\chapter{Lepton Hadron Interactions}

The photonuclear interaction of muons is currently the only process treated
in this category. 

\section{\it G4MuonNucleusProcess}

This class simulates the photonuclear interaction of muons in a material.
The muon interacts electromagnetically with a nucleus, exchanging a virtual
photon.  At energies above a few GeV, the photon interacts hadronically with 
the nucleus, producing hadronic secondaries.  

The outcome of the simulation depends heavily upon the interaction model 
chosen.  Hence the model-dependent part of the process is implemented in 
the {\it G4MuonNucleusInteractionModel} class, which can be easily replaced by
another model.

{\it G4MuonNucleusInteractionModel} calculates the cross section and final 
states of the muon and hadronic secondaries.  The final muon momentum is 
given by a double-differential cross section which depends on the 
photoabsorption cross sections for longitudinally and transversely polarized
photons.  The final hadronic state is determined by replacing the virtual 
photon with a charged pion of the same $Q^2$ and then allowing the pion to 
interact with the nucleus.  The charge of the pion is chosen at random.  The 
pion interactions with the nucleus are modeled by processes derived from the 
GHEISHA \cite{GHEISHA} package.  These processes are: \\

\begin{tabular}[t]{ll}
{\it G4LEPionPlusInelastic}, {\it G4LEPionMinusInelastic} & $E \leq 25$ GeV \\
{\it G4HEPionPlusInelastic}, {\it G4HEPionMinusInelastic} & $E > 25$ GeV \\
\end{tabular}
\\

\subsection{Cross Section Calculation}

The cross section for the above process in a material is given roughly by

\begin{eqnarray*}
\sigma_{\mu A} = A \sigma_{\mu N}
\end{eqnarray*}

\noindent where $A$ is the atomic mass number of the material and 
$\sigma_{\mu N}$ is the cross section for the process on a single nucleon:

\[
  \sigma_{\mu N} =
    \left\{ \begin{array}{ll}
      0.3               & (E \leq 30GeV) \\
      0.3 (E/30)^{0.25} & (E > 30GeV) \\
    \end{array} \right. [\mu b] .
\]

\noindent
The differential cross section, in terms of muon energy $E$ and emission solid
angle $\Omega$, can be expressed as:

\begin{eqnarray*}
  \frac{d\sigma}{d\Omega dE} =\Gamma\,(\sigma_T + \epsilon \sigma_L)
\end{eqnarray*}
where $\sigma_L$ and $\sigma_T$ are the photoabsorption cross sections for 
longitudinal and transverse photons, respectively.  $\Gamma$ is the transverse
photon flux and $\epsilon$ is the polarization of the intermediate photon.
The photoabsorption cross sections are parameterized as:
\begin{eqnarray*}
  \sigma_L &=& 0.3\,\left( 1 - \frac{1}{1.868} Q^2 \nu \right)\,\sigma_T \\
  \sigma_T &\sim& const = 0.12 mb \\
\end{eqnarray*}

\noindent while the flux and polarization are given by 
\begin{eqnarray*}
  \Gamma   &=& \frac{K \alpha}{2\pi} \frac{E^\prime}{E} \frac{1}{1-\epsilon} \\
  \epsilon &=& \left[ 1 + 2 \frac{Q^2 + \nu^2}{Q^2} tan^2 \frac{\theta}{2} \right]^{-1} . \\
\end{eqnarray*}

\noindent
$E$ and $E^{\prime}$ are the initial and final muon energies, $Q^2$ and $\nu$ 
are the scaling variables
\begin{eqnarray*}
  Q^2   &=& -q^2 = 2 (EE^{\prime} - PP^{\prime} cos \theta - m_\mu^2) \\
  \nu   &=& E - E^{\prime} , \\
\end{eqnarray*}
and $K$ is given using the Gilman convention
\begin{eqnarray*}
  K = \nu + \frac{Q^2}{2\nu} .
\end{eqnarray*}


\section{Status of this document}
 20.04.02  re-written by D.H. Wright \\
 23.10.98  created by M.Takahata \\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{GHEISHA} H.Fesefeldt
   {\em GHEISHA The Simulation of Hadronic Showers} 149
   {\em RWTH/PITHA 85/02} (1985)
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item H.Fesefeldt
   {\em GHEISHA The Simulation of Hadronic Showers} 149
   {\em RWTH/PITHA 85/02} (1985)
\end{enumerate}

\end{htmlonly}