Second-Order Relativistic Hydrodynamics

Transcription

1 Second-Order Relativistic Hydrodynamics Guy D. Moore, Mark York, Kiyoumars Sohrabi, Paul Romatschke, Pavel Kovtun Why do we want to do relativistic Hydro? Why second order hydro, and what are coefficients? Perturbative Calculation of Coefficients Kubo Relations for Coefficients Self-consistency: hydro s contrib. to hydro coeff. Conclusions Virginia, 12 April 2011: page 1 of 34

2 Heavy ion collisions Accelerate two heavy nuclei to high energy, slam together. Just before: Lorentz contracted nuclei Virginia, 12 April 2011: page 2 of 34

3 After the scattering: region where nuclei overlapped: Flat almond shaped region of q, q,g which scattered. 2 thousand random v quarks+gluons: isotropic in xy plane Virginia, 12 April 2011: page 3 of 34

4 Behavior IF no re-interactions (transparency) Just fly out and hit the detector. Detector will see xy plane isotropy Virginia, 12 April 2011: page 4 of 34

5 local CM motions Pressure contours Expansion pattern Anisotropy leads to anisotropic (local CM motion) flow. Virginia, 12 April 2011: page 5 of 34

6 Free particle propagation: System-average CM flow velocities v 2 x,cm > v2 y,cm Must have local CM p 2 x < p 2 y so total p 2 x = p 2 y Efficient Equilibration: System-average CM flow still has v 2 x,cm > v2 y,cm system changes locally towards Tlocal xx yy CM = Tlocal CM Adding these together, Ttot,labframe xx yy > Ttot,labframe Net Elliptic Flow v 2 p2 x p 2 y p 2 x +p2 y measures re-interaction Virginia, 12 April 2011: page 6 of 34

7 Elliptic flow is measured v (a) 200 GeV Au + Au (minimum bias) STAR data π± π K K 0 - s p p Λ Λ+Λ - Hydrodynamics results v 2 (%) 10 5 (b)130 GeV Au + Au (minimum bias) + STAR π + π- hydro EOS Q hydro EOS H p + p p T Transverse momentum (GeV/c) (GeV/c) p T STAR experiment, minimum bias.... We should try to understand it theoretically. First try: ideal hydrodynamics (works OK!) Virginia, 12 April 2011: page 7 of 34

8 Ideal Hydrodynamics Ideal hydro: stress-energy conservation µ T µν = 0 (4 equations, 10 unknowns) plus local equilibrium assumption: T µν = T µν eq = ǫu µ u ν + P(ǫ) µν, u µ u µ = 1, µν = g µν u µ u ν depends on 4 parameters (ǫ, 3 comp of u µ ): closed. works pretty well for heavy ions. But quantify corrections! Virginia, 12 April 2011: page 8 of 34

9 Nonideal Hydro Assume that ideal hydro is good starting point, look for small systematic corrections. Near equilibrium iff t therm t vary,l vary /v (so small) Allows expansion of corrections in gradients: T µν = T µν eq + Π µν [,ǫ,u] Π µν = O( u, ǫ) + O( 2 u,( u) 2,...) + O( 3...) For Conformal theory T µ µ = 0 = Π µ µ, 1-order term unique: Π µν = ησ µν, σ µν = µα ( νβ α u β + β u α 2 ) 3 g αβ u Coefficient η is shear viscosity. Virginia, 12 April 2011: page 9 of 34

10 So why not consider (Navier-Stokes) T µν = ǫu µ u ν + P µν ησ µν? Because in relativisitc setting, it is Acausal: shear viscosity is transverse momentum diffusion. Diffusion t P 2 P has instantaneous prop. speed. Müller 1967, Israel+Stewart 1976 Unstable: v > c prop + non-uniform flow velocity propagate from future into past, exponentially growing solutions. Hiscock 1983 Problem: short length scales, η σ P. Numerics must treat these scales (or there s numerical viscosity ) Virginia, 12 April 2011: page 10 of 34

11 Israel-Stewart approach Add one second order term: Π µν = ησ µν + ητ π u α α σ µν Make (1 st order accurate) ησ Π in order-2 term: τ π u α α Π µν τ Πµν π = ησ µν Π µν Relaxation eq driving Π µν towards ησ µν. Momentum diff. no longer instantaneous. Causality, stability are restored (depending on τ π ) But why only one 2 nd order term??? Virginia, 12 April 2011: page 11 of 34

12 Second order hydrodynamics It is more consistent to include all possible 2 nd order terms. Assume conformality and vanishing chem. potentials: 5 possible terms Baier et al, [arxiv: ] Π µν 2 ord. = ητ π [u α α σ µν σµν α u α ] +λ 2 [ 1 2 (σµ αω να + σ ν αω µα ) (trace) + λ 1 [σ µ ασ να (trace)] +λ 3 [Ω µ αω να (trace)] + κ (R µν...), Ω µν 1 2 µα νβ ( α u β β u α ) [vorticity]. Let s learn what we can about this theory, its 6 coeff s ] Virginia, 12 April 2011: page 12 of 34

13 Step 1: What do σ µν, Ω µν mean? Consider y v x 0 and x v y 0: Each pattern is shear-flow. But not purely shear flow! Virginia, 12 April 2011: page 13 of 34

14 Step 1: What do σ µν, Ω µν mean? Same-sign x v y = y v x Opposite-sign x v y = y v x Shear flow Vorticity Two basic local measures of flow nonuniformity. First order: Π µν = ησ µν as it s symmetric! Fluid pushing back against shear flow Virginia, 12 April 2011: page 13 of 34

15 τ π : if shear flow σ µν turns on, delay in Π µν turning on λ 2 : if shear makes Π µν 0, vorticity rotates Π µν axis from shear axis. Sensible sign if λ 2 < 0 (sorry) λ 1 : some nonlinearity. λ 3 : rotate about z axis T zz reduced Virginia, 12 April 2011: page 14 of 34

16 I would like to calculate these coefficients. Two cases: weak coupling, strong-coupled N=4SYM That failing, I want a rule for relating them to equilibrium field theory correlators (Kubo relation) In any case I want to understand consistency or limitations of 2 nd order hydro theory. Goals of remainder of the talk Virginia, 12 April 2011: page 15 of 34

17 Theories where I can calculate: Weakly coupled QCD (realistically, α s < 1/20!) N=4 SYM in infinite N c and coupling limit What I expect to find: There should be some equilibration time scale τ. A term with n deriv s should be P/τ n. Hence, certain ratios should be dimensionless. Use η as my standard and build dim less ratios. If ratios robust, use as priors : only fit η to data Virginia, 12 April 2011: page 16 of 34

18 QCD vs SYM comparison η/s behaves as 1/α 2 s ln(1/α s ) diverges at weak coupling. But ratios stay finite! Ratio QCD value SYM value ητ π (ǫ+p) η 2 5 to λ 1 (ǫ+p) η to λ 2 (ǫ+p) η 2 10 to κ(ǫ+p) η λ 3 (ǫ+p) η Good news: Not qualitatively different. Kinetic theory relation λ 2 = 2ητ π not actually general. Virginia, 12 April 2011: page 17 of 34

19 Kubo formulae We want expressions which relate the transport coefficients to equilibrium correlation functions in the plasma fluct-diss Would provide rigorous definition of η,λ 123,... Example: long known that η is given by η = lim ω 0 d dω d 3 x dt e iωt [ T xy (x,t), T xy (0, 0) ] Θ(t) Similar relations for second-order transport coefficients? Virginia, 12 April 2011: page 18 of 34

20 How to get Kubo relations Find framework where I can compute T µν using hydro or using field theory, both should be valid. Time-varying geometry does the job: Start at t 0 with flat-space, equilibrium thermal system ρ = e HT, g µν = η µν At some time t 0 < 0 start deforming metric g µν = η µν + h µν (x) in such a way as to force the system to experience shear and vorticity Choose h µν small and slowly varying so you stay near equilibrium and gradient expansion, hydro are valid Virginia, 12 April 2011: page 19 of 34

21 Give a hydro theorist h xy (z,t), h 0x (y) nonzero. Ask them what T µν (0) will be. Answer: T µν = (ǫ + P)u µ u ν + Pg µν + Π µν First, find ǫ,u: Hydro says µ T µν = 0 u µ = (1, 0, 0, 0) + O( 2 ). Then u µ = (1,h 0x, 0, 0), Γ x yt etc nonzero. They give rise to nonzero σ xy, Ω xy, etc: σ xy = t h xy, Ω xy = y h 0x /2 Other terms R xy, u σ xy found similarly. Virginia, 12 April 2011: page 20 of 34

22 T xy at O(h) and O( 2 ), for h xy 0: T xy = η t h xy + ητ π 2 t h xy κ 2 ( 2 t h xy + 2 zh xy ) and T xy at O( 2,h 2 ) for h xz (t), h yz (t), h x0 (z), h y0 (z) nonzero: Π xy = η t (h xz h yz ) + κ ) (h xz t 2 h yz +h yz t 2 h xz + λ 1 t h xz t h yz 2 +ητ π ( 1 2 th xz z h 0y th yz z h 0x t h xz t h yz h xz 2 t h yz h yz 2 t h xz ) λ 2 4 ( th xz z h 0y + t h yz z h 0x ) + λ 3 4 zh 0x z h 0y So at O(h) T xy depends on η,τ π,κ; at O(h 2 ), depends on all 6! Virginia, 12 April 2011: page 21 of 34

23 Give field theorist h xy (z,t), etc nonzero. Ask them what T xy will be. T µν (t) = Tr e HT e iht ˆT µν e iht, T µν = 2 g gl h µν with H = H[h(t )]! Schwinger-Keldysh in g µν = η µν + h µν : W ln D(Φ 1,Φ 2,Φ 3 ) e is 1[h 1,Φ 1 ] is 2 [h 2,Φ 2 ] S 3 [Φ 3 ] C = S 1 [h 1 ], S 2 [h 2 ] depend on independent fields and metrics! T 1 = 2iδW δh 1, T 2 = +2iδW δh 2 Virginia, 12 April 2011: page 22 of 34

24 Introduce average and difference variables: h r = h 1+h 2 2, h a = h 1 h 2, T r = T 1+T 2 2, T a = T 1 T 2 Note, due to signs e is 1 is 2, T r = 2iδW δh a, T a = 2iδW δh r δ/δh a T. Then set h a = 0, h r = h, expand in h: T µν h = G µν r (0) d 4 xg µν,αβ ra (0,x)h αβ (x). Take d 4 xd 4 yg µν,αβ,γδ raa (0,x,y)h αβ (x)h γδ (y) G µν,αβ... ra... (0,x...) ( i)n 1 ( 2i) n δ n W δg a,µν (0)δg r,αβ (x)... = ( i) n 1 T µν r gµν =η µν (0)Ta αβ (x)... + c.t. Virginia, 12 April 2011: page 23 of 34

25 Now equate the hydro, field theorist answers: T xy = η t h xy + ητ π t 2 h xy κ 2 = d 4 x G xy,xy ra (0,x)h xy (x) ( 2 t h xy + 2 zh xy ) Introduce Fourier transform G xy,xy ra (ω,k) = d 4 xe i(ωt kz) G xy,xy ra (0, x) and use that h slowly varying, find BRSSS η = i ω G xy,xy ra (ω,k) ω=0=k, κ = k 2 z G xy,xy ra (ω,k) ω=0=k, ητ π = 1 ( 2 2 ωg xy,xy ra (ω,k) k 2 z G xy,xy ω=0=k ra (ω,k)). Virginia, 12 April 2011: page 24 of 34

26 Repeat for T xy and O(h 2 ) terms: Π xy = η t (h xz h yz ) + κ 2 (h xz 2 t h yz +h yz 2 t h xz ) + λ 1 t h xz t h yz +ητ π ( 1 2 th xz z h 0y th yz z h 0x t h xz t h yz h xz 2 t h yz h yz 2 t h xz ) = λ 2 4 ( th xz z h 0y + t h yz z h 0x ) + λ 3 4 zh 0x z h 0y d 4 xd 4 y (G xy,xz,yz raa (0,x,y)h xz (x)h yz (y) +G xy,xz,0y raa h xz (x)h 0y (y) + Graa xy,yz,0x h yz (x)h 0x (y) ) +G xy,0x,0y raa (0,x,y)h 0x (x)h 0y (y) Virginia, 12 April 2011: page 25 of 34

27 Introduce Fourier transforms again: Graa xy,xz,0y (p,q) d 4 xd 4 ye i(p x+q y) G xy,xz,0y raa (0, x, y) etc Read off 2 nd order Kubo relations: λ 1 = ητ π lim p t,q t 0 λ 2 λ 3 = 2ητ π 4 lim p t,q 0 = 4 lim p,q 0 2 p t q t 2 p z q z 2 p t q z lim lim p,q 0 Gxy,xz,yz raa (p, q) lim p,q t 0 Gxy,xz,0y raa (p, q) p t,q t 0 Gxy,0x,0y raa (p,q). Virginia, 12 April 2011: page 26 of 34

28 Nature of κ and λ 3 κ and λ 3 have Kubo relations NOT involving t s. May (must!) set frequency ω = 0 from outset: 2 κ = lim q 0 qz 2 λ 3 = 6 lim p, q 0 G xy,xy ra ( q,ω = 0) 2 p y q y G xx,0x,0x raa ( p,ω p = 0, q,ω q = 0) But G ra... (ω = 0) = ( ) n 1 G E (ω E = 0) Euclidean func. Weak-coupling expansions: κ,λ 3 = T 2 (O(1) + O(g,g 2,...)) Leading weak-coupling values calculable and nonzero Virginia, 12 April 2011: page 27 of 34

29 But is hydro even consistent? We said Π µν = O( u) + O( 2 u, ( u) 2 ) +... based on assumption thermalization is local, microscopic. Hydro itself predicts long-lived shear,sound modes: ( 0 = µ T µν = (ǫ+p)u µ u ν + Pg µν ησ µν) fluctuations in u µ,ǫ: dispersion relations ω shear = i η ǫ + P k2, ω sound = ± k 3 + i 2η 3(ǫ + P) k2 Small k: long lived, dissipation not local,microscopic Virginia, 12 April 2011: page 28 of 34

30 Hydro Waves Contribute to Viscosity! Consider shear flow: Flow decays because x-momentum leaves (diffuses from) flowing region. One mechanism: propagation of hydro (sound) waves! Virginia, 12 April 2011: page 29 of 34

31 How to compute hydro contribution to hydro Above we found G xy,xy ra (ω) = P iηω + ητ π ω Calculate contrib. of hydro modes themselves to G xyxy. Feynman rules: T ij = (ǫ + P)u i u j + Pg ij, u i u j (k, ω) = [ ] γ η = η ǫ+p, γ η = 4 3 γ η T ǫ + P + T ǫ + P (δ ij ˆk iˆkj )2γ η k 2 (γ η k 2 iω)(γ η k 2 + iω) shearwave (ˆk iˆkj )2γ ηk 2 ω 2 (ω 2 k 2 /3) 2 + (γ ηk 2 ω) 2 soundwave Think of hydro as IR effective theory, η etc are Wilson coeff. Virginia, 12 April 2011: page 30 of 34

32 Computing G xy,xy ra (ω,k = 0) Straightforward application of Feynman rules: G xy,xy ra (ω)[hydro] = iω ( 17Tkmax 120π 2 γ η ) +(i+1)ω ( )3 3 2 T 2 240πγη 3/2 k max : k-scale above which hydro incorrect/inconsistent. iω term: extra contrib. to η iω 3/2 : effective ω dependence of η. ω 3/2 : like τ π but wrong ω dependence. Virginia, 12 April 2011: page 31 of 34

33 Lesson: η Small η: freer propagation of sound, shear modes. More efficient momentum transport, raising η. Depends on k max. Where does hydro break down? Scale where it s no longer self-consistent. Safe guess: k max < τ 1 π /2. In N=4 SYM, this is about 2T. N=4SYM: added η/s is 1/N 2 c. Weak coupling: η from hydro α 4 while η tot α 2 Real QCD: η s =.16: add η s =.08: add 0.036! Virginia, 12 April 2011: page 32 of 34

34 Lesson: τ π Weak coupling and large N c : comparing N 0 c α 3 T 5/2 ω 3/2 vs N 2 c α 4 T 2 ω 2 Deep IR, ω 3/2 term wins, 2-order hydro breaks. But scale where ω 3/2 term takes over is ω N 4 c α 14 T. Check that ω where they equal is more IR than your physics and then use 2-order hydro! N c = 3 = N f QCD, T = 200MeV, η s =.16: ω T 20 Safe! N c = 3 = N f QCD, T = 200MeV, η s =.08: ω 7T Problem! Virginia, 12 April 2011: page 33 of 34

35 Conclusions Hydro seems sensible framework in heavy ion coll. Need 2 nd order Hydro, 6 hydro coefficients! Pert. computation of 2 nd order Hydro: dim less ratios same order as N=4SYM, differ in detail Kubo relations for nonlinear coefficients found. κ,λ 3 special (really thermodynamic) Hydro waves contribute to hydro coefficients! Self-consistency issues if η too small, and very low freq. Virginia, 12 April 2011: page 34 of 34