Quasiparticle Second-Order Viscous Hydrodynamics from an Effective Boltzmann Equation

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Quasiparticle second-order viscous hydrodynamics from an effective Boltzmann equation

Radoslaw Ryblewski

THE HENRYK NIEWODNICZAŃSKI INSTITUTE OF NUCLEAR PHYSICS POLISH ACADEMY OF SCIENCES

in collaboration with: A. Jaiswal and L. Tinti

details: L. Tinti, A. Jaiswal, R.R., Phys. Rev. D95 5, 054007 (2017)

Radoslaw Ryblewski (IFJ PAN) 1 September 19, 2017 1 / 15 Hydrodynamic description of heavy-ion collisions

Time

Energy Stopping Hydrodynamic Initial state Hard Collisions Evolution Hadron Freezeout

• local thermal equilibrium fluid dynamics applicable ⇒ • order-by-order expansion around equilibrium state: • 0th order – perfect fluid dynamics • 1st order – Navier-Stokes theory (parabolic PDE, acausal, unstable) • 2nd order – Israel-Stewart theory (hyperbolic PDE, causal, stability not guaranteed) • universal existence of the dissipation, precision measurements of the flow • due to above arguments necessary to use relativistic 2nd order viscous hydrodynamics (VH) • multiple successes of the VH description of the space-time evolution of the QGP

Radoslaw Ryblewski (IFJ PAN) 2 September 19, 2017 2 / 15 Hydrodynamic modelling – complications • form of VH equations of motion (EOM) is not universal • most of successfull derivations of VH EOM (DNMR, Jaiswal) are based on simple relativistic kinetic theory (KT) – RTA Boltzmann equation for particles with m = const. G. S. Denicol, T. Koide, and D. H. Rischke, Phys. Rev. Lett. 105, 162501 (2010) G. S. Denicol, H. Niemi, E. Molnar and D. H. Rischke, Phys. Rev. D 85 , 114047 (2012) A. Jaiswal, Phys.Rev. C87 (2013) no.5, 051901

˙ = ( + ) θ Π θ + π : σ, E − E P − µ µ = ( + ) u˙ µ + Π u˙ µ µΠ + ∆ ∂ παβ ∇ P E P − ∇ α β ˙ Π Π = βΠθ δΠΠΠθ + λΠππ : σ − τΠ − − µν µν π µν µ ν γ µ ν γ µν µν π˙ h i = + βπ2σ + 2πγh ω i τπππγh σ i δπππ θ + λπΠΠσ − τπ − − • within VH the thermodynamic (equation of state (EOS)) and transport (shear viscosity η, bulk viscosity ζ) properties of the QGP need to be incorporated • in principle they should be extracted from the experimental data • highly non-trivial task – 10 PDE for T, uµ, Π, πµν, multidimensional ﬁt, numerically expensive calculations • nowadays some results of ab-initio calculations may be used

(T) =? (T) =? quite precise lQCD results for µB = 0 E P ← βΠτR = ζ(T) =? and βπτR = η(T ) =? scarce lQCD results ← (however see talk by A. Czajka) (T) =? no information ← Radoslaw Ryblewski (IFJ PAN) 3 September 19, 2017 3 / 15 Kinetic-theory-wise approach – equilibrium

• consider a system of ideal (non-interacting) uncharged massive particles of a single species m = const • the EOS of such a system depends parametrically only on the mass (m) of the particle • impossible to reproduce exactly the temperature (T) dependence of energy density ( ) and pressure ( ) given by lQCD E P example: Maxwell–Boltzmann distribution

u p gT 4z2 gT 4z2 feq = g exp · 0 = 3K2(z) + zK1(z) 0 = K2(z) − T ⇒ E 2π2 P 2π2 z m ≡ T Kn – modiﬁed Bessel functions

Radoslaw Ryblewski (IFJ PAN) 4 September 19, 2017 4 / 15 Imposing arbitrary EOS through quasiparticle mass

• to describe an arbitrary EOS consider T -dependent mass – notion of quasiparticles V. Goloviznin and H. Satz, Z. Phys. C57, 671 (1993) m = const m = m(T) → • physically sound when considering high-T QCD (m(T) = gsT) E. Braaten and R. D. Pisarski, Nucl. Phys. B337, 569 (1990) • in general these quasi-particles do not correspond to any real excitations of the underlying fundamental theory (QCD) – especially close to the crossover region!

Radoslaw Ryblewski (IFJ PAN) 5 September 19, 2017 5 / 15 Thermodynamic consistency violation

• introducing m = m(T ) violates basic thermodynamic identities • example: Maxwell-Boltzmann M. I. Gorenstein and S.-N. Yang, Phys. Rev. D52, 5206 (1995) distribution • the thermodynamic relation must be u p satisﬁed feq = g exp · − T d 0 0 + 0 P = E P 0 dT T S ≡ gT 4z2 0 = 3K2(z) + zK1(z) • within KT one has E 2π2 4 2 µν gT z 0 = uµTequν 0 = K2(z) E P 2π2 1 µν • if m = const m = m(T) 0 = ∆µνT P − 3 eq → 3 2 " # where d 0 gT z dm P = 4K2(z) + zK1(z) K1(z) Z dT 2π2 − dT µν µ ν Z Teq = dP p p feq 0 + 0 dm = E P m dPfeq T − dT ∆µν gµν uµuν | {z } R ≡ R d−4p 0 dP = 2 Θ(p t)(2π) δ(p2 m2) , (2π)4 · −

Radoslaw Ryblewski (IFJ PAN) 6 September 19, 2017 6 / 15 Restoring thermodynamic consistency

• restore thermodynamic consistency by • example: Maxwell-Boltzmann introducing additional effective mean distribution ﬁeld through a bag function B0(T) M. I. Gorenstein and S.-N. Yang, Phys. Rev. D52, 5206 (1995) u p feq = g exp · 0 0+B0 0 0 B0 − T E → E P → P − • again, one requires 4 2 d + gT z 0 0 0 0 = 3K2(z) + zK1(z) +B0 0 P = E P 2 S ≡ dT T E 2π gT 4z2 with 0 = K2(z) B0 P 2π2 − µν 1 µν 0 = uµT uν, 0 = ∆µνT • if m = const m = m(T) E eq P − 3 eq → 3 2 " # • B0(T) may be included in the Lorentz d 0 gT z dm P = 4K2(z) + zK1(z) K1(z) covariant way by modifying the dT 2π2 − dT deﬁnition of T µν eq dB0 S. Jeon, Phys. Rev. D52, 3591 (1995) S. Jeon and L. G. Yaffe, Phys. Rev. D53, 5799 (1996) − dT P. Chakraborty and J. I. Kapusta, Phys. Rev. C83, 014906 (2011) Z ! P. Romatschke, Phys. Rev. D85, 065012 (2012) 0 + 0 dB0 dm = E P + m dPfeq M. Albright and J. I. Kapusta, Phys. Rev. C93, 014903 (2016) T − dT dT | {z } µν R µ ν µν Teq = dP p p feq+B0(T) g =0!

Radoslaw Ryblewski (IFJ PAN) 7 September 19, 2017 7 / 15 Imposing lQCD EOS

• degeneracy factor g is obtained by reproducing correct Stefan–Boltzmann limit

4 π 2 g = 4(N 1) + 7NcNf 180 c −

Nc = 3, Nf = 3 • to solve EOM one has to impose EOS – necessary to fix the T dependence of • m(T) is determined from equilibrium entropy density, 0 = ( 0 + 0)/T the thermodynamic quantities S E P (independent of B0) by numerically • it is sufficient to define m(T) P. Romatschke, Phys. Rev. D85, 065012 (2012) solving • consider finite-T lQCD EOS at = 0 by ! ! µB g m(T) m(T ) 0(T ) K = S the Wuppertal-Budapest collaboration 2 3 3 S. Borsanyi et al., JHEP 11, 077 (2010) 2π T T T lQCD

• B0(T ) is given through the relation expressing thermodynamic consistency

3 2 dB0(T ) gT z dm = K1 (z) dT − 2π2 dT

Radoslaw Ryblewski (IFJ PAN) 8 September 19, 2017 8 / 15 Off-equilibrium mean ﬁeld

• for the non-equilibrium case one can have in general L. Tinti, A. Jaiswal, R.R., Phys.Rev. D95 (2017) no.5, 054007 • R the effective Boltzmann equation (BE) T µν = dP pµpν f + Bµν for m = m(T) reads ..., W. Florkowski, J. Hufner, S. P. Klevansky, and L. Neise, Annals • in equilibrium (f feq) we require Phys. 245, 445 (1996) → P. Romatschke, Phys. Rev. D85, 065012 (2012) µν µν B eq = B0g | ρ (p) • out of equilibrium (f feq + δf ) we split (p ∂)f + m (∂ m) ∂ f = [f ] → · ρ C µν µν µν • B = B0 g +δB we assume the collision kernel in the relaxation-time approximation (RTA) µν µν J. Anderson and H. Witting, Physica 74, 466 (1974) • δB is ﬁxed by requiring ∂µT = 0 Z (u p) µν ν [f ] = · δf ∂µB + m ∂ m dPf C − τR Z h (p) i + dP pν (p ∂)f + m (∂ρm) ∂ f = 0 · ρ

Radoslaw Ryblewski (IFJ PAN) 9 September 19, 2017 9 / 15 Ansatz for off-equilibrium ﬁeld

• the symmetry of T µν restricts δBµν to have 10 independent components • energy and momentum conservation µν becomes • ∂µT = 0 leads to only 4 constraints Z • we make an ansatz for δBµν of the form µν ν 1 µν ∂µδB + m ∂ m dP δf + uµδB = 0 τ µν µν µ ν µ ν R δB = b0 g + u b + b u • if δBµν = 0 it reduces to u b = 0 · 3 m (∂νm) Π = 0 − • at ﬁrst order in the gradient expansion µ • for m = m(T) the bulk pressure Π is non- one gets b0 = 0 and b = 0 vanishing which means that in general • up to second-order one has we must have δBµν , 0 2 µ µ b0 = 3 τR κc Π θ, b = 3 τR κ Π u˙ − s

Radoslaw Ryblewski (IFJ PAN) 10 September 19, 2017 10 / 15 Tensor decomposition of E-M tensor and VH EOM • bulk pressure Π and shear-stress tensor πµν are deﬁned as 1 Π ∆ : T Teq ≡ − 3 − µν αβ µν πµν ∆ T αβ T = ∆ T αβ ≡ αβ − eq αβ • T µν may be tensor decomposed into • in our case they read T µν = uµuν ( +Π)∆ µν + πµν E − P Z 1 α β Π = ∆ dPp p δf b0 • four-momentum conservation gives − 3 αβ − EOM for uµ and T Z µν = ∆µν dPpαpβ f π αβ δ ˙ = ( + ) θ Π θ + π : σ, E − E P − µ µ µ µ µ αβ • EOM of dissipative quantities are = ( + ) u˙ + Π u˙ Π + ∆α∂βπ ∇ P E P − ∇ obtained by applying the co-moving µν ˙ σµν ∆ αuβ derivative () (u ∂) ≡ αβ∇ G. S. Denicol, T. Koide,≡ and D.· H. Rischke, Phys. Rev. Lett. 105, ∆µν = 1 (∆µ∆ν + ∆µ∆ν 2 ∆µν∆ ) 162501 (2010) αβ 2 α β β α − 3 αβ Z ˙ 1 ˙ Π = (u ∂) dP (p ∆ p) δf b0 − 3 · · · − Z µν µν α β π˙ h i = ∆ (u ∂) dP ph p i δf αβ ·

X µν ∆µνX αβ X˙ µν ∆µνX˙ αβ h i ≡ αβ h i ≡ αβ Radoslaw Ryblewski (IFJ PAN) 11 September 19, 2017 11 / 15 Evolution equations for dissipative quantities ! 5 I1 1 = 1 m2 , c2 • explicit second-order EOM for the δΠΠ χ κ s − 9 − − I3,1 dissipative quantities are obtained  c2m2 using the Chapman-Enskog-like 1 βκ s  2 +  1 3cs βI2,1 I1,1 iterative solution of the BE for δf 3 βΠ  − − A. Jaiswal, Phys.Rev. C87 (2013) no.5, 051901 A. Jaiswal, R. Ryblewski, and M. Strickland, Phys. Rev. C90,   044908 (2014) 1 3 c2 m2 I + I  κ s β 0,1 1,1  ˙ Π − − − Π = βΠθ δΠΠΠθ + λΠππ : σ − τΠ − − ! β 2 I1,1 2 µν λΠ = 2I 7I 1 κm c µν π µν µ ν γ µ ν γ π 3 3,2 3,3 I s π˙ h i = + 2βπσ + 2πh ω i τπππh σ i βπ − − − 3,1 − τ γ − γ π 4 µν µν β δπππ θ + λπΠΠσ τππ = 2 I3,3 − − βπ

• the bulk and shear viscosities are given 5 7 β β 2 2 δππ = I3,3 κcs m I1,2 I1,1 by βΠτR = ζ and βπτR = η with 3 − 3 βπ − βπ − 2 5 2 2 2 λπΠ = χ βΠ = β I3 2 cs ( + ) +κcs m β I1 1 3 3 , − E P , − βπ = βI3,2 where

P. Romatschke, Phys. Rev. D85, 065012 (2012)  β  2 T dm χ =  1 3cs I3,2 I3,1 • κ β  − − ≡ m dT Π • in the limit κ 0 these match the  constant mass→ results 2 2  1 3κcs m I1,2 I1,1  − − −  Radoslaw Ryblewski (IFJ PAN) 12 September 19, 2017 12 / 15 Longitudinal Bjorken ﬂow solution • use Milne coordinates • consider transversely homogeneous t and purely-longitudinal boost-invariant 10 (Bjorken) expansion const J. D. Bjorken, Phys. Rev. D27, 140 (1983) = Η Τ=const µ z 5 u = γ(1, 0, 0, t )

• all quantities are independent of ς and z therefore unchanged when performing -10 -5 5 10 a Lorentz-boost – evolution only in τ!

• EOM reduce to -5

˙ 1 = ( + + Π πs) E − τ E P − -10 ˙ Π βΠ Π πs Π + = δΠΠ + λΠπ τΠ − τ − τ τ ! t = τ cosh ς πs 4 βπ 1 πs 2 Π π˙ s + = τππ + δππ + λπΠ xµ = (t x y z) z = τ sinh ς τπ 3 τ − 3 τ 3 τ , , ,

2 ςς ⇓ ⇓ p where πs τ π . µ 2 2 ≡ − x 0 = (τ, x, y, ς) τ = t z − • the relaxation time is 1 z ς = tanh− t η¯ 0 µ τπ = τΠ = τR = S • in x 0 the ﬂuid becomes static, βπ µ u 0 = (1, 0, 0, 0)

Radoslaw Ryblewski (IFJ PAN) 13 September 19, 2017 13 / 15 Longitudinal Bjorken ﬂow

L = 0 + Π πs P P − T = 0 + Π + πs/2 P P • study the evolution of viscous QCD 1.0 matter by numerically solving EOM supplemented with the lQCD EOS 0.9 • we choose LHC initial conditions T 50 P 0.8 L T(τi ) = 0.6 GeV, πs(τi ) = 0, Π(τi ) = 0 at D 40 4 P Η = 1 4Π

- S τ = 0.25 fm 30

i fm @ QaHydro 0.7 s 20 Π QvHydro • in addition τf = 500 fm, η/ = 1/(4π) 10 H L S vHydro 0 • we compare: 0.6 1 10 100 - quasiparticle second-order VH 1 10 100 Τ fm (QvHydro, this work) L. Tinti, A. Jaiswal, R.R., Phys.Rev. D95 (2017) no.5, 054007 0.006 @ D - standard second-order VH (vHydro) Η S = 1 4Π 0.004 A. Jaiswal, R.R., Strickland, Phys. Rev. C 90 , no. 4, 044908 (2014) QaHydro 0.002 QvHydro H L vHydro

- quasiparticle anisotropic 0 P hydrodynamics (QaHydro) 0.000 P M. Alqahtani, M. Nopoush, M. Strickland, Phys. Rev. C92, 054910 -0.002 (2015) -0.004 (see M. Strickland talk) -0.006 1 10 100 Τ fm

Radoslaw Ryblewski (IFJ PAN) 14 September@ D 19, 2017 14 / 15 Summary

• we have presented the ﬁrst derivation of the second-order VH for a system of quasiparticles of a single species from an effective BE

• we devised a thermodynamically-consistent framework to formulate second-order EOM for Π and πµν for quasiparticles with T-dependent masses

• the presented formulation is capable of accommodating an arbitrary EOS within the framework of KT

Radoslaw Ryblewski (IFJ PAN) 15 September 19, 2017 15 / 15 Thank you for your attention!

Radoslaw Ryblewski (IFJ PAN) 15 September 19, 2017 15 / 15 Backup slides

Radoslaw Ryblewski (IFJ PAN) 15 September 19, 2017 15 / 15 Imposing lattice QCD equation of state - details

• For numerical convenience, we use 4 analytic ﬁt to the lQCD results for the

interaction measure (trace anomaly) 3

4 0(T) h1 h2 T 2 I = exp + 0 T 4 − Tˆ Tˆ2 I h i " ˆ # 1 h f0 tanh(f1T + f2) + 1 0 + ˆ2 ˆ ˆ2 × 1 + h3T 1 + g1T + g2T 0 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 ˆ T GeV with T T/(0.2 GeV) and h0 = 0.1396, ≡ h1 = 0.18, h2 = 0.035, f0 = 2.76, • B (T ) is given through the relation − 0 @ D f1 = 6.79, f2 = 5.29, g1 = 0.47, − − expressing thermodynamic consistency g2 = 1.04, and h3 = 0.01 dB (T ) gT 3z2 Z T 0 dm 0(T) dT 0(T) = K1 (z) P = I dT − 2π2 dT 4 4 T 0 T T The above equation can be solved 0(T) 0(T) 0(T) numerically using the boundary E = 3 P + I 4 4 4 condition B0 = 0 at T 0. T T T '

Radoslaw Ryblewski (IFJ PAN) 15 September 19, 2017 15 / 15 Imposing lattice QCD equation of state in standard approach

• in the standard formulations of the second-order viscous fluid dynamics approaches the particle mass is treated as a constant parameter, m = const. • In this case there is however no • Using m(T) extracted in this way one unambiguous prescription how to may also express all the second-order impose the lattice QCD equation of transport coefficients and the state in the dynamical equations. relaxation times τR = η¯I3,1/(I3,2T). • The usual methodology, which we call • This method is however not applicable here standard, is to extract the m(T ) to the energy density and pressure and dependence by matching squared the first-order terms, which require the speed of sound knowledge of the degeneracy factor ! g. For that reason for these quantities m(T) c2(T) = c2 we use directly values from the lattice s lQCD s T ideal gas (m=const.) QCD calculations.

where on the right hand side we use 2 the expression for cs valid in the case of ideal Boltzmann gas of particles with constant mass.

Radoslaw Ryblewski (IFJ PAN) 15 September 19, 2017 15 / 15