
\section[Positron - Electron Annihilation into Hadrons]{Positron - Electron Annihilation into Hadrons}

\subsection{Introduction}
The process {\tt G4eeToHadrons} simulates the in-flight annihilation 
of a positron with an atomic electron into hadrons.  It is assumed here 
that the atomic electron is initially free and at rest.  Currently only 
two-pion production is available with a validity range up to 1 TeV. 

\subsection{Cross Section and Mean Free Path}
The annihilation of positrons and target electrons producing pion pairs in
the final state (${\rm e}^+{\rm e}^- \to \pi^+\pi^-$) may give an
appreciable contribution to electron-jet conversion at the LHC, and
for the increasing total number of muons produced in the beam pipe of
the linear collider \cite{anniToHad.mu}.  The threshold positron energy in the
laboratory system for this process with the target electron at rest is
\begin{equation}\label{he0}
E_{\rm th}=2m_\pi^2/m_e-m_e\approx 70.35\:{\rm GeV}\,,
\end{equation}
where $m_\pi$ and $m_e$ are the pion and electron masses, respectively.
The total cross section is dominated by the reaction
\begin{equation}\label{he1}
{\rm e}^+{\rm e}^- \to \rho\gamma\to\pi^+\pi^-\gamma, 
\end{equation}
where $\gamma$ is a radiative photon and $\rho(770)$ is a well known vector
meson. This radiative correction is essential, because it significantly 
modifies the shape of the resonance. Details of the theory are described in 
\cite{anniToHad.ben}, in which the main term and the leading $\alpha^2$
corrections are taken into account.

\subsection {Sampling the final state}

The final state of the $e+e-$ annihilation process \ref{he1} 
is simulated by first determining the kinematic limits of the photon energy
in the center of mass system
and then sampling the photon energy within those limits using the 
differential cross section.  Conservation of energy-momentum is then used to
determine the four-momentum of the pion final state.  Then 
the backward transformation to the laboratory system is performed.
 

\subsection{Status of this document}
 09.12.05 created by V.Ivanchenko    \\
   
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\begin{thebibliography}{99}
\bibitem{anniToHad.mu} A.G. Bogdanov et al., IEEE-NSS-33-179
conference Record, 2004, accepted by IEEE Transaction.
\bibitem{anniToHad.ben} M. Benayoun et al., Mod. Phys. Lett. A14, 
2605 (1999).
\end{thebibliography}

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\subsection{Bibliography}

\begin{enumerate}
\item A.G. Bogdanov et al., IEEE-NSS-33-179
conference Record, 2004, accepted by IEEE Transaction.
\item M. Benayoun et al., Mod. Phys. Lett. A14, 
2605 (1999).
\end{enumerate}

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