Shear viscosity of a hadronic gas mixture

1. Shear viscosity of a hadronic gas mixture Based on Phys. Rev. D77 (08) 014014 K. Itakura (KEK) with O. Morimatsu (KEK,Tokyo) & H. Otomo(Tokyo) Plan 1. Motivation 2. Theoretical framework 3. Numerical results 4. Summary Quark Matter 2008 at Jaipur, India

2. Two conjectures on “h/s ”s : entropy density • 1. h/s has universal lower bound?? • “KSS bound” Kovtun-Son-Starinets ’03~ 2. h/s has information of phase transition??Cernai-Kapusta-McLerran ’06 - has minimum at the critical temperature in various substances - valley with a jump  cusp  shallow (1st order  2nd order  crossover) liquid P=100 MPa solid P=22.06 MPa P=10 MPa gas Motivation (1/2) • QGP at RHIC is close to perfect liquid ? Shear viscosity coefficient h (or h /s) in the QCD matter is small ?  “sQGP” P 22.06MPa 600Pa T 273K 647K

3. T m Motivation (2/2) A typical cartoon of h/sin the QCD matter (from Cernai-Kapusta-McLerran) Solid : weak-coupling QGP (Arnold-Moore-Yaffe 03) Dashed : dilute pion gas (Prakash2 -Venugopalan-Welke 93) sQGP? • h/s will have valley structure • RHIC suggests small h/s ~ 0.1 – 0.3 at T above (but close to) Tc • h/s of a meson gas is small enough continuity at Tc and small h/s in sQGP We include nucleon degrees of freedom  h/s in a wide region of the phase diagram (mB dependence)  Cross sections “effectively” increase (thenh/s decreases?? How about the KSS bound??)

4. Theoretical framework (1/5) Shear viscosity coefficienth : the ability of momentum transfer small deviation of energy momentum tensor from thermal equilibrium Vmflow vector i, j : spatial coordinates traceless part z bulk viscosity We compute h in a dilute gas of pions and nucleons in the kinetic theory pion gas: Davesne, Dobado et al, Chen-Nakano, … pN gas : Prakash et al, Chen et al. Relativistic Boltzmann equations forfp(x,p), fN(x,p)  Solved for small deviation from equilibrium via Chapman-Enskog method fp= fp(0)+dfp ,fN= fN(0)+dfN gp =3,gN=2 degeneracy factor

5. Theoretical framework (2/5) Boltzmann eq. Collision terms: • Effects of statistics included (+ bosons, - fermions) • 22 elastic scattering amplitude in the vacuum (pp, pN, NN) s Scatt. Amp.  Use cross sections fitted to experimental data(a great merit compared to the calculation by the Kubo formula)

6. Theoretical framework (3/5) Comparison of elastic cross sections Fit performed for differential cross sections Low energy effective theory (ChPT, Finite range eff. theory) Phenomenological ~ exp data Fit region (mb) (mb) (mb)

7. Range of validityconstrained by fit Theoretical framework (4/5) Need to “map” the fit region for s onto T-m plane  Scattering of two particles in (almost) thermal distribution typical scattering energy is a function of T and m Dispersion : highest energy of the scatt. in thermal distr. Validity condition NN T Pheno. Cross sec OK Validity region of low energy effective theory is VERY narrow for NN scattering m

8. Validity of the Boltzmann description Theoretical Framework (5/5) Valid only for a dilute gas

9. Pheno Boltzmann (pheno) ChPT Numerical results (1/4) Pion gas • Entropy density evaluated for equilibrium states • Decreasing function of T (both h and s increase) • Small enough at higher T, but well above • the KSS bound if pheno. cross section is used • - LO-ChPT valid only for T < 70MeV • - Phenomenological valid for T < 170MeV • - Boltzmann (w. Pheno) valid for T < 140MeV

10. h/s T h/s m Numerical results (2/4) (only results of pheno. shown) Pion-nucleon gas T=100MeV m Total viscosity increases ! (pion contribution decreases) h/s decreases with increasing m but still above the KSS bound (flattening seems to occur at lower T with increasing m) h/s ~ 0.3 @ T > 100MeV, m ~900MeV Consistent with hadron cascade calc.(URASiMA)

11. T m Anything about phase transition? • We do not expect we can describe phase transition in the Boltzmann equation! • Only tendency towards the critical T/m will be quantitatively reliable. • Still, extrapolation of the results will give some qualitative information about phase transition. 140MeV 940MeV

12. T m Slope Approaching phase boundary (1/2) Numerical results (3/4) Pion gas (towards higher T) Extrapolate the Boltzmann description which is justified only up to 140MeV - Extrapolating results (of pheno. cross sec.) at T=140MeV towards higher T - Slope is zero at T ~ 170MeV = Tc-h/s ~ 0.9 at T=Tc h/s will have a minimum at T=Tc. For crossover transition, slope of h/s will vanish at Tc

13. T h/s m T=10MeV h/s Valley structure will correspond to the nuclear liquid-gas transition !! (But precise structure is not described well) Approaching phase boundary (2/2) Numerical results (4/4) p-N gas at low T and high m Non-monotonic behavior of h/s suggests the existence of phase transition • h/s minimum at m ~ 940MeV • = upper limit of Boltzmann description • (m > 940MeV is no more dilute gas) • valley becomes shallower as T increases

14. Summary • We studied T and m dependence of h and h/s in a pion-nucleon gas. • In a wide region of T-m plane, the ratio h/s is a decreasing function of T and m. • The ratio h/s becomes small enough at high m, but is still above the conjectured KSS bound. • The decrease of h/s with increasing m is mainly due to the enhancement of entropy (viscosity increases). • At low T (~10MeV) and high m ~ 940 MeV, the ratio h/s shows valley structure which will probably corresponds to the nuclear liquid-gas phase transition.

15. Classical kinetic theory of mixtures cf Kennard “Kinetic theory of gases” 1938 Pion-nucleon gas

16. T dependence of viscosity Pion gas pion-nucleon gas