\begin{align*}
\mathcal{L}^{t+1} -\mathcal{L}^t &\leq -\gamma\mu_h h^t - \left(\frac\gamma4-2\phi_z\overline{\beta}_{zx}\frac{\gamma^2}\rho -2\phi_v\overline{\beta}_{vx}\frac{\gamma^2}\rho - \phi_sP\gamma^2 \right)\bbE[\|D_x(z^t,v^t,x^t)\|^2]\\
&\qquad - \left(\phi_z\frac{\mu_G}8\rho - L_x^2\gamma - 8\phi_v\beta_{vz}\rho - \phi_s\beta_{sz}\rho^2\right)\delta^t_z\\
&\qquad - \left(\phi_v\frac{\mu_G}{16}\rho - L_x^2\gamma - 2\phi_zL_x''\gamma^2 - \phi_s\beta_{sv}\rho^2\right)\delta^t_v\\
&\qquad -\left(\phi_s\frac\Gamma2 - 5\phi_zL_z'\rho^2-5\phi_vL_v'\rho^2-L^hL_x'\gamma^2\right)S^t\enspace.
\end{align*}

We now try to find linear convergence, hence we add to this $c\mathcal{L}^t$ to get

\begin{align*}
\mathcal{L}^{t+1} - (1-c)\mathcal{L}^t &\leq -(\gamma\mu_h - c) h^t - \left(\frac\gamma4-2\phi_z\overline{\beta}_{zx}\frac{\gamma^2}\rho -2\phi_v\overline{\beta}_{vx}\frac{\gamma^2}\rho - \phi_sP\gamma^2 - c\right)\bbE[\|D_x(z^t,v^t,x^t)\|^2]\\
&\qquad - \left(\phi_z\frac{\mu_G}8\rho - L_x^2\gamma - 8\phi_v\beta_{vz}\rho - \phi_s\beta_{sz}\rho^2 - c\phi_z\right)\delta^t_z\\
&\qquad - \left(\phi_v\frac{\mu_G}{16}\rho - L_x^2\gamma - 2\phi_zL_x''\gamma^2 - \phi_s\beta_{sv}\rho^2 - c\phi_v\right)\delta^t_v\\
&\qquad -\left(\phi_s\frac\Gamma2 - 5\phi_zL_z'\rho^2-5\phi_vL_v'\rho^2-L^hL_x'\gamma^2 - c\phi_S\right)S^t\enspace.
\end{align*}

Hence, the set of inequations for  decrease becomes

\begin{align*}
c &\leq \gamma\mu_h\\
2\phi_z\overline{\beta}_{zx}\frac{\gamma^2}\rho  + 2\phi_v\overline{\beta}_{vx}\frac{\gamma^2}\rho + \phi_sP\gamma^2 + c &\leq\frac\gamma4\\
L_x^2\gamma + 8\phi_v\beta_{vz}\rho + \phi_s\beta_{sz}\rho^2 + \phi_zc&\leq\phi_z\frac{\mu_G}8\rho\\
L_x^2\gamma + 8\phi_zL_x''\gamma^2 + \phi_s\beta_{sv}\rho^2 + \phi_v c&\leq\phi_v\frac{\mu_G}{16}\rho\\
5\phi_zL_z'\rho^2+5\phi_vL_v'\rho^2+L^hL_x'\gamma^2 +\phi_s c &\leq \phi_s\frac\Gamma2\enspace.
\end{align*}

We see that it is more convenient to write $c = \gamma c'$. As previously, we write $\gamma = \gamma'N^{n_\gamma}$, $\rho = \rho' N^{n_\rho}$, $\phi_z = \phi_z'N^{n_z}$, $\phi_v = \phi_v'N^{n_v}$, $\phi_s = \phi_s'N^{n_s}$, $P = P'N$, $\Gamma = \Gamma'N^{-1}$, $\beta_{sx} = \beta_{sx}'N$ and $\beta_{sv} = \beta_{sv}'N$. The equations read:

\begin{align*}
c' &\leq \mu_h\\
2\phi_z'\overline{\beta}_{zx}\frac{\gamma'}{\rho'}N^{n_z+n_\gamma-n_\rho}  + 2\phi_v'\overline{\beta}_{vx}\frac{\gamma'}{\rho'}N^{n_v+n_\gamma-n_\rho} + \phi_s'P'\gamma'N^{n_s+1+n_\gamma} + c' &\leq\frac14\\
L_x^2\gamma'N^{n_\gamma} + 8\phi_v'\beta_{vz}\rho'N^{n_v+n_\rho} + \phi_s'\beta_{sz}'(\rho')^2N^{n_s+2n_\rho+1} + \phi_z' c'\gamma' N^{n_z+n_\gamma}&\leq\phi_z'\frac{\mu_G}8\rho'N^{n_\rho+n_z}\\
L_x^2\gamma'N^{n_\gamma} + 8\phi_z'L_x''(\gamma')^2N^{n_z+2n_\gamma} + \phi_s'\beta_{sv}'(\rho')^2N^{n_s+1+2n_\rho} + \phi_v' c'\gamma'N^{n_v+n_\gamma}&\leq\phi_v'\frac{\mu_G}{16}\rho'N^{n_v+n_\rho}\\
5\phi_z'L_z'(\rho')^2N^{n_z+2n_\rho}+5\phi_v'L_v'(\rho')^2N^{n_v + 2n_\rho}+L^hL_x'(\gamma')^2N^{2n_\gamma} +\phi_s' c'\gamma' N^{n_s+n_\gamma} &\leq \phi_s'\frac{\Gamma'}2N^{n_s - 1}\enspace.
\end{align*}

In order to ensure that the exponents on $N$ are lower in the left-hand-side than those on the right-hand-side, we take $n_z = n_v = 0$, $n_\rho = -\frac{2}{3}$, $n_\gamma = -1$ and $n_s =- \frac{1}{3}$. The Equations become

\begin{align*}
c' &\leq \mu_h\\
2\phi_z'\overline{\beta}_{zx}\frac{\gamma'}{\rho'}N^{-\frac13}  + 2\phi_v'\overline{\beta}_{vx}\frac{\gamma'}{\rho'}N^{-\frac13} + \phi_s'P'\gamma'N^{-\frac13} + c' &\leq\frac14\\
L_x^2\gamma'N^{-1} + 8\phi_v'\beta_{vz}\rho'N^{-\frac23} + \phi_s'\beta_{sz}'(\rho')^2N^{-\frac23} + \phi_z' c'\gamma' N^{-1}&\leq\phi_z'\frac{\mu_G}8\rho'N^{-\frac23}\\
L_x^2\gamma'N^{-1} + 8\phi_z'L_x''(\gamma')^2N^{-2} + \phi_s'\beta_{sv}'(\rho')^2N^{-\frac23} + \phi_v' c'\gamma'N^{-1}&\leq\phi_v'\frac{\mu_G}{16}\rho'N^{-\frac23}\\
5\phi_z'L_z'(\rho')^2N^{-\frac43}+5\phi_v'L_v'(\rho')^2N^{-2}+L^hL_x'(\gamma')^2N^{-2} +\phi_s' c'\gamma' N^{-\frac43} &\leq \phi_s'\frac{\Gamma'}2N^{-\frac43}\enspace.
\end{align*}

Now we have to find $\rho'$, $\gamma'$, $\phi_z'$, $\phi_v'$ and $\phi_s'$ that verifies the following conditions (which are a bit stronger than thoose in the previous Equations):

\begin{align*}
c' &\leq \mu_h\\
2\phi_z'\overline{\beta}_{zx}\frac{\gamma'}{\rho'} + 2\phi_v'\overline{\beta}_{vx}\frac{\gamma'}{\rho'} + \phi_s'P'\gamma' + c' &\leq\frac14\\
L_x^2\gamma'+ 8\phi_v'\beta_{vz}\rho' + \phi_s'\beta_{sz}'(\rho')^2 + \phi_z' c'\gamma' &\leq\phi_z'\frac{\mu_G}8\rho'\\
L_x^2\gamma' + 8\phi_z'L_x''(\gamma')^2 + \phi_s'\beta_{sv}'(\rho')^2 + \phi_v' c'\gamma'&\leq\phi_v'\frac{\mu_G}{16}\rho'\\
5\phi_z'L_z'(\rho')^2+5\phi_v'L_v'(\rho')^2+L^hL_x'(\gamma')^2 + \phi_s' c'\gamma'  &\leq \phi_s'\frac{\Gamma'}2\enspace.
\end{align*}

As previously, we take $\phi'_s = 1$ and we denote $\phi'_z = \phi_z''\frac{\rho'}{\gamma'}$ with $\phi_z'' = \frac1{32\overline{\beta}_{zx}}$ and $\phi'_z = \phi_z''\frac{\rho'}{\gamma'}$ with $\phi_v'' = \min\left(\frac1{32\overline{\beta}_{vx}}, \phi_z''\frac{\mu_G}{128\beta_{vz}}\right)$, the equations become

\begin{align*}
c' &\leq \mu_h\\
P'\gamma' + c' &\leq\frac18\\
L_x^2(\gamma')^2+ \beta_{sz}'(\rho')^2\gamma' + \phi_z'' c'\rho'\gamma' &\leq\phi_z''\frac{\mu_G}{16}(\rho')^2\\
L_x^2(\gamma')^2 + 8\phi_z''L_x''\rho'(\gamma')^2 + \beta_{sv}'(\rho')^2\gamma' + \phi_v'' c'\rho'\gamma'&\leq\phi_v''\frac{\mu_G}{16}(\rho')^2\\
5\phi_z''L_z'(\rho')^3+5\phi_v''L_v'(\rho')^3+L^hL_x'(\gamma')^3 + c'(\gamma')^2  &\leq \frac{\Gamma'}2\gamma'\enspace.
\end{align*}

Since $c'\leq \frac1{16}$ and $\gamma'\leq\frac1{16P'}$, the second equation is verified. With $\gamma'\leq \min\left(\sqrt{\frac{\phi_z''\mu_G}{48L_x^2}}\rho', \frac{\phi_z''\mu_G}{48\beta_{sv}}\right)$ and $c'\leq \frac{\mu_G\rho'}{48\gamma'}$ the third is verified. The conditions $\gamma'\leq\min\left(\sqrt{\frac{\phi_v''\mu_G}{64L_x^2}}\rho',\sqrt{\frac{\phi_v''\mu_G}{512\phi_z''L_x''\rho'}},\frac{\phi_v''\mu_G}{64\beta_{sv}'}\right)$ and $c'\leq\frac{\mu_G\rho'}{64\gamma'}$ ensure that the forth is verified. With $\gamma'\leq\sqrt{\frac{\Gamma'}{8L^hL_x'}}$ and $c'\leq\frac{\Gamma'}{8\gamma'}$, the fifth is simplified in

$$5\phi_z''L_z'(\rho')^3+5\phi_v''L_v'(\rho')^3 \leq \frac{\Gamma'}4\gamma'\enspace .$$

As in the proof of <ref>, let us denote $\gamma' = \xi\rho'$. To verify this equation and the previous bounds on $\gamma'$ and $c'$, we need

\begin{align*}
\gamma'&\leq\underbrace{\min\left(\sqrt{\frac{\phi_z''\mu_G}{48L_x^2}},\sqrt{\frac{\phi_v''\mu_G}{64L_x^2}},\sqrt{\frac{L_z'}{2L_x'\beta_{zx}}},\sqrt{\frac{L_v'}{2L_x'\beta_{zx}}}\right)}_{K_1}\rho'\enspace,\\
\gamma'&\leq \underbrace{\min\left(\sqrt{\frac{\mu_G}{64\beta_{zx}L_x''}},\sqrt{\frac{\mu_G}{128\beta_{vx}L_x''}},\sqrt{\frac{\beta_{vz}}{4L_x''\beta_{vx}}}\right)}_{K_2}\sqrt{\rho'}\enspace,\\
\gamma'&\leq \underbrace{\sqrt{\frac{\phi_v''\mu_G}{512\phi_z''L_x''}}}_{K_3}\frac1{\sqrt{\rho'}}\enspace,\\
\gamma'&\leq \underbrace{\min\left(\frac1{4L^h},\frac{L_x^2}{2L^hL_x''},\frac{\phi_z''\mu_G}{48\beta_{sv}},\frac{\phi_v''\mu_G}{64\beta_{sv}'},\frac1{16P'},\sqrt{\frac{\Gamma'}{8L^hL_x'}}\right)}_{K_4}\\
\gamma'&\geq \underbrace{\frac{20(\phi_z''L_z'+\phi_v''L_v')}{20}}_{K_5}(\rho')^3 \enspace,\\
c'&\leq \underbrace{\min\left(\mu_h, \frac1{16},\frac1{16P'}\right)}_{K_6}\enspace,\\
c'&\leq \underbrace{\frac{\mu_G}{64}}_{K_7}\frac1\xi\enspace,\\
\end{align*}

So, $\xi$, $\rho'$ and $c'$ must verify

\begin{align*}
\xi&\leq \underbrace{\min\left(K_1,\frac{K_7}{c'}\right)}_{K_1'}\enspace,\\
\xi&\leq K_2(\rho')^{-\frac12}\enspace,\\
\xi&\leq K_3(\rho')^{-\frac32}\enspace,\\
\xi&\leq \underbrace{\min\left(K_4,\frac{K_8}{c'}\right)}_{K_4'}(\rho')^{-1}\\
\xi&\geq K_5(\rho')^2 \enspace,\\
c'&\leq \underbrace{\min\left(\mu_h, \frac1{16},\frac1{16P'}\right)}_{K_6}\enspace,\\
\end{align*}

which is possible if

\begin{align*}
\rho'\leq\min\left(\sqrt{\frac{K_1'}{K_5}},\left(\frac{K_2}{K_5}\right)^{\frac25},\left(\frac{K_3}{K_5}\right)^{\frac27},\left(\frac{K_4'}{K_5}\right)^{\frac13}\right)\enspace.
\end{align*}

So let us take $c' = \min\left(\mu_h, \frac1{16},\frac1{16P'}\right) = \min\left(\mu_h, \frac1{16P'}\right)$,

\rho' = \min\left(\sqrt{\frac{K_1'}{K_5}},\left(\frac{K_2}{K_5}\right)^{\frac25},\left(\frac{K_3}{K_5}\right)^{\frac27},\left(\frac{K_4'}{K_5}\right)^{\frac13},\frac{\mu_G}{64L_z^2},\frac{\overline{\beta}_{zx}}{2\beta_{zx}},\frac{\mu_G}{128(L_v^2 + L_v'')},\frac{\beta_{vz}}{8(L_v^2+L_v'')},\frac{\overline{\beta}_{vx}}{2\beta_{vx}}\right)

\xi = \min(K_1, K_2(\rho')^{-\frac12}, K_3(\rho')^{-\frac32},K_4(\rho')^{-1})\enspace.

We have

\calL^{t+1}\leq(1-c)\calL^t

therefore, unrolling yields

\boxed{h^t - h^*\leq \calL^t\leq(1-c'\gamma)^t\calL^0.}

§ CONVERGENCE RATES WITH WEAKER REGULARITY ASSUMPTIONS

To get our rates, we need stronger assumptions than in the stochastic bilevel optimization literature [Ghadimi and Wang, 2018, Hong et al., 2021, Ji et al., 2021, Arbel and Mairal, 2022]. In this section, we shortly present the convergence rates we can expect if we replace Assumptions <ref> and <ref> by Assumptions <ref> and <ref>.

The function $F$ is differentiable. The gradient $\nabla F$ is Lipschitz continuous in $(z,x)$ with Lipschitz constants $L^F_1$.

The function $G$ is twice continuously differentiable on $\bbR^p\times\bbR^d$. For any $x\in\bbR^d$, $G(\,\cdot\,,x)$ is $\mu_G$-strongly convex. The derivatives $\nabla G$ are $\nabla^2 G$ are Lipschitz continuous in $(z,x)$ with respective Lipschitz constants $L^G_1$ and $L^G_2$.

With these assumptions, we are not ensured that $v^*$ is smooth, and so the descent lemmas take the form of <ref>.

Assume that $\rho\leq\frac{2}{\mu_G}$. We have:

\begin{align*}
\delta_z^{t+1}&\leq \left(1-\frac{\rho\mu_G}2\right)\delta_z^t + 2\rho^2V_z^t + 4\frac{L_*^2}{\mu_G} \frac{\gamma^2}{\rho}V_x^t \\
\delta_v^{t+1}&\leq \left(1-\frac{\rho\mu_G}4\right)\delta_v^t +\rho \beta_{vz}\delta_z^t + 2\rho^2V_v^t + 8\frac{L_*^2}{\mu_G} \frac{\gamma^2}{\rho}V_x^t
\end{align*}

where $L_*$ is the maximum between the Lipschitz constants of $z^*$ and $v^*$ (see <ref>) and $\beta_{vz} = \frac1{\mu_G^3}(L^F\mu_G+L^G_{2})^2$.

Inequality for $\delta_z$.

Instead of expanding the square as done in the proof of <ref> in <ref>, we use Young's inequality for some $a>0$

\begin{align}
\delta^{t+1}_z &\leq (1+a) \bbE[\|z^{t+1} - z^*(x^t)\|^2] + (1+a^{-1})\bbE[\|z^*(x^{t+1}) - z^*(x^t)\|^2]\enspace.
\end{align}

Treating $\bbE[\|z^{t+1} - z^*(x^t)\|^2]$ and $\bbE[\|z^*(x^{t+1}) - z^*(x^t)\|^2]$ as done in the proof of <ref> leads to

\begin{equation}
\delta^{t+1}_z \leq (1+a)\left[(1 - \rho\mu_G)\delta_z^t + \rho^2V_z^t\right] + (1+a^{-1})L^2_*\gamma^2V_x^t
\end{equation}

In order to keep a decrease in $\delta_z$, we might want to use $a = \frac12\rho\mu_G$, which gives the bound

\begin{equation}
\label{eq:bound_delta_z}
\boxed{
\delta^{t+1}_z \leq \left(1 - \frac{\rho\mu_G}{2}\right)\delta_z^t + 2\rho^2V_z^t +\beta_{zx} \frac{\gamma^2}{\rho}V_x^t
\end{equation}

with $\beta_{zx} = 4\frac{L_*^2}{\mu_G}$.
Indeed, this gives $(1 + \frac12\rho\mu_G)(1-\rho\mu_G) \leq 1 - \frac12\rho\mu_G$.
We have $a \leq 1$ since $\rho\leq\frac{2}{\mu_G}$, so $(1+a)\rho^2\leq 2\rho^2$. Finally, we also have $1+ a^{-1}\leq 2a^{-1} = \frac4{\rho\mu_G}$.

Inequality for $\delta_v$.   As for $\delta_z$, the difference with the proof of <ref> is that we use we use Young's inequality for some $b>0$ to get

\begin{align}
\delta^{t+1}_v &\leq (1+b) \bbE[\|v^{t+1} - v^*(x^t)\|^2] + (1+b^{-1})\bbE[\|v^*(x^{t+1}) - v^*(x^t)\|^2]\enspace.
\end{align}

The remaining part of the proof is similar to the proof of <ref>.

The main difference with <ref> is that we have $O(\frac{\gamma^2}{\rho})$ in factor of $V_x^t$ instead of $O(\gamma^2)$. As a consequence, we need that the ratio $\frac\gamma\rho$ goes to zero to get convergence, as in [Hong et al., 2021]. This prevent us in getting rates that match rates of single level algorithms.

Hence, for SOBA, we have to choose $\gamma = O(T^{-\frac35})$ and $\rho = O(T^{-\frac25})$ and we end up with a convergence rate in $O(T^{-\frac25})$. For SABA, we get a $O((n+m)\epsilon^{-1})$ sample complexity, which is actually the sample complexity of SOBA used with full batch estimated directions.


# Estimation of transport coefficients of dense hadronic and quark matter

Debashree Sen1, Naosad Alam1,2, and Sabyasachi Ghosh3 1Physics Group, Variable
Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, India  2Department
of Nuclear and Atomic Physics, Tata Institute of Fundamental Research, Homi
Bhabha Road, Mumbai 400005, India  3Indian Institute of Technology Bhilai, GEC
Campus, Sejbahar, Raipur 492015, Chhattisgarh, India
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We calculate the transport coefficients like shear viscosity and electrical
conductivity with respect to density of dense hadronic and quark matter. By
considering the simple massless limit for the quark matter and two different
effective models for the hadronic matter, we have estimated the transport
coefficients of the two phases separately. We have sketched the density
profile of the transport coefficients in two parts viz. the phase-space part
and the relaxation time part. By calculating the shear viscosity to density
ratio, we have also explored the nearly perfect fluid domain of the quark and
hadronic matter.

##  1 Introduction

At high density the composition and interaction of matter is still
inconclusive from experimental perspectives. Therefore the understanding of
the properties of dense matter is largely dependent on various theoretical
models. Theoretically, hadronic matter may undergo phase transition to quark
matter at high density or at high temperature. This hadron-quark transition
density largely depends on the various theoretical models and the interactions
considered. In the very low temperature and high baryon density domain of
quantum chromodynamics (QCD) phase diagram, the first-principle predictions
via lattice QCD (LQCD) calculations are missing due to the infamous sign
problem [1]. However, the high temperature and low/vanishing baryon density
domain of the QCD phase diagram is well studied for more than three decades
[2, 3]. One can look at [4] for its latest status. They conclude a cross-over
type phase transition, which has been alternatively realized from the
estimations of thermodynamic quantities at low and high temperature ranges
with hadron resonance gas (HRG) model [5] and finite temperature perturbative
QCD (pQCD) calculations [6, 7], respectively. Similar kind of mapping in the
low/vanishing temperature and high baryon density domain may be possible [8]
by fusing ultra-high density (approximately 10 times larger than the hadronic
saturation density) pQCD calculations [9, 10, 11] and low density hadronic
model calculations. In this context, the present work attempts to compare the
estimations of transport coefficients, obtained from the standard massless
outcome for the quark phase [12] and the hadronic phase using two relativistic
mean field (RMF) models viz. the effective chiral model [13, 14, 15, 16, 17,
18, 19, 20, 21] and the RMF model of [22, 23, 24, 25]. At high density, the
formation of heavier and strange baryons like the hyperons is theoretically
possible. However, for simplicity and also due to the uncertainty in the
hyperon couplings, we consider only the nucleons in the hadronic phase. Both
the hadronic models adopted in this work, have been explored thoroughly to
construct the equation of state (EoS) of dense matter and successfully
determine the structural properties of neutron stars (NSs) or hybrid stars
(HSs) in the light of recent constraints from various astrophysical
observations [13, 14, 15, 16, 17, 24, 25]. For the quark phase we consider the
massless two flavor quark matter with u and d quarks. Similar to the hadronic
phase we do not consider the strange degree of freedom (the s quark) in the
quark phase. The present work is aimed to compare the estimations of the
transport coefficients of both hadronic and quark matter in the high baryon
density and low/vanishing temperature domain of QCD phase diagram. The
motivation for this investigation comes from the equivalent pattern of LQCD
thermodynamics [3, 4] and normalized transport coefficients [26] in the high
temperature and low/vanishing baryon density domain of QCD phase diagram. If
we analyze the temperature ($T$) profile of the thermodynamical quantities
like pressure $P$, energy density $\epsilon$, entropy density $s$ and
transport coefficients like shear viscosity $\eta$, electrical conductivity
$\sigma$ for massless quark matter, then we can find the proportional
relations $P=\frac{1}{3}\epsilon=\frac{1}{4}TS\propto T^{4}$ and
$\eta\propto\tau T^{4}$, $\sigma\propto\tau T^{2}$, where $\tau$ is relaxation
time of massless quark matter. So their normalized values $P/T^{4}$,
$\epsilon/T^{4}$, $s/T^{3}$, $\eta/(\tau T^{4})$ and $\sigma/(\tau T^{2})$
will appear as horizontal line against $T$-axis and they can be marked as
their upper or massless or the Stefan-Boltzmann (SB) limits. At very high $T$,
these limiting values can be reached. As we go from high to low $T$, the
values of thermodynamical quantities and transport coefficients will decrease
and their maximum decrement will be noticed around quark-hadron transition
temperature [26]. Here we are interested to find similar kind of graphs along
baryon density axis.

Literature presents a long list of works [27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] that have concentrated on the
microscopic calculation of transport coefficients of the dense hadronic matter
system. Refs. [27, 28, 29, 30] are evident that the microscopic calculations
of transport coefficients of dense hadronic matter is quite an old topic of
research but it is still cultivated in recent time as reflected in [39, 40,
41, 42, 43, 44, 45, 46]. For the calculation of transport coefficients of
neutron star matter at high density, one also includes the electro-magnetic
interaction part, whose time scale far away from QCD time scale. Many works
[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46] can be found in this regard, many of which have considered the
contribution of the electrons and muons apart from the nucleons due to
consideration of beta stable neutron star matter [36, 47]. However, in the
present work we focus only on hadronic matter contribution and calculate the
transport properties of dense hadronic matter. In the present work the
transport coefficients estimations are obtained within QCD time scale (fm) as
we have considered only the strong interaction related to dense hadronic and
massless quark matter separately. The present work is focused within the QCD
time scale for comparing the hadronic and quark phases pattern in terms of
momentum and charge transportation due to QCD interactions. If one
concentrates on the hadronic matter contribution of the works [39, 40, 41, 42,
43, 44, 45, 46], then it is noticed that the main ingredients for the
calculation of the transport coefficients of hadronic matter are the effective
masses of the nucleons and the quasiparticle relaxation time. Both quantities
can have medium modification, which can be different in different models of
many-body approach [39]. Owing to this model-dependent estimation scope, in
the present work we have attempted to estimate the transport coefficients like
shear viscosity and electrical conductivity of dense hadronic matter by using
two different hadronic models. In this regard, it is just an alternative model
estimation but we have followed an unique presentation of the transport
coefficients along the density axis, which is generally adopted in the
scenario of heavy-ion collisions [26], where the normalized transport
coefficients are studied with respect to temperature to understand the nearly
perfect fluid nature of hadronic matter.

This article is organized as follows. Next in the Formalism section (2), we
have briefly addressed the two different models for the hadronic phase and the
MIT Bag model for the massless quark phase in three different subsections
(2.1), (2.2) and (2.3), respectively. Then in the Sec. (3), the formalism of
relaxation time approximation to calculate the transport coefficients is
briefly addressed. After obtaining the final expressions of transport
coefficients and the effective kinematic informations of the hadronic and the
quark phases in the formalism part, we have shown their variations with
respect to density in the result section (4) along with detailed discussion.
Finally in Sec (5), we have summarized our findings.

##  2 Formalism of density-dependent hadronic and quark phases

In this section, we will address briefly the two hadronic models calculations
and the standard massless or zero-mass quark matter (z-MQM) calculations
respectively. We have adopted two different hadronic models, whose brief
details are given in the next two subsections (2.1), (2.2). In the third
subsection (2.3), brief discussion of the z-MQM calculation for quark phase is
given.

###  2.1 Hadronic Phase: Hadronic model - 1

Considering only the nucleons as the baryonic degrees of freedom, the
Lagrangian density for the effective chiral model [19, 20, 21] is given by

$\displaystyle\mathcal{L}=\overline{\psi}\Biggl{[}\left(i\gamma_{\mu}\partial^{\mu}-g_{\omega}~{}\gamma_{\mu}\omega^{\mu}-\frac{1}{2}g_{\rho}~{}\overrightarrow{\rho_{\mu}}\cdot\overrightarrow{\tau}\gamma^{\mu}\right)-g_{\sigma}\left(\sigma+i\gamma_{5}\overrightarrow{\tau}\cdot\overrightarrow{\pi}\right)\Biggr{]}\psi$
$\displaystyle+\frac{1}{2}\left(\partial_{\mu}\overrightarrow{\pi}\cdot\partial^{\mu}\overrightarrow{\pi}+\partial_{\mu}\sigma~{}\partial^{\mu}\sigma\right)-{\frac{\lambda}{4}}\left(\chi^{2}-\chi_{0}^{2}\right)^{2}-\frac{\lambda
B}{6}(\chi^{2}-\chi_{0}^{2})^{3}-\frac{\lambda
C}{8}(\chi^{2}-\chi_{0}^{2})^{4}$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}{g_{\omega}}^{2}~{}\chi^{2}~{}\omega_{\mu}\omega^{\mu}-\frac{1}{4}~{}\overrightarrow{R_{\mu\nu}}\cdot\overrightarrow{R^{\mu\nu}}+\frac{1}{2}~{}m_{\rho}^{2}~{}\overrightarrow{\rho_{\mu}}\cdot\overrightarrow{\rho^{\mu}}~{},$
(1)

where, $\psi$ is the nucleon isospin doublet. The nucleons interact with each
other via the scalar $\sigma$ meson, the vector $\omega$ meson (783 MeV) and
the isovector $\rho$ meson (770 MeV) with corresponding coupling strengths
$g_{\sigma}$, $g_{\omega}$ and $g_{\rho}$, respectively. As mean field
treatment is considered, the pions do not contribute. The model is based on
chiral symmetry with the $\sigma$ and the pseudo-scalar $\pi$ mesons as chiral
partners and $\chi^{2}=(\pi^{2}+\sigma^{2})$. The $\sigma$ field attains a
vacuum expectation value (VEV) $\sigma_{0}=x_{0}$ with the spontaneous
breaking of the chiral symmetry at ground state [48]. The masses of the
nucleons ($m$) and the scalar and vector mesons can be expressed in terms of
$\chi_{0}$ as

$\displaystyle
m=g_{\sigma}\chi_{0},~{}~{}m_{\sigma}=\sqrt{2\lambda}~{}\chi_{0},~{}~{}m_{\omega}=g_{\omega}\chi_{0}~{},$
(2)

where, $\lambda=({m_{\sigma}}^{2}-{m_{\pi}}^{2})/(2{f_{\pi}}^{2})$ is derived
from chiral dynamics. The $f_{\pi}$, being the pion decay constant, relates to
the VEV of $\sigma$ field as $<\sigma>=\sigma_{0}~{}=f_{\pi}$ [19, 21]. Since
in mean field approximation, $<\pi>=0$ and the pion mass becomes $m_{\pi}=0$,
the explicit contributions of the pions do not play any role in the
interactions and also in the expression of $\lambda$. The term
$\frac{1}{2}{g_{\omega}}^{2}\chi^{2}\omega_{\mu}\omega^{\mu}$ in Eq. 1 implies
an explicit dependence of the nucleon effective mass on both the scalar and
the vector fields and this is one of the salient features of the present
model. The isospin triplet $\rho$ mesons are incorporated to account for the
asymmetric hadronic matter. An explicit mass term for the isovector $\rho$
meson
$\frac{1}{2}~{}m_{\rho}^{2}~{}\overrightarrow{\rho_{\mu}}\cdot\overrightarrow{\rho^{\mu}}$
is chosen following [49, 19, 20, 21]. The coupling strength of the $\rho$
mesons with the nucleons is obtained by fixing the symmetry energy coefficient
$J=32$ MeV at hadronic saturation density $\rho_{0}$. In terms of the baryon
density $\rho$ and the Fermi momentum $k_{F}=(6\pi^{2}\rho/{\gamma})^{1/3}$
($\gamma=2$ is degeneracy factor of nucleon), the isovector coupling strength
is related to $J$ as

$\displaystyle
J=\frac{C_{\rho}~{}k_{F}^{3}}{12\pi^{2}}+\frac{k_{F}^{2}}{6\sqrt{(k_{F}^{2}+m^{*2})}}$
(3)

where, $C_{\rho}=g^{2}_{\rho}/m^{2}_{\rho}$ and $m^{*}$ is the nucleon
effective mass. The scalar density is obtained as

$\displaystyle\rho_{S}=<\overline{\psi}\psi>=\frac{1}{\pi^{2}}\Bigg{[}\int^{k_{Fn}}_{0}dk_{n}~{}k_{n}^{2}\frac{m^{*}}{\sqrt{k_{n}^{2}+{m^{*}}^{2}}}+\int^{k_{Fp}}_{0}dk_{p}~{}k_{p}^{2}\frac{m^{*}}{\sqrt{k_{p}^{2}+{m^{*}}^{2}}}\Bigg{]}$
(4)

while the baryon density for asymmetric hadronic matter is given as

$\displaystyle\rho=<\psi^{\dagger}\psi>=\rho_{n}+\rho_{p}=\frac{1}{\pi^{2}}\Bigg{[}\int^{k_{Fn}}_{0}dk_{n}~{}k_{n}^{2}+\int^{k_{Fp}}_{0}dk_{p}~{}k_{p}^{2}\Bigg{]}.$
(5)

Symmetric nuclear matter (SNM) is defined as hadronic matter with equal number
of neutrons and protons (N=Z) while at high density matter becomes asymmetric
(N$>>$Z). The effective nucleon chemical potential is given as

$\displaystyle\mu_{B}=\sqrt{k_{F}^{2}+{m^{*}}^{2}}+g_{\omega}\omega_{0}+g_{\rho}I_{3B}\rho_{03}~{},$
(6)

where, $I_{3B}$ (with $B=n,p$) is the third component of isospin of the
individual nucleons and $\omega_{0}$ and $\rho_{03}$ are the mean field values
of the vector and isovector mesons, respectively. In Eq. 6 the chemical
potential of the individual nucleons in absence of the meson field terms is
given by the first term $\mu_{0}=\sqrt{k_{F}^{2}+{m^{*}}^{2}}$ which is
modified into the effective chemical $\mu_{B}$ due to the interaction between
the nucleons via exchange of the mesons as seen from the last two terms of Eq.
(6). So the effective chemical potential $\mu_{B}$ (where, $B=n,p$) of the
individual nucleons differs by the third component of isospin of the
individual nucleons $I_{3B}$ as seen from Eq. (6).

The scalar EoM of the scalar field is

$\displaystyle(1-Y^{2})-\frac{B}{C_{\omega}}(1-Y^{2})^{2}+\frac{C}{C_{\omega}^{2}}(1-Y^{2})^{3}+\frac{2~{}C_{\sigma}~{}C_{\omega}~{}\rho^{2}}{m^{2}~{}Y^{4}}-\frac{2~{}C_{\sigma}~{}\rho_{S}}{m~{}Y}=0$
(7)

Here $C_{i}={g_{i}}^{2}/{m_{i}}^{2}$ are the scaled couplings
($i=\sigma~{}\&~{}\omega$, $m_{i}$ being the mass of the mesons) and
$Y=m^{*}/m$. $B$ and $C$ are the coefficients of higher order scalar field
terms. The EoM of the vector field is given as

$\displaystyle\omega_{0}=\frac{\rho}{g_{\omega}\chi^{2}}$ (8)

while that of the isovector field is

$\displaystyle\rho_{03}=\sum_{B}\frac{g_{\rho}}{m_{\rho}^{2}}I_{3_{B}}\rho_{B}$
(9)

The five model parameters $C_{\sigma},C_{\omega},C_{\rho}$ and the
coefficients $B~{}\&~{}C$ of higher order scalar field terms are determined by
reproducing the properties of SNM at saturation density $\rho_{0}$. The
detailed procedure of obtaining these model parameters can be found in [21].
For the present work the parameter set is chosen from [21] and is presented in
table 1 below along with the SNM properties yielded by this parameter set.

Table 1: Model parameters chosen for the present work (adopted from [21]). $C_{\sigma}$ |  $C_{\omega}$ |  $C_{\rho}$ |  $B/m^{2}$ |  $C/m^{4}$ |
---|---|---|---|---|---
($\rm{fm^{2}}$) | ($\rm{fm^{2}}$) | ($\rm{fm^{2}}$) | ($\rm{fm^{2}}$) | ($\rm{fm^{2}}$) |
6.772 | 1.995 | 5.285 | -4.274 | 0.292 |
$m^{*}/m$ |  $K$ |  $B/A$ |  $J$ |  $L_{0}$ |  $\rho_{0}$
| (MeV) | (MeV) | (MeV) | (MeV) | ($\rm{fm^{-3}}$)
0.85 | 303 | -16.3 | 32 | 87 | 0.153

Since the nucleon effective mass $m^{*}$ for this model depends on both the
scalar and vector fields, it is quite high compared to well-known RMF models.
Also at high density, unlike RMF models, the value of $m^{*}$ increases after
a certain high value of density [20, 21, 13, 16]. This is due the dominance of
vector potential at such density. Moreover, at high density the higher order
terms of scalar field with coefficients $B$ and $C$ and the mass term of the
vector field of the present model also become highly non-linear and dominant
[20, 21, 13, 16]. The hadronic imcompressibility $K$ obtained with the chosen
parameter set, though consistent with the results of [50] but it is larger
than estimated in [51, 52]. The other SNM properties like the binding energy
per nucleon $B/A$, the symmetry energy $J$ and the saturation density
$\rho_{0}$ match well with the estimates of Refs. [53, 54]. The slope
parameter $L_{0}$ is also quite consistent with the range specified by [53,
55, 56]. The same model parameter set has also been adopted earlier in [14,
13, 15, 18, 16, 17, 57] to successfully investigate different properties of
NSs as well as HSs in the light of various constraints specified on their
structural properties.

###  2.2 Hadronic Phase: Hadronic model - 2

In the conventional RMF theory [58, 59, 60, 61, 62, 63, 64, 22, 23, 24, 25]
nucleons are treated as elementary particles and interactions between the
nucleons are mediated by the exchange of $\sigma$, $\omega$ and $\rho$ mesons.
The $\sigma$ mesons give rise to the strong attractive force, while the
$\omega$ mesons cause the strong repulsive force between the nucleons. In
addition to this, several self and cross interaction terms between the mesons
are also considered to yield the saturation properties correctly. The
Lagrangian density for the extended RMF model can be written as,

$\displaystyle{\cal L}={\cal L_{NM}}+{\cal L_{\sigma}}+{\cal L_{\omega}}+{\cal
L_{\mathbf{\rho}}}+{\cal L_{\sigma\omega\mathbf{\rho}}}~{},$ (10)

where, the Lagrangian ${\cal L_{NM}}$ describing the interactions of the
nucleons with mass $m$ through the mesons is,

$\displaystyle{\cal L_{NM}}$ $\displaystyle=$
$\displaystyle\sum_{B=n,p}\overline{\psi}_{i}[i\gamma^{\mu}\partial_{\mu}-(m-g_{\sigma}\sigma)-(g_{\omega}\gamma^{\mu}\omega_{\mu}+\frac{1}{2}g_{\mathbf{\rho}}\gamma^{\mu}\tau.\mathbf{\rho}_{\mu})]\psi_{i}.~{}~{}~{}~{}~{}~{}$
(11)

Here, the sum is taken over the neutrons and protons and $\tau$ are the
isospin matrices. The Lagrangians for the $\sigma$, $\omega$, and $\rho$
mesons including their self interaction terms can be written as,

$\displaystyle{\cal L_{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma-
m_{\sigma}^{2}\sigma^{2})-\frac{{\kappa_{3}}}{6m}g_{\sigma}m_{\sigma}^{2}\sigma^{3}-\frac{{\kappa_{4}}}{24m^{2}}g_{\sigma}^{2}m_{\sigma}^{2}\sigma^{4},$
$\displaystyle{\cal L_{\omega}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}+\frac{1}{24}\zeta_{0}g_{\omega}^{2}(\omega_{\mu}\omega^{\mu})^{2},$
$\displaystyle{\cal L_{\mathbf{\rho}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mathbf{\rho}_{\mu\nu}\mathbf{\rho}^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}\mathbf{\rho}_{\mu}\mathbf{\rho}^{\mu}$
(12)

The $\omega^{\mu\nu}$, $\mathbf{\rho}^{\mu\nu}$ are field tensors
corresponding to the $\omega$ and $\rho$ mesons, and can be defined as
$\omega^{\mu\nu}=\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$ and
$\mathbf{\rho}^{\mu\nu}=\partial^{\mu}\mathbf{\rho}^{\nu}-\partial^{\nu}\mathbf{\rho}^{\mu}$.
Here, $m_{\sigma}$, $m_{\omega}$ and $m_{\rho}$ are the masses of
$\sigma,\omega$, and $\mathbf{\rho}$ mesons, respectively. The cross
interactions of $\sigma,\omega$, and $\mathbf{\rho}$ mesons are described by
${\cal L_{\sigma\omega\rho}}$ which can be written as,

$\displaystyle{\cal L_{\sigma\omega\rho}}$ $\displaystyle=$
$\displaystyle\frac{\eta_{1}}{2m}g_{\sigma}m_{\omega}^{2}\sigma\omega_{\mu}\omega^{\mu}+\frac{\eta_{2}}{4m^{2}}g_{\sigma}^{2}m_{\omega}^{2}\sigma^{2}\omega_{\mu}\omega^{\mu}+\frac{\eta_{\rho}}{2m}g_{\sigma}m_{\rho}^{2}\sigma\rho_{\mu}\rho^{\mu}$
(13)
$\displaystyle+\frac{\eta_{1\rho}}{4m^{2}}g_{\sigma}^{2}m_{\rho}^{2}\sigma^{2}\rho_{\mu}\rho^{\mu}+\frac{\eta_{2\rho}}{4m^{2}}g_{\omega}^{2}m_{\rho}^{2}\omega_{\mu}\omega^{\mu}\rho_{\mu}\rho^{\mu}~{}.$

The field equations derived from the above Lagrangian can be solved self-
consistently by adopting mean-field approximation, i.e., the meson-field
operators are replaced by their expectation values. The effective mass of the
nucleon is

$\displaystyle m^{*}=m-g_{\sigma}\sigma$ (14)

and the equilibrium densities are defined as $\rho=\rho_{p}+\rho_{n}$ and
$\rho_{3}=\rho_{p}-\rho_{n}$.

The values of the coupling constants are usually determined in such a way that
they yield appropriate values for finite nuclei properties (e.g. binding
energy, charge radii) and various quantities associated with the hadronic
matter at the saturation density. The expression for the effective chemical
potential of the individual nucleons in this model is also same as that in HM1
(Eq. 6) and is given as

$\displaystyle\mu_{B}=\sqrt{k_{F}^{2}+{m^{*}}^{2}}+g_{\omega}\omega_{0}+g_{\rho}I_{3B}\rho_{03}$
(15)

with $I_{3B}$ being the isospin 3-component of nucleon. The fields involved in
the expression of effective mass (Eq. 14) and chemical potential (Eq. 15) of
nucleons can be obtained by solving the following field-equations self-
consistently,

$\displaystyle\sigma_{0}$ $\displaystyle=$
$\displaystyle-\frac{g_{\sigma}}{m_{\sigma}^{2}}\displaystyle{\sum_{i=n,p}\frac{1}{\pi^{2}}\int_{0}^{k_{F_{i}}}dk\,k^{2}\,\frac{m_{i}^{*}}{\sqrt{k^{2}+{m_{i}^{*}}^{2}}}}-\frac{1}{m_{\sigma}^{2}}\Bigl{[}\frac{{\kappa_{3}}}{2m}g_{\sigma}m_{\sigma}^{2}\sigma_{0}^{2}$
(16)
$\displaystyle+\frac{{\kappa_{4}}}{6m^{2}}g_{\sigma}^{2}m_{\sigma}^{2}\sigma_{0}^{3}-\frac{\eta_{1}}{2m}g_{\sigma}m_{\omega}^{2}\omega_{0}^{2}-\frac{\eta_{2}}{2m^{2}}g_{\sigma}^{2}m_{\omega}^{2}\sigma_{0}\omega_{0}^{2}$
$\displaystyle-\frac{\eta_{\rho}}{2m}g_{\sigma}m_{\rho}^{2}\rho_{03}^{2}-\frac{\eta_{1\rho}}{2m^{2}}g_{\sigma}^{2}m_{\rho}^{2}\sigma_{0}\rho_{03}^{2}\Bigr{]}~{},$
$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle\frac{g_{\omega}}{m_{\omega}^{2}}\rho-\frac{1}{m_{\omega}^{2}}\Bigl{[}\frac{1}{6}\zeta_{0}g_{\omega}^{2}\omega_{0}^{3}-\frac{\eta_{1}}{m}g_{\sigma}m_{\omega}^{2}\sigma_{0}\omega_{0}$
(17)
$\displaystyle-\frac{\eta_{2}}{2m^{2}}g_{\sigma}^{2}m_{\omega}^{2}\sigma_{0}^{2}\omega_{0}-\frac{\eta_{2\rho}}{2m^{2}}g_{\omega}^{2}m_{\rho}^{2}\rho_{03}^{2}\omega_{0}\Bigr{]},$
$\displaystyle\rho_{03}$ $\displaystyle=$
$\displaystyle\frac{g_{\rho}}{m_{\rho}^{2}}\rho_{3}+\frac{1}{m_{\rho}^{2}}\Bigl{[}\frac{\eta_{\rho}}{m}g_{\sigma}m_{\rho}^{2}\sigma_{0}\rho_{03}+\frac{\eta_{1\rho}}{2m^{2}}g_{\sigma}^{2}m_{\rho}^{2}\sigma_{0}^{2}\rho_{03}$
(18)
$\displaystyle+\frac{\eta_{2\rho}}{2m^{2}}g_{\omega}^{2}m_{\rho}^{2}\omega_{0}^{2}\rho_{03}\Bigr{]},$

We use the BSP parametrization [22] for our calculation, which describe the
properties of finite nuclei very well. This parameterization includes the
quartic order cross-coupling between $\omega$ and $\sigma$ mesons to model the
high density behavior of the EoS.

Table 2:  BSP parameter sets for the extended RMF model with the nucleon mass
$m=$ 939.2 MeV.

$g_{\sigma}/4\pi$ | $g_{\omega}/4\pi$ | $g_{\rho}/4\pi$ | $\kappa_{3}$ | $\kappa_{4}$ | $\eta_{1}$ | $\eta_{2}$ | $\eta_{\rho}$ | $\eta_{1\rho}$ | $\eta_{2\rho}$ | $\zeta_{0}$ | $m_{\sigma}/m$ | $m_{\omega}/m$ | $m_{\rho}/m$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0.8764 | 1.1481 | 1.0508 | 1.0681 | 14.9857 | 0.0872 | 3.1265 | 0.0 | 0.0 | 53.7642 | 0.0 | 0.5383 | 0.8333 | 0.8200

Table 3:  Properties of the hadronic matter at the saturation density. $m^{*}/m$ | $B/A$ | $\rho_{0}$ | $K$ | $J$ | $L_{0}$
---|---|---|---|---|---
| (MeV) | (fm-3) | (MeV) | (MeV) | (MeV)
0.60 | -15.9 | 0.149 | 230 | 28.83 | 50

In the present work we consider asymmetric matter motivated by the asymmetry
of dense neutron star matter. In typical dense neutron star matter the proton
content is $\sim$(10-15)% of the total matter. Therefore in the present work
we consider the proton fraction to be 15% in case of both the hadronic models
HM1 and HM2. However, as discussed in the Introduction section 1, we do not
consider the leptonic contribution in the hadronic sector of the present work
to calculate the transport coefficients.

###  2.3 Quark Phase

We consider the MIT bag model [12, 65] with u and d quarks to describe the
pure massless quark phase. The chemical potential of the quarks is
respectively given as

$\displaystyle\mu_{f}=\sqrt{{k_{F}}_{f}^{2}+m_{f}^{2}}$ (20)

where, $k_{f}$ is the Fermi momenta of individual flavors $f=$ u and d.
$m_{f}$ is the mass of individual quarks. We consider $m_{u}=$ 2.2 MeV and
$m_{u}=$ 4.7 MeV, which are low enough to consider the quark matter to be
massless in absence of the massive s quark. The total baryonic density is
given as

$\displaystyle\rho=\frac{\rho_{u}+\rho_{d}}{3}=\frac{1}{3}(2\times
3)\frac{1}{6\pi^{2}}\Bigg{[}{k_{Fu}^{3}}+{k_{Fd}^{3}}\Bigg{]},$ (21)

where spin, color degeneracy factors of quark matter are considered as 2, 3
respectively and $k_{Fu}$, $k_{Fd}$ are Fermi-momentum of the u and d quarks
respectively. The factor 1/3 comes to conserve the net baryon number since the
MIT Bag model is based on deconfinement of hadrons (nucleons) and the fact
that there are three quarks per nucleon, each having baryon number 1/3.

##  3 Framework of transport coefficients for hadrinic and quark phases

Here we will address the standard kinetic theory framework for obtaining the
transport coefficients like shear viscosity $\eta$ and electrical conductivity
$\sigma$, which are basically the part of the dissipation component of any
many body system or medium or fluid. Let us first derive the expression of
shear viscosity. Considering the matter as a dissipative fluid, one can
express energy-momentum tensor in macroscopic form as

$\displaystyle T^{\mu\nu}={T_{0}}^{\mu\nu}+{T_{D}}^{\mu\nu}~{},$ (22)

where, ideal (${T_{0}}^{\mu\nu}$) and dissipation (${T_{D}}^{\mu\nu}$) parts
in terms of fluid quantities like fluid four velocity $u^{\mu}$, energy
density $\epsilon$, pressure $P$ and metric tensor $g^{\mu\nu}$ can be
expressed as

$\displaystyle{T_{0}}^{\mu\nu}=-g^{\mu\nu}+(\epsilon+P)u^{\mu}u^{\nu}$ (23)

and

$\displaystyle
T_{D}^{\mu\nu}=\pi^{\mu\nu}+...=\eta^{\mu\nu\alpha\beta}~{}{\mathcal{U}}_{\alpha\beta}^{\eta}+..$
(24)

respectively. In the expression of the dissipative part of the energy-momentum
tensor, the viscous stress tensor is $\pi^{\mu\nu}$, velocity gradient tensor
is
${\mathcal{U}}^{\mu\nu}_{\eta}=D^{\mu}u^{\nu}+D^{\nu}u^{\mu}-\frac{2}{3}\Delta^{\mu\nu}\partial_{\rho}u^{\rho}$,
with $D^{\mu}=\partial^{\mu}-u^{\mu}u^{\sigma}\partial_{\sigma}$,
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$ and $\eta^{\mu\nu\alpha\beta}$ is
shear viscosity tensor. Also, the dissipative part can have components of
shear, bulk and thermal dissipation but here we are dealing only with the
shear part. The notation of $(+...)$ indicates the other dissipation
components. The picture of relativistic shear viscosity $\eta$ as a
proportional constant between viscous stress tensor $\pi^{\mu\nu}$ and
tangential fluid velocity gradient ${\mathcal{U}}^{\mu\nu}_{\eta}$ can be
compared with Newton-Stoke law, applicable in the non-relativistic domain,
where the shear viscosity is defined as the proportionality constant between
shear stress and velocity gradient.

Considering the hadronic matter to be composed of nucleons, the microscopic
expression of its energy-momentum tensor can be written in kinetic theory
framework as

$\displaystyle
T^{\mu\nu}=\gamma\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{\mu}k^{\nu}}{E}(f_{N}+f_{\bar{N}})$
(25)

where, $f_{N}$ and $f_{\bar{N}}$ are assumed as non-equilibrium distribution
functions for nucleons $N$ and anti-nucleons ${\bar{N}}$. Now splitting
$f_{N,{\bar{N}}}$ as the sum of equilibrium distribution $f_{N,{\bar{N}}}^{0}$
and a small deviation $\delta f_{N,{\bar{N}}}$ i.e,
$f_{N,{\bar{N}}}=f_{N,{\bar{N}}}^{0}+\delta f_{N,{\bar{N}}}$, one can
separately identify the microscopic expressions of ${T_{0}}^{\mu\nu}$ and
${T_{D}}^{\mu\nu}$ parts in terms of the quantities like particle’s four
momenta $k^{\mu}$, degeneracy factor $\gamma$ etc. as

$\displaystyle{T_{0}}^{\mu\nu}=\gamma\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{\mu}k^{\nu}}{E}(f^{0}_{N}+f^{0}_{\bar{N}})$
(26)

and

$\displaystyle
T_{D}^{\mu\nu}=\pi^{\mu\nu}+...=\gamma\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{\mu}k^{\nu}}{E}(\delta
f_{N}+\delta f_{\bar{N}})+..$ (27)

Now, with the help of the standard relaxation time approximation (RTA) methods
of Boltzmann transport equation, one can get [26]

$\displaystyle\delta
f_{N,{\bar{N}}}=\frac{k^{\alpha}k^{\beta}}{E}\tau_{N,{\bar{N}}}\beta
f^{0}_{N,{\bar{N}}}(1-f^{0}_{N,{\bar{N}}}){\mathcal{U}}_{\alpha\beta}^{\eta}~{},$
(28)

where, $\tau_{N,{\bar{N}}}$ are relaxation times and $E$ are energy of nucleon
and anti-nucleon. Now, using Eq. (28) in Eq. (27) and then comparing with Eq.
(24), we get the shear viscosity tensor as

$\displaystyle\eta^{\mu\nu\alpha\beta}=\gamma\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{\mu}k^{\nu}k^{\alpha}k^{\beta}}{E^{2}}\tau_{N,{\bar{N}}}[f^{0}_{N}(1-f^{0}_{N})+f^{0}_{\bar{N}}(1-f^{0}_{\bar{N}})]/T$
(29)

Using tensor identity [26, 66, 67], one can obtain the isotropic expression as

$\displaystyle\eta=\frac{\gamma}{15}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{4}}{E^{2}}\tau_{N,{\bar{N}}}[f^{0}_{N}(1-f^{0}_{N})+f^{0}_{\bar{N}}(1-f^{0}_{\bar{N}})]/T$
(30)

If we analyze Eq. (30), then we can find two components. One is relaxation
time $\tau_{N,{\bar{N}}}$, whose order of magnitude will fix the strength of
viscosity for the system. Another is the remaining part of the expression of
$\eta$, which can be called as phase-space part of $\eta$ and it will
basically decide the temperature and density dependence of $\eta$. The
expression for any transport coefficient will have these two component
structures.

Now, if we take our calculation from finite $T$ to the $T=0$ case, then the
equilibrium Fermi-Dirac distribution takes the form of a step function as

$\displaystyle f_{N}^{0}$ $\displaystyle=$ $\displaystyle 1~{}{\rm{if}}E<\mu$
$\displaystyle=$ $\displaystyle 0~{}{\rm{if}}E>\mu$ $\displaystyle
f_{\bar{N}}^{0}$ $\displaystyle=$ $\displaystyle 1~{}{\rm{if}}E<-\mu$ (31)
$\displaystyle=$ $\displaystyle 0~{}{\rm{if}}E>-\mu~{}.$

The above conditions imply that the anti-nucleons do not contribute anymore to
the positive energy for the $T=0$ case. We have to use the replacement

$\displaystyle\frac{\partial f^{0}_{N}}{\partial
E}=-f^{0}_{N}(1-f^{0}_{N})/T\rightarrow\frac{\partial}{\partial
E}\theta(\mu-E)=-\delta(E-\mu)~{}.$ (32)

Using these replacements in Eq. (30), we get

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{\gamma}{15}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{4}}{E^{2}}\tau_{N}\delta(E-\mu)$
(33) $\displaystyle=$
$\displaystyle\frac{\gamma}{30\pi^{2}}\tau_{N}\frac{(\mu^{2}-m^{*2})^{5/2}}{\mu}$

Here, $\mu$ is the weighted average of the effective chemical potentials of
neutron and proton as obtained from Eq. (6) and Eq. (15) for HM1 and HM2,
respectively.

Let us next come to another transport coefficient viz. the electrical
conductivity, which is basically proportionality constant between electric
current density and the field. This macroscopic definition comes from Ohm’s
law, which can be expressed in three dimensional notation as

$\displaystyle J^{i}=\sigma^{ij}E_{j}~{},$ (34)

where, the electrical conductivity tensor $\sigma^{ij}$ connect the electric
field $E_{j}$ and the electric current density $J^{i}$. Realizing current due
to electric field as a dissipation phenomenon, we can microscopically express
it as

$\displaystyle
J^{i}=q\gamma\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{i}}{E}\delta
f_{N}~{},$ (35)

where, $q$ is the electric charge of nucleon, i.e. $q=0,e$ for neutron and
proton. Here, the RTA methods of Boltzmann transport equation will again help
us to give the form of $\delta f_{N}$ [26]:

$\displaystyle\delta f_{N}=\frac{k^{j}}{E}\tau_{N}\beta
f^{0}_{N}(1-f^{0}_{N})qE_{j}~{},$ (36)

Using Eq. (36) in Eq. (35) and then comparing with Eq. (34), we get the
conductivity tensor as

$\displaystyle\sigma^{ij}=q^{2}\gamma\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{i}k^{j}}{E^{2}}\tau_{N}\beta
f^{0}_{N}(1-f^{0}_{N})~{},$ (37)

whose isotropic expression at $T=0$ will be

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{q^{2}\gamma}{3}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{2}}{E^{2}}\tau_{N}\delta(E-\mu)$
(38) $\displaystyle=$
$\displaystyle\frac{q^{2}\gamma}{6\pi^{2}}\tau_{N}\frac{(\mu^{2}-m^{*2})^{3/2}}{\mu}$

The final expressions of shear viscosity and electrical conductivity for
hadronic matter are given by Eqs. (33) and (38), respectively. The neutrons
will not contribute to the electrical conductivity given in Eq. (38). Hence
only protons contribute to the electrical conductivity and thus in Eq. (38) we
have $\mu=\mu_{p}$ as the effective proton chemical potential, calculated from
Eqs. (6) and (15) for HM1 and HM2, respectively. Moreover, in Eq. (38) only
protons will contribute to the relaxation time i.e., $\tau_{N}=\tau_{p}$.

When we apply these two expressions for quark phase, then nucleon effective
mass $m^{*}$ will be replaced by quark mass and the degeneracy factor $\gamma$
will also be replaced by the corresponding quark degeneracy factor $g$ (say).
Considering two flavor quark matter with u and d quarks, $g=3\times 2\times
2=12$ for Eq. (33) and $gq^{2}=3\times
2\Big{(}\frac{4e^{2}}{9}+\frac{e^{2}}{9}\Big{)}=\frac{10e^{2}}{3}$ for Eq.
(38). The values of $\eta$ and $\sigma$ for quark matter will be very close to
their massless limits

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{g}{30\pi^{2}}\tau_{Q}\mu^{4}$ $\displaystyle\sigma$
$\displaystyle=$ $\displaystyle\frac{gq^{2}}{6\pi^{2}}\tau_{Q}\mu^{2},$ (39)

whose, normalized values become constant. These massless limits may act as
reference line at $T=0$ and finite $\mu$ case.

The relaxation time can also be calculated microscopically. For the degenerate
scenario, the medium constituents will occupy all the energy levels from
${m^{*}}$ to $\mu$ and ideally they have zero probability to move outside
$\mu$. However, in reality, due to very low $T$ instead of exactly $T=0$, we
can have a small deviation from step function type distribution function. So
medium constituents, having energy nearly Fermi energy $\mu$ and velocity
nearly Fermi velocity $v_{F}=\sqrt{\mu^{2}-{m^{*}}^{2}}/\mu$ will participate
in the momentum transfer scattering process. So we can define relaxation time
as

$\displaystyle\tau_{c}$ $\displaystyle=$ $\displaystyle
1/[\sigma_{s}v_{F}\rho]$ (40) $\displaystyle=$
$\displaystyle\mu/[\sigma_{s}\sqrt{\mu^{2}-{m^{*}}^{2}}~{}\rho]~{},$

where, $\rho$ is density of the medium and $\sigma_{s}=4\pi a^{2}$ is cross
section with scattering length $a$. Now depending upon the system, we have to
use the inputs for the medium constituents to calculate their relaxation time.
We first normalize the quantities $\eta$ and $\sigma$ as
$\eta/(\tau_{c}\mu^{4})$ and $\sigma/(\tau_{c}\mu^{2})$ to obtain
dimensionless forms. For electrical conductivity we consider
$\tau_{c}=\tau_{cp}$ and $\mu=\mu_{p}$ for the scaling.

##  4 Result and Discussions

We obtain the transport coefficients of the hadronic and quark phases based on
the formalism discussed in the earlier section and study the variation of
these coefficients with respect to density. The phenomenon of phase transition
at vanishing temperature and the transition density is still inconclusive in
literature. So rather than achieving phase transition, in the present work we
compare the order of magnitudes of the transport coefficients for the hadronic
and quark phases.

Figure 1: Weighted average of the effective chemical potential of hadronic
matter with two different hadronic models HM1 and HM2 and zero-mass quark
matter (z-MQM (left) and the effective nucleon mass of hadronic matter with
HM1 and HM2 (right) vs normalized baryon density.

We calculate the shear viscosity of hadronic and quark matter using Eq. (33).
The electrical conductivity for hadronic matter is calculated using Eq.(38)
while that of quark matter is calculated using Eq. (39). We then normalize
these quantities as $\eta/(\tau_{c}\mu^{4})$ and $\sigma/(\tau_{c}\mu^{2})$ to
obtain dimensionless forms, where $\tau_{c}$ is the relaxation time of quark
or hadronic matter in general i.e. $\tau_{c}=\tau_{N}$ for hadronic matter and
$\tau_{c}=\tau_{Q}$ for quark matter. These quantities
$\eta/(\tau_{c}\mu^{4})$ and $\sigma/(\tau_{c}\mu^{2})$ are plotted against
the normalized density in left and right panels of Fig. (2), respectively. We
notice roughly 10-20 times difference between the hadronic model-1 (HM1) and
hadronic model-2 (HM2) in terms of the estimations of $\eta/(\tau_{c}\mu^{4})$
and $\sigma/(\tau_{c}\mu^{2})$. This is because of the difference in chemical
potential ($\mu$) and also the nucleon effective mass ($m^{*}$) between the
two hadronic models. Both chemical potential and specially the effective mass
are quite different for HM1 and HM2 as seen from Fig. (1). Re-looking at Eqs.
(33) and (38), one can understand the numerical and physical impact of Fig.
(1) on Fig. (2). The difference in the order of magnitudes between the two
hadronic models reflects a numerical band of transport coefficients for
hadronic matter along density axis. For both $\eta/(\tau_{c}\mu^{4})$ and
$\sigma/(\tau_{c}\mu^{2})$, we notice jump in the estimated values of the
hadronic models and the quark model (z-MQM). For shear viscosity, this jump is
from 0.001 (HM1) or 0.02 (HM2) to 0.05 (z-MQM) and for electrical conductivity
this jump is from 0.0003 (HM1) or 0.003 (HM2) to 0.005 (z-MQM). One may notice
the saturation trend of this normalized transport coefficients, which zoom in
basically its thermodynamical phase-space part (or in other words, the
thermodynamical probability of shear and electrical charge transportation).
The saturation profile of HM1 (red solid line) and HM2 (green dash line) have
a mild increasing trend with $\mu$ due to their dependence
$\eta\propto\tau_{c}\mu^{4}(1-m^{*2}/\mu^{2})^{5/2}$ and
$\sigma\propto\tau_{c}\mu^{2}(1-m^{*2}/\mu^{2})^{5/2}$. They saturate at
different values due to the difference in $m^{*}$ between the two hadronic
models.

Figure 2: Normalized shear viscosity (left) and electrical conductivity
(right) with respect to scaled baryon density of hadronic matter and zero-mass
quark matter (z-MQM).

On the other hand, the quark matter estimation follow the massless transport
relation along the entire density axis $\eta\propto\tau_{c}\mu^{4}$ and
$\sigma\propto\tau_{c}\mu^{2}$ since we have put exact massless limit for the
quark phase. In actual case one should sketch the estimations of HM1 or HM2 at
low density domain and the z-MQM estimation at high density domain and at some
intermediate density quark-hadron phase transition will take place, connecting
the estimations of the low and high density domains. However, due to lack of
proper knowledge regarding phase transition density (which is highly model
dependent), we restrict our focus only on the order of magnitude of the
transport coefficients for two phases.

These estimations of the transport coefficients viz. the shear viscosity and
the electrical conductivity at $T=0$ and $\mu\neq 0$ domain can be compared
with the estimations of the transport coefficients at $T\neq 0$ and $\mu=0$
domain, given in Ref. [26] and the references therein. The quark-hadron
transition at $T\neq 0$ and $\mu=0$ domain is understood by estimating the
thermodynamics of massless quark gluon plasma (QGP) and hot pion gas for high
and low temperature zones, respectively. However, lattice quantum
chromodynamics (LQCD) calculations [2, 3, 4] went deeper and unfolded the
crossover type nature of quark-hadron transition at $T\neq 0$ and $\mu=0$
domain. The values of normalized thermodynamical quantities like $P/T^{4}$,
$\epsilon/T^{4}$ etc. for massless QGP will act as reference line or upper
limits of QCD matter for the $\mu=0$ case. The values of thermodynamical
quantities, obtained from LQCD, remain quite lower than the massless QGP limit
in low (or hadronic matter) temperature domain, which can be realized as non-
perturbative aspects of QCD. Near transition temperature, their values
increase to reach towards the massless limits but in a smooth crossover way
instead of a first order phase transition kind of jump. Beyond the transition
temperature, LQCD thermodynamical values still remain little suppressed with
respect to their massless limits. This small suppression is also realized from
the direction of finite temperature perturbative quantum chromodynamics (pQCD)
theory, whose latest status can be found in [68]. Similar investigation of
thermodynamical variations with respect to density or chemical potential at
$T=0$ are attempted over a very long time for understanding the properties of
dense matter [69]. Due to problems in LQCD calculations at finite $\mu$, the
present knowledge of quark-hadron phase transition along $\mu$-axis is not
quite converging like the understanding of phase transition along $T$-axis. In
this context, in the present work we consider the effective hadronic models
and the Bag model to get the order of magnitudes for different transport
coefficients of the hadronic and quark phases along $\rho$ or $\mu$-axis. The
normalized transport coefficients $\eta/(\tau_{c}\mu^{4})$ and
$\sigma/(\tau_{c}\mu^{2})$ against $\rho$ or $\mu$-axis will exhibit
thermodynamical phase-space of transportation, which follow similar
suppression in hadronic $\mu$-domain (at $T=0$) as noticed for hadronic
$T$-domain (at $\mu=0$) in [26]. In [26], the normalized coefficients are
chosen as $\eta/(\tau_{c}T^{4})$ and $\sigma/(\tau_{c}T^{2})$, while in the
present work they are chosen as $\eta/(\tau_{c}\mu^{4})$ and
$\sigma/(\tau_{c}\mu^{2})$. In both cases, around $10^{1-2}$ suppression is
observed in hadronic $T$ or $\mu$ domain with respect to their massless
limits. The suppressed values in both the cases can be realized as the non-
pQCD effect in the phase-space of transportation. This equivalence between
normalized phase-space of transport coefficients along $T$-axis and
$\rho$-axis may be considered as an unique finding of present investigation.

Up to now, the transport coefficients are normalized by relaxation time. So we
basically see the phase space part of transport coefficients. However, the
actual estimation of relaxation time will be important to know the absolute
values of the transport coefficients. We now calculate the relaxation time for
the hadronic and massless quark matter using Eq. (40). We consider the
effective nucleon mass and two possible orders of cross-section
$\sigma_{s}=4\pi a^{2}\approx 5340~{}\rm{fm^{2}}$ and $12.56~{}\rm{fm^{2}}$
for hadronic matter case with scattering lengths $a\approx 20$ fm and $1$ fm.
The former value of $\sigma_{s}\approx 5340~{}\rm{fm^{2}}$ or $a\approx 20$ fm
is the isospin averaged cross section/scattering length of NN interactions,
taken from [70, 71]. This value is used for the calculation of relaxation time
of nucleons at finite temperature in [72]. Here, we have considered finite
density system. We also consider another small value of scattering length
$a=1$ fm.

Figure 3: Relaxation time with respect to scaled baryon density of hadronic
matter and zero-mass quark matter (z-MQM) for different values of cross-
section.

In Fig. (3), we have plotted the variation of $\tau_{c}$ for HM1 and HM2 and
z-MQM. For hadronic matter we consider two different values of scattering
lengths - one is guided by the vacuum scattering interaction strength and
another is a guess value by assuming that scattering length may decrease with
density. As a rough order of magnitude, we get $\tau_{c}\approx 10-0.1$ fm and
$\tau_{c}\approx 0.01-0.0002$ fm for scattering lengths $a\approx 1$ fm and
$a=20$ fm, respectively. One should notice that in this context the ranking of
$\tau_{c}$ is HM1 $>$ HM2 $>$ z-MQM. This is because of the inverse relation
of relaxation time with thermodynamic quantity like density $\rho$ as seen
from Eq. 40.

Once we obtain the relaxation time for each phase, we calculate the shear
viscosity and the electrical conductivity by using the density dependent
relaxation time values. The normalized estimations of $\eta/\rho$ and
$\sigma/\mu$ are presented in the left and right panels of Fig. (4),
respectively. Though $\eta/\mu^{3}$ is also possible dimensionless quantity
but $\eta/\rho$ is chosen for fluid property [73] measurement of quark and
hadronic matter in the dense sector. This ratio
$\eta/\rho=\eta\mu/(\epsilon+P)$ at $T=0$, $\mu\neq 0$ can have equivalent
role like viscosity to entropy density ratio $\eta/s$ for early universe or
RHIC/LHC environment with $T\neq 0$, $\mu=0$. From the Euler’s thermodynamical
relation, we get a general relation $Ts=\epsilon+P-\mu\rho$, which will be
transformed into $Ts=\epsilon+P$ for $T\neq 0$, $\mu=0$ and
$\mu\rho=\epsilon+P$ for $T=0$, $\mu\neq 0$ respectively. So, for measuring
the fluid property, we may choose dimensionless ratio either $\eta
T/(\epsilon+P)$ or $\eta\mu/(\epsilon+P)$ for intermediate $T$, $\mu$. Now,
when one goes from the intermediate $T$-$\mu$ domain to the limit of $T\neq
0$, $\mu=0$, then $\eta T/(\epsilon+P)$ will be better quantity [73] and in
limiting case, we can write $\eta T/(\epsilon+P)=\eta/s$. Similarly, when one
goes from the intermediate $T$-$\mu$ domain to the limit of $T=0$, $\mu\neq
0$, then $\eta\mu/(\epsilon+P)$ will be the better quantity to study and in
limiting case, we can write $\eta\mu/(\epsilon+P)=\eta/\rho$. Reader may find
good discussion on this fluidity quantity in [73, 87].

We know that the data of RHIC and LHC experiments [74, 75, 76, 77, 78]
indicate very small values of $\eta/s$, ever observed in nature [79]. On the
other hand, a string theory-based calculation [80] tells
$\frac{\eta}{s}\geq\frac{1}{4\pi}$, which may be considered as lower bound
conjecture of $\eta/s$. This lower bound is popularly known as the Kovtun-Son-
Starinet (KSS) bound [80]. The existence of a non-zero (but may not be equal
to $\frac{1}{4\pi}$) lower bound of $\eta/s$ can also be realized from quantum
aspects, which can prevent the classical possibility $\eta/s\rightarrow 0$.
Being roughly proportional to the ratio of mean free path to de-Broglie
wavelength of medium constituent, the ratio $\eta/s$ of any fluid can never
vanish because the mean free path of any constituent can never be lower than
its de-Broglie wavelength. It indicates that quantum fluctuations prevent the
existence of perfect fluid in nature and $\eta/s$ of any fluid should have
some lower bound. Along with the support from quantum aspect, this lower bound
of $\eta/s$ is also validated through supersymmetric Yang–Mills (SYM) theory
[83], which provide $\frac{\eta}{s}=\frac{1}{4\pi}$ in the infinite limit of
the ’t Hooft coupling. From experimental direction also, we noticed that
$\eta/s$ of all fluids, including super-fluid helium and even trapped Li atom
at strong coupling [84], remain well above the bound for the range of measured
temperatures and pressures. All these directions, ultimately support the lower
bound conjecture of $\eta/s$, which will be used as a gross reference point in
present work. The term ”gross” is used because we have to accept the
possibility of violation of the KSS bound [85, 86].

Figure 4: Shear viscosity to density ratio (left) and electrical conductivity
upon chemical potential (right) with respect to scaled baryon density of
hadronic matter and zero-mass quark matter (z-MQM) for different values of
cross-section.

Interestingly, the values of $\eta/s$ for RHIC and LHC matter are very close
to this quantum lower bound $1/(4\pi)$. So, a natural question arises -
whether this nearly perfect fluid nature is also expected along the $\mu$-axis
at $T=0$ like that along the $T$-axis at $\mu=0$ of the QCD phase diagram? The
present investigation aims in this direction for the first time but to get
more conclusive outcome, probably further (alternative) research is required
in future. The question becomes more important as recently [81] have
experimentally found the value $\eta/s\approx 1/(4\pi)$ for hadronic matter in
the low energy hadronic physics experiment. We have plotted $\eta/\rho$ of
hadronic and quark matter with two values of scattering length in the left
panel of Fig. (4). We notice that the results for $a\approx 20$ fm cross the
KSS values ($\approx 0.08$), which may not be considered as a physically
acceptable order of magnitude. Probably the magnitude of the cross
section/scattering length in vacuum may be largely modified in the finite
density picture, which is missing in the present calculation. In this regard,
our guess value scattering $a=1$ fm provides an acceptable range of
$\eta/\rho$ in the hadronic density range, which still crosses the KSS
boundary in the quark density range. So a safe zone may be considered as
scattering lengths $a<1$ fm for getting $\eta/\rho>1/(4\pi)$. For mathematical
guidance we can get a lower limit curve of the relaxation time $\tau$ as a
function of density or chemical potential for massless quark matter by
imposing the KSS limit. For finite temperature we get $\tau=5/(4\pi T)$ [82]
by imposing KSS limit $\eta/s=1/(4\pi)$. Similarly, for finite density, one
can easily find the corresponding expression $\tau=5/(4\pi\mu)$ by using
$\eta$ from Eq. (39), $\rho$ from Eq. (21) and then by imposing KSS limit
$\eta/\rho=1/(4\pi)$. In terms of density $\rho$, the KSS limit of relaxation
time for massless quark matter with $N_{f}=2$ flavor will be

$\displaystyle\tau=\frac{5}{4\pi}\Big{(}\frac{2}{\pi^{2}}\Big{)}^{1/3}\frac{1}{\rho^{1/3}}$
(41)

Another interesting outcome here is that we still see the jump in $\eta/\rho$
is possible in case of a possible scenario of hadron-quark phase transition as
in terms of $\eta/\rho$ the ranking is noticed as HM2 $>$ HM1 $>$ z-MQM. So
phase transition from hadronic to quark phase may imply a reduction in
$\eta/\rho$. However, for normalized electrical conductivity $\sigma/\mu$, we
observed that the ranking is HM2 $>$ z-MQM $>$ HM1. So the ranking of
magnitude of $\sigma/\mu$ for the two different phases may be model dependent.
We cannot get any conclusive picture in this regard. The present work reveals
this uncertainty. We also do not go into any comparative discussion of two
hadronic models, rather we intend to show that a possible numerical
uncertainty pertains to the estimations of $\sigma/\mu$ by using two different
hadronic models. Hence, future research with other existing hadronic models is
probably necessary for getting any conclusive or broader picture. Only an
order of magnitude difference in transport coefficients for two phases can be
considered as conclusive message in present study.

##  5 Summary and Conclusion

To summarize, we have attempted to visualize the variation of shear viscosity
and electrical conductivity of hadronic and quark matter along the density
axis. Inspiration of such calculation comes from the calculations of transport
coefficients of RHIC or LHC matter, where we get a cross-over type of quark-
hadron phase transition along temperature axis. The temperature dependence of
shear viscosity and electrical conductivity of RHIC or LHC matter mainly
contain two parts - one is thermodynamical phase-space part and the relaxation
time part. If we exclude the relaxation time part by normalizing it, then we
obtain their nice pattern along temperature axis as one notices for other
thermodynamical quantities like pressure, energy density etc. from lattice QCD
calculations. The pattern of normalized thermodynamical quantities and
transport coefficients is as follows. At high temperature, they reach close to
their massless limits and they are gradually suppressed as temperature goes
down. Similar kind of pattern is observed in the present work when we go from
high density quark phase to low density hadronic phase. The finite density
calculations of hadronic phase is obtained with two RMF models and massless
quark model for the quark phase. The equivalence of thermodynamic phase-space
profile of transportation along temperature axis and density axis is a very
interesting finding of the present work.

Instead of normalizing relaxation time, if we use its microscopic estimated
values, which is generally of the order of fm due to strong interaction, then
we can get the final profile of the transport coefficients. A long list of
references can be found for RHIC or LHC matter, where most of them found that
shear viscosity to entropy density ratio will decrease first then increase
with temperature. In this regard, we find shear viscosity to density ratio
decreases with density in the hadronic phase and may suddenly drop in a
possible scenario of hadron-quark phase transition at $T=0$. In the pure quark
phase, a mild decreasing trend with density is observed. This is the
qualitative trend of the fluidity measurement of quark and hadronic matter
along the density axis, observed in the present investigation. During the
computation of the quantitative values, we use the quantum lower bound of
fluidity or shear viscosity to density ratio as a physical reference point and
we found an interesting finding which is as follows. The experimental data for
standard nucleon-nucleon scattering length in vacuum is of the order of 20 fm
and using this as a hard-sphere scattering cross section, we find that the
fluidity values cross the lower bound. We also found that scattering length
less than 1 fm can only provide physical fluidity, beyond its lower bound. It
indicates that a good amount of in-medium modification will play an important
role in the dense sector, for which vacuum scattering length 20 fm will be
reduced to lower values, less than 1 fm. By using scattering length as an
input parameter, the present work just unfolds this issue but not well
equipped for the microscopic calculation of density dependent scattering
length, which may be explored in future. We believe that other hadronic
models, whose mass and chemical potential undergo almost similar kind of
modification in terms of order of magnitude, will face same problem of
fluidity calculation in the dense sector, which should be checked with other
hadronic models.

The authors thank Dr. Sandeep Chatterjee, Department of Physical Sciences,
IISER Berhampur for useful discussions. DS acknowledges the institute
postdoctoral funding and research facilities at Indian Institute of Science
Education and Research Berhampur, Odisha at the initial stage of this work. NA
acknowledges the funding support of IFCPAR/CEFIPRA under Project No. 5804-3.

## References

## References

  * [1] P. de Forcrand, Simulating QCD at finite density, PoS LAT 2009 (2009) 010.
  * [2] F. Karsch, Lattice QCD at finite temperature: a status report, Zeitschrift Fur Physik C38, 147 (1988).
  * [3] S. Borsanyi, G. Endrodi, Z. Fodor et al., The QCD equation of state with dynamical quarks, Journal of High Energy Physics, 2010, 77 (2010).
  * [4] J. N. Guenther, Overview of the QCD phase diagram, Eur. Phys. J. A57, 136 (2021).
  * [5] A. N. Tawfik, Equilibrium Statistical-Thermal Models in High-Energy Physics, Int. J. Mod. Phys. A29 (2014), 1430021.
  * [6] M. Strickland, J. O. Andersen, L. E. Leganger, N. Su, Hard-thermal-loop QCD Ther-modynamics, Prog. Theor. Phys. Suppl.187, 106 (2011).
  * [7] J. O. Andersen, M. Strickland, and N. Su, Three-loop HTL gluon thermodynamics at intermediate coupling, JHEP 1008, 113 (2010).
  * [8] E. Annala, T. Gorda, A. Kurkela, J. Nattila, and A. Vuorinen, Quark-matter cores in neutron stars, Nature Physics 16, 907 (2020).
  * [9] A. Kurkela, P. Romatschke and A. Vuorinen, Cold quark matter, Phys. Rev. D 81 (2010) 105021.
  * [10] A. Kurkela and A. Vuorinen, Cool Quark Matter Phys. Rev. Lett. 117 (2016) no.4, 042501.
  * [11] T. Gorda, A. Kurkela, P. Romatschke, M. Säppi and A. Vuorinen, Next-to-Next-to-Next-to-Leading Order Pressure of Cold Quark Matter: Leading Logarithm, Phys. Rev. Lett. 121 (2018) no.20, 202701.
  * [12] A. Chodos et al., New extended model of hadrons, Phys. Rev. D 9, 3471 (1974).
  * [13] D. Sen and T. K. Jha, Effects of hadron-quark phase transition on properties of Neutron Stars, J. Phys. G: Nucl. Part. Phys. 46 (2019) 015202.
  * [14] D. Sen and T. K. Jha, Deconfinement of nonstrange hadronic matter with nucleons and $\Delta$ baryons to quark matter in neutron stars, Int. J. Mod. Phys. D 28 (2019) no.02, 1950040.
  * [15] D. Sen et al., Properties of Neutron Stars with hyperon cores in parameterized hydrostatic conditions, Int.J.Mod.Phys. E27 (2018) 1850097.
  * [16] D. Sen, Nuclear matter at finite temperature and static properties of proto-neutron star, J.Phys. G48 (2021) 025201.
  * [17] D. Sen, Variation of the $\Delta$ baryon mass and hybrid star properties in static and rotating conditions, Phys.Rev.C 103 (2021) 4, 045804; arXiv:2103.14136 [nucl-th].
  * [18] D. Sen, Role of $\Delta$s in determining the properties of neutron stars in parameterized hydrostatic equilibrium, Int. J. Mod. Phys. D 28, No. 9 (2019) 1950122.
  * [19] P. K. Sahu and A. Ohnishi, SU(2) Chiral Sigma Model and Properties of Neutron Stars, Prog. Theor. Phys. 104, 1163 (2000).
  * [20] P. K. Sahu, T. K. Jha, K. C. Panda, and S. K. Patra, Hot Nuclear Matter in Asymmetry Chiral Sigma Model, Nucl. Phys. A733, 169 (2004).
  * [21] T. K. Jha and H. Mishra, Constraints on nuclear matter parameters of an effective chiral model, Phys. Rev. C 78 (2008) 065802.
  * [22] B. K. Agrawal, A. Sulaksono and P. -G. Reinhard, Optimization of relativistic mean field model for finite nuclei to neutron star matter, Nucl. Phys. A882, 1-20 (2012).
  * [23] A. Sulaksono, N. Alam and B. K. Agrawal, Core–crust transition properties of neutron stars within systematically varied extended relativistic mean-field model, Int. J. Mod. Phys. E 23, 1450072 (2014).
  * [24] N. Alam, A. Sulaksono, and B. K. Agrawal, Diversity of neutron star properties at the fixed neutron-skin thickness of ${}^{208}\mathrm{Pb}$, Phys. Rev. C 92, 015804 (2015).
  * [25] N. Alam, H. Pais, C. Providência, and B. K. Agrawal, Warm unstable asymmetric nuclear matter: Critical properties and the density dependence of the symmetry energy, Phys. Rev. C 95, 055808 (2017).
  * [26] C. A. Islam, J. Dey, S. Ghosh, Impact of different extended components of mean field models on transport coefficients of quark matter and their causal aspects, Phys. Rev. C103 (2021) 3, 034904
  * [27] L. Mestel and F. Hoyle, On the thermal conductivity in dense stars, Proc. Cambridge Philos. Soc. 46, 331 (1950).
  * [28] A. A. Abrikosov, The conductivity of strongly compressed matter, Sov. Phys. JETP 18, 1399 (1964).
  * [29] V. Canuto, Electrical conductivity and conductive opacity of a relativistic electron gas, Astrophys. J. 159, 641 (1970).
  * [30] E. Flowers and N. Itoh, Transport properties of dense matter, Astrophys. J. 206, 218 (1976).
  * [31] G. Baym, C. Pethick, and D. Pikes, Electrical Conductivity of Neutron Star Matter, Nature volume 224, pages674–675 (1969).
  * [32] D. G. Yakovlev and D. A. Shalybkov, Electrical conductivity of neutron star cores in the presence of a magnetic field, Astrophysics and Space Science volume 176, pages191–215 (1991).
  * [33] G. M. Ewart, R. A. Guyer, and G. Greenstein, Electrical conductivity and magnetic field decay in neutron stars, The Astrophysical Journal, 202:238-247, 1975.
  * [34] M. E. Raikh and D. G. Yakovlev, Thermal and electrical conductivities of crystals in neutron stars and degenerate dwarfs, Astrophysics and Space Science volume 87, pages193–203 (1982)
  * [35] A. Y. Potekhin et al., Transport properties of degenerate electrons in neutron star envelopes and white dwarf cores, Astron.Astrophys.346:345,1999.
  * [36] P. S. Shternin and D. G. Yakovlev, Shear viscosity in neutron star cores, Phys. Rev. D 78, 063006 (2008)
  * [37] S. Banik and D. Bandyopadhyay, Effect of shear viscosity on the nucleation of antikaon condensed matter in neutron stars, Phys.Rev.D82:123010,2010
  * [38] S. Banik, R. Nandi, and D. Bandyopadhyay, Shear viscosity and the nucleation of antikaon condensed matter in protoneutron stars Phys.Rev.C 84 (2011) 065804.
  * [39] A. Schmitt and P. Shternin, Reaction Rates and Transport in Neutron Stars, Astrophys.Space Sci.Libr. 457 (2018) 455-574.
  * [40] P. Shternin and M. Baldo, Transport coefficients of nucleon neutron star cores for various nuclear interactions within the Brueckner-Hartree-Fock approach, Phys. Rev. D 102, 063010 (2020).
  * [41] E. McLaughlin et al., Building a testable shear viscosity across the QCD phase diagram, arXiv:2103.02090 [nucl-th] (2021).
  * [42] P. Shternin, M. Baldo, and P. Haensel, Transport coefficients of nuclear matter in neutron star cores, Phys.Rev.C 88 (2013) 6, 065803.
  * [43] P. Shternin, M. Baldo, and H-J Schulze, Transport coefficients in neutron star cores in BHF approach. Comparison of different nucleon potentials, J. Phys.: Conf. Ser. 932, 012042 (2017).
  * [44] X. L. Shang, P. Wang, W. Zuo, and J. M. Dong, Role of nucleon-nucleon correlation in transport coefficients and gravitational-wave-driven r-mode instability of neutron stars, Phys. Lett. B, 811, 135963 (2020).
  * [45] R. Nandi, and S. Schramm, Calculation of the transport coefficients of the nuclear pasta phase, J.Astrophys.Astron. 39 (2018) 40.
  * [46] P. Shternin and I. Vidana, Transport Coefficients of Hyperonic Neutron Star Cores, Universe 7(6), 203 (2021).
  * [47] P. S. Shternin, Phys. Rev. D 98, 063015 (2018)
  * [48] V. Koch, Aspects of Chiral Symmetry, International Journal of Modern Physics E6, 203-250 (1997).
  * [49] P. K. Sahu, R. Basu, and B. Datta, High-Density Matter in the Chiral Sigma Model Astrophys. J. 416, 267 (1993).
  * [50] J. R. Stone and P. G. Reinhard, The Skyrme Interaction in finite nuclei and nuclear matter Prog. Part. Nucl. Phys. 58, 587 (2007).
  * [51] E. Khan and J. Margueron, Determination of the density dependence of the nuclear incompressibility, Phys. Rev. C 88, 034319 (2013).
  * [52] U. Garg and G. Colo, The compression-mode giant resonances and nuclear incompressibility, Prog. Part. Nucl. Phys. 101 (2018) 55.
  * [53] M. Dutra et. al., Relativistic mean-field hadronic models under nuclear matter constraints, Phys. Rev., C 90, 055203 (2014).
  * [54] J. R. Stone, N. J. Stone, S. A. Moszkowski, Incompressibility in finite nuclei and nuclear matter, Phys. Rev. C 89, 044316 (2014).
  * [55] F. J. Fattoyev et al., Neutron Skins and Neutron Stars in the Multimessenger Era, Phys. Rev. Lett. 120, 172702 (2018).
  * [56] Z-Y Zhu, E-P Zhou, A. Li Neutron star equation of state from the quark level in the light of GW170817, Astrophys.J. 862 (2018) no.2, 98.
  * [57] D. Sen, N. Alam, and G. Chaudhuri, Properties of hybrid stars with a density-dependent bag model, J. Phys. G: Nucl. Part. Phys. 48 (2021) 105201.
  * [58] J. D. Walecka, A theory of highly condensed matter, Annals of Physics, 83, 491 (1974).
  * [59] J. Boguta and A. R. Bodmer, Relativistic calculation of nuclear matter and the nuclear surface, Nucl. Phys. A292, 413 (1977).
  * [60] B. D. Serot and J. D. Walecka, Advances in Nuclear Physics, edited by J. W. Negele and E. Vogt (Plenum, New York, 1986), Vol. 16.
  * [61] R. J. Furnstahl, C. E. Price, and G. E. Walker, Systematics of light deformed nuclei in relativistic mean-field models, Phys. Rev. C 36, 2590 (1987).
  * [62] H. Muller and B. D. Serot, Relativistic mean-field theory and the high-density nuclear equation of state, Nucl. Phys. A 606, 508 (1996)
  * [63] G. A. Lalazissis, J. König, and P. Ring, New parametrization for the Lagrangian density of relativistic mean field theory, Phys. Rev. C 55, 540 (1997).
  * [64] B. D. Serot and J. D. Walecka, Recent Progress in Quantum Hadrodynamics, Int. J. Mod. Phys. E 6, 515 (1997).
  * [65] N. K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics, and General Relativity (Springer-Verlag, New York, 2000).
  * [66] P. Chakraborty, J.I. Kapusta, Quasi-Particle Theory of Shear and Bulk Viscosities of Hadronic Matter Phys.Rev.C 83 (2011) 014906
  * [67] S. Gavin, Transport Coefficients in ultra-relativistic heavy ion collision Nucl. Phys. A 435 (1985) 826.
  * [68] M. Strickland, J. O. Andersen, L. E. Leganger, N. Su, Hard-thermal-loop QCD Ther-modynamics, Prog. Theor. Phys. Suppl.187, 106 (2011).
  * [69] G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song, T. Takatsuka, From hadrons to quarks in neutron stars: a review, Reports on Progress in Physics 81 (2018) 056902.
  * [70] M. M. Nagelset al., Compilation of coupling constants and low-energy parameters Nucl. Phys. B147, 189 (1979).
  * [71] O. Dumbrajset al., Compilation of coupling constants and low-energy parameters. 1982-edition, Nucl. Phys. B216, 277 (1983).
  * [72] S. Ghosh, S. Ghosh, S. Bhattacharya, Phenomenological bound on the viscosity of the hadron resonance gas, Phys. Rev. C98 (2018) 045202.
  * [73] G.S. Denicol, C. Gale, S. Jeon, J. Noronha, Fluid behavior of a baryon-rich hadron resonance gas, Phys. Rev. C 88 (2013) 6, 064901
  * [74] PHENIX collaboration, S. S. Adler et al., Elliptic flow of identified hadrons in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV, Phys. Rev. Lett. 91 (2003) 182301,
  * [75] STAR collaboration, J. Adams et al., Azimuthal anisotropy in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV, Phys. Rev. C 72 (2005) 014904
  * [76] ALICE collaboration, K. Aamodt et al., Higher harmonic anisotropic flow measurements of charged particles in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV, Phys. Rev. Lett. 107 (2011) 032301.
  * [77] P. Romatschke and U. Romatschke, Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?, Phys. Rev. Lett. 99, 172301 (2007)
  * [78] U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Annu. Rev. Nucl. Part. Sci. 63 (2013) 123.
  * [79] M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30 (2005).
  * [80] P. Kovtun, D. T. Son, and O. A. Starinets, Phys. Rev. Lett. 94, 111601 (2005).
  * [81] D. Mondal et al. Phys. Rev. Lett. 118, 192501 (2017).
  * [82] J. Dey, S. Satapathy, P. Murmu, S. Ghosh Shear viscosity and electrical conductivity of relativistic fluid in presence of magnetic field: a massless case, Pramana 95 (2021) 3, 125
  * [83] A. Buchel, J. T. Liu and A. O. Starinets, Coupling constant dependence of the shear viscosity in N=4 supersymmetric Yang-Mills theory, Nucl. Phys. B 707, 56 (2005)
  * [84] T. Schafer, Phys. Ratio of shear viscosity to entropy density for trapped fermions in the unitarity limit, Rev. A 76, 063618 (2007).
  * [85] A. Dobado and F. J. Llanes-Estrada, On the violation of the holographic viscosity versus entropy KSS bound in non relativistic systems, Eur. Phys. J. C 51, 913 (2007).
  * [86] R. C. Myers, M. F. Paulos and A. Sinha, Holographic Hydrodynamics with a Chemical Potential, JHEP 0906, 006 (2009).
  * [87] J. Liao, V. Koch, Fluidity and supercriticality of the QCD matter created in relativistic heavy ion collisions, Phys. Rev. C 81, 014902 (2010).