\section{Introduction}

High temperature QCD is expected to asymptote a weakly coupled
Coulomb plasma albeit with still strong infrared divergences. The
latters cause its magnetic sector to be non-perturbative at all
temperatures. At intermediate temperatures of relevance to
heavy-ion collider experiments, the electric sector is believed to
be strongly coupled.

Recently, Shuryak and Zahed~\cite{SZ_newqgp} have suggested that
certain aspects of the quak-gluon plasma in range of temperatures
$(1-3)\,T_c$ can be understood by a stronger Coulomb interaction
causing persistent correlations in singlet and colored channels.
As a result the quark and gluon plasma is more a liquid than a gas
at intermediate temperatures. A liquid plasma should exhibit
shorter mean-free paths and stronger color dissipation, both of
which are supported by the current experiments at
RHIC~\cite{hydro}.

To help understand transport and dissipation in the strongly
coupled quark gluon plasma, a classical model of the colored
plasma was suggested in~\cite{gelmanetal}. The model consists of
massive quarks and gluons interacting via classical colored
Coulomb interactions. The color is assumed classical with all
equations of motion following from Poisson brackets. For the SU(2)
version both molecular dynamics simulations~\cite{gelmanetal} and
bulk thermodynamics~\cite{cho&zahed, cho&zahed2} were recently
presented including simulations of the energy loss of heavy
quarks~\cite{dusling&zahed}.

In this paper we extend our recent equilibrium analysis of the
static properties of the colored Coulomb plasma, to transport. In
section 2 we discuss the classical equations of motion in the
SU(2) colored phase space and derive the pertinent Liouville
operator. In section 3,  we show that the resolvent of the
Liouville operator obeys a hierarchy of equations in the SU(2)
phase space. In section 4 we derive an integral equation for the
time-dependent structure factor by introducing a non-local
self-energy  kernel in phase space. In section 5, we close the
Liouville hierarchy through a free streaming approximation on the
4-point resolvent and derive the self-energy kernel in closed form.
In section 6, we project the self-energy kernel and the non-static
structure factor onto the colorless hydrodynamical phase space.
 In section 7, we show that the sound and plasmon mode are the
 leading hydrodynamical modes in the SU(2) colored Coulomb plasma.
 In section we analyze the shear viscosity for the transverse sound mode
 for arbitrary values of $\Gamma$.  We show that a minimum forms at
$\Gamma\approx 5$ at the cross-over between the hydrodynamical
and single-particle regimes. In section 8, we analyze self-diffusion in
phase space, and derive an explicit expression for the diffusion
constant at strong coupling. Our conclusions and prospects are
in section 9.  In appendix A we briefly summarize our variables
in the SU(2) phase space. In appendix B we detail the projection
method for the self-energy kernel used in the text. In appendix C
we show that the collisional color contribution to the Liouville
operator drops in the self-energy kernel. In appendix D some useful
aspects of the hydrodynamical projection method are outlined.

\section{Colored Liouville Operator}

\renewcommand{\theequation}{II.\arabic{equation}}
\setcounter{equation}{0}

The canonical approach to the colored Coulomb plasma was discussed
in~\cite{gelmanetal}. In brief, the Hamiltonian for a single
species of constituent quarks or gluons in the SU(2)
representation is defined as

\begin{equation}
H=\sum_{i}^N\frac{\vec p^2_i}{2m_i}+\sum_{i>j=1}^N\frac{\vec
Q_i\cdot\vec Q_j}{|\vec r_i-\vec r_j|} \label{HAMILTON}
\end{equation}
The charge $g^2/4\pi$ has been omitted for simplicity of the
notation flow and will be reinserted in the pertinent physical
quantities by inspection.

The equations of motion in phase space follows from the classical
Poisson brackets. In particular

\begin{equation}
\frac{d\vec r_i}{dt}=-\{H,\vec r_i\}= \frac{\partial H}{\partial
\vec p_j}\frac{\partial \vec r_i}{\partial \vec r_j}=\frac{\vec
p_i}m \label{XPB}
\end{equation}
The Newtonian equation of motion is just the colored electric Lorentz
force

\begin{equation}
\frac{d\vec p_i}{dt}=-\{H,\vec p_i\}=- \frac{\partial H}{\partial
\vec r_j}\frac{\partial \vec p_i}{\partial \vec p_j}= Q_i^a\,\vec
E_i^a=\vec{F}_i \label{PPB}
\end{equation}
with the colored electric field and potentials defined as
($a=1,2,3$)

\begin{equation}
\vec{E}_i^a=-\vec\nabla_i\Phi^a_i=-\vec\nabla_i\sum_{j\neq
i}\frac{Q^a_j}{|\vec r_i-\vec r_j|} \label{XPB2}
\end{equation}
Our strongly coupled colored plasma is mostly electric following
the original assumptions in~\cite{gelmanetal,gelmanetal2}. The
equation of motion of the color charges is

\begin{equation}
\frac{dQ_i^a}{dt}=-\{H,Q_i^a\}= -\sum_{j,b}\frac{\partial
H}{\partial Q_i^b}\frac{\partial Q_i^a}{\partial Q_j^c} \{Q_j^b,
Q_j^c\}=\sum_{j\neq i}\frac{Q_iT^aQ_j}{|\vec r_i-\vec r_j|}
\label{QPB}
\end{equation}
for arbitrary color representation. For SU(2) the classical color
charge (\ref{QPB}) precesses around the net colored potential
$\Phi$ determined by the other particles as defined in
(\ref{XPB2}),

\begin{equation}
\frac{d\vec Q_i}{dt}=(\vec \Phi_i\times \vec Q_i)
\end{equation}
This equation was initially derived by Wong~\cite{wong}.
Some aspects of the SU(2) phase space are briefly recalled
in Appendix A.

The set (\ref{XPB}), (\ref{PPB}) and (\ref{QPB}) define the
canonical evolution in phase space. The time-dependent phase
distribution is formally given by

\begin{equation}
f(t,\vec r \vec p \vec Q)=\sum_{i=1}^N\delta(\vec r-\vec
r_i(t))\delta(\vec p-\vec p_i(t))\delta(\vec Q-\vec Q_i(t))\equiv
\sum_i\delta(\vec q-\vec q_i(t)) \label{FDIS}
\end{equation}
For simplicity $\vec q$ is generic for $\vec r,\vec p,\vec Q$.
Using the chain rule, the time-evolution operator on (\ref{FDIS})
obeys

\begin{equation}
\frac{d}{dt}=\frac{\partial}{\partial t}+ \frac{d\vec
r_i}{dt}\frac{\partial}{\partial \vec r_i}+ \frac{d\vec
p_i}{dt}\frac{\partial}{\partial \vec p_i}+ \frac{d\vec
Q_i}{dt}\frac{\partial}{\partial \vec Q_i} \equiv
\partial_t+i\mathcal{L} \label{LIOU}
\end{equation}
The last relation defines the Liouville operator

\begin{equation}
\mathcal{L}=\mathcal{L}_0+\mathcal{L}_I+\mathcal{L}_Q=-i\vec
v_i\cdot\nabla_{\vec r_i}-i\,\vec{F}_i\cdot\vec{\nabla}_{\vec p_i}
-i\vec \Phi_i \cdot( \vec Q_i\times\vec \nabla_{\vec Q_i})
\label{LIOU1}
\end{equation}
The last contribution in (\ref{LIOU1}) is genuily a 3-body force
because of the cross product (orbital color operator). It requires
3 distinct colors to not vanish. This observation will be
important in simplifying the color dynamics below.  Also
(\ref{LIOU1}) is hermitean.

Since (\ref{FDIS}) depends implicitly on time, we can write
formally

\begin{equation}
\frac{d}{dt}f(t,\vec r\vec p\vec Q)=i\mathcal{L}f(t,\vec r\vec
p\vec Q) \label{EVOLUTION}
\end{equation}
with a solution $f(t)=e^{i\mathcal{L}t}f(0)$. The formal relation
(\ref{EVOLUTION}) should be considered with care since the action
of the Liouville operator on the 1-body phase space distribution
(\ref{FDIS}) generates also a 2-body phase space distribution.
Indeed, while $\mathcal{L}_0$ is local in phase space

\begin{equation}
\mathcal{L}_0\sum_i\delta(\vec q-\vec q_i)=-i\vec
v\cdot\nabla_{\vec r}\sum_i\delta(\vec q-\vec q_i)=L_0(\vec
q)\sum_i\delta(\vec q-\vec q_i) \label{LO}
\end{equation}
the 2 other contributions are not. Specifically

\begin{eqnarray}
\mathcal{L}_I\sum_m\delta(\vec q-\vec q_m)&=&i\sum_{i\neq
j}\nabla_{\vec r_i} \frac{\vec Q_i\cdot\vec Q_j}{|\vec r_i-\vec
r_j|}
\cdot \nabla_{\vec p_i}\sum_m \delta(\vec q-\vec q_m) \nonumber \\
&=& i\int d\vec q'   \sum_{i\neq j , mn}\nabla_{\vec
r_i}\frac{\vec Q_i\cdot\vec Q_j}{|\vec r_i-\vec
r_j|}\cdot\nabla_{\vec p_i}\,\,\delta(\vec
q-\vec q_m) \delta(\vec q'-\vec q_n) \nonumber \\
&=&- \int d\vec q'L_I(\vec q,\vec q')\,\sum_{mn}\delta(\vec q-\vec
q_m)\delta(\vec q'-\vec q_n)
\label{L1}
\end{eqnarray}
with

\begin{equation}
L_I(\vec q,\vec q')=i\nabla_{\vec r} \frac{\vec Q\cdot\vec
Q'}{|\vec r-\vec r'|} \cdot (\nabla_{\vec p}- \nabla_{\vec p'})
\end{equation}
Similarly

\begin{eqnarray}
\mathcal{L}_Q\sum_m\delta(\vec q-\vec q_m)&=&-i\sum_{ j\neq i, m}
\frac{\vec Q_i\times\vec Q_j}{|\vec r_i-\vec r_j|}\cdot
\nabla_{\vec Q_i} \,\delta(\vec q-\vec q_m) \nonumber \\
&=& -i\int d\vec q'\sum_{j\neq i, mn}\frac{\vec Q_i\times \vec
Q_j}{|\vec r_i-\vec r_j|}\cdot \nabla_{\vec Q_i}\delta(\vec
q-\vec q_m) \delta(\vec q'-\vec q_n) \nonumber \\
&=&- \int d\vec q'L_Q(\vec q,\vec q') \sum_{mn}\delta(\vec q-\vec
q_m)\delta(\vec q'-\vec q_n) \label{LQ}
\end{eqnarray}
with

\begin{equation}
L_Q(\vec q,\vec q')=-i \frac{\vec Q\times\vec Q'}{|\vec r-\vec
r'|} \cdot (\nabla_{\vec Q}-\nabla_{\vec Q'})
\end{equation}
Clearly (\ref{LQ}) drops from 2-body and symmetric phase space
distributions. It does not for 3-body and higher.

\section{Liouville Hierarchy}

\renewcommand{\theequation}{III.\arabic{equation}}
\setcounter{equation}{0}

An important correlation function in the analysis of the colored
Coulomb plasma is the time dependent structure factor or 2-body
correlation in the color phase space

\begin{equation}
{\bf S}(t-t',\vec r-\vec r', \vec p\vec p', \vec Q\cdot \vec
Q')=\langle\delta f(t,\vec r\vec p\vec Q)\,\delta f(t',\vec r'\vec
p'\vec Q')\rangle \label{FF}
\end{equation}
with $\delta f=f-\langle f\rangle$ the shifted 1-body phase space
distribution.  The averaging in (\ref{FF}) is carried over the
initial conditions with fixed number of particles $N$ and average
energy or temperature $\beta=1/T$. Thus $\langle f\rangle=nf_0(p)$
which is the Maxwellian distribution for constituent quarks or
gluons. In equilibrium, the averaging in (\ref{FF}) is time and
space  translational invariant as well as color rotational
invariant.

Using the ket notation with $\vec 1\equiv \vec q\equiv {\vec
r}{\vec p}{\vec Q}$

\begin{equation}
|\delta f(t,\vec 1)\rangle=\bigg|\sum_{m}\delta(\vec q-\vec
q_m(t))-\langle \sum_{m}\delta(\vec q-\vec q_m(t))\bigg\rangle
\equiv |\delta f(t,\vec 1)-\langle f(t,\vec 1)\rangle>
\end{equation}
with also $\vec 2=\vec q'$, $\vec 3=\vec q''$, $\vec 4=\vec q'''$
and so on and the formal Liouville solution $\delta f(t,\vec
1)=e^{i\mathcal{L}t}\delta f(\vec 1)$ we can write (\ref{FF}) as

\begin{equation}
{\bf S}(t-t',\vec q,\vec q')=\langle\delta f(t,\vec 1)|\delta
f(t',\vec 2)\rangle= \langle\delta f(\vec 1)|e^{i{\cal
L}(t'-t)}|\delta f(\vec 2)\rangle
\end{equation}
The bra-ket notation is short for the initial or equilibrium
average. Its Laplace or causal transform reads

\begin{equation}
{\bf S}(z, \vec q, \vec q')=-i\int_{-\infty}^{+\infty} \,dt
\,\,\theta(t-t')\,\, e^{izt} \,\,{\bf S}(t-t',\vec q,\vec q')=
\langle\delta f(\vec 1)|\frac 1{z+\mathcal{L}}|\delta f(\vec
2)\rangle \label{RES}
\end{equation}
with $z=\omega+i0$. Clearly

\begin{equation}
z{\bf S}(z, \vec q, \vec q')+\langle\delta f(\vec
1)|\mathcal{L}\frac 1{z+\mathcal{L} }|\delta f(\vec
2)\rangle=\langle\delta f(\vec 1)|\delta f(\vec 2)\rangle
\label{RES1}
\end{equation}
Since ${\cal L }^\dagger={\cal L}$ is hermitian and using
(\ref{LO}), (\ref{L1}) and (\ref{LQ}) it follows that

\begin{equation}
\langle\delta f(\vec 1)|{\cal L}=\langle\delta f(\vec 1)|{
L}_0(\vec q)- \int \,d\vec q"\,L_{I+Q}(\vec q,\vec
q'')\,\langle\delta f(\vec 1)\delta f(\vec 3)|
\end{equation}
Thus

\begin{equation}
\Big(z-L_0(\vec q)\Big){\bf S}(z, \vec q,\vec q')-\int\,d\vec
q''L_{I+Q}(\vec q',\vec q''){\bf S}(z,\vec q\vec q'',\vec q')={\bf
S}_0(\vec q,\vec q') \label{RES2}
\end{equation}
where we have defined the 3-body phase space resolvent

\begin{equation}
{\bf S}(z, \vec q\vec q'', \vec q')=\,\langle\delta f(\vec
1)\delta f(\vec 3)|\frac 1{z+{\cal L}}\,|\delta f(\vec 2)>
\label{SS3}
\end{equation}
${\bf S}_0(\vec q,\vec q')$ is the static colored structure factor
discussed by us in~\cite{cho&zahed3}. Since $L_{I+Q}(\vec q',\vec
q)$ is odd under the switch $\vec q\leftrightarrow \vec q'$, and
since ${\bf S}(z, \vec q\vec q'',\vec q')={\bf S}(-z, \vec q,\vec
q'\vec q'')$ owing to the $t\leftrightarrow t'$ in (\ref{RES}),
then

\begin{equation}
\Big(z+L_0(\vec q')\Big){\bf S}(z, \vec q,\vec q')-\int\,d\vec
q''L_{I+Q}(\vec q',\vec q''){\bf S}(z,\vec q, \vec q' \vec
q'')={\bf S}_0(\vec q,\vec q') \label{RES2X}
\end{equation}
(\ref{RES2}) or equivalently (\ref{RES2X}) define the Liouville
hierarchy, whereby the 2-body phase space distribution ties to the
3-body phase space distribution and so on.  Indeed,  (\ref{RES2X})
for instance implies

\begin{equation}
\Big(z+L_0(\vec q'')\Big){\bf S}(z, \vec q\vec q', \vec
q'')-\int\,d\vec q'''L_{I+Q}(\vec q'',\vec q'''){\bf S}(z,\vec q
\vec q', \vec q''\vec q''')={\bf S}_0(\vec q\vec q',\vec q'')
\label{RES2XX}
\end{equation}
with the 4-point resolvent function

\begin{equation}
{\bf S}(z,\vec q \vec q', \vec q''\vec q''')= \,\langle\delta
f(\vec 1)\delta f(\vec 2)|\frac 1{z+{\cal L}}\,|\delta f(\vec
3)\delta f(\vec 4)> \label{SS3X}
\end{equation}
These are the microscopic kinetic equations for the color phase
space distributions.  They are only useful when closed, that is by
a truncation as we discuss below. These formal equations where
initially discussed
in~\cite{foster&martin,foster,mazenko1,mazenko3} in the context of
the one component Coulomb Abelian Coulomb plasma. We have now
generalized them to the multi-component and non-Abelian colored
Coulomb plasma.

\section{Self-Energy Kernel}

\renewcommand{\theequation}{IV.\arabic{equation}}
\setcounter{equation}{0}

In (\ref{RES2}) the non-local part of the Liouville operator plays
the role of a non-local self-energy kernel $\Sigma$ on the 2-body
resolvent.  Indeed, we can rewrite (\ref{RES2}) as

\begin{eqnarray}
(z-L_0(\vec q)){\bf S}(z,\vec q,\vec q')-\int dq''\Sigma (z,\vec
q,\vec q'') {\bf S}(z,\vec q'',\vec q')={\bf S}_0(\vec q,\vec q')
\label{SELF}
\end{eqnarray}
with the non-local self-energy kernel defined formally as

\begin{equation}
\int d\vec q''\Sigma(z,\vec q,\vec q'')\,{\bf S}(z,\vec q'',\vec
q')=\int d\vec q'' L_{I+Q}(\vec q,\vec q'')\,{\bf S}(z,\vec q\vec
q'',\vec q') \label{SELF1}
\end{equation}
The self-energy kernel $\Sigma$ can be regarded as the sum of a
static or z-independent contribution $\Sigma_S$ ans a non-static
or collisional contribution $\Sigma_C$,

\begin{equation}
\Sigma(z, \vec q, \vec q'')=\Sigma_S(\vec q,\vec
q'')+\Sigma_C(z,\vec q,\vec q'') \label{SUM}
\end{equation}
The stationary part $\Sigma_S$ satisfies

\begin{equation}
\int d\vec q'' \Sigma_S(\vec q,\vec q'')\,{\bf S}_0(\vec q'',\vec
q')= \int d\vec q''L_{I+Q}(\vec q,\vec q'')\,{\bf S}_0(\vec q,\vec
q',\vec q'') \label{SELF2}
\end{equation}
which identifies it with the sum of the 2- and 3-body part of the
Liouville operator $L_{I+Q}$.

The collisional part $\Sigma_C$ is more involved. To unwind it, we
operate with $(z+L_0(\vec q'))$ on both sides of (\ref{SELF1}),
and then reduce the left hand side contribution using
(\ref{RES2X}) and the right hand side contribution using
(\ref{RES2XX}).  The outcome reduces to

\begin{eqnarray}
\Sigma_C(z, \vec q, \vec q'')\,{\bf S}_0(\vec q'',\vec q')=&&
-\int d\vec q'''\,L_{I+Q}(\vec q,\vec q'')L_{I+Q}(\vec q',\vec
q'')
\,{\bf S}(z,\vec q\vec q'',\vec q'\vec q''')\nonumber\\
&&+\int\,dq'''\,\Sigma(z,\vec q,\vec q'')\,L_{I+Q}(\vec q',\vec
q'')\,{\bf S}(z, \vec q'', \vec q'\vec q''') \label{INT}
\end{eqnarray}
after using (\ref{SELF2}).  From (\ref{SELF1}) it follows formally
that

\begin{equation}
\Sigma(z,\vec q,\vec q'')=\int\,d\vec q'''\, L_{I+Q}(\vec q,\vec
q''')\,{\bf S}^{-1}(z,\vec q',\vec q'')\,{\bf S}(z, \vec q\vec
q''',\vec q') \label{FORMX}
\end{equation}
Inserting (\ref{FORMX}) into the right hand side of (\ref{INT})
and taking the $\vec q'$ integration on both sides yield

\begin{equation}
nf_0(p'')\,\Sigma_C(z,\vec q,\vec q'') =-\int d\vec q'' d\vec
q'''\,L_{I+Q}(\vec q,\vec q'')L_{I+Q} (\vec q',\vec q''')\,{\bf
G}(z,\vec q\vec q'',\vec q'\vec q''') \label{SELF5}
\end{equation}
with ${\bf G}$ a 4-point phase space correlation function

\begin{equation}
{\bf G}(z,\vec q\vec q_1',\vec q'\vec q_2')={\bf S}(z,\vec q\vec
q_1',\vec q'\vec q_2')-\int d\vec q_3d\vec q_4 {\bf S}(z, \vec
q\vec q_1',\vec q_3){\bf S}^{-1}(z,\vec q_3,\vec q_4){\bf
S}(z,\vec q_4,\vec q'\vec q_2') \label{G4}
\end{equation}
The collisional character of the self-energy $\Sigma_C$ is
manifest in (\ref{SELF5}).  The formal relation for the
collisional self-energy (\ref{SELF5}) was initially derived
in~\cite{mazenko1,mazenko3} for the one-component  and Abelian
Coulomb plasma. We now have shown that it holds for any
non-Abelian SU(N) Coulomb plasma.

Eq. (\ref{SELF5}) shows that the connected part of the self-energy
kernel is actually tied to a 4-point correlator in the colored
phase space. In terms of (\ref{SELF5}), the original kinetic
equation (\ref{RES2}) now reads

\begin{eqnarray}
&&\Big(z-L_0(\vec q)\Big){\bf S}(z, \vec q, \vec q')-\int\,d\vec
q''\Sigma_S(\vec q,\vec q''){\bf S}(z,\vec q'',\vec q')= {\bf
S}_0(\vec q,\vec q') \nonumber\\&& -\int\,d\vec q''\,d\vec q_1
d\vec q_2\,L_{I+Q}(\vec q,\vec q_1)L_{I+Q} (\vec q'',\vec
q_2)\,{\bf G}(z,\vec q\vec q_1,\vec q''\vec q_2)\,{\bf S}(z,\vec
q'',\vec q') \label{BOLZ}
\end{eqnarray}
which is a Boltzman-like equation. The key difference is that it
involves correlation functions and the Boltzman-like kernel in the
right-hand side is {\bf not} a scattering amplitude but rather a
reduced 4-point correlation function.  (\ref{BOLZ}) reduces to the
Boltzman equation for weak coupling. An alternative derivation of
(\ref{BOLZ}) can be found in Appendix C through a direct
projection of (\ref{SELF1}) in phase space.

\section{Free Streaming Approximation}

\renewcommand{\theequation}{V.\arabic{equation}}
\setcounter{equation}{0}

The formal kinetic equation (\ref{SELF5})  can be closed by
approximating the 4-point correlation function in the color phase
space by a product of 2-point correlation
function~\cite{mazenko3},

\begin{equation}
{\bf G}(t, \vec q\vec q_1, \vec q'\vec q_2)\approx \Big( {\bf
S}(t, \vec q,\vec q'){\bf S}(t, \vec q_1,\vec q_2)+{\bf S}(t, \vec
q, \vec q_2){\bf S}(t, \vec q', \vec q_1) \Big)\label{ST3}
\end{equation}
This reduction will be referred to as the free steaming
approximation. Next we substitue the colored Coulomb potentials in
the double Liouville operator $L_{1+Q}\times L_{1+Q}$ with a bare
Coulomb ${\bf V}(\vec r-\vec r',\vec Q\cdot\vec Q')=\vec
Q\cdot\vec Q'/|\vec r-\vec r'|$.

\begin{eqnarray}
& & L_{I+Q}(\vec q,\vec q_1)=i\nabla_{\vec r}{\bf V}(\vec r-\vec
r_1,\vec Q\cdot\vec
Q_1)\cdot (\nabla_{\vec p}-\nabla_{\vec p_1}) \nonumber \\
& &-i \Big(\vec Q\times\nabla_{\vec Q}{\bf V}(\vec r-\vec r_1,\vec
Q\cdot \vec Q_1)\cdot \nabla_{\vec Q}+\vec Q_1\times \nabla_{\vec
Q_1}{\bf V}(\vec r-\vec r_1,\vec Q\cdot \vec
Q_1)\cdot\nabla_{Q_1}\Big) \label{ST4}
\end{eqnarray}
times a dressed colored Coulomb potential ${\bf c}_D$ defined
in~\cite{cho&zahed3}

\begin{eqnarray}
& & L_{I+Q}^R(\vec q,\vec q_1)=-i\frac{1}{\beta}\nabla_{\vec
r}{\bf c}_D(\vec r-\vec r_1,\vec Q\cdot \vec Q_1)\cdot
(\nabla_{\vec p}-\nabla_{\vec p_1}) \nonumber \\
& & +i\frac{1}{\beta}\Big(\vec Q\times\nabla_{\vec Q}{\bf
c}_D(\vec r-\vec r_1,\vec Q\cdot \vec Q_1)\cdot \nabla_{\vec
Q}+\vec Q_1\times \nabla_{\vec Q_1}{\bf c}_D(\vec r-\vec r_1,\vec
Q\cdot \vec Q_1)\cdot\nabla_{Q_1}\Big) \label{ST5}
\end{eqnarray}
This bare-dressed or half renormalization was initially
suggested~\cite{wallenborn&baus} in the context of the
one-component Coulomb plasma to overcome the shortcomings of a
full or dressed-dressed renormalization initially suggested
in~\cite{mazenko1,mazenko3}. The latter was shown to upset the
initial conditions. Thus

\begin{equation}
L_{I+Q}(\vec q,\vec q_1)L_{I+Q}(\vec q',\vec q_2)\rightarrow
\frac{1}{2}\Big( L_{I+Q}(\vec q,\vec q_1)L_{I+Q}^R(\vec q',\vec
q_2)+L_{I+Q}^R(\vec q,\vec q_1)L_{I+Q}(\vec q',\vec q_2)\Big)
\label{eq004s}
\end{equation}
Combining (\ref{ST3}) and (\ref{eq004s}) in (\ref{SELF5}) yields

\begin{eqnarray}
&& n\,f_0(\vec p')\,\Sigma_C(t, \vec q, \vec q')\approx
-\frac{1}{2}\int d\vec q_1\,d\vec q_2\,\bigg(L_{I+Q}(\vec q,\vec
q_1)L_{I+Q}^R (\vec q',\vec q_2){\bf S}(t, \vec q,\vec q'){\bf
S}(t,\vec q_1,\vec q_2) \nonumber\\
&& +L_{I+Q}(\vec q,\vec q_1)L_{I+Q}^R (\vec q',\vec q_2){\bf S}(t,
\vec q,\vec q_2){\bf S}(t,\vec q', \vec q_1) + (\vec
q_1\leftrightarrow \vec q_2,\vec q\leftrightarrow \vec q' ) \bigg)
\label{ST6}
\end{eqnarray}

\noindent This is the half dressed but free streaming
approximation for the connected part of the self-energy for the
colored Coulomb plasma. Translational invariance in space and
rotational invariance in color space allows a further reduction of
(\ref{ST6}) by Fourier and Legendre transforms respectively.
Indeed, Eq. (\ref{ST6}) yields

\begin{eqnarray}
&&n\,f_0(\vec p')\,\Sigma_C(t, \vec q, \vec
q')\nonumber\\
&&\approx -\frac{1}{2} \int d\vec q_1\,d\vec q_2\,\bigg(L_{I}(\vec
q,\vec q_1)L_{I}^R (\vec q',\vec q_2){\bf S}(t, \vec q,\vec
q'){\bf S}(t,\vec q_1,\vec q_2) \nonumber\\
&& +L_{I}(\vec q,\vec q_1)L_{I}^R (\vec q',\vec q_2){\bf S}(t,
\vec q,\vec q_2){\bf S}(t,\vec q', \vec q_1) + (\vec
q_1\leftrightarrow \vec q_2,\vec q\leftrightarrow \vec q') \bigg) \nonumber \\
&&=-\frac{1}{2\beta} \int d\vec q_1\,d\vec q_2\,\bigg(
\nabla_{\vec r}{\bf c}_D(\vec r-\vec r_1,\vec Q\cdot \vec
Q_1)\cdot \nabla_{\vec p}\nabla_{\vec r'}{\bf V}(\vec r'-\vec
r_2,\vec Q'\cdot \vec Q_2)\cdot \nabla_{\vec
p'}{\bf S}(t, \vec q,\vec q'){\bf S}(t,\vec q_1,\vec q_2) \nonumber\\
&& +\nabla_{\vec r}{\bf c}_D(\vec r-\vec r_1,\vec Q\cdot \vec
Q_1)\cdot \nabla_{\vec p}\nabla_{\vec r'}{\bf V}(\vec r'-\vec
r_2,\vec Q'\cdot \vec Q_2)\cdot \nabla_{\vec p'}{\bf S}(t, \vec
q,\vec q_2){\bf S}(t,\vec q', \vec q_1) + (\vec q_1\leftrightarrow
\vec q_2,\vec q\leftrightarrow \vec q') \bigg) \nonumber \\
\label{eq006s}
\end{eqnarray}
where we note that the colored part of the Liouville operator
dropped from the collision kernel in the free streaming
approximation as we detail in Appendix C. Both sides of
(\ref{eq006s}) can be now Legendre transformed in color to give

\begin{eqnarray}
&& n\,f_0(\vec p')\,\sum_l\Sigma_{Cl}(t, \vec r\vec r', \vec p\vec
p')\frac{2l+1}{4\pi}P_l(\vec Q\cdot\vec Q') \nonumber \\
& & \approx -\frac 1{2\beta}\int d\vec r_1 d\vec p_1 d\vec
r_2d\vec p_2 \sum_{l}\frac{2l+1}{4\pi}\bigg(\frac{l+1}
{2l+1}P_{l+1}(\vec Q\cdot\vec Q')+\frac{l}{2l+1}P_{l-1}
(\vec Q\cdot\vec Q')\bigg) \nonumber \\
& & \times \bigg( \nabla_{\vec r}{\bf c}_{D1}(\vec r-\vec
r_1)\cdot \nabla_{\vec p}\nabla_{\vec r'} \frac{1}{|\vec r'-\vec
r_2|}\cdot \nabla_{\vec p'}{\bf S}_l(t, \vec r\vec r',\vec p\vec
p'){\bf S}_1(t, \vec r_1\vec r_2,\vec p_1\vec p_2) \nonumber \\
& & +\nabla_{\vec r}{\bf c}_{Dl}(\vec r-\vec r_1)\cdot
\nabla_{\vec p}\nabla_{\vec r'} \frac{1}{|\vec r'-\vec r_2|}\cdot
\nabla_{\vec p'}{\bf S}_1(t, \vec r\vec r_2,\vec p\vec p_2){\bf
S}_l(t, \vec r'\vec r_1,\vec p'\vec p_1) \nonumber \\
& & +\nabla_{\vec r'}{\bf c}_{Dl}(\vec r'-\vec r_2)\cdot
\nabla_{\vec p'}\nabla_{\vec r} \frac{1}{|\vec r-\vec r_1|}\cdot
\nabla_{\vec p}{\bf S}_l(t, \vec r\vec r_2,\vec p\vec p_2){\bf
S}_1(t, \vec r'\vec r_1,\vec p'\vec p_1) \nonumber \\
& & \nabla_{\vec r'}{\bf c}_{D1}(\vec r'-\vec r_2)\cdot
\nabla_{\vec p'}\nabla_{\vec r} \frac{1}{|\vec r-\vec r_1|}\cdot
\nabla_{\vec p}{\bf S}_l(t, \vec r\vec r',\vec p\vec p'){\bf
S}_1(t, \vec r_1\vec r_2,\vec p_1\vec p_2) \bigg) \label{eq007s}
\end{eqnarray}
Thus

\begin{eqnarray}
&& n\,f_0(\vec p')\,\Sigma_{Cl}(t, \vec r\vec r', \vec p\vec
p') \nonumber \\
& & \approx -\frac 1{2\beta}\int d\vec r_1 d\vec p_1 d\vec
r_2d\vec p_2 \bigg( \nabla_{\vec r}{\bf c}_{D1}(\vec r-\vec
r_1)\cdot \nabla_{\vec p}\nabla_{\vec r'} \frac{1}{|\vec
r'-\vec r_2|}\cdot \nabla_{\vec p'} \nonumber \\
& & \times \Big(\frac{l}{2l+1}{\bf S}_{l-1}(t, \vec r\vec r',\vec
p\vec p'){\bf S}_1(t, \vec r_1\vec r_2,\vec p_1\vec
p_2)+\frac{l+1}{2l+1}{\bf S}_{l+1}(t, \vec r\vec r',\vec p\vec
p'){\bf S}_1(t, \vec r_1\vec r_2,\vec p_1\vec
p_2)\Big) \nonumber \\
& & +\nabla_{\vec r}{\bf c}_{D1}(\vec r-\vec r_1)\cdot
\nabla_{\vec p}\nabla_{\vec r'} \frac{1}{|\vec r'-\vec r_2|}\cdot
\nabla_{\vec p'} \nonumber \\
& & \times \Big(\frac{l}{2l+1}{\bf S}_{1}(t, \vec r\vec r_2,\vec
p\vec p_2){\bf S}_{l-1}(t, \vec r'\vec r_1,\vec p'\vec p_1)+\frac{
l+1}{2l+1}{\bf S}_{1}(t, \vec r\vec r_2,\vec p\vec p_2){\bf
S}_{l+1}(t, \vec r'\vec r_1,\vec p'\vec p_1)\Big) \nonumber \\
& & +\nabla_{\vec r'}{\bf c}_{Dl}(\vec r'-\vec r_2)\cdot
\nabla_{\vec p'}\nabla_{\vec r} \frac{1}{|\vec r-\vec r_1|}\cdot
\nabla_{\vec p} \nonumber \\
& & \times \Big(\frac{l}{2l+1}{\bf S}_{l-1}(t, \vec r\vec r_2,\vec
p\vec p_2){\bf S}_1(t, \vec r'\vec r_1,\vec p'\vec p_1)+\frac{l+1}
{2l+1}{\bf S}_{l+1}(t, \vec r\vec r_2,\vec p\vec p_2){\bf
S}_1(t, \vec r'\vec r_1,\vec p'\vec p_1) \Big)  \nonumber \\
& & +\nabla_{\vec r'}{\bf c}_{D1}(\vec r'-\vec r_2)\cdot
\nabla_{\vec p'}\nabla_{\vec r} \frac{1}{|\vec r-\vec r_1|}\cdot
\nabla_{\vec p} \nonumber \\
& & \times \Big(\frac{l}{2l+1}{\bf S}_{l-1}(t, \vec r\vec r',\vec
p\vec p'){\bf S}_1(t, \vec r_1\vec r_2,\vec p_1\vec
p_2)+\frac{l+1}{2l+1}{\bf S}_{l+1}(t, \vec r\vec r',\vec p\vec
p'){\bf S}_1(t, \vec r_1\vec r_2,\vec p_1\vec p_2)\Big)  \bigg)
\nonumber \\ \label{eq008s}
\end{eqnarray}
with ${\bf S}_{l-1}\equiv0$ by definition. In the colored Coulomb
plasma the collisional contributions diagonalize in the color
projected channels labelled by $l$, with $l=0$ being the density
channel, $l=1$ the plasmon channel and so on.  In momentum space
(\ref{eq008s}) reads

\begin{eqnarray}
&& n\,f_0(\vec p')\, \Sigma_{Cl}(t, \vec k, \vec p\vec p')
\nonumber \\
& & = -\frac 1{2\beta} \int d\vec p_1 d\vec p_2 \int \frac{d\vec
l}{(2\pi)^3}\bigg( \vec l\cdot\nabla_{\vec p}\vec
l\cdot\nabla_{\vec p'} {\bf c}_{D1}(l) V_{\vec l} \nonumber
\\ & & \times \Big( \frac{l}{2l+1} {\bf S}_{l-1}(t, \vec k-\vec
l,\vec p\vec p'){\bf S}_1(t, \vec l,\vec p_1\vec
p_2)+\frac{l+1}{2l+1} {\bf S}_{l+1}(t, \vec k-\vec l,\vec p\vec
p'){\bf S}_1(t, \vec l,\vec p_1\vec p_2)\Big) \nonumber \\
& & + \vec l\cdot\nabla_{\vec p}(\vec k-\vec l)\cdot\nabla_{\vec
p'} {\bf c}_{Dl}(l) V_{\vec k-\vec l} \nonumber
\\ & & \times \Big( \frac{l}{2l+1} {\bf S}_{1}(t, \vec k-\vec
l,\vec p\vec p_2){\bf S}_{l-1}(t, \vec l,\vec p'\vec
p_1)+\frac{l+1}{2l+1} {\bf S}_{1}(t, \vec k-\vec l,\vec p\vec
p_2){\bf S}_{1+1}(t, \vec l,\vec p'\vec p_1)\Big) \nonumber \\
& & + (\vec k-\vec l)\cdot\nabla_{\vec p}\vec l\cdot\nabla_{\vec
p'} {\bf c}_{Dl}(l) V_{\vec k-\vec l} \nonumber
\\ & & \times \Big( \frac{l}{2l+1} {\bf S}_{l-1}(t, \vec
l,\vec p\vec p_2){\bf S}_{l}(t, \vec k-\vec l,\vec p'\vec
p_1)+\frac{l+1}{2l+1} {\bf S}_{l+1}(t, \vec l,\vec p\vec
p_2){\bf S}_{1}(t, \vec k-\vec l,\vec p'\vec p_1)\Big) \nonumber \\
& & +\vec l\cdot\nabla_{\vec p}\vec l\cdot\nabla_{\vec p'} {\bf
c}_{D1}(l) V_{\vec l} \nonumber \\
& & \times \Big( \frac{l}{2l+1} {\bf S}_{l-1}(t, \vec k-\vec
l,\vec p\vec p'){\bf S}_1(t, \vec l,\vec p_1\vec
p_2)+\frac{l+1}{2l+1} {\bf S}_{l+1}(t, \vec k-\vec l,\vec p\vec
p'){\bf S}_1(t, \vec l,\vec p_1\vec p_2)\Big) \bigg) \nonumber \\
\label{SFOURIER}
\end{eqnarray}
with $V_{\vec l}=4\pi/{\vec l}^2$. We note that for $l=0$ which is
the colorless density channel (\ref{SFOURIER}) involves only ${\bf
S}_1$ which is the time-dependent charged form factor due to the
Coulomb interactions.

\section{Hydrodynamical Projection}

\renewcommand{\theequation}{VI.\arabic{equation}}
\setcounter{equation}{0}

In terms of (\ref{SFOURIER}) , (\ref{SELF1}) and

\begin{equation}
\Sigma_l (z\vec k, \vec p\vec p_1)=\bigg(
\Sigma_{0l}+\Sigma_{Il}+\Sigma_{Ql}+\Sigma_{Cl}\bigg)(z\vec k,
\vec p\vec p_1) \label{KEY}
\end{equation}
the Fourier and Legendre transform of the kinetic equation
(\ref{RES2})  now read

\begin{equation}
z{\bf S}_l(z\vec k, \vec p\vec p')-\int\,d\vec p_1\Sigma_l(z\vec
k, \vec p\vec p_1){\bf S}_l(z\vec k, \vec p_1\vec p')={\bf
S}_{0l}(\vec k, \vec p\vec p') \label{K1}
\end{equation}
with $\Sigma_{0l}=L_0$ and $\Sigma_{Sl}=L_{(I+Q)l}$. Specifically

\begin{eqnarray}
&& \Sigma_{0l}(z\vec k,\vec p\vec p_1)=\vec k\cdot\vec v \delta(\vec p -\vec p_1)\nonumber\\
&& \Sigma_{Il}(z\vec k,\vec p\vec p_1)=-n\,f_0(p)\frac{\vec k\cdot\vec p}{m}\,{\bf c}_{Dl}(\vec k)\nonumber\\
&& \Sigma_{Ql}(z\vec k,\vec p\vec p_1)=0 \label{K2}
\end{eqnarray}
and $\Sigma_{Cl}$ is defined in (\ref{SFOURIER}). See also
Appendix B for an alternative but equivalent derivation using the
operator projection method.

(\ref{K1}) is the key kinetic equation for the colored Coulomb
plasma. It still contains considerable information in phase space.
A special limit of the classical phase space is the long
wavelength or hydrodynamical limit. In this limit, only few
moments of the phase space fluctuations $\delta f$ or equivalently
their correlations in ${\bf S}\approx \langle\delta f\delta
f\rangle$ will be of interest. In particular,

\begin{eqnarray}
&& \vec{n}(t, \vec r)=\int d\vec{p} d\vec Q\,
\, \delta f (t, \vec r, \vec p, \vec Q)\nonumber\\
&& \vec{p}(t, \vec r)=\int d\vec{p} d\vec Q\, \vec p
\, \delta f (t, \vec r, \vec p, \vec Q)\nonumber\\
&& {\bf e}(t, \vec r)=\int d\vec{p} d\vec Q \,\frac {p^2}{2m}\,
\delta f (t, \vec r, \vec p, \vec Q) \label{MOM}
\end{eqnarray}
The local particle density, 3-momentum and energy (kinetic).  The
hydrodynamical sector described by the macro-variables (\ref{MOM})
is colorless. An interesting macro-variable which carries charge
representation of SU(2) would be

\begin{equation}
{\bf n}_l (t, \vec r)=\frac 1{2l+1} \sum_m \int d\vec{r} d\vec Q
\,Y_l^m (\vec Q)\, \delta f (t, \vec r, \vec p, \vec Q)
\label{DENSL}
\end{equation}
which reduces to the $l$ color density with $l=0$ being the
particle density, $l=1$ the charged color monopole density,  $l=2$
the charged color quadrupole density and so on.  Because of color
rotational invariance in the SU(2) colored Coulomb plasma, the
constitutive equations for (\ref{DENSL}) which amount to charge
conservation hold for each $l$.

To project (\ref{K1}) onto the hydrodynamical part of the phase
space characterized by (\ref{DENSL}) and (\ref{MOM}), we define
the hydrodynamical projectors

\begin{equation}
{\cal P}_H=\sum_{i=1}^5|i\rangle\langle i|\qquad\qquad {\cal
Q}_H={\bf 1}_H-{\cal P}_H \label{PRO}
\end{equation}
with $1=$ l-density, $2,4,5=$ momentum and $3=$ energy as detailed
in Appendix D. When the $l=0$ particle density is retained in
(\ref{PRO}) the projection is on the colorless sector of the phase
space. When the $l=1$ charged monopole density is retained in
(\ref{PRO}) the projection is on the plasmon channel, and so on.
Most of the discussion to follow will focus on projecting on the
canonical hydrodynamical phase space (\ref{MOM}) with $l=0$ or
singlet representation. The inclusion of the $l\neq 0$
representations of SU(2)  is straightforward.

Formally (\ref{KEY}) can be viewed as a $\vec p\times \vec p_1$
matrix in momentum space

\begin{equation}
\left( z-\Sigma_l(z\vec k)\right)\,{\bf S}_l(z\vec k)={\bf
S}_{0l}(\vec k) \label{KEY1}
\end{equation}
The projection of the matrix equation (\ref{KEY1}) follows the
same procedure as in Appendix B. The result is

\begin{equation}
\left(z-{\cal P}_H\Sigma_l (z\vec k){\cal P}_H-{\cal
P}_H\Theta_l(z\vec k){\cal P}_H\right) {\cal P}_H{\bf S}_l(z\vec
k){\cal P}_H={\cal P}_H{\bf S}_{0l}(k){\cal P}_H \label{KEY2}
\end{equation}
with

\begin{equation}
\Theta_l=\Sigma_l(z\vec k) {\cal Q}_H(z-{\cal Q}_H\Sigma_H(z\vec
k){\cal Q}_H)^{-1} {\cal Q}_H\Sigma_l(z\vec k)
\end{equation}
If we define the hydrodynamical matrix elements

\begin{eqnarray}
&&{\bf G}_{lij}(z\vec k)=\langle i|{\bf S}_l(z\vec k)|j\rangle\nonumber\\
&&{\Sigma}_{lij}(z\vec k)=\langle i|{\Sigma }_l(z\vec k)|j\rangle\nonumber\\
&&{\Theta}_{lij}(z\vec k)=\langle i|{\Theta}_l(z\vec k)|j\rangle\nonumber\\
&&{\bf G}_{0lij}(z\vec k)=\langle i|{\bf S}_{0l}(\vec k)|j\rangle
\label{MAT}
\end{eqnarray}
then (\ref{KEY2}) reads

\begin{equation}
\left(z\delta_{ii'}-\Omega_{lij}(z\vec k)\right)\,{\bf
G}_{lji'}(z\vec k)={\bf G}_{0lii'}(\vec k) \label{DIS}
\end{equation}
with $\Omega_l=\Sigma_l+\Theta_l$. (\ref{DIS}) takes the form of a
dispersion for each color partial wave $l$ with the projection
operator (\ref{PRO}) set by the pertinent density (\ref{DENSL}).
The contribution $\Sigma_l$ to $\Omega_l$ will be referred to as
{\it direct} while the contribution $\Theta_l$ will be referred to
as {\it indirect}.

\section{Hydrodynamical Modes}

\renewcommand{\theequation}{VII.\arabic{equation}}
\setcounter{equation}{0}

The zeros of (\ref{DIS}) are the hydrodynamical modes originating
from the Liouville equation for the time-dependent structure factor. The
equation is closed under the free streaming approximation with
half renormalized vertices as we detailed above.

We start by analyzing the 2 transverse modes with $i=T$ in
(\ref{MAT}) and (\ref{DIS}). We note with~\cite{baus} that ${\bf
G}_{lTi}=0$ whenever $T\neq i$. The hydrodynamical projection (see
Appendix D) causes the integrand to be odd whatever $l$. The 2
independent transverse modes in (\ref{DIS}) decouple from the
longitudinal $i=L$, the (kinetic) energy $i=E$ and particle
density $i=N$ modes for all color projections. Thus

\begin{equation}
{\bf G}_{lT}(z\vec k)=\frac 1{z-\Omega_{lT}(z\vec k)} \label{TDIS}
\end{equation}
with $\Omega_{lT}=\langle T|\Omega_l|T\rangle$ and ${\bf
G}_{lT}=\langle T|{\bf G}_l|T\rangle$. The hydro-projected
time-dependent $l$ structure factor for fixed frequency
$z=\omega+i0$, wavenumber $k$ develops 2 transverse poles

\begin{equation}
z_l(\vec k)=\Omega_{lT}(z\vec k)\approx {\cal O}(k^2)
\label{TPOLE}
\end{equation}
The last estimate follows from O(3) momentum symmetry under
statistical averaging whatever the color projection.  We identify
the transverse poles in  (\ref{TPOLE}) with 2 shear modes of
consititutive dispersion

\begin{equation}
\omega+i\frac{\eta_l}{mn}k^2+{\cal O}(k^3)= 0
\label{VIS2}
\end{equation}
with $\eta_l$ the shear viscosity for the lth color
representation. Unlike conventional plasmas, the classical SU(2)
color Coulomb plasma admits an infinite hierarchy of shear modes
for each representation $l$.

\begin{figure}[!h]
\begin{center}
\subfigure{\label{radial:all}\includegraphics[width=0.495\textwidth]
{S00_alla.ps}}
\subfigure{\label{radial:G021}\includegraphics[width=0.495\textwidth]
{S01_allb.ps}}
\end{center}
\begin{center}
\subfigure{\label{radial:G066}\includegraphics[width=0.495\textwidth]
{S02_allc.ps}}
\subfigure{\label{radial:G128}\includegraphics[width=0.495\textwidth]
{S03_alld.ps}}
\end{center}
\caption{${\bf S}_{0l}(q)$ from SU(2) Molecular Dynamics.} \label{structure}
\end{figure}

The remaining 3 hydrodynamical modes $L,E,N$ are more involved as
they mix in (\ref{DIS}) and under general symmetry consideration.
Indeed current conservation, ties the L mode to the N mode for
instance.  Most of the symmetry arguments regarding the generic
nature of $\Omega_l$ in~\cite{baus} carry to our case for each
color representation. Thus,  for the 3 remaining non-transverse
modes (\ref{DIS}) reads in matrix form

\begin{equation}
\left(
\begin{array}{ccc}
{\bf G}_{l NN} &  {\bf G}_{lNL} & {\bf G}_{lNE} \\
{\bf G}_{lLN}  &{\bf G}_{lLL}   & {\bf G}_{lLE}  \\
{\bf G}_{lEN}&  {\bf G}_{lEL} &   {\bf G}_{lEE}
\end{array}
\right)= \left(
\begin{array}{ccc}
z  &  -\Omega_{lNL} & 0  \\
-\Omega_{lLN}  &z-\Omega_{lLL}   & -\Omega_{lLE}  \\
0  &  -\Omega_{lEL} &   z-\Omega_{lEE}
\end{array}
\right)^{-1} \left(
\begin{array}{ccc}
1+n\,{\bf h}_l &  0 & 0  \\
0  &1 & 0  \\
0  & 0 &  1
\end{array}
\right) \label{MAT1}
\end{equation}
The 3 remaining hydrodynamical modes are the zeros of the
determinant

\begin{equation}
\Delta_l=\left|
\begin{array}{ccc}
z  &  -\Omega_{lNL} (zk)& 0  \\
-\Omega_{lLN} (zk) &z-\Omega_{lLL} (zk)  & -\Omega_{lLE} (zk) \\
0  &  -\Omega_{lEL} (zk) &   z-\Omega_{lEE} (zk)
\end{array}
\right|=0 \label{MAT2}
\end{equation}
(\ref{MAT2}) admits infinitly many solutions $z_l(k)$.  We seek
the hydrodynamical solutions as analytical solutions in $k$ for
small $k$, ie. $z(k)=\sum_nz_{ln}k^n$ for each SU(2) color
representation $l$.  In leading order, we have

\begin{equation}
\Delta_l\approx z_{l0}\left(z_{l0}^2-\frac{k^2T}{m}\,{\bf
S}_{0l}^{-1}(k\approx 0)\right)\approx 0 \label{MAT3}
\end{equation}
after using the symmetry properties of $\Omega_l$ as
in~\cite{baus} for each $l$.  We have also made use of the
generalized Ornstein-Zernicke equations for each $l$
representation~\cite{cho&zahed3}
In Fig.~\ref{structure} we show the molecular dynamics simulation
results for 4 typical structure factors~\cite{cho&zahed3}

\begin{equation}
{\bf S}_{0l}(\vec k )=\left({\frac{4\pi}{2l+1}}\right)
\left<\left| \sum_{jm}\,e^{{i\vec k}\cdot{\vec
x_j(0)}}\,Y_l^m(\vec Q_i )\,\right|^2 \right>
\end{equation}
for $l=0,1,2,3$. We have made use of the dimensionless wavenumber
$q=k\,a_{WS}$ with $a_{WS}$ is the Wigner-size radius.  In
Fig.~\ref{fluctuation} we show the analytical result for ${\bf
S}_{01}$ which we will use for the numerical estimates below.  We
note that the $l=1$ structure factor which amounts to the monopole
structure factor vanishes at $k=0$. All other $l$'s are finite at
$k=0$ with $l=0$ corresponding to the density structure factor.

(\ref{MAT3}) displays 3 hydrodynamical zeros as $k\rightarrow 0$
for each $l$ representation. One is massless and we identify it
with the diffusive heat mode. The molecular dynamics simulations
of the structure factors in Fig.~\ref{structure} implies that all
$l\neq 0$ channels are sound dominated with two massless
modes, while the $l=1$ is plasmon dominated with two massive
longitudinal plasmon states. Thus

\begin{equation}
z_{l\pm}=\pm \omega_{p}^2\delta_{l1}
\end{equation}
with $\omega_{p}=k_D\sqrt{T/m}$ the plasmon frequency. The
relevance of this channel to the energy loss has been discussed in
\cite{cho&zahed5}. We used ${\bf S}_{01}(k\approx 0)\approx
k^2/k_D^2$ with $k_D^2$ the squared Debye momentum. All even
$l\neq 1$ are contaminated by the sound modes. The SU(2) classical
and colored Coulomb plasma supports plasmon oscillations even at
strong coupling. These modes are important in the attenuation of
soft monopole color oscilations.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.55\textwidth]{structure_tale.ps}
\end{center}
\caption{${\bf S}_{01}(q)$ for different $\Gamma$~\cite{cho&zahed3}}.
\label{fluctuation}
\end{figure}

\section{Shear Viscosity}

\renewcommand{\theequation}{VIII.\arabic{equation}}
\setcounter{equation}{0}

The transport parameters associated to the SU(2) classical and colored Coulomb plasma
follows from the hydrodynamical projection and expansion discussed above. This includes,
the heat diffusion coefficient, the transverse shear viscosity and the longitudinal plasmon
frequency and damping parameters. In this section, we discuss explicitly the shear viscosity
coefficient for the SU(2) colored Coulomb plasma.

Throughout,  we define  $\lambda=\frac{4}{3}\pi(3\Gamma)^{3/2}$,
the bare Coulomb interaction  $\bar{V}_l=k_0^2/l^2$ in units of the Wigner-size radius $k^{-1}_0=a_{
WS}$.  While varying the Coulomb coupling

\begin{equation}
\Gamma=\frac{g^2}{4\pi}\beta\frac{C_2}{a_{WS}}
\end{equation}
 all length scales will be measured in  $a_{WS}=(4\pi n/3)^{-1/3}$,
all times in the inverse plasmon frequency $1/\omega_p$ with
$\omega_p^2={\kappa_D^2}/{m\beta}={n}g^2C_2/m$.
All units of mass will be measured in $m$. The Debye
momentum is $\kappa_D^2=g^2n\beta C_2$ and the plasma density is $n$.
 for instance, the shear viscosity will be expressed in fixed dimensionless units
of $\eta_*=nm\omega_p a_{WS}^{2}$.

The transverse shear viscosity follows from (\ref{TDIS}) with $\Sigma_l$ contributing to the direct
or hydrodynamical part, and $\Theta_l$ contributing to the indirect or single-particle part. For $l=0$

\begin{equation}
\frac{\eta_{0}}{\eta^*}=\frac{\eta_{0\,
dir}}{\eta^*}+\frac{\eta_{0\, ind}}{\eta^*} \label{ETAFULL}
\end{equation}
respectively. The direct or hydrodynamical
contribution is likely to be dominant at strong coupling, while
the indirect or single-particle contribution is likely to take
over at weak coupling. We now  proceed to show that.

The indirect contribution to the viscosity follows from the
contribution outside the hydrodynamical subspace through
${\cal Q}_H$ and lumps the single-particle phase contributions.
It  involves the inversion of  ${\cal Q}_H\Sigma_{C0}{\cal Q}_H$
in (\ref{eq013e}) with

\begin{eqnarray}
\eta_{0\rm ind}=\lim_{k\to0}\frac {mn}{k^2} \frac{|\langle
t|\Sigma_0 |tl\rangle|^2}{\langle tl|i\Sigma_0|tl\rangle}
=\lim_{k\to0}\frac {mn}{k^2} \frac{|\langle
t|(\Sigma_{00}+\Sigma_{C0}) |tl\rangle|^2}{\langle
tl|i\Sigma_{C0}|tl\rangle} \label{VIS3}
\end{eqnarray}
In short we expand $\Sigma_{C0}$ in terms of generalized Hermite
polynomials, with the first term identified with the stress tensor
due to the projection operator (\ref{eq003h}). The inversion
follows by means of the first Sonine polynomial expansion.
Explicitly

\begin{equation}
\eta_{ind}=\frac{\eta_{0\,ind}}{\eta_*}=nm \lim_{k \to
0}\frac{1}{k^2}\frac{ |\langle t \vert\Sigma_{00}+\Sigma_{C0}(\vec
k, 0)\vert lt \rangle |^2 }{\langle lt \vert i\Sigma_{C0}(\vec k,
0)\vert lt \rangle} =\frac{(1+\lambda I_2)^2} {\lambda I_3}
\label{eq013v}
\end{equation}
with

\begin{eqnarray}
& & I_2=\frac{1}{60\pi^2}\frac{1}{(3\Gamma)^{1/2}}
\int_0^{\infty} dq \Big(2({\bf S}_{01}(q)^2-1)+(1-{\bf S}_{01}(q))\Big) \nonumber \\
& & I_3=\frac{1}{10\pi^{3/2}}\frac{1}{3\Gamma} \int_0^{\infty}dq q
(1-{\bf S}_{01}(q)) \label{eq014v}
\end{eqnarray}
with the dimensionless wave number $q=ka_{WS}$.

We recall that ${\bf S}_{01}$ is the monopole structure factor
discussed in~\cite{cho&zahed3} both analytically and numerically.
In Fig.~\ref{fluctuation} we show the behavior of the static
monopole structure factor from~\cite{cho&zahed3} for different
Coulomb couplings. The larger $\Gamma$ the stronger the first
peak, and the oscillations. These features characterize the onset
of the crystalline structure in the SU(2) colored Coulomb plasma.
A good fit to Fig.~\ref{fluctuation}  follows from the following
parametrization

\begin{equation}
1+ C_0 e^{-q/C_1}\sin{((q-C_2)/C_3)} \label{FIT}
\end{equation}
with 4 parameters $C_{0,1,2,3}$.  The fit following from
(\ref{FIT}) extends to $q\approx 100$  within $10^{-5}$ accuracy,
thanks to the exponent.

The direct contribution to the shear viscosity follows from similar arguments.
From (\ref{TDIS}) and (\ref{VIS2}), we have in the zero momentum limit

\begin{eqnarray}
\eta_{0\, dir}=\lim_{k\to0} \frac {mn}{k^2}\langle t|i\Sigma_0
|t\rangle=\lim_{k\to0}\frac {mn}{k^2}\langle t|i\Sigma_{C0}
(0,0)|t\rangle
\end{eqnarray}
with $\Sigma_0=\Sigma_{00}+\Sigma_{I0}+\Sigma_{C0}$ as defined in
(\ref{K2}) and (\ref{SFOURIER}). Only those nonvanishing
contributions after the hydrodynamical projection were retained in
the second equalities in (\ref{VIS3})  as we detail in Appendix D.
A rerun of the arguments yields

\begin{eqnarray}
\eta_{dir}^{*}&=&\eta_{0\,dir}/\eta_*=\lambda\frac{\omega_p}
{\kappa_D^3}\lim_{k\to0}\frac{1}{\vec k^2}\int\frac{d\vec
l}{(2\pi)^3} \int_{0}^{\infty}dt
n(\vec \epsilon\cdot\vec l)^2  \nonumber \\
& \times & \bigg( {\bf c}_{D1}(l){\cal G}_{n1}(\vec k-\vec
l,t){\cal G}_{n1}(\vec l,t)\bar{V}_{\vec l}-{\bf c}_{D0}(l){\cal
G}_{n1}(\vec k-\vec l,t){\cal G}_{n1}(\vec l,t)\bar{V}_{\vec
k-\vec
l} \bigg) \nonumber \\
\label{eq002v}
\end{eqnarray}
The projected non-static structure factor is

\begin{eqnarray}
{\cal G}_{n1}(\vec l,t) &=&\frac{1}{n}\int d\vec pd\vec p'\,{\bf
S}_{1}(\vec l,t;\vec p\vec p')={\overline{\cal G}}_{n1}(\vec l,
t)\,{\bf S}_{01}(\vec l)  \label{eq010v}
\end{eqnarray}
with the normalization ${\overline{\cal G}}_{n1}(\vec l, 0)=1$. As
in the one component Coulomb plasma studied
in~\cite{gould&mazenko} we will approximate the dynamical part by
its intermediate time-behavior where the motion is free. This
consists in solving (\ref{SELF}) with no self-energy kernel or
$\Sigma=0$,

\begin{equation}
{\cal G}_{n1}(\vec l, t)\approx e^{-(lt)^2/2m\beta}\,{\bf
S}_{01}(\vec l) \label{GFREE}
\end{equation}
Thus inserting (\ref{GFREE}) and performing the integrations with
$k\rightarrow 0$ yield the direct contribution to the shear
viscosity

\begin{equation}
\eta_{dir}^{*}
=\frac{\eta_{dir}}{\eta_0}=\frac{\sqrt{3}}{45\pi^{1/2}}\Gamma^{\frac{1}{2}}
\label{eq012v}
\end{equation}

The full shear viscosity result is then
\begin{equation}
\frac{\eta_{0}}{\eta^*}=\frac{\eta_{0\,
dir}}{\eta^*}+\frac{\eta_{0\, ind}}{\eta^*}=
\frac{\sqrt{3}}{45\pi^{1/2}}\Gamma^{\frac{1}{2}} +\frac{(1+\lambda
I_2)^2} {\lambda I_3} \label{ETAFULLX}
\end{equation}
after inserting (\ref{eq013v}) and (\ref{eq012v}) in
(\ref{ETAFULL}). The result (\ref{ETAFULLX}) for the shear
viscosity of the transverse sound mode is analogous to the result
for the sound velocity in the one component plasma derived
initially in~\cite{wallenborn&baus} with two differences: 1/ The
SU(2) Casimir in $\Gamma$; 2/ the occurrence of ${\bf S}_{01}$
instead of ${\bf S}_{00}$.  Since ${\bf S}_{01}$ is plasmon
dominated at low momentum, we conclude that the shear viscosity is
dominated by rescattering against the SU(2) plasmon modes in the
cQGP.

Using the fitted monopole structure factor (\ref{FIT}) in
(\ref{eq014v}) we can numerically assess (\ref{eq013v}) for
different values of $\Gamma$.  Combining this result for the
indirect viscosity together with (\ref{eq012v}) for the direct
viscosity yield the colorless or sound viscosity $\eta_0$.  The
values of $\eta_0$ are displayed  in Table I, and shown in
Fig.~\ref{viscosity} (black).  The SU(2) molecular dynamics
simulations in~\cite{gelmanetal}  which are parameterized as

\begin{equation}
\eta^{*}_{MD}\simeq0.001\Gamma+\frac{0.242}{\Gamma^{0.3}}+\frac{0.072}{\Gamma^2}
\label{eq011v}
\end{equation}
are also displayed in Table I and shown in Fig.~\ref{viscosity}
(red) for comparison. The sound viscosity dips at about
$\Gamma\approx 8$ in our analytical estimate. To understand the
origin of the minimum, we display in Fig. \ref{viscosity_fit} the
scaling with $\Gamma$ of the direct or hydrodynamical and the
indirect part of the shear viscosity. The direct contribution to
the viscosity grows like $\Gamma^{1/2}$, the indirect contribution
drops like $1/\Gamma^{5/2}$.  The latter dominates at weak
coupling, while the former dominates at strong coupling. This is
indeed expected, since the direct part is the contribution from
the hydrodynamical part of the phase space, while the indirect
part is the contribution from the non-hydrodynamical or
single-particle part of  phase space.  The crossing is at
$\Gamma\approx 4$.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{viscosity_separated.ps}
\includegraphics[width=0.49\textwidth]{viscosity_full.ps}
\end{center}
\caption{The direct and indirect part of the
viscosity}\label{viscosity}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{eta_dir.ps}
\includegraphics[width=0.49\textwidth]{eta_ind.ps}
\end{center}
\caption{The best fit of the direct and indirect part of the
viscosity}\label {viscosity_fit}
\end{figure}

\begin{table}[!h]
\caption{Reduced shear viscosity. See text.} \label{tb1}

\begin{center}
\begin{tabular}{cccccccccccc}
\hline

$\Gamma        $ & $    2  $ & $    4  $ & $   6   $ & $    8  $ & $   10  $ & $   12  $ & $   14  $ & $   16  $ & $   18  $  \\
\hline
$\eta^{*}_{QGP}$ & $ 0.286 $ & $ 0.092 $ & $ 0.067 $ & $ 0.066 $ & $ 0.070 $ & $ 0.076 $ & $ 0.081 $ & $ 0.087 $ & $ 0.092 $  \\
$\eta_{MD}     $ & $ 0.217 $ & $ 0.168 $ & $ 0.168 $ & $ 0.139 $ & $ 0.132 $ & $ 0.127 $ & $ 0.124 $ & $ 0.122 $ & $ 0.120 $  \\
\hline

\end{tabular}
\end{center}
\end{table}

The reduced sound velocity $\eta_*$ is dimensionless. To restore
dimensionality and compare with expectations for an SU(2) colored
Coulomb plasma, we first note that the particle density is about
$3\times\,0.244\,T^3=0.732\,T^3$. There are 3 physical gluons,
each carrying black-body density. The corresponding Wigner-Seitz
radius is then $a_{WS}=({3}/{4\pi n})^{{1}/{3}}\approx
{0.688}/{T}$. The Coulomb coupling is $\Gamma\approx
1.453\,({g^2N_c}/{4\pi})$. Since the plasmon frequency is
$\omega_p^2={\kappa_D^2}/{m\beta}={n}g^2N_c/m$, we get
$\omega_p^2\simeq3.066\,T^2({g^2N_c}/{4\pi})$ with $m\simeq3T$.
The unit of viscosity  $\eta_0=nm\omega_pa_{WS}^2$ translates to
$1.822\,T^3({g^2}N_c/4\pi)^{{1}/{2}}$.  In these units, the
viscosity for the SU(2) cQGP dips at about $0.066$ which is
$\eta^*_{QGP}\approx 0.066\,\eta_0\approx
0.120\,T^3\,(g^2N_c/4\pi)^{1/2}$. Since the entropy in our case is
$\sigma=6\,(4\pi^2/90)T^3$, we have for the SU(2) ratio
$\eta/\sigma|_{SU(2)}=0.046\,(g^2N_c/4\pi)^{1/2}$. The minimum in
the viscosity occurs at $\Gamma=1.453\,({g^2N_c}/{4\pi})\approx
8$, so that $({g^2N_c}/{4\pi})^{1/2}\approx 2.347$. Thus, our
shear viscosity to entropy ratio is
$\eta/\sigma|_{SU(2)}\simeq0.107$. A rerun of these estimates for
SU(3) yields $\eta/\sigma|_{SU(3)}\simeq0.078$ which is lower than
the bound $\eta/\sigma=1/4\pi\simeq0.0795$ suggested from
holography.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{viscosity_approx_separated.ps}
\includegraphics[width=0.49\textwidth]{viscosity_approx_full.ps}
\end{center}
\caption{Comparison with weak coupling. See text.}
\label{viscosity_approx}
\end{figure}

Finally, we show in Fig.~\ref{viscosity_approx} the shear
viscosity $\eta*_{S(q)}$ at low $\Gamma$ (a:green) and large
$\Gamma$ (b:black) assessed using the weak-coupling structure
factor $S(k)=k^2/(k^2+k_D^2)$.  The discrepancy is noticeable for
$\Gamma$ near the liquid point. The large discrepancy for small
values of $\Gamma$ reflects on the fact that the integrals in
(\ref{eq014v}) are infrared sensitive. The sensitivity is tamed by
our analytical structure factor and the simulations. We recall
that in weak coupling, the Landau viscosity $\eta_L$
is~\cite{ichimaru3}

\begin{equation}
\frac{\eta_L}{\eta^*}=\frac{5\sqrt{3\pi}}{18}\frac{1}{\Gamma^{5/2}}\frac 1{{\rm ln}(r_D/r_0)}
\label{LANDAU}
\end{equation}
which follows from a mean-field analysis of the kinetic equation with the
plasma dielectric constant set to 1. The logarithmic dependence in (\ref{LANDAU})
reflects on the infrared and ultraviolet  sensitivity of the mean-field approximation.
Typically $r_D=1/k_D$ and $r_0=(g^2C_2/4\pi)\beta$ which are the Debye length
and the the distance of closest approach. Thus

\begin{eqnarray}
\frac{\eta_L}{\eta^*}\approx\frac{5\sqrt{3\pi}}{27}\frac{1}{\Gamma^{5/2}}\frac 1{{\rm ln}(1/\Gamma)}
\label{LANDAU1}
\end{eqnarray}
or $\eta_L/\eta^*\approx 0.6/(\Gamma^{5/2}{\rm ln}(1/\Gamma))$ which is overall consistent
with our analysis.

The Landau or mean-field result  is smaller for the viscosity than the result
from perturbative QCD.  Indeed, the unscaled Landau viscosity (\ref{LANDAU1})
reads

\begin{equation}
\eta_L\approx \frac{10}{24}\frac{\sqrt{m}}{(\alpha_sC_2)^2\beta^{5/2}}\frac{1}{\alpha_s}
\label{LANDAU2}
\end{equation}
after restoring the viscosity unit $\eta^*=nm\omega_pa_{WS}^2$ and using
${\rm ln}(r_D/r_0)\approx 3{\rm ln}(1/\alpha_s)/2$ with $\alpha_s=g^2/4\pi$.
While our consituent gluons carry $m\approx \pi T$, in the mean field or weak
coupling we can set their masses to $m\approx gT$.  With this in mind, and setting $C_2=N_c=3$ in
(\ref{LANDAU2}) we obtain

\begin{equation}
\eta_L\approx \frac{5\sqrt{2}}{108\pi^{1/4}}\frac{T^3}{\alpha_s^{7/4}{\rm ln}(1/\alpha_s)}
\approx 0.05\frac{T^3}{\alpha_s^{7/4}{\rm ln}(1/\alpha_s)}
\label{LANDAU3}
\end{equation}
which is to be compared with the QCD weak coupling
result~\cite{heiselberg}

\begin{equation}
\eta_{QCD}\approx \frac{T^3}{\alpha_s^{2}{\rm ln}(1/\alpha_s)}
\label{LANDAU4}
\end{equation}
The mean-field result (\ref{LANDAU3}) is $\alpha_s^{1/4}\approx \sqrt{g}$ {\it smaller} in weak
coupling than the QCD perturbative result. The reason is the fact that in perturbative QCD the
viscosity is not only caused by collisions with the underlying parton constituents, but also
quantum recombinations and decays. These latter effects are absent in our classical QGP.

\section{Diffusion Constant}

\renewcommand{\theequation}{IX.\arabic{equation}}
\setcounter{equation}{0}

The calculation of the diffusion constant in the SU(2) plasma is
similar to that of the shear viscosity. The governing equation is
again (\ref{RES2}) with $\Sigma$ and ${\bf S}$  replaced by
$\Sigma_s$, ${\bf S}_s$. The label is short for single particle.
The difference between ${\bf S}$ and ${\bf S}_s$ is the
substitution of (\ref{FDIS}) by

\begin{equation}
f_s(\vec r\vec p \vec Q t)=\sqrt{N}\delta(\vec r-\vec
r_1(t))\delta(\vec p-\vec p_1(t))\delta(\vec Q-\vec Q_1(t))
\label{eq001t}
\end{equation}
The diffusion constant follows from the velocity auto-correlator

\begin{equation}
V_D(t)=\frac{1}{3}\langle\vec V(t)\cdot\vec V(0)\rangle
\label{eq003t}
\end{equation}
through

\begin{equation}
D=\int_{0}^{\infty}dt V_D(t)\label{eq002t}
\end{equation}
Solving (\ref{RES2}) using the method of one-Sonine polynomial
approximation as in~\cite{gould&mazenko} yields the Langevin-like
equation

\begin{equation}
\frac{dV_D(t)}{dt}=-\int_{0}^{t}dt'M(t')V_D(t-t') \label{eq004t}
\end{equation}
with the memory kernel tied to $\Sigma_{C0}^S$,

\begin{eqnarray}
&& n\,f_0(\vec p')\, \Sigma_{Cl}^S(t, \vec k, \vec p\vec p')
\nonumber \\
& & = -\frac{1}{\beta} \int d\vec p_1 d\vec p_2 \int \frac{d\vec
l}{(2\pi)^3} \vec l\cdot\nabla_{\vec p}\vec l\cdot\nabla_{\vec p'}
{\bf c}_{D1}(l) V_{\vec l} \nonumber \\
& & \times \Big( \frac{l}{2l+1} {\bf S}_{l-1}^S(t, \vec k-\vec
l,\vec p\vec p'){\bf S}_1(t, \vec l,\vec p_1\vec
p_2)+\frac{l+1}{2l+1} {\bf S}_{l+1}^S(t, \vec k-\vec l,\vec p\vec
p'){\bf S}_1(t, \vec l,\vec p_1\vec p_2)\Big) \nonumber \\
\label{eq005d}
\end{eqnarray}
and

\begin{eqnarray}
&& n\,f_0(\vec p')\, \Sigma_{C0}^S(t, \vec k=\vec 0, \vec p\vec
p')
\nonumber \\
& & = -\frac{1}{\beta} \int d\vec p_1 d\vec p_2 \int \frac{d\vec
l}{(2\pi)^3}\vec l\cdot\nabla_{\vec p}\vec l\cdot\nabla_{\vec p'}
{\bf c}_{D1}(l)V_{\vec l} {\bf S}_{1}^S(t,\vec l,\vec p\vec
p'){\bf S}_1(t, \vec l,\vec p_1\vec p_2) \label{eq006d}
\end{eqnarray}
therefore

\begin{equation}
M(t)=\frac{\beta}{3m}\int d\vec p d\vec p' \vec p\cdot\vec p'
\Sigma_{C0}^S(t,\vec k=\vec 0,\vec p\vec p')f_0(\vec p')
\label{eq005t}
\end{equation}
which clearly projects out the singlet color contribution. If we
introduce the dimensionless diffusion constant,
$D^{*}=D/w_pa_{WS}^2$,  then (\ref{eq002t}) together with
(\ref{eq004t}) yield

\begin{equation}
\frac{1}{D}=m\beta\int_{0}^{\infty}dtM(t) \rightarrow
\frac{1}{D^{*}}=3\Gamma\int_{0}^{\infty}w_p dt\frac{M(t)}{w_p^2}=
3\Gamma\int_{0}^{\infty} d\tau\bar{M}(\tau) \label{eq006t}
\end{equation}
Using similar steps as for the derivation of the viscosity, we can
unwind the self-energy kernel $\Sigma_s$ in (\ref{eq006t}) to give

\begin{equation}
\frac{1}{D^{*}}=-\Gamma\int\frac{d\vec l}{(2\pi
)^3}\int_{0}^{\infty}d\tau \vec l^2 {\bf c}_{D1}(l)V_{\vec l}
{\cal G}_{n1}^S(l,t)\,{\cal G}_{n1}(l,t) \label{eq007t}
\end{equation}
where we have used the same the half-renormalization method
discussed above for the viscosity.  The color integrations are
done by Legendre transforms. Here again, we separate the
time-dependent structure factors as ${\cal G}_{n1}(l,t)={\bf
S}_{01}(l)\bar{\cal G}_{n1}(l,t)$ and ${\bf
S}_{01}^S(l,t)=\bar{\cal G}_{n1}(l,t)$ in the free particle
approximation. Thus

\begin{equation}
\frac{1}{D^{*}}=\Gamma^{3/2}\Big(\frac{1}{3\pi}
\Big)^{\frac{1}{2}}\int_{0}^{\infty} dq q(1-{\bf S}_{01}(q))
\label{eq008t}
\end{equation}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.55\textwidth]{diffusion_full_half.ps}
\end{center}
\caption{Diffusion Constant (black, green) versus molecular dynamics
simulations (red). See text. } \label{diffusion}
\end{figure}

\begin{table}[!h]
\caption{Diffusion constant. See text.}
 \label{tb2}

\begin{center}
\begin{tabular}{cccccccccc}
\hline

$\Gamma       $ & $    2  $ & $   4   $ & $   6   $ & $    8  $ & $   10  $ & $   12  $ & $  14   $ & $   16  $ & $   18  $ \\
\hline
$D^{*}_{QGP}  $ & $ 0.410 $ & $ 0.115 $ & $ 0.055 $ & $ 0.034 $ & $ 0.024 $ & $ 0.017 $ & $ 0.014 $ & $ 0.012 $ & $ 0.010 $ \\
$D^{*}_{MD}   $ & $ 0.230 $ & $ 0.132 $ & $ 0.095 $ & $ 0.076 $ & $ 0.063 $ & $ 0.055 $ & $ 0.048 $ & $ 0.044 $ & $ 0.040 $ \\
\hline \label{DIFF}
\end{tabular}
\end{center}
\end{table}

The results following from (\ref{eq008t}) are displayed in
Table~\ref{DIFF} and in Fig.~\ref{diffusion} (black) from weak to
strong coupling. For comparison, we also show the the diffusion
constant measured using molecular dynamics simulations with an
SU(2) colored Coulomb plasma~\cite{gelmanetal}. The molecular
dynamics simulations are fitted to

\begin{equation}
D^{*}\simeq\frac{0.4}{\Gamma^{0.8}}
\end{equation}
For comparison, we also show the diffusion constant (\ref{eq008t})
assessed using the weak coupling or Debye structure factor $S(k)=k^2/(k^2+k_D^2)$
in Fig.~\ref{diffusion} (green).  The discrepancy between the analytical
results at small $\Gamma$ are similar to the ones we noted above
for the shear viscosity. In our correctly resummed structure factor
of Fig.~\ref{fluctuation}, the infrared behavior of the cQGP is controlled
in contrast to the simple Debye structure factor.

 Finally, a comparison of (\ref{eq008t}) to (\ref{eq014v}) shows that
$1/D^*\approx 1/\lambda I_3$ which is seen to grow like
$\Gamma^{3/2}$. Thus $D^*$ drops like $1/\Gamma^{3/2}$
which is close to the numerically generated result fitted in
Fig.~\ref{diffusion_fit} (left). The weak coupling self-diffusion
coefficient scales as $1/\Gamma^{5/2}$ as shown in Fig.~\ref{diffusion_fit} (right).
 More importantly, the diffusion constant in the SU(2) colored Coulomb
plasma is caused solely by the non hydrodynamical modes or single
particle collisions in our analysis.
It does not survive at strong coupling where
most of the losses are caused by the collective sound and/or
plasmon modes. This result is in contrast with the shear viscosity we
discussed above, where the hydrodynamical modes level it off at large $\Gamma$.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.495\textwidth]{diffusion_fit_half.ps}
\includegraphics[width=0.495\textwidth]{diffusion_free_fit.ps}
\end{center}
\caption{Fit to the diffusion constant. See text.}
\label{diffusion_fit}
\end{figure}

\section{Conclusions}

We have provided a general framework for discussing
non-perturbative many-body dynamics in the colored SU(2) Coulomb
plasma introduced in \cite{SZ_newqgp}. The framework extends the
analysis developed intially for one-component Abelian plasmas to
the non-Abelian case. In the latter, the Liouville operator is
supplemented by a color precessing contribution that contributes
to the connected part of the self-energy kernel.

The many-body content of the SU(2) colored Coulomb plasma are best
captured by the Liouville equation in phase space in the form of
an eigenvalue-like equation. Standard projected perturbation
theory like analysis around the static phase space distributions
yield a resummed self energy kernel in closed form. Translational
space invariance and rigid color rotational invariance in phase
space simplifies the nature of the kernel.

In the hydrodynamical limit, the phase space projected equations
for the time-dependent and resummed structure factor displays both
transverse and longitudinal hydrodynamical modes. The shear
viscosity and longitudinal diffusion constant are expressed
explicitly in terms of the resummed self-energy kernel. The latter
is directly tied with the interacting part of the Liouville
operator in color space. We have shown that in the free streaming
approximation and half-renormalized Liouville operators, the
transport parameters are finite.

We have explicitly derived the shear viscosity and longitudinal
diffusion constant of the SU(2) colored Coulomb plasma in terms of
the monopole static structure factor and the for all values of the
classical Coulomb parameter $\Gamma=V/K$, the ratio of the
potential to kinetic energy per particle. The results compare
fairly with molecular dynamics simulations for SU(2).

The longitudinal diffusion constant is found to drop from weak to
strong coupling like $1/\Gamma^{3/2}$. The shear viscosity is
found to reach a minimum for $\Gamma$ of about 8. The large
increase at weak coupling is the result of the large mean free
paths and encoded in the direct or driving part of the connected
self-energy. The minimum at intermediate $\Gamma$ is tied with the
onset of hydrodynamics which reflects on the liquid nature of the
colored Coulomb plasma in this regime.

At larger values of $\Gamma$ an SU(2) crystal forms as reported
in~\cite{SZ_newqgp}. Our current analysis should be able to
account for the emergence of elasticities, with in particular an
elastic shear mode. This point will be pursued in a future
investigation. The many body analysis presented in this work treats the color
degrees of freedom as massive constituents with a finite mass and
a classical SU(2) color charge. The dynamical analysis is fully
non-classical. In a way, quantum mechanics is assumed to generate
the constituent degrees of freedom with their assigned parameters.
While this picture is supported by perturbation theory at very
weak coupling, its justification at strong coupling is by no means
established.

\begin{acknowledgements}
This work was supported in part by US DOE grants DE-FG02-88ER40388
and DE-FG03-97ER4014.
\end{acknowledgements}

\newpage


\section{Introduction}

Central black holes (BHs) with masses ranging from $\sim 10^6$ to a few $\times
10^9$ $M_\odot$\ are an integral component of most, perhaps all, galaxies with
a bulge component (Kormendy 2004), and although rarer, at least some late-type
galaxies host nuclear BHs with masses as low as $\sim 10^5$ $M_\odot$\
(Filippenko \& Ho 2003; Barth et al. 2004; Greene \& Ho 2007a, 2007b).  It is
now widely believed that BHs play an important role in the life cycle of
galaxies (see reviews in Ho 2004).   To date, most of the observational effort
to investigate the relationship between BHs and their host galaxies have
focused on the stellar component of the hosts, especially the velocity
dispersion and luminosity of the bulge, which empirically seem most closely
coupled to the BH mass.  Although the gas content of inactive galaxies has
been extensively studied (e.g., Haynes \& Giovanelli 1984; Knapp et al. 1985;
Roberts et al.  1991; Bregman et al. 1992; Morganti et al. 2006),
comparatively little attention has been devoted to characterizing the
interstellar medium of active galaxies or of systems with knowledge of their
BH mass or accretion rate.

The gaseous medium of the host galaxy, especially the cold phase as traced
in neutral atomic or molecular hydrogen, offers a number of diagnostics
inaccessible by any other means.  Since cold gas constitutes the very raw
material out of which both the stars form and the BH grows, the cold gas
content of the host galaxy is one of the most fundamental quantities that can
be measured in the effort to understand the coevolution of BHs and galaxies.
At the most rudimentary level, we might naively expect the gas content of the
host to be correlated with the BH accretion rate or the luminosity of its
active galactic nucleus (AGN).  Likewise, the gas content should reflect
the particular evolutionary stage of the host galaxy.  Many current models
(e.g., Granato et al.  2004; Springel et al. 2005) invoke AGN feedback as a
key ingredient for galaxy formation and for coupling the BH to its host.
Depending on the violence with which the accretion energy is injected into the
host and the evolutionary state of the system, AGN feedback can wreck havoc on
the interstellar medium of the host.  For example, recent \ion{H}{1}\ absorption
observations of radio-loud AGNs detect substantial quantities of high-velocity
outflowing neutral gas, presumably in the midst of being expelled from the
host galaxy by the radio jet (Morganti et al. 2007).  Performing a careful,
systematic census of the cold gas content of AGN hosts will provide much
needed empirical guidance for AGN feedback models.  Apart from the sheer gas
mass, \ion{H}{1}\ and CO observations, even when conducted in spatially unresolved
mode, can provide other useful probes of the physical properties of the host,
and of its circumgalactic environment (e.g., Ho 2007a, 2007b).  For example,
the width of the integrated line profile, if it is sufficiently regular, gives
an estimate of the rotation velocity of the disk, and hence an additional
handle on the gravitational potential of the system.  Combining the line width
with the Tully-Fisher (1977) relation, we can infer immediately the total
luminosity of the host, independent of any contamination from the AGN.  The
degree of symmetry of the line profile furnishes useful, if crude, information
on the spatial distribution of gas within and around the host, as well as an
effective probe of possible dynamic disturbances due to neighboring galaxies.

The primary goal of this study is to quantify the \ion{H}{1}\ content of a large,
well-defined sample of active galaxies with uniformly measured BH masses and
optical properties, spanning a wide range in AGN properties.  Despite the
obvious importance of

\vskip 0.3cm
\figurenum{1}
\psfig{file=f1.eps,width=8.5cm,angle=0}
\figcaption[fig1.ps]{
The distribution of BH masses and Eddington ratios for the sample included
in this study.  The 101 newly surveyed objects for which \ion{H}{1}\ observations
were successfully obtained are plotted as circles, while the sample of 53
sources taken from the literature are marked as triangles.
\label{fig1}}
\vskip 0.3cm

\noindent
understanding the cold gas component of AGN host
galaxies, there has been relatively little modern work conducted with this
explicit goal in mind.  Although there have been a number of \ion{H}{1}\ surveys of
AGNs, most of them have focused on relatively low-luminosity Seyfert nuclei
(Allen et al. 1971; Heckman et al. 1978; Bieging \& Biermann 1983; Mirabel \&
Wilson 1984; Hutchings 1989; Greene et al. 2004) and radio-emitting elliptical
galaxies (Dressel et al. 1982; Jenkins 1983), with only limited attention
devoted to higher luminosity quasars (Condon et al.  1985; Hutchings et al.
1987; Lim \& Ho 1999).  This is in part due to sensitivity limitations
(quasars are more distant), but also due to the poor baselines of pre-upgrade
Arecibo\footnote{The Arecibo Observatory is part of the National Astronomy and
Ionosphere Center, which is operated by Cornell University under a cooperative
agreement with the National Science Foundation.} spectra.  With the new
Gregorian optics, $L$-band receiver, and modern
backend at Arecibo, the time is ripe to revisit the problem in a concerted
fashion.  In light of the scientific issues outlined above, the motivation has
never been stronger.  We are particularly keen to use the \ion{H}{1}\ line width as a
kinematic tracer of the host galaxy potential.  Since the rotation velocity of
the disk is correlated with the stellar velocity dispersion of the bulge (see
Ho 2007a, and references therein), the \ion{H}{1}\ line width can be used as a new
variable to investigate the correlation between BH mass and galaxy potential.
We are additionally interested in using the \ion{H}{1}\ spectra to obtain dynamical
masses for the host galaxies, to use the line shape to probe the nearby
environment and dynamical state of the hosts, and to evaluate possible
correlations between \ion{H}{1}\ content and AGN properties. These issues are
investigated in a companion paper (Ho et al. 2008).

\section{Observations and Data Reduction}

\subsection{Sample}

Our sample of AGNs was chosen with one overriding scientific motivation in
mind: the availability of a reliable BH mass estimate.  As we rely on
the virial mass method to estimate BH masses (Kaspi et al. 2000;
Greene \& Ho 2005b; Peterson 2007), this limits our targets to type 1 AGNs.
Sensitivity considerations with the current Arecibo system imposes a practical
redshift limit of $z$\lax0.1.  Apart from these two factors, and the visibility
restrictions of Arecibo (0$^{\circ}$\ {$\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$}\ $\delta$ {$\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$}\ 37$^{\circ}$), the targets were
selected largely randomly to fill the available schedule blocks of the
telescope.  The 113 newly observed objects, whose basic properties are
summarized in Table~1, contains two subsamples.  The first comprises 98 type 1
AGNs from the Fourth Data Release of the Sloan Digital Sky Survey (SDSS;
Adelman-McCarthy et al. 2006), which form part of an on-going study of
low-redshift AGNs by Greene (2006; see also Greene \& Ho 2004, 2005b, 2006a,
2006b, 2007a, 2007b).  Although the SDSS objects strictly do not form a
complete or unbiased sample, they are representative of low-redshift
broad-line AGNs of moderate to high luminosities.  With $M_g \approx -18.8$
to $-23.1$ mag, only $\sim 3-4$ objects satisfy the conventional luminosity
threshold of quasars\footnote{The canonical luminosity threshold of quasars,
$M_B = -23.0$ mag (Schmidt \& Green 1983), translates to $M_B = -22.1$ mag in
our distance scale, which assumes $H_0$ = 70 km s$^{-1}$~Mpc$^{-1}$, $\Omega_m
= 0.3$, and $\Omega_{\Lambda} = 0.7$.  For a power-law AGN spectrum of the
form $f_\lambda \propto \lambda^{-1.56}$ (Vanden~Berk et al. 2001), this
threshold is $M_g \approx -22.3$ mag.}, but most are very prominent Seyfert 1
nuclei.  Twenty-eight of the objects have broad H$\alpha$\ profiles with full-width
at half maximum (FWHM) less than 2000 km s$^{-1}$, and thus meet the formal line width
criterion of narrow-line Seyfert 1 galaxies (e.g., Osterbrock \& Pogge 1985).
The second subsample, in total 15 objects, were primarily chosen because they
have been studied with reverberation mapping (Kaspi et al. 2000; Peterson et
al. 2004); we deem these to be high-priority objects because they have
better-determined BH masses.  This subsample includes seven Palomar-Green (PG)
sources (Schmidt \& Green 1983), among them five luminous enough to qualify as
bona fide quasars, and two satisfying the line width criterion of narrow-line
Seyfert 1 galaxies (PG~0003+199 and PG~1211+143).

To augment the sample size and to increase its dynamic range in terms of BH
mass and AGN luminosity, we performed a comprehensive search of the literature
to compile all previously published \ion{H}{1}\ measurements of type~1 AGNs that
have sufficient optical data to allow estimation of BH masses.  The results of
this exercise yielded a sizable number of additional objects (53), the
details of which are documented in the Appendix.  Our final sample, now
totaling 166 and by far the largest ever studied, covers a wide range of BH
masses, from {$M_{\rm BH}$}\ $\approx\,10^5$ to $10^9$ $M_\odot$, and a significant
spread in Eddington ratios, from $\log L_{\rm bol}/L_{\rm Edd} \approx -2.7$
to 0.3 (Fig.~1), where $L_{\rm Edd} \equiv 1.26 \times 10^{38}
\left(M_{\rm BH}/M_{\odot}\right)$ ergs s$^{-1}$.  Although the sample definitely
contains predominantly low-luminosity AGNs, it covers at least 4 orders of
magnitude in nuclear luminosity (Fig.~2{\it a}), from $L_{\rm H\alpha} \approx
10^{40}$ to $10^{44}$ ergs s$^{-1}$\ (excluding the ultra-low-luminosity object
NGC~4395 at $L_{\rm H\alpha} \approx 10^{38}$ ergs s$^{-1}$), which in more familiar
units corresponds to $B$-band absolute magnitudes of $M_B \approx -15.5$ to
$-24.75$ mag (Fig.~2{\it b}).

\subsection{Arecibo Observations}

We observed the 21~cm spin-flip transition of neutral hydrogen (\ion{H}{1})
in our sample at the Arecibo radio telescope from

\vskip 0.3cm
\begin{figure*}[t]
\figurenum{2}
\centerline{\psfig{file=f2.eps,width=19.5cm,angle=-90}}
\figcaption[fig2.ps]{
The distribution of ({\it a}) H$\alpha$\ luminosity and ({\it b}) $B$-band absolute
magnitude of the AGN component for the sample objects with \ion{H}{1}\ data.  The
H$\beta$\ luminosities of the literature sample (Table~6) were converted to
H$\alpha$\ assuming H$\alpha$/H$\beta$\ = 3.5, as empirically determined by Greene \& Ho
(2005b).  To convert between H$\alpha$\ luminosity and $B$-band absolute magnitude,
we employ the correlation between H$\alpha$\ and 5100 \AA\ continuum luminosity of
Greene \& Ho (2005b), and then assume a continuum spectrum of the form
$f_\lambda \propto \lambda^{-1.56}$ (Vanden~Berk et al. 2001) to extrapolate
to 4400 \AA.
\label{fig2}}
\end{figure*}
\vskip 0.3cm

\noindent
November 2005 through April 2007.  The \ion{H}{1}\ observations were
conducted in four independently tracked 25~MHz bands centered on redshifted
\ion{H}{1} and OH (1420.405751786, 1612.2310, 1667.3590, and 1720.5300~MHz,
respectively).  Observations consisted of 5-minute position-switched scans,
with a calibration diode fired after each position-switched pair and spectral
records recorded every 6~seconds.  Typically sources were observed for
1--2~hours.  The autocorrelation spectrometer used 1024 channels and 9-level
sampling in two (subsequently averaged) polarizations.  Rest-frame velocity
resolutions ranged from 5.15~km~s$^{-1}$ ($z=0$) to 5.72~km~s$^{-1}$
($z=0.11$), but most spectra were Hanning smoothed, reducing the velocity
resolution roughly by a factor of 2.

Records were individually calibrated and bandpasses flattened using the
calibration diode and the corresponding off-source records.  Records and
polarizations were subsequently averaged, and a low-order polynomial baseline
was fit and subtracted.  Systematic flux calibration errors in these data are
of order $10\%$.  All data reduction was performed in AIPS++\footnote{The
AIPS++ (Astronomical Information Processing System) is freely available for
use under the Gnu Public License. Further information may be obtained from
{\tt http://aips2.nrao.edu}.}.  Some spectra showed standing waves due to
resonances within the telescope superstructure or strong continuum sources
($\gtrsim 300$~mJy) falling in the beam (either coincidentally or due to
strong radio emission from the target galaxy itself).  The expected \ion{H}{1}\
line widths are similar to the size of the standing wave features
($\sim1$~MHz), so the detectability of lines was severely impaired in a few
cases.

The observed bands were generally interference-free in the vicinity of the
observed lines, requiring little or no flagging, but some redshift ranges,
most prominently $z\simeq 0.066$--0.069 and $z\simeq0.051$--0.054, were
unobservable due to radio frequency interference (RFI).  Hence, the redshift
distribution of the sample has gaps.  \ion{H}{1} lines were detected in 66
galaxies in the sample, 35 galaxies were significant nondetections, and 12
galaxies are indeterminate due to standing waves in the bandpass or RFI. The
spectra for the detected sources are plotted in Figure~3, accompanied by
their optical images. No 18 cm OH lines were detected
in the sample (many lines were unobservable due to RFI).

The \ion{H}{1}\ properties of the sample are summarized in Table~2.  For each
detected source, we list the systemic velocity of the line in the barycentric
frame ($\upsilon_{\rm sys}$), defined to be the midpoint (mean) of the
velocities corresponding to the 20\% point of the two peaks in the \ion{H}{1}\
profile, and the line width $W_{20}$, the difference between these two
velocities.  The actual high and low velocities are obtained from an
interpolation between the two data points bracketing 20\% of peak flux.  For a
typical root-mean-square noise level of $\sim 0.3$ mJy, we estimate that the
uncertainty in the systemic velocity is $\sigma(\upsilon_{\rm sys}) \approx
3.4$ km s$^{-1}$; the uncertainty in the line width is $\sigma(W_{20}) = 2
\sigma(\upsilon_{\rm sys}) \approx 6.8$ km s$^{-1}$.  In practice, these formal
values underestimate the true errors for spectra affected by RFI or poor
bandpasses, or in instances when the line profile is not clearly
double-peaked.  Profiles that are single-peaked and/or highly
asymmetric are noted in Table~2.

To convert the raw line widths to ${\upsilon_m}$, the maximum rotational velocity, four
corrections must be applied to $W_{20}$: (1) instrumental resolution, which we
assume to be $W_{\rm inst} = 10$ or 5 km s$^{-1}$, depending on whether the spectrum
was Hanning smoothed or not, and that it can be removed by linear subtraction;
(2) redshift, which stretches the line width by a factor $(1+z)$; (3)
turbulent broadening, which for simplicity we assume to be $W_{\rm turb} = 22$
km s$^{-1}$\ for $W_{20}$ and can be subtracted linearly (Verheijen \& Sancisi 2001);
and (4) inclination angle.  We assume that the inclination of the \ion{H}{1}-emitting
disk to the line-of-sight can be approximated by the photometric inclination
angle of the optical disk, $i$ (see \S2.3).  The final maximum rotational
velocity is then

\begin{figure*}[t]
\centerline{\psfig{file=table1_p1.ps,height=0.99\textheight,angle=180}}
\end{figure*}
\clearpage
\begin{figure*}[t]
\centerline{\psfig{file=table1_p2.ps,height=0.99\textheight,angle=180}}
\end{figure*}
\clearpage
\begin{figure*}[t]
\hskip -0.4truein
\centerline{\psfig{file=table1_p3.ps,height=0.99\textheight,angle=180}}
\end{figure*}
\clearpage
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it a}}
\centerline{\psfig{file=f3a.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
\ion{H}{1}\ spectra and optical $g$-band SDSS images of the \ion{H}{1}-detected objects.
The velocity scale is given in the barycentric frame, and the velocity
range is chosen such that the lines have roughly comparable widths on the
plots.  Features suspected to be due to radio frequency interference are
labeled ``RFI.''  Each image subtends a physical scale of 50 kpc $\times$ 50
kpc, with north oriented up and east to the left.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it b}}
\centerline{\psfig{file=f3b.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it c}}
\centerline{\psfig{file=f3c.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it d}}
\centerline{\psfig{file=f3d.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it e}}
\centerline{\psfig{file=f3e.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it f}}
\centerline{\psfig{file=f3f.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it g}}
\centerline{\psfig{file=f3g.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}.  The image for Akn 120 comes from {\it HST}/PC2 (filter
F750LP) and subtends 22.4 kpc $\times$ 22.4 kpc.
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it h}}
\centerline{\psfig{file=f3h.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}, except that for MCG~+01-13-012, NGC~3227,
NGC~5548, and NGC~7469 the images are in the $B$ band taken from the
Digital Sky Survey.  The images for the four PG quasars come from {\it HST},
taken with the following detector and filter combinations and field sizes:
PG~0844+349 (ACS/F625W, 84.6 kpc $\times$ 84.6 kpc),
PG~1229+204 (PC2/F606W, 43.9 kpc $\times$ 43.9 kpc),
PG~1426+015 (PC2/F814W, 61.4 kpc $\times$ 61.4 kpc), and
PG~2130+099; (PC2/F450W, 43.9 kpc $\times$ 43.9 kpc).
\label{fig3}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{3{\it i}}
\centerline{\psfig{file=f3i.eps,width=19.5cm,angle=0}}
\figcaption[fig3.ps]{
Same as Fig.~3{\it a}, except that for RX~J0602.1+2828 and RX~J0608.0+3058
the images are in the $B$ band and were taken from the Digital Sky Survey.
\label{fig3}}
\end{figure*}
\vskip 0.3cm

\vskip 0.3cm
\begin{figure*}[t]
\figurenum{4{\it a}}
\centerline{\psfig{file=f4a.eps,width=19.5cm,angle=0}}
\figcaption[fig4.ps]{
Optical $g$-band SDSS images of the \ion{H}{1}\ nondetections.  Each image subtends a
physical scale of 50 kpc $\times$ 50 kpc, with north oriented up and east to
the left.
\label{fig4}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{4{\it b}}
\centerline{\psfig{file=f4b.eps,width=19.5cm,angle=0}}
\figcaption[fig4.ps]{
Same as Fig.~4{\it a}.
\label{fig4}}
\end{figure*}
\vskip 0.3cm
\vskip 0.3cm
\begin{figure*}[t]
\figurenum{4{\it c}}
\centerline{\psfig{file=f4c.eps,width=19.5cm,angle=0}}
\figcaption[fig4.ps]{
Same as Fig.~4{\it a}, except for PG~0003+199, which comes
from {\it HST}/PC2 (filter F606W) and subtends 21.9 kpc $\times$ 21.9 kpc.
\label{fig4}}
\end{figure*}
\vskip 0.3cm
\clearpage

\begin{figure*}[t]
\centerline{\psfig{file=table2_p1.ps,width=18.5cm,angle=0}}
\end{figure*}
\clearpage
\begin{figure*}[t]
\centerline{\psfig{file=table2_p2.ps,width=18.5cm,angle=0}}
\end{figure*}
\clearpage
\begin{figure*}[t]
\centerline{\psfig{file=table2_p3.ps,width=18.5cm,angle=0}}
\end{figure*}

\begin{figure*}[t]
\centerline{\psfig{file=table3_v1.ps,width=16.5cm,angle=0}}
\end{figure*}

\clearpage
\begin{figure*}[t]
\centerline{\psfig{file=table4_p1.ps,width=18.5cm,angle=0}}
\end{figure*}
\clearpage
\begin{figure*}[t]
\centerline{\psfig{file=table4_p2.ps,width=18.5cm,angle=0}}
\end{figure*}

\clearpage

\begin{equation}
\upsilon_m = {{(W_{20} - W_{\rm inst})/(1+z) - W_{\rm turb}}\over{2 \ {\rm sin}\ i}}.
\end{equation}

In the optically thin limit, the integrated line flux,
$\int S_\nu\,d\upsilon$, in units of Jy~km s$^{-1}$, is related to the \ion{H}{1}\ mass as
(Roberts 1962)

\begin{equation}
M_{{\rm H~{\tiny I}}} = 2.36\times10^5 \ D_{L}^2\ \int S_\nu\,d\upsilon
\, \, \, \, M_\odot,
\end{equation}

\noindent
where $D_{L}$ is the luminosity distance expressed in Mpc and $d\upsilon$
is the line width in the observer's frame.  We neglect any correction for
self-absorption, since this is controversial (see, e.g., Springob et al.
2005), and, in any case, depends on Hubble type, which is not well-known for
many of our sources (see \S2.3).  Upper limits for the integrated fluxes
and \ion{H}{1}\ masses are calculated using 3 times the root-mean-square noise level
and a rest-frame line width of 304 km s$^{-1}$, the median value for the 66 detected
objects.

Single-dish \ion{H}{1}\ observations always run the risk of source confusion,
especially for relatively distant samples such as ours.  At the median redshift
of $z = 0.05$ for our targets, Arecibo's telescope beam (FWHM $\approx$
3\farcm5) subtends a linear diameter of $\sim 200$ kpc.  We use the optical
images (from SDSS if available, or else from the Palomar Digital Sky Survey;
see \S2.3), in combination with the redshifts, to identify potential sources
of confusion within a search radius of 7\farcm5.  The intensity of the first
sidelobes of the Arecibo beam drops to $\sim$10\% of the peak at a distance of
5\farcm5 from the beam center, and by 7$^\prime$--8$^\prime$\ it becomes negligible
(Heiles et al. 2000).  We consider an object as a candidate confusing source
if it lies within the search radius and has a cataloged radial velocity
within $\pm 500$ km s$^{-1}$\ of that of the science target.  Only a few candidates
have been identified, and these are noted in Table~2.  The vast majority of
the objects in our survey are unaffected by source confusion.

Eight of the objects in our survey have published \ion{H}{1}\ data.  A comparison
of our measurements with those in the literature (Table~3) shows that in
general there is fairly good agreement.  The most noticeable exception is
PG~2130+099, for which both our line width and flux are lower than the
literature values by about a factor of 2.

\subsection{Optical Data}

We use both optical spectroscopic and imaging data to ascertain a number of
parameters that are central to our analysis.  For the SDSS objects, these
data were taken directly from the SDSS archives.  The spectra were analyzed
following the procedures previously described in Greene \& Ho (2004, 2005b;
see also Kim et al. 2006).  In brief, we obtain a pure emission-line spectrum
for each object by subtracting from the observed total spectrum a model
consisting of a stellar component, a featureless power-law component, and an
\ion{Fe}{2}\ ``pseudo-continuum.''   We then fit the resulting narrow and broad
emission lines using a combination of multi-component Gaussians.  The optical
emission-line parameters are collected in Table~4.  We also give (in Table~1),
where available, values of the central stellar velocity dispersion
and its associated uncertainty, derived using the technique of Greene \& Ho
(2006a).  If the data do not permit the stellar velocity dispersion to be
measured, we list instead the velocity dispersion of the [\ion{O}{2}]\ $\lambda$ 3727
line, which Greene \& Ho (2005a) have shown to be an effective substitute.
BH masses were estimated using the broad H$\alpha$\ method of Greene \& Ho
(2005b), using the FHWM and luminosities given in Table~1.  We further convert
the broad H$\alpha$\ luminosity to the AGN continuum luminosity at 5100 \AA, using
Equation 1 of Greene \& Ho (2005b), from which we deduce the bolometric
luminosity assuming that  $L_{\rm bol} = 9.8$ \ensuremath{L_{\rm{5100 \AA}}}
(McClure \& Dunlop 2004).

The non-SDSS objects were treated differently.  The majority of these, by
design, have BH masses directly measured from reverberation mapping, and we
simply adopt the values given in Peterson et al. (2004), from which continuum
luminosities at 5100 \AA\ were also taken.  Three of the non-SDSS
objects (MCG~+01-13-012, RX~J0602.1+2828, and RX~J0608.0+3058) only have
measurements for the H$\beta$\ line, but BH masses based on this line alone can
also be estimated with reasonable accuracy (Greene \& Ho 2005b).

The images provide five important pieces of information about the sources:
the total (AGN plus host galaxy) magnitude, morphological type, size,
inclination angle, and potential sources of confusion within the \ion{H}{1}\
beam.  For the SDSS objects, we choose the $g$ band as our
fiducial reference point, since it is closest to the more traditional
$B$ band on which most of the literature references are based.  In Figure~3,
we display the optical image of the sources detected in \ion{H}{1}; images of the
\ion{H}{1}\ nondetections are shown in Figure~4.  In a few cases we were able to
locate high-resolution images in the {\it Hubble Space Telescope (HST)}\
archives.  The size of each image has been scaled to a constant physical scale
of 50 kpc $\times$ 50 kpc to facilitate comparison of objects with very
different distances.

Inspection of Figures~3 and 4 shows that obtaining reliable morphological
types of the host galaxies is challenging for most of the sources, because of
their small angular sizes and the coarse resolution and shallow depth of the
SDSS images.  In assigning a morphological type, we must be
careful to give lower weight to the apparent prominence of the bulge,
since a substantial fraction of the central brightness enhancement presumably
comes from the AGN core itself.  The SDSS database provides quantitative
measurements of the Petrosian radius containing 50\% and 90\% of the
light, from which one can calculate the (inverse) ``concentration index,''
defined to be $C \equiv r_{\rm P50}/r_{\rm P90}$.   We use the correlation
between $C$ and morphological type index of Shimasaku et al. (2001) as an
additional guide to help us assign morphological types, again bearing in
mind that because of the AGN contamination the concentration index should be
viewed strictly as an upper limit to the true value.   We generally give less
weight to the classifications based on $C$.  (We have discovered a
few glaring examples where the SDSS-based concentration index gives an
egregiously erroneous morphological type.)  The most difficult classifications
are those that lie on the boundary between ellipticals and S0s, which is
sometimes ambiguous even for nearby, bright galaxies.  Unless the galaxy is
highly inclined, it is often just impossible to tell; we label these cases as
``E/S0.''  Another difficult situation arises when trying to discern whether a
disk galaxy truly possesses spiral arms.  Given the modest quality of the SDSS
images and the relatively large distances of the galaxies, again often no
clear-cut decision can be made, and we are forced to assign a classification
of ``S0/Sp.''   For a few of the objects, the image material is simply
inadequate to allow a classification to be made at all.

The SDSS photometry additionally provides values for the major axis ($a$)
and minor axis ($b$) isophotal diameters measured at a surface brightness
level of $\mu = 25$ mag~arcsec$^{-2}$, from which we can deduce the
photometric inclination angle using Hubble's (1926) formula
\begin{equation}
{\rm cos}^2  i = {{q^2 - q_0^2}\over{1-q_0^2}},
\end{equation}

\vskip 0.3cm
\figurenum{5}
\begin{figure*}[t]
\centerline{\psfig{file=f5.eps,width=19.0cm,angle=-90}}
\figcaption[fig5.ps]{
The distribution of ({\it a}) \ion{H}{1}\ masses and ({\it b}) \ion{H}{1}\
masses normalized to the $B$-band luminosity of the
host galaxy.  Limits are plotted as open histograms.
\label{fig5}}
\end{figure*}
\vskip 0.3cm

\noindent
where $q = b/a$.  The intrinsic thickness of the disk, $q_0$, varies by about a
factor of 2 along the spiral sequence; we adopt $q_0 = 0.3$, a value
appropriate for early-type systems (Fouqu\'e et al. 1990).  It is also
of interest to combine the galaxy's optical size ($D_{\rm 25}$, diameter
at $\mu = 25$ mag~arcsec$^{-2}$) with the \ion{H}{1}\ line width to compute a
characteristic dynamical mass.  From Casertano \& Shostak (1980),
\begin{equation}
M_{\rm dyn}  = 2\times 10^4
\left( {D_{L}}\over{{\rm Mpc}} \right)
\left( {D_{\rm 25}}\over{{\rm arcmin}} \right)
\left( {\upsilon_{m}}\over{{\rm km~s}^{-1}} \right)^2 \,\,\, \, M_\odot.
\end{equation}
\noindent
Because we have no actual measurement of the size of the \ion{H}{1}\ disk, this
formula yields only an approximate estimate of the true dynamical mass.
However, from spatially resolved observations we know that the sizes of \ion{H}{1}\
disks of spiral galaxies, over a wide range of Hubble types and luminosities,
scale remarkably well with their optical sizes.  From the studies of Broeils \&
Rhee (1997) and Noordermeer et al. (2005), $D_{\rm H~{\tiny I}}/D_{\rm 25}
\approx 1.7$ within 30\%--40\%.   Nevertheless, our values of $M_{\rm dyn}$
are probably much more accurate as a relative rather than an absolute
measure of the galaxy dynamical mass.

The optical photometry, albeit of insufficient angular resolution to yield a
direct decomposition of the host galaxy from the AGN core, nevertheless can be
used to give a rough, yet still useful, estimate of the host galaxy's
luminosity.  Following the strategy of Greene \& Ho (2004, 2007b), we obtain
the host galaxy luminosity by subtracting the AGN contribution, derived from
the spectral analysis, from the total Petrosian (galaxy plus AGN) luminosity
available from the photometry.  In the current application, we use the broad
H$\alpha$\ luminosity as a surrogate for the 5100 \AA\ continuum luminosity to
minimize the uncertainty of measuring the latter, since in some of our objects
there may be significant starlight within the 3$^{\prime\prime}$\ aperture of the SDSS
spectra  (see Greene \& Ho 2005b).  We extrapolate the flux density at 5100
\AA\ to the central wavelength of the $g$ filter (5120 \AA) assuming that the
underlying power-law continuum has a shape $f_\lambda \propto \lambda^{-1.56}$
(Vanden~Berk et al.  2001), adding the small offset to the photometric
zeropoint of the $g$-band filter recommended in the SDSS
website\footnote{\tt http://photo.astro.princeton.edu/\#data\_model}.  In a
few sources the host galaxy luminosity derived in this manner actually exceeds
the total luminosity.  This may reflect the inherent scatter introduced by our
procedure, or perhaps variability in the AGN.  For these cases, we adopt the
total luminosity as an upper limit on the host galaxy luminosity.

\section{Discussion and Summary}

We have used the Arecibo telescope to conduct the largest modern survey to
date for \ion{H}{1}\ emission in active galaxies.  The sample consists of 113
$z$ {$\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$}\ 0.11 galaxies with type~1 AGNs, selected from an extensive study
of SDSS sources for which BH masses can be reliably determined.  The new
observations were supplemented with an additional 53 type 1 AGNs assembled
from the literature, forming a final, comprehensive sample of 154 sources with
\ion{H}{1}\ detections or useful upper limits.
Among the newly observed galaxies, we detected \ion{H}{1}\ in  66 out of the 101
objects that were not adversely affected by RFI, for an overall detection rate
of 65\%.  The \ion{H}{1}\ masses for the detected sources range from $M_{{\rm H~I}}$\
$\approx\,10^9$ to $4\times10^{10}$ $M_\odot$, with an average value of
$8.6\times 10^9$ $M_\odot$, while upper limits for the undetected objects
generally hover around $M_{{\rm H~I}}$\ $\approx\,10^{10}$ $M_\odot$\ (Fig.~5{\it a}).
Adding in the literature sample does not appreciably change these values.  The
host galaxies of the current sample of type 1 AGNs are therefore quite rich in
neutral hydrogen.  For reference, recall that our Galaxy has a total \ion{H}{1}\ mass
of $5.5 \times 10^9$ $M_\odot$\ (Hartmann \& Burton 1997).
Since the \ion{H}{1}\ content of galaxies scales with the stellar luminosity in a
manner that depends on morphological type (e.g., Roberts \& Haynes 1994),
Figure~5{\it b}\ examines the \ion{H}{1}\ masses normalized to the $B$-band
luminosity of the host galaxy.  In the case of the SDSS objects, we converted
the host galaxy luminosities in the $g$ band (\S 2.3) to the $B$ band assuming
an average color of $g-B = -0.45$ mag, appropriate for an Sab galaxy (Fukugita
et al. 1995), roughly the average morphological type of our sample.  The
resulting distribution, ranging from $M_{{\rm H~I}}$/$L_B \approx 0.02$ to 4.5 with an
average value of 0.42, agrees well with the distribution of inactive spiral
galaxies of Hubble type Sa to Sb (e.g., Roberts \& Haynes

\vskip 0.3cm
\figurenum{6}
\psfig{file=f6.eps,width=8.5cm,angle=0}
\figcaption[fig6.ps]{
Distribution of radial velocity difference as measured in the optical
and in \ion{H}{1}, $\Delta \upsilon = \upsilon_{\rm opt} - \upsilon_{\rm sys}$.
Note the excess of objects toward negative values of $\Delta \upsilon$.
\label{fig6}}
\vskip 0.3cm

\noindent
1994).  This
reinforces the conclusion that the host galaxies of type 1 AGNs possess a
normal gas content, at least as far as neutral atomic hydrogen is concerned.

The implications of these detection statistics, along with an extensive
analysis of the \ion{H}{1}\ and AGN properties assembled here, are presented in
our companion paper (Ho et al. 2008).

Figure~6 compares the systemic radial velocity measured from \ion{H}{1}\ with the
published optical radial velocity, $\upsilon_{\rm opt} = cz$.  The velocity
difference, $\Delta \upsilon = \upsilon_{\rm opt} - \upsilon_{\rm sys}$,
shows a large spread, from $\Delta \upsilon \approx -300$ to $+250$ km s$^{-1}$,
but there is a noticeable excess at negative velocities.  On average,
$\langle \Delta \upsilon \rangle = -46 \pm 91$ km s$^{-1}$.  A similar effect was
previously reported by Mirabel \& Wilson (1984) and Hutchings et al. (1987);
in their samples, the mean offset is $\langle \Delta \upsilon \rangle \approx
-50$ km s$^{-1}$, essentially identical to our result.  Since our sources are
relatively bright, type~1 AGNs, the optical radial velocities are
predominantly derived from the narrow emission lines.  The systemic velocity
of the galaxy, on the other hand, is well anchored by the \ion{H}{1}\ measurement.
The negative value of $\langle \Delta \upsilon \rangle$ therefore implies
that on average the ionized gas in the narrow-line region has a general
tendency to be mildly outflowing.

\acknowledgements
The work of L.~C.~H. was supported by the Carnegie Institution of Washington
and by NASA grants HST-GO-10149.02 and HST-AR-10969 from the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., for NASA, under contract NAS5-26555.  Support for
J.~D. and J.~E.~G. was provided by NASA through Hubble Fellowship grants
HF-01183.01-A and HF-01196, respectively, awarded by the Space Telescope
Science Institute.  We made use of the databases in HyperLeda
({\tt http://leda.univ-lyon1.fr/}), the Sloan Digital Sky Survey, and
the NASA/IPAC Extragalactic Database ({\tt http://nedwww.ipac.caltech.edu/}),
which is operated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with NASA.  We thank Minjin Kim for help with
preparing the {\it HST}\ images and for analyzing the SDSS spectra shown in
Table~7, Aaron Barth for sending the {\it HST}\ image of PG~0844+349, and C. Motch
for making available his published spectra of RX~J0602.1+2828 and
RX~J0608.0+3058.  We thank the anonymous referee for helpful suggestions.