Bulk viscosity, spectra, and flow in heavy ion collisions Thomas - - PowerPoint PPT Presentation

Bulk viscosity, spectra, and flow in heavy ion collisions

Thomas Schaefer & Kevin Dusling, North Carolina State University

Why bulk viscosity?

0.1 0.15 0.2 0.25 0.3 0.35 0.4 100 150 200 250 300 350 400 450 500 T [MeV] cs

2

Ideal Gas Lattice

Real QCD is not scale invariant, and ζ = 0. Usually, this is treated as a nuisancance – it leads to uncertainties in the extraction of η. Here, I want to estimate ζ from data and see what (if anything) we can learn.

Viscosity and dissipative forces

Shear viscosity determines shear stress (“friction”) in fluid flow F = A η ∂vx ∂y Bulk viscosity controls non-equlibrium pressure P = P0 − ζ(∂ · v)

Shear and bulk viscosity in heavy ion collisions (first guess)

b

Ep dN d3p

• pz=0

= v0(p⊥) (1 + 2v2(p⊥) cos(2φ) + . . .) η suppresses v2, enhances v0 ζ suppresses v0, (typically) enhances v2 Note: v0 also sensitive to eos, freezeout, hadronic phase.

Differential elliptic flow from dissipative hydrodynamics

Spectra computed on freeze-out surface (“Cooper-Frye”) Ep dN d3p = 1 (2π)3

• σ

f(Ep)pµdσµ Write f = f 0 + δf and match to hydrodynamics δΠµν =

• dΩp pµpνδf(Ep)

Only moments of δf fixed by η, ζ. Need kinetic models.

0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 2.5 3 3.5 4 v2 (pT) pT [GeV] Ideal fo only fo + δf 0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 2.5 3 3.5 4 v2 (pT) pT [GeV] Ideal Quadratic Linear

Relaxation time approximation

Approximate collision term by single relaxation time C[δfp] ≃ δfp τ(Ep) fp = n0

p + δfp

Bulk viscosity second order in conformal breaking parameter δc2

s

ζ = 15η

• c2

s − 1

3 2

Weinberg (1972)

Distribution function is first order in conformal breaking δf ∼ f 0

p

η sT p2 T 2

• c2

s − 1

3

• (∂ · u)

Near conformal fluids: Bulk viscous correction dominated by δf

Bulk viscosity in kinetic theory

QCD: Elastic vs inelastic reactions

• g + g → g + g (m2

g ∼ g2T 2)

g + g → g + g + g Hadron gas: inelastic scattering, hadro-chemistry

• π + π → 4π

p + ¯ p → 5π

Distribution function in QGP

elastic 2 ↔ 2 can be written as Fokker-Planck equation (diffusion equation in momentum space) (∂ · u) p2 3 − c2

sEp

∂ (βEp) ∂β

• = TµA

np ∂ ∂pi

• np

∂ ∂pi δfp np

• + . . .

drag coefficient µA = g2CAm2

D

8π

log

• T

mD

• Find χB ∼

1

3 − c2 s

• χS and (pure glue)

ζ = 0.44α2

sT 3

log(α−1

s )

ζ ∼ 47.9 1 3 − c2

s

2 η

Arnold, Dogan, Moore (2006)

Distribution function in QGP

Shear Bulk

5 10 15 20 pT 0.5 1.0 1.5 2.0 2.5 3.0

Χp

Quarks Gluons

2 4 6 8 10 pT 0.2 0.4 0.6

Χp

Pure glue: shear vs bulk QGP: quarks vs gluons (bulk rescaled by δc2

s)

δfp = −np(1 ± np) [χS(p)ˆ piˆ pjσij + χB(p)(∂ · u)]

Pion gas

Pion gas: Bulk viscosity governed by chemical non-equilibration δfp = np(1 + np) δµ T + EpδT T 2

• = −np(1 + np)(χ0 + χ1Ep)(∂ · u)

More formal: χ0 is a “quasi zero mode” which dominates C−1 Inelastic rate determines χ0, energy conservation fixes χ1 χ0 = ζ F ζ = βF2 4Γ2π→4π where we have defined F =

• dΩp
• p2

3 − c2 sEp ∂(βEp) ∂β

• np(1 + np)

ζ ≃ 12285 f 8

π

m5

π

exp

• −2mπ

T

• Lu, Moore (2011)

Hadron resonance gas (model)

Hadron gas: Assume bulk viscosity dominated by chemical relaxation δf a

p = −np(1 ± np) (χa 0 − χ1Ep) (∂ · u)

χa

0 determined by rates, χ1 fixed by energy conservation

Slowest rate determines ζ, other rates fix δµa/δµπ. Simple model χa

0 ≃ χπ

   2 mesons 2.5 baryons inspired by µρ = 2µπ and 2µN = 5µπ. Find ζ/s = 0.05 ⇔ δµπ = 20 MeV

Bounds on ζ/s from differential v2 (here: Ks)

0.05 0.1 0.15 0.2 0.25 1 2 3 4 v2 (pT) pT [GeV] Ideal Shear only η/s=0.16 ζ/s=0.005 η/s=0.16 ζ/s=0.015

Pion/Proton pT spectra

10-3 10-2 10-1 100 101 102 103 0.5 1 1.5 2 2.5 3 3.5 1/(2π pT) dN/(dy dpT) [GeV-2] pT [GeV] PHENIX Data (2030%) Pions Protons ζ/s ≈ 0.01, η/s=0.16  0.24 ζ/s=0, η/s=0.08

Data: PHENIX nucl-ex/0307022. Hydro fit: Kevin Dusling (2012). LHC: Bozek & Wyskiel arxiv:1203.6513. Also: afterburners (Vishnu etc).

Pion/Proton differential v2(pT) spectra

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4

STAR Data (2030%) Pions Protons

ζ/s ≈ 0.01, η/s=0.16  0.24 ζ/s=0, η/s=0.08

Data: STAR, nucl-ex/0409033. Hydro fit: Kevin Dusling (2012)

Conclusions

Bulk viscous corrections dominated by freezeout distributions QGP: ζ controlled by momentum rearrangement Hadron gas: ζ determined by chemical non-equilibration A new way to look at fugacity factors in thermal fits? RHIC spectra seem to require ζ/s ∼ < 0.05 Bulk viscosity not zero: Spectra prefer δµ, fine structure of v2 improves

Extras: Second order hydrodynamics

0.2 0.4 0.6 0.8 1 2 4 6 8 10 12

• Π/ζ

τ [fm/c] ζ/s = 0.015

1 2 3 4 5 6 7 8 r [fm] 2 4 6 8 10 12 14 τ [fm/c]

gradient expansion (bulk stress) freeze out surface (w/o bulk viscosity)

Spectra and flow: Kaons and Lambdas

10-6 10-5 10-4 10-3 10-2 10-1 100 101 1 2 3 4 5 dN/(2 π pT dpT dy) [GeV-2] pT [GeV] Kaons Lambda

0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 v2 (pT) pT [GeV] Kaons Lambda

η/s = 0.16 ζ/s = 0.04

Flow: Interplay between shear and bulk viscosity

0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 v2 (pT) pT [GeV] Ideal η/s=0.16 ζ/s=0.00 η/s=0.24 ζ/s=0.008 η/s=0.32 ζ/s=0.010 η/s=0.40 ζ/s=0.012

Integrated v2 versus centrality

0.02 0.04 0.06 0.08 0.1 0.12 50 100 150 200 250 300 350 400 v2 Npart Integrated v2 of Pions Ideal Shear only Shear + Bulk

0.02 0.04 0.06 0.08 0.1 0.12 50 100 150 200 250 300 350 400 v2 Npart Integrated v2 of Protons Ideal Shear only Shear + Bulk

Distribution functions: Signs

Consider four-velocity uα with u2 = −1 (gαβ = (−1, 1, 1, 1)) δfp = −npχSpαpβ∂αuβ − npχB(∂ · u) Asymptotic behavior χS,B ∼ p2. Consider BJ flow: pαpβ∂αuβ ∼ − p2

T

τ and ∂ · u ∼ 1 τ .

δfp ∼ η s pT T 2 1 τT − ζ s pT T 2 1 τT Elliptic flow v2 =

• dφ [f(φ) + δf(φ)] cos(2φ)
• dφ [f(φ) + δf(φ)]

≃ v0

2 + δv2 − v0 2δv0

Elliptic flow: Shear vs bulk viscosity

Dissipative hydro with both η, ζ

200 400 600 800 1000 1200 1400 0.0 0.5 1.0 1.5

[ms] ideal = 0 ζ = 0

Elliptic flow: Shear vs bulk viscosity

Dissipative hydro with both η, ζ βη,ζ = (η, ζ)EF E 1 (3λN)1/3

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4

βη βζ

η ≫ ζ

Dusling, Schaefer (2010)