Shear Viscosity in an Anisotropically Expanding QGP

Transcription

1 Shear Viscosity in an Anisotroically Exanding QGP Steffen A. Bass Duke University QGP Proerties: key findings The sqgp: ro s and con s Anomalous Viscosity work in collaboration with M. Asakawa, B. Mueller & C. Nonaka M. Asakawa, S.A. Bass & B. Mueller: Phys. Rev. Lett. 96: 5301 (006) C. Nonaka & S.A. Bass: nucl-th/ M. Asakawa, S.A. Bass & B. Mueller: Prog. Theo. Phys. 116 in rint, he-h/ Steffen A. Bass Viscosity of an exanding QGP #1

2 Lattice Gauge Theory redictions: QCD bulk roerties Steffen A. Bass Viscosity of an exanding QGP #

3 Lattice: Phase Transition Proerties Forcrand, SEWM (F. Karsch, he-lat/ ) LGT redicts a hase-transition to a state of deconfined nearly massless quarks and gluons: QGP range for T C : MeV at finite baryon-density, there exists a tri-critical oint and a mixed hase Steffen A. Bass Viscosity of an exanding QGP #3

4 Lattice: BS-Correlations different DOF introduce different correlations between conserved charges: C BS BS B S BS = 3 = 3 S S S χ us QP-QGP: C BS =1 vs. BS-QGP: C BS 0.61 C BS : V. Koch, A. Majumder & J. Randru, PRL (005) BS-QGP: E. Shuryak, I. Zahed, PRC (004); PRD (004) calculable on the Lattice: 1 F 1 F = χ = V µ µ µ C BS ss u s V s χ = 1+ ds + χ χ Gavai & Guta, PRD 73, 14006, 006 ss us T>T C : C BS =1 mesonic as well as baryonic bound-states (involving quarks) above T C ruled out via χ BS and µ B -derivatives thereof Lattice confirms quasi-quark nature of flavor carriers in the deconfined hase Steffen A. Bass Viscosity of an exanding QGP #4

5 Relativistic Heavy-Ion Collisions: robing the QGP at RHIC Steffen A. Bass Viscosity of an exanding QGP #5

6 What is a jet? hadrons q hadrons Jet-Quenching: Basic Idea q leading article leading article fragmentation of hard scattered artons into collimated jets of hadrons + reactions rovide a calibrated robe, well described by QCD what haens if artons traverse a high energy density colored medium? hadrons hadrons artons loose energy and/or fragment differently than in the vacuum: radiative energy loss qˆ transort coefficient q is sensitive to density of (colored) charges Steffen A. Bass Viscosity of an exanding QGP #6 q = ρ q q dq q leading article suressed L dσ dq leading article suressed q g ρσ k = T µ λ f

7 suression can be exerimentally quantified in terms of R AA ratio: R AA d N dydt AA AA d N dydt Ncoll q-hat at RHIC sqgp? QGP RHIC data Baier lot Pion gas Cold nuclear matter RHIC data shows values for q-hat far larger than exected even for a QGP! Steffen A. Bass Viscosity of an exanding QGP #7

8 Relativistic Fluid Dynamics & Parton Recombination Steffen A. Bass Viscosity of an exanding QGP #8

9 Ellitic Flow at RHIC failure of ideal hydro: reco and QCD domain Mass slitting characteristic roerty of hydrodynamics Steffen A. Bass Viscosity of an exanding QGP #9

10 in leading order of v, recombination redicts: v ( ) M t v ( ) B t Recombination: Parton Number Scaling of v t v t 3v 3 qqq T,µ,v qq smoking gun for hadron formation from deconfined quarks measurement of artonic v Steffen A. Bass Viscosity of an exanding QGP #10

11 Hydro evolution from QGP to hadrons Full 3-d Hydrodynamics QGP evolution C. Nonaka & S.A. Bass: nucl-th/ see also Hirano and Nara Hadronization T C Cooer-Frye formula Monte Carlo low (no) viscosity T SW UrQMD hadronic rescattering t fm/c high viscosity good agreement with data for η QGP =0 Steffen A. Bass Viscosity of an exanding QGP #11

12 Current Picture of QGP Structure LGT & Parton Recombination: at T C, QGP degrees of freedom carry the quantum numbers of quarks and recombine to form hadrons Jet-Qenching & Ellitic Flow: QGP roduced at RHIC has very large oacity behaves like a nearly ideal fluid (vanishing viscosity) Conclusion: the strongly interacting QGP (sqgp) combine QP nature & large oacity: very high gluon densities ideal fluid: vanishing mean free ath of quarks and gluons (i.e. very large arton cross sections) Steffen A. Bass Viscosity of an exanding QGP #1

13 The sqgp Dilemma D. Molnar microscoic transort theory shows that assuming quasi-article q & g degrees of freedom would require unhysically large arton cross sections to match ellitic flow data even for λ 0.1 fm (close to uncertainty bound) dissiative effects are large gluon densities needed for jetquenching calculations may be too large comared to measured entroy does a small viscosity have to imly that matter is strongly interacting? Paradigm shift needed: consider effects of (turbulent) color fields Steffen A. Bass Viscosity of an exanding QGP #13

14 (Anomalous) Viscosity Steffen A. Bass Viscosity of an exanding QGP #14

15 What is is viscosity? shear and bulk viscosity are defined as coefficients in the exansion of the stress tensor in gradients of the velocity field: ( δ u ) η ( u u ) T = εuu + P + u + δ u + ς δ u ik i k ik i k i k k i 3 ik ik microscoically, η is given by the rate of momentum transort: η nλ = 1 3 f 3σ the unitary limit on cross sections suggests that η has a lower bound: tr σ 4π tr 3 η 1π Steffen A. Bass Viscosity of an exanding QGP #15

16 Anomalous Viscosity Anomalous Viscosity: any contribution to the shear viscosity not exlicitly resulting from momentum transort via a transort cross section Plasma hysics: A.V. = large viscosity induced in nearly collisionless lasmas by long-range fields generated by lasma instabilities. Astrohysics - dynamics of accretion disks: A.V. = large viscosity induced in weakly magnetized, ionized stellar accretion disks by orbital instabilities. Biohysics: A.V. = The viscous behavior of non-homogeneous fluids or susensions, e.g., blood, in which the aarent viscosity increases as flow or shear rate decreases toward zero. (From: htt:// Steffen A. Bass Viscosity of an exanding QGP #16

17 Can the QGP viscosity be anomalous? can the extreme oaqueness of the QGP be exlained without invoking suer-strong couling? exanding lasmas (such as the QGP at RHIC) always have anisotroic momentum distributions roerties of anisotroic lasmas: lasma turbulence: random, nonthermal excitation of coherent field modes with ower sectrum similar to the vorticity sectrum in a turbulent fluid [P(k) ~ 1/k ]. lasma turbulence arises naturally in lasmas with an anisotroic momentum distribution (Weibel-tye instabilities). soft color fields generate anomalous transort coefficients, which may give the medium the character of a nearly erfect fluid even at moderately weak couling. Steffen A. Bass Viscosity of an exanding QGP #17

18 Turbulent Color Fields Weibel Instabilities calculated in HTL formalism see for examle: Rebhan, Romatschke & Strickland, PRL 94: (005) Romatschke & Strickland, PRD 68: (003) Arnold, Lenaghan & Moore, JHEP 0308, 00 (003) Mrowczynski, PLB 314, 118 (1993) Mrowczynski, PLB 393, 6 (1997) Steffen A. Bass Viscosity of an exanding QGP #18

19 Anomalous Viscosity in QCD Steffen A. Bass Viscosity of an exanding QGP #19

20 Vlasov-Boltzmann Eqn for Color Charges v µ x µ f a a ( r, t) gf f ( r,, t) C [ f ] with arton distribution functions: f f a ( r,, t) = dq f ( r,, Q, t), + + = 0 a ( r,, t) = dqq f ( r,, Q, t) solution for Vlasov term is given by: color Lorentz force: a a a ( π ) = E + v B Steffen A. Bass Viscosity of an exanding QGP #0 F ( x x ) 4 ik a C dk 4 e b f (, x) ig d x U 4 ab ( x, x ) x Nc 1 v k+ iε ( ) f ( ) = F resulting Vlasov force term: ( π ) ( x x ) 4 ik a a C a dk 4 e b gf ( x ) f (, x) = ig F ( ) d xu 4 ab( x, x) Nc 1 x v k iε x F + ( ) f ( )

21 Turbulent Color Fields effect on color charges can be described via ensemble average (w/ ): ( ) U ( x, x ) ( ) f ( ) = ( ) U ( xx, ) ( ) ( ) F x F x F x F x f a b a b i ab j i ab j correlation function of fields deend only on x-x and fall off raidly: (el) (el) ( x) Uab ( x, x ) ( x ) = Φτ ( t t ) Φ σ ( x x ) (mag) (mag) ( x) Uab ( x, x ) ( x ) = Φτ ( t t ) Φ σ ( x x ) E E E E a b a a i j i j B B B B E a b a a i j i j a i b ( x) U ( xx) B ( x ) ab, = 0 with Gaussian correlators Φ el and Φ mag j solution for Vlasov force term: a a gc el a a mag a a gf x f = τm EE + τm BB v v N 1 i c i j with the memory time: (electric and magnetic fields are uncorrelated) ( ) ( ) ( ) f ( ) D τ i j i j ( ) f ( ) 1 e l/mag ( el/mag) g) m = d Φ τ σ ( ) (el/ma t Φ v ( ) Steffen A. Bass Viscosity of an exanding QGP #1 f f j ( ) t t t t

22 Shear Viscosity from Linear Resonse ensemble-average: diffusive Vlasov-Boltzmann (Fokker-Planck) Equation µ v f ( r,, t) (,, ) [ ] D f r t + C f = 0 µ x Chaman-Enskog formalism: small erturbation of thermal equilibrium f 0 : f ( r, ) = f0( ) + δ f ( r, ) = f0( ) 1+ f1 ( r, )( 1± f0( ) ) with local erturbation: ( r) ( ) ( ) f = u 1, i j ij ET () measures magnitude of deviation from equilibrium 3 d ik, 3 i k f ( ) extract viscosity via comarison of E- tensor: T = r ( π ) E with macroscoic definition of the viscous stress: ( ) T = T + δt = Pδ + εuu η u ςδ u (0) ik ik ik ik i k ik ik with: ressure P, energy density ε, shear viscosity η and bulk visc. ζ η = ( 3 ) 15T ( π ) E E Steffen A. Bass Viscosity of an exanding QGP # comarison yields for shear viscosity: 1 d f

23 Diffusive Vlasov-Boltzmann Eqn: solution drift term: v D µ µ v f ( r,, t) (,, ) [ ] D f r t + C f = 0 µ x x µ force term: f 0 = ± 1 m τ i j ( ) f ( 1 f ) ( u) el 0 0 ij D m ET 3 T E E ( ε / T ) el a a mag a a ( ) f ( ) = τ EE + τ BB ( v ) ( v ) f ( ) collision term: [ ] gc N c ( π ) 1 m i j m i j i i j [see Arnold, Moore & Yaffe: JHEP 0011,1 (000)] Steffen A. Bass Viscosity of an exanding QGP #3 [ ] [ ] j C f I f I f = 1 = 1 3 dk 1 = 3 1vrel f0 ( ) f0 ( k) f1 ( ) + f1 ( k) f1 ( ) f1 ( k ) vanishes in equilibrium, 1 st contribution linear in f 1 : I f dσ ( ): effective color conductivity and differential cross section dσ (leading log: use ME s w/ highest IR div.)

24 Examle: transverse color magnetic fields additional aroximation: Drift term: Force term: v µ x µ D f a a 1 a a BB i j = ( δij δizδ jz ) B ; EE i j 0 i j ( ) = f ( 1± f ) ( u) ( ) f ( ) Steffen A. Bass Viscosity of an exanding QGP #4 3C = E T g ( Nc 1) ( Nc 1) B ij τ mag m E T E T ( ) = mag 3C g B τ m ( 1± ) ( ) f f u 0 0 Ignoring the collision term, the contribution of the Vlasov term yields: momentum aninsotroy: anomalous viscosity: η (gluon) A ( )( N ) 6 16ς 6 c 1 T = mag π Nc g B τ m η (quark) A ( ) 6ς 6 N N f i j ij 6 c = mag π g B τ m T

25 Comlete Shear Viscosity Calculation solution of the full Vlasov-Boltzmann eqn. coincides w/ minimum of a quadratic functional W[f 1 ] and can be obtained via a variational rincile: ( ) 3 d µ f0 1 ( ) ( ( ) ) W f 1 f 3 1 v + D δ f + I f 1 = min µ ( π ) x minimum defines otimal solution for f 1 or () in transort eqn simle ansatz: = A / T, A= A, A r ( ) ( q g) minimization results in set of linear eqn. for A q & A g : with: ( ) ( ) 3ς 5 = 3π Nc 1 15 NN c 8 f ( ) a + a A = r [in absence of turbulent fields, this reduces to result of Arnold, Moore & Yaffe: JHEP 0011,1 (000)] Steffen A. Bass Viscosity of an exanding QGP #5 a A ( ) 3ς 4 = π g B τ A mag m N 3 5 T 0 ( ) C c 0 7 N 8 π N 1 c 0 c N π N f Nc 1 1 ac = g ln g ( Nc + N f ) N 18 c 1 1 f N 8 f

26 Shear Viscosity: full exression use result from linear resonse, insert ansatz for () and erform momentum integration: ( ) 3 4ς 5 T 15 3 ( c 1) Ag c f q η = N N N A r A 3π + = 8 4 result of variational calculation then yields: 3 η = r ( a ) 1 A + ac r 4 full exression of shear viscosity can be rewritten as: η 1 1 η 1 A C = η + (additivity of relaxation rates) η A 3 1 = r aa r 4 with and 3 1 = r ac r 4 Steffen A. Bass Viscosity of an exanding QGP #6 η C

27 Anomalous Viscosity Estimate follow work by Romatschke & Strickland in PRD 68: (003): erturbation of equilib. distribution: connection to momentum anisotroy : comarison with linear resonse yields: η A T = c0 s g u 3/5 ( ) case of transverse B-fields, make ansatz: ξ z Steffen A. Bass Viscosity of an exanding QGP #7 f 1 = ET 3 u ξ = T 15η u η u ξ = = 10 T s T g B 00 n = b g T ξ memory time determined by coherence length of fields: self-consistent anomalous shear viscosity: τ m = d gt ηc 5 < collisional viscosity 4 1 s g ln g 0 ξ (HTL weak-couling limit)

28 initial state Collisional vs. Anomalous Viscosity QGP and hydrodynamic exansion hadronic hase and freeze-out re-equilibrium η η A viscosity:?? hadronization η η C η η HG relaxation rates are additive sumrule for viscosities: η = η + smaller viscosity dominates in system w/ viscosities! η A C temerature evolution: η A < ηc ηa < ηc ηa > ηc HG η Steffen A. Bass Viscosity of an exanding QGP #8

29 Summary Heavy-Ion collisions at RHIC have roduced a state of matter which behaves similar to an ideal fluid a small viscosity does not necessarily imly strongly interacting matter! (turbulent) color fields induce an anomalous viscosity, which kees the total shear-viscosity small during the QGP evolution erfect liquidity in the weak couling limit note that the early Universe with its much slower isotroic exansion most likely did not osses an anomalous viscosity RHIC Big Bang! Steffen A. Bass Viscosity of an exanding QGP #9

30 The End Steffen A. Bass Viscosity of an exanding QGP #30