
\section[Bremsstrahlung]{Bremsstrahlung} \label{secmubrem}

Bremsstrahlung dominates other muon interaction processes in the region of
catastrophic collisions ($v \geq 0.1$ ), that is at "moderate" muon 
energies above the kinematic limit for knock--on electron production. 
At high energies ($E \geq 1$ TeV) this process contributes about
40\% of the average muon energy loss.

\subsection{Differential Cross Section}

The differential cross section for muon bremsstrahlung (in units of
${\rm cm}^{2}/(\mbox{g GeV})$)
can be written as
\begin{eqnarray}
\label{mubrem.a}
\frac{d \sigma (E,\epsilon,Z,A)}{d \epsilon}&=&
\frac{16}{3} \alpha N_A (\frac{m}{\mu} r_{e})^2
\frac{1}{\epsilon A} Z(Z \Phi_n + \Phi_e)(1-v+\frac{3}{4} v^2)\nonumber\\
&=&0 \quad {\rm if} \quad \epsilon \geq \epsilon_{\rm max} = E-\mu ,
\end{eqnarray}
where $\mu$ and $m$ are the muon and electron masses, $Z$ and $A$ are the 
atomic number and atomic weight of the material, and $N_{A}$ is Avogadro's 
number.  If $E$ and $T$ are the initial total and kinetic energy of the
muon, and $\epsilon$ is the emitted photon energy, then 
$\epsilon = E - E'$ and the relative energy transfer $v = \epsilon /E$.

$\Phi_{n}$ represents the contribution of the nucleus and can be expressed 
as 
\begin{eqnarray*}
 \Phi_{n} &=& \ln \frac {BZ^{-1/3}(\mu + \delta (D_{n}' \sqrt{e} -2))}
    {D_{n}'(m+ \delta \sqrt{e}BZ^{-1/3})} ; \\
   &=&  0 \quad {\rm if} \quad {\rm negative}.
\end{eqnarray*}
$\Phi_{e}$ represents the contribution of the electrons and can be expressed 
as
\begin{eqnarray*}
  \Phi_{e} &=& \ln \frac {B'Z^{-2/3} \mu }
   {\left(1+ \displaystyle\frac{\delta \mu}{m^{2} \sqrt{e}}\right)(m+ \delta
\sqrt{e}
B'Z^{-2/3})
};  \\
   &=&  0 \quad {\rm if} \quad \epsilon \geq \epsilon'_{\rm max} = E/(1+
\mu^{2}/2mE); \\
   &=&  0 \quad {\rm if} \quad {\rm negative}.    
\end{eqnarray*}
In $\Phi_n$ and $\Phi_e$, for all nuclei except hydrogen,
\begin{eqnarray*}
\delta &=& \mu^{2} \epsilon /2EE' = \mu^{2} v/2(E- \epsilon);\\
D'_{n} &=& D_{n}^{(1-1/Z)}, \quad D_{n}= 1.54A^{0.27}; \\
B &=& 183, \quad B'=1429, \quad \sqrt{e}=1.648(721271).
\end{eqnarray*}
%
For hydrogen ($Z$=1) $B = 202.4,\: B' = 446, \: D_{n}' = D_{n}$.


These formulae are taken mostly from Refs. \cite{brem.kel95} and 
\cite{brem.kel97}.  They include improved nuclear size corrections in 
comparison with Ref. \cite{brem.petr68} in the region $v \sim 1$ and low $Z$. 
Bremsstrahlung on atomic electrons (taking into account target recoil and 
atomic binding) is introduced instead of a rough substitution $Z(Z+1)$. 
A correction for processes with nucleus excitation is also 
included \cite{brem.andr94}.

\subsubsection{Applicability and Restrictions of the Method}

The above formulae assume that: \\
1. $E \gg \mu $, hence the ultrarelativistic approximation is used; \\
2. $E \leq 10^{20}$ eV; above this energy, LPM suppression can be expected;\\
3. $v \geq 10^{-6}$ ; below $10^{-6}$ Ter-Mikaelyan suppression takes place.
However, in the latter region the cross section of muon bremsstrahlung is 
several orders of magnitude less than that of other processes.\\
The Coulomb correction (for high $Z$) is not included. However, existing
calculations \cite{brem.andr97} show that for muon bremsstrahlung this
correction is small.

\subsection{Continuous Energy Loss}

The restricted energy loss for muon bremsstrahlung $(d E/ dx)_{\rm rest}$
with relative transfers $v = \epsilon / (T+ \mu) \leq v_{\rm cut}$
can be calculated as follows :
$$
\left(\frac{d E}{d x}\right)_{\rm rest}
= \int_{0}^{\epsilon_{\rm cut}} \epsilon\,\sigma (E,\epsilon )\,d\epsilon
=
(T+\mu ) \int_{0}^{v_{\rm cut}}\epsilon\,\sigma (E,\epsilon)\,dv\,.
$$
% 
If the user cut $v_{\rm cut} \geq v_{\rm max}=T/(T+ \mu)$, the total 
average energy loss is calculated.  Integration is done using Gaussian
quadratures, and binning provides an accuracy better than about 0.03\% for 
$T = 1$ GeV, $Z=1$.  This rapidly improves with increasing $T$ and $Z$.


\subsection{Total Cross Section}

The integration of the differential cross section over $d\epsilon$ gives the
total cross section for muon bremsstrahlung:
\begin{equation}
\label{mubrem.b}
\sigma_{\rm tot} (E,\epsilon_{\rm cut})
= \int_{\epsilon_{\rm cut}}^{\epsilon_{\rm max}}\sigma (E,\epsilon ) 
d \epsilon  =
 \int_{\ln v_{\rm cut}}^{\ln v_{\rm max}}\epsilon \sigma (E,\epsilon) 
d(\ln v) ,
\end{equation}
where $v_{\rm max}=T/(T+ \mu)$.  If $v_{\rm cut} \geq v_{\rm max}$ ,
$\sigma_{\rm tot}=0$.

\subsection{Sampling}

The photon energy $\epsilon_{p}$ is found by numerically solving the 
equation :
$$ P \:= \int_{\epsilon_{p}}^{\epsilon_{\rm max}}  \sigma (E,\epsilon,Z,A
) \, d \epsilon  
 \left/ \int_{\epsilon_{\rm cut}}^{\epsilon_{\rm max}}  
 \sigma (E,\epsilon,Z,A ) \, d \epsilon\right. .
$$
Here $P$ is the random uniform probability, $\epsilon_{\rm max}=T$, and
$\epsilon_{\rm cut}=(T+\mu) \cdot v_{\rm cut}$.  $v_{min.cut}=10^{-5}$ is 
the minimal relative energy transfer adopted in the algorithm.

For fast sampling, the solution of the above equation is tabulated at 
initialization time for selected $Z$, $T$ and $P$.  During simulation, this
table is interpolated in order to find the value of $\epsilon_{p}$ 
corresponding to the probability $P$.

The tabulation routine uses accurate functions for the differential cross 
section.  The table contains values of 
\begin{equation}
\label{mubrem.c}
 x_p = \ln (v_p / v_{\rm max})/\ln (v_{\rm max}/v_{\rm cut}) ,
\end{equation}
where $v_{p} = \epsilon_{p}/(T+ \mu)$ and $v_{\rm max} = T/(T+ \mu)$. 
Tabulation is performed in the range 
$1 \leq Z \leq 128$, $1 \leq T \leq 1000$~PeV, $10^{-5} \leq P \leq 1$
with constant logarithmic steps.  Atomic weight (which is a required 
parameter in the cross section) is estimated here with an iterative solution of the approximate relation:
$$ A = Z\,(2+0.015\,A^{2/3}).$$
For $Z=1$, $A=1$ is used.

To find $x_{p}$ (and thus $\epsilon_{p}$) corresponding to a given 
probability $P$, the sampling method performs a linear interpolation in 
$\ln Z$ and $\ln T$, and a cubic, 4 point Lagrangian interpolation 
in $\ln P$.  For $P \leq P_{\rm min}$, a linear interpolation in $(P,x)$ 
coordinates is used, with $x = 0$ at $P = 0$.  Then the energy 
$\epsilon_p$ is obtained from the inverse transformation of \ref{mubrem.c} :
%
$$\epsilon_{p} = (T+ \mu ) v_{\rm max} (v_{\rm max}/v_{\rm cut})^{x_{p}} $$
%
The algorithm with the parameters described above has been
tested for various $Z$ and $T$. It reproduces the differential cross section
to within 0.2 -- 0.7 \% for $T \geq 10$~GeV.  The average total energy loss 
is accurate to within 0.5\%.  While accuracy improves with increasing $T$, 
satisfactory results are also obtained for $1 \leq T \leq 10$~GeV.

It is important to note that this sampling scheme allows the generation of
$\epsilon_{p}$ for different user cuts on $v$ which are above
$v_{\rm min.cut}$.  To perform such a simulation, it is sufficient to
define a new probability variable
%
$$P' = P \: \sigma_{\rm tot} \: (v_{\rm user.cut}) / \sigma_{\rm tot} (v_{\rm 
min.cut})$$
%
and use it in the sampling method.  Time consuming re-calculation of the 
3-dimensional table is therefore not required because only the tabulation
of $\sigma_{\rm tot}(v_{\rm user.cut})$ is needed.

The small-angle, ultrarelativistic approximation is used for the simulation
(with about 20\% accuracy at $\theta\le\theta^*\approx1$) of the angular
distribution of the final state muon and photon.  Since the target recoil is
small, the muon and photon are directed symmetrically (with equal transverse momenta and coplanar with the initial muon):
\begin{equation}
p_{\perp \mu} = p_{\perp \gamma}, \quad {\rm where} \quad p_{\perp \mu}=
E' \theta_{\mu}, \quad p_{\perp \gamma} = \epsilon \theta_{\gamma} .
\end{equation}
$\theta_{\mu}$ and $\theta_{\gamma}$ are muon and photon emission angles.
The distribution in the variable
$r=E\theta_{\gamma}/\mu$ is given by
\begin{equation}
f(r) dr \sim r dr/(1+r^2)^2 .
\end{equation}
Random angles are sampled as follows:
\begin{equation}
\theta_{\gamma} = \frac{\mu}{E} r \quad
\theta_{\mu} = \frac{\epsilon}{E'} \theta_{\gamma} ,
\end{equation}
where
$$
r=\sqrt{\frac{a}{1-a}}\,,\quad a=\xi\,\frac{r_{\rm \max}^2}{1+r_{\rm max}^2}\,,\quad
r_{\max}=\min(1,E'/\epsilon)\cdot E\,\theta^*/\mu\,,$$
and $\xi$ is a random number uniformly distributed between 0 and 1.

\subsection{Status of this document}

09.10.98 created by R.Kokoulin and A.Rybin\\
17.05.00 updated by S.Kelner, R.Kokoulin and A.Rybin\\
30.11.02 re-written by D.H. Wright\\

\begin{latexonly}

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CERN SCAN-9510048.
\bibitem{brem.kel97}
  S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Phys. Atomic Nuclei, {\bf 60} (1997) 576.
\bibitem{brem.petr68}
  A.A.Petrukhin, V.V.Shestakov. Canad.J.Phys., {\bf 46} (1968) S377.
\bibitem{brem.andr94}
  Yu.M.Andreyev, L.B.Bezrukov, E.V.Bugaev. Phys. Atomic Nuclei, {\bf 57} (1994)
2066.
\bibitem{brem.andr97}
  Yu.M.Andreev, E.V.Bugaev, Phys. Rev. D, {\bf 55} (1997) 1233. 
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\subsection{Bibliography}

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\item A.A.Petrukhin, V.V.Shestakov. Canad.J.Phys., {\bf 46} (1968) S377.
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(1994) 2066.
\item Yu.M.Andreev, E.V.Bugaev, Phys. Rev. D, {\bf 55} (1997) 1233. 
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\end{htmlonly}
