1. 1 Michael Daugherity for the STAR Collaboration Graduate Student - University of Texas Angular Correlations in STAR Fluctuations and Correlations Workshop Firenze, July 2006

2. Daugherity – Fluctuations and Correlations 2Outline Relating fluctuations and correlations Making a correlation measure from scratch Angular correlations in STAR Charge-dependent angular correlations

3. Daugherity – Fluctuations and Correlations 3 Event-by-Event Fluctuations Au-Au 130 GeV PRC 71 064906 It all started by looking at event-wise mean p t looking for anomalous events similar to z-score in statistics, counts number of σ’s away from mean Data mixed event reference 14% increase Distribution is smooth, contrary to some phase-transition model predictions... …but it’s broader than expected. A measurement of non-statistical fluctuations But what causes fluctuations? How do we quantify and interpret the result? it turns out that measuring p t fluctuations is fairly difficult…

4. Daugherity – Fluctuations and Correlations 4 Fluctuation Measures A large number of multiplicity, net charge, and transverse momentum fluctuation measures have been used at SPS at RHIC: ν +-,dyn, ν(Q), Φ q, D, Δσ 2 nch, Δσ 2 q, Φ pt, Σ pt, F pt, σ 2 pt,dyn, Δσ pt:n, etc. Not much agreement on how to quantify fluctuations, but the essential common feature is an integral of a covariance Cov = - = mean of products - product of means = object - reference Now we can take Pearson’s Correlation Coefficient: the gold-standard correlation measure for the last 100 years. We can understand fluctuations by measuring 2-particle correlations Easier to interpret and relate to physical processes Must use all pairs equally, no high-p t trigger requirement Zero covariance means =, thus is our uncorrelated reference

5. Daugherity – Fluctuations and Correlations 5 J Phys G 31 809-824 fluctuation sum over bins 2D binning function correlation Defined as variance - reference Written as covariance between bins a and b Integral of correlation hep-ph/0506173 PRC 71, 064906 at full STAR acceptance Scale (bin size) dependence Correlation invert integrate Fluctuation measure More on fluctuations and inversion this afternoon The Big Picture A formal relationship between fluctuation, covariance, and correlation: STAR Preliminary

6. Daugherity – Fluctuations and Correlations 6 Correlation Measures Number Correlations Or, defining Δρ as a histogram, bin (a,b) can be written as: We calculate this as a function of (η Δ = η 1 –  η 2, Φ Δ =  1 –  2 ), separation in pseudorapidity and azimuth (axial momentum space) Normalize is a per-particle measure 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 ) This measure comes from a direct application of the standard correlation function, and all we have to do is count pairs Covariance Δρ = object - reference ε = bin width, converts density to bin counts

7. Daugherity – Fluctuations and Correlations 7 Correlation Measures What is ? reference: acceptance and efficiency corrected, ~ flat from azimuthal symmetry and longitudinal expansion, provides per particle normalization ratio: explicitly cancels out acceptance and some sys error ρ sib dominated by η Δ acceptance + permil corr signal The terminology: correlation – measured as a function of variable x for each particle, e.g. (x 1,x 2 ) autocorrelation – transformed to relative variable x Δ = x 1 – x 2 by averaging along x Σ = x 1 + x 2, requires stationarity along x Σ joint autocorrelation – autocorrelation as function of two different relative variables, e.g. (x Δ,y Δ ) The joint autocorrelation (η Δ = η 1 –  η 2, Φ Δ =  1 –  2 ) compactly represents the entire axial space ηΔηΔ ΦΔΦΔ

8. Daugherity – Fluctuations and Correlations 8 Correlation Analysis A quick recap before moving on Fluctuation measures all depend in some way on covariance (correlations) of particles, but no agreement on normalization and other factors Relating fluctuation to correlations places the results in a larger context Correlations can be defined with straightforward statistics, and have a direct physics interpretation By looking at all possible pairs we measure correlations that are minimum-bias, model-independent, and require no high-p t trigger Next up, two examples of correlation analysis Proton-Proton the essential reference before tackling Au-Au well known and described physics in terms of soft transverse strings and semi- hard scattering Hijing what changes from p-p to Au-Au, and what changes with centrality? does quenching describe the data well?

9. Daugherity – Fluctuations and Correlations 9Proton-Proton We expect to see STRINGS (soft, Lund-model) and MINIJETS (semi- hard, back-to-back scattering ) “away-side” ridge MINIJET “same-side” jet cone STRING 1D Gaussian proton-proton 200 GeV axial minimum-bias; i.e. no high-pt trigger STAR Preliminary y t1 y t2 We can even separate them Spectrum on transverse rapidity using two-component model Correlation on transverse rapidity y t ~ ln p t p t ~ 0.5 p t ~ 1.0 p t ~ 2.0 soft hard STAR Preliminary

10. Daugherity – Fluctuations and Correlations 10Proton-Proton STRINGS MINIJETS hep-ph/0506172 y t1 y t2 This is a minimum-bias jet, no trigger particle required we can see jets down to 0.5 GeV same-side – small opening angle away-side – Φ Δ ~ π HBT string fragments – 1D Gaussian on η Δ STAR Preliminary

11. Daugherity – Fluctuations and Correlations 11 H IJING central – quench on peripheral (70-80%) http://www.rhip.utexas.edu/~daugherity/analysis/hijing/index.html mid (40-50%)central (0-5%) Quench Off We can do the same soft/hard cuts and see the same string and minijet components as in p-p Hijing predicts very little change with centrality, soft component a bit smaller in central, but no major modifications The jet quenching does reduce the hard component, but again no modifications to correlation structures proton-proton Au-Au 200 GeV

12. Daugherity – Fluctuations and Correlations 12 Features: peak at small relative angles cos(   ) - momentum conservation at low p t cos(2   ) - elliptic anisotropy ~300k events 0.15 < p t <2 GeV/c |  |<1.3, full  merging & HBT cuts applied Au-Au 130 GeV 40-70%0-5%17-40%5-17% PRC, in press (nucl-ex/0411003) p-p 200 GeV ? Now remove the (η Δ -independent) sinusoids to isolate the small-angle peak

13. Daugherity – Fluctuations and Correlations 13 Au-Au 130 GeV 40-70%0-5%17-40%5-17% sinusoids removed 130 GeV Au-Au mid-central p-p elongation along η Δ narrowing along Φ Δ Widths σηση σΦσΦ

14. Daugherity – Fluctuations and Correlations 14 proton-proton Correlation structure evolves smoothly from p-p to central Au-Au We see strings disappearing and minimum-bias jets being modified ηΔηΔ ΦΔΦΔ ηΔηΔ ΦΔΦΔ 90-100% 30-40% 80-90%70-80%60-70%50-60% 20-30%10-20%5-10%0-5% Au-Au 62 GeV STAR Preliminary

15. Daugherity – Fluctuations and Correlations 15 Au-Au 200 GeV ηΔηΔ ΦΔΦΔ ηΔηΔ ΦΔΦΔ Similar to 62 GeV, but strings damp out more quickly, and broadening along η Δ is more dramatic 90-100% 30-40% 80-90%70-80%60-70%50-60% 20-30%10-20%5-10%0-5% STAR Preliminary

16. Daugherity – Fluctuations and Correlations 16 Possible interpretation… Soft, away-side recoil, cos(   ) Au minijet Interaction with longitudinally expanding medium carries radiated gluons and hadron fragments along pseudorapidity Gluon bremsstrahlung/ medium dragging calculations: (Armesto, Salgado, Wiedemann, hep-ph/0405301) 100 GeV jet Fragmentation asymmetry reverses from p-p to Au-Au z    Hubble expansion p-p HI dramatic evolution with centrality 130 GeV Au-Au mid-central p-p

17. Daugherity – Fluctuations and Correlations 17 Axial Correlations Recap The dominant feature is a jet-like correlation that broadens with centrality –consistent with coupling to longitudinally expanding medium Minimum-bias correlations reveal dynamics of low-Q 2 partons – new access to non-perturbative interactions These correlations have significant energy and centrality dependence This rich structure drives observed multiplicity fluctuations –Measuring the correlations directly gives new insight into the physics behind the fluctuations Up Next: measuring charge-dependent correlations

18. Daugherity – Fluctuations and Correlations 18 We can access additional dynamics by considering the relative charge of particle pairs: Like Sign (LS = ++ and --) pairs include quantum interference correlations and boson enhancement from identical particles Unlike Sign (US = +- or -+) pairs are produced nearby from quark-antiquark pairs and resonance decays We expect to see a short-range enhancement of US pairs. Charge-ordering In string fragmentation models, the charge-ordered particles are also ordered in η:     02-2 η + - + - + - + - Charge-Dependent Correlations from PLB 407 174: “Observation of Charge-Ordering in Particle Production in Hadronic Z 0 Decay”

19. Daugherity – Fluctuations and Correlations 19 CD References = Gaussian on η Δ No structure on Φ Δ p-p shows charge-ordering signal as Gaussian on η Δ with no structure on Φ Δ Hijing also shows charge-ordering along η Δ and no change with centrality Proton-Proton HIJING peripheralmidcentral STAR Preliminary US LS CI CD

20. Daugherity – Fluctuations and Correlations 20 STAR 130 GeV Charge-Dependent PLB 634 347 most peripheralcentral Same plots viewed from above… The 130 GeV data show changes in structure with centrality, need finer centrality bins to see more…

21. Daugherity – Fluctuations and Correlations 21 proton-proton ηΔηΔ ΦΔΦΔ ηΔηΔ ΦΔΦΔ 90-100% 30-40% 80-90%70-80%60-70%50-60% 20-30%10-20%5-10%0-5% Au-Au 62 GeV Good agreement between p-p and peripheral bin Smooth evolution to symmetric exponential signal STAR Preliminary

22. Daugherity – Fluctuations and Correlations 22 ηΔηΔ ΦΔΦΔ ηΔηΔ ΦΔΦΔ 90-100% 30-40% 80-90%70-80%60-70%50-60% 20-30%10-20%5-10%0-5% Au-Au 200 GeV Similar to 62 GeV results 1-D Gaussian on η Δ disappears more quickly STAR Preliminary

23. Daugherity – Fluctuations and Correlations 23 proton-protoncentral Au-Au peripheral Au-Aumid Au-Au Smooth evolution all the way from proton-proton to central Au-Au Evidence for charge-ordering moving from one-dimensional string to a surface -The 1-D signal becomes symmetric on η Δ and Φ Δ in central Au-Au - Inconsistent with resonance gas or string fragments Evidence for attenuation through an opaque medium The change from Gaussian to exponential implies pair loss increasing with opening angle, consistent with attenuation through a medium The largest correlation amplitude observed at RHIC Charge-Dependent Summary

24. Daugherity – Fluctuations and Correlations 24 Summary: Angular Correlations net-charge correlations minijet correlations Au-Au 200 GeV peripheralcentral minijet ‘ string ’ p t < 0.5 GeV p t > 0.5 GeV peripheralcentral elongation 1D2D p-p 200 GeV no p t cut LS - US charge-ordering

25. Daugherity – Fluctuations and Correlations 25Conclusions Fluctuations and correlations provide different manifestations of underlying dynamics; correlations are more readily interpreted. Correlations show that multiplicity and fluctuations at RHIC are driven by minijets, while net-charge fluctuations are related to charge-ordering String fragmentation and minimum-bias jet correlations smoothly and dramatically evolve from p-p to central Au-Au. Our observations are consistent with the following interpretation: –semi-hard processes measured in p-p are embedded in an increasingly dense and thick longitudinally expanding medium in Au-Au. –hadronization via longitudinal strings in p-p becomes insignificant in Au- Au where the bulk medium hadronizes isotropically along the axial surface.

26. Daugherity – Fluctuations and Correlations 26 The Big Picture We have developed a general and powerful method for measuring two- particle correlations These number correlations were found by counting pairs, but covariance derivation allows for easy extension to any arbitrary function …so we can directly measure the correlations relating to any non- statistical fluctuation Results are model independent and minimum-bias, includes important measurements of low-Q 2 dynamics –other correlation measurements done at RHIC require jet hypothesis and trigger bias or are limited in phase-space The Bottom Line: A lot of work has been invested on integral measures of fluctuations, but differential measures of correlations show dramatic novel behavior and access new physics