V2 and hydrodynamics ~ Has ideal hydrodynamical limit reached at RHIC ? ~ Hiroshi Masui Lawrence Berkeley National Laboratory ATHIC 2008, Oct. 13 - 15 Tsukuba, Japan Thanks to Y. Bai, Y. Lu, S. Shi, R. Snellings, A. Tang and N. Xu
H. Masui / LBNL

dN φ : azimuthal angle of emitted particles Ψ RP : azimuthal angle of the reaction plane v2 = 〈cos (2[φ — Ψ rp])〉 ● Anisotropic “collective” particle emission in non- central A-A collisions ● Sensitive to early stage ✔ Interactions among particles ➡ Pressure gradient + spatial anisotropy → V2 ✔ v2 saturate within a few fm/c ➡ A probe to explore early time dynamics of heavy ion collisions 22
dN φ : azimuthal angle of emitted particles Ψ RP : azimuthal angle of the reaction plane v2 = 〈cos (2[φ — Ψ rp])〉 ● Anisotropic “collective” particle emission in non- central A-A collisions ● Sensitive to early stage ✔ Interactions among particles ➡ Pressure gradient + spatial anisotropy → V2 ✔ v2 saturate within a few fm/c ➡ A probe to explore early time dynamics of heavy ion collisions 22
● Mass ordering of V2 ✓ Radial flow ● Sizable v₂ for multi- strange hadrons ✓ sensitive to partonic stage ●  Number of Quark (NQ) scaling √  vs (m - mass)/nq ➡ Partonic collectivity ➡ Deconfinement PHENIX:  Phys. Rev. Lett. 98, 162301 (2007) STAR: Phys. Rev. Lett. 99, 112301 (2007), Phys. Rev. C77, 054901 (2008) Hydro results: Pasi Huovinen, private communication; P. Huovinen and P. V. Ruuskanen, Annu. Rev. Nucl. Part. Sci. 56, 133 (2006)
S. Voloshin, QM2006 Hvdro curves are obtained from calculations Kolb. Sollfrank. Heinz. PRC62 (2000) 054909. made at b=7 fm in Pb + Pb s and rescaled by ‘optical’ eccentricity value. ntrality dependence is not fully reflected by these curves, as it is more ‘flat’ at each given collision energy (very roughly indicated by strait lines' ●  V₂/ɛ  scales (1/S)dN/dy ✓ scaling holds well from AGS, SPS to RHIC, among different systems and methods ●  At RHIC, has v₂ reached hydrodynamical limit ?! 4
● Ideal hydrodynamical calculations ✓ for pions in Pb + Pb collisions ● Particle composition could change v² for inclusive charged hadrons ✔ Heavier hadrons have larger v₂ due to radial flow ● Compare v₂ for identified hadrons ● Extract hydrodynamical limit from the data ●♦ A transport model approach, extract hydrodynamical limit from identified hadron v₂ measurements
Forward TPC |η| = 2.5 - 4 ● Different methods to measure v2 ✓ 4-particle cumulant v₂{4}, ZDC-SMD event plane from directed flow v₂{ZDC-SMD} for unidentified charged hadrons ✔ For identified hadrons, standard event plane method at the main or forward TPC were used
● Knudsen number ✓ degrees of equilibration ● Transport model ✓ Reach hydro limit when K → 0 ● Simple formula to describe V2(K) ✓ Reduces hydro limit (K → 0) and low density limit (K >> 1) ✓ Reproduce transport model calculations hydro / Vンjhydro |(ε × K/Ko) 、 V շ ( K  ≫  1 )
σ : partonic cross section Cs : speed of sound S : transverse area Fit unidentified charged hadrons ✓ Extract parton cross section -  Note: one can only extract the product Koσcs ● Fit all identified hadrons simultaneously ✔ σ is fixed from fitting result for unidentified hadron ●  Ko = 0.7 and cs² = 1/3 (fixed) ✓ K o value is determined so as to reproduce the transport model calculation (K o = 0.7 ± 0.03)*   *C. Gombeaud  and J.-Y. Ollitrault, PRC77, 054904 (2008) Only Glauber initial condition is considered
●  V2{4} and  V2{ZDC- SMD} ✓ sensitive to the true reaction plane ✓ scaled by standard eccentricity ● Parton cross section ✓ σ = 4.1 ± 1.2 mb ➡ Extract hydro limit for identified hadrons with Ko = 0.7, Cs² = 1/3 and σ = 4.1 mb
● Transport model fit well describe the data ● “Fitting” hydro limits (arrows) increase as a function of mass ● Ideal hydro v₂/ɛ ✔ ε from “optical” glauber model PHENIX π, K and p: preliminary, nucl-ex/0604011v1 STAR Kºs, Λ, Ξ : Phys. Rev. C77, 054901 (2008) STAR φ : Phys. Rev. Lett. 99, 112301 (2007) Ideal Hydro. : P. Huovinen and P. V. Ruuskanen, Annu. Rev. Nucl. Part. Sci. 56, 163 (2006) and private communication
ār preliminary PHENIX π, K and p: preliminary, nucl-ex/0604011v1 STAR Λ, Ξ : Phys. Rev. C77, 054901 (2008) STAR φ : Phys. Rev. Lett. 99, 112301 (2007) Ideal Hydro. : P. Huovinen and P. V. Ruuskanen, Annu. Rev. Nucl. Part. Sci. 56, 163 (2006) and private communication Ideal hydro : Average over all centrality bins ● “Fitting” hydro limits increase as a function of mass ✓ Mass dependence is s similar to that from ideal hydro ➡ Radial flow at partonic stage ✓ Early decoupling for (multi-)strange hadrons ? 11
ār preliminary PHENIX π, K and p: preliminary, nucl-ex/0604011v1 STAR Λ, Ξ : Phys. Rev. C77, 054901 (2008) STAR φ : Phys. Rev. Lett. 99, 112301 (2007) Ideal Hydro. : P. Huovinen and P. V. Ruuskanen, Annu. Rev. Nucl. Part. Sci. 56, 163 (2006) and private communication Ideal hydro : Average over all centrality bins ● “Fitting” hydro limits increase as a function of mass ✓ Mass dependence is s similar to that from ideal hydro ➡ Radial flow at partonic stage ✓ Early decoupling for (multi-)strange hadrons ? 11
K=1  S$Tar preliminarv ~ 10 % errors on hydro limit from the fit K = 1 → (1/S) dN/dy ~ 17 K = 0.6 → (1/S) dN/dy ~ 30 1/S dN/dy (fm²) ● Universal curve ✓ different particles, methods, systems and experiments → Hydrodynamical limit has not been reached at RHIC within the transport model approach
K=1   Star  preliminarv 1/S dN/dy (fm²) ● Universal curve ✓ different particles, methods, systems and experiments → Hydrodynamical limit has not been reached at RHIC within the transport model approach ~ 10 % errors on hydro limit from the fit K = 1 → (1/S) dN/dy ~ 17 K = 0.6 → (1/S) dN/dy ~ 30
STAR  preliminary ~ 10 % errors on hydro limit from the fit K = 1 → (1/S) dN/dy ~ 17 K = 0.6 → (1/S) dN/dy ~ 30 1/S dN/dy (fm²) ● Universal curve ✓ different particles, methods, systems and experiments → Hydrodynamical limit has not been reached at RHIC within the transport model approach
STAR  preliminary ~ 10 % errors on hydro limit from the fit K = 1 → (1/S) dN/dy ~ 17 K = 0.6 → (1/S) dN/dy ~ 30 1/S dN/dy (fm²) ● Universal curve ✓ different particles, methods, systems and experiments → Hydrodynamical limit has not been reached at RHIC within the transport model approach
K=1   Star  preliminarv ~ 10 % errors on hydro limit from the fit K = 1 → (1/S) dN/dy ~ 17 K = 0.6 → (1/S) dN/dy ~ 30 1/S dN/dy (fm²) ● Universal curve ✓ different particles, methods, systems and experiments → Hydrodynamical limit has not been reached at RHIC within the transport model approach
small Knudsen number large η/s (1/4π) s = 4n for a classical relativistic gas D. Teaney, PRC68, 034913 (2003) Temperatures from PHENIX direct γ measurements, nucl-ex/0804.4168 centrality ● σCs ↔ from fit,   n (density) ↔ centrality ✓ η/ѕ ~ 5/4π in 0 - 40 % centrality ✓ Closer to ideal hydro in more central collisions
● Extract hydrodynamical limits for identified hadrons ✓ Fitting results well describe the centrality dependence of v 2 -   Simultaneous fit by the simple formula motivated from the transport model ✓ Universal curve is obtained -   for different methods, particle species, systems and experiments ✓ “Fitting” hydrodynamical limits increase as a function of mass -   similar mass dependence to that from ideal hydro calculation ✓ Simultaneous fit for identified hadrons gives a constraint on the shear viscosity as a function of centrality -  η/ѕ ~ 5/4π at in 0 - 40 % centrality, close to hydro limit in more central collisions → Ideal hydrodynamical limit has not been reached within the transport model approach (with Glauber initial conditions)

2-particle correlation
• Event plane method : v₂{EP₂}, v₂{BBC}, v²{FTPC}, v²{etaSub} • Two particle cumulant : v2{2} • Scalar product method : v²{uQ} ➡ Sensitive to “Participant Plane” ➡ Scaled by participant eccentricity εpart
Multi-particle correlation
• ZDC-SMD event plane : v₂{ZDC-SMD} • q-distribution method: v2{q} • 4 or higher order cumulant : v₂{n}, n ≥ 4 • Lee-Yang zero method : v²{LYZ} • Bessel-Transform method : v₂{BT} ➡ Sensitive to “Reaction Plane” ➡ Scaled by standard eccentricity εstd or ε{4} − σ²)² + 4σ² - {x2} = Estd ({x} = {y} = {xy} = 0) Epart + {x2} } y² } σ 2 = {x 2 } − {x} 2 ,   σ 2 = {y 2 } − {y} 2 ,   σxy = {xy} − {x}{y} ● Methods based on multi-particle correlation are sensitive to the true reaction plane ✔ should be scaled by standard eccentricity εstd
● In more central collisions ✓ Stronger v²/ε and peak position shift toward higher pτ ✔ Applicable range for hydro extends to higher p т
Yuting Bai, SQM2008 ●  v2{4} and v2{ZDCSMD} ✓ sensitive to reaction plane -   scaled by standard eccentricity ● Knudsen number is constrained by simultaneous fit for charged hadron v₂{4}, V2{ZDCSMD} as well as identified hadrons √ K ~ 0.3 - 1 from central to peripheral collisions
Hydro, Boltzmann calculations:J-Y. Ollitrault v4 å K-1 for small K-1 V2 X K- U4 decreases with K-1 、U2 | Yuting Bai, SQM2008 ●  V4/(V2)² reach a minimum when K → 0 ● Comparison with a transport model ✓ Model calculations approach hydro limit when K → 0 ✓ The data is consistent with the transport model calculation when K > 0.5 - Knudsen number is consistent with that from identified hadron v2 →Hydro limit has not reached at RHIC !
small Knudsen number large 10 20 30 40 50 60 70 80 90 centrality ● σCs ↔ from fit,   n (density) ↔ centrality ✓ Average η/s ~ 0.22 for T = 224 MeV ✓ η/s decrease in more central collisions → stronger flow T = 0.316 TCsN s = 4n for a classical relativistic gas from PHENIX direct γ measurements, nucl-ex/0804.4168