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Conservation laws & femtoscopy of small systems

Conservation laws & femtoscopy of small systems. Zbigniew Chajecki & Mike Lisa Ohio State University. Outline. Introduction / Motivation intriguing pp versus AA [reminder] data features not under control: Energy-momentum conservation? SHD as a diagnostic tool [reminder]

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Conservation laws & femtoscopy of small systems

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  1. Conservation laws & femtoscopy of small systems Zbigniew Chajecki & Mike Lisa Ohio State University ma lisa - Momentum conservation effects - WPCF 2006

  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 MCIC • Experimentalists’ recipe: Fitting correlation functions [in progress] • Conclusion ma lisa - Momentum conservation effects - WPCF 2006

  3. STAR preliminary mT (GeV) mT (GeV) femtoscopy in p+p @ STAR Z. Chajecki WPCF05 • p+p and A+A measured in same experiment • great opportunity to compare physics • what causes pT-dependence in p+p? • same cause as in A+A? ma lisa - Momentum conservation effects - WPCF 2006

  4. Ratio of (AuAu, CuCu, dAu) HBT radii by pp pp, dAu, CuCu - STAR preliminary Surprising („puzzling”) scaling • p+p and A+A measured in same experiment • great opportunity to compare physics • what causes pT-dependence in p+p? • same cause as in A+A? HBT radii scale with pp Scary coincidence or something deeper? ma lisa - Momentum conservation effects - WPCF 2006

  5. Ratio of (AuAu, CuCu, dAu) HBT radii by pp pp, dAu, CuCu - STAR preliminary Surprising („puzzling”) scaling • p+p and A+A measured in same experiment • great opportunity to compare physics • what causes pT-dependence in p+p? • same cause as in A+A? A. Bialasz (ISMD05): I personally feel that its solution may provide new insight into the hadronization process of QCD HBT radii scale with pp Scary coincidence or something deeper? ma lisa - Momentum conservation effects - WPCF 2006

  6. STAR preliminary d+Au peripheral collisions Gaussian fit Clear interpretation clouded by data features Non-femtoscopic q-anisotropic behaviour at large |q| does this structure affect femtoscopic region as well? ma lisa - Momentum conservation effects - WPCF 2006

  7. QLONG Q  QOUT  QSIDE Spherical harmonic decomposition of CF • Cartesian-space (out-side-long) naturally encodes physics, but is “inefficient” representation • Harmonic Moments -- 1::1 connection to source geometry [Danielewicz,Pratt: nucl-th/0501003] • ~immune to acceptance • full information content at a glance[thanks to symmetries] This new method of analysis represents a real breakthrough. ...(should) become a standard tool in all experiments. - A. Bialas, ISMD 2005 Chajecki., Gutierrez, MAL, Lopez-Noriega, nucl-ex/0505009 ma lisa - Momentum conservation effects - WPCF 2006

  8. Decomposition of CF onto Spherical Harmonics Au+Au: central collisions C(Qout) C(Qside) C(Qlong) Z.Ch., Gutierrez, MAL, Lopez-Noriega, nucl-ex/0505009 Pratt, Danielewicz [nucl-th/0501003] STAR preliminary ma lisa - Momentum conservation effects - WPCF 2006

  9. d+Au: peripheral collisions Decomposition of CF onto Spherical Harmonics non-femtoscopic structure (not just “non-Gaussian”) Z.Ch., Gutierrez, MAL, Lopez-Noriega, nucl-ex/0505009 Pratt, Danielewicz [nucl-th/0501003] STAR preliminary ma lisa - Momentum conservation effects - WPCF 2006

  10. Just push on....? • ... no! • Irresponsible to ad-hoc fit (often the practice) or ignore (!!) & interpret without understanding data • no particular reason to expect non-femtoscopic effect to be limited to non-femtoscopic (large-q) region • not-understood or -controlled contaminating correlated effectsat low q ? • A possibility: energy-momentum conservation? • must be there somewhere! • but how to calculate / model ?(Upon consideration, non-trivial...) ma lisa - Momentum conservation effects - WPCF 2006

  11. statistics: “density of states” larger particle momentum more available states P conservation Induces “trivial” correlations (i.e. even for M=1) energy-momentum conservation in n-body states spectrum of kinematic quantity  (angle, momentum) given by n-body Phasespace factor Rn ma lisa - Momentum conservation effects - WPCF 2006

  12. Example of use of total phase space integral • In absence of “physics” in M : (i.e. phase-space dominated) • single-particle spectrum of : • “spectrum of events”: F. James, CERN REPORT 68-15 (1968) ma lisa - Momentum conservation effects - WPCF 2006

  13. ALL EVENTS ARE EQUAL, BUT SOME EVENTS ARE MORE EQUAL THAN OTHERS Genbod:phasespace sampling w/ P-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 ma lisa - Momentum conservation effects - WPCF 2006

  14. 6 particles ALL EVENTS ARE EQUAL, BUT SOME EVENTS ARE MORE EQUAL THAN OTHERS larger particle momentum more available states “Rounder” events: higher WT ma lisa - Momentum conservation effects - WPCF 2006

  15. ALL EVENTS ARE EQUAL, BUT SOME EVENTS ARE MORE EQUAL THAN OTHERS 30 particles larger particle momentum more available states “Rounder” events: higher WT ma lisa - Momentum conservation effects - WPCF 2006

  16. Genbod:phasespace sampling w/ P-conservation • Treat identical to measured events • use WT directly • MC sample WT • Form CF and SHD ma lisa - Momentum conservation effects - WPCF 2006

  17. Effect of varying frame & kinematic cuts Watch the green squares --  ma lisa - Momentum conservation effects - WPCF 2006

  18. N=18 <K>=0.9 GeV; LabCMS Frame - no cuts ma lisa - Momentum conservation effects - WPCF 2006

  19. N=18 <K>=0.9 GeV; LabCMS Frame - ||<0.5 kinematic cuts have strong effect! ma lisa - Momentum conservation effects - WPCF 2006

  20. N=18 <K>=0.9 GeV, LCMS - no cuts kinematic cuts have strong effect! as does choice of frame! ma lisa - Momentum conservation effects - WPCF 2006

  21. N=18 <K>=0.9 GeV; LCMS - ||<0.5 kinematic cuts have strong effect! as does choice of frame! ma lisa - Momentum conservation effects - WPCF 2006

  22. N=18 <K>=0.9 GeV; PRF - no cuts kinematic cuts have strong effect! as does choice of frame! ma lisa - Momentum conservation effects - WPCF 2006

  23. N=18 <K>=0.9 GeV; PRF - ||<0.5 kinematic cuts have strong effect! as does choice of frame! ma lisa - Momentum conservation effects - WPCF 2006

  24. Effect of varying multiplicity & total energy Watch the green squares --  ma lisa - Momentum conservation effects - WPCF 2006

  25. GenBod : 6 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  26. increasing mult reduces P.S. constraint GenBod : 9 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  27. increasing mult reduces P.S. constraint GenBod : 15 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  28. increasing mult reduces P.S. constraint GenBod : 18 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  29. increasing mult reduces P.S. constraint GenBod : 18 pions, <K>=0.7 GeV/c increasing s reduces P.S. constraint ma lisa - Momentum conservation effects - WPCF 2006

  30. increasing mult reduces P.S. constraint GenBod : 18 pions, <K>=0.9 GeV/c increasing s reduces P.S. constraint ma lisa - Momentum conservation effects - WPCF 2006

  31. So... • Momentum Conservation Induced Correlations (MCIC) “resemble” our data • So, MCIC... on the right track... • But what to do with that? • Sensitivity to s, Mult 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 MCICs using data!! • Danielewicz et al, PRC38 120 (1988) • Borghini, Dinh, & Ollitraut PRC62 034902 (2000) we generalize their 2D pT considerations to 4-vectors ma lisa - Momentum conservation effects - WPCF 2006

  32. Distributions w/ phasespace constraints single-particle distribution w/o P.S. restriction k-particle distribution (k<N) with P.S. restriction ma lisa - Momentum conservation effects - WPCF 2006

  33. N.B. relevant later Using central limit theorem (“large N-k”) k-particle distribution in N-particle system (*) For simplicity, I from now on assume identical particles (e.g. pions). I.e. all particles have the same average energy and RMS’s of energy and momentum. Similar results (esp “experimentalist recipe) but more cumbersome notation otherwise ma lisa - Momentum conservation effects - WPCF 2006

  34. Effects on 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, really ma lisa - Momentum conservation effects - WPCF 2006

  35. Dependence on “parent” distrib f vanishes, except for energy/momentum means and RMS k-particle correlation function 2-particle correlation function (1st term in 1/N expansion) ma lisa - Momentum conservation effects - WPCF 2006

  36. 2-particle correlation function (1st term in 1/N expansion) “The pT term” “The E term” “The pZ term” Names used in the following plots ma lisa - Momentum conservation effects - WPCF 2006

  37. Effect of varying multiplicity & total energy 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 ma lisa - Momentum conservation effects - WPCF 2006

  38. GenBod : 6 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  39. GenBod : 9 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  40. GenBod : 15 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  41. GenBod : 18 pions, <K>=0.5 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  42. GenBod : 18 pions, <K>=0.7 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  43. GenBod : 18 pions, <K>=0.9 GeV/c ma lisa - Momentum conservation effects - WPCF 2006

  44. Findings • 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, but don’t nail it. Why? • neither (N-k) nor s is infinite • however, probably more important... [next slide]... ma lisa - Momentum conservation effects - WPCF 2006

  45. relevant quantities are average over the (unmeasured) “parent” distribution, not the physical distribution Remember... of course, the experimentalist never measures all particles (including neutrinos) or <pT2> anyway, so maybe not a big loss ma lisa - Momentum conservation effects - WPCF 2006

  46. The experimentalist’s recipe • Treat the not-precisely-known factors as fit parameters (4 of them) • values determined mostly by large-|Q|; should not cause “fitting hell” • look, you will either ignore it or fit it ad-hoc anyway (both wrong) • this recipe provides physically meaningful, justified form ma lisa - Momentum conservation effects - WPCF 2006

  47. 18 pions, <K>=0.9 GeV ma lisa - Momentum conservation effects - WPCF 2006

  48. ma lisa - Momentum conservation effects - WPCF 2006

  49. The COMPLETE experimentalist’s recipe femtoscopic function of choice fit this... ...or image this... ma lisa - Momentum conservation effects - WPCF 2006

  50. 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 MCICs quantified by Alm’s • stronger effects for small mult and/or s • Analytic calculation of MCIC • 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 MCIC • 4 parameters, determined @ large |Q| • parameter are “physical” - values may be guessed ma lisa - Momentum conservation effects - WPCF 2006

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