1. Universal Centrality and Collision Energy Trends for v 2 Measurements From 2D Angular Correlations Dave Kettler for the STAR Collaboration Hot Quarks Estes Park, CO August, 2008

2. Dave KettlerTwo-Particle Correlations2 Agenda Overview of correlation analysis 62 and 200 GeV 2D angular correlations Fit components v 2 trends on centrality and energy

3. Dave KettlerTwo-Particle Correlations3 Au-Au Collisions at STAR Potentially hundreds of tracks Massive amount of data Subtle signal How do we get a human- interpretable signal out while preserving as much information as possible? One Event

4. Dave KettlerTwo-Particle Correlations4 Autocorrelations I One event described by a probability distribution  which is sampled by observed particles On azimuth for single event: Single-particle distribution: Two-particle distribution: Define: Project two-particle distribution to difference variable Loop over all particle pairs, make a histogram of their angular difference: Exact same form as the autocorrelation in conventional signal analysis histogram Step-by-step procedure for large N measure tracks in TPC ‘sample’ probability distribution with observed particles

5. Dave KettlerTwo-Particle Correlations5 Autocorrelations II Properties of the autocorrelation: Translation Invariance: Summations: Standard (event plane) v 2 analysis: Line up single-particle distributions according to reaction plane Reaction plane is estimated from particles in the event. Subject to nonflow effects What does a single event angular distribution look like? Global v 2 structure? Varying substructure (minijets, resonances, momentum cons., etc) varies event by event We measure all structure Average the autocorrelations, not single-particle distributions for better statistics need many events

6. Dave KettlerTwo-Particle Correlations6 Multivariable Correlations LSUS      If the structure is all in the difference variable then you can project without loss of information 2D Angular Autocorrelation Make use of   dependences of different structures p-p 200 GeV Minijets? No trigger Later consider 6D space:

7. Dave KettlerTwo-Particle Correlations7 Δρ as a histogram on bin (a,b): Normalize measures number of correlated pairs per final state particle ρ( p 1,p 2 ) = 2 particle density in momentum space Event 1 Event 2 ρ sibling ( p 1,p 2 ) ρ reference ( p 1,p 2 ) ε = bin width, converts density to bin counts Start with a standard definition in statistics: Correlation Measure Pearson’s Correlation Coefficient

8. Dave KettlerTwo-Particle Correlations8 84-93% 28-38% 74-84% 18-28% 64-74%55-64%46-55% 9-18% 5-9%0-5% proton-proton note: 38-46% not shown We observe the evolution of several correlation structures from peripheral to central Au+Au ηΔηΔ φΔφΔ ηΔηΔ φΔφΔ Analyzed 1.2M minbias 200 GeV Au+Au events; included all tracks with p t > 0.15 GeV/c, |η| < 1, full φ STAR Preliminary 200 GeV Au-Au Data CI=LS+US M. Daugherity

9. Dave KettlerTwo-Particle Correlations9 84-95% 28-37% 75-84% 18-28% 65-75%56-65%46-56% 9-18%5-9%0-5% note: 37-46% not shown Analyzed 13M 62 GeV Au+Au minbias events; included all tracks with p T > 0.15 GeV/c, |η| < 1, full φ 62 GeV Au-Au Data A similar evolution appears but with quantitative differences compared to the 200 GeV data. STAR Preliminary CI=LS+US M. Daugherity

10. Dave KettlerTwo-Particle Correlations10 y t1 y t2 p-p transverse correlations ηΔηΔ φΔφΔ p-p axial correlations semi-hard component ηΔηΔ φΔφΔ soft component ηΔηΔ φΔφΔ Longitudinal Fragmentation: 1D Gaussian on η Δ HBT peak at origin, LS pairs only Minijets: 2D Gaussian at origin plus broad away-side peak: -cos( φ Δ ) Proton-Proton Components STAR Preliminary

11. Dave KettlerTwo-Particle Correlations11 Proton-Proton fit function =+ STAR Preliminary longitudinal fragmentation 1D gaussian HBT, e+e- 2D exponential ηΔηΔ φΔφΔ ηΔηΔ φΔφΔ ηΔηΔ φΔφΔ Au-Au fit function Use proton-proton fit function + cos(2φ Δ ) quadrupole term (~flow). This gives the simplest possible way to describe Au+Au data. Note: from this point on we’ll include entire momentum range instead of using soft/hard cuts ηΔηΔ φΔφΔ dipole quadrupole cos(2φ Δ ) Fit Function (5 easy pieces) Same-side “Minijet” Peak, 2D gaussian Away-side -cos(φ) “soft”“hard”

12. Dave KettlerTwo-Particle Correlations12 Quadrupole Centrality Systematics dashed curves : all have common shape – amplitudes follow linear dependence on 2D autocorrelation model fits primary 2D measurements transform star preliminary

13. Dave KettlerTwo-Particle Correlations13 Quadrupole Energy Systematics A new QCD phenomenon at RHIC? saturation? squeezeout  per-pair Bevalac AGS SPS RHIC low-x glue quadrupole star preliminary nucleon hydro AGS Bevalac SPS RHIC per-particle

14. Dave KettlerTwo-Particle Correlations14 A-A Eccentricity Minbias N-N interactions are not point-like objects acting at a distance The W-S distribution may better describe low-x glue point-like objects acting at a distance N-N minbias: interacting spheres point-like nucleon structure we use the optical Glauber eccentricity Optical Glauber parametrization

15. Dave KettlerTwo-Particle Correlations15 Universal Centrality and Energy Trends is this hydro-inspired format relevant to data? universal trends represent all A-A systems for energies above 12 GeV quadrupole represented by initial conditions (b, s 1/2 ); no medium properties, EoS, viscosity, hydro v 2   does not describe data star preliminary

16. Dave KettlerTwo-Particle Correlations16 Deviations from binary scaling represent new physics unique to heavy ion collisions Binary scaling: Kharzeev and Nardi model 200 GeV 62 GeV small increase before transition constant widths STAR Preliminary Peak AmplitudePeak η WidthPeak φ Width Same-side 2D gaussian – binary scaling Note the absence of a transition point in the quadrupole: v 2 & elliptic flow STAR Preliminary Statistical and fitting errors as shown peripheralcentral Systematic error is 9% of correlation amplitude M. Daugherity L. Ray Gaussian parameters

17. Dave KettlerTwo-Particle Correlations17 Conclusions Simultaneous measurement of quadrupole (~flow) and other structures (~nonflow) Depending on centrality, 20-100% of naïvely measured v 2 (uncorrected v 2 {2}, 1D projections, etc) appears to be due to ‘nonflow’ Minijet peak scales with binary collisions until a transition point, then increases dramatically The quadrupole component has no equivalent transition Accurate quadrupole measurements reveal simple trends on b and s 1/2, no dependence on EoS