1. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Measures of charged particle fluctuations (in high energy heavy-ion collisions) Joakim Nystrand University of Bergen Bergen, Norway Charged particle fluctuation measures. Behaviour of these measures in various non-dynamical scenarios. Simple dynamical (toy) models. Conclusions and comparison with some experimental results.

2. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand I assume that the physics motivation for studying fluctuations in these quantities is well known to this audience, if not, see 1. S. Jeon, V. Koch Phys. Rev. Lett. 85 (2000) 2076. 2. M. Asakawa, U. Heinz, B. Müller Phys. Rev. Lett. 85 (2000) 2072. 3. H. Heiselberg, A.D. Jackson Phys. Rev. C 63 (2001) 064904. 4. M. Gazdzicki, Eur. Phys. J. C 8 (1999) 131. 5. S. Mrowczynski, Phys. Lett. B 459 (1999) 13. … Event-by-event fluctuations of charged particles: Example variable 1. Net-chargeQ = n + – n – 2. StrangenessK/π 3. Baryon number(p+p)/(π + + π – ) 4. Multiplicityn ch = n + + n –

3. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Net charge fluctuations Measured in one event:n + positive and n – negative particles n ch = n + + n –, Q = n + – n –, R = n + /n – V(Q), V(R) – variance of Q and R. - average over events 4 measures of fluctuations: S.Jeon, V.Koch PRL 85 (2000) 2076. S. Mrowczynski, Phys. Rev. C 66 (2002) 024904; a modification of the Φ measure C.Pruneau, S.Gavin, S.Voloshin, Phys. Rev. C 66 (2002) 044904.

4. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Goal of this study  Investigate the behaviour of these measures in a set of scenarios (simplified → semirealistic) First scenario (simplest possible) : Generate N ch particles, + or – with equal prob. and assume that we detect a fraction p a of them. Expectations v(Q) = Γ = 1 v(R) = 4 asymptotically v = 0 ● N ch = 1000 □ N ch = 200

5. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Add charge conservation  Assume first that N + = N – = N ch /2 (these are the global multiplicities) Expectations v(Q), Γ, v(R): → 0 as p a → 1 (no fluctuations if we measure the whole event) In general, prev. res.  (1 – p a ) ν = – 4/N ch (indep. of p a )

6. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Add excess of positive particles (but no charge conservation)  Let N + = p + N ch and N – = p – N ch and introduce ε = p + – p – Expectations v(Q) = 1 – p a ε 2 V(R) = 4 + 16ε + O(ε 2 ) Γ = 1 – ε 2 ν = 0  Major problem when studying fluctuations in a ratio, strong, linear dependence of R on ε (applies to n + /n – and to K/π) ● ε = 0 □ ε = 0.1 ∆ ε = 0.2

7. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Combine charge conservation with excess of positive particles, and add fluctuations in the global multiplicity.  N ch = 1000, N ch = 900-1100, N ch = 800-1200 ; ε = 0.2 Expectations Only v(Q) depends on the variation in global mult. v(Q) = 1 – p a + p a ε 2 v(N ch ) Slight shift in ν with charge conservation and ε  0 ν = –4 / (1- ε 2 )  Dependence of v(Q) on global multiplicity variation problematic.

8. Experimental efficieny and background tracks For a detector that on average finds a fraction p e of the tracks passing through it, the probability to detect exactly n d tracks out of n true is given by a binomial distribution.  In this context, equivalent to reducing the acceptance p a → p a · p e. Introducing a certain fraction, f bg, of uncorrelated background tracks. Keeping charge conservation and with N ch =1000 and ε=0. Reduces the slopes of v(Q), v(R), and Γ by a factor (1-f bg ). Constant reduction for ν; ν = – 4 (1-f bg ) /. ● f bkg = 0 □ f bkg = 0.2 ∆ f bkg = 0.4

9. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Summary non-dynamical scenarios 1. A variation of the exp. acceptance, p a, or efficiency, p e  A reduction in v(Q), v(R), and Γ by a factor (1 – p a · p e )  ν is unaffected by p a and p e. 2. v(R) has a complicated behavior for small acceptances and depends strongly on the asymmetry, or  1. 3. All 4 measures are affected by an uncorrelated background.

10. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Add some dynamics  decay of neutral resonances, for example  0 → π + π –. Generate particles with one phase space variable , which could correspond to azimuthal angle,  = φ/2π. Assume acceptance in other variables (y, p T ) is complete, or at least wide. Let f res be the fraction of particles emanating from resonance decays and Δ  =1/24 be the opening angle of the daughters. Varying f res ● f res = 0□ f res = 0.3 ∆ f res = 0.6

11. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Varying Δ  Conclusions: * Clear deviations from the purely stochastic behavior are seen in all variables when resonance correlations are included. * Deviations are seen at small acceptances, but note that this requires that the coverage in the other variables (y, p T ) is complete (or at least wide). ● Δ  = 1/12□ Δ  = 1/24 ∆ Δ  = 1/72

12. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand A simplified model of the hadronization from a Quark-Gluon Plasma could be done along the same lines: Hadronization only into pions (π + π – π 0 ). Angular distribution between pions from same quark pair distributed with width Δ . Dividing out the (1-p a ) dependence ● Δ  = 1/6□ Δ  = 1/12 ∆ Δ  = 1/36

13. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Relevance of these studies to real data (?) 0-5% 10-20% 30-40% 70-80% Au+Au @ s NN 1/2 = 62 GeV STAR Data (preliminary/work in progress), C. Pruneau 2 nd Wkshp on the Critical Point (Bergen 2005)

14. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand The large-scale variations for certain variables are often dominated by statistical, non-dynamical fluctuations. K. Adcox et al. (PHENIX) Phys. Rev. Lett. 89 (2002) 082301. n ch is used here to select the centrality.

15. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand Conclusions Avoid studying fluctuations in particle ratios (n + /n – ) or (K/π), or at least be extremely careful. The variation of ν with Δφ sensitive probe of dynamical fluctuations. Hard to directly interpret measured ν vs. Δφ but Any reasonable resonance cocktail (  0, f 0, K 0 S, …) with correct momentum distributions (including collective flow) should be able to reproduce the behavior of ν vs. Δφ.

16. Some notes and references 1.The net-charge results are from J. Nystrand, E. Stenlund, H. Tydesjö Phys. Rev. C 68 (2003) 034902. 2.Some more details and derivations of the analytical expectations in H. Tydesjö Lic. and Phd. Theses, available at http://www.hep.lu.se/staff/tydesjo/theses.html 3.Similar studies by J. Zaranek Phys. Rev. C 66 (2002) 044902; S. Mrowczynski, Phys. Rev. C 66 (2002) 024904. Some quotes about statistics 1.He uses statistics as a drunken man uses lamp-posts – for support rather than illumination. (A. Lang) 2.If your experiment needs statistics, you ought to have done a better experiment. (E. Rutherford) 3.Then there is the man who drowned crossing a stream with an average depth of six inches. (W.I.E. Gates).

17. Correlations and Fluctuations in Relativistic Nuclear Collisions, Florence, Italy, July 7-9 2006 Joakim Nystrand