# Presentation: 134973

## Shear viscosity of a hadronic 

**gas mixture**

***K. Itakura ******(KEK) ***
***   ******with                       ******      ******O. Morimatsu ******(KEK,Tokyo)******             & H. Otomo****** ******(Tokyo)******          ***

*** ******Quark Matter 2008 at Jaipur, India***

*** *****Plan**
**1. Motivation**
**2. Theoretical framework**
**3. Numerical results**
**4. Summary**

**Based 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”

## 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)

* ***/*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

-    Relativistic Boltzmann equations for *f***(*x,p*)*, f**N*(*x,p*)* *

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

- 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 method**
***f******f******f********f******f******f**N*

*  **g***=3, *g*** =2
- degeneracy factor

## Theoretical framework (2/5)

_**Theoretical framework (2/5)**_

_**Collision terms**_:
- _Effects of statistics included_ (+ bosons, - fermions)
- _2____2 elastic scattering amplitude in the vacuum_  (***, ****N, NN*)

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

_**Boltzmann eq.**_

**

## Theoretical framework (3/5)

**Phenomenological ~ exp data**

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

-  **Fit region**

**Comparison of elastic cross sections**

- (mb)

- (mb)

- (mb)

**Fit performed for **_**differential**_** cross sections**

**Theoretical ****framework (3/5)**

## 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**

- Pheno. Cross sec OK

- Validity region of low energy
- effective theory is VERY narrow
- for *NN* scattering

*T*

**

*NN*

## Theoretical  Framework  (5/5)

- Valid only for a dilute gas

_**Validity of the Boltzmann description**_

## 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

## Numerical results (2/4)

**Pion-nucleon gas**

- Total viscosity increases !
- (pion contribution decreases)

- (only results of pheno. shown)

**

********/*****s***** decreases with increasing ********
- but still above the KSS bound
- (flattening seems to occur at lower T with increasing **)
**/*s *~ 0.3 @ T > 100MeV,  ** ~900MeV
- Consistent 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.

- 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

- /s will have a minimum at T=Tc.
- For crossover transition, slope of /s will vanish at Tc

_**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 

**Numerical results (4/4)**

_**Approaching phase boundary (2/2)**_

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

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

## Summary

**We studied **_**T and **__******__** dependence**_** of ******** and ********/s in a pion-nucleon 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.

## Classical kinetic theory of mixtures

_**Classical kinetic theory of mixtures**_

- cf Kennard “Kinetic theory of gases” 1938

## T dependence of viscosity

- Pion gas                             pion-nucleon gas