\vspace*{-1cm} \begin{flushright} {{\sl Version 20.1(\today)}} \end{flushright} \vspace*{-0.3cm} Quantum Opacity, the RHIC HBT Puzzle, and the Chiral Phase Transition
John G. Cramer
Gerald A. Miller
Jackson M. S. Wu
We present a relativistic quantum mechanical treatment of opacity and refractive effects that allows reproduction of observables measured in two-pion (HBT) interferometry and pion spectra at RHIC. The inferred emission duration %$\Delta\tau$ is substantial. The results are consistent with the emission of pions from a system that has a restored chiral symmetry.
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\begin{document}

%\preprint{NT@UW-02-017}
\title{
\vspace*{-1cm}
\begin{flushright}
{{\small \sl Version 20.1(\today)}}
\end{flushright}
\vspace*{-0.3cm}
Quantum Opacity, the RHIC HBT Puzzle, and the Chiral Phase Transition}

\author{John G. Cramer, Gerald A. Miller, Jackson M. S. Wu}
\author{Jin-Hee Yoon}\altaffiliation{Inha University, Republic of Korea.}
\affiliation{Department of Physics, University of Washington\\
  Seattle, WA 98195-1560}
%E-mail: miller@phys.washington.edu}

\date{\today}

\begin{abstract}
We present a relativistic quantum mechanical treatment of opacity and refractive effects 
that allows reproduction of observables measured in two-pion 
(HBT) interferometry and pion spectra at RHIC. The inferred emission duration 
%$\Delta\tau$ 
is substantial.
 The 
results are
consistent with the emission of pions from a system that has a restored chiral symmetry.
\end{abstract}

\pacs{25.75.-q}
\maketitle


\vskip 0.5in

The space-time structure of the 
``fireball'' produced
in the collision between two heavy ions moving relativistically 
is studied by measuring  the two-particle momentum correlations
between pairs of identical particles. The quantum statistical 
effects of symmetrization cause an enhancement of the 
two-boson coincidence rate at small momentum differences
that can be related to the space-time size of the particle source.
This method, called HBT interferometry,  has been applied extensively 
in recent experiments at the Relativistic
 Heavy Ion Collider (RHIC) by the
STAR and PHENIX 
collaborations \cite{Kolb:2003dz}.

The invariant ratio of the 
 cross section for the production of two 
  pions of momenta $\bfp_1,\bfp_2$
to the product of single particle production cross sections is analyzed as
the correlation function $C(\bfp_1,\bfp_2)$.
 We define
 $\bfq$=$\bfp_1$--$\bfp_2$
and $\bfK$=$(\bfp_1$+$\bfp_2)/2$, with $\bfK_T$ as 
the component perpendicular to the beam direction. 
(We focus on mid-rapidity data, where 
$\bfK=\bfK_T$.)
The correlation function
is  parameterized
for small $\bfq$ as 
$%\bea
C(\bfq,\bfK)$ $\approx$$1 +
\lambda(1-R_o^2q_o^2-R_s^2q_s^2-R_l^2q_l^2),$%\eea
where $o,s,l$ %refer to Bertsch-Pratt momentum coordinates \cite{BP-HBT}
%and 
represent directions parallel to $\bfK_T$, perpendicular
to $\bfK_T$ and the beam direction, 
and parallel to the beam direction\cite{BP-HBT}. 
Early \cite{Rischke:1996em} and recent \cite{Kolb:2003dz} hydrodynamic 
calculations predicted  that  a fireball evolving through a quark-gluon-hadronic phase 
transitions would emit pions
over a long time period, causing a large ratio $R_o/R_s$.
The puzzling experimental
result that $R_o/R_s\approx 1$ 
\cite{Adler:2001zd} is
part of what has been  %%gm816
 called ``the RHIC HBT puzzle'' \cite{Heinz:2002un}. % \cite{Heiselberg:1997vh}  

Data shows the medium produced by 200 GeV Au+Au collisions to be very dense.
Consequently, pions should emerge from
an opaque source\cite{Heiselberg:1997vh}. 
Analyticity tells us that opacity implies accompanying refractive effects.
Our purpose here is to derive and apply a relativistic quantum--mechanical
treatment of opacity and refractive effects that
simulataneously reproduces the values of $R_o, R_s, R_l$ and 
 the pion spectrum 
for central
RHIC Au+Au
collisions\cite{STARHBT,STARspec}.

The experimental observables depend on an emission function, % $S(x,K)$, 
the Wigner transform of the density matrix for the 
currents that emit pions. This emission function has often been
modelled (see  review \cite{Wiedemann:1999qn}),
 as a function  $S_0$ 
having  the form of a hydrodynamic source parameterization  
with approximately boost-invariant longitudinal 
dynamics: %~\cite{Tomasik:1997eq,CL96a,H96}
% (for particles of 0 rapidity observed by the  STAR detector).
\bea
&&S_0(x,K)={\cal S}_0(\tau,\eta)\;B_\eta(\bfb, \bfK_T)/(2\pi)^{3}\label{fact}\\
&&{\cal S}_0(\tau,\eta)\equiv 
\frac{ \cosh\eta}{\sqrt{2\pi(\Delta \tau)^2}}  
            \exp \left[- {(\tau-\tau_0)^2 \over 2(\Delta \tau)^2}
           -{\eta^2\over2\Delta\eta^2}\right]\;\\
&&B_\eta(\bfb, \bfK_T)\equiv M_{T}{1\over \exp({K\cdot u-\mu_\pi \over T})-1  }
\rho(b),
%\exp\left({-b^2\over 2R^2}\right),
\label{S0}\eea 
in which 
components of  $x^\mu$ are expressed using the
variables $\tau=\sqrt{t^2-z^2},~\eta=\frac{1}{2}\log{t+z\over t-z},$  $\bfb \equiv (x_1,x_2),$ and
  $K\cdot u= M_{T} \cosh\eta\cosh \eta_t(b) 
                - K_{T} \, \sinh \eta_t(b) \,\cos \phi$,
where $\phi$ is the angle between $\bfK_T$ and $\bfb$,
%${\cal S}_0(\tau,\eta)\equiv 
%\frac{1}{\sqrt{2\pi(\Delta \tau)^2}}  
%            \exp [- {(\tau-\tau_0)^2 \over 2(\Delta \tau)^2}
%           -{\eta^2\over2\Delta\eta^2}],$
%$\bfx_\perp$,$ %\;\eta={1\over 2}\ln{t+z\over t-z},\;\tau=\sqrt{t^2-z^2}, 
 $ M_T=\sqrt{K_{T}^2+m_\pi^2}$, and $\mu_\pi$ is the pion chemical potential.
The function $\rho(b)$ represents the cylindrically symmetric 
 transverse source density. We use
$\rho(b)= [1/(\exp((r-R_{WS})/a_{WS})+1)]^2$, reflecting superimposed nuclear densities.
%%%here
      Eq.~(\ref{fact})
represents  the transverse flow rapidity using 
 a linear radial profile of strength $\eta_f$:
$
  \eta_t(b) = \eta_f {b \over R_{WS}}
$. Eq.~(\ref{fact}) incorporates the finite lifetime and size of the source:
 pions are
emitted for a duration controlled by the parameters $\Delta\tau$ and $\Delta\eta$.
Using Eq.~(\ref{fact}) is not sufficient  to capture all of the physics. As
the basis of the blast wave parameterization,  it gives $\Delta \tau\approx0$ and does not
predict the magnitude of the pion spectrum\cite{BW}. 

The salient feature 
of  the 200  GeV data is the high density of the produced matter, so we
treat  the effects of pion 
interactions with the dense medium.
We adopt  a single-channel  approach that 
uses 
the  
interaction--distorted incoming wave
%function
$\Psi_{\bfp_1}^{(-)*}(x_1)$ 
in which\cite{GKW79}:
\bea  S(x,K) & = & \int d^4K' S_0(x,K')\int {d^4x'\over(2\pi)^4}\;e^{-i K'\cdot x'} \label{convo}  \\
&&\times ~\Psi_{\bfp_1}^{(-)}(x+x'/2) 
\Psi_{\bfp_2}^{(-)*}(x-x'/2).\nonumber\eea
One could imagine a generalization in which $\Psi$ would have two or more components 
(corresponding {\it e.g.} to a $\rho$ final state in addition to a pion final state). 
Our task here is 
to compute pion correlation functions, so that it is sufficient to use a single channel wave function.
One obtains the single-pion emission function from Eq.~(\ref{convo}) by 
using the same momentum (either $\bfp_1$ or
 $\bfp_2$) to compute the wave function $\Psi^{(-)}_\bfp$.

Using Eq.~(\ref{convo})  requires  evaluating  an     eight-dimensional integral,
and modelling the interactions that determine $\Psi_{\bfp_1}^{(-)}$. We use  
 symmetries to reduce the number of integrals and obtain a tractable treatment
of the interactions. First, note that 
$ \Psi_\bfp^{(-)}(x)$  
is an energy-eigenfunction \cite{GKW79}:
$ \Psi_\bfp^{(-)}(x)=e^{-i \omega_p\;x^0}\Psi_\bfp^{(-)}({\bf x}).$
 We assume that the 
matter formed in the central region of the collision 
is cylindrically symmetric with a very long axis, 
so that 
\bea
\Psi_{\bfp_{1,2}}^{(-)}({\bf x})& = &
e^{\mp iq_l z/2}\psi_{\bfp_{1,2}}^{(-)}({{\bf x_\perp}=\bfb}),\label{psidef}\\
\quad \bfp_{1,2} & = & \bfK\pm \bfq_T/2\pm {\bf\widehat{z}}\;q_l/2, \nonumber
\eea 
with $\psi_\bfp^{(-)}({\bfb})$  obtained by solving
a two-dimensional  Klein-Gordon  %%gm816 wave 
equation
\bea \left(-\nabla_\perp^2 +U(b)\right)\psi_\bfp^{(-)*}({\bfb})=p^2
\psi_\bfp^{(-)*}({\bfb}).
\label{wave}\eea
The ``optical potential'' $U$ is a complex, azimuthally-symmetric function 
depending on pion momentum and local density that represents
the strength of the interaction between a pion and the medium.  Within our
formalism the influence of time-dependent effects in $U$ introduced by
the time-dependent source $S_0$ is incorporated in the energy dependence of
the optical potential. However, we note that the pions-medium interaction
time is %rather short, as 
restricted by $S_0$.

At large values of $K$ the solution of Eq.~(\ref{wave}) 
reduces to well-known semi-classical (eikonal) expression, but
our procedure is more general. For example
at $K$=0 there is no distinction between the out and side directions, so
$R_o$=$R_s$. This constraint is violated if the eikonal approximation to (\ref{wave}) is used. 
%violates the constraint that $R_o=R_s$ for  $K=0$. 
We use a partial wave expansion: 
 \bea
\psi_\bfp^{(-)*}({\bfb})=f_0(p,b)+2\sum_{m=1,\infty}f_m(p,b){(-i)}^m\;\cos{m\phi}\label{pwe}\eea
 to   solve Eq.(\ref{wave}) exactly  and maintain the constraint.

The optical potential accounts for %%%
situations in which the pion changes energy or disappears entirely due to its
interactions with the dense medium.
Suppose, {\it e.g.}, that 
the medium is a gas of pions. Then 
$\pi\pi$ scattering would be  the origin of $U$.
In the impulse approximation, the central optical potential would be
$U_0=-4\pi f\rho_0$, where $f$ is the complex 
forward scattering amplitude and $\rho_0$ the central density. 
For low energy pion-pion interactions, $4\pi$Im$[f(p)]=p\sigma,$
with $\sigma\approx 1\; $mb. At a %%%pion 
momentum $p=1~\rm{fm}^{-1}$, %%gm816
 %%%
using a pion density about
ten times the baryon density of ordinary nuclear matter, %%% we have
%%% $\rho_0\approx 1.5 \;\rm {fm}^{-3}$, so  %%gm816
%%comment: we use rho(b) normalized to one at b=0. thus need different notation here 
Im$[U(0)]\approx -0.15~\rm{fm}^{-2}$, representing significant opacity.
Furthermore, if two interacting 
pions each have less energy than half  of the rho meson mass, the 
final state interactions would be strongly attractive.

We  model the interaction $U$ for a situation in which 
the medium is dense, but we have not  otherwise specified  its  nature.
Now we consider chiral symmetry, which 
 gives a general form for the dispersion relation of low energy 
pions in nuclear matter:
%Now consider the
%interesting possibility that, in the dense medium of a RHIC collision,
 % chiral symmetry is restored and
 %the quark condensate vanishes, and consequently the pion mass changes\cite{Koch:1997ei}.
%%%Since the pion mass 
%%is proportional to the quark condensate, it should also change\cite{Koch:1997ei}. Indeed, %%gm816 
%\cite{Son:2002ci}:
 \bea
\omega^2=v^2(\hat{p}^2 +m^2_\pi(T)).\label{mode}\eea
 Here $m_\pi(T) $ is the $\pi$ screening mass, $v$ is the
so-called $\pi$ velocity, $\hat{p}$ is the momentum operator, and the
 pole mass is $m_\pi(T)v$. Son and Stephanov\cite{Son:2002ci}
argued   that $v$ and the pion pole mass 
decreases at 
temperatures near the critical temperature, while the screening
mass increases. Several papers have used specific low-order 
model calculations that challenge
 this result\cite{boy,sas}. Here we start with the general form (\ref{mode}) that appears
in both Refs.~\cite{Son:2002ci,boy} and use data to determine the parameters.  
%Defining $t\equiv(T-T_c)/T_c$, 
%SS find $m_\pi^2(T)\sim t^{\beta-\nu}$ and $v^2\sim t^\beta$, 
%with $\beta<\nu$, e.g.,
%$\nu=.73,~\beta=.38$ \cite{Baker:1977hp}. These equations 
%are valid for temperatures $T$  near (but not at) 
%the chiral critical point $T_c$.  
We first define an equivalent $U$ by using
$\omega^2\equiv p^2+m_\pi^2$, Eq.~(\ref{mode}), and the Klein Gordon equation
$
\hat{p}^2 +U=p^2$,
%%gm816 Subtract (\ref{opt}) from (\ref{mode}) 
to  obtain:
\bea U=-(m_\pi^2-v^2m_\pi^2(T)) -(1-v^2)\hat{p}^2. \label{uttt}\eea 
%get strong attraction.

For matter of finite size, Eq.~(\ref{uttt}) suggests a
 potential with terms constant and proportional to $\hat{p}^2$. The latter 
has 
the form $w_2p^2 -\nabla\cdot w_K\nabla$, where $w_{2,K}$ are complex constants. 
For the 200 GeV data, 
 the results prefer a small value of $w_K$, so we take $w_K=0$.
The optical potential is therefore modelled as:
\bea
U_p(b)= -(w_0 + w_2 p^2)\rho(b) ,\label{uopt}
\eea %%gm816
with $w_0$ real (no opacity at p=0).
The density profile $\rho(b)$ is that specified in
$S_0(x,K)$. This simple form is sufficient to account
for the data we study. 

Eqns.(\ref{fact},\ref{uopt}) are the  essence of our model. 
These may be compared
with the Buda-Lund model, which is  an 
efficient representation of the data\cite{Buda} in which
 the temperature and fugacity are taken as position-dependent functions appearing
in a Boltzmann distribution. The effects of our optical potential could provide
an explanation of those deduced dependencies. 

The use of Eqs.~(\ref{psidef},\ref{fact}) in Eq.~(\ref{convo}) gives
%\bea  S(x,K)= %{1\over(2\pi)^4}
%\int{ d^4K'_\perp \over(2\pi)^2}S_0(x;K^0,\bfK'_\perp)
%\int \tau d\tau d\eta e^{-iq^0t-iq_L z}
%\int d^2b'
% e^{-i\bfK_\perp'\cdot 
%\bfb'}
%\psi_{\bfp_1}^{(-)}(\bfb+\bfb'/2) 
%\psi_{\bfp_2}^{(-)*}(\bfb      -\bfb'     /2), \label{dwa1}%\nonumber\\
%\eea
%To compute radii it is sufficient
%to expand the exponentials involving $q^0,q_L$ to terms of second order. Then  
% the integrals over $\tau$ and $\eta$ can be done analytically.
%%%%here
%With Eqs.~(\ref{dwa1},\ref{fact}), the correlation function,
%$C(q,K)-1={\vert\int d^4x S(x,K)\vert^2}/{\int d^4x S(x,p_1)
%\int d^4x S(x,p_2)}$ becomes a function of $\bfq$ and $\bfK$.
\bea
S(x,K) & = & {1\over (2\pi)^2}{\cal S}_0(\tau,\eta) e^{iq^0t-iq_l z}%e^{i(\omega_1-\omega_2)t}\;
%\;\widetilde{B}^{\bfp_1\bfp_2}(\bfx_\perp,\bfK_\perp)\\
%\widetilde{B}^{\bfp_1\bfp_2}(\bfx_\perp,\bfK_\perp)
%\equiv
\int d^2b' \widetilde{B}_\eta(\bfb,\bfb')\nonumber\\
%*\exp\left[i\vec{q}_\perp\cdot\vec{r}\;'\right]
&& \times~\psi_{\bfp_1}^{(-)}(\bfb      +\bfb'     /2) 
\psi_{\bfp_2}^{(-)*}(\bfb      -\bfb'     /2),\label{cvo}\eea
%\psi^*_1(\vec{r}+\vec{r}\;'/2)\psi_2(\vec{r}-\vec{r}\;'/2)
where $
\widetilde{B}_\eta(\bfb,\bfb')\equiv \int d^2K'_{T}\;B_\eta(\bfb      ,\bfK'_{T}     )
\exp\left[-i\bfK_T'\cdot\bfb'     \right]. $
The range
of the variable $\bfb'$ appearing 
in $\widetilde{B}_\eta(\bfb,\bfb')$ 
is controlled by a size, $1/T$ that is much smaller than the source size. 
Thus it is reasonable to 
 ignore the $\pm\bfb'/2 $ appearing in (\ref{cvo}). 
Maintaining the phases is important,  %%gm816 However, one needs to maintain the correct  phases,
so we use 
$ \psi_\bfp^{(-)*}(\bfb\pm\bfb'/2)\equiv
e^{-i\bfp\cdot(\bfb       \pm\bfb'/2)}
\chi_\bfp(\bfb\pm\bfb'/2)\to 
e^{-i\bfp\cdot(\bfb\pm\bfb'/2)}\chi_\bfp(\bfb)$.
This approximation is exact in several different limits: plane wave, %%gm816 limit,
short wavelength, 
 long wave length, %%gm816 limit, 
and taking   $(R_{WS}T\to\infty)$.

 
We expand the exponentials in (\ref{cvo}) involving $q^0,q_l$ to 2nd order,
 expand the Bose-Einstein function
in a series of Boltzmann functions of temperature $T_n=T/n$, and analytically
evaluate integrals over   
 $\tau$ and $\eta$ 
  to obtain:
\bea  &&C(\bfq,\bfK_T)  =  1-q_l^2R_l^2 -q^2_o\beta^2\widetilde{\Delta\tau}^2+
\frac{\vert\Phi_{12}\vert^2}{\Phi_{11}
\Phi_{22}}, \\
&&\Phi_{ij}  =\sum_n  \int d^2bf_0(\xi_n(b)) 
\psi^{(-)}_{\bfp_i}(\bfb)\psi_{\bfp_j}^{(-)*}(\bfb)
B_n(\bfb,\bfK_T)\nonumber \\
&&B_n(\bfb,\bfK_T)=\exp({\mu_\pi+K_T\sinh\eta_t(b)\cos\phi\over T_n})M_T\rho(b)\nonumber\\
%&&\Phi_{ii}  \equiv  \int d^2bf_0(\xi) %%gm816beta,\Delta\eta)
%\exp\left(-\bfx_\perp^2\over 2R^2\right) 
%{{\vert}}\psi^{(-)}_{\bfp_i}(\bfb){{\vert}}^2B(\bfb,\bfp_i)\label{phidefs}\\
&&R_l^2  =  {(3\Delta\tau^2+\tau_0^2)F_1(K)\over F_0(K)}\nonumber\\
&&\widetilde{\Delta\tau}^2=(3\Delta\tau^2+\tau_0^2){F_3(K_T)\over F_0(K_T)}-
%(\tau_0+{\Delta\tau^2\over\tau_0})^2
\left\vert
{(\tau_0^2+\Delta\tau^2) F_2(K_T)\over \tau_0F_0(K_T)}\right\vert^2\nonumber\\
&&F_m(K)  =\sum_n  \int d^2bB_n(\bfb,\bfK_T)
f_m(\xi_n(b)){{\vert}}\psi^{(-)}_{\bfK_T}(\bfb){{\vert}}^2%F(K,b)
\nonumber
\eea
%\\
%\times
%\left[\sum_{m,n=0}^\infty\epsilon_m\epsilon_n f_m(p_i,b)f_n^*(p_i,b)(i)^{n-m}
% (I_{m+n}(p\sinh{\eta_t(b)}/T)+I_{m-n}(p\sinh{\eta_t(b)}/T))\right]\\
%&&F_1(K)  =    \int d^2bB(\bfb,\bfK_T) f_1(\xi) %%gm816 beta,\Delta\eta)
%{{\vert}}\psi^{(-)}_{\bfK_T}(\bfb){{\vert}}^2\eea
where $\xi_n(b)=M_T\cosh\eta_t(b)/T_n+1/\Delta\eta^2,\;
f_1(\xi)\equiv 2 K_0(\xi)/\xi+4K_1(\xi)/\xi^2,\;
f_0(\xi)\equiv 2K_1(\xi),\;f_2(\xi)=K_0(\xi)+K_2(\xi),\;f_3(\xi)=
2(K_1(\xi)+K_2(\xi)/\xi)$, 
and $K_j$ are modified Bessel functions.
The pion spectrum is  given by:
\bea
\left\langle
{dN\over 2\pi M_{T}  dM_{T} dY}\right\rangle_{\vert Y\vert<0.5}
=~{\tau_0\over 8\pi^3}e^{1\over\Delta\eta} F_0(\bfK_T).\label{spectrum}\eea
The angular  integrals are  performed 
analytically  %partial wave 
Eq.~(\ref{pwe}). 
 The transverse HBT radii are 
 $R_i(K_T)=\sqrt{1-C(\Delta q_i,K_T)}/\Delta q_i$, with $i=o,s$ and $\Delta q_i\approx K_T/40$.


The values of the parameters 
$T,\eta_f,\Delta\tau,R_{WS},a_{WS},$
$w_{0,2},\tau_0,
\Delta \eta$ and $\mu_\pi$ 
are varied to reproduce the STAR 200 GeV data for $R_o, R_s, R_l$ \cite{STARHBT} and
the magnitude and shape of the pion spectrum \cite{STARspec}.
%%gm816 already said w_0 is real 
The resulting parameters are
displayed in Table I, with the variances that produce a $\chi^2$ increase of one unit.  
The agreement of the curves with STAR HBT radii \cite{STARHBT} in Fig. 1 
and with the STAR pion spectrum \cite{STARspec} in Fig. 2
 is quite good, giving a $\chi^2\approx$ 3.7 per data point and 5.6 per degree of freedom.  


\begin{figure}
\includegraphics[width=10 cm]{OpacFig1.eps}% Import EPS figure
%\includegraphics[width=8 cm]{OpacFig1_nd.eps}% Import EPS figure
\caption{\label{one}  (Color online) HBT Radii $R_s, R_o,$, $R_l$ and the ratio $R_o/R_s$; 
Data \cite{STARHBT}): $\nabla$ (green) $\Rightarrow \pi^{+}\pi^{+}$; $\triangle$ (red) $\Rightarrow \pi^{-}\pi^{-}$.  
Curves: solid (black) $\Rightarrow$ full calculation; 
dotted (green) $\Rightarrow\eta_f=0$ (no flow); dashed (red) $\Rightarrow$ Re$[U]$=0 (no refraction); dot-dashed (blue) $\Rightarrow$ 
$U$=0 (no potential), double-dot-dashed (magenta) $\Rightarrow$ substituting Boltzmann for Bose-Einstein thermal distribution.
Insets show predictions of low-$K_T$ resonance behavior in $R_o$ and $R_s$.}
\end{figure}

\small
\begin{table*}
  \centering
  \caption{Parameters of the calculation with variances}
  \vspace{0.1cm}
  \begin{tabular}{|rrrrrllrrr|}\hline
   $~~T(MeV)$ & $~~\eta_f~~$ & $\Delta\tau(fm/c)$ & $R_{WS}(fm)$ & $~~a_{WS}(fm)$& $~~w_0(fm^{-2})$& $~~~~~~~~w_2$
& ~~~$\tau_{0}(fm/c)$ &  ~~~$\Delta\eta$~~~ & $\mu_\pi(MeV)$\\
\hline
173.2~~   & ~~1.314 & 2.852~~~ & 11.728~ & 0.725~~ & ~~~0.137 & 0.582~+~$i$~0.121 & 8.23~~~ &  1.063~~~& 123.2~~~\\
~~$\pm$1.6~~   & ~~$\pm$0.025 & $~\pm$0.067~~~ & ~$\pm$0.056~~~ & $\pm$0.015~~ & ~$\pm$0.046 & $\pm$0.014~~$\pm$0.002~~~ & ~$\pm$0.10~~~ &  $\pm$0.032~~~& $\pm$1.1~~~\\

      \hline
      \end{tabular}
      \end{table*}
\normalsize


\begin{figure}
\includegraphics[width=8.5 cm]{OpacFig2.eps}% Import EPS figure
\caption{\label{three} (Color online) 
Pion momentum spectrum.
Data \cite{STARspec}: $\nabla$ (green) $\Rightarrow \pi^{+}$; $\triangle$ (red) $\Rightarrow \pi^{-}$. 
Inset is low-$K_T$ prediction.}

\end{figure}


We need to assess and interpret the values of our parameters.
The temperature ($T$=173 MeV) is comparable to the $T_c$ expected for a chiral phase
 transition ($\approx$160 MeV).
The  
flow rapidity ($\eta_f$=1.31)  corresponds to a maximum flow velocity of 0.85 c,
 which is not unreasonable.  The source size is large ($R_{WS}$=11.7 fm),
 4.4 fm larger than a gold nucleus ($R$=7.3 fm).  If the system
 requires an expansion time $\tau_0$= 8.22 fm/c to expand by this
 amount, the average expansion velocity would be about 0.5 c, which is reasonable. 
 The emission duration ($\Delta\tau$=2.9 fm/c) is comfortably smaller than $\tau_0$. The longitudinal width ($\Delta\eta$=1.063) is smaller than expected, but
 is large enough that
the system's axial length ($\sim 2\tau_0 \Delta \eta\sim 17.5$ fm) is sufficient for the
approximate validity of Eq.~(\ref{psidef}). 

We assess the strength of $U$ by examining it at $p \approx 1$ fm$^{-1}$. At the system center, the
%%%gm
 momentum-dependent term of $U$
is --(0.58+0.12$i$) fm$^{-2}$. To understand the imaginary part, which is comparable to
 our estimate given above (and consistent with the presence of high density
matter), consider the equivalent classical mean free path: 
 $p/\vert Im[U]\vert \approx 8~\rm{fm}\ll 2R_{WS}$. 
Thus, the imaginary potential is large enough to 
restrict emission of pions from regions 
deep inside the medium. To understand the real part,
note that the strength of the attraction is greater than $m_\pi^2=0.49 \;{\rm fm}^{-2}$. 
Thus inside the medium,  the pion acts as if it has no mass. This is why we
assert that a chiral phase transition has occured. 

We further 
recall Eq.~(\ref{uttt}), and 
note that a momentum-dependent term 
 of -0.58 fm$^{-2}$ corresponds $v= 0.65$. The
 momentum independent term of $U$ (-.14 fm$^{-2}$) corresponds to %%%a pion pole mass 
of $vm_\pi(T)=0.6$ fm$^{-1}$. These values are comparable to the estimates of \cite{Son:2002ci},
but not too far from the lowest value $v=0.83$ of Ref.~\cite{sas}.
%$Re [U]<-m_\pi^2=-0.5~\rm{fm}^{-2}$ in the center of the system, but vanishes
% outside.
%Thus, in the emission regions $Re[U] +m_\pi^2$ nearly vanishes, as  
%expected \cite{Son:2002ci} for a chiral phase transition.

The chemical potential and Bose-Einstein distribution have modest effects on the radii
 but are very important for the normalization and slope of the pion spectrum.
 
%which can be taken as another indication that a chiral phase transition has occurred.  

%The computed radii $R_s, R_o, R_l$, and the ratio $R_s/R_o$ are displayed vs.~$K_T$
%in Fig.~\ref{one}.  The computed pion spectrum is shown
%in  Fig.~\ref{three}.
%In each case, an excellent description of the data is achieved.
%\begin{figure}
%\includegraphics[width=8 cm]{OpacFig2-11.eps}% Import EPS figure
%\caption{{\label{two} The ratio $R_o/R_s$}}
%\end{figure}
Figs.~\ref{one},~\ref{three} show the importance of both the real and imaginary parts of the
optical potential: omitting either drastically changes the predictions. 
The attractive  real potential is the critical element needed to reproduce the
$K_T$ dependence of $R_o$ and $R_S$.

We predict peaks in $R_o$ and $R_s$ at low momentum ($p\approx$ 15--65 MeV/c)
and  a rapid rise and peaking in the low-momentum spectrum.  
This is a pionic version of the
Ramsauer effect, in which cross sections show peaks when the scattered wave
is in phase with the incident plane wave. We 
confirm the computed existence of such peaks through 
analytic calculations for a purely real
attractive square well potential. %The STAR and 
%PHENIX detectors  cannot detect pions
%of such low momenta, but 
The PHOBOS detector 
(and perhaps the BRAHMS detector) at RHIC could,
in principle,
confront these low--momentum predictions of structure.

Our 
relativistic quantum-mechanical treatment of refractive and opacity effects
on two-pion correlations and $\pi$ spectra
has surmounted a number of technical problems 
associated with previous semi-classical approximations, while achieving an 
excellent description of the 200 GeV STAR data. The results are consistent with the
interpretation that the dense medium has undergone a chiral phase transition.  
The
definitive predictions of interesting momentum dependences at small
 momenta should be testable in present and future RHIC
experiments.
 


%\section*{Acknowledgments}
This work is partially supported by the USDOE grants DE-FG-02-97ER41014 and DE-FG-02-97ER41020. GAM thanks LBL, TJNAF and BNL for their
hospitality during the course of this work. We thank J.~Draper, S.~Reddy and D. Son for useful
discussions. %, and T.~Hallman and the STAR Collaboration for use of the HBT data.
\begin{thebibliography}{99}

%\cite{Kolb:2003dz}
\bibitem{Kolb:2003dz}
P.~F.~Kolb and U.~Heinz,
%``Hydrodynamic description of ultrarelativistic heavy-ion collisions,''
%%gm816 no need to put arXiv. prl doesnt require it:
nucl-th/0305084 
(in {\it Quark Gluon Plasma 3}, Editors: R.C. Hwa and X.-N. Wang, World Scientific, Singapore)
%%CITATION = NUCL-TH 0305084;%%

\bibitem{BP-HBT}
S.~Pratt,
%``Pion Interferometry For Exploding Sources,''
Phys.\ Rev.\ Lett.\  {\bf 53}, 1219 (1984).
%%CITATION = PRLTA,53,1219;%%
	G. F. Bertsch {\it et al.}, % M. Gong, and M. Tohyama, 
Phys. Rev. {\bf C37}, 1896 (1988).
%	G. F. Bertsch, Nucl. Phys. {\bf A498}, 173c (1989).
%%CITATION = PHRVA,C37,1896;%%

%\cite{Rischke:1996em}
\bibitem{Rischke:1996em}
D.~H.~Rischke and M.~Gyulassy,
%``The time-delay signature of quark-gluon plasma formation in  relativistic nuclear collisions,''
Nucl.\ Phys.\ A {\bf 608}, 479 (1996)
%[arXiv:nucl-th/9606039].
%%CITATION = NUCL-TH 9606039;%%

%\cite{Adler:2001zd}
\bibitem{Adler:2001zd}
C.~Adler {\it et al.}  [STAR Collaboration],
%``Pion interferometry of s(NN)**(1/2) = 130-GeV Au + Au collisions at  RHIC,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 082301 (2001);
%[arXiv:nucl-ex/0107008].
%%CITATION = NUCL-EX 0107008;%%
%\cite{Adcox:2002uc}
%\bibitem{Adcox:2002uc}
K.~Adcox {\it et al.}  [PHENIX Collaboration],
%``Transverse mass dependence of two-pion correlations in Au + Au  collisions at s(NN)**(1/2) = 130-GeV,''
Phys.\ Rev.\ Lett.\  {\bf 88}, 192302 (2002);
%%CITATION = NUCL-EX 0201008;%%
%\cite{Enokizono:2002dn}
%\bibitem{Enokizono:2002dn}
A.~Enokizono  [PHENIX Collaboration],
%``Two-particle correlations measured by PHENIX in Au Au collisions at  s(NN)**(1/2) = 200-GeV,''
Nucl.\ Phys.\ A {\bf 715}, 595 (2003).
%[arXiv:nucl-ex/0209026].
%%CITATION = NUCL-EX 0209026;%%



\bibitem{Heinz:2002un}
U.~W.~Heinz and P.~F.~Kolb,
%``Two RHIC puzzles: Early thermalization and the HBT problem,''
hep-ph/0204061.
%%CITATION = HEP-PH 0204061;%%


%\cite{Heiselberg:1997vh}
\bibitem{Heiselberg:1997vh}
H.~Heiselberg and A.~P.~Vischer,
%``Bose-Einstein correlations from opaque sources,''
Eur.\ Phys.\ J.\ C {\bf 1}, 593 (1998);
%%CITATION = EPHJA,C1,593;%%
%\bibitem{Tomasik:1998qt}
%B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two particle interferometry. II: Opacity effects,''
%nucl-th/9805016;
%%CITATION = NUCL-TH 9805016;%%
%\bibitem{Tomasik:1999kz}
B.~Tomasik and U.~W.~Heinz,
%``Potentials and limits of Bose-Einstein interferometry,''
Acta Phys.\ Slov.\  {\bf 49}, 251 (1999);
%%CITATION = NUCL-TH 9901006;%%
%L.~McLerran and S.~S.~Padula,
%``R(out)/R(sid) and opacity at RHIC,''
%%gm816arXiv:
%nucl-th/0205028.
%%CITATION = NUCL-TH 0205028;%%
D.~Molnar and M.~Gyulassy,
%``Differential freezeout and pion interferometry at RHIC from covariant  transport theory,''
Phys.\ Rev.\ Lett.\  {\bf 92}, 052301 (2004).
%%CITATION = NUCL-TH 0211017;%%
%\cite{Pratt:wm}

\bibitem{STARHBT}J.~Adams, {\it et al.}  [STAR Collaboration],
Phys.\ Rev.\ C (submitted for publication, 2004);nucl-ex/0411036
%M.~L\'{o}pez Noriega (STAR Collaboration), 
%Proc. of 2nd Warsaw Meeting on 
%Particle Correlations and Resonances in Heavy Ion
%Collisions, to be published in {\it  Nukelonika}, Supplement 1, (2004).

\bibitem{STARspec}
J.~Adams, {\it et al.} [STAR Collaboration], Phys. Rev. Lett. {\bf 92}, 112301 (2004).

\bibitem{Wiedemann:1999qn}
U.~A.~Wiedemann and U.~W.~Heinz,
%``Particle interferometry for relativistic heavy-ion collisions,''
Phys.\ Rept.\  {\bf 319}, 145 (1999). %[arXiv:nucl-th/9901094].
%%CITATION = NUCL-TH 9901094;%%



\bibitem{BW} 
F.~Reti\`{e}re and M.~A.~Lisa,
%``Observable implications of geometrical and dynamical aspects of freeze-out
%in heavy ion collisions,''
arXiv:nucl-th/0312024.
%%CITATION = NUCL-TH 0312024;%%

\bibitem{GKW79}
M.~Gyulassy, S.~K.~Kauffmann and L.~W.~Wilson,
%``Pion Interferometry Of Nuclear Collisions. I. Theory,''
Phys.\ Rev.\ C {\bf 20}, 2267 (1979).
%%CITATION = PHRVA,C20,2267;%%

%\cite{Son:2002ci}
\bibitem{Son:2002ci}
D.~T.~Son and M.~A.~Stephanov,
%``Real-time pion propagation in finite-temperature QCD,''
Phys.\ Rev.\ D {\bf 66}, 076011 (2002);
%%CITATION = HEP-PH 0204226;%%has a convergent partial wave expansion. \section{Moment Analysis}
D.~T.~Son and M.~A.~Stephanov,
%``Pion propagation near the QCD chiral phase transition,''
Phys.\ Rev.\ Lett.\  {\bf 88}, 202302 (2002)
%%CITATION = HEP-PH 0111100;%%
%\bibitem{Pratt:wm}
%S.~Pratt,
%``Two Particle And Multiparticle Measurements For The Quark - Gluon Plasma,''
% in Hwa, R.C. (ed.): Quark-gluon plasma, vol.2, page 700-748.

\bibitem{boy}
D.~Boyanovsky, H.~J.~de Vega and S.~Y.~Wang,
%``Nonequilibrium pion dynamics near the critical point in a constituent quark
%model,''
Nucl.\ Phys.\ A {\bf 741}, 323 (2004).
%%CITATION = HEP-PH 0312185;%%
\bibitem{sas}C.~Sasaki,
%``Pion velocity near the chiral phase transition,''
arXiv:hep-ph/0404079.
%%CITATION = HEP-PH 0404079;%%

\bibitem{Buda}
M.~Csanad, T.~Csorgo, B.~Lorstad and A.~Ster,
%``Indication of quark deconfinement and evidence for a Hubble flow in 130-GeV
%and 200-GeV Au + Au collisions,''
J.\ Phys.\ G {\bf 30}, S1079 (2004);
%%CITATION = NUCL-TH 0403074;%%
T.~Csorgo and B.~Lorstad,
%``Bose-Einstein Correlations for Three-Dimensionally Expanding, Cylindrically
%Symmetric, Finite Systems,''
Phys.\ Rev.\ C {\bf 54}, 1390 (1996);
%%CITATION = HEP-PH 9509213;%%
M.~Csanad, T.~Csorgo and B.~Lorstad,
%``Buda-Lund hydro model for ellipsoidally symmetric fireballs and the elliptic
%flow at RHIC,''
Nucl.\ Phys.\ A {\bf 742}, 80 (2004).
%%CITATION = NUCL-TH 0310040;%%



%\cite{Heinz:2002un}
%\cite{Heiselberg:1997vh}
%\bibitem{Koch:1997ei}
%V.~Koch,
%%``Aspects of chiral symmetry,''
%Int.\ J.\ Mod.\ Phys.\ E {\bf 6}, 203 (1997).
%%CITATION = NUCL-TH 9706075;%%

%\cite{Baker:1977hp}
%\bibitem{Baker:1977hp}
%G.~A.~Baker, B.~G.~Nickel and D.~I.~Meiron,
%``Critical Indices From Perturbation Analysis Of The Callan-Symanzik
%Equation,''
%Phys.\ Rev.\ B {\bf 17}, 1365 (1978).
%%CITATION = PHRVA,B17,1365;%%

%\bibitem{Tomasik:1997eq}
%B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two-particle interferometry: 
%Effects from opacity and  temperature gradients in the source,''
%Eur.\ Phys.\ J.\ C {\bf 4}, 327 (1998).
%[arXiv:nucl-th/9707001].
%%CITATION = NUCL-TH 9707001;%%
%\cite{Tomasik:1998qt}
%B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two particle interferometry. II: Opacity effects,''
%arXiv:nucl-th/9805016.
%\bibitem{CL96a}
%  T. Cs\"org\H o and B. L\"orstad, Phys. Rev. C {\bf 54}, 1390 (1996)

%\bibitem{Tomasik:1998qt}
%B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two particle interferometry. II: Opacity effects,''
%arXiv:nucl-th/9805016.
%%CITATION = NUCL-TH 9805016;%%
%\bibitem{H96}
%  U. Heinz, in: {\it Correlations and Clustering Phenomena in Subatomic
%  Physics}, edited by M.N. Harakeh, O. Scholten, and J.H. Koch, NATO ASI 
%  Series B, (Plenum, New York, 1997)  
%  nucl-th/9609029
%\bibitem{CSH95b} 
 % S. Chapman, P. Scotto, and U. Heinz, Heavy Ion Physics {\bf 1}, 1 (1995)
%\bibitem{WSH96}
 % U.A. Wiedemann, P. Scotto and U. Heinz, Phys. Rev. C{\bf 53}, 918 (1996) 







\end{thebibliography}
\end{document}

%\cite{Kolb:2003dz}
\bibitem{Kolb:2003dz}
P.~F.~Kolb and U.~Heinz,
%``Hydrodynamic description of ultrarelativistic heavy-ion collisions,''
nucl-th/0305084.
%%CITATION = NUCL-TH 0305084;%%

\bibitem{BP-HBT}
	G. F. Bertsch, M. Gong, and M. Tohyama, Phys. Rev. {\bf C37}, 1896 (1988).
	G. F. Bertsch, Nucl. Phys. {\bf A498}, 173c (1989).

%\cite{Rischke:1996em}
\bibitem{Rischke:1996em}
D.~H.~Rischke and M.~Gyulassy,
%``The time-delay signature of quark-gluon plasma formation in  relativistic nuclear collisions,''
Nucl.\ Phys.\ A {\bf 608}, 479 (1996)
%[arXiv:nucl-th/9606039].
%%CITATION = NUCL-TH 9606039;%%

%\cite{Adler:2001zd}
\bibitem{Adler:2001zd}
C.~Adler {\it et al.}  [STAR Collaboration],
%``Pion interferometry of s(NN)**(1/2) = 130-GeV Au + Au collisions at  RHIC,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 082301 (2001);
%[arXiv:nucl-ex/0107008].
%%CITATION = NUCL-EX 0107008;%%
%\cite{Adcox:2002uc}
%\bibitem{Adcox:2002uc}
K.~Adcox {\it et al.}  [PHENIX Collaboration],
%``Transverse mass dependence of two-pion correlations in Au + Au  collisions at s(NN)**(1/2) = 130-GeV,''
Phys.\ Rev.\ Lett.\  {\bf 88}, 192302 (2002);
%%CITATION = NUCL-EX 0201008;%%
%\cite{Enokizono:2002dn}
%\bibitem{Enokizono:2002dn}
A.~Enokizono  [PHENIX Collaboration],
%``Two-particle correlations measured by PHENIX in Au Au collisions at  s(NN)**(1/2) = 200-GeV,''
Nucl.\ Phys.\ A {\bf 715}, 595 (2003).
%[arXiv:nucl-ex/0209026].
%%CITATION = NUCL-EX 0209026;%%


%\cite{Heiselberg:1997vh}
\bibitem{Heiselberg:1997vh}
H.~Heiselberg and A.~P.~Vischer,
%``Bose-Einstein correlations from opaque sources,''
Eur.\ Phys.\ J.\ C {\bf 1}, 593 (1998);
%%CITATION = EPHJA,C1,593;%%
%\bibitem{Tomasik:1998qt}
B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two particle interferometry. II: Opacity effects,''
arXiv:nucl-th/9805016;
%%CITATION = NUCL-TH 9805016;%%
%\bibitem{Tomasik:1999kz}
B.~Tomasik and U.~W.~Heinz,
%``Potentials and limits of Bose-Einstein interferometry,''
Acta Phys.\ Slov.\  {\bf 49}, 251 (1999);
%%CITATION = NUCL-TH 9901006;%%
L.~McLerran and S.~S.~Padula,
%``R(out)/R(sid) and opacity at RHIC,''
arXiv:nucl-th/0205028.
%%CITATION = NUCL-TH 0205028;%%
D.~Molnar and M.~Gyulassy,
%``Differential freezeout and pion interferometry at RHIC from covariant  transport theory,''
arXiv:nucl-th/0211017.
%%CITATION = NUCL-TH 0211017;%%
%\cite{Pratt:wm}

%\cite{Son:2002ci}
\bibitem{Son:2002ci}
D.~T.~Son and M.~A.~Stephanov,
%``Real-time pion propagation in finite-temperature QCD,''
Phys.\ Rev.\ D {\bf 66}, 076011 (2002)
%%CITATION = HEP-PH 0204226;%%has a convergent partial wave expansion. \section{Moment Analysis}
D.~T.~Son and M.~A.~Stephanov,
%``Pion propagation near the QCD chiral phase transition,''
Phys.\ Rev.\ Lett.\  {\bf 88}, 202302 (2002)
%%CITATION = HEP-PH 0111100;%%
\bibitem{Pratt:wm}
S.~Pratt,
``Two Particle And Multiparticle Measurements For The Quark - Gluon Plasma,''
 in Hwa, R.C. (ed.): Quark-gluon plasma, vol.2, page 700-748.
\bibitem{Wiedemann:1999qn}
U.~A.~Wiedemann and U.~W.~Heinz,
%``Particle interferometry for relativistic heavy-ion collisions,''
Phys.\ Rept.\  {\bf 319}, 145 (1999). %[arXiv:nucl-th/9901094].
%%CITATION = NUCL-TH 9901094;%%
\bibitem{GKW79}
M.~Gyulassy, S.~K.~Kauffmann and L.~W.~Wilson,
%``Pion Interferometry Of Nuclear Collisions. I. Theory,''
Phys.\ Rev.\ C {\bf 20}, 2267 (1979).
%%CITATION = PHRVA,C20,2267;%%
%\cite{Heiselberg:1997vh}
\bibitem{Koch:1997ei}
V.~Koch,
%``Aspects of chiral symmetry,''
Int.\ J.\ Mod.\ Phys.\ E {\bf 6}, 203 (1997).
%%CITATION = NUCL-TH 9706075;%%
\bibitem{Tomasik:1997eq}
B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two-particle interferometry: Effects from opacity and  temperature gradients in the source,''
Eur.\ Phys.\ J.\ C {\bf 4}, 327 (1998).
%[arXiv:nucl-th/9707001].
%%CITATION = NUCL-TH 9707001;%%
%\cite{Tomasik:1998qt}
\bibitem{CL96a}
  T. Cs\"org\H o and B. L\"orstad, Phys. Rev. C {\bf 54}, 1390 (1996)

\bibitem{Tomasik:1998qt}
B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two particle interferometry. II: Opacity effects,''
arXiv:nucl-th/9805016.
%%CITATION = NUCL-TH 9805016;%%
\bibitem{H96}
  U. Heinz, in: {\it Correlations and Clustering Phenomena in Subatomic
  Physics}, edited by M.N. Harakeh, O. Scholten, and J.H. Koch, NATO ASI 
  Series B, (Plenum, New York, 1997) (Los Alamos eprint archive 
  nucl-th/9609029)
\bibitem{CSH95b} 
  S. Chapman, P. Scotto, and U. Heinz, Heavy Ion Physics {\bf 1}, 1 (1995)
\bibitem{WSH96}
  U.A. Wiedemann, P. Scotto and U. Heinz, Phys. Rev. C{\bf 53}, 918 (1996) 



%\cite{Baker:1977hp}
%\bibitem{Baker:1977hp}
%G.~A.~.~Baker, B.~G.~Nickel and D.~I.~Meiron,
%``Critical Indices From Perturbation Analysis Of The Callan-Symanzik
%Equation,''
%Phys.\ Rev.\ B {\bf 17}, 1365 (1978).
%%CITATION = PHRVA,B17,1365;%%
\end{thebibliography}
\end{document}
One suggestion 
 \cite{Heiselberg:1997vh} 
 %%gm816
%%One suggested explanation  \cite{Heiselberg:1997vh} 
is that an opaque source leads to a 
surface--dominated emission process in which
the  %%gm816
 spatial contribution to $R_o^2$ 
is much smaller than that for 
$R_s^2$. Then including the effects of emission duration could bring the
ratio  $R_o/R_s$ up to unity.  But computed effects seem too small to explain the data \cite{Kolb:2003dz}.
Previous treatments of opacity have used the eikonal (short wavelength)
%%gm816 semi-classical) 
approximation in which the intensity 
of the outgoing particle decreases exponentially
 as it transverses a linear path in the medium.
The pions with the largest influence on
HBT radii have long wavelengths, calling into question the eikonal approximation and mandating 
a quantum mechanical treatment. 
Moreover, previous opacity treatments have used only the imaginary (absorptive) part of the complex 
interaction between the emitted particle and the medium, while 
the interaction must also have a real part.
If the dense system formed in a heavy ion collision 
is near a chiral phase transition, 
a substantial attractive, real interaction is expected \cite{Son:2002ci}. 
%%gm816 Our aim 
a gas of mesons, or a quark-gluon plasma, or
a mixture also containing baryons and anti-baryons. 
%%%In particular, 
A loss of pion flux %%%through opacity 
is represented by the imaginary (absorptive)
part of the optical potential. a gas of mesons, or a quark-gluon plasma, or
a mixture also containing baryons and anti-baryons. 
%%%In particular, 
A loss of pion flux %%%through opacity 
is represented by the imaginary (absorptive)
part of the optical potential. 

nucl-th0411031.extracted_bib
\begin{thebibliography}{99}

%\cite{Kolb:2003dz}
\bibitem{Kolb:2003dz}
P.~F.~Kolb and U.~Heinz,
%``Hydrodynamic description of ultrarelativistic heavy-ion collisions,''
%%gm816 no need to put arXiv. prl doesnt require it:
nucl-th/0305084 
(in {\it Quark Gluon Plasma 3}, Editors: R.C. Hwa and X.-N. Wang, World Scientific, Singapore)
%%CITATION = NUCL-TH 0305084;%%

\bibitem{BP-HBT}
S.~Pratt,
%``Pion Interferometry For Exploding Sources,''
Phys.\ Rev.\ Lett.\  {\bf 53}, 1219 (1984).
%%CITATION = PRLTA,53,1219;%%
	G. F. Bertsch {\it et al.}, % M. Gong, and M. Tohyama, 
Phys. Rev. {\bf C37}, 1896 (1988).
%	G. F. Bertsch, Nucl. Phys. {\bf A498}, 173c (1989).
%%CITATION = PHRVA,C37,1896;%%

%\cite{Rischke:1996em}
\bibitem{Rischke:1996em}
D.~H.~Rischke and M.~Gyulassy,
%``The time-delay signature of quark-gluon plasma formation in  relativistic nuclear collisions,''
Nucl.\ Phys.\ A {\bf 608}, 479 (1996)
%[arXiv:nucl-th/9606039].
%%CITATION = NUCL-TH 9606039;%%

%\cite{Adler:2001zd}
\bibitem{Adler:2001zd}
C.~Adler {\it et al.}  [STAR Collaboration],
%``Pion interferometry of s(NN)**(1/2) = 130-GeV Au + Au collisions at  RHIC,''
Phys.\ Rev.\ Lett.\  {\bf 87}, 082301 (2001);
%[arXiv:nucl-ex/0107008].
%%CITATION = NUCL-EX 0107008;%%
%\cite{Adcox:2002uc}
%\bibitem{Adcox:2002uc}
K.~Adcox {\it et al.}  [PHENIX Collaboration],
%``Transverse mass dependence of two-pion correlations in Au + Au  collisions at s(NN)**(1/2) = 130-GeV,''
Phys.\ Rev.\ Lett.\  {\bf 88}, 192302 (2002);
%%CITATION = NUCL-EX 0201008;%%
%\cite{Enokizono:2002dn}
%\bibitem{Enokizono:2002dn}
A.~Enokizono  [PHENIX Collaboration],
%``Two-particle correlations measured by PHENIX in Au Au collisions at  s(NN)**(1/2) = 200-GeV,''
Nucl.\ Phys.\ A {\bf 715}, 595 (2003).
%[arXiv:nucl-ex/0209026].
%%CITATION = NUCL-EX 0209026;%%



\bibitem{Heinz:2002un}
U.~W.~Heinz and P.~F.~Kolb,
%``Two RHIC puzzles: Early thermalization and the HBT problem,''
hep-ph/0204061.
%%CITATION = HEP-PH 0204061;%%


%\cite{Heiselberg:1997vh}
\bibitem{Heiselberg:1997vh}
H.~Heiselberg and A.~P.~Vischer,
%``Bose-Einstein correlations from opaque sources,''
Eur.\ Phys.\ J.\ C {\bf 1}, 593 (1998);
%%CITATION = EPHJA,C1,593;%%
%\bibitem{Tomasik:1998qt}
%B.~Tomasik and U.~W.~Heinz,
%``Fine-tuning two particle interferometry. II: Opacity effects,''
%nucl-th/9805016;
%%CITATION = NUCL-TH 9805016;%%
%\bibitem{Tomasik:1999kz}
B.~Tomasik and U.~W.~Heinz,
%``Potentials and limits of Bose-Einstein interferometry,''
Acta Phys.\ Slov.\  {\bf 49}, 251 (1999);
%%CITATION = NUCL-TH 9901006;%%
%L.~McLerran and S.~S.~Padula,
%``R(out)/R(sid) and opacity at RHIC,''
%%gm816arXiv:
%nucl-th/0205028.
%%CITATION = NUCL-TH 0205028;%%
D.~Molnar and M.~Gyulassy,
%``Differential freezeout and pion interferometry at RHIC from covariant  transport theory,''
Phys.\ Rev.\ Lett.\  {\bf 92}, 052301 (2004).
%%CITATION = NUCL-TH 0211017;%%
%\cite{Pratt:wm}

\bibitem{STARHBT}J.~Adams, {\it et al.}  [STAR Collaboration],
Phys.\ Rev.\ C (submitted for publication, 2004);nucl-ex/0411036
%M.~L\'{o}pez Noriega (STAR Collaboration), 
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\end{thebibliography}