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\begin{document}
\title{Influence of Multiple Scattering on Particle Decays in a Medium}
\tocauthor{A. V. Koshelkin}
\author{A.V.Koshelkin\thanks{koshelkn@gpd.mephi.msk.su}.\\
\small\em Moscow Engineering Physics Institute,\\
\small\em Kashirskoye sh.31, Moscow 115409, Russia.}
\date{}
\maketitle

\begin{abstract}
{\small Impact of  scattering in matter on particle decays at high energy
is studied. The consistent method of the calculation of the decay probability
of the particles scattered  multiply elastically in an equilibrium  medium is
developed. The decay probability of such particles is obtained.  The Dalitz
decay of a pion in the matter is studied.}
\end{abstract}

PACS : 25.15.-g; 12.39.Fe; 12.38.Mh; 12.40.Vv.

\section{Introduction.}

The need to investigate of influence of  scattering in a medium on particle decays
arises in the study of states of the nuclear matter produced in the result of
collisions of high energy heavy ions~\cite{1,2,3,4,5,6,7,8,9,10}.
Photons and leptons produced in the  medium  are very
important source of the information about the state of the nuclear matter
because ones are electro-weak interacting particles only and therefore they
carry information about the primordial state of the nuclear matter studied.
In this way one of the most important  channel of production of such particles
is  decays in the matter.

Impact of a medium on damping rates is studied in the
papers~\cite{2,5,9,11,12,13,14,15,16}. On the basis of the resummation method
of infinite sets of diagrams which has been developed by E.Braaten and
R.D.Pisarski~\cite{17,18},  in-matter polarization effects ( renormalization of quark
and pion masses ) in QCD~\cite{5,18} is studied. In the
papers~\cite{11,12,13,14,15,16} the method of the hard thermal loops~\cite{17,18}
has been used and developed for the calculation of  self energy due to electron
photon  interaction in QED. In the papers~\cite{2,19} impact of the occupancy of final states
in the photon bath on decoupling constants has been investigated.

But the influence of multiple scattering of  particles on their decay are not studied
anywhere in the papers above mentioned. In this way  if energy fluctuations of the decaying
particle because of multiple scattering of this particle in matter is of the
order of the medium temperature impact  of multiple scattering in the matter
on the particle decay  is strong. This means that  in-medium effects of multiple
scattering  should be taken into account in calculations of decay damping
widths of the particles in the matter in this case.

We study decays of the particles undergoing multiple elastic collisions in an
equilibrium medium in this paper. The model-independent method of calculation
of the decay of such particles  is developed. Obtained decay
probability of particles depends strongly just as on the matter  temperature,
so on the parameters characterizing  particle scattering in the medium.
On the basis of the developed method the Dalitz decay of a pion in an equilibrium
pion gas is studied.

\section{Decay probability of a particle in a scattering medium.}

Let us consider  in-mater particles  with the spin  $s$ and the mass  $M$.
We suppose that the particles undergo multiple elastic
collisions in the medium which is in  thermodynamic equilibrium.
We  consider the  decay of the  particle according to the processes
$A \to A_1 + \dots + A_N   )$.

In the quasi-classical case when the wave length of a particle less than the
mean distance between them, the decoupling constant $\gamma (x, p) $ is given
by the well known expression ( see, for example~\cite{20} ) :
\begin{eqnarray}
&& \gamma (x, p) =- 2 Im \Biggl\{  Tr \Bigl\{ {\Sigma}^{ret} \Bigr\} \Biggr\},
\end{eqnarray}
where ${\Sigma}^{ret}$ is the retarded self energy in the Keldysh diagram
technique~\cite{21}, $x = (t, {\vec r}) $ and $  = (p^0, {\vec p}) $ are
a four-coordinate and a four-momentum.

Let us suppose that the influence of  particle scattering on the  decay
vertex and on the interaction of decay products in the final state is
negligible. Then we can write

\begin{eqnarray}
&&{\Sigma}^{ret} = \pm {\Sigma^{\alpha \beta}}_{vac}(p) G^{- +}_{\alpha \beta}(x, p)
\prod_{j=1}^N (1 \pm n_j ),
\end{eqnarray}
where ${\Sigma}^{\alpha \beta}_{vac}(p)$ is the self energy of the decaying
particle in vacuum; $G^{- +}_{\alpha \beta}(x, p)$ is the Green's function of the
particle in the matter; $n_j $ are the occupancy number of the decay products
( plus and minus signs correspond to the Boson and Fermi
statistics, respectively); $\alpha$ and  $\beta $ are the spin variables.

To obtain the decay damping width $\Gamma$, which would be observed, we should average
expression (1) over four-coordinates and four-momentums. Using the relation
between $G^{- +}_{\alpha \beta}(x, p)$ and particle density
$n(t, {\vec r}, {\vec p})$ at the thermodynamics
equilibrium~\cite{20,22} and integrating over $d^4 x $ and $d^4 p $  in the
formula (1), we obtain following expression for the damping width of an on-shell
particle :

%\begin{eqnarray}
\begin{equation}
%&& 
\Gamma =  {2 \pi \over t_0} \prod_{j=1}^N (1 \pm n_j )
Im \Biggl\{ \int\limits_0^{t_0} dt \int {d^3
{\vec p} \over \varepsilon }  Tr \Bigl\{ {\Sigma}^{\alpha \beta}_{vac}(p)
{\varrho}_{\alpha \beta} \Bigr\} \int d^3 {\vec r} \cdot n(t, {\vec r}, {\vec p})
\Biggr\}
%\end{eqnarray}
\end{equation}
where  ${\varrho}_{\alpha \beta}$ is the polarization density matrix of the
decaying particle; $t_0$ is the time during which the decay is observed ( the
observation time ); $p^0 \equiv \varepsilon ={\sqrt {{\vec p}^2   + M^2 } }$.
We set the normalization  volume $V = 1$.

Let us suppose that times of relaxation in the matter $\tau \ll
min\{ t_0, {\Gamma}^{-1}  \} $. This means that the static approximation~\cite{23}
for the description of the decaying particle movement in the medium can be
used. In this case the density $n(t, {\vec r}, {\vec p})$ satisfies the standard
transport equation~\cite{24}. We note that the function $n(t, {\vec r}, {\vec p})$
appears in Eq.(3) should be integrated over the particle coordinates ${\vec r}$.
Then, integrating transport equation~\cite{24} over $d^3 {\vec r} $ , we find
the following equation for the function $f( \varepsilon , {\vec p}, t) $
being  the integral of the density $n(t, {\vec r}, {\vec p})$ over the particle
coordinate :

\begin{eqnarray}
&& {\partial f (p ;t ) \over \partial t } =
\int d^3 {\vec p}_1 d{\sigma} (p - p^{\prime} ;p ; p_1 ) v f_{eq}(p_1 )
\Biggl\{ f(p^{\prime} ; t) - f(p; t ) \Biggr\} ,
\end{eqnarray}
where $d{\sigma} (p - p^{\prime} ;p ; p_1 ) $ is the cross section of a single
scattering of the particle in the medium; $f_{eq}(p_1 ) $ is an equilibrium
distribution function of the particles in the matter; \break
$v = {\sqrt{s(s - 4 m^2)}} / 2 \varepsilon  \varepsilon_1;
p=(\varepsilon , {\vec p}); p_1=(\varepsilon_1 , {\vec p}_1);
p^{\prime}=(\varepsilon^{\prime} , {\vec p}^{\prime}); s=(p + p_1 )^2 $.

Let us note that in the case of homogeneous and isotropic matter the
expression \break $ Tr \Bigl\{ {\Sigma}^{\alpha \beta}_{vac}(p)
{\varrho}_{\alpha \beta} \Bigr\} $ appears in the Eq.(3) depends only on
the particle energy but does not depend on the momentum of the decaying particle.
This circumstance allows us to rewrite the formula (3) as follows :


\begin{eqnarray}
&& \Gamma =  {1 \over t_0 } \int\limits_0^{t_0} dt \int\limits_0^{+ \infty} p^2 dp
{\Gamma}_{vac} (\varepsilon ) F( \varepsilon , t )
\end{eqnarray}

Here we introduce

\begin{eqnarray}
\Gamma_{vac}(p) = {2 \pi \over \varepsilon } \prod_{j=1}^N (1 \pm n_j )
Im \Biggl\{ Tr \Bigl\{ {\Sigma}^{\alpha \beta}_{vac}(p)
{\varrho}_{\alpha \beta} \Bigr\} \Biggr\}
\end{eqnarray}

\begin{eqnarray}
F( \varepsilon , t ) = \int d \Omega_{{\vec p}} f( \varepsilon , {\vec p} ),
\end{eqnarray}
where $d{\Omega}_{{\vec p}}$ is the solid angle in the direction of the vector
${\vec p}$. Function $F( \varepsilon , t )$ satisfies following equation obtained
by means of the integration of the expression (4) over all directions of the
vector ${\vec p}$ :

\begin{equation}
%\begin{eqnarray}
%&& 
{\partial F (\varepsilon ; t ) \over \partial t } \!=\!\!\!
\int\limits_{\varepsilon - E_{max}}^{\varepsilon + E_{max}} \!\!\!\!
d {\varepsilon}^{\prime}
\!\int \!\!d^3 {\vec p}_1 \!\int\! 
d{\Omega}_{\vec p}  {d{\sigma} ({\varepsilon} - {\varepsilon}^{\prime};
{\varepsilon}; p_1 )
\over d\varepsilon^{\prime} d{\Omega}_{\vec p} } v f_{eq}(p_1 )
\Biggl\{ F ({\varepsilon}^{\prime} ;t )  - F (\varepsilon ;t ) \Biggr\} ,
%\end{eqnarray}
\end{equation}
where $E_{max}$ is the maximum energy imparted in the result of the individual
elastic collision of two particles of the matter. The energy $E_{max}$ depends
on the energies $\varepsilon$ and $\varepsilon^{\prime}$ of the colliding
particles.

In the case when the matter is an equilibrium one and when elastic collisions
occur only the energy transmitted is small as compared with the matter
temperature. Then we can expand the collision integral in right hand of the
last expression over small parameter $\vert \varepsilon - \varepsilon^{\prime}
\vert / \varepsilon \ll 1 $. In the result we get :

\begin{eqnarray}
&& {\partial F (\varepsilon ;t ) \over \partial t } =
\alpha {{\partial}^2 F (\varepsilon ;t ) \over \partial {\varepsilon}^2 },
\end{eqnarray}
where $\alpha$ is the energy imparted to particle per unit time period.
When obtaining the last equation we use the fact that the cross section is the
even function of the transmitted energy. The parameter $\alpha$ is defined
by the following expression :

\begin{eqnarray}
&& \alpha = \int d^3 {\vec p}_1 d{\Omega}_{\vec p}
 \int\limits_{0}^{ E_{max}} d {\varepsilon}^{\prime}
 {d{\sigma} ({\varepsilon} - {\varepsilon}^{\prime};
{\varepsilon}; p_1 ) \over d\varepsilon^{\prime}  d{\Omega}_{\vec p}}
v f_{eq}(p_1 ) E^2
\end{eqnarray}

To solve the Eq. (4) we should specify the initial condition for the function
$F (\varepsilon ;  t )$. We suppose that some equilibrium distribution of the
particle is given at time $t=0$ :

\begin{eqnarray}
&& F (\varepsilon ; t=0 ) = f_{eq} (\varepsilon) \eta (\varepsilon ),
\end{eqnarray}
where $\eta (\varepsilon ) $ is the standard unit function ( $\eta (\varepsilon ) =1$,
 when $\varepsilon  \geq 0$, but $\eta (\varepsilon ) =0 $ at $\varepsilon  < 0$ );
$\varepsilon ={\sqrt {M^2 + {\vec p}^2 }}$ .

The parameter $\alpha$ determined by the Eq.(10) depends on the energy particle
$\varepsilon$. Except that in the case of an equilibrium matter the average value of the
particle energy is constant but the  energy imparted in the result of an
individual collision of  particles is  small as compared with the matter
temperature. Therefore we can set   $\varepsilon = \varepsilon_1 $ in the
formula (10) and think that $\alpha$ is constant approximately.
Then, solving the equation (9) with the initial condition (11) we get :

\begin{equation}
 F (\varepsilon ; t) = {\eta (\varepsilon ) \over {\sqrt {4 \pi \alpha t }}}
\int\limits_0^{+ \infty} d \varepsilon^{\prime} \Biggl\{
\exp \Bigl\{-{(\varepsilon - \varepsilon^{\prime})^2 \over 4 \alpha t }\Bigr\} +
\exp \Bigl\{-{(\varepsilon + \varepsilon^{\prime})^2 \over 4 \alpha t }\Bigr\}
\Biggr\} f_{eq} (\varepsilon^{\prime})
\end{equation}

Substituting obtained function $F (\varepsilon ; t)$ into the Eq.(5) we find
probability $ W $ of the particle decay in the scattering matter :

\begin{eqnarray}
&& W = \Gamma \cdot t_0 = \int\limits_0^{t_0} {dt \over {\sqrt{4 \pi \alpha}} }
\int\limits_0^{+ \infty } p^2 d p \Gamma_{vac} (\varepsilon (p))
\int\limits_0^{+ \infty} d\varepsilon^{\prime} \nonumber \\
&& \Biggl\{
\exp \Bigl\{-{(\varepsilon - \varepsilon^{\prime})^2 \over 4 \alpha t }\Bigr\} +
\exp \Bigl\{-{(\varepsilon + \varepsilon^{\prime})^2 \over 4 \alpha t }\Bigr\}
\Biggr\} f_{eq} (\varepsilon^{\prime})
\end{eqnarray}

The last formula determines the decay probability of the particle undergoing
multiple elastic collisions in an equilibrium matter.
The obtained expression for the decay probability does not
depend on the interaction model of the  particle  in the scattering medium
at all. The influence of the matter on the probability $W$ is only due to
the observable parameters such as the matter temperature (by means of the
equilibrium distribution function $f_{eq}$ ),  the mean square
energy imparted  by the particle in the result of an individual collision
$\alpha$ and the observation time $t_0$.

We should note that the density of the states of the particle in an
equilibrium matter has a sharp maximum at the energy $\varepsilon \sim T$ where
$T$ is the matter temperature. On the other hand it follows from the formula (13)
that the width of this maximum is determined by the relation
between the value of the energy fluctuations of the particle due to
the multiple scattering and the matter temperature. In this way the energy
fluctuation during the observation time $t_0$ is given by the magnitude of
the parameter $\alpha t_0 $. If the $\alpha t_0  \ge T $ influence of the
multiple scattering the particle decay is strong. In opposite case
$\alpha t_0  \ll T $ the impact is negligible.

Let us next consider applications of the developed method for the calculation
of decay probability of the particles in a scattering medium.

\section{Dalitz decay of a pion in a scattering medium.}

The  decay of a pion   ${\pi}^0 \to e^{+} e^{-} \gamma $ ( Dalitz decay )
is of interest in the connection of the study of dilepton pair production
in the nuclear matter arising in the result of collisions of  high energy heavy
ions~\cite{7,8,25}. Contribution of this decay into the rate of dilepton
pair production in the absence of multiple scattering of decaying
particle in  matter is studied in the paper~\cite{26}. As this
electromagnetic decay is also strong one virtually we use the quark model for
the calculation of its probability. In this way the calculation result depends
on both the quark and the strong interaction constant. We note that presently
the same combination of these parameters appears in the formula for
the width ${\Gamma}_{\pi^0 \to 2 \gamma }$ of the decay ${\pi^0 \to 2 \gamma }$.
Therefore we change above mentioned parameters by the expression for the
width ${\Gamma}_{\pi^0 \to 2 \gamma }$ in the final equation for the decay
probability of the pion in matter.

Ignoring  the distinction between masses of $u$ and $d$ quarks in the pion quark
model we write the following relativistic invariant  expression for the matrix
element of the decay $\pi^0 \to e^+ e^- \gamma $:

\begin{eqnarray}
&& {\cal M } = {4 \pi g e^{3/2} \over q^2 {m_q}^2 } {\epsilon}_{ijsr}
(v^{*} )^i (k) k^j q^s  {\bar u} (- p_+ ) \gamma^r u (p_- ) , \nonumber \\
&& q = p_+  +  p_-
\end{eqnarray}
where $p_- =(E_- , {\vec p}_- ), p_+ =(E_+ , {\vec p}_+ ),
k =(\omega , {\vec k}) $ are 4-momentum of an electron, a positron and
a photon, respectively; $u(p)$ and $\gamma^r$ are the Dirac spinor and the Dirac
matrixes ; $v^j (k)$ is the photon polarization vector; ${\epsilon}_{ijsr} $ is
the antisymmetric tensor; $e$ is the electron charge; $m_q$ is a quark mass;
$g$ is the constant of strong interaction which corresponds to the vertex of
the pion quark loop.

Calculating the decay rate according to the standard rules we find the
following expression for $\Gamma_{vac} (p)$ being inserted in the
Eq.(5) for the decay probability in matter :

\begin{eqnarray}
&& {d{\Gamma}_{vac}  \over dM_l }  =  { 4  e^2  \over 3 \pi \varepsilon
{M_l}^4 M^5 } (M^2 - {M_l}^2 )^3 ({M_l}^2 + 2 m^2 ) ( {M_l}^2 - 4 m^2 )^{1/2}
 {\Gamma}_{\pi^0 \to 2 \gamma } \nonumber \\
&& {\Gamma}_{\pi^0 \to 2 \gamma } = {27 \pi g^2 e^2 M^3 \over {m_q}^2 },
\end{eqnarray}
where ${\Gamma}_{\pi^0 \to 2 \gamma } = 8 eV $ is the width of the decay of
a rest pion~\cite{27}, $M$ and $m$ are  the masses of a pion and an electron,
respectively; $\varepsilon$ is the pion energy.
$M_l = \vert p_+  +  p_- \vert $ is the invariant mass of a
dilepton pair. When we have obtained the last expression we suppose that the
occupation numbers of decay products are equal to zero. We note that
in the case of nonrelativistic pions the last formula coincides with expression
for the Dalitz decay rate~\cite{26}.

Let us consider the Dalitz pion decay in an equilibrium pion gas. In the real
situation the temperature $T$ of the pion  gas arising in the result of
collisions of high energy heavy ions is about $T \sim 200-300 MeV$~\cite{3,25}.
This means that we can think approximately that the pions are ultrarelativistic
ones. In this case the energy imparted in the result of an individual elastic
collision of the particles is small as compared with the particle energy which
is of the order of the matter temperature. Estimating from the conservation laws
we find that  the maximum energy imparted $E_{max}$ is of the order of
$E_{max} \sim T (M / T )^2 \ll T$.

Then, substituting obtained expression for $d{\Gamma}_{vac} / dM_l $ into the
general Eq. (5) for the decay probability in matter and assuming that
distribution $f_{eq} (\varepsilon) $ is Boltzmann one we obtain
\begin{eqnarray}
&& {dW \over d M_l } =  { 4  e^2  \over 3 {\pi}^{3/2}  T  {M_l}^4 M^5 }
(M^2 - {M_l}^2 )^3 ({M_l}^2 + 2 m^2 ) ( {M_l}^2 - 4 m^2 )^{1/2}
{\Gamma}_{\pi^0 \to 2 \gamma } \nonumber \\
&&\Biggl\{\! 
{2 {\alpha}^{1/2} {t_0}^{3/2} \over 3 T } + { T^2 \over \alpha } \cdot
\Bigl[ \exp \bigl( {\alpha t_0 \over T^2 } \bigr) \cdot \!\!\!\int
\limits_{{{\sqrt{\alpha t_0}} \over T }} ^{+ \infty} \!\!\!dx \exp (- x^2 ) -
{{\sqrt{\pi}} \over 2} + {{\sqrt{\alpha t_0 }} \over T} \Bigr] \Biggr\},
\end{eqnarray}
where $T$ is the matter temperature, $t_0$ is the observation time which can
be estimated as time life of the equilibrium state of the matter which is
of the order of  $ \sim 10 Fm $~\cite{10}.

Let us study the obtained expression for the Dalitz decay of the pion in the
equilibrium hadronic matter.

When the energy fluctuation due to elastic scattering is small as compared
with the matter temperature, the parameter $({\sqrt{\alpha t_0 }} / T) \ll 1$.
Then, calculating the integral at small lower limit in Eq.(16) we find:

\begin{eqnarray}
&& {dW \over d M_l } =  { 2  e^2 t_0 \over 3 \pi  T  {M_l}^4 M^5 }
(M^2 - {M_l}^2 )^3 ({M_l}^2 + 2 m^2 ) ( {M_l}^2 - 4 m^2 )^{1/2}
{\Gamma}_{\pi^0 \to 2 \gamma } \times
\nonumber \\
&&\Biggl\{ 1 + {\alpha t_0 \over 2 T^2 } +
  O \bigl( (\alpha t_0 /  T^2 )^{3/2} \bigr) \Biggr\},
\end{eqnarray}

It follows from the last formula that  the small energy fluctuations due to
multiple elastic collisions lead to the insignificant increasing of the
decay probability as compared with the situation of the particle decay in the
matter in the absence of scattering.

In opposite limit case $({\sqrt{\alpha t_0 }} / T) \ge 1 $ when the energy
fluctuations are large  we obtain from  Eq. (16):

\begin{eqnarray}
&& {dW \over d M_l } =  { 4  e^2 t_0 \over 3 \pi^{3/2}   T  {M_l}^4 M^5 }
(M^2 - {M_l}^2 )^3 ({M_l}^2 + 2 m^2 ) ( {M_l}^2 - 4 m^2 )^{1/2}
{\Gamma}_{\pi^0 \to 2 \gamma } \times
\nonumber \\
&&\Biggl\{  {2 {\sqrt{\alpha t_0}} \over 3 T } +
 O \bigl(  T / {\sqrt{ \alpha t_0}} \bigr) \Biggr\},
\end{eqnarray}

The last means that the strong energy fluctuation results in the considerable
increasing of the decay probability by the factor $({\sqrt{\alpha t_0}}/ T ) \gg 1$.

Let us discuss obtained results from the point of view of the possibility of
the observation of increasing of the Dalitz pion decay in the equilibrium matter
due to multiple elastic scattering. According to the Eq. (10) the parameter
$\alpha $ being the mean square fluctuation of the particle energy, can be
estimated as the product of the collision frequency $\nu$ and the square of the maximum energy
${E_{max}}^2$ imparted in the result of an individual collision  of the particles.  Because
$E_{max} \sim T (M/ T)^2 $ the parameter $({\sqrt{\alpha t_0 }} / T) \sim
{\sqrt{\nu t_0 }} (M/T)^2 =  (M/ T)^2 \cdot {\sqrt N}$, where $N \gg 1$ is the number
of collisions of the particle in the matter during life time of the equilibrium
state of the medium. The last means  when number of the particle collisions is
more than the ratio $(T / M )^2 \gg 1 $  the considerable increasing of the pion decay
probability occurs. In the real situation~\cite{3,10,25} $T \sim 200 - 300
MeV; \nu \sim 1.33 - 4 {Fm}^{-1}, respectively; t_0 \sim 10 Fm $ that is
${\sqrt{\nu t_0 }} (M/T)^2 \sim 1.38 - 1.79 $. This means that the multiple
elastic scattering have to influence strongly on the pion Dalitz in the
matter. In conclusion we should note that as  the dielectron yield due to
the Dalitz decay of pions are not suppressed by any other processes of the
dielectron production at low invariant mass of a dielectron pair~\cite{25}.
That is we can say about the possibility to observe the increasing
of the pion decay probability  due to multiple scattering of the pions
in hadron matter.

\section{Conclusion.}

The decay of particles undergoing multiple elastic collisions in
an equilibrium scattering medium is studied in this paper.
The decay probability of such particles is calculated. The developed
method of the calculation of the decay probability does not depend on the
dynamic model of the particle scattering  in the medium. In this way the impact
of the medium on the decay probability is determined by the energy
fluctuation of the decaying particle due to multiple scattering and the matter
temperature. It is  shown when the energy fluctuation is of the order of the
medium temperature multiple scattering results in considerable changes of the decay
probability as compared with the case of scattering lack.

We study the Dalitz decay of pions undergoing multiple elastic collisions in an
equilibrium pion gas. We  shown that the multiple scattering leads to the
increasing of the decay probability in any cases. In this way when
the energy fluctuations are of the order of the magnitude of the matter
temperature multiple scattering leads to the strong broadening of the in-matter
pion in the Dalitz decay channel. We discuss the possibility to observe the
increasing of the Dalitz decay probability due to the multiple scattering of a
pion in matter arising in the result of collisions of heavy ions of high energy.
According to the estimations made impact of multiple scattering may be
considerable in the equilibrium pion gas at temperature $T \sim 200 - 300 MeV$.

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