Shear viscosity of a hadronic Shear viscosity of a hadronic gas mixturegas mixture

K. Itakura (KEK)

　　　with 　 　 O. Morimatsu (KEK,To

kyo) & H. Otomo (Tokyo)

Quark Matter 2008 at Jaipur, India

Plan1. Motivation

2. Theoretical framework3. Numerical results

4. Summary

Based on Phys. Rev. D77 (08) 014014Based on Phys. Rev. D77 (08) 014014

Motivation (1/2)• QGP at RHIC is close to perfect liquid ? Shear viscosity coefficient (or /s) in the QCD matter is small ? “sQGP”

P

T273K 647K

600Pa

22.06MPa

gas

solid

liquid

P=10 MPa

P=22.06 MPa

P=100 MPa

2. /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)

41

s

• Two conjectures on “ /s ” s : entropy density

1. /s has universal lower bound?? “KSS bound” Kovtun-Son-Starinets ’03~

Motivation (2/2)A typical cartoon of /s in the QCD matter (from Cernai-Kapusta-McLerran)

Solid : weak-coupling QGP (Arnold-Moore-Yaffe 03)Dashed : dilute pion gas (Prakash2 -Venugopalan-Welke 93)

T

• /s will have valley structure • RHIC suggests small /s ~ 0.1 – 0.3 at T above (but close to) Tc• /s of a meson gas is small enough continuity at Tc and small /s in sQGP

We include nucleon degrees of freedom /s in a wide region of the phase diagram (B dependence) Cross sections “effectively” increase (then /s decreases?? How about the KSS bound??)

sQGP?

Theoretical framework (1/5)

Shear viscosity coefficient : the ability of momentum transfer small deviation of energy momentum tensor from thermal equilibrium V flow vector i, j : spatial coordinates

traceless part bulk viscosity vvijvvij

ijjiij VV

VVT

31

22

Relativistic Boltzmann equations for f(x,p), fN(x,p)

We compute in a dilute gas of pions and nucleons in the kinetic theory

3 3i j i jij N N

N

p p p pT g d p f g d p f

E E

pion gas: Davesne, Dobado et al, Chen-Nakano, … N gas : Prakash et al, Chen et al.

Solved for small deviation from equilibrium via Chapman-Enskog methodffffffN

g=3, g =2 degeneracy factor

N

Theoretical framework (2/5)

Collision terms:

• Effects of statistics included (+ bosons, - fermions) • 22 elastic scattering amplitude in the vacuum (, N, NN)

],[],[ ffCffCfvtf

dtdf

N

],[],[ NNNN ffCffC

dtdf

)()()(1)(1)(1)(1)()(],[ 321321 pfkfkfkfpfkfkfkfdffC BABABABABA

)()2()2)(2( 321

)4(43

2

3,2,1

pkkkE

Mdkd

iii

Scatt. Amp. Use cross sections fitted to experimental data (a great merit compared to the calculation by the Kubo formula)

Boltzmann eq.

)(MeVs )(GeVs

Phenomenological ~ exp data

Low energy effective theory (ChPT, Finite range eff. theory)

)(MeVs

Fit region

Comparison of elastic cross sections

(mb)

(mb) (mb)

Fit performed for differential cross sections

Theoretical framework (3/5)

GeV 30.2:

GeV 2.00 :

GeV 15.1 :

sNN

sN

s

Theoretical framework (4/5)

Need to “map” the fit region for s onto T- plane Scattering of two particles in (almost) thermal distribution

typical scattering energy is a function of T and

Dispersion

: highest energy of the scatt. in thermal distr.

Validity condition

Range of validity constrained by fit

)()(

)()(),(

20

10

21

20

10

2121

pfpfdpdp

pfpfppsdpdps

22),( ssT

(max)ss

s

s

s

Pheno. Cross sec OK

(max)ss

Validity region of low energy effective theory is VERY narrowfor NN scattering

T

NN

Theoretical Framework (5/5)

Valid only for a dilute gas

Validity of the Boltzmann description

1range)on (interactipath) free(mean

d

md /1~

n

1~

Numerical results (1/4)Pion gas

• Entropy density evaluated for equilibrium states• Decreasing function of T (both 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

ChPT

Boltzmann (pheno)Pheno

Numerical results (2/4)Pion-nucleon gas

N

Total viscosity increases !(pion contribution decreases)

(only results of pheno. shown)

/s

T/s

/s decreases with increasing but still above the KSS bound(flattening seems to occur at lower T with increasing )

/s ~ 0.3 @ T > 100MeV, ~900MeVConsistent with hadron cascade calc.(URASiMA)

T=100MeV

Anything about phase transition?

• We do not expect we can describe phase transition in the Boltzmann equation!

• Only tendency towards the critical T/ will be quantitatively reliable.

• Still, extrapolation of the results will give some qualitative information about phase transition.

T

140MeV

940MeV

Numerical results (3/4)

- Extrapolating results (of pheno. cross sec.) at T=140MeV towards higher T - Slope is zero at T ~ 170MeV = Tc - /s ~ 0.9 at T=Tc

T

/s will have a minimum at T=Tc.

For crossover transition, slope of /s will vanish at Tc

Slope

Approaching phase boundary (1/2)Pion gas (towards higher T) Extrapolate the Boltzmann description

which is justified only up to 140MeV

-N gas at low T and high T

Numerical results (4/4)Approaching phase boundary (2/2)

Non-monotonic behavior of /s suggests the existence of phase transition

T=10MeV

/s

)(MeV

/s• /s minimum at ~ 940MeV = upper limit of Boltzmann description (> 940MeV is no more dilute gas)• valley becomes shallower as T increases

Valley structure will correspond tothe nuclear liquid-gas transition !!

(But precise structure is not described well)

Summary• We studied T and dependence of and /s in a pion-nu

cleon gas.

• In a wide region of T- plane, the ratio /s is a decreasing function of T and .

• The ratio /s becomes small enough at high , but is still above the conjectured KSS bound.

• The decrease of /s with increasing is mainly due to the enhancement of entropy (viscosity increases).

• At low T (~10MeV) and high ~ 940 MeV, the ratio /s shows valley structure which will probably corresponds to the nuclear liquid-gas phase transition.

2/11

2/11

00

mmnnmm

nn N

NN

N

N

N

NNN

Pion-nucleon gas

,1

'3

0

i

ij i

jij

i

i i

ii

i

iii

nnl

lvnm 2

/1 ji

ii

ijij

mm

componentth -i of gas pure a ofosity shear visc :0i

componentth -i ofdensity number :incomponentsth -j andth -i btwsection cross :ij

30 p

NNN

p

3

0

Classical kinetic theory of mixturescf Kennard “Kinetic theory of gases” 1938

T dependence of viscosity

Pion gas pion-nucleon gas