QGP Hydrodynamics. Mahnaz Q. Haseeb Department of Physics CIIT, Islamabad

Transcription

1 QGP Hydrodynamics Mahnaz Q. Haseeb Department of Physics CIIT, Islamabad First School on LHC Physics, NCP, Islamabad Oct 28,

2 Outline QGP Evolution Centrality Why Hydrodynamics? What is a flow? Percolation in QGP 2

3 Study of QGP QGP is mainly defined theoretically by lattice QCD. Fascinating phenomena discovered and studied already Quantitative estimate of some fundamental quantities. Models are used to map the T, S, viscosity, size, time dependence onto observables. Hydrodynamics is a good start 3

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5 0 fm/c 2 fm/c 7 fm/c Diagram from Peter Steinberg >7 fm/c 5

6 Details in Heavy Ion Collision 5 0. Gluon dominant nuclei Colliding Ions Collision!! Hard Collisions Thermalization QGP Plasma instabilities?? 2. QGP expands and cools down Perfect fluid?? 3. Hadronization Fragmentation vs recombination 4. Particle abundances fixed 5. Particle freeze out free streaming Original figure by T. Chujo s, modified 6

7 Energy Density Δy 0.5 ε Energy density (Bjorken): = de A dz T T = 1 πr 2 τ de dy T Particle streaming from origin z = vz = tanh y t dz = τ cosh y dy 1/3 R = 1.18 A 7 fm τ τ SPS RHIC 1 fm/c fm/c τ = Estimate ε for RHIC: de T /dy ~ 720 GeV Time estimate from hydro: 0.6 fm/c ε ~ 8 GeV/fm T initial ~ MeV 7 3

8 QGP and hydrodynamic expansion hadronic phase and freeze-out initial state pre-equilibrium hadronization high-p t and early times: manifestations of pre-equilibrium jet production and quenching [photons & leptons] 8

9 participants spectators 9

10 Centrality The most central collision, the most dense matter Spectators Participants Spectators Very central collisions Very peripheral collisions 10

11 Centrality: impact parameter In heavy ion collisions the volume and energy of the fireball is determined (at given beam energy) mostly by the number of participating nucleons N part, which in turn depends on the impact parameter b x = beam axis (y,z) = transverse plane 11

12 Centrality: number of collisions The average number of N-N collisions at impact parameter b is: <ν(b)> = k=1 AB kp(k,b) = ABσ 0 T AB (b) Under the assumption of an inelastic A-B collision, the average number N coll (b) of N-N collisions is the same as <ν(b)> except for very large b: N coll (b) = k=1 AB kp(k,b) / k=1 AB P(k,b) = ABσ 0 T AB (b) / σ AB (b) 12

13 Centrality: number of participants The number of participants (wounded) nucleons from both nuclei A and B is on average: N W (b)) = N A (b)+n B (b) = [A/σ AB (b)] T A (s)σ B (b-s)d 2 s + [B/[ B/σ AB (b)] T B (b-s)σ A (s)d 2 s A T A (s){1-[1-t B (b-s)σ 0 ] B }d 2 s + B T B (b-s){1- [1-T A (s)σ 0 ] A }d 2 s N W is the number of nucleons having suffered at least one inelastic collision; there are other ways to count participants, which can lead to different numbers: N part = A + B N (spectators), e.g.: N pro part = A(1-E F /E beam ) N part from a dynamical simulation may (or may not) include rescattering with produced particles 13

14 QGP and hydrodynamic expansion hadronic phase and freeze-out initial state pre-equilibrium hadronization low-p t and intermediate times: creation and evolution of the QGP Hydrodynamics and anisotropic flow Thermalization 14

15 Why Hydrodynamics? Energy-momentum: Static EoS from Lattice QCD Finite T, μ field theory Critical phenomena Chiral property of hadron Conserved number: Dynamic Phenomena in HIC Expansion, Flow Space-time evolution of thermodynamic variables 15

16 Hydrodynamic quark model Hydrodynamics provides a direct link between the equation of state te (EOS) of the expanding fluid and the flow pattern manifested in the emitted hadron spectra. A quantitative determination of the EOS requires both precision flow data and systematic theoretical studies of the influence of the initial conditions - equation of state, non-ideal transport effects and the final decoupling kinetics on the observed hadron spectra. Theoretically limited by the difficulty of computations in viscous relativistic hydrodynamics. Determination of the equation of state also a big issue. Note that the hydrodynamics model also breaks down for more peripheral collisions, lower energy collisions etc. 16

17 Landau Hydrodynamics Landau, Izv. Akad. Nauk SSSR 17, 51 (1953) Nuovo Ciment, Suppl. 3, (1956) pp collision Initial condition initial entropy of the system adiabatic hydrodynamic motion constant total entropy - constant number of particles longitudinal expansion followed by transverse expansion has successfully explained 1. total number of produced charged particles 2. rapidity distribution dn / dy 17

18 Hydrodynamic equations Energy momentum tensor Longitudinal expansion Transverse expansion T μ T μν T t T t T t 02 μν = J = ( ε + T + z 01 T + z ν 11 T + x 22 P) u = 0 = 0 μ = 0 u ν Pg μν constraints: μ μ (nu (su u μ u μ μ μ ) ) = = = baryon number conservation entropy conservation flow velocity normalization 18

19 Relativistic (Ideal) Hydrodynamics conservation of energy and momentum and conserved currents (baryon-number) 4 1 With baryon current 5 equations for 6 fields close the system by supplying an equation of state, e.g. - EOS I : ultrarelativistic, ideal gas, P = ε/3 - EOS H: interacting resonance gas, P ~ 0.15 ε - EOS Q: Maxwell construction of those two: critical temperature T crit = MeV bag constant B 1/4 = 0.23 GeV latent heat ε lat =1.15 GeV/fm 3 19

20 QGP and hydrodynamic expansion hadronic phase and freeze-out initial state pre-equilibrium hadronization Intermediate-p t and late(r) times: dynamics of hadronization Recombination & Fragmentation Recombination + Fragmentation Model Results: spectra, ratios and elliptic flow Challenges: correlations, entropy balance & gluons 20

21 Armesto et al, nucl-ex/

22 Viscosity Fx A v x = η y Think of a not-quite quite-ideal ideal fluid: not-quite-ideal supports a shear stress Viscosity η is defined as Fx v x = η A y η ( momentum density) ( mean n p mfp = n p 1 nσ = p σ free path) 22

23 The event geometry in complicated events Degree of overlap Central Orientation with respect to overlap Peripheral Reaction Plane 23

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28 Elliptic flow Due to rapid expansion along the beam axis, an anisotropy in momentum space develops The elliptic flow is a measure of the anisotropy for the number of particles produced with respect to φ. It arises from the elliptical shape of the overlapping region in colliding nuclei and is usually parameterized with dependencies on parameters v 2 (p T ) and φ. The angular dependence is well known so elliptical flow is often used to mean the elliptic flow coefficient v 2 (p T ), - a measure of the small differences between the p t spectra with momenta pointing into and perpendicular to the reaction plane. 28

29 Isotropic expansion nano-kelvin gas of 6 Li atoms magnetic trap small scattering length leads to viscous hydrodynamics isotropic expansion when trapping field dropped Ken O Hara (Penn. St.) 29

30 Anisotropic expansion resonance tuned for large scattering length nearly ideal hydrodynamics anisotropic expansion when trapping field dropped Julia Velkovska 30

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32 Ollitrault ( 92) How does a system respond to spatial anisotropy? No secondary interaction y INPUT φ x Spatial Anisotropy Hydro behavior dn/dφ Interaction among produced particles OUTPUT dn/dφ Momentum Anisotropy 2v 2 0 φ 2π 0 φ 2π 32

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34 The coefficients ν 1 and v 2 Picture: UrQMD X p v = x 1 p = cos φ t ( Φ ) R Z b r v 2 2 px = 2 pt p p = cos ( φ Φ R ) 2 y 2 2 t p = p + t 2 x p 2 y XZ the reaction plane φ = tan 1 p y p x Anisotropic flow correlations with respect to the reaction plane 3 d N dp dy dφ t 2 d N 1 = ( 1+ 2v1 cos(φ ) + 2v2 cos( 2φ) +...) dp dy 2π t Directed flow Elliptic flow 34

35 Collision Geometry: Elliptic Flow Reaction plane y elliptic flow (v 2 ): z x gradients of almond-shape surface will lead to preferential emission in the reaction plane asymmetry out- vs. in-plane emission is quantified by 2 nd Fourier coefficient of angular distribution: v 2 calculable with fluid-dynamics The application of fluid-dynamics implies that the medium is in local thermal equilibrium! Note that fluid-dynamics cannot make any statements how the medium reached the equilibrium stage y x p y 35 p x

36 Hydrodynamics (for further reading) Ultrarelativistic Heavy Ion Collisions Author: Ramona Vogt Elsevier (2007) Hydrodynamic Models for Heavy Ion Collisions Authors: P. Huovien, P.V. Ruuskanen An invited review for Nov edition of Annual Review of Nuclear and Particle Physics; nucl-th/ Hydrodynamic Approaches to Relativistic Heavy Ion Collisions Author: Tetsufumi Hirano, invited talk given at XXXIV International Symposium on Multiparticle Dynamics, Sonoma, USA, July 26 - August 1, 2004 Journal-ref: Acta Phys.Polon. B36 (2005) ; nucl-th/

37 Percolation Parton percolation is a geometric, pre- equilibrium form of deconfinement an essential prerequisite for QGP production is cross-talk between the partons from different nucleons Size of the biggest cluster Low parton density High parton density parton density n c H. Satz, hep-ph v1 37

38 Percolation Model: geometrical transition In Central collisions nucleons undergo several interactions and, since each collision establishes a string, we will obtain a spaghetti like of intertwined overlapping QCD strings. Deconfinement is expected when there is enough internetting between nucleons. Deconfinement is a function of string size (QCD) and deconfinement string density H. Satz, M. Nardi 38

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42 Deconfinement and coalescence Believe that there is a very good chance that the effect of the light nuclei emission in heavy ion collisions may be one of the accompanying effects of percolation cluster formation and decay. that light nuclei could be formed as a result of coalescence mechanism. 42

43 Jets in heavy ion collisions Studying deconfinement with jets Fragmentation key QCD prediction: jets are quenched heavy nucleus radiated gluons quark di-quark Interaction at the quark (parton) level Models of jet suppression Various approaches; main points: ΔE med is independent of parton energy. X.-N. Wang and M. Gyulassy, Phys. Rev. Lett. 68 (1992) 1480 soft beam jet jet p TOT p L The same interaction at the hadron level ΔE med depends on length of medium, L. ΔE med gives access to gluon density dn g /dy or transport coefficient q ˆ = λ Leads to a deficit of high p t hadrons compared to p+p collisions (no medium). k T 2 43 p T Multiple soft scattering: Weidemann et al. Opacity expansion: Gyulassy et al. Twist expansion: Wang et al.

44 Jet Suppr. - Nuclear Modification Factor We can study jet suppression using leading hadrons We define a nuclear modification factor, R AA, in terms of the ratio of the p t spectra in nucleus-nucleus collisions divided by the p t spectra in p+p collisions R AA = 1 dn / dηd 2 AA p t pp T T AB dn pp / dηd 2 AA = N bin / σ inelastic p t We also define a nuclear modification factor, R CP, in terms of the ratio of the p t spectra in central nucleus-nucleus collisions divided by the p t spectra in peripheral nucleus-nucleus collisions R CP = ( d 2 N / dηdp t / N bin ) central b ( d 2 N / dηdp t / N bin ) peripheral Participant Binary Collisions With binary scaling, these factors as a function of p t are =1 44

45 Utilization of Hydro Results Jet quenching J/psi suppression Heavy quark diffusion Recombination Coalescence Thermal radiation (photon/dilepton) Meson J/psi c bar c Baryon Information along a path Information on surface Information inside medium 45

46 Using transparency function the rate of yields can be calculated R = n R = n 1 2 n n 1 2 (here e.g. n 1 and n 2 could be heavy flavor particles yields with fixed values of and) as a function of centrality, the masses and energy, it is expected to get the necessary information on the properties of the nuclear matter. With percolation model and experimental data on the behaviour of the nuclear modification factors it is possible to get information on the appearance of the anomalous nuclear transparency as a signal of formation of the percolation cluster. 46

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48 High energy limit of QCD A universal form of matter at high energy Color Glass Condensate (CGC)!! Gluons Gluons have have color color created created from from frozen random color color source, source, that that evolves slowly slowly compared to to natural natural time time scale scale High High density density!! occupation number ~ 1/α 1/α s at s at saturation higher energy Dilute gas CGC: high density gluons 48

49 The Color Glass Condensate and Glasma What is the high energy limit of QCD? What are the possible form of high energy density matter? How do quarks and gluons originate in strongly interacting particles? Art due to Hatsuda and S. Bass CGC Initial Singularity Glasma sqgp Hadron Gas 49

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51 Extra Slides 51

52 High p T Particle Production High p T (> 2.0 GeV/c) hadron production in pp collisions ~ Jet: A localized collection of hadrons which come from a fragmenting parton Parton Distribution Functions Hard-scattering cross-section Fragmentation Function a hadrons hadrons d c leading particle b d dyd p had = z p c, z <1 energy needed to create quarks from vacuum h 0 σ pp 2 2 dσ Dh / c = K dxadxb f a ( xa, Q ) f b ( xb, Q ) ( ab cd ) 2 pt abcd dtˆ πzc Collinear factorization 52

53 dn dyd f S h AB 2 pt a/ A g( k A D 0 h/ c ( x a ( x a a, Q ) g( k, Q c 2 a * ( zc, Q πz ABK b ) S c 2 c ) ) B abcd b/ B ( x dσ ( ab cd) dtˆ * 1 zc dεp( ε) 0 z High p T Particle Production in A+A = 2 ) f ( x Parton Distribution Functions Intrinsic k T, Cronin Effect b dx b, Q 2 b a, Q ) dx 2 ) b d 2 Known from pp and pa k Shadowing, EMC Effect Hard-scattering cross-section Partonic Energy Loss Fragmentation Function a d 2 k b * p c = p c * z c = z c (1 ε ) /(1 ε ) hadrons a d c b leading particle suppressed 53

54 Expect to get a result which would demonstrate the changing of absorption properties of medium depending on the kinematical characteristics of heavy particles. A comparison of yields in different ion systems by using nuclear modification factors such as R CP (involving Central and Peripheral collisions) should provide information on hadronization. R CP highlights the particle type dependence at intermediate p T as suggested by coalescence models - -- hadrons result from the coalescence of quarks in the dense medium. At high p T, jet fragmentation becomes the dominant process to explain the hadron formation. Thus, the quark constituents may be the relevant degrees of freedom for the description of the collision. 54

55 p t limit for hydrodynamics Particles with very large transverse momenta (jets) are never expected to suffer sufficiently many interactions with the fireball medium to fully thermalize before escaping; hence a hydrodynamic approach can never work at very high p t. However, we can turn this inescapable failure of hydrodynamics in small collision systems and at high p to our favour : since ideal fluid dynamics appears to work well in near-central collisions, and at low p t GeV/c (...), we can study its gradual breakdown at larger impact parameters, rapidities and transverse momenta in order to learn something about the mechanisms for the approach h to thermal equilibrium at the beginning of the collision and the decay of thermal equilibrium near the end of the expansion stage, Hence about the transport properties of the early quark gluon plasma and the late hadron resonance gas created in these collisions. Breakdown also consistent with the expectated behavior of v 2 with varying shear viscosity. 55

56 Elliptic flow y x v n = cos n φ x y p y z p x Look at non-central collisions Overlap region is not symmetric in coordinate space Almond shaped overlap region Larger pressure gradient in x-z plane than in y direction Spatial anisotropy -> > momentum anistropy Process quenches itself -> > sensitive to early time in the evolution of the system Sensitive to the equation of state Perform a Fourier decomposition of the momentum space particle distributions in the x-y plane v n is the n th harmonic Fourier coefficient of the distribution of particles with respect to the reaction plane v 1 : directed flow v 2 : elliptic flow 56