1. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20071 Conservation laws & femtoscopy of small systems Zbigniew Chajeçki & Mike Lisa Ohio State University nucl-th/0612080

2. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20072 Outline Introduction –RHI, femtoscopy, & collectivity in bulk matter –including: new representation of correlation functions the p+p “reference” –intriguing features pp versus AA –non-femtoscopic effects Global conservation effects (EMCICs) –Restricted phasespace calculations: GenBod (FOWL) –Analytic EMCIC calculations –Experimentalists’ recipe: –Fitting correlation functions [in progress] Summary

3. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20073 Introduction: Heavy ions & bulk matter

4. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20074 RHIC BRAHMS PHOBOS PHENIX STAR AGS TANDEMS NuclearParticle

5. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20075 Why heavy ion collisions? NuclearParticle STAR ~500 Collaborators

6. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20076 2 types (?) of collisions... looks like fun... looks like a mess...

7. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20077 A dynamical but crude view of the collision In this model, insufficient re-interactions c/o UrQMD Collaboration, Frankfurt

8. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20078 A dynamical but crude view of the collision In this model, insufficient re-interactions

9. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 20079 R.H.I.C. defined... collision of nuclei sufficiently large that nuclear details unimportant –distinct from nuclear or particle physics –“Geranium on Linoleum” sufficiently large for meaningful bulk & thermodynamic quantities non-trivial spatial scales & geometry drive bulk dynamics (e.g. flow) how big is “sufficient” ? –important reference issue

10. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200710 phase structure of bulk system: driving symmetries long-range collective behaviour “new” physics [superfluidity in l-He] relevance of meaningful EoS

11. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200711 Generating a deconfined state Nuclear Matter (confined) Hadronic Matter (confined) Quark Gluon Plasma deconfined ! Present understanding of Quantum Chromodynamics (QCD) heating compression  deconfined color matter !

12. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200712 Expectations from Lattice QCD  /T 4 ~ # degrees of freedom confined: few d.o.f. deconfined: many d.o.f. T C ≈ 173 MeV ≈ 2  10 12 K ≈ 130,000  T[Sun’s core]

13. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200713 The four elements from 400 BC to 2000AD 400 BC : all creation EarthFire Air Water ?

14. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200714 RHIC energies: the first quantitative success of hydro direct access to EoS (phase transitions, lattice, etc.) EoS: P versus  versus n (Heinz et al)

15. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200715 MicroexplosionsFemtoexplosions  10 17 J/m 3 5 GeV/fm 3 = 10 36 J/m 3 ss0.1  J 1  J T10 6 K200 MeV = 10 12 K rate10 18 K/sec10 35 K/s energy quickly deposited enter plasma phase expand hydrodynamically cool back to original phase do geometric “postmortem” & infer momentum

16. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200716 MicroexplosionsFemtoexplosions ss0.1  J 1  J  10 17 J/m 3 5 GeV/fm 3 = 10 36 J/m 3 T10 6 K200 MeV = 10 12 K rate10 18 K/sec10 35 K/s energy quickly deposited enter plasma phase expand hydrodynamically cool back to original phase do geometric “postmortem” & infer momentum

17. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200717 Impact parameter & Reaction plane b = 0  “central collision” many particles produced “peripheral collision” fewer particles produced

18. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200718 Impact parameter & Reaction plane b = 0  “central collision” many particles produced “peripheral collision” fewer particles produced

19. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200719 How do semi-central collisions evolve? 1) Superposition of independent p+p: momenta pointed at random relative to reaction plane

20. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200720 How do semi-central collisions evolve? 1) Superposition of independent p+p: 2) Evolution as a bulk system momenta pointed at random relative to reaction plane high density / pressure at center “zero” pressure in surrounding vacuum Pressure gradients (larger in-plane) push bulk “out”  “flow” more, faster particles seen in-plane

21. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200721 How do semi-central collisions evolve? 1) Superposition of independent p+p: 2) Evolution as a bulk system momenta pointed at random relative to reaction plane Pressure gradients (larger in-plane) push bulk “out”  “flow” more, faster particles seen in-plane N  -  RP (rad) 0  /2   /43  /4 N  -  RP (rad) 0  /2   /43  /4

22. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200722 Azimuthal distributions at RHIC STAR, PRL90 032301 (2003) b ≈ 4 fm “central” collisions b ≈ 6.5 fm midcentral collisions

23. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200723 Azimuthal distributions at RHIC STAR, PRL90 032301 (2003) b ≈ 4 fm b ≈ 6.5 fm b ≈ 10 fm peripheral collisions

24. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200724 Beyond press releases Nature of EoS under investigation ; agreement with data may be accidental ; viscous hydro under development ; assumption of thermalization in question sensitive to modeling of initial state, presently under study The detailed work now underway is what can probe & constrain sQGP properties It is probably not press-release material......but, hey, you’ve already got your coffee mug

25. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200725 Femtoscopy

26. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200726 Beyond press releases: Access to the dynamically-generated geometric substructure? The feature of collectivity: space is globally correlated with momentum

27. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200727 probing source geometry through interferometry Measurable! F.T. of pion source Creation probability  (x,p) = U * U 5 fm 1 m  source  (x) r1r1 r2r2 x1x1 x2x2 p1p1 p2p2 The Bottom line… if a pion is emitted, it is more likely to emit another pion with very similar momentum if the source is small experimentally measuring this enhanced probability: quite challenging

28. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200728 Correlation functions for different colliding systems C 2 (Q inv ) Q inv (GeV/c) STAR preliminary p+p R ~ 1 fm d+Au R ~ 2 fm Au+Au R ~ 6 fm Different colliding systems studied at RHIC Interferometry probes the smallest scales ever measured ! Au+Au: central collisions C(Q out ) C(Q side ) C(Q long ) 3 “radii” by using 3-D vector q

29. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200729 Femtoscopic information xaxa xbxb papa pbpb xaxa xbxb papa pbpb femtoscopic correlation at low |q| must vanish at high |q|. [indep “direction”] Au+Au: central collisions C(Q out ) C(Q side ) C(Q long ) 3 “radii” by using 3-D vector q

30. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200730 Femtoscopic information - Spherical harmonic representation femtoscopic correlation at low |q| must vanish at high |q|. [indep “direction”] Au+Au: central collisions C(Q out ) C(Q side ) C(Q long ) 3 “radii” by using 3-D vector q Q OUT   Q SIDE Q LONG Q This new method of analysis represents a real breakthrough....(should) become a standard tool in all experiments. - A. Bialas, ISMD 2005 nucl-ex/0505009

31. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200731 Femtoscopic information - Spherical harmonic representation femtoscopic correlation at low |q| must vanish at high |q|. [indep “direction”] A LM (Q) =  L,0 Au+Au: central collisions C(Q out ) C(Q side ) C(Q long ) 3 “radii” by using 3-D vector q L=0 L=2 M=0 L=2 M=2 nucl-ex/0505009

32. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200732 Why do the radii fall with increasing momentum ??

33. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200733 It’s collective flow !! Direct geometrical/dynamical evidence for bulk behaviour!!! Amount of flow consistent with p-space nucl-th/0312024 Huge, diverse systematics consistent with this substructure nucl-ex/0505014 Why do the radii fall with increasing momentum ??

34. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200734 p+p reference

35. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200735 p+p: A clear reference system?

36. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200736 STAR preliminary m T (GeV) Z. Chajecki QM05 nucl-ex/0510014 femtoscopy in p+p @ STAR Decades of femtoscopy in p+p and in A+A, but... for the first time: femtoscopy in p+p and A+A in same experiment, same analysis definitions... unique opportunity to compare physics ~ 1 fm makes sense, but... p T -dependence in p+p? (same cause as in A+A?)

37. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200737 Surprising („puzzling”) scaling HBT radii scale with pp Scary coincidence or something deeper? On the face: same geometric substructure pp, dAu, CuCu - STAR preliminary Ratio of (AuAu, CuCu, dAu) HBT radii by pp A. Bialasz (ISMD05): I personally feel that its solution may provide new insight into the hadronization process of QCD

38. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200738 BUT... Clear interpretation clouded by data features STAR preliminary d+Au peripheral collisions Gaussian fit Non-femtoscopic q-anisotropic behaviour at large |q| does this structure affect femtoscopic region as well?

39. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200739 STAR preliminary d+Au peripheral collisions Gaussian fit Decomposition of CF onto Spherical Harmonics Z.Ch., Gutierrez, MAL, Lopez-Noriega, nucl-ex/0505009 non-femtoscopic structure (not just “non-Gaussian”)

40. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200740 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 effects at low q ? A possibility: energy-momentum conservation? –must be there somewhere! –but how to calculate / model ? (Upon consideration, non-trivial...)

41. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200741 Genbod

42. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200742 energy-momentum conservation in n-body states spectrum of kinematic quantity  (angle, momentum) given by n-body Phasespace factor R n statistics: “density of states” larger particle momentum  more available states P  conservation Induces “trivial” correlations (i.e. even for M=1)

43. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200743 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)

44. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200744 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 ALL EVENTS ARE EQUAL, BUT SOME EVENTS ARE MORE EQUAL THAN OTHERS

45. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200745 “Rounder” events: higher WT ALL EVENTS ARE EQUAL, BUT SOME EVENTS ARE MORE EQUAL THAN OTHERS larger particle momentum  more available states 6 particles

46. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200746 “Rounder” events: higher WT ALL EVENTS ARE EQUAL, BUT SOME EVENTS ARE MORE EQUAL THAN OTHERS larger particle momentum  more available states 30 particles

47. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200747 Genbod:phasespace sampling w/ P- conservation Treat identical to measured events use WT directly MC sample WT Form CF and SHD

48. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200748 Effect of varying frame & kinematic cuts Watch the green squares --

49. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200749 N=18 =0.9 GeV; LabCMS Frame - no cuts Watch the green squares

50. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200750 N=18 =0.9 GeV; LabCMS Frame - |  |<0.5 kinematic cuts have strong effect! Watch the green squares

51. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200751 N=18 =0.9 GeV, LCMS - no cuts kinematic cuts have strong effect! as does choice of frame! Watch the green squares

52. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200752 N=18 =0.9 GeV; LCMS - |  |<0.5 kinematic cuts have strong effect! as does choice of frame! Watch the green squares

53. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200753 N=18 =0.9 GeV; PRF - no cuts kinematic cuts have strong effect! as does choice of frame! Watch the green squares

54. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200754 N=18 =0.9 GeV; PRF - |  |<0.5 kinematic cuts have strong effect! as does choice of frame! Watch the green squares

55. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200755 Effect of varying multiplicity & total energy Watch the green squares --

56. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200756 GenBod : 6 pions, =0.5 GeV/c

57. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200757 GenBod : 9 pions, =0.5 GeV/c increasing mult reduces P.S. constraint

58. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200758 GenBod : 15 pions, =0.5 GeV/c increasing mult reduces P.S. constraint

59. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200759 GenBod : 18 pions, =0.5 GeV/c increasing mult reduces P.S. constraint

60. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200760 GenBod : 18 pions, =0.7 GeV/c increasing mult reduces P.S. constraint increasing  s reduces P.S. constraint

61. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200761 GenBod : 18 pions, =0.9 GeV/c increasing mult reduces P.S. constraint increasing  s reduces P.S. constraint

62. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200762 So... Energy & Momentum Conservation Induced Correlations (EMCICs) “resemble” our data So... on the right track... But what to do with that? –Sensitivity to  s, Mult 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) –Chajecki & MAL, nucl-th/0612080 - generalization of the above

63. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200763 Distributions w/ phasespace constraints single-particle distribution w/o P.S. restriction

64. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200764 Distributions w/ phasespace constraints single-particle distribution w/o P.S. restriction k-particle distribution (k<N) with P.S. restriction

65. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200765 Using central limit theorem (“large N-k”) (*) 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 k-particle distribution in N-particle system N.B. relevant later

66. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200766 Effects on single-particle distribution

67. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200767 k-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)

68. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200768 2-particle correlation function (1 st term in 1/N expansion) “The p T term” “The p Z term” “The E term” Names used in the following plots

69. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200769 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

70. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200770 GenBod : 6 pions, =0.5 GeV/c

71. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200771 GenBod : 9 pions, =0.5 GeV/c

72. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200772 GenBod : 15 pions, =0.5 GeV/c

73. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200773 GenBod : 18 pions, =0.5 GeV/c

74. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200774 GenBod : 18 pions, =0.7 GeV/c

75. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200775 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, but don’t nail it. Why? –neither (N-k) nor  s is infinite –however, probably more important... [next slide]...

76. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200776 Remember... relevant quantities are average over the (unmeasured) “parent” distribution, not the physical distribution of course, the experimentalist never measures all particles (including neutrinos) or anyway, so maybe not a big loss

77. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200777 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

78. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200778 18 pions, =0.9 GeV

79. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200779 Highly correlated EMCIC parameters

80. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200780 The COMPLETE experimentalist’s recipe femtoscopic function of choice fit this......or image this...

81. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200781 Summary H.I.C raison d’etre & distinction from p+p –bulk, thermalized, collective matter Momentum-space and femtoscopic probes support bulk collectivity The p+p “reference” - [femtoscopic apples-to-apples for the first time] –is it understood? –is it really different than a “small A+A” ? –full understanding must be high priority

82. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200782 Famous picture from famous article by famous guy Energy loss of energetic partons in quark-gluon plasma: Possible extinction of high p T jets in hadron-hadron collisions J.D. Bjorken, 1982 b

83. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200783 Summary H.I.C raison d’etre & distinction from p+p –bulk, thermalized, collective matter Momentum-space and femtoscopic probes support bulk collectivity The p+p “reference” - [femtoscopic apples-to-apples for the first time] –is it understood? –is it really different than a “small A+A” ? –full understanding must be high priority Interpretation complicated by non-femtoscopic correlations May be largely due to P.S. restrictions (EMCICs) –numerical studies track data –analytic formulation of EMCIC projection onto femtoscopic analysis presented –highly non-trivial interplay between cuts, frames, and conserved components (pz, pT, E) –first-order expansion --> experimentalists’ formula –application to STAR underway Hotter spotlight due to impending LHC (December 2007!!)

84. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200784

85. ma lisa - Femtoscopy in small systems & EMCICs - Kent State University - January 200785 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]