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? 0.4 0.35 Ideal Gas 0.3 2 c s 0.25 Lattice 0.2 0.15 0.1 100 150 200 250 300 350 400 450 500 T [MeV] 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 ∂v x ∂y Bulk viscosity controls non-equlibrium pressure P = P 0 − ζ ( ∂ · v )

Shear and bulk viscosity in heavy ion collisions (first guess) b � dN � E p = v 0 ( p ⊥ ) (1 + 2 v 2 ( p ⊥ ) cos(2 φ ) + . . . ) � d 3 p � p z =0 η suppresses v 2 , enhances v 0 ζ suppresses v 0 , (typically) enhances v 2 Note: v 0 also sensitive to eos, freezeout, hadronic phase.

Differential elliptic flow from dissipative hydrodynamics Spectra computed on freeze-out surface (“Cooper-Frye”) dN 1 � f ( E p ) p µ dσ µ E p d 3 p = (2 π ) 3 σ Write f = f 0 + δf and match to hydrodynamics � δ Π µν = d Ω p p µ p ν δf ( E p ) Only moments of δf fixed by η, ζ . Need kinetic models. 0.5 0.5 Ideal Ideal 0.4 0.4 f o only Linear 0.3 0.3 v 2 (p T ) v 2 (p T ) f o + δ f 0.2 0.2 Quadratic 0.1 0.1 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 p T [GeV] p T [GeV]

Relaxation time approximation Approximate collision term by single relaxation time δf p f p = n 0 C [ δf p ] ≃ p + δf p τ ( E p ) Bulk viscosity second order in conformal breaking parameter δc 2 s � 2 � s − 1 c 2 ζ = 15 η 3 Weinberg (1972) Distribution function is first order in conformal breaking p 2 � � η s − 1 δf ∼ f 0 c 2 ( ∂ · u ) p T 2 sT 3 Near conformal fluids: Bulk viscous correction dominated by δf

Bulk viscosity in kinetic theory QCD: Elastic vs inelastic reactions � � g + g → g + g ( m 2 g ∼ g 2 T 2 ) g + g → g + g + g Hadron gas: inelastic scattering, hadro-chemistry � � π + π → 4 π p + ¯ p → 5 π

Distribution function in QGP elastic 2 ↔ 2 can be written as Fokker-Planck equation (diffusion equation in momentum space) � p 2 � � � δf p �� ∂ ( βE p ) = Tµ A ∂ ∂ 3 − c 2 ( ∂ · u ) s E p n p + . . . ∂p i ∂p i ∂β n p n p drag coefficient µ A = g 2 C A m 2 � � T log D 8 π m D � 1 3 − c 2 � Find χ B ∼ χ S and (pure glue) s � 2 ζ = 0 . 44 α 2 s T 3 � 1 3 − c 2 ζ ∼ 47 . 9 η s log( α − 1 s ) Arnold, Dogan, Moore (2006)

Distribution function in QGP Χ � p � Χ � p � 3.0 0.6 2.5 Quarks Bulk 2.0 0.4 1.5 Shear 0.2 Gluons 1.0 0.5 10 p � T 2 4 6 8 20 p � T 5 10 15 Pure glue: shear vs bulk QGP: quarks vs gluons (bulk rescaled by δc 2 s ) δf p = − n p (1 ± n p ) [ χ S ( p )ˆ p i ˆ p j σ ij + χ B ( p )( ∂ · u )]

Pion gas Pion gas: Bulk viscosity governed by chemical non-equilibration � δµ T + E p δT � δf p = n p (1 + n p ) = − n p (1 + n p )( χ 0 + χ 1 E p )( ∂ · u ) T 2 More formal: χ 0 is a “quasi zero mode” which dominates C − 1 Inelastic rate determines χ 0 , energy conservation fixes χ 1 β F 2 χ 0 = ζ ζ = F 4Γ 2 π → 4 π � � p 2 ∂ ( βE p ) 3 − c 2 � where we have defined F = d Ω p s E p n p (1 + n p ) ∂β ζ ≃ 12285 f 8 � � − 2 m π π exp m 5 T π Lu, Moore (2011)

Hadron resonance gas (model) Hadron gas: Assume bulk viscosity dominated by chemical relaxation δf a p = − n p (1 ± n p ) ( χ a 0 − χ 1 E p ) ( ∂ · u ) χ a 0 determined by rates, χ 1 fixed by energy conservation Slowest rate determines ζ , other rates fix δµ a /δµ π . Simple model  2 mesons  χ a 0 ≃ χ π 0 2 . 5 baryons  inspired by µ ρ = 2 µ π and 2 µ N = 5 µ π . Find ζ/s = 0 . 05 ⇔ δµ π = 20 MeV

Bounds on ζ/s from differential v 2 (here: K s ) 0.25 0.2 0.15 v 2 (p T ) 0.1 Ideal Shear only 0.05 η /s=0.16 ζ /s=0.005 η /s=0.16 ζ /s=0.015 0 0 1 2 3 4 p T [GeV]

Pion/Proton p T spectra 10 3 PHENIX Data (20  30%) ζ /s ≈ 0.01, η /s=0.16  0.24 10 2 Pions ζ /s=0, η /s=0.08 1/(2 π p T ) dN/(dy dp T ) [GeV -2 ] 10 1 10 0 Protons 10 -1 10 -2 10 -3 0 0.5 1 1.5 2 2.5 3 3.5 p T [GeV] Data: PHENIX nucl-ex/0307022. Hydro fit: Kevin Dusling (2012). LHC: Bozek & Wyskiel arxiv:1203.6513. Also: afterburners (Vishnu etc).

Pion/Proton differential v 2 ( p T ) spectra 0.2 STAR Data (20  30%) ζ /s ≈ 0.01, η /s=0.16  0.24 ζ /s=0, η /s=0.08 0.15 Pions 0.1 Protons 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 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? < 0 . 05 RHIC spectra seem to require ζ/s ∼ Bulk viscosity not zero: Spectra prefer δµ , fine structure of v 2 improves

Extras: Second order hydrodynamics 1 ζ /s = 0.015 14 0.8 12 10 0.6 τ [fm/c] - Π / ζ 8 0.4 6 4 0.2 2 0 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 r [fm] τ [fm/c] gradient expansion freeze out surface (bulk stress) (w/o bulk viscosity)

Spectra and flow: Kaons and Lambdas 0.3 10 1 Kaons Kaons Lambda 10 0 Lambda 0.25 dN/(2 π p T dp T dy) [GeV -2 ] 10 -1 0.2 10 -2 v 2 (p T ) 0.15 10 -3 0.1 10 -4 10 -5 0.05 10 -6 0 0 1 2 3 4 5 0 1 2 3 4 5 p T [GeV] p T [GeV] η/s = 0 . 16 ζ/s = 0 . 04

Flow: Interplay between shear and bulk viscosity η /s=0.24 ζ /s=0.008 0.3 η /s=0.32 ζ /s=0.010 Ideal η /s=0.40 ζ /s=0.012 0.25 η /s=0.16 0.2 ζ /s=0.00 v 2 (p T ) 0.15 0.1 0.05 0 0 1 2 3 p T [GeV]

Integrated v 2 versus centrality 0.12 0.12 Ideal Ideal Shear only Shear only 0.1 Shear + Bulk 0.1 Shear + Bulk 0.08 0.08 0.06 0.06 v 2 v 2 0.04 0.04 Integrated v 2 of Pions 0.02 Integrated v 2 of Protons 0.02 0 0 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 N part N part

Distribution functions: Signs Consider four-velocity u α with u 2 = − 1 ( g αβ = ( − 1 , 1 , 1 , 1) ) δf p = − n p χ S p α p β � ∂ α u β � − n p χ B ( ∂ · u ) Asymptotic behavior χ S,B ∼ p 2 . Consider BJ flow: p α p β � ∂ α u β � ∼ − p 2 τ and ∂ · u ∼ 1 τ . T � 2 1 � 2 1 δf p ∼ η � p T τT − ζ � p T s T s T τT Elliptic flow � dφ [ f ( φ ) + δf ( φ )] cos(2 φ ) ≃ � v 0 2 � + � δv 2 � − � v 0 � v 2 � = 2 �� δv 0 � � dφ [ f ( φ ) + δf ( φ )]

Elliptic flow: Shear vs bulk viscosity Dissipative hydro with both η, ζ ideal � � 1.5 � � = 0 1.0 0.5 ζ � = 0 0.0 0 200 400 600 800 1000 1200 1400 � [ms]

Elliptic flow: Shear vs bulk viscosity Dissipative hydro with both η, ζ β η,ζ = ( η, ζ ) E F 1 (3 λN ) 1 / 3 E 0.4 0.3 β η 0.2 η ≫ ζ 0.1 0.0 0.0 0.1 0.2 0.3 0.4 β ζ Dusling, Schaefer (2010)