Investigation of the linear and mode-coupled flow harmonics in Au+Au collisions at s NN − − − − √ = 200 GeV
STAR Collaboration, J. Adam, L. Adamczyk, J. R. Adams, J. K. Adkins, G. Agakishiev, M. M. Aggarwal, Z. Ahammed, I. Alekseev, D. M. Anderson, A. Aparin, E. C. Aschenauer, M. U. Ashraf, F. G. Atetalla, A. Attri, G. S. Averichev, V. Bairathi, K. Barish, A. Behera, R. Bellwied, A. Bhasin, J. Bielcik, J. Bielcikova, L. C. Bland, I. G. Bordyuzhin, J. D. Brandenburg, A. V. Brandin, J. Butterworth, H. Caines, M. Calderón de la Barca Sánchez, D. Cebra, I. Chakaberia, P. Chaloupka, B. K. Chan, F-H. Chang, Z. Chang, N. Chankova-Bunzarova, A. Chatterjee, D. Chen, J. H. Chen, X. Chen, Z. Chen, J. Cheng, M. Cherney, M. Chevalier, S. Choudhury, W. Christie, X. Chu, H. J. Crawford, M. Csanád, M. Daugherity, T. G. Dedovich, I. M. Deppner, A. A. Derevschikov, L. Didenko, X. Dong, J. L. Drachenberg, J. C. Dunlop, T. Edmonds, N. Elsey, J. Engelage, G. Eppley, R. Esha, S. Esumi, O. Evdokimov, A. Ewigleben, O. Eyser, R. Fatemi, S. Fazio, P. Federic, J. Fedorisin, C. J. Feng, Y. Feng, P. Filip, E. Finch, Y. Fisyak, A. Francisco, L. Fulek, C. A. Gagliardi, T. Galatyuk, F. Geurts, A. Gibson, K. Gopal, D. Grosnick, W. Guryn, A. I. Hamad, A. Hamed, S. Harabasz, J. W. Harris, S. He, W. He, X. H. He, S. Heppelmann, S. Heppelmann, N. Herrmann, E. Hoffman, L. Holub, Y. Hong, S. Horvat, Y. Hu, et al. (265 additional authors not shown)
aa r X i v : . [ nu c l - e x ] J un Investigation of the linear and mode-coupled flow harmonicsin Au+Au collisions at √ s N N = 200 GeV
J. Adam , L. Adamczyk , J. R. Adams , J. K. Adkins , G. Agakishiev ,M. M. Aggarwal , Z. Ahammed , I. Alekseev , , D. M. Anderson , A. Aparin ,E. C. Aschenauer , M. U. Ashraf , F. G. Atetalla , A. Attri , G. S. Averichev ,V. Bairathi , K. Barish , A. Behera , R. Bellwied , A. Bhasin , J. Bielcik ,J. Bielcikova , L. C. Bland , I. G. Bordyuzhin , J. D. Brandenburg , ,A. V. Brandin , J. Butterworth , H. Caines , M. Calder´on de la Barca S´anchez ,D. Cebra , I. Chakaberia , , P. Chaloupka , B. K. Chan , F-H. Chang , Z. Chang ,N. Chankova-Bunzarova , A. Chatterjee , D. Chen , J. H. Chen , X. Chen ,Z. Chen , J. Cheng , M. Cherney , M. Chevalier , S. Choudhury , W. Christie ,X. Chu , H. J. Crawford , M. Csan´ad , M. Daugherity , T. G. Dedovich ,I. M. Deppner , A. A. Derevschikov , L. Didenko , X. Dong , J. L. Drachenberg ,J. C. Dunlop , T. Edmonds , N. Elsey , J. Engelage , G. Eppley , R. Esha ,S. Esumi , O. Evdokimov , A. Ewigleben , O. Eyser , R. Fatemi , S. Fazio ,P. Federic , J. Fedorisin , C. J. Feng , Y. Feng , P. Filip , E. Finch , Y. Fisyak ,A. Francisco , L. Fulek , C. A. Gagliardi , T. Galatyuk , F. Geurts , A. Gibson ,K. Gopal , D. Grosnick , W. Guryn , A. I. Hamad , A. Hamed , S. Harabasz ,J. W. Harris , S. He , W. He , X. H. He , S. Heppelmann , S. Heppelmann ,N. Herrmann , E. Hoffman , L. Holub , Y. Hong , S. Horvat , Y. Hu ,H. Z. Huang , S. L. Huang , T. Huang , X. Huang , T. J. Humanic , P. Huo ,G. Igo , D. Isenhower , W. W. Jacobs , C. Jena , A. Jentsch , Y. JI , J. Jia , ,K. Jiang , S. Jowzaee , X. Ju , E. G. Judd , S. Kabana , M. L. Kabir ,S. Kagamaster , D. Kalinkin , K. Kang , D. Kapukchyan , K. Kauder , H. W. Ke ,D. Keane , A. Kechechyan , M. Kelsey , Y. V. Khyzhniak , D. P. Kiko la ,C. Kim , B. Kimelman , D. Kincses , T. A. Kinghorn , I. Kisel , A. Kiselev ,M. Kocan , L. Kochenda , L. K. Kosarzewski , L. Kramarik , P. Kravtsov ,K. Krueger , N. Kulathunga Mudiyanselage , L. Kumar ,R. Kunnawalkam Elayavalli , J. H. Kwasizur , R. Lacey , S. Lan , J. M. Landgraf ,J. Lauret , A. Lebedev , R. Lednicky , J. H. Lee , Y. H. Leung , C. Li , W. Li ,W. Li , X. Li , Y. Li , Y. Liang , R. Licenik , T. Lin , Y. Lin , M. A. Lisa ,F. Liu , H. Liu , P. Liu , P. Liu , T. Liu , X. Liu , Y. Liu , Z. Liu ,T. Ljubicic , W. J. Llope , R. S. Longacre , N. S. Lukow , S. Luo , X. Luo ,G. L. Ma , L. Ma , R. Ma , Y. G. Ma , N. Magdy , R. Majka , D. Mallick ,S. Margetis , C. Markert , H. S. Matis , J. A. Mazer , N. G. Minaev ,S. Mioduszewski , B. Mohanty , M. M. Mondal , I. Mooney , Z. Moravcova ,D. A. Morozov , M. Nagy , J. D. Nam , Md. Nasim , K. Nayak , D. Neff ,J. M. Nelson , D. B. Nemes , M. Nie , G. Nigmatkulov , T. Niida ,L. V. Nogach , T. Nonaka , A. S. Nunes , G. Odyniec , A. Ogawa , S. Oh ,V. A. Okorokov , B. S. Page , R. Pak , A. Pandav , Y. Panebratsev , B. Pawlik ,D. Pawlowska , H. Pei , C. Perkins , L. Pinsky , R. L. Pint´er , J. Pluta ,J. Porter , M. Posik , N. K. Pruthi , M. Przybycien , J. Putschke , H. Qiu ,A. Quintero , S. K. Radhakrishnan , S. Ramachandran , R. L. Ray , R. Reed ,H. G. Ritter , J. B. Roberts , O. V. Rogachevskiy , J. L. Romero , L. Ruan ,1. Rusnak , N. R. Sahoo , H. Sako , S. Salur , J. Sandweiss , S. Sato ,W. B. Schmidke , N. Schmitz , B. R. Schweid , F. Seck , J. Seger , M. Sergeeva ,R. Seto , P. Seyboth , N. Shah , E. Shahaliev , P. V. Shanmuganathan ,M. Shao , F. Shen , W. Q. Shen , S. S. Shi , Q. Y. Shou , E. P. Sichtermann ,R. Sikora , M. Simko , J. Singh , S. Singha , N. Smirnov , W. Solyst ,P. Sorensen , H. M. Spinka , B. Srivastava , T. D. S. Stanislaus , M. Stefaniak ,D. J. Stewart , M. Strikhanov , B. Stringfellow , A. A. P. Suaide , M. Sumbera ,B. Summa , X. M. Sun , X. Sun , Y. Sun , Y. Sun , B. Surrow , D. N. Svirida ,P. Szymanski , A. H. Tang , Z. Tang , A. Taranenko , T. Tarnowsky ,J. H. Thomas , A. R. Timmins , D. Tlusty , M. Tokarev , C. A. Tomkiel ,S. Trentalange , R. E. Tribble , P. Tribedy , S. K. Tripathy , O. D. Tsai , Z. Tu ,T. Ullrich , D. G. Underwood , I. Upsal , , G. Van Buren , J. Vanek ,A. N. Vasiliev , I. Vassiliev , F. Videbæk , S. Vokal , S. A. Voloshin , F. Wang ,G. Wang , J. S. Wang , P. Wang , Y. Wang , Y. Wang , Z. Wang , J. C. Webb ,P. C. Weidenkaff , L. Wen , G. D. Westfall , H. Wieman , S. W. Wissink ,R. Witt , Y. Wu , Z. G. Xiao , G. Xie , W. Xie , H. Xu , N. Xu , Q. H. Xu ,Y. F. Xu , Y. Xu , Z. Xu , Z. Xu , C. Yang , Q. Yang , S. Yang , Y. Yang ,Z. Yang , Z. Ye , Z. Ye , L. Yi , K. Yip , H. Zbroszczyk , W. Zha , C. Zhang ,D. Zhang , S. Zhang , S. Zhang , X. P. Zhang , Y. Zhang , Y. Zhang ,Z. J. Zhang , Z. Zhang , Z. Zhang , J. Zhao , C. Zhong , C. Zhou , X. Zhu ,Z. Zhu , M. Zurek , M. Zyzak (STAR Collaboration) Abilene Christian University, Abilene, Texas 79699 AGH University of Science and Technology, FPACS, Cracow 30-059, Poland Alikhanov Institute for Theoretical and Experimental Physics NRC ”Kurchatov Institute”, Moscow117218, Russia Argonne National Laboratory, Argonne, Illinois 60439 American University of Cairo, New Cairo 11835, New Cairo, Egypt Brookhaven National Laboratory, Upton, New York 11973 University of California, Berkeley, California 94720 University of California, Davis, California 95616 University of California, Los Angeles, California 90095 University of California, Riverside, California 92521 Central China Normal University, Wuhan, Hubei 430079 University of Illinois at Chicago, Chicago, Illinois 60607 Creighton University, Omaha, Nebraska 68178 Czech Technical University in Prague, FNSPE, Prague 115 19, Czech Republic Technische Universit¨at Darmstadt, Darmstadt 64289, Germany ELTE E¨otv¨os Lor´and University, Budapest, Hungary H-1117 Frankfurt Institute for Advanced Studies FIAS, Frankfurt 60438, Germany Fudan University, Shanghai, 200433 University of Heidelberg, Heidelberg 69120, Germany University of Houston, Houston, Texas 77204 Huzhou University, Huzhou, Zhejiang 313000 Indian Institute of Science Education and Research (IISER), Berhampur 760010 , India Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India Indian Institute Technology, Patna, Bihar 801106, India Indiana University, Bloomington, Indiana 47408 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu 730000 University of Jammu, Jammu 180001, India Joint Institute for Nuclear Research, Dubna 141 980, Russia Kent State University, Kent, Ohio 44242 University of Kentucky, Lexington, Kentucky 40506-0055 Lawrence Berkeley National Laboratory, Berkeley, California 94720 Lehigh University, Bethlehem, Pennsylvania 18015 Max-Planck-Institut f¨ur Physik, Munich 80805, Germany Michigan State University, East Lansing, Michigan 48824 National Research Nuclear University MEPhI, Moscow 115409, Russia National Institute of Science Education and Research, HBNI, Jatni 752050, India National Cheng Kung University, Tainan 70101 Nuclear Physics Institute of the CAS, Rez 250 68, Czech Republic Ohio State University, Columbus, Ohio 43210 Institute of Nuclear Physics PAN, Cracow 31-342, Poland Panjab University, Chandigarh 160014, India Pennsylvania State University, University Park, Pennsylvania 16802 NRC ”Kurchatov Institute”, Institute of High Energy Physics, Protvino 142281, Russia Purdue University, West Lafayette, Indiana 47907 Rice University, Houston, Texas 77251 Rutgers University, Piscataway, New Jersey 08854 Universidade de S˜ao Paulo, S˜ao Paulo, Brazil 05314-970 University of Science and Technology of China, Hefei, Anhui 230026 Shandong University, Qingdao, Shandong 266237 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 Southern Connecticut State University, New Haven, Connecticut 06515 State University of New York, Stony Brook, New York 11794 Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Chile Temple University, Philadelphia, Pennsylvania 19122 Texas A&M University, College Station, Texas 77843 University of Texas, Austin, Texas 78712 Tsinghua University, Beijing 100084 University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan United States Naval Academy, Annapolis, Maryland 21402 Valparaiso University, Valparaiso, Indiana 46383 Variable Energy Cyclotron Centre, Kolkata 700064, India Warsaw University of Technology, Warsaw 00-661, Poland Wayne State University, Detroit, Michigan 48201 Yale University, New Haven, Connecticut 06520
Abstract
Flow harmonics ( v n ) of the Fourier expansion for the azimuthal distributions of hadronsare commonly employed to quantify the azimuthal anisotropy of particle production rel-ative to the collision symmetry planes. While lower order Fourier coefficients ( v and v ) are more directly related to the corresponding eccentricities of the initial state, thehigher-order flow harmonics ( v n> ) can be induced by a mode-coupled response to thelower-order anisotropies, in addition to a linear response to the same-order anisotropies.These higher-order flow harmonics and their linear and mode-coupled contributions canbe used to more precisely constrain the initial conditions and the transport properties ofthe medium in theoretical models. The multiparticle azimuthal cumulant method is usedto measure the linear and mode-coupled contributions in the higher-order anisotropicflow, the mode-coupled response coefficients, and the correlations of the event plane an-gles for charged particles as functions of centrality and transverse momentum in Au+Aucollisions at nucleon-nucleon center-of-mass energy √ s NN = 200 GeV. The results arecompared to similar LHC measurements as well as to several viscous hydrodynamic cal-culations with varying initial conditions. Keywords:
Collectivity, correlation, shear viscosity
PACS:
1. Introduction
Experimental studies of heavy-ion collisions at the Relativistic Heavy Ion Collider(RHIC) indicate that a state of matter predicted by Quantum Chromodynamics (QCD),called Quark-Gluon Plasma (QGP), is formed in these collisions. Many of the ongoingstudies are aimed at characterizing the transport properties (particularly, the specificshear viscosity: the ratio of shear viscosity to entropy density η/ s ) of the QGP. Theazimuthal anisotropy of particle production relative to the collision symmetry planes,known as anisotropic flow, is a key observable in many such studies because it displays Preprint submitted to Phys. Lett. B June 29, 2020 he viscous hydrodynamic response to the initial spatial distribution created in the earlystages of the collision [1–14].The anisotropic flow can be characterized by the Fourier expansion [15] of the particleazimuthal angle ( φ ) distributions, dNdφ = N π X n =1 V n e − i n φ ! , (1)where V n = v n e i n Ψ n is the n-th complex anisotropic flow vector, v n and Ψ n represent thevector magnitude and direction, respectively. The flow coefficient v is commonly termedas directed flow, v is the elliptic flow, and v is the triangular flow. Anisotropic flowstudies of higher-order flow harmonics v n> [10, 16–22], correlation between different flowharmonics [20, 23–27] and flow fluctuations [18, 28–30] have led to a deeper understandingof the initial conditions [31] and the properties of the matter created in heavy-ioncollisions.In the hydrodynamic models, anisotropic flow arises from the evolution of the mediumin the presence of initial-state energy density anisotropies, characterized by the complexeccentricity vectors [24, 32–35]: E n ≡ ε n e i n Φ n ≡ − R d r ⊥ r n e i n ϕ ρ e (r , ϕ ) R d r ⊥ r n ρ e (r , ϕ ) , ( n > , (2)where ρ e ( r, ϕ ) is the initial anisotropic density profile, ε n = D |E n | E / represents theeccentricity vectors magnitude and Φ n denotes the azimuthal direction of the eccentricityvector [35–37].The elliptic and triangular flow harmonics are, to a reasonable approximation, linearlyproportional to the initial-state anisotropies, ε and ε , respectively [7, 24, 38–44]: v n = k n ε n , (3)where k n is the proportionality factor that encodes the medium response, and is expectedto be sensitive to η/ s and the system lifetime [45]. Therefore, the ratio v n /ε n (for n = 2 ,
3) could be used as a tool to probe η/ s of the QGP [17]. In contrast, the higher-order flow harmonics are expected to arise from a mode-coupled (nonlinear) response tothe lower-order eccentricities, ε and/or ε [12, 36, 37] in addition to linear response tothe same-order initial-state anisotropies [46]: V = V L4 + V mc4 = V L4 + χ , V V , (4) V = V L5 + V mc5 = V L5 + χ , V V , (5)where V Ln and V mc n represents the linear and the mode-coupled contributions to the flowvector V n respectively. The χ , and χ , are the mode-coupled response coefficientswhich define the magnitude of the V mc n> measured with respect to the lower-order sym-metry plane angle(s). Also, the mode-coupled contribution of V n is expected to reflectthe correlation between different order flow symmetry planes, Ψ n , which could shed lighton the initial stage dynamics [23, 27, 36, 47–53].5he v and v harmonics are sensitive to the respective influence of the initial-stateeccentricity and the final-state viscous attenuation, which have proven difficult to dis-entangle. The mode-coupled coefficients show characteristically different dependencieson the viscous attenuation and the initial-state eccentricity [44]. Therefore, they canbe used in conjunction with measurements for the v and v harmonics to leverage addi-tional unique constraints for initial-state models, as well as reliable extraction of transportcoefficients.In this paper we report new differential and integral measurements of v and v andtheir mode-coupled response coefficients, obtained with the two- and multiparticle cu-mulant methods described in Section 2. Measurements of these quantities as functions ofcollision centrality and charged particle transverse momentum, p T , in Au+Au collisionsat √ s NN = 200 GeV, are reported in Section 3. The presented results and conclusionsare summarized in Section 4.
2. Experimental setup and analysis method
The data reported in this analysis were collected with the STAR detector at RHICusing a minimum-bias trigger [54] in 2011. Charged particle tracks, measured in pseu-dorapidity range | η | < . .
52. Tracks used in this study are restrictedto transverse momentum 0 . < p T < c . Events are chosen with vertex positionswithin ±
30 cm from the TPC center (along the beam direction), and within ± z -vertex positions defined by the TPC and Vertex PositionDetector is required to be less than 3 cm to decrease beam-induced background andpileup.The systematic uncertainties associated with the measurements presented in this workare estimated by changing different parameters of the analysis and comparing the resultswith their baseline values. The systematic uncertainty associated with the event selectionis estimated by using more restrictive requirements for the vertex positions determinedby the TPC along the beam direction ( −
30 to 0 cm or 0 to 30 cm instead of the nominalvalue of ±
30 cm). The systematic uncertainty arising from track selection is evaluated byemploying more strict requirements: (i) Distance of Closest Approach (DCA) is changedto be less than 2 cm instead of the standard value of 3 cm, and (ii) number of TPC spacepoints from more than 15 points to more than 20 points. The systematic uncertaintyassociated with the nonflow effects, due to Bose-Einstein correlations, resonance decaysand the fragments of individual jets, is estimated by investigating the impact of a pseu-dorapidity gap, ∆ η = η − η , for the track pairs used in the measurements. Studieswere performed for ∆ η values of 0.6, 0.7, and 1.0.6able 1 shows the systematic uncertainties evaluated for this work. The overallsystematic uncertainty was calculated by combining uncertainties from different sourcesin quadrature. In the ensuing figures, the overall systematic uncertainties (which do notinclude those from ∆ η variation) are shown as open boxes; statistical uncertainties areshown as vertical lines.Variations of Quantities Minimum value Maximum valueEvent 2% 4%Track 3% 6%∆ η
3% 8%
Table 1: The contributions to the total systematic uncertainties from various sources.
The two- and multiparticle cumulant techniques are used in this work. The frameworkfor the cumulant method is described in Refs. [47, 57], which was extended to the caseof subevents in Refs. [58, 59]. In this work, the two- and multiparticle correlations wereconstructed using the two-subevents cumulant method [59], with particle weights, e.g.weighted with the particles acceptance correction, and ∆ η > . A and B ( i.e. , η A > .
35 and η B < − . v Inclusive k = hh cos( k ( ϕ A − ϕ B )) ii / , (6) C k,nm = hh cos( kϕ A − nϕ B − mϕ B ) ii , (7) h v n v m i = hh cos( nϕ A + mϕ A − nϕ B − mϕ B ) ii , (8)where hh ii indicates the average over all particles in a single event and then the averageover all events, k = n + m , n = 2, m = 2 or 3, and ϕ i is the azimuthal angle of the i -thparticle.Using Eqs. (6)-(8), the mode-coupled contribution in higher-order anisotropic flowharmonics, v and v , can be expressed as [37, 60]: v mc4 = C , p h v v i , (9) ∼ h v cos(4Ψ − − ) i ,v mc5 = C , p h v v i , (10) ∼ h v cos(5Ψ − − ) i , and the linear contribution to v and v can be given as: v L = q ( v Inclusive4 ) − ( v mc4 ) , (11) v L = q ( v Inclusive5 ) − ( v mc5 ) . v and v are independent [37, 61]. The ratios of the mode-coupled contribution to the inclusive v and v are expected to measure the correlations between different order flow symmetryplanes [62] and are expressed as ρ , and ρ , , respectively. The ρ , and ρ , can begiven as: ρ , = v mc4 v Inclusive4 = h cos(4Ψ − − ) i , (12) ρ , = v mc5 v Inclusive5 = h cos(5Ψ − − ) i . (13)The mode-coupled response coefficients, χ , and χ , , which quantify the contri-butions of the mode-coupling to the the higher-order anisotropic flow harmonics, aredefined as χ , = v mc4 p h v v i (14) χ , = v mc5 p h v v i . (15)In Eq.(15) for the differential χ , , this work further makes the approximation h v v i ∼h v i h v i [36]. These dimensionless ratios that represent the mode-coupled coefficients inEq.(4) are expected to be weakly sensitive to viscous effects [44].
3. Results and discussion
In A+A collisions, short-range nonflow correlations contribute to the measured three-particle correlators C , and C , [61]. However, such correlations can be reduced byusing subevents cumulant methods [59]. Figure 1 compares the C , and C , valuesobtained from the standard ( i.e. , the three particles are selected using the entire detectoracceptance) and the two-subevents cumulant methods as a function of centrality in therange 0 . < p T < . c for Au+Au collisions at √ s NN = 200 GeV. The magnitudesof the measured C , and C , from the standard cumulant method are larger than thosefrom the subevents cumulant method, compatible with the expectation that the subeventscumulant method can further reduce the nonflow correlations. The shaded bands in Fig. 1indicate viscous hydrodynamic model predictions [63, 64], as summarized in Table 2.Note that these model predictions include an influence from changes in the initial- andfinal-state assumptions incorporated in model calculations. The model predictions, whichwere generated with the standard cumulant method, show good qualitative agreementwith both C , and C , . However, Hydro − b with no hadronic cascade gives a betterdescription of the data for C , and C , obtained with the two-subevents cumulantmethod.The centrality dependence of the inclusive, linear and mode-coupled v and v in the p T range from 0 . . c for Au+Au collisions at √ s NN = 200 GeV are shown inFig. 2. They indicate that the linear mode of v and v depends weakly on the collisioncentrality and constitutes the dominant contribution to the inclusive v and v in centralcollisions. These results are compared to similar LHC measurements in the p T range from8ydro − − a/b [64] η/ s Table 2: Summary description of the hydrodynamic simulations, Hydro − − a/b [64]. × -4 (a)n = 2, m = 2Au+Au 200 GeV Centrality (%) C n + m , n m Two-subevent methodStandard method
0 20 40 60 ( × (b)n = 2, m = 3 Centrality (%)
Hydro-1Hydro-2 a Hydro-2 b Figure 1: Comparison of the p T -integrated three-particle correlators, C , and C , , for Au+Au col-lisions at √ s NN = 200 GeV, obtained with the standard (red squares) and the two-subevents cumulant(blue circles) methods. The respective systematic uncertainties, that do not include the nonflow contri-butions, are shown as open boxes. The vertical lines represent the statistical errors. The shaded bandsindicate hydrodynamic model predictions Hydro − − a and Hydro − b [64].
0 20 40 60 (a)Inclusive v (b)Mode-coupled (c)Linear Pb+Pb 2.76 TeVAu+Au 200 GeV
0 20 40 60(d)Inclusive
Centrality (%)
0 20 40 60(e)Mode-coupled
Centrality (%)
0 20 40 60(f)Linear
Centrality (%)
Figure 2: Comparison of the inclusive mode-coupled and linear higher-order flow harmonics v and v obtained with the two-subevents cumulant method, as a function of centrality in the p T range 0 . − . c for Au+Au collisions at √ s NN = 200 GeV. The systematic uncertainties, that do not includethe nonflow contributions, are shown as open boxes. The solid diamonds indicate LHC measurementsfor the p T range from 0 . − . c for Pb+Pb collisions at √ s NN = 2.76 TeV [62]. . . c and pseudorapidity range | η | < . √ s NN =2.76 TeV [62]. The comparison indicates strikingly similar patterns for the RHIC and9
0 20 40 60 (a)n = 2 and m = 2 χ n + m , n m Pb+Pb 2.76 TeVAu+Au 200 GeV (b)n = 2 and m = 3
Hydro-1Hydro-2 a Hydro-2 b (c)n = 2 and m = 2 Centrality (%) ρ n + m , n m
0 20 40 60 (d)n = 2 and m = 3
Centrality (%)
Figure 3: Results as a function of centrality in the p T range from 0 . . c for Au+Au collisionsat √ s NN = 200 GeV. Panels (a) and (b) shows the mode-coupled response coefficients, χ , and χ , ,and panels (c) and (d) show the correlations of event plane angles, ρ , and ρ , . The results wereobtained with the two-subevents cumulant method; the open boxes indicate the systematic uncertainties.The closed-symbols represents similar LHC measurements in the p T range from 0 . . c forPb+Pb collisions at √ s NN = 2.76 TeV [62]. The shaded bands indicate hydrodynamic model predictionsHydro − − a and Hydro − b [64].
0 1 2 3 4 × -2 (a) v Inclusive Mode-coupled Linear
Au+Au 200 GeV10-40% (b) (c) p T (GeV/c) χ ρ
0 1 2 3 4 (d) p T (GeV/c) χ ρ Figure 4: Results as a function of p T for 10-40% central Au+Au collisions at √ s NN = 200 GeV.Panels (a) and (b) present the inclusive, linear and mode-coupled higher-order flow harmonics v and v obtained with the two-subevents cumulant method. Panel (c) presents the χ , and ρ , , whilepanel (d) presents the χ , and ρ , . The open boxes indicate the systematic uncertainties. LHC measurements, albeit with a difference in the magnitude of the measurements. This10bserved difference could result from a sizable difference in the h p T i for the p T -integrated v and v measurements at RHIC and the LHC, respectively. Here, it is noteworthy thateven though the p T range for both measurements is similar, the inverse slopes of thehadron p T spectra are larger at the LHC than at RHIC. Subtleties related to a differencein the viscous properties of the medium created at RHIC and LHC energies could alsocontribute to the observed difference in the magnitude of the measurements [63].The centrality dependence of the mode-coupled response coefficients, χ , and χ , ,for Au+Au collisions, is presented in Fig. 3(a) and (b) for the range 0 . < p T < . c . They show a weak centrality dependence, akin to the patterns observedfor similar measurements at the LHC for Pb+Pb collisions at √ s NN = 2.76 TeV [62](closed symbols). These patterns suggests that (i) the centrality dependence observedfor the mode-coupled v and v ( cf. , Figs. 2(b) and (e)) stems from the lower-orderflow harmonics and (ii) the mode-coupled response coefficients are dominated by initial-state eccentricity couplings which have a weak dependence on beam energy. The shadedbands in Figs. 3(a) and (b) show that the predictions from the viscous hydrodynamicmodels [63, 64] summarized in Table 2, give a good qualitatively description of the χ , and χ , data. However, the predictions from Hydro − − b (cf. Table 2),give the overall closest description to χ , and χ , .Figures 3(c) and (d) show the centrality dependence of the correlations of the eventplane angles, ρ , and ρ , , for 0 . < p T < . c in Au+Au collisions at √ s NN =200 GeV. The data suggest stronger event plane correlations in peripheral than in centralcollisions. This centrality dependent pattern is also captured by the viscous hydrody-namic model predictions [63, 64] indicated by the shaded bands in the figure. The LHC ρ , and ρ , measurements for Pb+Pb collisions at √ s NN = 2.76 TeV [62] (closed sym-bols), also indicate magnitudes and trends similar to those for the Au+Au collisions.This observation could be an indication that the correlation of event plane angles aredominated by initial-state effects.The p T dependence of the inclusive, linear and mode-coupled higher-order flow har-monics, v and v , for 10-40% central Au+Au collisions, are compared in Figs. 4(a)and (b). They show that the p T -dependent trends of the linear and mode-coupled con-tributions are similar to the inclusive v and v , as previously measured by the STARcollaboration [10, 19]. This observation suggests that the linear and mode-coupled con-tributions are driven by the same p T -dependent physics processes. The correspondingmode-coupled response coefficients χ , and χ , and the correlations of event planeangles ρ , and ρ , are shown in Figs. 4 (c) and (d). They indicate little, if any, p T dependence for the centrality selection presented. These trends suggest that bothdimensionless coefficients are dominated by initial-state effects.
4. Summary
In summary, we have presented new differential measurements of the charge-inclusive,linear and mode-coupled contributions to the higher-order anisotropic flow coefficients v and v , mode-coupled response coefficients χ , and χ , and the correlations of theevent plane angles ρ , and ρ , , for Au+Au collisions at √ s NN = 200 GeV. The p T -integrated measurements indicate a sizable centrality dependence for the mode-coupledcontributions of v and v , whereas the linear contributions, that dominate the centralcollisions, show a weak centrality dependence. The v and v results are compared with11imilar LHC measurements which show larger magnitude that could be driven by thedifference in the viscous effects and the mean p T between RHIC and LHC energies. The χ , and χ , show a weak centrality dependence, however the ρ , and ρ , increasefrom central to peripheral collisions. These dimensionless coefficients show magnitudesand trends which are similar to those observed for LHC measurements, suggesting thatthe correlations of event plane angles as well as the mode-coupled response coefficientsare dominated by initial-state effects. This is further supported by the observed p T inde-pendence of the χ , , χ , , ρ , and ρ , . Viscous hydrodynamic model comparisonsto the data indicate good qualitatively agreement. However, none of the models providea simultaneous description of the three-particle correlations, the mode-coupled responsecoefficients, and the correlations of event plane angles. These higher-order flow mea-surements could provide additional stringent constraints to discern between initial statemodels and aid precision extraction of the transport properties of the medium producedin the collisions. Acknowledgments
We thank the RHIC Operations Group and RCF at BNL, the NERSC Center atLBNL, and the Open Science Grid consortium for providing resources and support. Thiswork was supported in part by the Office of Nuclear Physics within the U.S. DOE Officeof Science, the U.S. National Science Foundation, the Ministry of Education and Sci-ence of the Russian Federation, National Natural Science Foundation of China, ChineseAcademy of Science, the Ministry of Science and Technology of China and the ChineseMinistry of Education, the Higher Education Sprout Project by Ministry of Educationat NCKU, the National Research Foundation of Korea, Czech Science Foundation andMinistry of Education, Youth and Sports of the Czech Republic, Hungarian NationalResearch, Development and Innovation Office, New National Excellency Programme ofthe Hungarian Ministry of Human Capacities, Department of Atomic Energy and De-partment of Science and Technology of the Government of India, the National ScienceCentre of Poland, the Ministry of Science, Education and Sports of the Republic ofCroatia, RosAtom of Russia and German Bundesministerium fur Bildung, Wissenschaft,Forschung and Technologie (BMBF), Helmholtz Association, Ministry of Education, Cul-ture, Sports, Science, and Technology (MEXT) and Japan Society for the Promotion ofScience (JSPS).
ReferencesReferences [1] U. Heinz, P. Kolb, Early thermalization at RHIC, Nucl. Phys. A702 (2002) 269–280.[2] T. Hirano, U. W. Heinz, D. Kharzeev, R. Lacey, Y. Nara, Hadronic dissipative effects on elliptic flowin ultrarelativistic heavy-ion collisions, Phys.Lett. B636 (2006) 299–304. arXiv:nucl-th/0511046 , doi:10.1016/j.physletb.2006.03.060 .[3] P. Huovinen, P. F. Kolb, U. W. Heinz, P. V. Ruuskanen, S. A. Voloshin, Radial and elliptic flow atRHIC: Further predictions, Phys. Lett. B503 (2001) 58–64.[4] T. Hirano, K. Tsuda, Collective flow and two pion correlations from a relativistic hydrodynamicmodel with early chemical freeze out, Phys. Rev. C66 (2002) 054905. arXiv:nucl-th/0205043 , doi:10.1103/PhysRevC.66.054905 .
5] P. Romatschke, U. Romatschke, Viscosity Information from Relativistic Nuclear Collisions: HowPerfect is the Fluid Observed at RHIC?, Phys.Rev.Lett. 99 (2007) 172301. arXiv:0706.1522 , doi:10.1103/PhysRevLett.99.172301 .[6] M. Luzum, Flow fluctuations and long-range correlations: elliptic flow and beyond, J. Phys. G38(2011) 124026. arXiv:1107.0592 , doi:10.1088/0954-3899/38/12/124026 .[7] H. Song, S. A. Bass, U. Heinz, T. Hirano, C. Shen, 200 A GeV Au+Aucollisions serve a nearly perfect quark-gluon liquid, Phys. Rev. Lett. 106(2011) 192301, [Erratum: Phys. Rev. Lett.109,139904(2012)]. arXiv:1011.2783 , doi:10.1103/PhysRevLett.106.192301,10.1103/PhysRevLett.109.139904 .[8] J. Qian, U. W. Heinz, J. Liu, Mode-coupling effects in anisotropic flow in heavy-ion collisions, Phys.Rev. C93 (2016) 064901. arXiv:1602.02813 , doi:10.1103/PhysRevC.93.064901 .[9] N. Magdy, Beam energy dependence of the anisotropic flow coefficients v n , PoS CPOD2017 (2018)005.[10] N. Magdy, Viscous Damping of Anisotropic Flow in 7.7 to 200 GeV Au+Au Collisions, J. Phys.Conf. Ser. 779 (2017) 012060. doi:10.1088/1742-6596/779/1/012060 .[11] B. Schenke, S. Jeon, C. Gale, Anisotropic flow in √ s = 2 .
76 TeV Pb+Pb collisions at the LHC,Phys.Lett. B702 (2011) 59–63. arXiv:1102.0575 , doi:10.1016/j.physletb.2011.06.065 .[12] D. Teaney, L. Yan, Nonlinearities in the harmonic spectrum of heavy ion collisionswith ideal and viscous hydrodynamics, Phys. Rev. C86 (2012) 044908. arXiv:1206.1905 , doi:10.1103/PhysRevC.86.044908 .[13] F. G. Gardim, F. Grassi, M. Luzum, J.-Y. Ollitrault, Anisotropic flow in event-by-event idealhydrodynamic simulations of √ s NN = 200 GeV Au+Au collisions, Phys.Rev.Lett. 109 (2012)202302. arXiv:1203.2882 , doi:10.1103/PhysRevLett.109.202302 .[14] R. A. Lacey, D. Reynolds, A. Taranenko, N. N. Ajitanand, J. M. Alexander, F.-H. Liu, Y. Gu,A. Mwai, Acoustic scaling of anisotropic flow in shape-engineered events: implications for ex-traction of the specific shear viscosity of the quark gluon plasma, J. Phys. G43 (2016) 10LT01. arXiv:1311.1728 , doi:10.1088/0954-3899/43/10/10LT01 .[15] A. M. Poskanzer, S. A. Voloshin, Methods for analyzing anisotropic flow in relativistic nuclear colli-sions, Phys. Rev. C58 (1998) 1671–1678. arXiv:nucl-ex/9805001 , doi:10.1103/PhysRevC.58.1671 .[16] N. Magdy, Beam-energy dependence of the azimuthal anisotropic flow from RHIC. arXiv:1909.09640 .[17] J. Adam, et al., Azimuthal Harmonics in Small and Large Collision Systems at RHIC Top Energies,Phys. Rev. Lett. 122 (2019) 172301. arXiv:1901.08155 , doi:10.1103/PhysRevLett.122.172301 .[18] N. Magdy, Collision system and beam energy dependence of anisotropic flow fluctuations, Nucl.Phys. A982 (2019) 255–258. arXiv:1807.07638 , doi:10.1016/j.nuclphysa.2018.09.027 .[19] L. Adamczyk, et al., Azimuthal anisotropy in Cu+Au collisions at √ s NN = 200 GeV, Phys. Rev.C98 (2018) 014915. arXiv:1712.01332 , doi:10.1103/PhysRevC.98.014915 .[20] L. Adamczyk, et al., Harmonic decomposition of three-particle azimuthal correlations at en-ergies available at the BNL Relativistic Heavy Ion Collider, Phys. Rev. C98 (2018) 034918. arXiv:1701.06496 , doi:10.1103/PhysRevC.98.034918 .[21] B. Alver, G. Roland, Collision geometry fluctuations and triangular flow in heavy-ion colli-sions, Phys. Rev. C81 (2010) 054905, [Erratum: Phys. Rev.C82,039903(2010)]. arXiv:1003.0194 , doi:10.1103/PhysRevC.82.039903,10.1103/PhysRevC.81.054905 .[22] S. Chatrchyan, et al., Measurement of higher-order harmonic azimuthal anisotropy in PbPbcollisions at √ s NN = 2.76 TeV, Phys. Rev. C89 (2014) 044906. arXiv:1310.8651 , doi:10.1103/PhysRevC.89.044906 .[23] J. Adam, et al., Correlation Measurements Between Flow Harmonics in Au+Au Collisions at RHIC,Phys. Lett. B783 (2018) 459–465. arXiv:1803.03876 , doi:10.1016/j.physletb.2018.05.076 .[24] Z. Qiu, U. W. Heinz, Event-by-event shape and flow fluctuations of relativistic heavy-ion collisionfireballs, Phys. Rev. C84 (2011) 024911. arXiv:1104.0650 , doi:10.1103/PhysRevC.84.024911 .[25] A. Adare, et al., Measurements of Higher-Order Flow Harmonics in Au+Au Colli-sions at √ s NN = 200 GeV, Phys. Rev. Lett. 107 (2011) 252301. arXiv:1105.3928 , doi:10.1103/PhysRevLett.107.252301 .[26] G. Aad, et al., Measurement of event-plane correlations in √ s NN = 2 .
76 TeV lead-leadcollisions with the ATLAS detector, Phys. Rev. C90 (2014) 024905. arXiv:1403.0489 , doi:10.1103/PhysRevC.90.024905 .[27] G. Aad, et al., Measurement of the correlation between flow harmonics of different order in lead-lead collisions at √ s NN =2.76 TeV with the ATLAS detector, Phys. Rev. C92 (2015) 034903. arXiv:1504.01289 , doi:10.1103/PhysRevC.92.034903 .
28] B. Alver, et al., Importance of correlations and fluctuations on the initial source eccentric-ity in high-energy nucleus-nucleus collisions, Phys. Rev. C77 (2008) 014906. arXiv:0711.3724 , doi:10.1103/PhysRevC.77.014906 .[29] B. Alver, et al., Non-flow correlations and elliptic flow fluctuations in gold-gold collisions at √ s NN =200 GeV, Phys. Rev. C81 (2010) 034915. arXiv:1002.0534 , doi:10.1103/PhysRevC.81.034915 .[30] J.-Y. Ollitrault, A. M. Poskanzer, S. A. Voloshin, Effect of flow fluctuations and nonflow on ellipticflow methods, Phys. Rev. C80 (2009) 014904. arXiv:0904.2315 , doi:10.1103/PhysRevC.80.014904 .[31] W. Busza, K. Rajagopal, W. van der Schee, Heavy Ion Collisions: The Big Picture,and the Big Questions, Ann. Rev. Nucl. Part. Sci. 68 (2018) 339–376. arXiv:1802.04801 , doi:10.1146/annurev-nucl-101917-020852 .[32] B. H. Alver, C. Gombeaud, M. Luzum, J.-Y. Ollitrault, Triangular flow in hydrodynamics and trans-port theory, Phys. Rev. C82 (2010) 034913. arXiv:1007.5469 , doi:10.1103/PhysRevC.82.034913 .[33] H. Petersen, G.-Y. Qin, S. A. Bass, B. Muller, Triangular flow in event-by-event ideal hydrodynamicsin Au+Au collisions at √ s NN = 200 A GeV, Phys. Rev. C82 (2010) 041901. arXiv:1008.0625 , doi:10.1103/PhysRevC.82.041901 .[34] R. A. Lacey, R. Wei, J. Jia, N. N. Ajitanand, J. M. Alexander, A. Taranenko, Initial eccentricityfluctuations and their relation to higher-order flow harmonics, Phys. Rev. C83 (2011) 044902. arXiv:1009.5230 , doi:10.1103/PhysRevC.83.044902 .[35] D. Teaney, L. Yan, Triangularity and Dipole Asymmetry in Heavy Ion Collisions, Phys. Rev. C83(2011) 064904. arXiv:1010.1876 , doi:10.1103/PhysRevC.83.064904 .[36] R. S. Bhalerao, J.-Y. Ollitrault, S. Pal, Characterizing flow fluctuations with moments, Phys. Lett.B742 (2015) 94–98. arXiv:1411.5160 , doi:10.1016/j.physletb.2015.01.019 .[37] L. Yan, J.-Y. Ollitrault, ν , ν , ν , ν : Nonlinear hydrodynamic response versus LHC data, Phys.Lett. B744 (2015) 82–87. arXiv:1502.02502 , doi:10.1016/j.physletb.2015.03.040 .[38] H. Niemi, G. S. Denicol, H. Holopainen, P. Huovinen, Event-by-event distributions of az-imuthal asymmetries in ultrarelativistic heavy-ion collisions, Phys. Rev. C87 (2013) 054901. arXiv:1212.1008 , doi:10.1103/PhysRevC.87.054901 .[39] F. G. Gardim, J. Noronha-Hostler, M. Luzum, F. Grassi, Effects of viscosity on the mapping ofinitial to final state in heavy ion collisions, Phys. Rev. C91 (2015) 034902. arXiv:1411.2574 , doi:10.1103/PhysRevC.91.034902 .[40] J. Fu, Centrality dependence of mapping the hydrodynamic response to the initial geometry inheavy-ion collisions, Phys. Rev. C92 (2015) 024904. doi:10.1103/PhysRevC.92.024904 .[41] H. Holopainen, H. Niemi, K. J. Eskola, Event-by-event hydrodynamics and ellipticflow from fluctuating initial states, Phys. Rev. C83 (2011) 034901. arXiv:1007.0368 , doi:10.1103/PhysRevC.83.034901 .[42] G.-Y. Qin, H. Petersen, S. A. Bass, B. Muller, Translation of collision geometry fluctuationsinto momentum anisotropies in relativistic heavy-ion collisions, Phys.Rev. C82 (2010) 064903. arXiv:1009.1847 , doi:10.1103/PhysRevC.82.064903 .[43] C. Gale, S. Jeon, B. Schenke, P. Tribedy, R. Venugopalan, Event-by-event anisotropic flow in heavy-ion collisions from combined Yang-Mills and viscous fluid dynamics, Phys. Rev. Lett. 110 (2013)012302. arXiv:1209.6330 , doi:10.1103/PhysRevLett.110.012302 .[44] P. Liu, R. A. Lacey, Acoustic scaling of linear and mode-coupled anisotropic flow; implica-tions for precision extraction of the specific shear viscosity, Phys. Rev. C 98 (2018) 021902. arXiv:1802.06595 , doi:10.1103/PhysRevC.98.021902 .[45] U. Heinz, R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Ann. Rev.Nucl. Part. Sci. 63 (2013) 123–151. arXiv:1301.2826 , doi:10.1146/annurev-nucl-102212-170540 .[46] F. G. Gardim, F. Grassi, M. Luzum, J.-Y. Ollitrault, Mapping the hydrodynamic response tothe initial geometry in heavy-ion collisions, Phys. Rev. C 85 (2012) 024908. arXiv:1111.6538 , doi:10.1103/PhysRevC.85.024908 .[47] A. Bilandzic, C. H. Christensen, K. Gulbrandsen, A. Hansen, Y. Zhou, Generic framework foranisotropic flow analyses with multiparticle azimuthal correlations, Phys. Rev. C89 (2014) 064904. arXiv:1312.3572 , doi:10.1103/PhysRevC.89.064904 .[48] J. Adam, et al., Correlated event-by-event fluctuations of flow harmonics in Pb-Pb col-lisions at √ s NN = 2 .
76 TeV, Phys. Rev. Lett. 117 (2016) 182301. arXiv:1604.07663 , doi:10.1103/PhysRevLett.117.182301 .[49] Y. Zhou, Review of anisotropic flow correlations in ultrarelativistic heavy-ion collisions, Adv. HighEnergy Phys. 2016 (2016) 9365637. arXiv:1607.05613 , doi:10.1155/2016/9365637 .[50] Z. Qiu, U. Heinz, Hydrodynamic event-plane correlations in Pb+Pb collisions at √ s = 2 . arXiv:1208.1200 , doi:10.1016/j.physletb.2012.09.030 .
51] D. Teaney, L. Yan, Event-plane correlations and hydrodynamic simulations of heavy ion collisions,Phys. Rev. C90 (2014) 024902. arXiv:1312.3689 , doi:10.1103/PhysRevC.90.024902 .[52] H. Niemi, K. J. Eskola, R. Paatelainen, Event-by-event fluctuations in a perturbative QCD + satura-tion + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavy-ioncollisions, Phys. Rev. C93 (2016) 024907. arXiv:1505.02677 , doi:10.1103/PhysRevC.93.024907 .[53] Y. Zhou, K. Xiao, Z. Feng, F. Liu, R. Snellings, Anisotropic distributions in a multiphase transportmodel, Phys. Rev. C93 (2016) 034909. arXiv:1508.03306 , doi:10.1103/PhysRevC.93.034909 .[54] E. G. Judd, et al., The evolution of the STAR Trigger System, Nucl. Instrum. Meth. A902 (2018)228–237. doi:10.1016/j.nima.2018.03.070 .[55] M. Anderson, et al., The Star time projection chamber: A Unique tool for studying high mul-tiplicity events at RHIC, Nucl. Instrum. Meth. A499 (2003) 659–678. arXiv:nucl-ex/0301015 , doi:10.1016/S0168-9002(02)01964-2 .[56] B. Alver, M. Baker, C. Loizides, P. Steinberg, The PHOBOS Glauber Monte Carlo. arXiv:0805.4411 .[57] A. Bilandzic, R. Snellings, S. Voloshin, Flow analysis with cumulants: Direct calculations, Phys.Rev. C83 (2011) 044913. arXiv:1010.0233 , doi:10.1103/PhysRevC.83.044913 .[58] K. Gajdoˇsov´a, Investigations of anisotropic collectivity using multi-particle correlations in pp, p–Pband Pb–Pb collisions, Nucl. Phys. A967 (2017) 437–440. doi:10.1016/j.nuclphysa.2017.04.033 .[59] J. Jia, M. Zhou, A. Trzupek, Revealing long-range multiparticle collectivity in small col-lision systems via subevent cumulants, Phys. Rev. C96 (2017) 034906. arXiv:1701.03830 , doi:10.1103/PhysRevC.96.034906 .[60] R. S. Bhalerao, J.-Y. Ollitrault, S. Pal, Event-plane correlators, Phys. Rev. C88 (2013) 024909. arXiv:1307.0980 , doi:10.1103/PhysRevC.88.024909 .[61] N. Magdy, O. Evdokimov, R. A. Lacey, A method to test the coupling strength of the lin-ear and nonlinear contributions to higher-order flow harmonics via Event Shape Engineering. arXiv:2002.04583 .[62] S. Acharya, et al., Linear and non-linear flow modes in Pb-Pb collisions at √ s NN = 2.76 TeV, Phys.Lett. B773 (2017) 68–80. arXiv:1705.04377 , doi:10.1016/j.physletb.2017.07.060 .[63] P. Alba, V. Mantovani Sarti, J. Noronha, J. Noronha-Hostler, P. Parotto, I. Portillo Vazquez,C. Ratti, Effect of the QCD equation of state and strange hadronic resonances on multipar-ticle correlations in heavy ion collisions, Phys. Rev. C98 (2018) 034909. arXiv:1711.05207 , doi:10.1103/PhysRevC.98.034909 .[64] B. Schenke, C. Shen, P. Tribedy, Multiparticle and charge-dependent azimuthal correlationsin heavy-ion collisions at the Relativistic Heavy-Ion Collider, Phys. Rev. C99 (2019) 044908. arXiv:1901.04378 , doi:10.1103/PhysRevC.99.044908 ..