rm T}^{\rm a}-p_{\rm
T}^{\rm t}|$ with ART OFF (Par5 vs Par2). This is consistent with
$v_{2}\\{2\\}/v_{2}[2]$, as expected since $r_{2}$ is basically the double-
differential $v_{2}\\{2\\}/v_{2}[2]$. The hadronic phase also shows an effect
of around 10% at high $|p_{\rm T}^{\rm a}-p_{\rm T}^{\rm t}|$ across all
centralities for 1.5 mb (Par2 vs Par1) and 3.0 mb (Par5 vs Par4). The ALICE
measurements favour the model with a 1.5 mb cross section, a = 0.3, b = 0.15
and ART ON (Par1), although the model overestimates the breaking of
factorization as $|p_{\rm T}^{\rm a}-p_{\rm T}^{\rm t}|$ increases.

Figure 5: The $p_{\rm T}$-dependence of the flow magnitude fluctuations,
$M_{2}^{\mathrm{f}}$, in different centrality ranges from the AMPT model with
different configurations: $\sigma=$1.5 mb and ART ON (Par1), $\sigma=$1.5 mb
and ART OFF (Par2), $\sigma=$1.5 mb, ART ON, $a=0.5$ and $b=0.9$ (Par3),
$\sigma=$3.0 mb and ART ON (Par4), and $\sigma=$3.0 mb and ART OFF (Par5).
ALICE data points are shown as black circles for comparison [50].

In fig. 4, the flow angle fluctuations $A_{2}^{\mathrm{f}}$ are shown as a
function of transverse momentum in centrality ranges 0-5% to 40-50%. A
deviation of $A_{2}^{\mathrm{f}}$ from unity is observed for all the model
calculations, with the majority showing deviations for $p_{\rm T}>2.5$ GeV/c
and the Par3 calculation with $a=0.5,b=0.9$ showing deviation from unity for
$p_{\rm T}>1.5$ GeV/c. The deviation from unity indicates that the $p_{\rm
T}$-dependent flow angle, $\Psi_{n}(p_{\rm T})$ fluctuates around the $p_{\rm
T}$-integrated flow angle $\Psi_{n}$ in the AMPT model, and the fluctuations
are strongest in the central collisions, where the initial state density
fluctuations dominate. However, significant fluctuations of up to 50% for Par3
and 30-40% for the other configurations can be seen across the centralities.
The flow angle fluctuations are highly sensitive to the initial conditions.
Changing the Lund string parameters for the initial conditions affects the
$A^{\rm f}_{2}$ by up to 70% in the 20-30% centrality range as shown in 4
bottom left (Par3 vs Par2). The large uncertainties at high $p_{\rm T}$ in
central collisions do not allow for a quantitative statement about the effect,
but at lower $p_{\rm T}$, the effect is up to 50%. The flow angle fluctuations
show no sensitivity to the partonic cross section within the uncertainties
whether ART is ON (Par4 vs Par1) or OFF (Par5 vs Par2). No significant effect
of the hadronic rescattering is observed at high $p_{\rm T}$ due to the large
uncertainties. At lower $p_{\rm T}$, the effect is $\sim 20\%$ for the 1.5 mb
calculations (Par2 vs Par1) and even less for the 3.0 mb calculations (Par5 vs
Par4). Therefore, the fluctuating flow angles are most likely driven mainly by
the event-by-event fluctuations in the initial state. The AMPT model with 1.5
mb cross section, Lund string parameters $a=0.3$ and $b=0.15$, and with
hadronic interactions (Par1) shows the most accurate description of the ALICE
measurements, only slightly overestimating the deviation from unity at $p_{\rm
T}>3.0$ GeV/c in the 0-5% and 5-10% central collisions and at $p_{\rm T}>2.0$
GeV/c in centralities 30-40% and 40-50%. The flow angle fluctuations, which
have been observed in data and hydrodynamic models, are also reproduced in the
AMPT model.

The flow magnitude fluctuations $M_{2}^{\mathrm{f}}$ are shown in fig. 5 as a
function of transverse momentum in centrality ranges 0-5% to 40-50%. As
$M_{2}^{\mathrm{f}}$ is normalized to the baseline $p_{\rm T}$-integrated flow
fluctuations $\langle v_{n}^{4}\rangle/\langle v_{n}^{2}\rangle$, it does not
depend on the magnitude of the flow coefficient itself. The flow magnitude
fluctuations are strongest in the 0-5% central collisions and then decrease as
the centrality increases. The largest deviation from unity is observed for the
Par3 calculation, which also deviates from unity at lower values of $p_{\rm
T}$ compared to the other calculations. The large deviation from unity
suggests a strong sensitivity of the flow magnitude fluctuations to the
initial conditions by changing the Lund string parameters with an effect of up
to 20% when comparing Par3 vs Par1. The flow magnitude fluctuations
$M_{2}^{\mathrm{f}}$ show no sensitivity to the partonic cross section within
uncertainties, regardless of whether ART is enabled (Par4 vs Par1) or not
(Par5 vs Par2). The puzzling $\eta/s$-dependence shown in [50] based on the
models from [49] is not observed in the low $p_{\rm T}$ region even though the
change in $\eta/s$ is larger in the AMPT calculations. The AMPT calculations
show $M^{\rm f}_{2}$ = 1 at low $p_{\rm T}$ and the results are compatible for
different $\sigma$, as expected. The hadronic rescattering plays a small role
in the flow magnitude fluctuations with $\sim$10% difference between the
calculations with and without ART in the second-highest $p_{\rm T}$ bin, which
has smaller uncertainties. Both the 1.5 mb (Par4 vs Par1) and 3.0 mb (Par5 vs
Par2) calculations are equally affected by the hadronic phase within
uncertainties. The Par1 AMPT model with $\sigma=1.5$ mb, a = 0.3, b = 0.15 and
ART ON describes the ALICE data reasonably well in all centralities. The
calculation slightly underestimates the data at $p_{\rm T}>3.0$ GeV/c in the
0-5% and 5-10% central collisions and at $p_{\rm T}>2.0$ GeV/c in the 30-40%
and 40-50% central collisions. The flow magnitude fluctuations are reproduced
in the transport model, and the data vs model comparison enable a further
opportunity to tune the initial conditions in AMPT and, in general, the use of
ALICE data to constrain the initial conditions of the heavy-ion collisions.

##  VI Summary

In this paper, the AMPT calculations of $p_{\rm T}$-dependent flow vector,
flow angle and flow magnitude fluctuations in Pb–Pb collisions at
$\sqrt{s_{{}_{\rm NN}}}$ = 5.02 TeV are presented. The different
configurations of the AMPT model probe the sensitivity of the fluctuations to
1) the initial conditions, 2) the partonic cross section, and 3) the hadronic
rescattering to pinpoint the source of these fluctuations. It is found that
the observables are highly sensitive to the changes in the initial conditions.
The observables show weak to no dependence on changes in the partonic cross
section and the hadronic rescattering. The high sensitivity to changes in the
initial state suggests that the fluctuations mainly originate from the event-
by-event fluctuations of the initial state and are only slightly dependent on
the transport properties of the QGP. The presented studies on the $p_{\rm
T}$-dependent flow vector fluctuations and comparison between models and data
can help constrain the initial stage of the QGP formation.

###### Acknowledgements.

This work is supported by a research grant (00025462) from VILLUM FONDEN.

##  Appendix A Model ratios

In order to compare the different configurations against each other the ratios
of the model calculations are presented here. Fig. 6 shows the comparison for
$v_{2}\\{2\\}/v_{2}[2]$. The $r_{2}$ comparison is shown in fig. 7. Finally,
the ratios for the flow angle fluctuations $A^{\rm f}_{2}$ and the flow
magnitude fluctuations $M^{\rm f}_{2}$ are shown in figures 8 and 9,
respectively.

##  Appendix B Factorization ratio

The factorization ratio of the second-order flow harmonic, $r_{2}$, is shown
in fig. 10 as a function of associate particle transverse momentum, $p_{\rm
T}^{\rm a}$, in centrality ranges 5-10%, 20-30% and 40-50% and different
trigger particle transverse momentum, $p_{\rm T}^{\rm t}$, ranges. These are
the centrality ranges not presented in the main body of this paper.

Figure 6: The ratio of the AMPT calculations of $v_{2}\\{2\\}/v_{2}[2]$ with
different configurations; ART OF vs ART ON with 1.5 mb (Par2/Par1), ART OF vs
ART ON with 3.0 mb (Par5/Par3), 3.0 mb vs 1.5 mb with ART OFF (Par5/Par2), 3.0
mb vs 1.5 mb with ART ON (Par4/Par1), and $a=0.5,b=0.9$ vs $a=0.3,b=0.15$ with
1.5 mb and ART OFF (Par3/Par2). Figure 7: The ratio of the AMPT calculations
of $r_{2}$ with different configurations; ART OF vs ART ON with 1.5 mb
(Par2/Par1), ART OF vs ART ON with 3.0 mb (Par5/Par3), 3.0 mb vs 1.5 mb with
ART OFF (Par5/Par2), 3.0 mb vs 1.5 mb with ART ON (Par4/Par1), and
$a=0.5,b=0.9$ vs $a=0.3,b=0.15$ with 1.5 mb and ART OFF (Par3/Par2). Figure 8:
The ratio of the AMPT calculations of $A^{\rm f}_{2}$ with different
configurations; ART OF vs ART ON with 1.5 mb (Par2/Par1), ART OF vs ART ON
with 3.0 mb (Par5/Par3), 3.0 mb vs 1.5 mb with ART OFF (Par5/Par2), 3.0 mb vs
1.5 mb with ART ON (Par4/Par1), and $a=0.5,b=0.9$ vs $a=0.3,b=0.15$ with 1.5
mb and ART OFF (Par3/Par2). Figure 9: The ratio of the AMPT calculations of
$M^{\rm f}_{2}$ with different configurations; ART OF vs ART ON with 1.5 mb
(Par2/Par1), ART OF vs ART ON with 3.0 mb (Par5/Par3), 3.0 mb vs 1.5 mb with
ART OFF (Par5/Par2), 3.0 mb vs 1.5 mb with ART ON (Par4/Par1), and
$a=0.5,b=0.9$ vs $a=0.3,b=0.15$ with 1.5 mb and ART OFF (Par3/Par2). Figure
10: The $p_{\rm T}$-dependence of the factorization ratio $r_{2}$ in
centralities 5-10%, 20-30% and 40-50% with different trigger particle
transverse momentum ranges from the AMPT model with different partonic cross
sections, different Lund string parameters and with and without a hadronic
rescattering phase. ALICE data points are shown as black circles for
comparison.

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# Resolving conclusions about the early Universe requires accurate nuclear
measurements

Cyril Pitrou,1 Alain Coc,2 Jean-Philippe Uzan,1 Elisabeth Vangioni1 1Institut
d’Astrophysique de Paris, CNRS UMR 7095, 98 bis Bd Arago, 75014 Paris, France
Sorbonne Université, Institut Lagrange de Paris, 98 bis Bd Arago, 75014 Paris,
France
2IJCLab, CNRS IN2P3, Université Paris-Saclay, Bâtiment 104, F-91405 Orsay
Campus France<EMAIL_ADDRESS>
###### Abstract

Nuclear physics experiments give reaction rates that, via modelling and
comparison with primordial abundances, constrain cosmological parameters. The
error bars of a key reaction, D(p,$\gamma)^{3}$He, were tightened in 2020,
bringing to light discrepancies between different analyses and calling for
more accurate measurements of other reactions.

One of the key parameters of the standard cosmological model is its density of
baryons, a quantity that can be inferred by analysing Planck Collaboration et
al. (2020) the cosmic microwave background (CMB) or independently by studying
Big Bang nucleosynthesis (BBN), the set of reactions that produced atomic
nuclei from these baryons in the primordial Universe. The relevant
measurements have dramatically improved in the past decades, and been
complemented by new observational probes. As a consequence, the error bars on
cosmological parameters have been reduced, leading to the era of precision
cosmology. However, tensions between different observations remain. One of the
key steps in BBN is the D(p,$\gamma)^{3}$He reaction — that is, the fusion of
deuterium with a proton resulting in a helium-3 nucleus and a photon, ${\rm
d}+p\rightarrow{}^{3}{\rm He}+\gamma$ — which partially controls the abundance
of deuterium in the first minutes after the Big Bang. The accuracy of the
cross section of the D(p,$\gamma)^{3}$He reaction was recently improved
NatureDPG , relaunching a debate on the baryon density and divergences between
its estimation from BBN and the CMB. We wish to highlight the importance for
cosmology of refined measurements and statistical analyses of three nuclear
rates, D(p,$\gamma)^{3}$He, D(d,n)3He and D(d,p)3H, and draw attention to the
reasons of recent diverging conclusions by different teams.

The reanalysis of BBN theoretical predictions after the new
D(p,$\gamma)^{3}$He measurement NatureDPG  has led to two different
conclusions. On the one hand, a series of works Pisanti et al. (2020); Yeh et
al. (2020) claim that it confirms the perfect agreement of BBN with other
estimates of the baryon density, $\Omega_{{\rm b}}h^{2}$, where $h$ is the
reduced Hubble constant. On the other hand, our analysis Pitrou et al. (2020)
argues that a new, albeit mild, tension exists between the value this
parameter deduced from BBN analysis and the one derived from the CMB and large
scale structure data.

All the reaction rates required for the prediction of the early-Universe
abundance of low-mass nuclei are measured in accelerators. Assuming the
Copernican principle, the theory describing BBN has only two free parameters,
namely $\Omega_{{\rm b}}h^{2}$ and the effective number of relativistic
particles $N_{\rm eff}$. Prior to WMAP (the Wilkinson Microwave Anisotropy
Probe) in 2003, both were adjustable but $\Omega_{{\rm b}}h^{2}$ is now
independently determined with high accuracy from the CMB. Hence, one can
either fix $\Omega_{{\rm b}}h^{2}$ from CMB and $N_{\rm eff}$ from particle
physics (thus making BBN a parameter-free model) or instead constrain the two
parameters from BBN (by confronting the predicted abundances to the measured
ones) and assess the agreement with other probes. Therefore, the comparison of
BBN and CMB at the percent level offers an important test of consistency of
the standard cosmological model.

Of the low-mass nuclei produced in the early Universe, not all put strong
constraints on $\Omega_{{\rm b}}h^{2}$. Today, the prediction of helium-4
abundance reaches the percent level and is in full agreement with its observed
value. Given its mild dependence on $\Omega_{{\rm b}}h^{2}$ and the accuracy
of its measurement, further study of it will not tighten the constraints on
baryon density. Helium-3 is not very constraining because it is both produced
and destroyed in stars so that the evolution of its abundance in time is not
very precise. Lithium-7 exhibits a factor $\sim 3$ discrepancy, which is
usually discarded quietly, the consensus being that it cannot arise from the
nuclear sector. Finally, deuterium is a very fragile isotope that can only be
destroyed after BBN. The most recent recommended observed value Cooke et al.
(2018) is ${\rm D}/{\rm H}=(2.527\pm 0.030)\times 10^{-5}$ at a redshift
$z\sim 2.5-3.1$. Deuterium is thus the most constraining BBN observable
because both its observational measurement and its theoretical prediction
reach 1% accuracy.

Theoretical predictions of abundances depend both on the baryon abundance and
the adopted values of nuclear reactions rates. When the baryon abundance is
set to its CMB-determined value, if the D(p,$\gamma)^{3}$He rate from the
NACRE–II database Xu et al. (2013) is used, an abundance ${\rm D}/{\rm
H}=(2.57\pm 0.13)\times 10^{-5}$ is predicted Fields et al. (2020), whereas
using the new D(p,$\gamma)^{3}$He rate NatureDPG  (the so-called ‘LUNA’ rate)
yields a lower value ${\rm D}/{\rm H}=(2.51\pm 0.11)\times 10^{-5}$ (Ref. Yeh
et al. (2020)). Similarly, using the PArthENoPE code,the new
D(p,$\gamma)^{3}$He (LUNA) rate leads to ${\rm D}/{\rm H}=(2.52\pm 0.07)\times
10^{-5}$ in Ref. NatureDPG , and ${\rm D}/{\rm H}=(2.54\pm 0.07)\times
10^{-5}$ with a re-evaluated LUNA-based rate in Ref. Pisanti et al. (2020),
whereas our independent analysis using the code PRIMAT Pitrou et al. (2018)
and the LUNA rate leads to ${\rm D}/{\rm H}=(2.439\pm 0.037)\times 10^{-5}$.
Consequently, the first two series of works Yeh et al. (2020); Pisanti et al.
(2020) argue that BBN is in agreement with CMB, whereas Ref. Pitrou et al.
(2020) concludes that there is a tension, which can equivalently be considered
as a $1.8\sigma$ tension on $\Omega_{{\rm b}}h^{2}$ between the BBN and CMB-
inferred values.

The production of deuterium mostly depends on 4 nuclear reactions. First,
deuterium is produced through 1H(n,$\gamma)^{2}$H, the cross-section of which
is obtained from effective field theory computations (Ando et al., 2006) and
is reliable at the 1%-level. Then, the three nuclear reactions
D(p,$\gamma)^{3}$He, D(d,n)3He and D(d,p)3H are the main sources of nuclear
uncertainty for the prediction of the primordial deuterium. Matching the
accuracy of observational data requires a percent accuracy on the predictions,
which in turn requires reaching a percent level accuracy on the cross sections
of these reactions. Achieving such accuracy is a tremendous challenge for
nuclear physics, requiring a tight control of systematic errors. Because none
of them have resonances, the problem reduces to determining the slowly varying
energy-dependent $S$-factors and to the precise determination of their
absolute scale. Because all works and codes consider the same physics and the
same data, the differences lie in the modelling of this $S$-factor, the
selection of datasets, the statistical analysis of uncertainties and on
assumptions to fit it to the raw nuclear data. (Existing codes also differ in
their treatments of weak interaction rates, which interconvert neutrons and
protons, but differences never exceed 0.2% in the relevant range of
temperature and are thus a subdominant source of disagreement.)

Because discrepancies exist between calculations that use the same rate for
D(p,$\gamma)^{3}$He, it follows that the source of these discrepancies is the
D(d,n)3He and D(d,p)3H rates Pisanti et al. (2020). For these two reactions,
PRIMAT relies on the rates based on theoretical, ab initio energy dependences,
re-normalized to a selection of experimental data for which estimations of
systematic errors are provided. The updated PArthENoPE code instead uses a
polynomial expansion of the $S$-factor and excludes fewer datasets in fitting
its coefficients. Both choices and methods are reasonable and well motivated,
and it is a priori impossible with the current data to determine which one is
the most appropriate. Hence the disagreement between the adopted nuclear rates
can only be reduced with new and accurate nuclear data, for which the effect
of these fitting choices will be effectively irrelevant.

In other words, nuclear data are at the crux of the debate. Because
differences between BBN codes are attributed to different choices made when
modelling the nuclear cross-sections, and not on weak rates, it is important
to control their accuracy at least at the percent level and to take into
account the latest data. Given the recent LUNA data, the next step is to
obtain higher precision measurements and analyses of the D(d,n)3He and
D(d,p)3H rates in order to control the accuracy of the different proposed
$S$-factors. We also note that the D/H measurement rests heavily on Ref. Cooke
et al. (2018) and more observations are needed not only to decrease
statistical errors but also to eventually reveal subtle systematics.

This debate on the status of BBN illustrates the interplay between laboratory
nuclear physics and astrophysics. It shows that pushing cosmological
predictions at the percent level requires more data and a better control of
choices that had no significant effect at lower precision. Only then will one
be able to conclude if the mild tension we note is the seed of a sharper
compatibility issue between CMB or BBN, or if it only reflects the limit of
BBN predictions.

## References

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