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Event-by-Event Fluctuations in Heavy Ion Collisions

Event-by-Event Fluctuations in Heavy Ion Collisions. M. J. Tannenbaum Brookhaven National Laboratory Upton, NY 11973 USA. 2nd International Workshop on the Critical Point and Onset of Deconfinement Bergen, Norway April 1, 2005.

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Event-by-Event Fluctuations in Heavy Ion Collisions

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  1. Event-by-Event Fluctuations in Heavy Ion Collisions M. J. Tannenbaum Brookhaven National Laboratory Upton, NY 11973 USA 2nd International Workshop on the Critical Point and Onset of Deconfinement Bergen, Norway April 1, 2005 Critical Point-Onset April 2005

  2. or I don’t know much about Statistical Mechanicsbut I’m really good at Statistics! M. J. Tannenbaum Brookhaven National Laboratory Upton, NY 11973 USA 2nd International Workshop on the Critical Point and Onset of Deconfinement Bergen, Norway April 1, 2005 Critical Point-Onset April 2005

  3. A Quick Course in Statistics • A Statistic is a quantity computed from a sample (which is drawn at random from a population). A statistic is any function of the observed sample values. • In physics we also call a population a probability density function, typically f(x) • Two of the most popular statistics are the sum and the average where xi are the results of n repeated independent trials from the same population. • Another popular statistic is the sample variance Critical Point-Onset April 2005

  4. note that average, , is a property of the sample • mean, , is a property of the population A Quick Course in Probability - I • It is important to distinguish probability--which refers to properties or functions of the population, from statistics--which refers to properties or functions of the sample, although this distinction is often blurred (but not by statisticians). • The probability density functions f(x) must be normalized so that the total probability for all possible outcomes is 1. • The most popular probability computation is the expectation value or the mean: Critical Point-Onset April 2005

  5. note the difference • where is the variance of the population Probability--II • The mean or expectation value of a Statistic is often discussed: • Of note is the biased expectation value of the sample variance: is the standard deviation • and the mean of the population is • and the variance of the average is Critical Point-Onset April 2005

  6. Probability-III-sumsconvolutions • From the theory of mathematical statistics, the probability distribution of a random variable S(n) which is itself the sum of n independent random variables with a common distribution function f(x): is given by fn(x), the n-fold convolution of the distribution f(x): The mean, n=<S(n)> and standard deviation, n, of the n-fold convolution obey the familiar rule where  and  are the mean and standard deviation of the distribution f(x). Critical Point-Onset April 2005

  7. Example-ET distributions • ET is an event-by-event variable which is a sum (S(n)) • The sum is over all particles emitted on an event into a fixed but large solid angle (which is different in every experiment) • Measured in hadronic and electromagnetic calorimeters and even as the sum of charged particles i |pTi| • Uses Gamma distribution as the pdf for ET on 1 collision=2 participants • If ETadds independently for n collisions, participants, etc, the pdf is the n-fold convolution of f(x): pnp bb Critical Point-Onset April 2005

  8. NA5 (CERN) (1980) First ET dist. pp UA1 (1982) (C.Rubbia) s=540 GeV. No Jets because ET is like multiplicity (n), composed of many soft particles near <pT> ! CERN-EP-82/122. OOPS UA2 discovers jets 5 orders of magnitude down ET distribution! NA5 300 GeV PLB 112, 173 (1980) 2, -0.88<y<0.67 NO JETS! s=23.7 GeV Fit (by me) is  dist p= 2.39 ± 0.06 Critical Point-Onset April 2005

  9. First RHI data NA35 (NA5 Calorimeter) CERN 16O+Pb sNN=19.4 GeVmidrapidity p+Au is a  dist p=3.36 Upper Edge of O+Pb is 16 convolutions of p+Au. WPNM!! PLB 184, 271 (1987) WPN=Wounded Projectile Nucleon=projectile participant Critical Point-Onset April 2005

  10. E802-O+Au, O+Cumidrapidity at AGS sNN=5.4GeVWPNM works in detail PLB 197, 285 (1987) ZPC 38, 35 (1988) • Maximum energy in O+Cu ~ same as O+Au--Upper edge of O+Au identical to O+Cu d/dE * 6 • Indicates large stopping at AGS 16O projectiles stopped in Cu so that energy emission (mid-rapidity) ceases • Full O+Cu and O+Au spectra described in detail by WPNM based on measured p+Au Critical Point-Onset April 2005

  11. E802-AGSMidrapidity stopping!pBe & pAu have same shape at midrapidity over a wide range of  PRC 63, 064602 (2001) • confirms previous measurement PRC 45, 2933 (1992) that pion distribution from second collision shifts by > 0.8 units in y, out of aperture. Explains WPNM. Critical Point-Onset April 2005

  12. participants spectators 10-15% 5-10% 0-5% Collision Centrality MeasurementZeroDegreeCalorimeter PHENIXat RHIC Au+Au-ZDC is biased WA80 O+Au CERN Critical Point-Onset April 2005

  13. Extreme-Independent or Wounded Nucleon Models • Number of Spectators (i.e. non-participants) Ns can be measured directly in Zero Degree Calorimeters (more complicated in Colliders) • Enables unambiguous measurement of (projectile) participants = Ap -Ns • For symmetric A+A collision Npart=2 Nprojpart • Uncertainty principle and time dilation prevent cascading of produced particles in relativistic collisions  h/mπc > 10fm even at AGS energies: particle production takes place outside the Nucleus in a p+A reaction. • Thus, Extreme-Independent models separate the nuclear geometry from the dynamics of particle production. The Nuclear Geometry is represented as the relative probability per B+A interaction wn for a given number of total participants (WNM), projectile participants (WPNM), wounded projectile quarks (AQM), or other fundamental element of particle production. • The dynamics of the elementary underlying process is taken from the data: e.g. the measured ET distribution for a p-p collision represents, 2 participants, 1 n-n collision, 1 wounded projectile nucleon, a predictable convolution of quark-nucleon collisions. Critical Point-Onset April 2005

  14. RA= <n>pA/ <n>pp= (1+<v>) / 2 <Npart>pA <Npart>pp WA80 proof of Wounded Nucleon Model at 60, 200 A GeV using ZDC Original Discovery by W. Busza, et al at FNAL <n>pA vs <> (Ncoll) PRD 22, 13 (1980) PRC 44, 2736 (1991) = <Npart> Critical Point-Onset April 2005

  15. ISR-BCMOR-pp,dd, sNN=31GeV WNM FAILS! WNM, AQM T.Ochiai, ZPC35,209(86) PLB168, 158 (86) Note WNM edge is parallel to p-p data! Critical Point-Onset April 2005

  16. But-Gamma Dist. fits uncover Scaling in the mean over10 decades?? p-p p=2.50±0.06 - p=2.48±0.05 Is it Physics or a Fluke? Critical Point-Onset April 2005

  17. Summary of Wounded Nucleon Models • The classical Wounded Nucleon (Npart) Model (WNM) of Bialas, Bleszynski and Czyz (NPB 111, 461 (1976) ) works only at CERN fixed target energies, sNN~20 GeV. • WNM overpredicts at AGS energies sNN~ 5 GeV (WPNM works at mid-rapidity)--this is due to stopping, second collision gives only few particles which are far from mid-rapidity. E802 • WNM underpredicts for sNN ≥ 31 GeV---is it Additive Quark Model? BCMOR • This is the explanation of the ‘famous’ kink, well known as p+A effect since QM87+QM84 Critical Point-Onset April 2005

  18. i.e. The kink is a p+A effect well known since 1987-seen at FNAL,ISR,AGS Critical Point-Onset April 2005

  19. ET systematics beyond the “kink” • In generic terms, dET/d implies a measurement corrected for: • Hadronic response---correct to E-mN for baryons, E+mN for antibaryons and E for all other hadrons. • ET corrected to =2, =1.0, scaling linearly in  x  • For fixed target dET/dy=dET/d • For collider at mid-rapidity dET/dy=1.2 x dET/d • “Central collisions” varies from 2.5%-ile to 0.5%-ile in different experiments--try to correct to average 0-5%-ile (PHENIX definition) Critical Point-Onset April 2005

  20. NA35-->NA49 Pb+Pb sNN=17 GeV PRL 75, 3814 (1995) ET(2.1-3.4)--> dET/d=405 GeV@sNN=17 GeV Critical Point-Onset April 2005

  21. PHENIX preliminary PHENIX and E802 ET compared  = 22.5o = 2 x 22.5o = 3 x 22.5o = 4 x 22.5o = 5 x 22.5o E802 dET/d=128 GeV E877 dET/d=200 GeV@sNN=4.8 GeV PHENIX dET/d~606 GeV@sNN=200 GeV Critical Point-Onset April 2005

  22. Au+Au ET spectra at AGS and RHIC are the same shape!!! Critical Point-Onset April 2005

  23. dET/dy vs sNN for “central collisions” Bj GeV/fm3 • Lines are pp s dependence. Lots of systematic issues but still kinky. • Note that Bj at sNN=20 GeV is the same in O+Au and Pb+Pb Critical Point-Onset April 2005

  24. ET has a dimension.Let’s now consider number distributions which are more typical of statistics Critical Point-Onset April 2005

  25. What you have to remember • The mean and standard deviation of an average of nindependent trials from the same population obey the rules: where  is the mean and x (or ) is the standard deviation of the population x . Critical Point-Onset April 2005

  26. Moments instead of distributions • Sometimes I will discuss the probability distribution functions in detail, e.g. Binomial, Negative Binomial, Gamma Distribution • More often I, as well as most others, will just use the first two moments, the mean and standard deviation (or variance=std2) • It will become important to use combinations of moments which vanish for the case of zero correlation. The second “normalized binomial cumulant” or vanishes for a poisson distribution, with no correlations. • Most people use the normalized variance which is 1 for a poisson. It has its purpose, but not what everybody thinks. Critical Point-Onset April 2005

  27. NBD Poisson Binomial Charged particle number fluctuations - + All “Particle number fluctuations in a canonical ensemble” V.V. Begun et al, PRC70, 034901 (2004) NA49-BariConf-JPConf 5 (2005) 74 Critical Point-Onset April 2005

  28. Binomial Distribution • A Binomial distribution is the result of repeated independent trials, each with the same two possible outcomes: success, with probability p, and failure, with probability q=1-p. The probability for m successes on n trials (m,n 0) is: • The moments are: • Example: distributing a total number of particles N onto a limited acceptance. Note that if p 0 with =np=constant we get a Critical Point-Onset April 2005

  29. Moments: Poisson Distribution • A Poisson distribution is the limit of the Binomial Distribution for a large number of independent trials, n, with small probability of success p such that the expectation value of the number of successes =<m>=npremains constant, i.e. the probability of m counts when you expect . • Example: The Poisson Distribution is intimately linked to the exponential law of Radioactive Decay of Nuclei, the time distribution of nuclear disintegration counts, giving rise to the common usage of the term “statistical fluctuations” to describe the Poisson statistics of such counts. The only assumptions are that the decay probability/time of a nucleus is constant, is the same for all nuclei and is independent of the decay of other nuclei. Critical Point-Onset April 2005

  30. Moments: Negative Binomial Distribution • For statisticians, the Negative Binomial Distribution represents the first departure from statistical independence of rare events, i.e. the presence of correlations. There is a second parameter 1/k, which represents the correlation: NBD  Poisson as k , 1/k0 • The n-th convolution of NBD is an NBD with k  nk,   n such that /k remains constant. Hence constant 2/ vs Npart means multiplicity added by each participant is independent. • Example: Multiplicity Distributions in p+p are Negative Binomial Critical Point-Onset April 2005

  31. UA5--Multiplicity Distributions in (small) intervals ||<c around mid-rapidity are NBD UA5 PLB 160, 193,199 (1985); 167, 476 (1986) Distributions are Negative Binomial, NOT POISSON: implies correlations s=540 GeV Critical Point-Onset April 2005

  32. k vs =2c ands • Distributions are never poisson at any s and  • Something fishy with NA49 p+p result Critical Point-Onset April 2005

  33. NBD in O+Cu central collisions at AGS vs  central collisions defined by zero spectators (ZDC)Correlations due to to B-E don’t vanish PRC 52, 2663 (1995) • No studies yet at RHIC. Also centrality cut not as good at collider Critical Point-Onset April 2005

  34. k() linear with non-zero intercept in p+p and Light Ion reactions. Also see MJT PLB 347, 431(1995) • This killed “intermittency” but dont ask, see E802 PRC52,2663 (1995) Critical Point-Onset April 2005

  35. NBD Poisson Binomial Charged particle number fluctuations - + All “Particle number fluctuations in a canonical ensemble” V.V. Begun et al, PRC70, 034901 (2004) NA49-BariConf-JPConf 5 (2005) 74 • This is the right way to do it but more work is needed! Critical Point-Onset April 2005

  36. But Net-Charge fluctuations are studied Instead • I really dislike net charge fluctuations compared to -,+, all. • Because net-charge Q=N+ - N- is conserved. You have to do some work to make it fluctuate--distribute the net charge on small intervals • But then you just get binomial statistics: • To make matters worse, ok interesting, a theorist who obviously never took a statistics course proposed to study the variable R=n+/n- • However, statisticians NEVER take <1/n->, which is divergent if there is any finite probability, no matter how infinitesimal, that n-=0. This is especially dumb since you have to go to small p (n-=N-p0) to get some flucuations. • See e.g. the work of our chairman for further details. J. Nystrand, E. Stenlund, H. Tydesjo, PRC 68, 034902 (2003) Critical Point-Onset April 2005

  37. The idea of net charge fluctuations as a QGP signature didn’t work • The idea was that fractional charges represent more particles fluctuating than unit charged hadrons so that the normalized variance ~1/n should be smaller. All experiments just see the standard random binomial unit-charged hadron fluctuations, with a small effect due to correlations from resonances, e.g. ++- PHENIX PRL89, 082301(2002) NA49 PRC 70, 064903 (2004) CERES JPhysG30, S1371(2004) Critical Point-Onset April 2005

  38. Event-by-Event Average pT • For events with n charged particles of transverse momentum pTi, MpTis just the sum divided by a constant and so has most of the same properties as ET distributions including being described by the convolutions of a Gamma Distribution. • By its definition <MpT>=<pT> but you must work hard to make sure that your data has this property to <<< 1%. • The random background is usually defined by mixed events. You must ensure that your mixed event sample is produced with exactly the samen distribution as the data events. Also no two tracks from the same event can appear in a mixed event. Critical Point-Onset April 2005

  39. dN/x dx p < 2 p=2 p > 2 x = Inclusive pT spectra are Gamma Distributions Critical Point-Onset April 2005

  40. NA49-First Measurement of MpT distribution NA49 Pb+Pb central measurement PLB 459, 679 (1999) • Points=data; hist=mixed; minimal, if any, difference • Very nice paper, gives all the relevant information Critical Point-Onset April 2005

  41. Statistics at Work--Analytical Formula for MpT for statistically independent Emission It depends on the 4 semi-inclusive parameters: b, p of the pT distribution (Gamma) <n>, 1/k (NBD), which are derived from the quoted means and standard deviations of the semi-inclusive pT and multiplicity distributions. The result is in excellent agreement with the NA49 Pb+Pb central measurement PLB 459, 679 (1999) See M.J.Tannenbaum PLB 498, 29 (2001) Critical Point-Onset April 2005

  42. 0-5 % Centrality It’s not a Gaussian…it’s a Gamma distribution! Black Points = Data Blue curve = Gamma distribution derived from inclusive pT spectra From one of Jeff Mitchell’s talks: “Average pT Fluctuations” PHENIX Critical Point-Onset April 2005

  43. PHENIX MpT vs centrality 200 GeV Au+Au PRL 93, 092301 (04) • compare Data to Mixed events for random. • Must use exactly the same n distribution for data and mixed events and match inclusive <pT> to <MpT> • best fit of real to mixed is statistically unacceptable • deviation expressed as: • FpT= MpTdata/ MpTmixed-1 ~ few % MpT (GeV/c) MpT (GeV/c) Critical Point-Onset April 2005

  44. Large Improvement at sNN= 200 GeV Compared to sNN= 130 GeV results PRL 93, 092301 (2004) • 3 times larger solid angle • better tracking • more statistics sNN=130 GeV PRC 66 024901 (2002) Critical Point-Onset April 2005

  45. Fluctuation is a few percent of MpT : Interesting variation with Npart and pTmax Errors are totally systematic from run-run r.m.s variations n >3 0.2 < pT < 2.0 GeV/c 0.2 GeV/c < pT < pTmax PHENIX nucl-ex/0310005 PRL 93, 092301 (2004) Critical Point-Onset April 2005

  46. Npart and pTmax dependences explained by jet correlations with measured jet suppression Other explanations proposed include percolation of color strings E.G.Ferreiro, et al, PRC69, 034901 (2004) 20-25% centrality Critical Point-Onset April 2005

  47. What e-by-e tells you that you don’t learn from the inclusive average • e-by-e averages separate classes of events with different average properties, for instance 17% of events could be all kaons, and 83% all pions---see C. Roland QM2004, e-by-e K/ consistent with random. • A nice example I like is by R. Korus, et al, PRC 64, 054908 (2004): The temperature T~1/b varies event by event with T and T. Critical Point-Onset April 2005

  48. Assuming all fluctuations are from T/T Very small and relatively constant with sNN CERES tabulation H.Sako, et al, JPG 30, S1371 (04) Where is the critical point? T/T Critical Point-Onset April 2005

  49. What Have We Learned • In central heavy ion collisions, the huge correlations in p-p collisions are washed out. The remaining correlations are: • Jets • Bose-Einstein correlations • These correlations saturate the fluctuation measurements. No other sources of non-random fluctuations are observed. This puts a severe constraint on the critical fluctuations that were expected for a sharp phase transition but is consistent with the present expectation from lattice QCD that the transition is a smooth crossover. Critical Point-Onset April 2005

  50. What e-by-e tells you that you don’t learn from the inclusive average Critical Point-Onset April 2005

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