aa r X i v : . [ nu c l - t h ] A p r Transverse momentum dependent decorrelation in Pb-Pb collisions at LHC
De-Xian Wei ∗ School of Science, Guangxi University of Science and Technology, Liuzhou, 545006, China (Dated: April 8, 2020)Based on A Multi-Phase Transport (AMPT) model simulations, the transverse momentum dependent decor-relation has been studied in Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV, respectively. It has found that theFactorization ratio r m,n value deviates significantly from unity in non-central collisions. Such effect becomesstronger with an increasing the p T difference p aT − p bT . These decorrelation is not only for the same orderharmonic flow but also for the difference order harmonic flow. It has also found that the correlations involvinghigher powers of the flow vector yield stronger decorrelation, r m | n ;3 < r m | n ;2 < r m | n ;1 ( m = 2 , , exceptfor the weighted factorization ratio r | k . PACS numbers: 25.75.Ld, 25.75.Gz
I. INTRODUCTION
The primary goal of ultra relativistic heavy-ion collisions is to understand the matter properties of Quark-Gluon Plasma (QGP),whose is produced in extreme conditions has been predicted by the Quantum Chromodynamics [1]. Anisotropic harmonics flowplays a major role in probing the properties of the QGP at the Relativistic Heavy Ion Collider (RHIC) at BNL [2] and LargeHadron Collider (LHC) at CERN [3]. The realization of higher order harmonics flow and its fluctuations [4], the correlationbetween the magnitude and phase of different order harmonics [5–7] and the transverse momentum and pseudorapidity depen-dence of event plane angles [8] has led to a good understanding of the initial fluctuating states and the properties of the strongQGP. Furthermore, the higher order harmonics (n >
3) can arise from initial fluctuating anisotropies in the same order harmonic(denoted linear response) or can be driven by lower order harmonics (denoted non-linear response) [9–12]. These mixed higherorder harmonics v { Ψ } , v { Ψ } , v { Ψ } , v { Ψ } , and the non-linear response coefficients χ , χ , χ , χ areweakly sensitive to the initial-state conditions and transport properties of the QGP [13–16].The experiment has been indicated that the flow vector fluctuations was observed by the decomposition of Fourier harmonicsof the two-particles azimuthal correlations [17]. To test the flow vector fluctuations, a useful observable is the factorization ratio, r n,n , which encodes the correlations of flow harmonics at different transverse momenta or pseudorapidities [8, 18–26]. Thesecorrelation revealed that the factorization ratio is sensitive to fluctuations in the initial states and not strongly dependent on theviscosity of the system [27].Hypothetically, the factorization ratio can be broken down for different order harmonics correlations as a consequence of theinitial fluctuations driven decorrelation between the higher order harmonics with its’ lower order harmonics and the same lowerorder harmonic. Following this idea, the main purpose of this paper is to illustrate a particular picture on the initial fluctuationdriven mixed harmonics flows decorrelation (denoted mixed order factorization ratio breaking) in Pb-Pb collisions at LHC. II. MATERIALS AND METHODS
Starting from the V n estimators studied in Ref. [9, 13, 15], the harmonic flow can be expressed as a sum of the linear andnon-linear modes, V = V L + χ V ,V = V L + χ V V . (1)where V nL denotes the linear part of V n ( n = 4 , that is not induced by lower-order harmonics [11], and the χ are thenonlinear response coefficients, characterizing the non-linear flow mode induced by the lower order harmonics. More higherorder V n ( n > are also shown in Ref. [15]. ∗ Electronic address: [email protected]
The mixed higher-order harmonics in each p T range are extracted using the scalar-product method as shown in Ref. [14],which describe by a Q-vector, as V { Ψ } ( p T ) = Re h Q ( p T ) Q ∗ B Q ∗ B i p Re h Q A Q A Q ∗ B Q ∗ B i ,V { Ψ } ( p T ) = Re h Q ( p T ) Q ∗ B Q ∗ B i p Re h Q A Q A Q ∗ B Q ∗ B i . (2)Here, Q nA and Q nB are vectors from two different parts of a single event with particles range in a positive or negative pseudo-rapidity region, Q n is the vector from charged particles in each p T range within a mid-pseudorapidity region, and angle bracketsdenote the average over all events within a given centrality range.Similar to the mixed higher-order flow harmonics, the non-linear response coefficients in each p T range can be expressedas [14], χ { Ψ } ( p T ) = Re h Q ( p T ) Q ∗ B Q ∗ B i Re h Q Atrk Q Atrk Q ∗ B Q ∗ B i ,χ { Ψ } ( p T ) = Re h Q ( p T ) Q ∗ B Q ∗ B i Re h Q Atrk Q Atrk Q ∗ B Q ∗ B i . (3)Where Q nAtrk is chosen the same pseudorapidity region with Q n .Correlations between Q n of different harmonics represent higher order correlations which can provide crucial information onthe initial-state and its’ fluctuations of the medium. One observable to probe the p T dependent flow vector fluctuations is thefactorization ratio, r n,n [18, 19, 24]. It can be calculated using the two-particle Fourier harmonic by the same order. To test themixed harmonics flows decorrelation, a mix-order factorization ratio r m,n , are expressed as r m,n ( p aT , p bT ) = V m,n ( p aT , p bT ) q V m,m ( p aT , p aT ) V n,n ( p bT , p bT ) ,V m,n ( p aT , p bT ) = h Q m ( p aT ) Q ∗ n ( p bT ) i . (4)Where V m,n is the m th - and n th -order Fourier harmonic of the two-particle azimuthal correlations of the triggered and associatedparticles from p aT and p bT . To avoid self-correlation, the triggered particles (denoted p aT ) are always selected from the positivepseudorapidity region and the associated particles (denoted p bT ) are from the negative pseudorapidity region. A psedorapiditygap is applied between p aT and p bT that to suppress non-flow effects. In that case r m,n ( p aT , p bT ) ≤ means that the harmonic flowat the transverse momenta p aT and p bT is partially decorrelated. This decorrelation can be due to the flow vector fluctuations bothof flow magnitude and flow phase decorrelation [8] were generated by initial event-by-event geometry fluctuation.Correlators of higher powers of the same order flow in two different p T bins has been calculated by Hydrodynamic [24].Naturally, a mix-order factorization ratio weighted with different powers of Q n can be defined as r m | n ; k ( p aT , p bT ) = V m | n ; k ( p aT , p bT ) q V m | m ; k ( p aT , p aT ) V n | n ; k ( p bT , p bT ) ,V m | n ; k ( p aT , p bT ) = h Q m ( p aT ) k Q ∗ n ( p bT ) k i . (5)For k = 1 one recovers the factorization ratio Eq. (4) r m | n ;1 ( p aT , p bT ) = r m,n ( p aT , p bT ) .In this proceeding, the p T -dependent factorization ratio is investigated in Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV forthe produced charged particles with the AMPT model [28], respectively. Base on AMPT event-by-event simulation [28], thispaper present the factorization ratio by used the scalar-product method [14]. III. RESULTS
Fig. 1 depicts the estimated magnitude of harmonic flow v n ( n = 4 , as a function of p T in 20-60% Pb-Pb collisions at2.76 and 5.02 TeV from AMPT event-by-event simulations (colored band), respectively. Here, Q nA ( Q nB ) are particles rangein pseudorapidity region . < η < . ( − . < η < . ), Q n is charged particles range in pseudorapidity region | η | < . .It shows that the results of AMPT calculations on Pb-Pb systems are agree with the CMS [14] data with error bars. In Fig. 1, AMPT { } p T (GeV/c) (a) AMPT | | < 2.4(b) { } p T (GeV/c) FIG. 1: (Color online) The magnitude of harmonic flows v n ( n = 4 , as a function of p T in 20-60% Pb-Pb collisions at 2.76 and 5.02 TeVfrom AMPT simulations (colored band), respectively. AMPT results are compared with the CMS [14] data (red points and black points). AMPT | | < 2.4 { } p T (GeV/c) (a) AMPT | | < 2.4(b) { } p T (GeV/c) FIG. 2: (Color online) Non-linear response coefficients χ n ( n = 4 , as a function of p T in 20-60% Pb-Pb collisions at 2.76 and 5.02 TeVfrom AMPT simulations (colored band), respectively. AMPT results are compared with the CMS [14] data (red points and black points). the effect of v n ( n = 4 , increases with transverse momentum increasing, is understood as a consequence of the degree ofinteraction in the proceeding of transport.Fig. 2 shows that the non-linear response coefficients χ n ( n = 4 , as a function of p T in 20-60% Pb-Pb collisions at 2.76and 5.02 TeV from AMPT event-by-event simulations (colored band), respectively. AMPT calculations on Pb-Pb systems arecompatible with the CMS [14] data for the presented centrality class. In Fig. 2, the χ n are increases with transverse momentumincreasing. The p T dependent χ n is quite different than the results of p T independent χ n in Ref. [15] where are calculated bydifference sub-events in the scalar-product method.One study for p T dependent flow vector fluctuations can be via the observable of the factorization ratio, r m,n . The results of r m,n are presented in Fig. 3 as a function of the p T difference p aT − p bT with |△ η | > Ta < 1.5 GeV/c r , ( p T a , p Tb ) Ta < 2 GeV/c Pb+Pb b=40-50%
CMS data
AMPT 2.76 TeV
AMPT 5.02 TeV Ta < 2.5 GeV/c Ta < 3 GeV/c r , ( p T a , p Tb ) r , ( p T a , p Tb ) r , ( p T a , p Tb ) r , ( p T a , p Tb ) r , ( p T a , p Tb ) r , ( p T a , p Tb ) p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 2.0 p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 2.0 p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 2.0 2.5 p Ta -p Tb (GeV/c) FIG. 3: (Color online) Factorization ratio r m,n as a function of p T in 40-50% Pb-Pb collisions at 2.76 and 5.02 TeV from AMPT simulations(colored band), respectively. The simulate results are compared with the 2.76 TeV on CMS [8] data (red points). k=1 k=2 k=3(a) r (b) r r r Pb+Pb 2.76 TeV (c) 2.4 < p Ta < 3 GeV/c (d) (e) p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 (f)Pb+Pb 5.02 TeV p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 (g) p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 2.0 (h) p Ta -p Tb (GeV/c) FIG. 4: (Color online) The weighted factorization ratio r | n ; k as a function of p T in 40-50% Pb-Pb collisions at 2.76 and 5.02 TeV, respectively.Up panels: 2.76 TeV for Pb-Pb collisions. Down panels: 5.02 TeV for Pb-Pb collisions. (a) (b) Pb+Pb 2.76 TeV (c) 2.4 < p Ta < 3 GeV/c (d) r r r p Ta -p Tb (GeV/c) k=1 k=2 k=3 (e) Pb+Pb 5.02 TeV p Ta -p Tb (GeV/c) 0.0 0.5 1.0 1.5 2.0 (f) p Ta -p Tb (GeV/c) FIG. 5: (Color online) Similar distributions as shown in Fig. 4, but for the weighted factorization ratio r | n ; k . r m,n is significantly deviate from unity in non-central collisions.This effect has becomes stronger with an increasing the p T difference p aT − p bT . It is indicated that p T dependent flow vectorare significant fluctuating in the presented p T range. From Fig. 3, it shows that the Factorization ratio broken effect is notonly for the same order harmonic flow but also for the difference order harmonic flow. These decorrelation can be found inthe difference order harmonic flow, e.g. r , , r , and r , . Note that the higher order harmonic flow are defined by the linearsame order harmonic and the non-linear lower order harmonics in Eq. (1), as a result, correlations of the higher order harmonicwith its non-linear lower order harmonic and the same lower order harmonic can be decorrelate due to the initial fluctuations.Furthermore, the decorrelation is weakly dependence on the collision energy.Fig. 4 and Fig. 5 shows that the weighted factorization ratio r m | n ; k as a function of p T in 40-50% Pb-Pb collisions at 2.76and 5.02 TeV from AMPT event-by-event simulations, respectively. In Fig. 4, the up panels are results of r | n ; k for 2.76 TeV onPb-Pb collisions and the down panels are results of r | n ; k for 5.02 TeV on Pb-Pb collisions, respectively. The charged triggerparticles are chosen region in . < p aT < . GeV/c. It has found that both ratios not agree with unity over the presented p aT − p bT range. The correlations involving higher powers of the flow vector yield stronger decorrelation, r | n ;3 < r | n ;2 < r | n ;1 where is shown in Fig. 4. Similar distributions for the weight factorization coefficient r | n ; k is also shown in Fig. 5. FromFig. 5, the correlations involving higher powers of the flow vector yield also stronger decorrelation, r | n ;3 < r | n ;2 < r | n ;1 ,except for the weighted factorization ratio r | k . IV. SUMMARY
Base on AMPT event-by-event calculations, this paper has carried out the p T dependent v n ( n = 4 , , χ n ( n = 4 , and r m,n by AMPT simulations in non-central Pb-Pb collisions at √ s NN = 2.76 and 5.02 TeV, respectively. The results of AMPTcalculations on Pb-Pb systems are compatible with the CMS data within error bars. By AMPT simulations, it has found thatthe Factorization ratio r m,n value deviates significantly from unity in non-central collisions. Such effect becomes stronger withan increasing the p T difference p aT − p bT . These Factorization ratio broken effect is not only for the same order harmonic flow,but also for the difference order harmonic flow, as a result of initial fluctuations driven decorrelation between the higher orderharmonic with its’ non-linear lower order harmonic and the same lower order harmonic. It has also found that the correlationsinvolving higher powers of the flow vector yield stronger decorrelation, r m | n ;3 < r m | n ;2 < r m | n ;1 ( m = 2 , , except for theweighted factorization ratio r | k . Acknowledgements
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