1 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]

$$\begin{aligned} P(\varphi )\propto \sum _{n=-\infty }^\infty V_ne^{-in\varphi }, \end{aligned}$$
(1)

which is a real function of the azimuthal angle of the produced particles, \(\varphi \), as the \(n\textrm{th}\) order flow vector \(V_n = v_ne^{in\Psi _n}\) has the property \(V_{-n}=V^*_n\). The flow vector consists of both the flow magnitude and flow angle. The flow magnitude \(v_n\) is the \(n\textrm{th}\) order flow coefficient, and the flow angle \(\Psi _n\) is the corresponding 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 of the experimental data has 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 \(p_{\textrm{T}}\)-differential studies of the flow coefficient \(v_n(p_{\textrm{T}})\) and its fluctuations, as they contain additional information about the dynamics of the expansion and the initial conditions of the heavy-ion collisions [30].

The event-by-event fluctuations of the position of the nucleons within the nuclei at fixed impact parameter in the initial state lead to event-by-event fluctuations of the final state observables [32,33,34]. As a consequence of these fluctuations, the flow vectors develop differently for particles within different kinematic ranges, which can be either the transverse momentum, \(p_{\textrm{T}}\), or pseudorapidity, \(\eta \). When the correlation between two flow vectors in different kinematic regions does not factorize into a product of the two single-particle distributions, we say that the flow vector is decorrelated in those regions. In this paper, we focus on decorrelation between the \(p_{\textrm{T}}\)-integrated flow vector and the \(p_{\textrm{T}}\)-differential flow vector as these are the most accessible in experiments. The decorrelation of the flow vector with rising \(p_{\textrm{T}}\) is predicted in hydrodynamical simulations [35, 36]. This has been measured at the LHC [37,38,39,40], and has been shown to provide a good probe of the event-by-event fluctuations of the initial state in heavy-ion collisions. However, the previous measurements of the flow vector decorrelation could not show whether it was due to decorrelation of the flow magnitude, flow angle or both. Recent theoretical [41, 42] and experimental [43] works have shown that the decorrelation of both the flow angle and flow magnitude have a unique sensitivity to the event-by-event fluctuation of the initial state. Thus, they can provide tight constraints on the initial conditions in heavy-ion collisions. Most recent studies show that they even help to probe the shape of quadropole deformed nuclei [44], and could play a crucial role in future runs at the LHC.

In this paper, the transverse momentum dependent decorrelation of the flow vector, flow angle and flow magnitude is studied in Pb–Pb collisions at \(\sqrt{s_{_{\textrm{NN}}}}\) = 5.02 TeV with A Multi-Phase Transport model (AMPT) [45, 46]. The AMPT model is well suited to the study of anisotropic flow observables. It reproduces measurements of anisotropic flow at RHIC [47, 48] and the LHC [49, 50]. The flow vector, flow angle and flow magnitude decorrelation observables are studied 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. We compare these observables to ALICE measurements across a wide range of centrality classes. In Sect. 2, details of the model and the different configurations are presented. Section 3 describes the observables and the method used to extract information about the decorrelation of the flow vector. The results are presented and discussed in Sect. 4. Finally, the work is summarized in Sect. 5. Ratio plots between the different AMPT configurations for all observables can be found in Appendix A.

2 The model

In this paper, AMPT [45, 46] 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 [51], 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 initial phase-space distribution of produced particles depends, in particular, on a and b in the Lund fragmentation function [52].

$$\begin{aligned} f(z) = z^{-1}(1-z)^{a}exp(-bm_\textrm{T}^2/z) \end{aligned}$$
(2)

where z is the light-cone momentum fraction of the produced hadron with respect to the fragmented string and \(m_\textrm{T}\) is the transverse mass of the hadron. The quark–antiquark pair production probability from fragmented strings is proportional to \(exp(-\pi m_\textrm{T}/\kappa )\), where the string tension, \(\kappa \), i.e. the energy per unit length of string, is approximately related to a and b by

$$\begin{aligned} \kappa \propto \frac{1}{b(2+a)} \end{aligned}$$
(3)

The final spectrum of produced particles thus depends on a and b and e.g. a smaller b will lead to a harder \(p_\textrm{T}\) spectrum [52, 53]. The interaction of partons is treated with ZDC [54], which only includes two-body scatterings. The cross sections are obtained from pQCD with screening masses

$$\begin{aligned} \sigma =\frac{9\pi \alpha _s^2}{2\mu ^2}, \end{aligned}$$
(4)

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 [53]. Partons are converted into hadrons with a quark coalescence model, combining two quarks into mesons, and three quarks into baryons [55]. Finally, the hadronic rescattering is handled by ART [56, 57], where the dynamics of the hadronic matter are described by a hadron cascade. In total, five different sets of parameters are used for AMPT in this study, as seen in Table 1. In order to study the effects of the event-by-event fluctuations in the initial state on the decorrelation of the flow vector, the Lund string parameters a and b are varied. In order to isolate the effect, we keep the other model parameters fixed and compare calculations with the Par2 tuning to those with the Par3 tuning. The former has the default values of \(a=0.3\) and \(b=0.15\) [50, 53], while the latter has \(a=0.5\) and \(b=0.9\), which are the default HIJING values [58, 59]. The effect of the QGP properties is studied by varying the partonic cross section using different values of the screening mass \(\mu \). The values \(\mu = 3.2042~\textrm{fm}^{-1}\) and \(\mu = 2.2814~\textrm{fm}^{-1}\) correspond to cross sections of \(\sigma \approx 1.5\) mb and \(\sigma \approx 3\) mb, respectively, and the effect is seen in the deviation between Par1 and Par4 results, as well as Par2 and Par5 results. A smaller cross section in AMPT corresponds to a larger viscosity in viscous hydrodynamics [60, 61]. The specific shear viscosity can be approximated by [49]

$$\begin{aligned} \eta /s \approx \frac{3\pi }{40\alpha _s^2}\frac{1}{\left( 9+\frac{\mu ^2}{T^2}\right) ln\left( \frac{18+\mu ^2/T^2}{\mu ^2/T^2}\right) -81}, \end{aligned}$$
(5)

where T is the initial temperature. At the LHC, the initial temperature is approximately 468 MeV [49], 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). We explore this effect for both cross section values by comparing results with the Par1 (Par4) tuning to those with the Par2 (Par5) tuning. Additionally, The Par4 settings correspond to the tuning used for comparison with ALICE measurements of \(p_{\textrm{T}}\)-differential and \(p_{\textrm{T}}\)-integrated \(v_n\) [62]. The results in this paper are presented in different centrality intervals. The centrality is determined by the impact parameter b as in [49]. 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.

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. (4)
Full size table

3 Observables

The \(p_{\textrm{T}}\)-dependent flow vector decorrelation has traditionally been measured with two-particle correlations [35,36,37,38,39]. The regular \(p_{\textrm{T}}\)-differential flow coefficient is defined as

$$\begin{aligned} v_n\{2\}&= \frac{\langle \langle \cos (n([\varphi _1^\textrm{POI}-\varphi _2])\rangle \rangle }{\sqrt{\langle \langle \cos (n[\varphi _1-\varphi _2])\rangle \rangle }}\nonumber \\&=\frac{\langle v_n(p_{\textrm{T}})v_n\cos (n[\Psi _n(p_{\textrm{T}})-\Psi _n])\rangle }{\sqrt{\langle v_n^2\rangle }}, \end{aligned}$$
(6)

where \(\varphi ^\textrm{POI}\) is the azimuthal angle of a particle of interest (POI) chosen from a narrow \(p_{\textrm{T}}\) range and \(\varphi \) is the azimuthal angle of a reference particle (RP) chosen from a wide \(p_{\textrm{T}}\) 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 \(p_{\textrm{T}}\)-differential flow and is sensitive to decorrelation in both the flow magnitude and flow angle. Another \(p_{\textrm{T}}\)-differential flow observable was proposed in [35]

$$\begin{aligned} v_n[2]&= \sqrt{\langle \langle \cos (n[\varphi _1^\textrm{POI}-\varphi _2^\textrm{POI}])\rangle \rangle }\nonumber \\&= \sqrt{\langle v_n^2(p_{\textrm{T}})\rangle }. \end{aligned}$$
(7)

which is unaffected by any decorrelation of the flow magnitude or the flow angle, as it only measures the flow magnitude within a narrow \(p_{\textrm{T}}\)  range. Thus the decorrelation of the flow vector at different transverse momentum ranges can be probed by taking the ratio of \(v_n\{2\}\) and \(v_n[2]\):

$$\begin{aligned} \frac{v_n\{2\}}{v_n[2]} = \frac{\langle v_n(p_{\textrm{T}})v_n\cos (n[\Psi _n(p_{\textrm{T}})-\Psi _n])\rangle }{\sqrt{\langle v_n^2(p_{\textrm{T}})\rangle }\sqrt{\langle v_n^2\rangle }}. \end{aligned}$$
(8)

A value of \(v_n\{2\}/v_n[2]<1\) indicates that the flow vector at a certain narrow \(p_{\textrm{T}}\)  is decorrelated with respect to the \(p_{\textrm{T}}\)-integrated flow vector. Similarly, the double-differential factorization ratio [36], which tests the factorization of the two-particle correlation into the product of single-particle flow coefficients, is a measure of the flow vector decorrelation. It provides more detailed information about the correlation structure.

$$\begin{aligned} r_n(p_{\textrm{T}}^{\textrm{a}})&= \frac{\langle \langle \cos (n[\varphi _1^a-\varphi _2^t])\rangle \rangle }{\sqrt{\langle \langle \cos (n[\varphi _1^a-\varphi _2^a])\rangle \rangle \langle \langle \cos (n[\varphi _1^t-\varphi _2^t])\rangle \rangle }}\nonumber \\&= \frac{\langle v_n(p_{\textrm{T}}^{\textrm{a}})v_n(p_{\textrm{T}}^{\textrm{t}})\cos (n[\Psi _n(p_{\textrm{T}}^{\textrm{a}})-\Psi _n(p_{\textrm{T}}^{\textrm{t}})])\rangle }{\sqrt{\langle v_n^2(p_{\textrm{T}}^{\textrm{a}})\rangle }\sqrt{\langle v_n^2(p_{\textrm{T}}^{\textrm{t}})\rangle }}. \end{aligned}$$
(9)

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 \(p_{\textrm{T}}^{\textrm{a}}\) and \(p_{\textrm{T}}^{\textrm{t}}\), respectively. The \(p_{\textrm{T}}^{\textrm{t}}\)-range plays the role of a parameter and is fixed for all values of \(p_{\textrm{T}}^{\textrm{a}}\). Again, a value of \(r_n(p_{\textrm{T}}^{\textrm{a}})<1\) indicates that factorization is broken and that decorrelation of the flow vector occurs.

Both Eqs. (8) and (9) are based on two-particle correlations and may be affected by decorrelation of the flow magnitude and/or flow angle. In experiments, the flow angle, \(\Psi _n\), is not accessible. The dependence of the flow angle on the transverse momentum [36, 41, 63,64,65] results in a decorrelation between the integrated symmetry plane, \(\Psi _n\), and the \(p_{\textrm{T}}\)-differential plane, \(\Psi _n(p_{\textrm{T}})\). This potentially affects measurements that rely on the assumption of a common symmetry plane for particles at high transverse momentum, as they will be highly decorrelated with the bulk of the produced particles, which will be predominantly produced with low \(p_{\textrm{T}}\). In order to separate the flow angle and flow magnitude components, the second moment of the correlation between flow vectors in different transverse momentum bins is needed, as proposed in [66]. The flow angle decorrelation is given by

$$\begin{aligned} A_n^\textrm{f}&= \frac{\langle \cos (n[\varphi _1^\textrm{POI}+\varphi _2^\textrm{POI}-\varphi _3-\varphi _4])\rangle }{\langle \cos (n[\varphi _1^\textrm{POI}+\varphi _2-\varphi _3^\textrm{POI}-\varphi _4])\rangle }\nonumber \\&= \frac{\langle v_n^2(p_{\textrm{T}})v_n^2\cos (2n[\Psi _n(p_{\textrm{T}})-\Psi _n])\rangle }{\langle v_n^2(p_{\textrm{T}})v_n^2\rangle }\nonumber \\&\approx \langle \cos (2n[\Psi _n(p_{\textrm{T}})-\Psi _n])\rangle _{w}, \end{aligned}$$
(10)

where the last equality holds if the non-flow is approximately the same in numerator and denominator and then \(A^\textrm{f}_n\) becomes a weighted average, denoted by the subscript w, of the flow angle fluctuations with a weight roughly equal to \(v_2^4\) [42]. Values of \(A^\textrm{f}_n<1\) indicate decorrelation of the flow angle. The decorrelation of the flow magnitude is given by the double ratio

$$\begin{aligned} M_n^\textrm{f}&= [\langle \langle \cos (n[\varphi _1^\textrm{POI}+\varphi _2-\varphi _3^\textrm{POI}-\varphi _4])\rangle \rangle /\nonumber \\&\quad \, \left( \langle \langle \cos (n[\varphi _1^\textrm{POI}-\varphi _3^\textrm{POI}])\rangle \rangle \langle \langle \cos (n[\varphi _2-\varphi _4])\rangle \rangle \right) ]/\nonumber \\&\quad \, \, [\langle \langle \cos (n[\varphi _1+\varphi _2-\varphi _3-\varphi _4])\rangle \rangle /\langle \langle \cos (n[\varphi _1-\varphi _2]\rangle \rangle ^2])\nonumber \\&= \frac{\langle v_n^2(p_{\textrm{T}})v_n^2\rangle /(\langle v_n^2(p_{\textrm{T}})\rangle \langle v_n^2\rangle }{\langle v_n^4\rangle /\langle v_n^2\rangle ^2}, \end{aligned}$$
(11)

where the normalisation with the baseline \(\langle v_n^4\rangle /\langle v_n^2\rangle ^2\) ensures that the decorrelation does not depend on the overall fluctuations of the \(p_{\textrm{T}}\)-integrated \(v_2\). Any deviation from unity indicates that the flow magnitude in the corresponding \(p_{\textrm{T}}\) bin is decorrelated with respect to the \(p_{\textrm{T}}\)-integrated flow magnitude. Hydrodynamical models have been able to reproduce a significant decorrelation of both the flow angle and flow magnitude [41, 42] and have been successful in describing the ALICE data [43]. However, the model calculations of the flow magnitude decorrelation show a puzzling trend when using large values of \(\eta /s\). At low \(p_{\textrm{T}}\), where the expectation is to see little to no decorrelation, these calculations instead show an unusually high degree of decorrelation. This should be investigated further in order to determine whether the flow magnitude decorrelation is sensitive to the transport properties of the QGP.

4 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 decorrelation of the flow angle and flow magnitude as in Eqs. (10) and (11). While the method has been used for the measurements in [43], a detailed explanation of the method has not been available before this paper.

4.1 Standard multi-particle correlations

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

$$\begin{aligned} Q_{n,p} = \sum _{k=1}^Mw_k^pe^{in\varphi _k}, \end{aligned}$$
(12)

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

$$\begin{aligned} \langle m\rangle _{n_1,\ldots ,n_m} = \frac{\sum _{k_1\ne \ldots \ne k_m}^M w_{k_1}\ldots w_{k_m}e^{i(n_1\varphi _{k_1}+\ldots +n_m\varphi _{k_m})}}{\sum _{k_1\ne \ldots \ne k_m}^M w_{k_1}\ldots w_{k_m}}, \end{aligned}$$
(13)

The numerator and denominator in Eq. (13) are trivially related

$$\begin{aligned} \text {N}\langle m\rangle _{n_1,\ldots ,n_m}&= \displaystyle \sum _{k_1\ne \ldots \ne k_m}^M w_{k_1}\ldots w_{k_m}e^{i(n_1\varphi _{k_1}+\ldots +n_m\varphi _{k_m})} \end{aligned}$$
(14)
$$\begin{aligned} \text {D}\langle m\rangle _{n_1,\ldots ,n_m}&= \displaystyle \sum _{k_1\ne \ldots \ne k_m}^M w_{k_1}\ldots w_{k_m} \nonumber \\&= \text {N}\langle m\rangle _{0,\ldots ,0} \end{aligned}$$
(15)

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 \(\textrm{N}\langle 2\rangle _{n_1,n_2}\) and \(\textrm{N}\langle 4\rangle _{n_1,n_2,n_3,n_4}\) 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

$$\begin{aligned} \langle \langle 2\rangle \rangle _{n_1,n_2}&= \frac{\sum _{\textrm{events}}\textrm{N}\langle 2\rangle _{n_1,n_2}}{\sum _{\textrm{events}}\textrm{D}\langle 2\rangle _{n_1,n2}}, \end{aligned}$$
(16)
$$\begin{aligned} \langle \langle 4\rangle \rangle _{n_1,n_2,n_3,n_4}&= \frac{\sum _{\textrm{events}}\textrm{N}\langle 4\rangle _{n_1,n_2,n_3,n_4}}{\sum _{\textrm{events}}\textrm{D}\langle 4\rangle _{n_1,n_2,n_3,n_4}}, \end{aligned}$$
(17)

where the double brackets refer to an average over all particles and events. \(\textrm{D}\langle m\rangle _{n_1,n_2,\ldots ,n_m}\) is used as an event weight in order to reduce the effect of the varying multiplicity [70].

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

$$\begin{aligned} p_{n,p} = \sum _{k=1}^{M_\textrm{POI}}w_k^pe^{in\varphi }, \end{aligned}$$
(18)

where \(M_\textrm{POI}\) is the multiplicity of the POIs in the selected kinematic region. The p-vector enables the calculation of various \(p_{\textrm{T}}\)-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 \(p_{\textrm{T}}\)-ranges. The \(p_{\textrm{T}}\)-differential two-particle correlations are then

$$\begin{aligned} \textrm{N}\langle 2'\rangle _{n_1,n_2}&= Q_{n_1,1}p_{n_2,1}-q_{n_1+n_2,2}, \end{aligned}$$
(19)
$$\begin{aligned} \textrm{N}\langle 2''\rangle _{n_1,n_2}&= p_{n_1,1}p_{n_2,1}-p_{n_1+n_2,2}, \end{aligned}$$
(20)
$$\begin{aligned} \textrm{N}\langle 2''\rangle _{n_1,n_2}^{a,t}&= p_{n_1,1}^ap_{n_2,1}^t, \end{aligned}$$
(21)

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. Equation (19) takes one POI and correlates it with the reference particles, whereas Eq. (20) correlates two POIs from the same \(p_{\textrm{T}}\)-range. Equation (21) is a special case of Eq. (20) when the two POIs are taken from different \(p_{\textrm{T}}\)-ranges. The event-averaged \(p_{\textrm{T}}\)-differential correlations then follow similarly to Eqs. (16) and (17)

$$\begin{aligned} \langle \langle 2'\rangle \rangle _{n_1,n_2}&= \frac{\sum _{\textrm{events}}\textrm{N}\langle 2'\rangle _{n_1,n_2}}{\sum _{\textrm{events}}\textrm{D}\langle 2'\rangle _{n_1,n2}}, \end{aligned}$$
(22)
$$\begin{aligned} \langle \langle 2''\rangle \rangle _{n_1,n_2}&= \frac{\sum _{\textrm{events}}\textrm{N}\langle 2''\rangle _{n_1,n_2}}{\sum _{\textrm{events}}\textrm{D}\langle 2''\rangle _{n_1,n2}}, \end{aligned}$$
(23)
$$\begin{aligned} \langle \langle 2''\rangle \rangle _{n_1,n_2}^{a,t}&= \frac{\sum _{\textrm{events}}\textrm{N}\langle 2\rangle _{n_1,n_2}^{a,t}}{\sum _{\textrm{events}}\textrm{D}\langle 2\rangle _{n_1,n2}^{a,t}}, \end{aligned}$$
(24)

4.2 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.

$$\begin{aligned} \textrm{N}\langle 2\rangle _{n1,n2}^{|\Delta \eta |}&= Q_{n_1,1}^A\cdot Q_{n_2,1}^{B}, \end{aligned}$$
(25)
$$\begin{aligned} \textrm{N}\langle 4\rangle _{n_1,n_2,n_3,n_4}^{|\Delta \eta |}&= \textrm{N}\langle 2\rangle _{n_1,n_2}^A\cdot \textrm{N}\langle 2\rangle _{n_3,n_4}^B. \end{aligned}$$
(26)

Here the four-particle correlation is constructed as the product of two-particle correlations in different subevents, which yields the correct terms for the four-particle correlation. Note that particles within the same subevent should enter the equations with the same sign (+, + or \(-,-\)) so that there is no short-range correlations within the subevents themselves. For \(p_{\textrm{T}}\)-differential correlators the subevents are implemented analogous to Eq. (25)

$$\begin{aligned} \textrm{N}\langle 2'\rangle _{n_1,n_2}^{|\Delta \eta |}&= p_{n_1,1}^AQ_{n_2,1}^{B}, \end{aligned}$$
(27)
$$\begin{aligned} \textrm{N}\langle 2''\rangle _{n_1,n_2}^{|\Delta \eta |}&= p_{n_1,1}^Ap_{n_2,1}^{B}, \end{aligned}$$
(28)
$$\begin{aligned} \textrm{N}\langle 2''\rangle _{n_1,n_2}^{a,t,|\Delta \eta |}&= p_{n_1,1}^{a,A}p_{n_2,1}^{t,B}. \end{aligned}$$
(29)
Fig. 1
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)

Full size image

The four-particle correlations, which are needed to calculate Eqs. (10) and (11), are obtained with the subevent method to reduce the possible non-flow contamination. Two subevents are sufficient to suppress the non-flow in correlations of the type presented here [72]. 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.

$$\begin{aligned} \textrm{N}\langle 4'\rangle _{n_1,n_2,n_3,n_4}^\textrm{SS} =&(p_{n_1,1}p_{n_2,1}-p_{n_1+n_2,2})^A\nonumber \\&\cdot (Q_{n_3,1}Q_{n_4,1}-Q_{n_3+n_4,2})^{B}, \end{aligned}$$
(30)
$$\begin{aligned} \textrm{N}\langle 4'\rangle _{n_1,n_2,n_3,n_4}^\textrm{OS} =&(p_{n_1,1}Q_{n_2,1}-q_{n_1+n_2,2})^A\nonumber \\&\cdot (p_{n_3,1}Q_{n_4,1}-q_{n_3+n_4,2})^{B}. \end{aligned}$$
(31)

The event average of the correlators described in Eqs. (19)–(31) are obtained by replacing the correlators in Eqs. (16) and (17). The four-particle correlations presented here are identical to the method used in the ALICE measurements [43]. By construction, the four-particle correlations are 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. Comparison of \(A^{\textrm{f}}_2\) and \(M^{\textrm{f}}_2\) with pseudorapidity gaps larger than 0.0, 0.4 and 0.8 show no statistically significant difference suggesting that a gap of \(|\Delta \eta |>0\) between the two subevents is sufficient to suppress the non-flow in the four-particle correlations. Additionally, the non-flow can be tested by performing the analysis with only particles with the same sign charge, also known as like-sign method. In AMPT, the measurements of \(A^{\textrm{f}}_2\) and \(M^{\textrm{f}}_2\) with only positive or only negative particles show no statistically significant difference to the measurements done with all charged particles. Finally, the analysis has been performed with HIJING, a model which only has non-flow. The HIJING results show that the four-particle correlations in equations (30) and (31) have similar non-flow, which at most constitutes \(2\%\) of the signal. These studies suggests that the observables \(A^{\textrm{f}}_2\) and \(M^{\textrm{f}}_2\) are robust against non-flow and thus only probe the decorrelation of the flow angle and flow magnitude that occurs when particles with different radial flow experience different anisotropic flow.

Fig. 2
figure 2

The \(p_{\textrm{T}}\)-differential flow, \(v_n(p_{\textrm{T}})\), with \(|\Delta \eta |>0.8|\) 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 [73]

Full size image

With the above definitions, the observables to probe the \(p_{\textrm{T}}\)-dependent decorrelation of the flow vector can easily be calculated. The \(p_{\textrm{T}}\)-differential flow coefficient \(v_n\{2\}\) is

$$\begin{aligned} v_n\{2\}&=\frac{\langle \langle 2'\rangle \rangle _{n,-n}}{\sqrt{\langle \langle 2\rangle \rangle _{n,-n}}}, \end{aligned}$$
(32)

and the alternative \(p_{\textrm{T}}\)-differential flow coefficient \(v_n[2]\) becomes

$$\begin{aligned} v_n[2]&= \sqrt{\langle \langle 2''\rangle \rangle _{n,-n}}. \end{aligned}$$
(33)

The factorization ratio \(r_n(p_{\textrm{T}}^{\textrm{a}})\) is given by

$$\begin{aligned} r_n(p_{\textrm{T}}^{\textrm{a}})&= \frac{\langle \langle 2''\rangle \rangle _{n,-n}^{a,t}}{\sqrt{\langle \langle 2''\rangle \rangle _{n,-n}^a\langle \langle 2''\rangle \rangle _{n,-n}^t}}. \end{aligned}$$
(34)

Finally, the observables based on four-particle correlations, \(A_n^\textrm{f}\) and \(M_n^\textrm{f}\), which describe the decorrelation of the flow angle and flow magnitude, respectively, are given by

$$\begin{aligned} A_n^\textrm{f}&= \frac{\langle \langle 4'\rangle \rangle _{n,n,-n,-n}^\textrm{SS}}{\langle \langle 4'\rangle \rangle _\textrm{OS}}, \end{aligned}$$
(35)
$$\begin{aligned} M_n^\textrm{f}&= \frac{\langle \langle 4'\rangle \rangle _{n,n,-n,-n}^\textrm{OS}/(\langle \langle 2''\rangle \rangle _{n,-n}\langle \langle 2\rangle \rangle _{n,-n})}{\langle \langle 4\rangle \rangle _{n,n,-n,-n}/\langle \langle 2\rangle \rangle _{n,-n}^2}. \end{aligned}$$
(36)
Fig. 3
figure 3

The \(p_{\textrm{T}}\)-dependence of \(v_2\{2\}/v_2[2]\) with \(|\Delta \eta |>0.8\) 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

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As the non-flow in the four-particle correlations seems to be well suppressed with an pseudorapidity gap of \(|\Delta \eta |>0\) between the subevents, we opt for this smaller \(\eta \)-gap to increase the available statistics. For the two-particle correlations, a gap of \(|\Delta \eta |>0.8\) between the subevents is used to suppress non-flow. No particle weights are used to correct detector inefficiencies and non-uniform acceptance for the generated AMPT events.

5 Results and discussion

5.1 Transverse momentum dependent flow

The transverse momentum dependent flow, \(v_2(p_{\textrm{T}})\), with pseudorapidity gap \(|\Delta \eta |>0.8\) is shown in Fig. 2 in centrality classes 0–5% to 40–50%. These measurements will provide the reference for how the various configurations of AMPT affect the over all \(p_{\textrm{T}} \)-differential flow. A reasonable description of the ALICE data is found for the Par4 configuration below \(\approx \)2 GeV/c. Above this the model severely underestimates the \(p_{\textrm{T}} \)-differential flow. However, it should be noted that the goal of this paper is not to tune the AMPT model to the data, but rather to gauge the source of the decorrelation effects observed in heavy-ion collisions. It is therefore not too important to match the exact shape of \(v_n(p_{\textrm{T}})\) as this is mostly cancelled out in the ratio observables in any case. As expected the \(v_2(p_{\textrm{T}})\) increases with larger cross section, which corresponds to larger \(\eta /s\). Additionally, the flow signal also increases when giving the hadronic interactions longer time to evolve with ART. Finally, changing the Lund string parameters a and b changes the \(v_2(p_{\textrm{T}})\) quite dramatically, as this will modify the initial energy prior to the parton-parton interactions leading to a different \(p_{\textrm{T}} \)-spectrum. These different configurations thus provide quite varied descriptions and will allow us to probe the sources of the \(p_{\textrm{T}}\)-dependent flow decorrelation.

5.2 Transverse momentum dependent flow vector decorrelation

Fig. 4
figure 4

The \(p_{\textrm{T}}\)-dependence of the factorization ratio \(r_2\) with \(|\Delta \eta |>0.8\) 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

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Fig. 5
figure 5

The decorrelation of the flow angle, \(A_2^\textrm{f}\approx \langle \cos (2n[\Psi _n(p_{\textrm{T}})-\Psi _n])\rangle _{w}\), with \(|\Delta \eta |>0\) as a function of \(p_{\textrm{T}}\) 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 [43]

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Fig. 6
figure 6

The decorrelation of the flow magnitude, \(M_2^\textrm{f}\), with \(|\Delta \eta |>0\) as a function of \(p_{\textrm{T}}\) 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 [43]

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The \(p_{\textrm{T}}\)-dependence of the \(v_2\{2\}/v_2[2]\) ratio with \(|\Delta \eta |>0.8\) calculated from the different configurations of the AMPT model is shown in Fig. 3 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. This shows that the flow vector decorrelation is highly sensitive to the initial state prior to the formation of the hot, partonic medium.

The ratio, \(v_2\{2\}/v_2[2]\), 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 [33]. This expectation is consistent with what is observed in Fig. 3: 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 \(p_{\textrm{T}}\) and is at most 10% in the presented \(p_{\textrm{T}}\)-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 \(p_{\textrm{T}}\) bin.

The above comparisons suggest that the decorrelation of the flow vector is highly sensitive to the initial state of heavy-ion collisions and, unlike the \(v_2\) flow coefficient itself, are less sensitive to the transport properties of the created matter. Measurements of \(v_2\{2\}/v_2[2]\) from the ALICE experiment in Pb–Pb collisions at \(\sqrt{s_{_{\textrm{NN}}}}\) = 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 \(p_{\textrm{T}}\) data point. These are different settings to the ones used for the tuned AMPT model (namely Par4) to describe the \(p_{\textrm{T}}\)-differential \(v_{n} (p_{\textrm{T}})\) [62]. The ratio \(v_{n}\{2\} /v_{n} [2]\) possibly carries additional information, which could be used to tune the model. Additionally, the combination of high sensitivity to the initial conditions and the high precision of the experimental measurements shows that \(v_{n}\{2\} /v_{n} [2]\) could be used as an input in a Bayesian analysis to further constrain the state-of-the-art understanding of the initial conditions of heavy-ion collisions.

The factorization ratio of the second-order flow harmonic, \(r_2\), with \(|\Delta \eta |>0.8\) is shown in Fig. 4 as a function of associate particle transverse momentum, \(p_{\textrm{T}}^{\textrm{a}}\), in centrality ranges 0–5%, 10–20% and 30–40% and different trigger particle transverse momentum, \(p_{\textrm{T}}^{\textrm{t}}\), 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 \(|p_{\textrm{T}}^{\textrm{a}}-p_{\textrm{T}}^{\textrm{t}} |\), which has also been observed in both hydrodynamics [36] and data [39, 74]. 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 \(|p_{\textrm{T}}^{\textrm{a}}-p_{\textrm{T}}^{\textrm{t}} |\). However, for the lower \(p_{\textrm{T}}^{\textrm{t}}\)-ranges, \(p_{\textrm{T}}^{\textrm{t}} <1.5\) GeV/c, the effect does not increase for \(p_{\textrm{T}}^{\textrm{a}} > 3\) GeV/c for central collisions and \(p_{\textrm{T}}^{\textrm{a}} > 4\) GeV/c for non-central collisions. In non-central collisions, the effect is up to 30% for large values of \(|p_{\textrm{T}}^{\textrm{a}}-p_{\textrm{T}}^{\textrm{t}} |\).Changing the cross section from 1.5 mb to 3.0 mb affects \(r_2(p_{\textrm{T}}^{\textrm{a}})\) by up to 10% at high \(|p_{\textrm{T}}^{\textrm{a}}-p_{\textrm{T}}^{\textrm{t}} |\) with ART OFF (Par5 vs Par2). This is consistent with \(v_2\{2\}/v_2[2]\), as expected, since \(r_2(p_{\textrm{T}}^{\textrm{a}})\) is basically the double-differential \(v_2\{2\}/v_2[2]\). The hadronic phase also shows an effect of around 10% at high \(|p_{\textrm{T}}^{\textrm{a}}-p_{\textrm{T}}^{\textrm{t}} |\) across all centralities for 1.5 mb (Par2 vs Par1) and 3.0 mb (Par5 vs Par4).

Although the factorization ratio carries more information about the detailed correlation structure, it does not show higher sensitivity to the partonic cross section than \(v_{n}\{2\} /v_{n} [2]\), which suggests that most of the sensitivity is cancelled out in the ratio. The ALICE measurements favour the model with a 1.5 mb cross section, a = 0.3, b = 0.15 and ART ON (Par1), however the breaking of factorization as \(|p_{\textrm{T}}^{\textrm{a}}-p_{\textrm{T}}^{\textrm{t}} |\) increases is overestimated with these settings of the AMPT calculation. As with \(v_{n}\{2\} /v_{n} [2]\), the factorization ratio also carries information that could constrain the model parameters further.

5.3 Transverse momentum dependent flow angle decorrelation

In Fig. 5, \(A_2^\textrm{f}\) with \(|\Delta \eta |>0\) is shown as a function of transverse momentum in centrality ranges 0–5% to 40–50%. A deviation of \(A_2^\textrm{f}\) from unity is observed for all the model calculations, with the majority showing deviations for \(p_{\textrm{T}} > 2.5\) GeV/c and the Par3 calculation with \(a=0.5,b=0.9\) showing deviation from unity for \(p_{\textrm{T}} >1.5\) GeV/c. The deviation from unity indicates that the \(p_{\textrm{T}}\)-dependent flow angle, \(\Psi _n(p_{\textrm{T}})\) fluctuates around the \(p_{\textrm{T}}\)-integrated flow angle \(\Psi _n\) in the AMPT model, and the decorrelation is strongest in the central collisions, where the initial state density fluctuations dominate. However, a significant decorrelation of up to 50% for Par3 and 30–40% for the other configurations can be seen across the centralities.

The decorrelation of the flow angle is highly sensitive to the initial conditions. Changing the Lund string parameters for the initial conditions affects the \(A^{\textrm{f}}_2\) by up to 70% in the 20–30% centrality range as shown in Fig. 5 bottom left (Par3 vs Par2). The large uncertainties at high \(p_{\textrm{T}}\) in central collisions do not allow for a quantitative statement about the effect, but at lower \(p_{\textrm{T}}\), the effect is up to 50%. The decorrelation of the flow angle shows 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 \(p_{\textrm{T}}\) due to the large uncertainties. At lower \(p_{\textrm{T}}\), 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).

The above observations suggests the fluctuating flow angles are most likely driven mainly by the event-by-event fluctuations in the initial state. Comparison of \(A^{\textrm{f}}_2\) to hydrodynamical models have also shown a similar sensitivity to the initial state [43]. 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 \(p_{\textrm{T}} > 3.0\) GeV/c in the 0–5% and 5–10% central collisions and at \(p_{\textrm{T}} > 2.0\) GeV/c in centralities 30–40% and 40–50%. The flow angle decorrelation, which has been observed in data and hydrodynamic models, is thus also reasonably described in the AMPT model.

5.4 Transverse momentum dependent flow magnitude decorrelation

The decorrelation of the flow magnitude, \(M_2^\textrm{f}\), with \(|\Delta \eta |>0\) is shown in Fig. 6 as a function of transverse momentum in centrality ranges 0–5% to 40–50%. As \(M_2^\textrm{f}\) is normalized to the baseline \(p_{\textrm{T}}\)-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 decorrelation of the flow magnitude is 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 \(p_{\textrm{T}} \) compared to the other calculations. The large deviation from unity suggests a strong sensitivity of the flow magnitude decorrelation to the initial conditions. This is observed by changing the Lund string parameters with an effect on \(M_2^\textrm{f}\) of up to 20% when comparing Par3 vs Par1. \(M_2^\textrm{f}\) shows 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 [43] based on the models from [42] is not observed in the low \(p_{\textrm{T}}\) region even though the change in \(\eta /s\) is larger in the AMPT calculations. The AMPT calculations show \(M^{\textrm{f}}_2\) = 1 at low \(p_{\textrm{T}}\) and the results are compatible for different \(\sigma \), as expected. The hadronic rescattering plays a small role in the decorrelation of the flow magnitude with \(\sim \)10% difference between the calculations with and without ART in the second-highest \(p_{\textrm{T}}\) 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 flow magnitude decorrelation measured with \(M^{\textrm{f}}_2\) shows similar order of sensitivity as the other decorrelation observables: Initial state \(\gg \) hadronic interactions \(\ge \) partonic cross section. 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 \(p_{\textrm{T}} > 3.0\) GeV/c in the 0–5% and 5–10% central collisions and at \(p_{\textrm{T}} > 2.0\) GeV/c in the 30–40% and 40–50% central collisions. The decorrelation of the flow magnitude is reproduced in the transport model, and the data to 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.

6 Summary

In this paper, the AMPT calculations of \(p_{\textrm{T}}\)-dependent decorrelation of the flow vector, flow angle and flow magnitude in Pb–Pb collisions at \(\sqrt{s_{_{\textrm{NN}}}}\) = 5.02 TeV are presented. The different configurations of the AMPT model probe the sensitivity of the decorrelation to (1) the initial conditions, (2) the partonic cross section, and (3) the hadronic rescattering to pinpoint the source of the decorrelation. It is found that the observables are highly sensitive to the changes in the initial conditions, and conversely 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 decorrelation mainly originates from the event-by-event fluctuations of the initial state and only slightly depends on the transport properties of the QGP. The presented studies on the \(p_{\textrm{T}}\)-dependent decorrelation of the flow vector and the comparison between models and data can help constrain the initial stage of the QGP formation.