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\section{Muon Photonuclear Interaction}

%muon photonuclear interaction
%(not deep inelastic scattering)

The inelastic interaction of muons with nuclei is important at high muon 
energies ($E \geq 10$~ GeV), and at relatively high energy transfers $\nu$ 
($\nu / E \geq 10^{-2}$).  It is especially important for light materials
and for the study of detector response to high energy muons, muon propagation
and muon-induced hadronic background.  The average energy loss for this 
process increases almost lineary with energy, and at TeV muon energies 
constitutes about 10\% of the energy loss rate.\\

\noindent 
The main contribution to the cross section $\sigma ( E, \nu $) and energy 
loss comes from the low $Q^{2}$--region ( $Q^{2} \ll 1~{\rm GeV}^{2}$).
In this domain, many simplifications can be made in the theoretical
consideration of the process in order to obtain convenient and simple
formulae for the cross section.  Most widely used are the expressions given 
by Borog and Petrukhin \cite{munu.bor75}, and Bezrukov and Bugaev 
\cite{munu.bez81}.  Results from these authors agree within 10\% for the 
differential cross section and within about 5\% for the average energy loss,
provided the same photonuclear cross section, $\sigma_{\gamma N}$, is used 
in the calculations.

\subsection{Differential Cross Section}

The Borog and Petrukhin formula for the cross section is based on:
\begin{itemize}
\item Hand's formalism \cite{munu.hand63} for inelastic muon scattering,
\item a semi-phenomenological inelastic form factor, which is a Vector 
Dominance Model with parameters estimated from experimental data, and 
\item nuclear shadowing effects with a reasonable theoretical 
parameterization \cite{munu.brod72}.
\end{itemize}

\noindent
For $E\geq10$~GeV, the Borog and Petrukhin cross section 
\mbox{(${\rm cm}^{2}$/g GeV)}, differential in transferred energy, is
\begin{equation}
\label{munu.1}
\sigma(E,\nu ) = \Psi(\nu ) \Phi( E,v ) ,
\end{equation}

\begin{equation}
\label{munu.2}
\Psi(\nu) = \frac{\alpha}{\pi} \frac{A_{\rm eff} N_{AV}}{A}
 \sigma_{\gamma N}(\nu) \frac{1}{\nu} ,
\end{equation}

\begin{equation}
\label{munu.3}
 \Phi(E,v) = v-1 + \left[1-v+\frac{v^{2}}{2} 
 \left(1+\frac{2\mu^{2}}{\Lambda^{2}}\right)\right]
 \ln\frac{\displaystyle\frac{E^{2}(1-v)}{\mu^{2}}\left(1+\displaystyle\frac{\mu^{2}
v^{2}}{\Lambda^{2}(1-v)}\right)}
{\displaystyle
1+\frac{Ev}{\Lambda}\left(1+\frac{\Lambda}{2M}+\frac{Ev}{\Lambda}\right)},
\end{equation}
%
where $\nu$ is the energy lost by the muon, $v = \nu /$E, and $\mu$ and $M$ 
are the muon and nucleon (proton) masses, respectively.  $\Lambda$ is a 
Vector Dominance Model parameter in the inelastic form factor which is 
estimated to be $\Lambda^{2}=0.4 \;{\rm GeV}^{2}$. \\

\noindent
For $A_{\rm eff}$, which includes the effect of nuclear shadowing, the
parameterization \cite{munu.brod72}
\begin{equation}
\label{munu.4}
  A_{\rm eff} = 0.22 A + 0.78 A^{0.89}
\end{equation}
is chosen. \\

\noindent
A reasonable choice for the photonuclear cross section, $\sigma_{\gamma N}$,
is the parameterization obtained by Caldwell et al. \cite{munu.cald79}
based on the experimental data on photoproduction by real photons:
\begin{equation}
\label{munu.5}
 \sigma_{\gamma N} = ( 49.2 + 11.1 \ln K + 151.8/ \sqrt{K} ) \cdot 10^{-30} {\rm cm}^{2} \quad K~~\mbox{in GeV} .
\end{equation}
The upper limit of the transferred energy is taken to be 
$\nu_{\rm max}$ = $E - M/2$.  The choice of the lower limit $\nu_{\rm min}$ 
is less certain since the formula \ref{munu.1}, \ref{munu.2}, \ref{munu.3} is 
not valid in this domain.  Fortunately, $\nu_{\rm min}$ influences the total 
cross section only logarithmically and has no practical effect on the average 
energy loss for high energy muons.  Hence, a reasonable choice for 
$\nu_{\rm min}$ is 0.2~GeV. \\

\noindent
In Eq.~\ref{munu.2}, $A_{\rm eff}$ and $\sigma_{\gamma N}$ appear as
factors.  A more rigorous theoretical approach may lead to some dependence 
of the shadowing effect on $\nu$ and $E$;  therefore in the differential 
cross section and in the sampling procedure, this possibility is forseen 
and the atomic weight $A$ of the element is kept as an explicit parameter. \\
        
\noindent         
The total cross section is obtained by integration of Eq.~\ref{munu.1} 
between $\nu_{\min}$ and $\nu_{\max}$;  to facilitate the computation, 
a $\ln (\nu )$--substitution is used.

\subsection{Sampling}

\subsubsection{Sampling the Transferred Energy}

The muon photonuclear interaction is always treated as a discrete process 
with its mean free path determined by the total cross section.  The total
cross section is obtained by the numerical integration of Eq.~\ref{munu.1} 
within the limits $\nu_{\min}$ and $\nu_{\max}$.  The process is considered 
for muon energies $1 \rm{GeV} \leq T \leq 1000 \rm{PeV}$, though it should 
be noted that above 100~TeV the extrapolation (Eq.~\ref{munu.5}) of 
$\sigma_{\gamma N}$ may be too crude. \\

\noindent 
The random transferred energy, $\nu_{p}$, is found from the numerical 
solution of the equation :
\begin{equation}
\label{munu.6}
P = \int_{\nu_{p}}^{\nu_{\rm max}}  \sigma(E,\nu ) d \nu 
\left/ \int_{\nu_{\rm min}}^{\nu_{\rm max}} \sigma(E,\nu) d\nu \right. .
\end{equation}
%
Here $P$ is the random uniform probability, with $\nu_{\rm max}= E-M/2$
and $\nu_{\rm min}=0.2$~GeV. \\ 

\noindent
For fast sampling, the solution of Eq.~\ref{munu.6} is tabulated at 
initialization time.  During simulation, the sampling method returns a value
of $\nu_{p}$ corresponding to the probability $P$, by interpolating the 
table.  The tabulation routine uses Eq.~\ref{munu.1} for the differential 
cross section.  The table contains values of
\begin{equation}
\label{munu.7}
 x_p = \ln (\nu_p / \nu_{\rm max})/\ln (\nu_{\rm max}/\nu_{\rm min}),
\end{equation}
calculated at each point on a three-dimensional grid with constant spacings 
in $\ln (T)$, $\ln(A)$ and $\ln(P)$ .  The sampling uses linear 
interpolations in $\ln(T)$ and $\ln(A)$, and a cubic interpolation in 
$\ln(P)$.  Then the transferred energy is calculated from the inverse 
transformation of Eq.~\ref{munu.7},
$\nu_{p}=\nu_{\rm max}(\nu_{\rm max}/\nu_{\rm min})^{x_{p}}$. 
Tabulated parameters reproduce the theoretical dependence to better than 
2\% for $T > 1$~GeV and better than 1\% for $ T >10$~GeV.

\subsubsection{Sampling the Muon Scattering Angle}

According to Refs.~\cite{munu.bor75, munu.bor77}, in the region where the 
four-momentum transfer is not very large ($Q^{2} \leq 3 {\rm GeV}^{2}$),
the $t$~--~dependence of the cross section may be described as:
\begin{equation}
\label{munu.8}
 \frac{d \sigma }{dt} \sim  \frac{(1- t / t_{\rm max}) }
{t (1+t/ \nu^{2})(1+t/m^{2}_{0})} [(1-y)(1-t_{\rm min}/t)+y^{2}/2] , 
\end{equation} 
where $t$ is the square of the four-momentum transfer,
$Q^{2} = 2 (EE' - PP'\cos\theta  - \mu^{2})$.  Also,
$t_{\rm min} = (\mu y)^{2} / (1-y)$, $y=\nu /E$ and
$t_{\rm max} = 2M\nu $.  $\nu = E-E'$ is the energy lost by the muon and 
$E$ is the total initial muon energy.  $M$ is the nucleon (proton) mass and
$m_{0}^{2} \equiv \Lambda^{2} \simeq 0.4\;{\rm GeV}^{2}$ is 
a phenomenological parameter determing the behavior of the inelastic form
factor.  Factors which depend weakly, or not at all, on $t$ are omitted. \\

\noindent
To simulate random $t$ and hence the random muon deflection angle, it is 
convenient to represent Eq.~\ref{munu.8} in the form :
%
\begin{equation}
\label{munu.9}
 \sigma( t ) \sim f(t) g(t), 
\end{equation}
where
\begin{eqnarray}
\label{munu.10}
f(t) = \frac{1}{t(1+ t/t_{1} )} , \\
g(t) = \frac{1-t/t_{\max}}{1+t/t_{2}} \cdot
\frac{(1-y)(1-t_{\min}/t)+y^{2}/2}
{(1-y)+y^{2}/2}, \nonumber
\end{eqnarray}
and
\begin{equation}
\label{munu.11}
 t_{1} = {\min} (\nu^{2}, m_{0}^{2}) \quad
  t_{2} ={\max} (\nu^{2}, m_{0}^{2}) .
\end{equation}
  
\noindent 
$t_{P}$ is found analytically from Eq.~\ref{munu.10} :
$$t_{P} = \frac{t_{\max}t_{1}}
{\displaystyle(t_{\max}+t_{1})\left[\frac{t_{\max}(t_{\min}+t_{1})}
{t_{\min}(t_{\max}+t_{1})}\right]^{P}-t_{\max}} ,
$$  
%
where $P$ is a random uniform number between 0 and 1, which is accepted with 
probability $g(t)$.  The conditions of Eq.~\ref{munu.11} make use of the 
symmetry between $\nu^{2}$ and $m_{0}^{2}$ in Eq.~\ref{munu.8} and allow 
increased selection efficiency, which is typically $\geq 0.7$.  The polar 
muon deflection angle $\theta$ can easily be found from 
\footnote{This convenient formula has been shown to the authors by 
D.A. Timashkov.}

$$\sin^{2}(\theta /2) = \frac {t_{P} - t_{\rm min}}
{4\,(EE'- \mu^{2}) - 2\,t_{\rm min}}.  $$
The hadronic vertex is generated by the hadronic processes taking
into account the four-momentum transfer.

\subsection{Status of this document}

12.10.98 created by R.Kokoulin, A.Rybin.\\
18.05.00 edited by S.Kelner, R.Kokoulin, and A.Rybin. \\
07.12.02 re-worded by D.H. Wright \\
30.08.04 correction of eq. 8.24 (to 1/sqrt) from H. Araujo \\
%

\begin{latexonly}

\begin{thebibliography}{99}

\bibitem{munu.bor75} V.V.Borog and A.A.Petrukhin,
               Proc. 14th Int.Conf. on Cosmic Rays, Munich,1975,
               {\bf vol.6}, p.1949.
\bibitem{munu.bez81} L.B.Bezrukov and E.V.Bugaev,
               Sov. J. Nucl. Phys., {\bf 33}, 1981, p.635.
\bibitem{munu.hand63} L.N.Hand. Phys. Rev., {\bf 129}, 1834 (1963).
\bibitem{munu.brod72} S.J.Brodsky, F.E.Close and J.F.Gunion,
                Phys. Rev. {\bf D6}, 177 (1972).
\bibitem{munu.cald79} D.O. Caldwell et al., 
               Phys. Rev. Lett., {\bf 42}, 553 (1979).
\bibitem{munu.bor77} V.V.Borog, V.G.Kirillov-Ugryumov, A.A.Petrukhin,
               Sov. J. Nucl. Phys., {\bf 25}, 1977, p.46.


\end{thebibliography}

\end{latexonly}

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\subsection{Bibliography}

\begin{enumerate}
\item V.V.Borog and A.A.Petrukhin,
               Proc. 14th Int.Conf. on Cosmic Rays, Munich,1975,
               {\bf vol.6}, p.1949.
\item L.B.Bezrukov and E.V.Bugaev,
               Sov. J. Nucl. Phys., {\bf 33}, 1981, p.635.
\item L.N.Hand. Phys. Rev., {\bf 129}, 1834 (1963).
\item S.J.Brodsky, F.E.Close and J.F.Gunion,
                Phys. Rev. {\bf D6}, 177 (1972).
\item D.O. Caldwell et al., 
               Phys. Rev. Lett., {\bf 42}, 553 (1979).
\item V.V.Borog, V.G.Kirillov-Ugryumov, A.A.Petrukhin,
               Sov. J. Nucl. Phys., {\bf 25}, 1977, p.46.
\end{enumerate}

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