1. η/s of a Relativistic Hadron Gas at RHIC Nasser Demir in collaboration with Steffen A. Bass Annual 2009 RHIC & AGS Users’ Meeting June 1, 2009 N. Demir and S.A. Bass, PRL 102, 172302 (2009), arXiv:0812.2422 [nucl-th]

2. 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 in and out of chem. equil. Summary/Outlook

3. Low Viscosity Matter at RHIC initial state pre-equilibrium QGP and hydrodynamic 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.

4. 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.

5. 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 T c on hadronic side  microscopic transport model can help here! Next Question: How do we compute transport coefficients?

6. 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 P yx V x = v 1 V x = v 2 Shear Viscosity Coefficient: Green-Kubo: compute linear transport coefficients by examining correlations near kinetic equilibrium! Linear Transport Coefficients & Green-Kubo Relations

7. 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.

8. “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!

9. 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 MeV μ = 0 Τ π = 136 +/- 11 fm/c

10. Entropy Scaling For system with fixed volume in equilibrium: Compute entropy using Gibbs formula :

11. Viscosity Results (μ~0): In full Chem and Kin Equil - η/s decreases with increasing T in hadronic phase, but levels for T>100 MeV. - (η/s) min (T~160 MeV)~0.9.

12. Fugacities in Hadronic Phase Chemical Freezeout: T chem ~160 MeV. Kinetic Freezeout: T kin ~130 MeV. In hydro evol.: –introduce non-unit fugacities for species (pions, kaons). (Equiv. to λ i =exp(μ i /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 above-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:

13. - 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 strong T-dependence of η/s in HIC, with possible sharp rise near T c. Effect of Finite Baryochem Pot & chem. non-equil

14. Outlook - Need precise parametrization of η/s as function of T,λ i ’s for vRFD calculations. - Assess systematic uncertainties associated with neglecting multiparticle processes. - Calculations of ς/s from UrQMD.

15. 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: –Minimum hadronic η/s ~(0.9-1.0) in full chem and kinetic equil. –η/s reduced at non-unit fugacities, and this suggests strong T-dep. of η/s as it evolves from T≈T c + ! Outlook : - Map out η /s trajectory probed by collision in hadronic phase. - Complete calculations for hadronic ς/s and possible effect of system evolving out of chemical equilibrium. - Assess systematic uncertainty associated with neglecting multiparticle processes. - Trajectory of viscositie(s) in a HIC crucial input to T-dep vRFD calculations and vRFD+micro models.

16. Backup Slides

17. 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.

18. Entropy Considerations Method I: Gibbs formula for entropy: (extract μ B for our system from SHAREv2, P and ε known from UrQMD.) Denote as s Gibbs. 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 & μ B /T! Denote as s specific