η/s of a Relativistic Hadron Gas at RHIC: Approaching the AdS/CFT bound?

Nasser Demir in collaboration with Steffen A. Bass

Quark Matter 2009 : 21st International Conference on Ultrarelativistic Nucleus-Nucleus Collisions

April 3, 2009N. Demir and S.A. Bass: arXiv:0812.2422 [nucl-th]

Overview• Motivation: “Low Viscosity Matter” at RHIC

& Consequences• Theory: Kubo Formalism for Transport

Coefficients• Analysis/Results: Equilibriation: Thermal

vs. Chemical Freezeout, Results for η/s• Summary/Outlook

Low Viscosity Matter at RHIC

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronic phase

freeze-out

•QGP-like phase at RHIC observed to behave very much like ideal fluid: success of ideal hydro description of bulk-evolution

•viscous hydro calculations of elliptic flow indicate low values of η/s• (late) hadronic phase resembles dilute gas: large value of η/s expected•separation of thermal and kinetic freeze-out: hadronic phase evolves out of chemical equilibrium

low η/s

large η/s

A comprehensive analysis of the viscosity of QCD matter at RHIC requires a systematic calculation of η/s in the hadronic phase in and out of chemical equilibrium.

Constraining η/s with viscous hydro

Csernai, Kapusta, McLerran:nucl-th/0604032 PRL 97. 152303 (2006)

Pert. Theory N/A here.

Viscous hydro needs η/s~(1-3)/4π , depending upon choice of initial condition.

M.Luzum & P. Romatschke: Phys. Rev. C78: 034915, 2008

NOTE: T- dependence of η/s neglected in vRFD.

What do we know thus far?

• Determining hadronic viscosity necessary to constrain viscosity of QGP.

• vRFD neglects T-dep. of η/s.• Perturbative methods not well trusted near Tc on hadronic side microscopic

transport model can help here!Next Question: How do we compute transport coefficients?

Phenomenological Transport Equation: thermodynamic/mechanical flux linearly proportional to applied field in small field limit.

Examples of transport coefficients: thermal conductivity, diffusion, shear viscosity.

y

x

y=a

y=0

Pyx

Vx= v1

Vx= v2

Shear Viscosity Coefficient:

Green-Kubo: compute linear transport coefficients by examining correlations near kinetic equilibrium!

Linear Transport Coefficients & Green-Kubo Relations

Modeling the Hadronic Medium:UrQMD (Ultrarelativistic Quantum Molecular Dynamics)

- Transport model based on Boltzmann Equation:

-Hadronic degrees of freedom.-Particles interact only through scattering. ( cascade )-Classical trajectories in phase space.

- Values for σ of experimentally measurable processes input from experimental data.

• 55 baryon- and 32 meson species, among those 25 N*, Δ* resonances and 29 hyperon/hyperon resonance species

• Full baryon-antibaryon and isospin symmetry:- i.e. can relate nn cross section to pp cross section.

“Box Mode” for Infinite Hadronic Matter & Equilibriation• Strategy: PERIODIC BOUNDARY

CONDITIONS! • Force system into equilibrium, and

PREVENT FREEZEOUT.Equilibrium Issues :- Kinetic Equilibrium: Compute TEMPERATURE by fitting to Boltzmann distribution!- Chemical equilibrium: DISABLE multibody decays/collisions. RESPECT detailed balance!

Kubo Formalism: Calculating Correlation Functions

NOTE: correlation function found to empirically obey exponential decay.

Exponential ansatz used in Muronga, PRC 69:044901,2004

T=67.9 +/- 0.7 MeVμ = 0

Τπ = 136 +/- 11 fm/c

Entropy Scaling

For system with fixed volume in equilibrium:

Compute entropy using Gibbs formula:

Viscosity Results (μ~0)

- η/s decreases with increasing T in hadronic phase, but levels off at higher hadronic T.- (η/s)min(T~160 MeV)~0.9.

Fugacities in Hadronic Phase• Chemical Freezeout: Tchem~160 MeV.

• Kinetic Freezeout: Tkin~130 MeV.

• In hydro evol.: – introduce non-unit fugacities for species (pions, kaons). (Equiv.

to λi=exp(μi/T)>1)

NOTE: Hadronic phase acquires increasingly non-unit fugacities as system evolves out of chem. equil in HIC!

c.f. PCE scheme (T. Hirano & K. Tsuda: Nucl. Phys. A715, 821 (2003)P.F. Kolb & R. Rapp: Phys. Rev. C67, 044903 (2003))

• Initialize matter w/ equilibrium distributions,but off-equilibrium yields, corresponding to desired fugacities.

•Perform viscosity measurement before system relaxes into equilibrium.•Verify fugacities at time of measurement w/statistical model analysis.

Chemical Non-equilibrium in HIC w/ UrQMD:

- η/s reduced, and result can be understood classically.- (η/s)min(T~160 MeV) ~(0.4-0.5) (reduced by factor of 2!)- Suggests possible complicated structure of evolution of η/s in HIC.

Effect of Finite Baryochem Pot & chem. non-equil

Where do we go from here?

- Need precise parametrization of η/s as function of T,λi ’s for vRFD calculations.- Future progress to be expected from hybrid vRFD+micro models. - Trajectory of η/s in a HIC as a function of temperaturemay have complicated structure.

Summary/Outlook Use Green-Kubo formalism to calculate η/s of hadronic matter:– UrQMD to model hadronic matter.– box mode to ensure kinetic equilibrium, then calculate viscosity

both at unit and non-unit fugacities. Verified entropy calculation via scaling law.

Results:– Hadronic η /s satisfies viscosity bound from AdS/CFT.– η/s reduced at non-unit fugacities, and this suggests complicated

structure of η/s as it evolves from T≈Tc+ !

Outlook:- Map out η /s trajectory probed by collision in hadronic phase

- Complete calculations for hadronic bulk viscosity and possible effect of system evolving out of chemical equilibrium.

- Trajectory of viscositie(s) in a HIC crucial input to T-dep vRFD calculations and vRFD+micro models.

Backup Slides

Analyzing effect of chemical non-equilibrium in HIC w/ UrQMD.

• Initialize matter w/ equilibrium distributions,but off-equilibrium yields, corresponding to desired fugacities.

• Perform viscosity measurement before system relaxes into equilibrium.• Verify fugacities at time of measurement

w/statistical model analysis.

Entropy ConsiderationsMethod I: Gibbs formula for entropy:(extract μB for our system from SHAREv2,P and ε known from UrQMD.) Denote assGibbs.

SHARE v2: Torrieri et.al.,nucl-th/0603026 -Tune particles/resonances to those in UrQMD.

Method II: Weight over specific entropies of particles, where s/n is a function of m/T & μB/T! Denote as sspecific