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Conservation Laws in low-multiplicity collisions

Conservation Laws in low-multiplicity collisions. Zbigniew Chaj ę cki and Michael A. Lisa The Ohio State University. Outline. Introduction / Motivation Non-femtoscopic correlations in low-multiplicity collisions : OPAL, NA22, STAR, …

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Conservation Laws in low-multiplicity collisions

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  1. Conservation Lawsin low-multiplicity collisions Zbigniew Chajęcki and Michael A. LisaThe Ohio State University WPCF 2007 - Aug. 1-3, 2007

  2. Outline • Introduction / Motivation • Non-femtoscopic correlations in low-multiplicity collisions : OPAL, NA22, STAR, … * data features not under control: Energy-momentum conservation? • Analytic calculation of Energy and Momentum Conservation Induced Correlations for • single particle spectra • two-particle correlations • Experimentalists’ recipe: Fitting correlation functions • Minv correlation function & background subtraction • V2 • Two-particle correlations • Resonance contribution to non-femtoscopic correlations - (π+,π-) • (π+,π-) correlations in p+p(p) at 200 GeV collisions from PYTHIA • Conclusion WPCF 2007 - Aug. 1-3, 2007

  3. Non-femtoscopic correlations : OPAL 1D projections of 3D CF Qx<0.2 GeV/c Femtoscopic correlations should go to the constant number at large Q(no directional dependence!) OPAL, CERN-PH-EP/2007-025(submitted to Eur. Phys. J. C.) WPCF 2007 - Aug. 1-3, 2007

  4. Non-femtoscopic correlations : NA22 NA22, Z. Phys. C71 (1996) 405 1D projections of 3D CF WPCF 2007 - Aug. 1-3, 2007

  5. Non-femtoscopic correlations : STAR Non-femtoscopic q-anisotropic behaviour at large |q| does this structure affect femtoscopic region as well? d+Au: peripheral collisions Clear interpretation clouded by data features STAR preliminary Qx<0.12 GeV/c STAR, NPA 774 (2006) 599 WPCF 2007 - Aug. 1-3, 2007

  6. Spherical harmonic decomposition QLONG Q  QOUT  QSIDE  : [0,2p]  : [0,p] Z.Ch., Gutierrez, Lisa, Lopez-Noriega, nucl-ex/0505009 WPCF 2007 - Aug. 1-3, 2007

  7. Non-femtoscopic correlations : STAR Baseline problem is increasing with decreasing multiplicity STAR preliminary WPCF 2007 - Aug. 1-3, 2007

  8. Common approaches to „remove” non-femtoscopic correlations • MC simulations • ‘ad-hoc’ parameterizations • OPAL, NA22, … • “zeta-beta” fit by STAR [parameterization of non-femtoscopic correlations in Alm’s] • A possibility: energy-momentum conservation? • must be there somewhere! • but how to calculate / model ?(Upon consideration, non-trivial...) WPCF 2007 - Aug. 1-3, 2007

  9. GenBod Phase-Space Event Generator WPCF 2007 - Aug. 1-3, 2007

  10. GenBod: Phase-space sampling with energy/momentum conservation Energy-momentum conservation in n-body system Events generated randomly, but each has an Event Weight P conservation Induces “trivial” correlations (i.e. even for M=1) WT ~ probability of event to occur • 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 ! WPCF 2007 - Aug. 1-3, 2007

  11. N=9, K=0.5 GeV, LCMS Frame - no cuts • The shape of the CF is sensitive to: • kinematic cuts • frame • particle multiplicity • total energy : √s WPCF 2007 - Aug. 1-3, 2007

  12. 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 p1 and p2 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 pT considerations to 4-vectors WPCF 2007 - Aug. 1-3, 2007

  13. k-particle distributions w/ phase-space constraints k-particle distribution (k<N) with P.S. restriction single-particle distribution w/o P.S. restriction observed P - total 4-momentum WPCF 2007 - Aug. 1-3, 2007

  14. Central Limit Theorem Can we assume that E and p are not correlated ? For simplicity we will assume that all particles are identical (e.g. pions) and that they share the same parent distribution (same RMS of energy/momentum) Then, we can apply CLT (the distribution of averages from any distribution approaches Gaussian with increase of N) k-particle distribution in N-particle system WPCF 2007 - Aug. 1-3, 2007

  15. E - p correlations? WPCF 2007 - Aug. 1-3, 2007

  16. EMCICs in single-particle distribution What if all events had the same “parent” distribution f(p), and all multiplicity (centrality) dependence of spectra was due just to loosening of P.S. restrictions as N increased? WPCF 2007 - Aug. 1-3, 2007

  17. EMCIC’s in spectra For N WPCF 2007 - Aug. 1-3, 2007

  18. EMCICs: Ratio of particle spectra Ratio of pT spectra in p+p@STAR for the lowest and the highest multiplicity events p+p @ 200GeV, STAR pT spectra from GenBod Simulations: Ratio of pT spectra for N=9 and N=18. Phys. Rev. D 74 (2006) 032006 WPCF 2007 - Aug. 1-3, 2007

  19. k-particle correlation function Dependence on “parent” distrib f vanishes, except for energy/momentum means and RMS 2-particle correlation function (1st term in 1/N expansion) 2-particle correlation function WPCF 2007 - Aug. 1-3, 2007

  20. 2-particle CF (1st term in 1/N expansion) “The pT term” “The E term” “The pZ term” Names used in the following plots WPCF 2007 - Aug. 1-3, 2007

  21. EMCICs An example of EMCICs:Effect of varying multiplicity • Same plots as before, but now we look at: • pT (), pz () and E () first-order terms • full () versus first-order () calculation • simulation () versus first-order () calculation WPCF 2007 - Aug. 1-3, 2007

  22. N=9, K=0.9 GeV, LabCMS Frame - no cuts WPCF 2007 - Aug. 1-3, 2007

  23. N=18, K=0.9 GeV, LabCMS Frame - no cuts WPCF 2007 - Aug. 1-3, 2007

  24. Findings • CF from GenBod (as well as EMCICs) depends on • multiplicity • frame • energy of the collisions • first-order and full calculations agree well for N>9 • will be important for “experimentalist’s recipe” • Non-trivial competition/cooperation between pT, pz, E terms • all three important • pT1•pT2 term does affect “out-versus-side” (A22) • pz term has finite contribution to A22 (“out-versus-side”) • calculations come close to reproducing simulation for reasonable (N-2) and energy WPCF 2007 - Aug. 1-3, 2007

  25. N=12,NK=3,Np=3, K=0.9 GeV, LCMS Frame - no cuts WPCF 2007 - Aug. 1-3, 2007

  26. The Experimentalist’s Recipe - average of X over # of pairs for each Q-bin Fitting formula: WPCF 2007 - Aug. 1-3, 2007

  27. EMCIC’s FIT: N=18, K=0.9GeV, LCMS WPCF 2007 - Aug. 1-3, 2007

  28. The Complete Experimentalist’s Recipe or image this … 9 fit parameters - 4 femtoscopic - normalization - 4 EMCICs or any other parameterization of CF Fit this …. WPCF 2007 - Aug. 1-3, 2007

  29. Minv distribution w/ background subtraction N=18 WPCF 2007 - Aug. 1-3, 2007

  30. EMCICs contribution to v2 no contribution to v2 from 1/N term contribution to v2 from 1/N2 term contribution to v2 from 1/N3 term for v2 n=2 WPCF 2007 - Aug. 1-3, 2007

  31. Non-id correlations (Resonance contrib.) WPCF 2007 - Aug. 1-3, 2007

  32. Non-id correlations (PYTHIA@200 GeV) WPCF 2007 - Aug. 1-3, 2007

  33. Summary • understanding particle spectra, two-particle correlations,v2, resonances in small systems • important physics-wise • should not be attempted until data fully under control • Restricted P.S. due to energy-momentum conservation • sampled by GenBod event generator • generates EMCICs [femtoscopy : quantified by Alm’s] • stronger effects for small multiplicities 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 pT, pz, E conservation effects • Physically correct “recipe” to fit/remove EMCICs [femtoscopy] • 4 new parameters, determined @ large |Q| WPCF 2007 - Aug. 1-3, 2007

  34. 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] • Mark Baker (BNL) [local momentum conservation] • Dariusz Miskowiec (GSI) [multiply (don’t add) correlations] WPCF 2007 - Aug. 1-3, 2007

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