Review of Analysis Methods for Correlations and Fluctuations Tom Trainor Firenze July 7, 2006

Trainor2 Agenda Fluctuations on binned spaces Pearson’s normalized covariance Scale-dependent variance differences Inverting fluctuations to correlations Autocorrelations from pair counting Relating n and p t fluctuations to physics Trompe l’oeil and measure design

Trainor3 Summary of Charge Fluctuation Measures Jeff Mitchell summary

Trainor4 Summary of Event-by-event Fluctuation Measures  pT  pT  pT,n F pT   pT,dyn Goal of the observables: State a comparison to the expectation of statistically independent particle emission. Jeff Mitchell summary

Trainor5 those slides are nice catalogs, but… Some similarity relations are violated for relevant conditions (e.g., low multiplicity) Some statistical measures exhibit misleading behaviors and/or strong biases Random variables in denominators lead to undesirable properties

Trainor6 Fluctuations on Binned Spaces x1x1 x2x2 a b a b covariance variance xx xx x a b p t, n multiple events single-particle space two-particle space particle number n or scalar p t sum in bin na-nana-na nb-nbnb-nb bb aa 0 0   mixed-pair reference na-nana-na 0 marginal variances project variances for bins a, b covariance between a and b dependence on bin size or scale bin size = scale so, back to basics…

Trainor7 Pearson’s Normalized Covariance r ab = 1 11 > 00< 0 covariance relative to marginal variances a b correlatedanticorrelated   Karl Pearson, uncorrelated geometric mean of marginal variances normalized covariance – the basic correlation measure e.g., forward-backward correlations 11 22

Trainor8 Variance Difference normalized number covariancenormalized p t covariance Poisson normalized (scaled) variances, covariances: Pearson a = b scale-dependent variance difference omit  2 factor in denominator to facilitate n vs p t comparisons differential measure: CLT

Trainor9 Correlations and Fluctuations running integral a covariance distribution measures correlations the scale integral of a covariance distribution measures fluctuations each bin is a mean of normalized covariances increasing scale angular autocorrelations contain averages of normalized covariances in 4D bins of (         ) variance difference macrobin size microbin separation autocorrelation

Trainor10 Fluctuation Inversion 2D scale integral single point statistical reference data 2D scale inversion scale dependence autocorrelation STAR acceptance Rosetta stone for fluctuation and correlation analysis

Trainor11 Derivation – Short Form x1x1 xx x2x2 xx M 0 0 macrobin average x1,ax1,a xx x2,bx2,b xx M m 0 0 xx k microbin average x1x1 x2x2 xx xx space invariance? average often true kxkx x1x1 x2x2 kxkx a+k a xx xx bin method strip method k autocorrelation methods: covariance distribution variance difference integration pair counting

Trainor12 Inversion Precision: Comparisons Pythia pair counting inversion CICD number autocorrelations autocorrelations from pair counting or by inversion of fluctuation scale dependence Jeff Duncan

Trainor13 ‘direct’ look at autocorrelation, computationally expensive: 200 GeV p-p joint  autocorrelations n ptpt Autocorrelations from Pair Counting equivalent ratios for n and p t k   l   average over a,b on k th, l th diagonals STAR preliminary  √  ref  √  ref (GeV 2 /c 2 ) autocorrelations the hard way

Trainor14 Projecting Two-point Momentum Space (y t1,y t2 ) correlations (         ) correlations  √  ref angular autocorrelations (y t,  ) 1  (y t,  ) 2             not an autocorrelation n - number ptpt with ‘hard’ p t cut away-side same-side unlike-sign pairs rapidity correlations  √  ref  √  ref (GeV 2 /c 2 ) intra-jet inter-jet

Trainor15 Two Correlation Types na-nana-na 0 0   na-nana-na 0 na-nana-na 0 0   nb-nbnb-nb common events exceptional events soft hard rare events relative frequency amplitude correlated and anticorrelated bins mixed reference Au-Au p-p log nb-nbnb-nb mixed reference  √  ref (GeV 2 /c 2 )  √  ref

Trainor16 The Physics Behind n Fluctuations CLT fluctuation scale (bin size) dependence integrates these two-particle correlations p-p Au-Aup-p Au-Au CD (negative) CI (positive) varies with bin size ,  4D hypercube: (a,b,  ) HBT

Trainor17 The Physics Behind p t Fluctuations fluctuation scale (bin size) dependence integrates these two-particle correlations p-p Au-Au p-p Au-Au ptpt n varies with bin size  x CLT

Trainor18 Trompe l’Oeil

Trainor19 v 2 and Elliptic Flow conventional abscissa and ordinate total variance difference multipoles extracted from p t (not number) autocorrelations 1/N part 1/  N part v2v2 quadrupole = mean participant path length p t correlations number correlations same 11 points elliptic flow compared directly to minijet correlations per-pair per-particle /  N part v 2 gives a misleading impression of flow centrality and p t trends

Trainor20 The Balance Function true width variation fixed amplitude autocorrelation balance functionBF rms width acceptance factor acceptance width p-p 200 GeVAu-Au 130 GeV project HI peak volume variation acceptance invert 02  CD angular autocorrelations STAR HBT the BF width measures amplitudes, not correlation lengths rms width yy

Trainor21  pt : the real DaVinci Code  s NN 8-12  CERES STAR running average integral inversion 2.5  SSC: ~ 3 acceptance dependence  pt  s NN  ref direct 2-point correlations A B B A centrality STAR estruct CERES ebye claim 130 GeV 22 C C assumes global thermalization

Trainor22 Summary Pearson’s normalized covariance provides the basis for fluctuation/correlation measure design An integral equation connects fluctuation scale dependence to angular autocorrelations Fluctuations are physically interpretable by means of two-particle correlations Optimal projections of two-particle momentum space to lower-dimensional spaces are defined Some F/C measures can be misleading

Trainor23  2 Dynamical –. fluctuation inversion correlations fluctuations biased measure Hijing 200 GeV Au-Au 65-85% central doesn’t tolerate low multiplicities physical unphysical scale dependence auto- correlations  ref ** (GeV/c) 2  ref (GeV/c) 2 unphysical

Trainor24 Fragmentation a new view of fragmentation p-p 200 GeV e + -e  91 GeV p-p 200 GeV

Trainor25 02    ref    CERES STAR running average  s NN C

Trainor26 Fussing about Fragmentation a new view of fragmentation p-p 200 GeV e + -e  91 GeV p-p 200 GeV