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23rd WWND, Big Sky, MT - Feb. 12-17, 2007 1 Conservation Laws and Femtoscopy of Small Systems Zbigniew Chajęcki and Michael A. Lisa The Ohio State University.

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Presentation on theme: "23rd WWND, Big Sky, MT - Feb. 12-17, 2007 1 Conservation Laws and Femtoscopy of Small Systems Zbigniew Chajęcki and Michael A. Lisa The Ohio State University."— Presentation transcript:

1 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 1 Conservation Laws and Femtoscopy of Small Systems Zbigniew Chajęcki and Michael A. Lisa The Ohio State University [nucl-th/0612080]

2 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 2 Outline Introduction / Motivation –intriguing pp versus AA [reminder] –data features not under control: Energy-momentum conservation? SHD as a diagnostic tool [reminder] Phase-space event generation: GenBod Analytic calculation of Energy and Momentum Conservation Induced Correlations Experimentalists’ recipe: Fitting correlation functions [in progress] Conclusion

3 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 3 Id. pion femtoscopy in p+p @ STAR STAR preliminary m T (GeV) Z. Ch. (for STAR) QM05, NP A774:599-602,2006 For the first time: femtoscopy in p+p and A+A measured in same experiment, same analysis definitions, …. great opportunity to compare physics what causes m T -dependence in p+p? same cause as in A+A?

4 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 4 Ratio of femtoscopic radii  All p T (m T ) dependences of HBT radii observed by STAR scale with pp although it’s expected that different origins drive these dependences Femtoscopic radii scale with pp Scary coincidence or something deeper? pp, dAu, CuCu - STAR preliminary Ratio of (AuAu, CuCu, dAu) HBT radii by pp Z. Ch. (for STAR) QM05, NP A774:599-602,2006

5 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 5 Clear interpretation clouded by data features d+Au: peripheral collisions STAR preliminary Non-femtoscopic q-anisotropic behaviour at large |q| does this structure affect femtoscopic region as well? Q x <0.12 GeV/c

6 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 6 Spherical Harmonic Decomposition of the Correlation Function

7 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 7 Spherical Harmonic Decomposition of CF Q OUT   Q SIDE Q LONG Q Cartesian-space (out-side-long) naturally encodes physics, but is poor/inefficient representation Recognize symmetries of Q-space -- decompose by spherical harmonics! Direct connection to source shapes [Danielewicz,Pratt: nucl-th/0501003] – decomposition of CF on cartesian harmonics ~immune to acceptance full information content at a glance [thanks to symmetries]  : [0,2  ]  : [0,  ] Z.Ch., Gutierrez, Lisa, Lopez-Noriega, nucl-ex/0505009 This new method of analysis represents, in my opinion, a real breakthrough. […] it has a good chance to become a standard tool in all experiments. A. Bialas, ISMD 2005

8 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 8 Decomposition of CF onto Spherical Harmonics Au+Au: central collisions C(Q out ) C(Q side ) C(Q long ) Z.Ch., Gutierrez, Lisa, Lopez-Noriega, [nucl-ex/0505009] Pratt, Danielewicz [nucl-th/0501003] Q x <0.03 GeV/c

9 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 9 Z.Ch., Gutierrez, Lisa, Lopez-Noriega, [nucl-ex/0505009] Pratt, Danielewicz [nucl-th/0501003] Decomposition of CF onto Spherical Harmonics d+Au: peripheral collisions STAR preliminary

10 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 10 Multiplicity dependence of the baseline Baseline problem is increasing with decreasing multiplicity STAR preliminary

11 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 11 GenBod Phase-Space Event Generator

12 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 12 GenBod: Phase-space sampling with energy/momentum conservation F. James, Monte Carlo Phase Space CERN REPORT 68-15 (1 May 1968) Sampling a parent phasespace, conserves energy & momentum explicitly –no other correlations between particles ! Events generated randomly, but each has an Event Weight WT ~ probability of event to occur P  conservation Induces “trivial” correlations (i.e. even for M=1) Energy-momentum conservation in n-body system

13 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 13 Sampling MC Events Low probability (PS weight) High probability (PS weight) 30 particles To treat MC events identical to measured events we have to sample them according to WT (PS weight) Then we can construct CF

14 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 14 CF from GenBod Varying frame and kinematic cuts

15 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 15 N=18, =0.9 GeV, LabCMS Frame - no cuts

16 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 16 N=18, =0.9 GeV, LabCMS Frame - |  | =0.9 GeV, LabCMS Frame - |  |<0.5 The shape of the CF is sensitive to kinematic cuts

17 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 17 N=18, =0.9 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame

18 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 18 N=18, =0.9 GeV, LCMS Frame - |  | =0.9 GeV, LCMS Frame - |  |<0.5 The shape of the CF is sensitive to kinematic cuts frame

19 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 19 GenBod Varying multiplicity and total energy

20 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 20 N=6, =0.5 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame

21 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 21 N=9, =0.5 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame particle multiplicity

22 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 22 N=15, =0.5 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame particle multiplicity

23 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 23 N=18, =0.5 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame particle multiplicity

24 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 24 N=18, =0.7 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame particle multiplicity total energy : √s

25 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 25 N=18, =0.9 GeV, LCMS Frame - no cuts The shape of the CF is sensitive to kinematic cuts frame particle multiplicity total energy : √s The shape of the CF is sensitive to: kinematic cuts frame particle multiplicity total energy : √s

26 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 26 Findings Energy and Momentum Conservation Induced Correlations (EMCICs) “resemble” our data so, EMCICs... on the right track... But what to do with that? –Sensitivity to  s, multiplicity of particles of interest and other particles –will depend on p 1 and p 2 of particles forming pairs in |Q| bins  risky to “correct” data with Genbod... Solution: calculate EMCICs using data!! –Danielewicz et al, PRC38 120 (1988) –Borghini, Dinh, & Ollitraut PRC62 034902 (2000) we generalize their 2D p T considerations to 4-vectors

27 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 27 k-particle distributions w/ phase-space constraints single-particle distribution w/o P.S. restriction k-particle distribution (k<N) with P.S. restriction

28 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 28 Central Limit Theorem k-particle distribution in N-particle system For simplicity we will assume that all particles are identical (e.g. pions) -> they have the same average energy and RMS’s of energy/momentum Then, we can apply CLT (the distribution of averages from any distribution approaches Gaussian with increase of N) Can we assume that E and p are not correlated ?

29 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 29 E - p correlations?

30 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 30 EMCICs in single-particle distribution ? What if all events had the same “parent” distribution f, and all centrality dependence of spectra was due just to loosening of P.S. restrictions as N increased? in this case, the index i is only keeping track of particle type

31 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 31 k-particle correlation function k-particle correlation function

32 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 32 2-particle correlation function 2-particle correlation function Dependence on “parent” distrib f vanishes, except for energy/momentum means and RMS 2-particle correlation function (1 st term in 1/N expansion)

33 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 33 2-particle CF (1st term in 1/N expansion) “The p T term” “The p Z term” “The E term” Names used in the following plots

34 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 34 EMCICs Effect of varying multiplicity & total energy Same plots as before, but now we look at: p T ( ), p z (  ) and E (  ) first-order terms full (  ) versus first-order (  ) calculation simulation ( ) versus first-order (  ) calculation

35 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 35 N=6, =0.5 GeV, LabCMS Frame - no cuts

36 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 36 N=9, =0.5 GeV, LabCMS Frame - no cuts

37 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 37 N=15, =0.5 GeV, LabCMS Frame - no cuts

38 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 38 N=18, =0.5 GeV, LabCMS Frame - no cuts

39 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 39 N=18, =0.7 GeV, LabCMS Frame - no cuts

40 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 40 N=18, =0.9 GeV, LabCMS Frame - no cuts

41 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 41 Findings first-order and full calculations agree well for N>9 –will be important for “experimentalist’s recipe” Non-trivial competition/cooperation between p T, p z, E terms –all three important p T1 p T2 term does affect “out-versus-side” (A 22 ) p z term has finite contribution to A 22 (“out-versus-side”) calculations come close to reproducing simulation for reasonable (N-2) and energy

42 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 42 The Experimentalist’s Recipe Fitting formula: - average of X over # of pairs for each Q-bin

43 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 43 EMCIC’s FIT: N=18, =0.9GeV, LCMS

44 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 44 Fit contours

45 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 45 The Complete Experimentalist’s Recipe or any other parameterization of CF 9 fit parameters - 4 femtoscopic - normalization - 4 EMCICs Fit this …. or image this …

46 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 46 Summary understanding the femtoscopy of small systems –important physics-wise –should not be attempted until data fully under control SHD: “efficient” tool to study 3D structure Restricted P.S. due to energy-momentum conservation –sampled by GenBod event generator –generates EMCICs quantified by A lm ’s –stronger effects for small mult and/or  s Analytic calculation of EMCICs –k-th order CF given by ratio of correction factors –“parent” only relevant in momentum variances –first-order expansion works well for N>9 –non-trivial interaction b/t p T, p z, E conservation effects Physically correct “recipe” to fit/remove EMCICs –4 new parameters, determined @ large |Q| –parameters are “physical” - values may be guessed

47 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 47 Thanks to: Alexy Stavinsky & Konstantin Mikhaylov (Moscow) [suggestion to use Genbod] Jean-Yves Ollitrault (Saclay) & Nicolas Borghini (Bielefeld) [original correlation formula] Adam Kisiel (Warsaw) [don’t forget energy conservation] Ulrich Heinz (Columbus) [validating energy constraint in CLT]

48 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 48 Some properties of A lm coefficients  A lm = 0 for l or m odd – identical particle correlations (for non-id particles, odd l encodes shift information)  A 00 (Q) ≈ one-dimensional “CF(Qinv)” (bump ~ 1/R)  A lm (Q) =  l,0 where correlations vanish  A l≠0,m (Q) ≠ 0  anisotropy in Q space  Im[A lm ] = 0

49 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 49 Long-range correlations : JETS ? Jets as a origin of the baseline problem ?? The idea was to try to eliminate pions coming from jet fragmentation from data sample. It can be done by applying an event cut which accepts only events that have no high-pt tracks (jets). HBT analyses where done for three classes of events  all - all events accepted – as a reference  soft – only events without high-p T tracks ( highest-p T < 1.2 GeV/c was chosen)  hard - only events with least one track with p T > 2 GeV/c

50 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 50 Simple, Gaussian source calculations ~acceptance free R L < R T R L > R T R O < R S R O > R S

51 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 51 NA22 parametrization of CF ?

52 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 52 NA22: 1D projections of 3D CF NA22, Z. Phys. C71 (1996) 405

53 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 53 NA22 parametrization of CF STAR preliminary d+Au peripheral collisions NA22 fit d+Au peripheral collisions

54 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 54 CLT? distribution of N uncorrelated numbers (and then scaled by N, for convenience) Note we are not starting with a very Gaussian distribution!! “pretty Gaussian” for N=4 (but  2 /dof~2.5) “Gaussian” by N=10

55 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 55 Schematic: How GenBod works 1/3

56 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 56 Schematic: How GenBod works 2/3

57 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 57 Schematic: How GenBod works 3/3

58 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 58 Remember relevant quantities are average over the (unmeasured) “parent” distribution, not the physical distribution

59 23rd WWND, Big Sky, MT - Feb. 12-17, 2007 59 Reconstruction of CF from Alm’s

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