Investigation of the linear and mode-coupled flow harmonics in Au + Au collisions at s " " = 200 GeV

Abstract Flow harmonics ( v n ) of the Fourier expansion for the azimuthal distributions of hadrons are commonly employed to quantify the azimuthal anisotropy of particle production relative to the collision symmetry planes. While lower order Fourier coefficients ( v 2 and v 3 ) are more directly related to the corresponding eccentricities of the initial state, the higher-order flow harmonics ( v n > 3 ) can be induced by a mode-coupled response to the lower-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 can be used to more precisely constrain the initial conditions and the transport properties of the medium in theoretical models. The multiparticle azimuthal cumulant method is used to measure the linear and mode-coupled contributions in the higher-order anisotropic flow, the mode-coupled response coefficients, and the correlations of the event plane angles for charged particles as functions of centrality and transverse momentum in Au+Au collisions at nucleon-nucleon center-of-mass energy s N N = 200 GeV. The results are compared to similar LHC measurements as well as to several viscous hydrodynamic calculations with varying initial conditions.


Introduction
The Solenoidal Tracker At RHIC

Ø Time Projection Chamber
Tracking of charged particles with: ü Full azimuthal coverage ü | | < 1 coverage 3 The two-and three-particle correlations: The four-particle correlations: Analysis Method The two-and three-particle correlations: The four-particle correlations: Analysis Method The two-and three-particle correlations: The four-particle correlations: hand the used four-particle correlations, ⟨v 4 2 ⟩ and ⟨v 2 2 v 2 3 ⟩ could be extracted from ween the four-particle cumulants constructed using the standard cumulant method eights, (i.e. the quadruplets are selected using the entire detector acceptance, the two-particle cumulants constructed using the two subevent method as: d non-linear modes are independent [28], then by using the Eqs. 1,2,7 the nonhigher order anisotropic flow, v k=4,5 , can be given as: v Non−Linear where n = 2 and/ or 3.

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The approximation is correct if the correlation between lower v n (  (2017) The two-and three-particle correlations: The four-particle correlations: hand the used four-particle correlations, ⟨v 4 2 ⟩ and ⟨v 2 2 v 2 3 ⟩ could be extracted from ween the four-particle cumulants constructed using the standard cumulant method eights, (i.e. the quadruplets are selected using the entire detector acceptance, the two-particle cumulants constructed using the two subevent method as: d non-linear modes are independent [28], then by using the Eqs. 1,2,7 the nonhigher order anisotropic flow, v k=4,5 , can be given as: v Non−Linear = ⟨v k v 2 v n cos(kΨ k − nΨ n − 2Ψ 2 )⟩ ⟨v 2 2 v 2 n ⟩ , ∼ ⟨v k cos(kΨ k − nΨ n − 2Ψ 2 )⟩, 3 where n = 2 and/ or 3.

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The approximation is correct if the correlation between lower v n ( Non−Linear k to v Inclusive k , expressed as ρ k,2n , can be calculated as: 12) measure the correlations between different order flow symmetry planes. ar mode coefficients χ k,2n in Eqs. (7 and 8) which quantify the contributions of de to the the higher order anisotropic flow harmonics, v k=4,5 , are defined as: where n = 2 and/ or 3.

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The approximation is correct if the correlation between lower 94 flow coefficients is weak. Then we can define the linear terms 95 flow, v k=4,5 , as: The ratio of v Non−Linear k to v Inclusive k , expressed as ρ k,2n , can be c The ρ k,2n Eq.(12) measure the correlations between differen 100 Also the non-linear mode coefficients χ k,2n in Eqs. (7 and (2017) Three-particle correlations, C .,,, and C /,,-for Au+Au collisions at "" = 200 GeV using the AMPT model Ø Two-subevents reduce the short-range non-flow effects in the three-particle correlation measurements The short-range non-flow contributions in the three-particle correlations
Ø Two-subevents reduce the short-range non-flow effects in the three-particle correlation measurements

STAR Preliminary STAR Preliminary
.,,, = cos(4 + C − 2 , E − 2 -E ) Three-particle correlations, C .,,, and C /,,-for Au+Au collisions at All of the introduced observables are based on the standard and/ or subevent two-and multiarticle cumulant technique introduced in Ref. [41]. In equation 13 and for the differential χ 5,23 ese analyses further make the approximation ⟨v 2 2 v 2 3 ⟩ ∼ v 2 v 3 , which is valid if the magnitudes f v 2 and v 3 are uncorrelated.

. Results and discussion
In A + A collisions, the short-range non-flow correlations have a significant contribution to e measured three-particle correlations C n+m,n,m . Such an effect could be reduced by using the ubevent cumulants method [41]. Figure. 1 compares the C 2+n,2,n (n = 2 and 3) values obtained from the standard and twoubevent cumulants methods for Au+Au collisions at √ s NN = 200 GeV. The measured threearticle correlations show larger values for the results extracted via the standard cumulants ethod which confirm the expectation that the standard method has more short-range non-flow ontributions. The three-particle correlations are also compared with different hydrodynamic imulations [43,44] summarized in Tab. 1. However, both models agree well with the measured n {2} they need additional constraints to describe the C 2+n,2,n (n = 2 and 3).  Ø However both models fit the 6 they need additional constrains in order to describe the 3-particle correlations .,,, = cos(4 + C − 2 , The p n -differential dependence of the inclusive, linear and non-linear . The p n -differential dependence of the non-linear mode-coupling coefficients, .,,, and the EP angular correlations .,,, for Au+Au collisions at "" = 200 GeV are shown Ø .,,, shows a weak p n dependence Ø .,,, shows a weak p n dependence ü Dynamical final-state effects are significantly less than the initial-state effect?
The .,,, shows a weak sensitivety to q , selections

Conclusion
The linear and mode-coupled contributions to the higher-order anisotropic flow coefficients . and / , have been studied using two-and multi-particle correlations in Au+Au collisions at "" = 200 GeV 15

Conclusion
The linear and mode-coupled contributions to the higher-order anisotropic flow coefficients . and / , have been studied using two-and multi-particle correlations in Au+Au collisions at "" = 200 GeV v Two-subevents reduce the short-range non-flow effect on in the three-particle correlations v The linear 6 (n=4,5) terms dominate in central collisions v The †,0 ‡ show a weak centrality dependence (weak viscous effects) v The .,,, and .,,, show a weak p n dependence