# A study of shear viscosity coefficient of quark matter in Polyakov quark
meson model for three flavors

Ritesh Ghosh<EMAIL_ADDRESS>Theory Division, Saha Institute of
Nuclear Physics, HBNI,
1/AF, Bidhannagar, Kolkata 700064, India  Homi Bhabha National Institute,
Anushaktinagar,
Mumbai, Maharashtra 400094, India

###### Abstract

We have computed shear viscosity coefficient for hot and dense QCD matter
within Polyakov loop extended quark meson(PQM) model for three flavors.
Viscosity coefficient is estimated using Boltzmann equation within the
relaxation time approximation. Relaxation times for quarks are calculated from
elastic scattering of quarks and antiquarks via gluon exchange. Also the mean
free paths of strange and light quarks are estimated for this cases. The
calculations are performed at finite chemical potential. Thermodynamic
properties like energy, pressure, entropy densities are also computed and
compared with lattice QCD data.

##  I Introduction

A state of matter of quarks and gluons i.e. quark-gluon plasma (QGP) is formed
in relativistic heavy ion collisions. Different signatures like jet quenching
Cao and Wang (2021), elliptic flow Csernai and Rohrich (1999), quarkonia
suppression Brambilla  _et al._ (2011), photon/dilepton production Rapp
(2003); Shuryak and Xiong (1993) etc from high energy heavy-ion collision
experiments at Relativistic Heavy-Ion Collider (RHIC) and at Large Hadron
Collider (LHC) suggest the creation of such deconfined QGP state. The produced
hot and dense deconfined matter expands and cools down and then undergoes
through a phase transition to the hadronic final state. Transport coefficients
are interesting quantities in this context. The temperature and chemical
potential dependency of transport coefficients draws relevant impact on the
location of phase transition. Elliptic flow Romatschke and Romatschke (2007);
Gale  _et al._ (2013) study describes the collectivity of the hot and dense
quark-gluon matter at high temperature and indicates the smallest viscosity to
entropy density ratio $(\eta/s)$. These studies suggest that the QGP produced
in heavy-ion collisions interacts strongly and behaves as a nearly perfect
relativistic fluid Schäfer and Teaney (2009). For hydrodynamic simulations,
the transport coefficients are also input parameters to explain the evolution
of the hot and dense matter Csernai  _et al._ (2006); Heinz and Snellings
(2013).

Transport coefficients can be computed from the QCD using Kubo formulas Kubo
(1957); Akemann (2003); Lang  _et al._ (2012). But it is complicated to
calculate from first principles. Lattice QCD Borsanyi  _et al._ (2014);
Nakamura and Sakai (2005) may be a good approach to calculate transport
coefficient at zero chemical potential. But it is very challenging to deal
with finite chemical potential due to the well known fermion sign problem for
QCD. Hence, at higher chemical potential several investigations have been done
to calculate shear viscosity of strongly interacting matter using effective
model calculations like Nambu-Jona- Lasinio (NJL) Marty  _et al._ (2013); Deb
_et al._ (2016) , quasiparticle model (QPM ) Sasaki and Redlich (2009); Bluhm
_et al._ (2011); Soloveva  _et al._ (2020), chiral perturbation theory Dobado
_et al._ (2011); Itakura  _et al._ (2008), the linear sigma model Chakraborty
and Kapusta (2011); Heffernan  _et al._ (2020), the functional renormalization
group approach Pawlowski (2007); Bagnuls and Bervillier (2001); Gies (2012).
To explore the QGP phase transition experimentally at medium and higher
chemical potential, the Beam Energy Scan (BES) program at RHIC Odyniec (2019)
has carried out several experiments. The near future experimental program of
FAIR (Facility for Antiproton and Ion Research) Senger (2020) at GSI and the
NICA(Nuclotron-based Ion Collider Facility) facility at Joint Institute for
Nuclear Research (JINR) Sissakian and Sorin (2009) are also likely to explore
more physics at moderate and relatively high chemical potential.

Effective models like NJL, quark meson (QM) model can explain the QCD chiral
phase diagram. But they can not address the confinement-deconfinement
transition scenario. Both the picture can be achieved by incorporating
Polyakov loop in NJL or QM type models. Polyakov NJL (PNJL) Ciminale  _et al._
(2008); Fukushima (2008); Fu  _et al._ (2008) or Polyakov quark meson (PQM)
Mao  _et al._ (2010); Stiele and Schaffner-Bielich (2016) models for three
flavors can agree better with lattice data. In the present article we have
investigated the shear viscosity coefficient within the Polyakov loop extended
quark meson model for $(2+1)$ flavors at finite temperature and chemical
potential. Shear viscosity coefficient is calculated within the relaxation
time approximation of the Boltzmann equation. Medium dependent particle masses
are used from the PQM model. The relaxation time is estimated from the
scattering rate of quark quark and quark antiquark via gluon exchange. In Ref.
Abhishek  _et al._ (2018); Singha  _et al._ (2019), the authors calculated
transport coefficients for quarks and hadronic matters within two flavor PQM
model.

The article is organized as follows. In Sec. II, the formalism of three flavor
Polyakov loop quark meson model is addressed. We get the medium dependent
masses of particles which are involved in calculation of transport
coefficients. Thermodynamic properties like energy, pressure, entropy
densities are also discussed in Sec. III. Relaxation time and mean free path
is obtained by studying the quark-quark and quark-antiquark scattering for
three quark flavors. We have derived shear viscosity in Sec. IV. Numerical
results are described in Sec. V and finally the work is summarized in Sec. VI.

##  II Polyakov Quark-meson model

###  II.1 Polyakov loop

Three flavor PQM model Schaefer  _et al._ (2010) is a generalization of two
flavor PQM model Schaefer  _et al._ (2007) combining the chiral symmetry
restoration and the feature of confinement-deconfinement transition. Polyakov
loop operator is defined as Wilson loop in the temporal direction Polyakov
(1978)

$\displaystyle\mathcal{P}(\vec{x})$ $\displaystyle=$ $\displaystyle
P\exp\Big{(}i\int_{0}^{\beta}d\tau A_{0}(\vec{x},\tau)\Big{)},$ (1)

where $P$ is the path ordering and $A_{0}$ is the temporal component of
Euclidean gauge field $A_{\mu}$. Polyakov loop variables are

$\displaystyle\Phi(\vec{x})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\langle\operatorname{\text{Tr}}\mathcal{P}(\vec{x})\rangle,$
(2) $\displaystyle\bar{\Phi}(\vec{x})$ $\displaystyle=$
$\displaystyle\frac{1}{N_{c}}\langle\operatorname{\text{Tr}}\mathcal{P}(\vec{x})^{\dagger}\rangle,$
(3)

where trace is taken over color space. $N_{c}$ is color degrees of freedom.
Mean value of $\Phi$ and $\bar{\Phi}$ are related to the free energy of
infinitely heavy, static quarks and antiquarks. $\Phi$ and $\bar{\Phi}$ vanish
in the confined phase as infinite energy is required to put a static quark in
that phase. $\Phi$ and $\bar{\Phi}$ are finite in deconfined phase. So $\Phi$
works as the order parameter of confinement-deconfinement phase transition.

The PQM Lagrangian contains quark-meson contribution and Polyakov loop
potential. The form of PQM Lagrangian reads as Schaefer  _et al._ (2010)

$\displaystyle L_{PQM}$ $\displaystyle=$
$\displaystyle\overline{\psi}(i\,/\\!\\!\\!\\!D-g\phi_{5})\psi+L_{m}-U(\phi,\overline{\phi}),$
(4)

where $\psi\equiv(u,d,s)$ indicates quark filed for $N_{f}=3$ flavors.
Covariant derivative is defined as $D_{\mu}=\partial_{\mu}-iA_{\mu}$ with
$A_{\mu}=\delta_{\mu 0}A^{0}$. Here $A_{\mu}=g_{s}A_{\mu}^{a}\lambda^{a}/2$,
where $\lambda^{a}$ with $a=1,...,N_{c}^{2}-1$ are usual Gell-Mann matrices
and $g_{s}$ is the $SU(N)$ gauge coupling. Quark meson interaction is
incorporated by the flavor blind Yukawa coupling $g$ and the meson matrix

$\displaystyle\phi_{5}=T_{a}(\sigma_{a}+i\gamma_{5}\pi_{a}).$ (5)

$\sigma_{a}$ and $\pi_{a}$ are nine scalar and pseudoscalar mesons
respectively. $T_{a}=\lambda_{a}/2$ represent the nine generators of the
$U(3)$ symmetry. The generators obey the usual $U(3)$ algebra.

Pure mesonic contribution can be written as

$\displaystyle L_{m}$ $\displaystyle=$
$\displaystyle\operatorname{\text{Tr}}(\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi)-m^{2}\operatorname{\text{Tr}}(\phi^{\dagger}\phi)-\lambda_{1}[\operatorname{\text{Tr}}(\phi^{\dagger}\phi)]^{2}-\lambda_{2}\operatorname{\text{Tr}}(\phi^{\dagger}\phi)^{2}$
(6) $\displaystyle+$ $\displaystyle
c[\det(\phi)+\det(\phi^{\dagger})]+\operatorname{\text{Tr}}[H(\phi+\phi^{\dagger})],$

where $\phi$ is a $3\times 3$ matrix defined as

$\displaystyle\phi=T_{a}\phi_{a}=T_{a}(\sigma_{a}+i\pi_{a}).$ (7)

Chiral symmetry is explicitly broken by the term $H=T_{a}h_{a}$, where $h_{a}$
are nine external parameters. $m$ is the tree level mass of meson fields.
$\lambda_{1}$ and $\lambda_{2}$ are two quartic coupling constant and $c$ is
cubic coupling constant which models the $U_{A}(1)$ axial anomaly of the QCD
vacuum.

###  II.2 Potential for the Polyakov loop

Polyakov loop potential for $SU(3)$ pure gauge theory has been proposed to
reproduce the lattice thermodynamic results. There are several forms of the
potential in literature. Here we adopt the polynomial parametrization form of
the potential as Ratti  _et al._ (2006); Schaefer  _et al._ (2010)

$\displaystyle
U_{p}(\Phi,\bar{\Phi},T)=T^{4}\bigg{[}-\frac{b_{2}}{4}\big{(}|\Phi|^{2}+|\bar{\Phi}|^{2}\big{)}-\frac{b_{3}}{6}\big{(}\Phi^{3}+\bar{\Phi}^{3}\big{)}+\frac{b_{4}}{16}\big{(}|\Phi|^{2}+|\bar{\Phi}|^{2}\big{)}^{2}\bigg{]},$
(8)

with the temperature dependent coefficient

$\displaystyle
b_{2}(T)=a_{0}+a_{1}\bigg{(}\frac{T_{0}}{T}\bigg{)}+a_{2}\bigg{(}\frac{T_{0}}{T}\bigg{)}^{2}+a_{3}\bigg{(}\frac{T_{0}}{T}\bigg{)}^{3},$
(9)

governing the confinement-deconfinement phase transition. At low temperature,
the potential $U_{p}(\Phi,\bar{\Phi},T)$ has one minimum at $\Phi=0$ in
confined phase. Above the critical temperature, due to the spontaneous
breaking of $Z(3)$ central symmetry, the potential has three degenerate global
minima at non zero $\Phi$. Here we mention the value of the temperature
independent parameters:

$\displaystyle
a_{0}=6.75,\,\,a_{1}=-1.95,\,\,a_{2}=2.625,\,\,a_{3}=-7.44,\,\,b_{3}=0.75,\,\,b_{4}=7.5.$

###  II.3 Thermodynamic Potential

To investigate the thermodynamic properties of the PQM model the thermodynamic
potential is used in light of mean field approximation. Consisting the mesonic
$\mathcal{U}(\sigma_{x},\sigma_{y})$, quark-antiquark
$\Omega_{q\overline{q}}(\Phi,\bar{\Phi})$ and Polyakov loop contribution
$U_{p}(\Phi,\bar{\Phi})$, the thermodynamic potential can be given by,

$\displaystyle\Omega(T,\mu)$ $\displaystyle=$
$\displaystyle\mathcal{U}(\sigma_{x},\sigma_{y})+U_{p}(\Phi,\bar{\Phi})+\Omega_{\overline{q}q}(\Phi,\bar{\Phi},\sigma_{x},\sigma_{y}).$
(10)

The mesonic contribution reads as Tawfik and Diab (2015); Schaefer and Wagner
(2009); Gupta and Tiwari (2010),

$\displaystyle\mathcal{U}(\sigma_{x},\sigma_{y})$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{2}(\sigma_{x}^{2}+\sigma_{y}^{2})-h_{x}\sigma_{x}-h_{y}\sigma_{y}-\frac{c}{2\sqrt{2}\sigma_{x}^{2}\sigma_{y}}+\frac{\lambda_{1}}{2}\sigma_{x}^{2}\sigma_{y}^{2}$
(11) $\displaystyle+$
$\displaystyle\frac{1}{8}(2\lambda_{1}+\lambda_{2})\sigma_{x}^{4}+\frac{1}{8}(\lambda_{1}+2\lambda_{2})\sigma_{y}^{4}.$

Minimizing the grand potential with respect to
$\sigma_{x}$,$\sigma_{y}$,$\Phi$ and $\bar{\Phi}$, we obtain four coupled
equations. There are six input parameters
$m^{2},c,\lambda_{1},\lambda_{2},h_{x},h_{y}$ and they are fitted with known
quantities: pseudoscalar masses i.e. kaon mass $m_{k}$, pion mass $m_{\pi}$,
$m_{\eta}^{2}+m_{\eta^{\prime}}^{2}$ , pion and kaon decay constant $f_{\pi}$,
$f_{K}$ and mass of scalar $\sigma$ meson $m_{\sigma}$. These parameters are
fitted to produce the observed pion mass in vacuum.

The fermionic part of the thermodynamic potential is written as

$\displaystyle\Omega_{\overline{q}q}(\Phi,\bar{\Phi},\sigma_{x},\sigma_{y})$
$\displaystyle=$
$\displaystyle-2T\sum_{f=u,d,s}\int\frac{d^{3}p}{(2\pi)^{3}}\bigg{\\{}\ln
g_{f}^{+}+\ln g_{f}^{-}\bigg{\\}},$ (12)

where $g_{f}^{+}$ and $g_{f}^{-}$ are defined as

$\displaystyle g_{f}^{+}$ $\displaystyle=$
$\displaystyle\bigg{[}1+3(\Phi+\bar{\Phi}e^{-(E_{f}-\mu_{f})/T})e^{-(E_{f}-\mu_{f})/T}+e^{-3(E_{f}-\mu_{f})/T}\bigg{]}$
(13) $\displaystyle g_{f}^{-}$ $\displaystyle=$
$\displaystyle\bigg{[}1+3(\bar{\Phi}+\Phi
e^{-(E_{f}+\mu_{f})/T})e^{-(E_{f}+\mu_{f})/T}+e^{-3(E_{f}+\mu_{f})/T}\bigg{]}.$
(14)

where $E_{f}=\sqrt{p^{2}+m_{f}^{2}}$ is the flavor dependent single particle
energy with the quark masses

$\displaystyle m_{u,d}=g\sigma_{x}/2,\,\,\,m_{s}=g\sigma_{y}/\sqrt{2}\,.$ (15)

$\mu_{f}$ is the chemical potential for a quark flavor $f$. In the present
work, uniform quark chemical potential $\mu=\mu_{u}=\mu_{d}=\mu_{s}$ is
considered. Finally, the behavior of quark condensate $\sigma_{x},\sigma_{y}$
and the Polyakov loop expectation values $\Phi,\bar{\Phi}$ can be obtained by
minimizing the total effective potential $\Omega$ for a given value of
temperature and chemical potential i.e.

$\displaystyle\frac{\partial\Omega}{\partial\sigma_{x}}=\frac{\partial\Omega}{\partial\sigma_{y}}=\frac{\partial\Omega}{\partial\Phi}=\frac{\partial\Omega}{\partial\bar{\Phi}}\Bigg{|}_{\sigma_{x}=\langle\sigma_{x}\rangle,\sigma_{y}=\langle\sigma_{y}\rangle,\Phi=\langle\Phi\rangle,\bar{\Phi}=\langle\bar{\Phi}\rangle}=0.$
(16)

##  III QCD Thermodynamics

Figure 1: Variation of $\sigma_{x}$, $\sigma_{y}$ condensate (left panel ) and
the expectation value of Polyakov loop $\langle\Phi\rangle$ (right panel) as
functions of temperature for $\mu_{f}=0$

By solving the coupled equations in (16) one can extract the temperature
dependence of strange condensate ($\sigma_{y}$), nonstrange condensate
$(\sigma_{x})$ and also the expectation value of the Polyakov loop. The
temperature variation of $\sigma_{x}$ and $\sigma_{y}$ are plotted in the left
panel and the expectation value of Polyakov loop $\langle\Phi\rangle$ is shown
in the right panel of Fig. 1 at zero chemical potential ($\mu=0$) in PQM
model. The quark condensates $\sigma_{x}$ and $\sigma_{y}$ are regarded as
order parameters of chiral phase transition in $(2+1)$ flavor PQM model. From
the left panel of Fig. 1 it is shown that chiral symmetry is restored at high
temperature. On the other hand $\langle\Phi\rangle$ increases from
$\langle\Phi\rangle=0$ at $T=0$ to about $\langle\Phi\rangle=1$ at high
temperature. The detailed discussion can be found in Ref. Schaefer  _et al._
(2010); Schaefer and Wagner (2009).

We have studied the thermodynamic quantities energy, pressure and entropy
density. Pressure density is defined as

$\displaystyle P(T)=-\Omega(T).$ (17)

By differentiating the thermodynamic potential, other quantities are obtained.
The entropy density(s) and energy density $(\epsilon)$ are defined as

$\displaystyle s(T)$ $\displaystyle=$
$\displaystyle-\frac{\partial\Omega(T)}{\partial T}$ (18)

and

$\displaystyle\epsilon(T)=-P(T)+Ts(T).$ (19)

At high temperature, they show the behavior $P,\epsilon\sim T^{4}$ and $s\sim
T^{3}$. In Fig. 2, the scaled quantities $s/T^{3},P/T^{4}$ and
$\epsilon/T^{4}$ are plotted with the variation of temperature. We have
plotted the graphs for zero chemical potential. They grow with the temperature
and tend to saturate at high temperature. We have also compared the
thermodynamic quantities with the LQCD results from Ref. Borsanyi  _et al._
(2010). The lattice data is shown for temporal extent $N_{t}=10$. We found our
results are in agreement with corresponding LQCD results. Here we want to
mention that, implicit dependence of the order parameter on temperature as
well as chemical potential should be taken into account to calculate various
thermodynamic quantities. After solving the Eq. 16, one can get the
thermodynamic quantities by numerical differentiation of of the order
parameters. To avoid the inaccuracy in the results, semianalytic approach
should be taken Ghosh  _et al._ (2014).

Figure 2: Energy density (left), entropy density (right) and pressure (bottom)
as functions of temperature for $\mu_{f}=0$

##  IV Transport coefficients

We have calculated shear viscosity coefficients for quark matter in relaxation
time approximation. Medium dependent particle masses coming from the PQM model
are considered. The viscosity coefficients are estimated for finite chemical
potential. The shear viscosity coefficient is given by Deb  _et al._ (2016);
Marty  _et al._ (2013)

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{1}{15T}\sum_{a}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{p^{4}}{E_{a}^{2}}\tau
f_{a}^{0}(1-f_{a}^{0}),$ (20)

where $f_{a}^{0}$ is equilibrium Fermi-Dirac distribution function and
$\tau_{a}$ is the relaxation time for particle ‘a’. The sum is over the three
quark species including their antiparticles. The relaxation time is calculated
from elastic scattering of quarks and antiquarks via gluon exchange in
perturbative QCD.

###  IV.1 Relaxation time estimation from elastic scattering in perturbative
QCD

Thermal relaxation time for a single species can be written as Rehberg  _et
al._ (1996); De Groot (1980)

$\displaystyle\tau_{f}^{-1}=\sum_{f^{\prime}}\rho_{f^{\prime}}\bar{w}_{ff^{\prime}},$
(21)

where $\bar{w}_{ff^{\prime}}$ is the sum of transition rates of all
species.The averaged transition rate is defined as

$\displaystyle\bar{w}=\int_{\text{Th}}^{\infty}ds\sqrt{(p_{1}p_{2})^{2}-(m_{1}m_{2})^{2}}\sigma(s)P(s,T,\mu),$
(22)

with threshold $\text{Th}=Max[(m_{1}+m_{2})^{2},(m_{3}+m_{4})^{2}]$. The
weight function is written as

$\displaystyle P(s,T,\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{\rho_{1}(T,\mu)\rho_{2}(T,\mu)}\frac{1}{16\pi^{4}}\int_{m_{1}}^{\infty}dE_{1}f^{0}(\beta
E_{1},\mu)\int_{m_{2}}^{\infty}dE_{2}f^{0}(\beta E_{2},\mu)$ (23)
$\displaystyle\times$
$\displaystyle\Theta\bigg{(}4|\vec{p}_{1}|^{2}|\vec{p}_{2}|^{2}-(s-m_{1}^{2}-m_{2}^{2}-2E_{1}E_{2})^{2}\bigg{)},$

where the quark density $\rho_{f}$ is defined as

$\displaystyle\rho_{f}(T,\mu)=\int\frac{d^{3}p}{(2\pi)^{3}}f^{0}\bigg{(}\beta\sqrt{\vec{p}^{2}+m_{f}^{2}},\mu\bigg{)},$
(24)

with $\beta=1/T$. Now we need to calculate scattering cross section
$\sigma(s,T)$ to obtain the relaxation time. To be specific we confine
ourselves by taking the contribution from the elastic scattering of quarks and
antiquarks via gluon exchange. Including the fermi blocking factors, the total
cross section in terms of differential cross section looks like Zhuang  _et
al._ (1995)

$\displaystyle\sigma(s)=\int dt\frac{d\sigma}{dt}\bigg{[}1-f^{0}(\beta
E_{3},\mu)\bigg{]}\bigg{[}1-f^{0}(\beta E_{4},\mu)\bigg{]},$ (25)

where $E_{3,4}^{2}=p_{3,4}^{2}+m_{3,4}^{2}$.

For quark-quark scattering quarks can interact through $t$ and $u$ channel.
There are four independent processes out of total six different possibilities
for quark-quark scattering Rehberg  _et al._ (1996). All six processes are

$\displaystyle uu\rightarrow uu,\,\,\,$ $\displaystyle ss\rightarrow
ss,\,\,\,dd\rightarrow dd,$ $\displaystyle ud\rightarrow ud,\,\,\,$
$\displaystyle us\rightarrow us,\,\,\,ds\rightarrow ds.$ (26)

The differential cross section for this case is Rehberg  _et al._ (1996);
Cutler and Sivers (1978)

$\displaystyle\frac{d\sigma}{dt}(q_{\alpha}q_{\beta}\rightarrow
q_{\alpha}q_{\beta})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\frac{1}{[s-(m_{\alpha}-m_{\beta})^{2}][s-(m_{\alpha}+m_{\beta})^{2}]}\frac{1}{36}\sum_{spin,color}|\mathcal{M}_{t}+\delta_{\alpha\beta}\mathcal{M}_{u}|^{2},$
(27)

with color and spin summed square of matrix element

$\displaystyle\frac{1}{36}\sum_{spin,color}|\mathcal{M}_{t}+\delta_{ff^{\prime}}\mathcal{M}_{u}|^{2}$
$\displaystyle=$
$\displaystyle\frac{64\pi^{2}\alpha_{s}^{2}}{9}\bigg{[}\frac{2(s-m_{\alpha}^{2}-m_{\beta}^{2})^{2}+t^{2}+2st}{(t-m_{g}^{2})^{2}}+\delta_{\alpha\beta}\frac{2(s-2m_{\alpha}^{2})^{2}+u^{2}+2su}{(u-m_{g}^{2})^{2}}$
(28) $\displaystyle-$
$\displaystyle\delta_{\alpha\beta}\frac{2}{3}\frac{(s-4m_{\alpha}^{2})^{2}-4m_{\alpha}^{2}}{(t-m_{g}^{2})(u-m_{g}^{2})}\bigg{]}.$

$\mathcal{M}_{t}$ and $\mathcal{M}_{u}$ are matrix elements for $t$ and $u$
channel respectively.

For quark-antiquark scattering, there are fifteen possible interaction
processes. But from these fifteen possibilities, one get only seven
independent processes. Here all the fifteen possibilities are

$\displaystyle u\bar{d}\rightarrow u\bar{d},\,\,\,\,d\bar{u}\rightarrow
d\bar{u},\,\,\,\,s\bar{u}\rightarrow s\bar{u},$ $\displaystyle
u\bar{s}\rightarrow u\bar{s},\,\,\,\,d\bar{d}\rightarrow
d\bar{s},\,\,\,\,s\bar{d}\rightarrow s\bar{d},$ $\displaystyle
u\bar{u}\rightarrow u\bar{u},\,\,\,\,d\bar{d}\rightarrow
d\bar{s},\,\,\,\,s\bar{d}\rightarrow d\bar{d},$ $\displaystyle
u\bar{u}\rightarrow d\bar{d},\,\,\,\,d\bar{d}\rightarrow
s\bar{s},\,\,\,\,s\bar{s}\rightarrow u\bar{u},$ $\displaystyle
u\bar{u}\rightarrow s\bar{s},\,\,\,\,d\bar{d}\rightarrow
u\bar{u},\,\,\,\,s\bar{s}\rightarrow s\bar{s}.$ (29)

Quark and antiquark interact through $s$ and $t$ channel and the differential
cross section is given by Rehberg  _et al._ (1996); Cutler and Sivers (1978)

$\displaystyle\frac{d\sigma}{dt}(q_{\alpha}\bar{q}_{\beta}\rightarrow
q_{\delta}\bar{q}_{\gamma})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\frac{1}{[s-(m_{\alpha}-m_{\beta})^{2}][s-(m_{\alpha}+m_{\beta})^{2}]}$
(30) $\displaystyle\times$
$\displaystyle\frac{64\pi^{2}\alpha_{s}^{2}}{9}\bigg{[}\delta_{\alpha\beta}\delta_{\gamma\delta}\frac{2(u-m_{\alpha}^{2}-m_{\delta}^{2})^{2}+s^{2}+2su}{(s-m_{g}^{2})^{2}}$
$\displaystyle+$
$\displaystyle\delta_{\alpha\delta}\delta_{\beta\gamma}\frac{2(u-m_{\alpha}^{2}-m_{\beta}^{2})^{2}+t^{2}+2tu}{(t-m_{g}^{2})^{2}}-\frac{2}{3}\delta_{\alpha\beta}\delta_{\beta_{\delta}}\delta_{\gamma\delta}\frac{(u-4m_{\alpha}^{2})^{2}-4m_{\alpha}^{4}}{(s-m_{g}^{2})(t-m_{g}^{2})}\bigg{]}.$

$\alpha_{s}$ is QCD coupling strength. To avoid the infrared divergent,
effective gluon mass,

$\displaystyle m_{g}^{2}=2\pi\alpha_{s}(1+\frac{1}{6}N_{f})T^{2},$ (31)

is used as a regulator Shuryak (1980); Rehberg  _et al._ (1996).

Figure 3: Plot of thermal relaxation time for light and strange quark as a
function of temperature.

##  V Results

Figure 4: Plot of mean free path for light and strange quark as a function of
temperature

Relaxation time is calculated numerically for quarks. Variation of relaxation
times for up quark ($\tau_{u}$) and strange quark $(\tau_{s})$ with
temperature are shown in Fig. 3. $\tau_{d}$ for down quark is same as
$\tau_{u}$. As quark density is low at higher temperature, relaxation time is
large in this temperature region. $\tau_{s}$ is greater than $\tau_{u}$. From
Fig. 1 it is shown that the condensate $\sigma_{y}$ is greater than
$\sigma_{x}$. Now the effective masses of strange and nonstrange quarks are
related via Eq (15). So it is clear that the effective mass of strange quarks
are larger than the light quarks. Due to the larger effective mass, strange
quark would take more time to relax than lighter quark. It should be noted
that the greater temperature region than the Mott temperature is physically
interested.

Here we define mean free path of particle. It is related to the relaxation
time for each flavor as

$\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\tau_{f}\frac{6}{\rho_{f}}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{p}{E_{f}}f^{0}(E_{f}/T).$
(32)

The term multiplied with the relaxation time in eq. (32) is the mean velocity.
In fig. 4, mean free paths of quark flavors are shown. As $\tau_{s}$ is
greater the $\tau_{u}$, it is expected that $\lambda_{s}$ is also higher than
$\lambda_{u}$.

Figure 5: viscosity over entropy ratio as a function of temperature for zero
and finite chemical potential

We have calculated shear viscosity numerically from eq. (20) using the
relaxation time for different chemical potential. In fig. 5, specific shear
viscosity $(\eta/s)$ is plotted as a function of temperature $(T)$ for zero
and finite chemical potential $(\mu=100\,\,\text{MeV})$. $\eta/s$ falls
rapidly near the transition temperature. We are not getting true minimum near
critical temperature. One needs to include mesonic contribution to the
relaxation time coming through the interaction. At finite chemical potential,
$\eta/s$ is lower than the value of the same at vanishing $\mu$. Value of
$\eta/s$ is well above the Kovtun-Son-Starinets (KSS) bound Kovtun  _et al._
(2005) of $1/4\pi$.

##  VI Summary

In present study, we have attempted to investigate the temperature behavior of
shear viscosity coefficients which is one relevant input for hydrodynamic
evolution of hot and dense QCD medium. The viscosity coefficient is estimated
within the Polyakov loop extended quark meson model for three quark flavors.
The approach uses the relaxation time approximation of the Boltzmann equation
where the particle masses are medium dependent. The expression of the shear
viscosity coefficient is positive definite as it should be.

Medium dependent quark masses are evaluated within mean field approximation.
QCD thermodynamics of PQM model for three flavors ($u,d,s$) is also discussed.
The thermodynamic quantities are also compare with lattice result for temporal
extent $N_{t}=10$ .The relaxation time for strange and non-strange quarks are
obtained from the quark quark and quark-antiquark scattering. The mean free
paths for both types of quarks are also discussed. Relaxation time as well as
the mean free path for strange particle is higher than the non-strange
particle. In this study we are limited only in quark scatterings. The shear
viscosity coefficient is decreasing rapidly near critical temperature. In
future a more rigorous study should be done by considering also the quark
meson scatterings to get more improved results of viscosity coefficients.

##  VII Acknowledgement

RG is funded by University Grants Commission (UGC). The author acknowledges
Munshi G. Mustafa for immense encouragement and support. The author also like
to thank Arghya Mukherjee for helpful discussions.

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