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Hydrodynamic scaling and analytically solvable models
- 1. Hydrodynamic scaling in an exactly solvable model Based on 1407.5952 with Yoshitaka Hatta,Bowen Xiao, Jorge Noronha G.Torrieri
- 2. What we think we know High pT distributions determined by tomography in dense matter Low pT distributions determined by hydrodynamics Missing: A connection of this to a change in the degrees of freedom (onset of deconfinement): How do opacity, η/s , EoS etc. change at that point? Hydrodynamics can be used as a tool to connect statistical physics (more or less understood) to particle distributions
- 3. A phase transition and/or a cross-over implies scaling violations η/s~Nc 2 dip (crossover) η/s~0.1 −2 Resonances? Hagedorn η λ 2 /s~ ~Ln(T) At T0 ≃ Tc speed of sound experiences a dip (not to 0,as its a cross-over,but a dip). Above Tc, η/s ∼ N0 c , below Tc, η/s ∼ N2 c . We should expect...
- 4. life phase Initial T Initial µ Phase 2 Phase 1 Data across 1/2 s , A,Npart Transition/ threshold Sdydy dN dN <N> (Or , ,...) (Intensive quantities) v2 An change in v2 as the system goes from the viscous hadron gas regime via a kink in the speed of sound to the sQGP regime.
- 5. 140 145 150 155 160 165 170 175 180 185 190 1 10 100 1000 T[MeV] A p-p C-C Si-Si Pb-Pb √ sNN = 17.2 GeV Not a hit of this is seen! Why?
- 6. 140 145 150 155 160 165 170 175 180 185 190 1 10 100 1000 T[MeV] A p-p C-C Si-Si Pb-Pb √ sNN = 17.2 GeV lots of correlated parameters (Qs, η/s, T0(y), µ0(y),freezeout,... ) Need 3D viscous hydro to investigate interplay between: EoS,η/s,τΠ,transverse initial conditions,longitudinal initial conditions,pre-existing flow,freeze-out dynamics, jet showers in-medium, fragmentation outside the medium .... . No jump clearly seen! In which parameters is the phase transition hiding?
- 7. The problem! η/s Equation of state Rapidity dependence Initial flow "With enough parameters you can fit..." vn from ALICE fits well with a NAIVE model with 5 parameters We understand the equation of state and hopefully the viscosity from first principles. But initial conditions and their dependence in energy, and transport coefficients, and jets, and freezeout... Even when you are trying to fit lots of data simultaneusly, a model with many correlated parameters can describe nealry any physical system
- 8. Some people think that this will always be with us The system we are studying is so complicated that models with lots of parameters will always be necessary and well never have a “smoking gun” link between theory and experiment. Perhaps, but I would not give up just yet!
- 9. • By decreasing energy Tinitial,final decreases, µB increases Lifetime increases Flow etc has more time to develop Phases change Intensive parameters change (η/s,,opacity, EoS ) Boost-invariance breaks down (regions at different rapidities talk) • By decreasing system size (pA at high √ s is an extreme example) Tfinal increases, Lifetime decreases Gradients go up , driving up Knudsen number lmfp/R ≃ η/(sTR) Thermalization/medium “turns off” • By varying rapidity Initial density decreases (Phase changes? ) (pA also effectively more ”forward” than AA at central rapidity) All these need to be compared against intensive variable 1 S dN dy ?
- 10. Buckingam’s theorem (How to do hydro, circa 19th century) Any quantitative law of nature expressible as a formula f(x1, x2, ..., xn) = 0 can be expressed as a dimensionless formula F(π1, π2, ..., πn−k) = 0 where πi = xλi i , λi = 0 Widely applied within hydrodynamics in the 19th century: Knudsen’s number, Reynolds number, Rayleigh’s number, etc. Since we are varying a whole slew of experimental (y, pT , Npart, √ s, A) And theoretical (T, µ, η, s, ˆq, τ0, τlife) parameters it would be nice to represent heavy ion observables this way
- 11. This is how hydrodynamics was done in the 19th century!!!! The idea: when you have a pipe and you make it twice as big does your variable of interest grow asn 2 ? What is n?
- 12. s,A,Npart,y dN/dy,<pT>,vn η/s,Cs,... Heavy ion−specific dimensionless number "O" life phase Initial T Initial µ Phase 2 Phase 1 Data across 1/2 s , A,Npart Transition/ threshold Sdydy dN dN <N> (Or , ,...) (Intensive quantities) <O> µ,εT,<R>,life, So, when you double size (or initial temperature, or whatever) how does vn, pT , ... change? Given enough variable conditions, a scaling dimensionless number makes it straight-forward to look for scaling violations
- 13. “And the theorist says.... Consider a spherical elephant in a vacuum” η/s Initial flow
- 14. The shortest course possible on hydro I:Evolution The 5 energy momentum conservation equations ∂µTµν = 0 have 10 unknowns. They can be closed by assuming approximate isotropy Tµν = (p + ρ)uµ uν + pgµν + η∆µναβ∂α uβ + ζ∆µνα α ∂βuβ And thermodynamic equations for p, η, ζ in terms ofρ . Once closed these equations can be integrated from initial conditions
- 15. The shortest course possible on hydro I:Freezeout At a critical condition (here critical T ) the fluid has to convert into particles. Energy-momentum and entropy conservation, plus ”fast” conversion, force the Cooper-Frye formula E dN d3p = 1 pT dN dpT dydφ = pµ dΣµf(pµ uµ, T) If Σµ is the locus of constant T , parametrized by t(x, y, z, T) then dΣµ = ǫµαβγ dΣα dx dΣβ dy dΣγ dz In this formalism vn = cos(nφ) dN dpT dydφ dφ
- 16. A ”semi-realistic” but solvable model: A deformed Gubser solution Gubser flow includes Viscosity , finite Knudsen number Transverse flow with ”Conformal” setup We add Inhomogeneities parametrized by dimensionless ǫn Freeze-out isothermal Cooper-Frye
- 17. The basic idea Conformal invariance of the solution constrains flow to be, in addition to the usual Bjorken u⊥ ∼ 2τx⊥ L2 + τ2 + x2 ⊥ , uz ∼ z t plugging this into the Relativistic Navier-Stokes equation gives you something you can solve ENS = λT4 NS = 1 τ4 λC4 (cosh ρ)8/3 1 + η0 9λC (sinh ρ)3 2F1 3 2 , 7 6 , 5 2 ; − sinh2 ρ 4 where sinh ρ = − L2 − τ2 + x2 ⊥ 2Lτ NB: issues at ρ ≪ −1 (negative temperature!) Physically this reflects implicit non-causality of NS limit, see 1307.6130 (Noronha et al) to fix this
- 18. Not (yet!) the real world: • Strictly conformal EoS (s ∼ T3 , e ∼ T4 ) and viscosity (η ∼ s ≡ η0s ) • Azimuthally symmetric • Transversely much more uniform than your “average” Glauber • “Small times”, or temperature becomes negative (Israel-Stewart needed). Temperature becomes negative (i.e., the solution becomes unphysical) for τL L or x⊥ ≫ η sC 3/2 Where C is an overall normalization constant ∼ dN/dy . NB limitation of the solution ansatz!
- 19. Azimuthal asymmetries: The Zhukovsky transform x → x⊥ + a2 x⊥ cos (nφ) , y → x⊥ − a2 x⊥ sin (φ) In two dimensions this is a conformal transformation, so it transforms a solution into a solution up to a calculable rescaling up to a volume rescaling. This can be neglected to O a2 /x2 ⊥, τa2 /x3 ⊥ (Again, early freezeout )
- 20. To first order in a/L (i.e., ǫn ≪ 1 ) we get E ≈ λC4 τ4/3 (2L)8/3 (L2 + x2 ⊥)8/3 1 − η0 2λC L2 + x2 ⊥ 2Lτ 2/3 4 × 1 − 4ǫn 1 + η0 2λC L2 + x2 ⊥ 2Lτ 2/3 2Lx⊥ L2 + x2 ⊥ n cos nφ , Deformation breaks down at τ ≃ L
- 21. this can be solved for an expression of an isothermal surface, ready for freeze-out T3 = C3 (2L)2 τ(L2 + x2 ⊥)2 1 − η0 2λC L2 + x2 ⊥ 2Lτ 2/3 3 × 1 − 3ǫn 1 + η0 2λC L2 + x2 ⊥ 2Lτ 2/3 2Lx⊥ L2 + x2 ⊥ n cos nφ ≡ C3 B3 (2L)3 , C: overall multiplicity. B Lifetime of the system NB: Need B ≫ 1, so lifetime ≪ L, “early” freezeout w.r.t. size. .
- 22. Now we are set f(p) = dN pT dpT dydφ = dσµpµ exp − uµpµ T 1 + Πµν pµpν 2(e + P)T2 χ(p) where χ(p) = 1, πµν = (gµα − uµ uα ) ∂αuν , σµ = T3 ǫµναβ dxν dT dxα dT dxβ dT and dN dy = dpT pT dφf(p), pT = dpT p2 T dφf(p), vn = dpT pT dφf(p) cos (2nφ) we can analytically map L, T, ǫn, η s , B ⇔ dN dy , pT , vn
- 23. After quite a bit of Algebra... (2π)3 dN dY pT dpT dφp ≡ J1 + J2 + J3 . J1 = 4πmT K1(mT /T ) ∞ 0 dx⊥x⊥τ0 I0(z) (1 − βπ)+ δτ τ0 In(z)ǫn cos nφp +(1− βπ) pT 2T δu⊥ − δuφ x⊥ In−1(z) + δu⊥ + δuφ x⊥ In+1(z) ǫn cos nφp J2 = −4πpT K0(mT /T ) ∞ 0 dx⊥x⊥τ0 ∂τ0 ∂x⊥ I1(z) 1 − βπ + ∂τ0 ∂x⊥ δτ τ0 I′ n(z)ǫn cos nφp +(1−βπ) ∂τ0 ∂x⊥ pT 2T δu⊥ − δuφ x⊥ I′ n−1(z) + δu⊥ + δuφ x⊥ I′ n+1(z) + ∂δτ ∂x⊥ I′ n(z) ǫn cos nφp J3 = −4πpT K0(mT /T ) ∞ 0 dx⊥τ0 n2δτ z In(z)(1−βπ)ǫn cos nφp ≡ δJ3ǫn cos nφp . where Jn = Jn0 + δJnǫn z ≡ pT u⊥0 T = 2x⊥pT (2L)5 T B3(L2+x2 ⊥ )3 (1 − α) .,
- 24. Expanding linearly in ǫn and pT /(TB3 ), In(x) ∼ xn /2n n! J0 1 = 4πmT K1(mT /T )16L3 B3 1 − κx2 ⊥max 64L2 6 + m2 T 2T 2 K3−K1 K1 − p2 T T 2 , J0 2 = 4πK0(mT /T ) 215L3p2 T T B9 1 21 − κ 640 12 + m2 T T 2 K2−K0 K0 − p2 T T 2 , δJ1 = 4π mT T K1(mT /T )Γ(3n) Γ(4n) 9·26nL3pn T B3(n+1)T n−1 ×(n−1) 2(3n+2) 4n+1 − nκ 8(3n−1) 6n + 6 + m2 T 2T 2 K3−K1 K1 − p2 T T 2 δJ2 = 4πK0(mT /T )Γ(3n) Γ(4n) 9·26nL3pn T B3(n+1)T n−1 ×2n 6n2−6n−5 4n+1 − (6n2−10n+1)κ 48(3n−1) 6n + m2 T T 2 K2−K0 K0 − p2 T T 2 , δJ3 = 4πK0(mT /T )Γ(3n) Γ(4n) 9·26nL3pn T B3(n+1)T n−1 ×2n 1 − (4n−1)κ 48(3n−1) 6n + m2 T T 2 K2−K0 K0 − p2 T T 2 ,
- 25. Low pT vn pT /(TB3 ) ≪ 1 , but B ≫ 1 vn(pT ) ǫn = 9(n − 1) 32 Γ(3n) Γ(4n) 64pT B3T n 2(3n + 2) 4n + 1 − nκ 8(3n − 1) 6n + 9 + 2mT T − p2 T T2 The v2 and “Knudsen number” for this solution: vn ǫ ∼ O pT T n (1 − K) , K ∼ η s L τ 2/3 A bit different from Gomebaud et al, Lacey et al vn n ∼ n T R Sensitivity to form of solution , Interplay of L, τ NB: vn(pT ) ∼ pn T phenomenologically important general prediction (Depends on azimuthal integral, independent of approxuimations!
- 26. vn ∼ pn T : A robust prediction All it requires is that vn ∼ dφ cos φ (1 − tf cos(φ) exp [γ (E − vT (φ)pT )]) ∼ In O pT T ∼ pT T n This is much more robust than the assumptions of Gubser flow
- 27. A large momentum region, pT ≫ TB3 is also possible, In(z) ≈ ez √ 2πz ∼ exp pT T 2x⊥(2L)5 B3(x2 ⊥ + L2)3 (1 − α) . The x⊥-integral can be evaluated by doing the saddle point at x∗ ⊥ = L/ √ 5. The result is vn(pT ) ≈ ǫn 2 pT T δu∗ ⊥0 = ǫn 500pT 27TB3 √ 5 3 n−1 n − 1 − 27κ 200 n . but jet contamination likely. Experimental opportunity to see how scaling ofvn(pT ) changes with n, pT ∼ pn T @low pT , ∼ pT @High pT . NB: High, low w.r.t. T×Size/Lifetime≫ 1
- 28. The role of bulk viscosity Plugging in the 14-moment correction of the distribution function δfbulk feq = 12T2 m2 12 + 8 T uµpµ + 1 T2 (uµpµ )2 ∇µuµ T ζ S , and assuming early time ∂µuµ ∼ 1/τ , we carry these terms to be δvbulk n ≈ 81 128 128 B3 n n2 (n − 1)Γ(3n) Γ(4n) Γ2 n 2 (3n + 2)2 4(4n + 1) x2 max L2 − 3n 3n − 1 B2 ζ CS ǫn , Shear and bulk viscosity compete with terms which may be of opposite sign and non-trivial contribution, Confirming the numerical work of Noronha- Hostler et al vn videal n − 1 ∼ ±n2 T2 m2 κbulk ,
- 29. Now we fix K, C, B in terms of bulk obvservables These are dominated by soft regions, so can calculate dN dY = 1 (2π)2 dpT pT (J0 1 + J0 2 ) ≈ 4C3 π pT ≡ dN dY −1 pT dpT dN dY dpT ≈ 3πT 4 = 3πCB 8L Therefore C ∼ dN dY 1/3 , 1 B3 ∼ 1 pT 3L3 dN dY .
- 30. As for azimuthal coefficients, these are vn(pT ) ǫn 1/n ∼ pT A 3/2 ⊥ pT 4 dN dY (1−nκ) , vn ǫn 1/n ∼ 1 A 3/2 ⊥ pT 3 dN dY (1−nκ) , Note that vn ∼ pn T robust against assumptions we made, should survive for realistic scenarios where the “knudsen number” is κ ∼ B2 C η S ∼ A⊥ pT 2 dN/dY η S , , A⊥ ∼ L2 , A 3/2 ⊥ ∼ Npart NB: this is a bit different from Bhalerao et al , as well as GT,1310.3529 v2 ǫ2 ∼ f(τ) (const. − O (κ))
- 31. Plugging in some more empirical formulae dN dY ∼ Npart( √ s)γ , pT ∼ F 1 N 2/3 part dN dY ∼ F N 1/3 part( √ s)γ , where γ ≈ 0.15 in AA collisions and γ ≈ 0.1 in pA and pp collisions, and F is a rising function of its argument, we get vn ǫn 1/n ∼ ( √ s)γ G N 1/3 part( √ s)γ (1−nκ) , κ ∼ H N 1/3 part( √ s)γ η S , where G(x) = F−3 (x) and H(x) = F2 (x)/x.
- 32. Flow... the experimental situation 0 10 20 30 40 50 60 pT -0.5 0 0.5 1 1.5 2 2.5 3 v2 (pT )/<v2 > 0-10% 10-20% 20-30% 30-40% 40-50% 0 5 10 15 20 pT (GeV) -2 0 2 4 6 v2 (pT )/<v2 > CMS 0-5% 60-70% PHENIX 0-10% 50-60% BRAHMS,NPA 830, 43C (2009) pT CMS 1204.1850 CMS 1204.1409 PHENIX PRL98, 162301 (2007) PHENIX PRL98:162301,2007 CMS PRL109 (2012) 022301 NPA830 (2009) PHOBOS STAR 1206.5528 Here is what we know experimentally v2 ≃ ǫ(b, A)F(pT ), v2 ≃ dpT F(pT )f pT , pT y,A,b, √ s F(pT ) universal for all energies , f(pT ) tracks mean momentum, ∼ 1 S dN dy This is an experimental statement, as good as the error bars. Very different from our scaling!
- 33. knew this: for years and we Wrong power w.r.t. vn(pT ) ǫn 1/n ∼ pT A 3/2 ⊥ pT 4 dN dY (1−nκ) , vn ǫn 1/n ∼ 1 A 3/2 ⊥ pT 3 dN dY (1−nκ) , but since κ ∼ 1 A⊥ dN dy , it is enough to “naively extrapolate” from B2 ∼ O (1) to B2 ∼ O (L/τ). Extra A1/2 power enough for scaling but Need Realistic hydrodynamics to test this extrapolation
- 34. pT BRAHMS,0907.4742v2 nucl−ex/0608033 PHENIX Au−Au,Cu−Cu Low energy scan, STAR 1206.5528 v2(pT ) constant (at least at high pT ). Definitely not dependent on pT as in vn(pT ) ǫn 1/n ∼ pT A 3/2 ⊥ pT 4 dN dY (1 − nκ) unless κ depends funnily on dN/dy . Problem also with realistic calculations.
- 35. LHC vn(pT ) data allows us to test vn ∼ pn T a robust prediction, based on In ≃ (z/2)n /n! , independent of lifetime. Not bad, not ideal! Can experimentalists constrain this further? 0 1 2 3 4 pT (GeV) 0 0,5 1 1,5 2 Ratio v3 (pT )/v2 (pT ) v4 (pT )/v2 (pT ) v5 (pT )/v2 (pT ) 0 1 2 3 4 pT (GeV) 0 0,1 0,2 0,3 0,4 0,5 Ratio v3 (pT )/v2 (pT ) v4 (pT )/v2 (pT ) v5 (pT )/v2 (pT ) 0 1 2 3 4 pT (GeV) 0 0,5 1 1,5 2 Ratio v3 (pT )/v2 (pT ) v4 (pT )/v2 (pT ) v5 (pT )/v2 (pT ) Data from ALICE 1105.3865
- 36. PRL107 032301 (2011) vn from ALICE eccentricities from Glauber model vn actually fit quite well with Glauber model ǫn , but see my intro... this is not how one checks this model is realistic
- 37. What we learned • A simplified exactly solvable model incorporating vn yields some very simple scaling patters – vn(pT ) ∼ pn T – vn ∼ A −3/2 ⊥ for early freezeout – vn(pT ) ∼ pT −1 dN dy – Given a constant η/s , κ ∼ A⊥ pT 2 (dN dy )−1 – ... • These scaling patters Can be compared to experiment! provided different system sizes, energies, rapidities compared! . This way no free parameters! What else can we do?
- 38. More detailed correlations... Mixing between ǫn and ǫ2n Lets put in two eccentricities v2n(pT ) ≈ pT 2TB3 10 3 3 √ 5 3 2n−1 (2n−1)ǫ2n+ 1 2 pT 2TB3 2 10 3 6 √ 5 3 2n−2 (n−1)2 For integrated v2 it becomes v2n → v2n ǫ2n + O(n2 ǫ2 n) ǫ2n Can be tested by finding v3 in terms of centrality
- 39. More generally v2n(pT ) ≈ pT 2TB3 10 3 3 √ 5 3 2n−1 (2n−1)ǫ2n+ 1 2 pT 2TB3 2 10 3 6 √ 5 3 2n−2 (n−1)2 together with the definition of the two-particle correlation function dN dpT 1dpT 2d(φ1 − φ2) ∼ n vn (pT 1) vn (pT 2) cos (n (φ1 − φ2)) Predicts a systematic rotation of the reaction plane that can be compared with data
- 40. A hydrodynamic outlook Calculate the same things we had with realistic hydro simulations • Long life • Realistic transverse initial conditions dN/dy pT vn = ... ... ... ... ... ... ... ... ... η/s,cs,τπ,... × Tinitial L ǫn →Npart,A, √ s Finding a scaling variable ≡ finding a basis to diagonalize this
- 41. Should hydrodynamic scaling persist in tomographic regime? NO! Take, as an initial condition, an elliptical distribution of opaque matter at a given ǫn , run jets through it and calculate vn . Now increase R while mantaining ǫn constant. vn ǫn tomo → Surface V olume → 0, vn ǫn hydro → constant Role of “size” totally different in tomo vs hydro regime . Probe by comparing vn in Cu-Cu vs Au-Au, Pb-Pb vs Ar-Ar collisions of Same multiplicity!
- 42. Can we investigate this both quantitatively and generally? When we study a jet traversing in the medium, we assume • Fragments outside the medium phadron T ∼ f(pparton T ) • Comes from a high-energy parton, T/pT ≪ 1 • Travels in an extended hot medium, (Tτ)−1 ≪ 1 When we expand any jet energy loss model, f (pT /T, Tτ) around T/pT , (Tτ)−1
- 43. The ABC-model! dE dx = κpa Tb τc + O T pT , 1 Tτ A phenomenological way of keeping track of every jet energy loss model: c = 0 Bethe Heitler c = 1 LPM c > 2 AdS/CFT “falling string” Conformal invariance, weakly or strongly coupled, implies a + b − c = 2
- 44. Embed ABC model in Gubser solution And calculate v2(pT ≫ ΛQCD) as a function of pT , L, T . 0 10 20 30 40 50 60 pT -0.5 0 0.5 1 1.5 2 2.5 3 v2 (pT )/<v2 > 0-10% 10-20% 20-30% 30-40% 40-50% 0 5 10 15 20 pT (GeV) -2 0 2 4 6 v2 (pT )/<v2 > CMS 0-5% 60-70% PHENIX 0-10% 50-60% CMS 1204.1850 CMS 1204.1409 PHENIX PRL98, 162301 (2007) v2 at low and high pT look remarkably similar.
- 45. Conclusions: heavy ions beyond fitting Choose observable O and your favorite theory, try to determine a, b, c, ... O ≃ La dN dy b ǫc n... compare a,b,c with all experimental data We did this with a highly simplified analytically solvable hydro model . Calculations fro ”real” hydro and tomography also possible.