9 0 0 2 n a J A parton re ombination approa h to heavy ion 3 2 ollisions at RHIC and LHC ] h t - l Diploma thesis c u n [ 1 v 7 by Daniel Krieg 1 De ember 2008 6 3 . 1 0 9 0 : v i X r a Für Daniela Contents I Introdu tion 1 I. 1 Curiosity and doubt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 I. 2 The fundamental building blo ks . . . . . . . . . . . . . . . . . . . . . . . 2 I. 3 Ba k to the Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 II Heavy Ion Collisions 5 II. 1 Phase diagram of the strong for e . . . . . . . . . . . . . . . . . . . . . . . 5 II. 2 Theoreti al des riptions of heavy ion ollision . . . . . . . . . . . . . . . . 6 II. 2.1 Statisti al thermal model . . . . . . . . . . . . . . . . . . . . . . . 6 II. 2.2 Quantum hromodynami s (QCD) . . . . . . . . . . . . . . . . . . . 6 II. 2.2.1 Fragmentation (pQCD) . . . . . . . . . . . . . . . . . . . 7 II. 2.2.2 MIT bag model . . . . . . . . . . . . . . . . . . . . . . . 7 II. 2.3 Transport theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 II. 2.4 Hydrodynami s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 II. 2.4.1 Collision geometry . . . . . . . . . . . . . . . . . . . . . . 9 III Hadronization from a QGP 11 III. 1Colinear re ombination of quarks . . . . . . . . . . . . . . . . . . . . . . . 11 III. 1.1Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 III. 1.2Non-relativisti model . . . . . . . . . . . . . . . . . . . . . . . . . 12 III. 1.3Relativisti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 III. 2Freeze-out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 III. 2.1Integration measure . . . . . . . . . . . . . . . . . . . . . . . . . . 17 III. 2.2Surfa e (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 III. 3Blast Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 III. 3.1Azimuthal momentum asymmetry . . . . . . . . . . . . . . . . . . 21 III. 3.2Comparing asymmetry parameters . . . . . . . . . . . . . . . . . . 22 III. 3.2.1 Expansion velo ity . . . . . . . . . . . . . . . . . . . . . . 23 III. 4Observables from re ombination . . . . . . . . . . . . . . . . . . . . . . . . 25 III. 4.1Invariant yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 III. 4.1.1 Central ollisions . . . . . . . . . . . . . . . . . . . . . . . 28 III. 4.1.2 Peripheral ollisions . . . . . . . . . . . . . . . . . . . . . 28 III. 4.2Flow omponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ii Contents III. 4.2.1 Ellipti (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . . 28 III. 4.2.2 Hexade upole (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . 29 p T III. 4.2.3 High- (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . 29 III. 4.3Constituent quark number s aling . . . . . . . . . . . . . . . . . . 29 III. 4.3.1 Ellipti (cid:29)ow s aling . . . . . . . . . . . . . . . . . . . . . 29 III. 4.3.2 Hexade upole (cid:29)ow s aling . . . . . . . . . . . . . . . . . . 32 III. 4.4The breaking of the CQNS . . . . . . . . . . . . . . . . . . . . . . 32 IV Results and predi tions from re ombination 35 IV. 1Overview of the parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 35 IV. 1.1Re ombination parameters . . . . . . . . . . . . . . . . . . . . . . . 35 IV. 1.1.1 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . 35 p T IV. 1.1.2 High- damping . . . . . . . . . . . . . . . . . . . . . . 36 IV. 1.2Freeze-out hypersurfa e parameters . . . . . . . . . . . . . . . . . . 36 IV. 1.2.1 Transverse freeze-out area . . . . . . . . . . . . . . . . . . 36 IV. 1.2.2 Impa tparameter . . . . . . . . . . . . . . . . . . . . . . . 36 IV. 1.2.3 Freeze-out e entri ity . . . . . . . . . . . . . . . . . . . . 36 IV. 1.2.4 Time dependent hypersurfa e . . . . . . . . . . . . . . . . 37 IV. 1.3Blast wave parameters . . . . . . . . . . . . . . . . . . . . . . . . . 37 IV. 1.3.1 Phase boundary . . . . . . . . . . . . . . . . . . . . . . . 37 IV. 1.3.2 Transverse expansion . . . . . . . . . . . . . . . . . . . . 38 IV. 2In(cid:29)uen e of the parameters and parameterisations. . . . . . . . . . . . . . 40 IV. 2.1Delta-shaped wavefun tions . . . . . . . . . . . . . . . . . . . . . . 40 IV. 2.2E(cid:27)e ts of the blast wave parameters . . . . . . . . . . . . . . . . . 40 IV. 2.2.1 Baryo- hemi al potential . . . . . . . . . . . . . . . . . . 40 IV. 2.2.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 40 IV. 2.2.3 Transverse (cid:29)ow rapidity . . . . . . . . . . . . . . . . . . . 41 IV. 2.2.4 Radial pro(cid:28)le of the transverse rapidity . . . . . . . . . . 41 IV. 2.3Separating (cid:29)ow and non-(cid:29)ow e(cid:27)e ts . . . . . . . . . . . . . . . . . 44 IV. 2.3.1 Geometri al ontributions . . . . . . . . . . . . . . . . . . 45 IV. 2.3.2 Flow ontributions . . . . . . . . . . . . . . . . . . . . . . 45 IV. 2.3.3 Relative strength of both ontributions . . . . . . . . . . 46 τ ρ IV. 2.4Correlation between and . . . . . . . . . . . . . . . . . . . . . . 49 IV. 3Transverse momentum spe tra . . . . . . . . . . . . . . . . . . . . . . . . 51 IV. 3.1Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 IV. 3.2Mean transverse momentum . . . . . . . . . . . . . . . . . . . . . . 51 IV. 3.3Hadron ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 v v 2 4 IV. 4Flow oe(cid:30) ients and . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 IV. 4.1Mean (cid:29)ow oe(cid:30) ients . . . . . . . . . . . . . . . . . . . . . . . . . 57 IV. 4.2E entri ity dependen e . . . . . . . . . . . . . . . . . . . . . . . . 57 IV. 4.2.1 Flu tuations . . . . . . . . . . . . . . . . . . . . . . . . . 59 IV. 4.3Flow ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 τ ρ IV. 4.3.1 In(cid:29)uen e of a − orrelation . . . . . . . . . . . . . . . 62 Contents iii IV. 4.3.2 In(cid:29)uen e of the radial rapidity pro(cid:28)le . . . . . . . . . . . 64 IV. 4.4Di(cid:27)erential (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 IV. 4.5Deuteron (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 IV. 4.6Heavy quark (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 IV. 4.7Centrality dependen e . . . . . . . . . . . . . . . . . . . . . . . . . 73 √s IV. 4.8 dependen e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 IV. 4.9Analyzing the negative ellipti (cid:29)ow . . . . . . . . . . . . . . . . . . 78 V Con lusion 81 Bibliography 83 A Analyti al derivations 89 A. 1 MIT bag model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 A. 2 Integrals of the (cid:29)ow oe(cid:30) ients . . . . . . . . . . . . . . . . . . . . . . . . 91 B A knowledgements 95 C Legal stu(cid:27) 97 C. 1 Erklärung der Selbstständigkeit (De laration) . . . . . . . . . . . . . . . . 97 C. 2 Zusammenfassung (german abstra t) . . . . . . . . . . . . . . . . . . . . . 99 List of Figures II.1 The (assumed) phase diagram of the strong for e . . . . . . . . . . . . . 5 II.2 Collision geometry in the transverse plane . . . . . . . . . . . . . . . . . 10 III.1 The two extreme ases for the expansion of (cid:28)reball. . . . . . . . . . . . 19 III.2 Comparison of the impa t parameter dependent e entri ity . . . . . . . 22 III.3 Study of the breaking of onstituent quark number s aling . . . . . . . . 34 T µ B IV.1 Temperature and baryo- hemi al potential . . . . . . . . . . . . . 38 IV.2 Parameterisation of the transverse (cid:29)ow rapidity . . . . . . . . . . . . . . 39 r δ IV.3 Relative deviations of delta-shaped wavefun tions . . . . . . . . . . . 40 IV.4 The invariant yield of pions and protons for di(cid:27)erent transverse (cid:29)ow β T velo ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 IV.5 The ellipti (cid:29)ow of pions and protons for di(cid:27)erent transverse (cid:29)ow velo - β T ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 IV.6 The invariant yield of pions andprotons for di(cid:27)erent radial pro(cid:28)les of the transverse rapidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 IV.7 The ellipti (cid:29)ow of pions and protons for di(cid:27)erent radial pro(cid:28)les of the transverse rapidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 IV.8 Purely geometri al ontributions to the ellipti (cid:29)ow from an ellipti al freeze-out area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 IV.9 S aled ellipti and hexade upole (cid:29)ow for di(cid:27)erent hadrons . . . . . . . . 47 IV.10 Ellipti and hexade upole (cid:29)ow for light quarks . . . . . . . . . . . . . . 47 IV.11 Comparison of the freeze-out e entri ity . . . . . . . . . . . . . . . . . 48 r (dN) τ ρ λ IV.12 Therelativedeviations oftheinvariantyieldswith − orrelations 50 ∆ (v ) τ ρ λ 2 IV.13 The deviations of the ellipti (cid:29)ow with − orrelations . . . . 50 √s = IV.14 Transverse momentum spe tra of hadrons for Au+Au ollisions at 200 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 IV.15 Charged hadron yield for di(cid:27)erent radial pro(cid:28)les of the transverse rapidity 53 IV.16 Mean transverse momentum and velo ity of hadrons as a fun tion of the hadron mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 P T IV.17 Invariant yield ratios for di(cid:27)erent hadrons as a fun tion of . . . . . . 55 v ε n IV.18 Mean (cid:29)ow h i as a fun tion of the e entri ity . . . . . . . . . . . . . 58 v n IV.19 Mean (cid:29)ow h i as a fun tion of the e entri ity from an ellipti freeze-out 58 IV.20 The relative (cid:29)ow (cid:29)u tuations for a ir ular freeze-out . . . . . . . . . . 60 IV.21 The relative (cid:29)ow (cid:29)u tuations for an ellipti freeze-out . . . . . . . . . . 61 v /(v )2 4 2 IV.22 Comparison of the ratio from an ellipsoidal freeze-out for di(cid:27)er- ent strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 v /(v )2 4 2 IV.23 The (cid:29)ow ratio for an ellipsoidal freeze-out ompared to STAR τ ρ and PHENIX data with di(cid:27)erent − orrelations . . . . . . . . . . . . 63 v /(v )2 4 2 IV.24 The (cid:29)ow ratio for an ellipsoidal freeze-out ompared to STAR and PHENIX data with di(cid:27)erent radial pro(cid:28)les of the transverse rapidity 65 p c = 0.0 T e IV.25 Ellipti (cid:29)ow of harged hadrons as a fun tion of for and c = 0.73 e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 p c = 0.73 T e IV.26 Ellipti (cid:29)ow of identi(cid:28)ed hadrons as a fun tion of with and λ = 0.3 − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 p c = 0.0 λ = 0 T e IV.27 Ellipti (cid:29)ow as a fun tion of with and . . . . . . . . . 68 p c = 0.73 λ = 0.3 T e IV.28 Hexade upole (cid:29)ow as a fun tion of with and − for di(cid:27)erent hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 v 2 IV.29 Ellipti (cid:29)ow of deuterons from 6 re ombining quarks or s aled proton 70 J/ψ IV.30 Estimating the harm (cid:29)ow from the ellipti (cid:29)ow of . . . . . . . . . 71 D v 0 2 IV.31 Ele tron ellipti (cid:29)ow data from heavy (cid:29)avor de ay omapared to 72 IV.32 Ellipti (cid:29)ow of heavy mesons with harm and bottom quark ontent at RHIC and LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 IV.33 Mean ellipti (cid:29)ow for harged hadrons as a fun tion of the entrality . . 74 p c = 0.73 λ = 0.3 T e IV.34 Hexade upole (cid:29)ow as a fun tion of with and − for di(cid:27)erent entralities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 v v √s= 100 2 4 IV.35 Comparison of h i and h i for di(cid:27)erent hadrons from GeV to 10 TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 v 2 IV.36 Comparison of h i for harged hadrons as a fun tion of enter of mass energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 v p 2 T IV.37 Ellipti (cid:29)ow of harged hadrons for a (cid:28)xed as a fun tion of enter of mass energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 v 4 IV.38 Comparison of h i for harged hadrons as a fun tion of enter of mass energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 v / v 2 4 2 IV.39 The ratio h i h i for harged hadrons as a fun tion of enter of mass energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 List of Tables v (ε) = aε+bε2 n IV.1 Fit values for the fun tion h i . . . . . . . . . . . . . . . 57