Searches for transverse momentum dependent flow vector fluctuations in Pb-Pb and p-Pb collisions at the LHC

The measurement of azimuthal correlations of charged particles is presented for Pb-Pb collisions at $\sqrt{s_{\rm NN}}=$ 2.76 TeV and p-Pb collisions at $\sqrt{s_{\rm NN}}=$ 5.02 TeV with the ALICE detector at the CERN Large Hadron Collider. These correlations are measured for the second, third and fourth order flow vector in the pseudorapidity region $|\eta|<0.8$ as a function of centrality and transverse momentum $p_{\rm T}$ using two observables, to search for evidence of $p_{\rm T}$-dependent flow vector fluctuations. For Pb-Pb collisions at 2.76 TeV, the measurements indicate that $p_{\rm T}$-dependent fluctuations are only present for the second order flow vector. Similar results have been found for p-Pb collisions at 5.02 TeV. These measurements are compared to hydrodynamic model calculations with event-by-event geometry fluctuations in the initial state to constrain the initial conditions and transport properties of the matter created in Pb-Pb and p-Pb collisions.


Introduction
The primary goal of ultrarelativistic heavy-ion collisions is to study the properties of the Quark-Gluon Plasma (QGP), a state of matter predicted by Quantum Chromodynamics to exist at high temperatures and energy densities [1,2]. An important experimental observable used to accomplish this goal is the azimuthal anisotropy of particles emitted in the transverse plane. In non-central heavy-ion collisions, the overlap region of the Lorentzcontracted nuclei is roughly almond-shaped. Nucleons contained in such anisotropic overlap region interact with each other and give rise to a system of high energy density which expands anisotropically. These interactions convert the initial spatial asymmetry into a final-state momentum anisotropy of the produced particles, a phenomenon referred to as collective anisotropic flow [3][4][5]. Anisotropic flow is characterised using a Fourier decomposition of the azimuthal distribution of particles with respect to the flow symmetry planes [6,7] where N is the number of produced particles, E is the energy, p the momentum, p T the transverse momentum, ϕ the azimuthal angle and η the pseudorapidity of the particle. The n th order flow (vector) V n is defined as: V n ≡ v n e inΨn , where v n is the flow coefficient, and Ψ n represents the azimuth of V n in momentum space (flow angle). For a uniform matter distribution in the initial stage of a heavy-ion collision, Ψ n for n ≥ 1 coincides with the reaction plane defined by the beam direction and impact parameter. Due to event-by-event fluctuations of the participating nucleons distribution inside the overlap region, the Ψ n may The traditional approach used to measure anisotropic azimuthal correlations is as follows: first, the flow coefficient of reference particles (RPs), called reference flow, is determined over a wide kinematic range, and then the transverse momentum differential flow coefficient is calculated by correlating the particles of interest (POIs) with respect to the reference flow obtained in the first step. Usually a pseudorapidity gap |∆η| is applied between the two correlated particles to suppress non-flow effects, which comprise azimuthal correlations not associated with flow symmetry planes, e.g. resonance decays and jet contributions. This approach has commonly been used to measure the anisotropic flow at the LHC [20,25,28]. Considering possible p T -dependent flow angle and/or magnitude fluctuations and neglecting non-flow contributions, the flow coefficient from p T interval a measured with 2-particle correlations can be expressed as Here, a single set of angular brackets denotes averaging over events, and a double set indicates averaging over both particles and events. The ϕ ref and ϕ a represent the azimuthal angle of RPs and POIs, respectively. The v ref n stands for the reference flow, and Ψ n (p a T ) denotes the p T differential symmetry plane angle at p a T , which might fluctuate around the p T integrated symmetry plane angle Ψ n . The cosine term cos [ n (Ψ n (p a T ) − Ψ n )] shows the effects of the difference between Ψ n (p T ) and Ψ n , due to the p T -dependent flow angle fluctuations. Additionally, v n (p a T ) v ref A new type of two-particle azimuthal correlations from p a T , denoted as v n [2](p a T ), is proposed in [33]: v n [2](p a T ) = cos [ n (ϕ a 1 − ϕ a 2 )] = cos [ n(ϕ a 1 − Ψ n (p a T )) − n (ϕ a 2 − Ψ n (p a T ))] = v n (p a T ) 2 . (2. 2) The difference between v n {2}(p a T ) and v n [2](p a T ) is that the former takes the flow of RPs from a wide p T range and the POIs from a certain p T interval, while the latter is essentially the reference flow calculated within a narrow p T range. The ratio of v n {2} and v n [2] allows p T -dependent flow vector fluctuations v n {2} v n [2] (p a When the correlations are dominated by flow, a ratio value smaller than unity shall indicate the presence of p T -dependent flow vector fluctuations.

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Another observable to probe the p T -dependent flow vector fluctuations is the factorisation ratio r n [32,33]. It can be calculated using the two-particle Fourier harmonic as , is the n th -order Fourier harmonic of the two-particle azimuthal correlations of triggered and associated particles from p t T and p a T , and is calculated as where Ψ n (p a T ) and Ψ n (p t T ) represent the flow angles at p a T and p t T , respectively. The subscript ∆ indicates that a pseudorapidity gap is usually applied to minimise contamination from non-flow effects. If both triggered and associated particle are from the same p T interval p t T , eq. (2.5) reduces to In the end r n is equivalent to It can be seen that r n = 1 does not always hold true, i.e. most of the known sources of non-flow effects do not factorise at low p T , which is confirmed by Monte Carlo studies [48]. In a flow-dominated system, r n ≤ 1 due to the Cauchy-Schwarz inequality. Factorisation implies r n = 1, while r n < 1 shows the breaking of factorisation, suggesting the presence of p T -dependent flow vector fluctuations [32,33]. Note that eqs. (2.3) and (2.8) look very similar. The ratios v n {2}/v n [2] include the p T integrated information and probe the p T -differential flow vector with respect to the p T integrated flow vector. The r n carries more detailed information on the 2-particle correlation structure for triggered and associated particle from narrow p T intervals, and probe the fluctuations of flow vector at p a T and p t T ; however, it also has larger statistical uncertainties. If the triggered particles are selected from a very wide kinematic range, the observable r n becomes identical with v n {2}/v n [2]. In this paper, we study v n {2}/v n [2] up to n = 4 and r n up to n = 3.

Experimental setup
A Large Ion Collider Experiment (ALICE) [49] is the dedicated heavy-ion experiment at the LHC designed to study strongly interacting matter at extreme energy densities. It was built to cope with the large charged-particle multiplicity density in central Pb-Pb collisions at -4 -JHEP09(2017)032 the LHC, with several thousand tracks per unit of pseudorapidity. The ALICE apparatus consists of a central barrel that measures hadrons, electrons, muons and photons, and a forward spectrometer for the identification of muons. Several smaller detectors in the forward region are used for triggering and global event characterization. The central barrel is located inside a solenoidal magnet that provides a magnetic field of up to 0.5 T. Charged tracks are reconstructed using the Time Projection Chamber (TPC) [49,50] and the Inner Tracking System (ITS) [49,51] with a track momentum resolution better than 2% for the momentum range 0.2 < p T < 5.0 GeV/c [52]. The TPC is the main tracking detector of the central barrel, sufficient with full azimuthal coverage in the range of |η| < 0.9. The ITS consists of six layers of silicon detectors placed at radii between 3.9 cm and 43 cm and matching the pseudorapidity acceptance of the TPC. Three different technologies are employed in the ITS: the two innermost layers are equipped with Silicon Pixel Detectors (SPD), the following two layers have Silicon Drift Detectors (SDD) and the two outer layers are double-sided Silicon Strip Detectors (SSD). The V0 detector [49,53] was used for triggering and the determination of the event centrality. It consists of two arrays called V0-A and V0-C, each built from 32 scintillator counters and providing full azimuthal coverage, positioned on each side of the interaction point. The V0-A is situated at z = 3.4 m (2.8 < η < 5.1) and the V0-C is located at z = −0.9 m (−3.7 < η < −1.7). Each V0 counter provides the signal amplitude and timing information with a time resolution better than 1 ns [49,53]. Two Zero Degree Calorimeters (ZDCs) [49] were used in the offline event selection. The ZDCs are a pair of hadronic calorimeters, one for detecting non-interacting neutrons (ZN) and one for spectator protons (ZP), located at 112.5 m on either side of the interaction point.

Event and track selection
The data samples analyzed in this article were recorded by ALICE during the 2010 Pb-Pb and 2013 p-Pb runs of the LHC at centre-of-mass energies of √ s NN = 2.76 TeV and √ s NN = 5.02 TeV, respectively. The Pb-Pb run had equal beam energies, while the p-Pb run had beam energies of 4 TeV for protons and 1.58 TeV per nucleon for lead nuclei, which resulted in a rapidity shift of −0.465 of the centre-of-mass system with respect to the ALICE laboratory system. In the following, all kinematic variables are reported in the laboratory system. Minimum bias Pb-Pb and p-Pb events were triggered by the coincidence of signals in both V0 detectors. The trigger efficiency is 99.7% for non-diffractive Pb-Pb collisions [54] and 99.2% for non-single-diffractive p-Pb collisions [55]. Beam background events were rejected in an offline event selection for all data samples using the timing information from the V0 and ZDC detectors and by correlating reconstructed SPD clusters and tracklets. The remaining beam background was found to be smaller than 0.1% and was neglected. More details about the offline event selection can be found in [52]. The fraction of pile-up events in the data sample is found to be negligible after applying dedicated pile-up removal criteria [52]. Only events with a reconstructed primary vertex within |z vtx | < 10 cm with respect to the nominal interaction point were selected. The position of the primary vertex was estimated using tracks reconstructed by the ITS and -5 -

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TPC. The Pb-Pb collision centrality was determined from the measured V0 amplitude distribution [54]. The dataset of p-Pb collisions is divided into several multiplicity classes defined as fractions of the analysed event sample, based on the charge deposition in the V0-A detector. These multiplicity classes are denoted as "0-20%", "20-40%", "40-60%", and "60-100%", from the highest to the lowest multiplicity. About 13 million Pb-Pb and 92 million p-Pb minimum bias events passed all event selection criteria.
This analysis used tracks that were reconstructed based on the combined information from the TPC and ITS detectors. Primary charged tracks were required to have a distance of closest approach to the primary vertex in the longitudinal (z) direction and transverse (xy) plane smaller than 3.2 cm and 2.4 cm, respectively. Tracks with 0.2 < p T < 5.0 GeV/c were selected in the pseudorapidity range |η| < 0.8, in order to exclude non-uniformities due to the detector boundaries. Additional track quality cuts were applied to remove secondary particles (i.e. particles originating from weak decays, photon conversions and secondary hadronic interactions in the detector material) while maintaining good track reconstruction efficiency. Tracks were required to have at least 70 TPC space points out of the maximum of 159. The χ 2 of the track fit per degree of freedom in the TPC reconstruction was required to be below 2.

Systematic uncertainties
The evaluation of systematic uncertainties was performed by varying the event and track selection cuts and by studying the detector response with Monte Carlo (MC) simulations. For Pb-Pb, the track selection criteria were changed to only require tracks reconstructed in the TPC alone. This led to a significant difference in most of the observables (up to 10 %), which was taken into account in the estimation of the systematic uncertainties. Altering the number of TPC space points from 70 to 80, 90 and 100 resulted in a maximum 0.5% variation of v n results. The variation of the v n results when using other detectors, e.g. the SPD or TPC, to determine the centrality, is less than 0.5%. No significant variation of the v n results was seen when altering the polarity of the magnetic field of the ALICE detector, or when narrowing the nominal |z vtx | range from 10 cm to |z vtx | < 7, 8, and 9 cm. The contribution from pileup events to the final systematic uncertainty was found to be negligible. Systematic uncertainties due to detector inefficiencies were investigated using HIJING [56] and AMPT [57] MC simulations. The calculations for a sample at the event generator level (i.e. without invoking either the detector geometry or the reconstruction algorithm) were compared with the results of the analysis of the output of the full reconstruction with a GEANT3 [58] detector model, in a procedure referred to as an MC closure test. A difference of up to 4% for v n is observed, which is included in the final systematic uncertainty. Most of the systematic uncertainties described above cancelled out for v n {2}/v n [2] and r n as indicated in table 2.
For p-Pb collisions, the approach used to evaluate the systematic uncertainty is similar. Different track quality cuts are applied, including varying the number of TPC space points, and using tracks reconstructed with the required TPC detector only instead of combined information from TPC and ITS. This leads to a systematic uncertainty of up to Table 1. Summary of systematic uncertainties of v n for Pb-Pb collisions.  Table 3. Summary of systematic uncertainties for p-Pb collisions.

Pb-Pb sources
6% depending on the multiplicity and p T range. It was also found that varying the event selection, which includes the cut on the |z vtx |, and the cuts to reject pileup events, yields negligible contributions to the final systematic uncertainty. The analysis was repeated using the energy deposited in the neutron ZDC (ZNA) which is located at 112.5 m from the interaction point, instead of using V0-A for the event classes determination. The observed differences with respect to the one using V0-A for event class determination is not included as systematic uncertainty, following the previous paper [36]. In addition, the MC closure is investigated with DPMJET simulations [59] combined with GEANT3; this leads to a systematic uncertainty of less than 9% for p T < 0.8 GeV/c and 2% for higher p T . The dominant sources of systematic uncertainty are summarized in tables 1, 2 and 3. The systematic uncertainties evaluated for each of the sources mentioned above were added in quadrature to obtain the total systematic uncertainty of the measurements.
6 Results and discussion   [2] with three different pseudorapidity gaps, for centrality classes from 0-5% to 70-80%. The analysed events are divided into two sub-events A and B, separated by a pseudorapidity gap. Note that |∆η| > 0 suggests that there is no separation in pseudorapidity between the two sub-events. Short-range correlations, one of the main sources of non-flow effects, are expected to be suppressed  Figure 1. v 2 {2} with |∆η| > 0 (circles), |∆η| > 0.4 (diamonds) and |∆η| > 0.8 (squares) for various centrality classes in Pb-Pb collisions at √ s NN = 2.76 TeV. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33], with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively. when using a large pseudorapidity gap. It is observed that v 2 {2} and v 2 [2] using various pseudorapidity gaps do not change significantly for central and semi-central collisions. The decrease of v 2 with larger pseudorapidity gaps is more prominent in the most peripheral collisions, mainly due to the suppression of non-flow effects. The results are also compared to the original predictions within the VISH2+1 hydrodynamic framework with: 1) Monte Carlo Glauber (MC-Glauber) initial conditions and η/s = 0.08; 2) Monte Carlo Kharzeev-Levin-Nardi (MC-KLN) initial conditions and η/s = 0.20 [33]. In addition, the comparisons to recently released calculations from the iEBE-VISHNU hydrodynamic framework with: 1) Trento initial conditions, temperature dependent shear and bulk viscosities, η/s(T) and ζ(T); and 2) AMPT initial conditions with η/s = 0.08 [60] are also presented. These combinations of various initial conditions and η/s are chosen due to the fact that they give the best descriptions of the particle spectra and the integrated flow measurements [60,61].
with |∆η| > 0 (circles), |∆η| > 0.4 (diamonds) and |∆η| > 0.8 (squares) for various centrality classes in Pb-Pb collisions at √ s NN = 2.76 TeV. Hydrodynamic calculations with MC-Glauber initial conditions [33] and η/s = 0.08, with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dashed and orange dashed curves, and magenta and grey shaded areas, respectively. different centrality classes in figure 3. This ratio is consistent with unity up to p T ≈ 2 GeV/c and starts to deviate from unity in the higher p T region in the most central collisions. The deviations from unity are weak and within 10% in non-central collisions in the presented p T range. To better understand whether such deviations from unity are caused by non-flow effects, the like-sign technique, which suppresses contributions from resonance decays by correlating only particles with same charge, is applied. The differences of the measured v 2 {2, |∆η| > 0.8}/v 2 [2, |∆η| > 0.8] from like-sign and all charged particles are found to be less than 0.5%. This shows that deviations of v 2 {2, |∆η| > 0.8}/v 2 [2, |∆η| > 0.8] from unity cannot be explained solely by non-flow effects from resonance decays. It is also seen in figure 3 that the hydrodynamic calculations with MC-KLN, Trento and AMPT initial conditions describe the data fairly well for all centrality classes except for the most peripheral collisions, while MC-Glauber calculations reproduce the data only for mid-central and peripheral collisions. This indicates that hydrodynamic calculations with AMPT and MC-KLN initial conditions and η/s = 0.20 not only generate reasonable v 2 values, but also reproduce the measured The higher order anisotropic flow coefficients, which were first measured in [20], are shown to be more sensitive to the initial conditions and η/s [12]. In figures 4 and 5, The different panels show the centrality evolution of the measurements. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33], with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dashed and orange dashed curves, and magenta and grey shaded areas, respectively.
trend as the pseudorapidity gap increases, in particular in more peripheral collisions. Only a weak centrality dependence is observed for both v 3 {2} and v 3 [2]. The comparison to hydrodynamic calculations demonstrates that although hydrodynamic calculations with MC-Glauber and MC-KLN initial conditions roughly describe v 2 {2} and v 2 [2], they cannot describe v 3 {2} and v 3 [2] over the full p T range and for all centrality classes, and tend to overpredict or underpredict the data. Similar as v 2 , the hydrodynamic calculation with Trento initial conditions overestimates both v 3 {2} and v 3 [2] measurements, while the one with AMPT initial conditions quantitatively describe the measured v 3 for presented p T and centrality intervals.
shown together with hydrodynamic calculations in figure 6. Wider p T intervals were used for the ratio than for the individual v 3 measurements in order to suppress statistical fluctuations. It was found that the ratio agrees with unity over a wide p T range, as opposed to  Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33], with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively.  Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33] and with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively. The different panels show the centrality evolution of the measurements. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33] and with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively. especially the one with MC-KLN initial conditions, overestimate the possible p T -dependent V 3 flow vector fluctuations, despite the good description for the second harmonic. A good agreement between data and hydrodynamic calculations from iEBE-VISHNU is found for all centrality intervals. This is expected for AMPT initial conditions as the calculations quantitatively reproduce both measured v 3 {2} and v 3 [2] as discussed above. However, the calculations with Trento initial conditions, which overestimate both v 3 {2} and v 3 [2], are consistent with the measured v 3 {2, |∆η| > 0.8}/v 3 [2, |∆η| > 0.8] ratio. This accidental agreement needs further investigations in the iEBE-VISHNU framework to understand the physics mechanism responsible for this behaviour. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33], with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively.  Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33] and with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively. The different panels show the centrality evolution of the measurements. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33] and with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively. and 11 as a function of p t T and p a T with |∆η| > 0.8 for three centrality classes in Pb-Pb collisions at √ s NN = 2.76 TeV. By construction, r n = 1 when the triggered and associated particles are from the same p T interval. In contrast to the previous analysis [34], there is no p t T ≥ p a T cut applied here to avoid auto-correlations (taking the same particle as both triggered and associated particles in the two-particle azimuthal correlations). The triggered particles are always selected from the negative pseudorapidity region and the associated particles are from the positive pseudorapidity region. The r 2 value deviates significantly from unity for the most central collisions. This effect becomes stronger with an increasing difference between p t T and p a T . The previous results indicated that factorisation holds approximately for n ≥ 2 and p T below 4 GeV/c, while deviations emerging at higher p T were ascribed to recoil jet contributions [34]. This analysis, however, shows that factorisation breaks down at lower p T when the more sensitive observable, r 2 , is used. The deviation reaches 10% for the lowest p a T in the 0-5% centrality range, for 2.5 < p t T < 3 GeV/c. One explanation from [32] is that this deviation is due to the p T -dependent V 2 flow vector fluctuations, which originate from initial event-by-event geometry fluctuations. Hydrodynamic calculations [33] are compared to data for the presented centrality classes and for selected p T bins. Both hydrodynamic calculations from VISH2+1 and iEBE-VISHNU frameworks qualitatively predict the trend of r 2 , while the data agree quantitatively better with iEBE-VISHNU. In addition, the CMS measurements [41,47] are consistent with our measurements.
For r 3 , the results are compatible with unity, and can be described by hydrodynamic calculations from both VISH2+1 and iEBE-VISHNU frameworks, albeit with large statistical  [41]. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33] and with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively.
uncertainties. The factorisation is valid over a wider range of p a T , p t T and centrality ranges, as opposed to r 2 . The possible breakdown of factorisation, if it exists, is within 10% when both p a T and p t T are below 3 GeV/c. The CMS measurements [41,47] are consistent with the r 3 results presented here despite the fact that the pseudorapidity gaps are different between the two measurements. Better agreements with hydrodynamic calculations are observed with VISH2+1.  Figure 11. The factorisation ratio r 3 , as a function of p a T in bins of p t T for 0-5%, 20-30% and 40-50% centralities in Pb-Pb collisions at √ s NN = 2.76 TeV, is presented (solid circles). CMS measurements [41] are presented by open squares. Hydrodynamic calculations with MC-Glauber initial conditions and η/s = 0.08 [33] and with MC-KLN initial conditions and η/s = 0.20 [33], with Trento initial conditions and temperature dependent η/s [60] and AMPT initial conditions and η/s = 0.08 [60] are shown in green dot-dash, orange dashed curves, and magenta and grey shaded areas, respectively.  unity above p T ≈ 2 GeV/c, but the statistical uncertainties are still too large to draw a firm conclusion. The DPMJET model [59], which is an implementation of the two-component Dual Parton Model for the description of interactions involving nuclei, and contains no collective effects, has been used as a benchmark to study the influence of non-flow in p-Pb collisions [38]. The calculations based on DPMJET simulations are compared to data. It is observed in figure 12 that DPMJET overestimates v 2 significantly for the presented multiplicity classes, and generates higher v 2 coefficients in lower multiplicity regions. Meanwhile, figure 13 shows that for v 2 {2}/v 2 [2] the agreement between data and DPMJET is better in low multiplicity p-Pb collisions, where no evidence of anisotropic collectivity is achieved from previous measurements [36,38]. In addition, the hydrodynamic calculations [62] from MUSIC v2.0 using a modified MC-Glauber initial state and η/s = 0.08 are also presented in figures 12 and 13. These calculations in general underpredict the measured v 2 coefficients but agree better with the data in high multiplicity than in low multiplicity classes. It should be emphasized that in contrast to hydrodynamic calculations, the measured v 2 {2} and v 2 [2] increase (albeit very slightly in particular when the |∆η| gap is applied) from 0-20% to 40-60% multiplicity classes, which indicates that non-flow effects might play a more important role in low multiplicity events. This could explain the increasing deviation between data and hydrodynamic calculations with p T and towards lower multiplicity classes, shown in figure 12. The hydrodynamic calculations reproduce the v 2 {2}/v 2 [2] measurements in the 0-20 % multiplicity class, which seems to indicate that hydrodynamic collectivity is present in high multiplicity p-Pb collisions. However, it is still unclear at the moment why the measured ratio is still reproduced by hydrodynamic calculations for multiplicity class above 20%, where no significant flow signal is expected to be produced [38]. The agreement might be accidental since the DPMJET and hydrodynamic calculations also agree with each other in this class.

p-Pb collisions
The   deviation increasing as a function of p T . The relative influence of non-flow effects appears to be stronger in v 3 than in v 2 measurements. A similar qualitative behaviour was observed for p T -integrated two-particle cumulants c 2 {2} and c 3 {2} in p-Pb collisions, measured as functions of multiplicity for different |∆η| gaps [36]. It might be worth noting that part of the remaining non-flow contamination with |∆η| > 0.8, the recoil jet ridge, has a positive sign contribution for v 2 and a negative sign one for v 3 for p T > 2 GeV/c. In addition, it is found that hydrodynamic calculations describe the data better at high multiplicity than at low multiplicity, while DPMJET generates negative (v 3 [2]) 2 values for all multiplicity classes and thus cannot be shown here for comparison. Furthermore, the deviations between v 3 {2, |∆η| > 0.8} and v 3 [2, |∆η| > 0.8] are not observed for the presented p T region.
There is no indication of p T -dependent V 3 flow vector fluctuations in p-Pb collisions. Figure 15 shows r 2 (|∆η| > 0.8) measurements as a function of p a T in three p t T intervals for multiplicity classes 0-20%, 20-40% and 40-60% in p-Pb collisions at √ s NN = 5.02 TeV.
The r 2 (|∆η| > 0.8) deviates from unity when the p t T and p a T are well away from each other (most pronouncedly in the lowest and highest p t T bins) with the trend being similar for all multiplicity classes. As mentioned earlier, the deviation is more significant at high multiplicity. In overlapping p t T and p a T intervals, the r 2 measurements in the highest -18 -JHEP09(2017)032 multiplicity p-Pb events are consistent with those made by CMS collaboration [47]. The breakdown of factorisation is more pronounced in high multiplicity p-Pb collisions than in the 40-50% centrality class in Pb-Pb collisions (see figure 10). The DPMJET calculations are presented for comparison. It is clearly seen that DPMJET overestimates the deviations of r 2 from unity in the high multiplicity region, nevertheless, the calculation describes the data better in low multiplicity events in which non-flow effects are dominant. At the same time, these measurements are found to be compatible with hydrodynamic calculations using modified MC-Glauber initial conditions and η/s = 0.08. When selecting the triggered particles from 0.6 < p t T <1.0 GeV/c or 1.0 < p t T < 1.5 GeV/c, the trend of r 2 looks similar to that of v 2 {2}/v 2 [2], mainly because the mean p T of charged particles is within 0.6 < p T < 1.0 GeV/c [63].

Summary
Searches for p T -dependent flow vector fluctuations are performed by measuring v n {2}/v n [2] and r n in Pb-Pb collisions at √ s NN = 2.76 TeV and p-Pb collisions at √ s NN = 5.02 TeV.
In Pb-Pb collisions, both v 2 {2}/v 2 [2] and r 2 show deviations from unity, and the r 2 results are consistent with previous measurements from the CMS collaboration. These effects are more pronounced in the most central collisions and cannot be explained solely by non-flow effects. Therefore, these results suggest the presence of possible V 2 vector fluctuations in Pb-Pb collisions. It further implies that the traditional v 2 {2} results should be interpreted precisely as the correlations of the azimuthal angle of produced particles with respect to the p T integrated flow vector over a certain kinematic region. Future comparisons between theoretical calculations and experimental measurements should be based on the same kinematic conditions. These comparisons, performed under carefully defined precisely matching kinematic conditions, are crucial to constrain the initial conditions and precisely extract the transport properties of the produced matter, without possible bias from additional p Tdependent flow vector fluctuations. Meanwhile, no significant deviation of v 3 {2}/v 3 [2] or v 4 {2}/v 4 [2] from unity was observed, meaning that there is no indication of p T -dependent V 3 and V 4 vector fluctuations. The comparison to hydrodynamic calculations shows only the calculations from iEBE-VISHNU with AMPT initial conditions could describe the data quantitatively. The measurements presented in this paper provide a unique approach to constrain the initial conditions and transport properties, e.g. shear viscosity over entropy density ratio η/s of the QGP, complementing the previous anisotropic flow measurements.
The results therefore bring new insights into the properties of the QGP produced in relativistic heavy ion collisions at the CERN Large Hadron Collider. Similar studies were performed in various multiplicity classes in p-Pb collisions. Deviations of v 2 {2}/v 2 [2] and r 2 from unity are observed, although with relatively large statistical fluctuations. For the highest p-Pb multiplicity class, the deviations are significantly overestimated by DPMJET; however, they are compatible with hydrodynamic calculations using modified MC-Glauber initial conditions and η/s = 0.08. Meanwhile for low multiplicity p-Pb collisions, the data sits between calculations from DPMJET and hydrodynamics. Neither the DPMJET model, which does not incorporate anisotropic flow, DPMJET calculations are presented by pink shaded areas. Hydrodynamic calculations (MUSIC) with modified MC-Glauber initial conditions and η/s = 0.08 are shown as magenta lines [62]. nor the hydrodynamic model, which does not include non-flow contributions, could provide a quantitative description of the data. Future theoretical developments together with comparisons to high-precision measurements are crucial to give a certain answer concerning p T -dependent vector V n fluctuations in p-Pb collisions.