Collaborators: Aleksas Mazeliauskas (Heidelberg) & Derek Teaney (Stony Brook) Refs: , /25

Transcription

1 Collaborators: Aleksas Mazeliauskas (Heidelberg) & Derek Teaney (Stony Brook) Refs: , /25

2 1. Introduction 2/25

3 Ultra-relativistic heavy-ion collisions and the Bjorken expansion Heavy-ion collision is an overwhelmingly complicated system Assumption that hydrodynamics is applicable in the collision center Bjorken expansion is a simple and characteristic hydrodynamic solution An approximate boost invariance along the beam axis u z = z t ( z t) Bjorken expansion is a one-dimensional Hubble expansion 3/25

4 Hydrodynamics with noise Kovtun-Yaffe (03), Gavin-AbdelAziz (06), Kovtun-Moore-Romatschke (11), Kapusta-Muller-Stephanov (12), Young-Kapusa-Gale-Jeon-Schenke (15), Murase-Hirano (16), Yan-Grönqvist (16), Gavin-Moschelli-Zin (16), YA-Mazeliauskas-Teaney (17), Thermal fluctuations for example Landau-Lifshitz N gg (t, k) g i (t, k)g j (t, k) =(e + p)tδ momentum, g i T 0i equilibrium Conceptually important (required by the FDT) Larger in smaller systems: N particle in the heavy-ions Essential near a critical point How do thermal fluctuations evolve during a Bjorken expansion? How do thermal fluctuations change the Bjorken expansion? ij 4/25

5 Kinetic regime of hydrodynamic fluctuations anewscalek 1. For hydrodynamic fluctuations with wavenumber k: Equilibration rate γη k 2 (γ η η/(e + p)) Expansion rate ω 1/τ for a Bjorken expansion 2. Compete at a critical scale: ω k 3. Derivative expansion controlled by ϵ γ η ω 1 γ η ω k c s ω c s ϵ k is hard! We derive an effective Hydro-kinetic theory for the kinetic regime k 5/25

6 Renormalization in hydrodynamics with noise 1. Modes above the cutoff contribute to local parameters, e.g. p 0, ζ 0 2. Modes below the cutoff are dynamical 3. Usual p and ζ are defined in the thermodynamic limit a! We will show how the cut-off dependence arises in Hydro-kinetic theory 6/25

7 Brief review of our results for conformal fluid (c 2 s =1/3,ζ =0) 1. Renormalization based on Hydro-kinetic theory Reproduced previous diagrammatic calculation on a trivial background Kovtun-Yaffe (03), Kovtun-Moore-Romatschke (11) p = p 0 (Λ) + Λ3 T 6π 2, e = e 0(Λ) + Λ3 T 2π 2, η = η 0(Λ) + 17ΛT 120π 2 e 0 + p 0 η 0 2. Fractional order correction to the Bjorken expansion (long-time tail) T zz = p 4η ( ) 1 3/ T +(λ 1 ητ π ) 8 3τ 4πγ η τ 9τ ideal 2 + 1st order 1.5th order (long-time tail) 2nd order Hard k = q! h xy (!) T xy =(e + p)u x u y Soft! sound propagator What about nonconformal fluid from Hydro-kinetic theory? 7/25

8 Renormalization for nonconformal fluid 1. Bulk viscosity renormalization: Renormalization proportional to conformal breaking Cζ,η 2 [ ] ζ(t )=ζ 0 (T ;Λ)+ T Λ C 2 ζ0 18π 2 +4 C2 η 0 γ ζ0 2γ η0 C ζ (T ) 1+ 3T 2 2. Pressure renormalization: dc 2 s dt 3c2 s, p(t )=p 0 (T ;Λ)+ 3. Energy density renormalization: ( 1+ T 2 C η (T ) 1 3c 2 s dc 2 ) s0 T Λ 3 dt 6π 2 e(t )=e 0 (T ;Λ)+ T Λ3 2π 2 Let us see how these results are obtained by Hydro-kinetic theory 8/25

9 2. Hydrodynamic fluctuations in the equilibrium 9/25

10 Hydro-kinetic theory: An analogy with Brownian motion 1. Langevin equation dp dt = γp + ξ drag noise, ξ(t)ξ(t ) =2TMγδ(t t ) 2. Calculate how p 2 (t) evolves through Langevin process d dt p2 = 2γ [ p 2 MT ] equilibration Follow the same steps for hydrodynamic fluctuations 10 / 25

11 Hydro-Langevin equation in equilibrium 1. Linearized fluctuations: e = e 0 + δe, g =(e 0 + p 0 ) v φ(t, k) (c s δe, g) 2. Hydro-Langevin equation for φ(t, k) φ(t, k)= 3. Fluctuation-dissipation theorem ilφ + Dφ ideal ic s k viscous γk 2 + ξ noise ξ(t, k)ξ(t, k ) =2T (e 0 + p 0 )Dδ( k k )δ(t t ) The Hydro-Langevin equation is similar to the Brownian motion 11 / 25

12 The matrices in the equilibrium L = ( ) ( ) 0 cs k i 0 0, D = k 2 c s k j 0 0 γ ζ0 δij L + γ η0 δij T γ ζ0 = ζ η 0 e 0 + p 0, γ η0 = η 0 e 0 + p 0 12 / 25

13 Hydro-kinetic equations for the hydro fluctuations 1. Four eigenmodes of L: φ +, φ, φ T1, φ T2 left moving sound φ λ = c s k right moving sound φ + λ + = c s k 2. Analyze the two-point functions e.g. transverse modes φ T λ T =0 N ++ (t, k) φ + (t, k)φ +(t, k) Neglect off-diagonal components (rotating wave approximation) 3. Hydro-kinetic equations for N AA N ++/ = γ ζ0 k 2 [ N ++/ T (e 0 + p 0 ) ] N T1 T 1 /T 2 T 2 = 2γ η0 k 2 [ N T1 T 1 /T 2 T 2 T (e 0 + p 0 ) ] equilibration The Hydro-kinetic equations describe equilibration (FDT) 13 / 25

14 Nonlinear contribution to energy-momentum tensor 1. Equilibrium fluctuations using N eq AA = T (e 0 + p 0 ) g i ( x )g j ( y ) eq = T (e 0 + p 0 )δ ij δ( x y ) 2. Nonlinear contributions (for conformal fluid) T xx eq = p 0 +(e 0 + p 0 ) u x u x eq = p 0 + gx g x eq e 0 + p 0 3. Renormalization (for conformal fluid) Λ d 3 k p = p 0 (Λ) + T (2π) }{{ 3 } p phys,independentofλ Background pressure and energy density need to be renormalized 14 / 25

15 3. Hydrodynamic fluctuations under an external forcing 15 / 25

16 Expanding systems 1. Analyze in a comoving frame v 0 =0 2. Examples: Tensor/scalar metric perturbations (h(t) =he iωt, h 1) ds 2 = dt 2 +(1+h(t))dx 2 +(1+h(t))dy 2 +(1 2h(t))dz 2 ds 2 = dt 2 +(1+h(t))dx 2 +(1+h(t))dy 2 +(1+h(t))dz 2 Bjorken expansion ds 2 = dτ 2 + dx 2 + τ 2 dη 2 What is the effect of expansions for hydrodynamic fluctuations? 16 / 25

17 Hydro-Langevin equation in an expanding system 1. Hydro-Langevin equation for φ =(c s δe, g) φ(t, k)=ilφ ideal + Dφ + ξ + Pφ viscous + noise expansion ξ(t, k)ξ(t, k ) =2T 0 (e 0 + p 0 )Dδ( k k )δ(t t ) 1 g P = ḣ diag ( 3 2 ( 1+c 2 s0 + T Repeat the same steps with the equilibrium case 2.1 Find 4 eigenmodes of L: φ +,φ,φ T1,φ T2 2.2 Kinetic equation for each mode N AA = V 1 φ A φ A 2.3 Nonlinear fluctuations in T µν The expanding background perturbs the fluctuations by P dc 2 ) ) s0, 2, 2, 2 dt 0 17 / 25

18 Hydro-kinetic equation in a scalar metric perturbation 1. Uniform background solution is time-dependent e 0 (t) =ē 0 3h(t) 2 (ē 0 + p 0 ) 2. Hydro-kinetic equations for N L = N ++/ and N T = N T1 T 1 /T 2 T 2 The scalar gravitational field does not distinguish T1 and T 2 Ṅ L = γ ζ0 k 2 [N L N eq (t)] ḣ [ 3c 2 s0 + 3T 0 dc 2 ] s0 +7 N L 2 2 dt 0 Ṅ T = 2γ η0 k 2 [N T N eq (t)] 4ḣN T equilibration external forcing N eq (t) = T 0(e 0 + p 0 ) g Equilibration and disturbance balance at k k ω/γ 18 / 25

19 Nonlinear contribution to T µν 1. Solve the kinetic equation in the linear order of h: N = N eq (t)+δn(t), δn(ω) iωh(ω) iω + γ 0 k 2 2. Compute fluctuation contribution to T µν in the linear order of h: T tt g 2 = e 0 + e 0 + p 0 T ii = 3(1 h)p ζ 0ḣ + 1 h [ g 2 + 3T 0 dc 2 ] s0 c 2 e 0 + p 0 2 dt s0δe 2 +O(h 2 ) 0 ideal viscous fluctuations Tfluct ii 3. Contribution from the fluctuations: g 2 = Λ g d 3 k [N L +2N T ], c 2 s0δe 2 = Λ g d 3 kn L Neq Λ 3 p 0, δn ḣλ ζ 0? Nonlinear fluctuations require renormalization of bulk viscosity and / 25

20 Absorb divergences in T tt Temperatureshift 1. T tt does not have a counter term to absorb a divergence ḣλ T tt = e 0 (T 0 ;Λ)+ T 0Λ 3 + O(ḣΛ) + 2π2 2. Temperature depends on the cutoff To account for off-equilibrium fluctuations near the cutoff T 0 (t;λ)=t(t)+ T(t;Λ) 3. Express T tt in terms of T : T tt e 0 (T ;Λ)+ T Λ3 }{{ 2π 2 } e phys (T ) + de 0 T + O(ḣΛ) + } dt {{} =0, by which T ḣλ fixed T tt is made finite by the temperature shift T Λ( u) 20 / 25

21 Absorb divergences in T ii Renormalization 1. Temperature shift in p 0 : 2. Compute isotropic stress: p 0 (T 0 ;Λ) p 0 (T ;Λ)+ dp 0 dt 0 T p ḣλ T ii = 3(1 h)p 0 (T 0 ) 9 2 ζ 0ḣ + Tfluct ii, Tfluct ii = ideal fluctuations viscous = 3(1 h)p 0 (T )+TN ii eq 9 2 ζ ii 0ḣ + 3(1 h) p + TδN (1 h)(p 0 +#Λ 3 ) ḣ(ζ 0 +#Λ)+long-timetail Temperature shift affects how bulk viscosity is renormalized T N ii eq + T (1 h)λ 3 ii δn ḣλ 21 / 25

22 Renormalization of bulk viscosity, pressure, and energy density 1. Bulk viscosity renormalization: Renormalization proportional to conformal breaking Cζ,η 2 c.f. Diagrammatic computation for cold Fermi gas [Martinez-Schaefer (17)] ζ(t )=ζ 0 (T ;Λ)+ T Λ 18π 2 C ζ (T ) 1+ 3T 2 2. Pressure renormalization: dc 2 s dt 3c2 s, p(t )=p 0 (T ;Λ)+ 3. Energy density renormalization: [ C 2 ζ0 γ ζ0 +4 C2 η 0 2γ η0 ( 1+ T 2 ] C η (T ) 1 3c 2 s dc 2 ) s0 T Λ 3 dt 6π 2 e(t )=e 0 (T ;Λ)+ T Λ3 2π 2 Bulk viscosity renormalization is derived for the first time! 22 / 25

23 Bulk viscosity from thermal fluctuations 1. Bulk viscosity is finite even if ζ 0 (Λ )=0at some scale Λ [ ζ s = s ( η Λ /T s/t π T dc 2 ) 2 ] s 4 2 dt 3c2 s + 2(1 3c 2 s) 2 2. Lattice EoS by Hot QCD Collaboration ζ/s Λ=2T 4T η/s =1/(4π) T MeV Bulk viscosity is enhanced near transition temperature T 150MeV 23 / 25

24 4. Summary & Outlook 24 / 25

25 Hydro-kinetic equation for k, advantageous in expanding systems Universal renormalization of energy density, pressure, and viscosities Background-dependent long-time tails ω 3/2,τ 3/2 Bulk viscosity enhanced due to scale symmetry breaking ζ(t )=ζ 0 (T ;Λ) + T Λ 18π 2 ( 1+ 3T 2 ) 2 dt e0 + p 0 3c2 s0 ζ η 0 dc 2 s0 +4 ( 1 3c 2 ) 2 e 0 + p 0 s0 2η 0 Application to the critical dynamics [YA-Teaney-Yan-Yin, in preparation] Kibble-Zurek scaling for critical fluctuations in a Bjorken expansion 25 / 25