\chapter{Leading Particle Bias} 

\section{Overview}
{\it G4Mars5GeV} is an inclusive event generator for hadron\,(photon)
interactions with nuclei, and translated from the MARS code system\,(MARS13\,(98)).
To construct a cascade tree, only a fixed
number of particles are generated at each vertex.
A corresponding statistical weight is assigned to each secondary particle
according to its type and phase-space. Rarely-produced particles 
or interesting phase-space region can be enhanced. \\ [2mm]
%%
N.B. This inclusive simulation is implemented in Geant4 partially 
for the moment, not completed yet.  \\ [3mm]
%% 
{\bf MARS Code System} \\
MARS is a set of Monte Carlo programs for inclusive simulation 
of particle interactions, and high multiplicity or rare events 
can be simulated fast with its sophisticated biasing techniques.
For the details on the MARS code system, see \cite{MARS98, MARSWWW}.  \\

\section{Method}
In {\it G4Mars5GeV}, three secondary hadrons are generated 
in the final state of an hadron(photon)-nucleus inelastic interaction, 
and a statistical weight is assigned to each particle
according to its type, energy and emission angle.
%% what really affects on weights?
%%
In this code, energies, momenta and weights of the secondaries are sampled, 
and the primary particle is simply terminated at the vertex.
The allowed projectile kinetic energy is $E_0 \leq $~5~GeV, and 
following particles can be simulated;
$$
        p,~n,~\pi^+,~\pi^-,~K^+,~K^-,~\gamma,~\bar{p}~.
$$

Prior to a particle generation, a Coulomb barrier is considered
for projectile charged hadrons\,($p$, $\pi^+$, $K^+$ and $\bar{p}$) 
with kinetic energy of less than 200 MeV. The coulomb potential
$V_{\rm coulmb}$ is given by 
\begin{equation}
        V_{\rm columb} = 1.11\times10^{-3} \times Z/A^{1/3}~~~~{\rm (GeV)},
\end{equation}
where $Z$~and~$A$ are atomic and mass number, respectivelly. \\ [2mm]
%%
%%      secondary particle generation
%%
\subsection{Inclusive hadron production}
%%
The following three steps are carried out in a sequence
to produce secondary particles:
\begin{itemize}
\item nucleon production,
\item charged pion/kaon production and
\item neutral pion production. 
\end{itemize}
%%
These processes are performed independently, i.e. 
the energy and momentum conservation law is broken at each event, however,
fulfilled on the average over a number of events simulated. \\ [5mm]
%%
%% -- nucleon production
{\bf nucleon production} \\ [1mm]
Projectiles $K^\pm$ and $\bar{p}$ are
replaced with $\pi^\pm$ and $p$, individually to generate
the secondary nucleon.
Either of neutron or proton is selected randomly as the secondary 
except for the case of gamma projectiling.
The gamma is handled as a pion. \\ [5mm]
%% reason??
%%
%% -- nucleon production
{\bf charged pion/kaon production} \\ [1mm]
If the incident nucleon does not have enough energy to produce 
the pion\,($> 280$ MeV), charged and neutral pions are not produced.
A charged pion is selected with the equal probability, and a bias is eliminated
with the appropriate weight which is assigned taking into account the difference between 
$\pi^+$ and $\pi^-$ both for production probability and for inclusive spectra.
It is replaced with a charged kaon a certain fraction 
of the time, that depends on the projectile energy if $E_0 > 2.1$~GeV.
The ratio of kaon replacement is given by
\begin{equation}
        R_{\rm kaon} = 1.3 \times \biggl\{ C_{\rm min} + ( C - C_{\rm min} )
        \frac{ \log(E_0/2) }{ \log(100/2) } \biggr\}~~~~~
        ( 2.1 \leq E_0 \leq 5.2~{\rm GeV} ),
\end{equation}
where $C_{\rm min}$ is 0.03\,(0.08) for nucleon\,(others) projectiling, and \\
{\it ~\hspace*{35mm}Produced particle~\hspace*{5mm}
Projectile particle}
\begin{equation}
C = \left\{
\begin{array}{ll}
0.071 & (~\pi^+~) \\
0.083 & (~\pi^-~)
\end{array} \right\}
\times
\left\{
\begin{array}{ll}
1.3 & (~\pi^\pm~) \\
2.0 & (~K^\pm~) \\
1.0 & (~{\rm others}~)
\end{array} \right\}
\end{equation}
A similar strangeness replacement is not considered for nucleon production.
\\ [3mm]
%%
%% -- nucleon production
%%
%%

\subsection{Sampling of energy and emission angle of the secondary}
%
The energy and emission angle of the secondary particle depends on 
projectile energy. There are formulae depending on whether or not 
the interaction particle\,(IP) is identical to the secondary\,(JP). \\ [2mm]
%%
%%
For IP $\neq$ JP, the secondary energy $E_2$ is simply given by
\begin{equation}
        E_2 = E_{\rm th} \times \biggl(\frac{E_{\rm max}}
        {E_{\rm th}}\biggr)^\epsilon~~~~~~~~~~~~~~{\rm (MeV)}~,
\end{equation}
where $E_{\rm max} = max\,(E_0,0.5\,{\rm MeV})$, $E_{\rm th} = 1$ MeV, 
and $\epsilon$ is a uniform random between 0 and 1. \\ [2mm]
%%
%%
For IP = JP,
\begin{equation}\label{eq:energy}
E_2 = \left\{
\begin{array}{l l l}
E_{\rm th} + \epsilon\,(E_{\rm max} - E_{\rm th}) &
        & E_0 < 100\,E_{\rm th}~{\rm MeV} \\
E_{\rm th}\times\,e^{\epsilon\,(\beta+99)} & {\rm (MeV)}
        & E_0 \geq 100\,E_{\rm th}~{\rm and}~\epsilon < \eta \\
E_0\times\bigl( \beta\,(\epsilon - 1) + 1 + 99\epsilon )/100  &
        & E_0 \geq 100\,E_{\rm th}~{\rm and}~\epsilon \geq \eta
\end{array} \right.
\end{equation}
Here, $\beta = \log(E_0/100\,E_{\rm th})$ and 
$\eta = \beta/(99 + \beta)$.
If resulting $E_2$ is less than 0.5 MeV, nothing is generated. \\ [5mm]
%
%
{\bf Angular distribution} \\ [1mm]
%
The angular distribution is mainly determined by the energy ratio
of the secondary to the projectile\,(i.e. 
the emission angle and probability of the occurrence
increase as the energy ratio decreases).
The emission angle of the secondary particle with respect to the incident direction
is given by 
\begin{equation}\label{eq:angle}
        \theta = - \log \bigl( 1 - \epsilon ( 1 - e^{-\pi\,\tau} ) \bigr)/
        \tau~,
\end{equation}
where $\tau = E_0/5(E_0 + 1/2)$. \\
%%
%%
\subsection{Sampling statistical weight}
The kinematics of the secondary particle are determined randomly
using the above formulae\,(\ref{eq:energy},\ref{eq:angle}).
A statistical weight is calculated and assigned to each generated particle
to reproduce a true inclusive spectrum in the event.
The weight is given by 
\begin{equation}
D2N = V10({\rm JP}) \times DW\,(E) \times DA\,(\theta) \times 
V1\,(E,\theta,{\rm JP}),
\end{equation}
%%
where \\
$\bullet~V10$ is the statistical weight for the production rate
based on neutral pion production\,($V10 = 1$).
\begin{equation}
V10 = \left\{
\begin{array}{ll}
2.0\,(2.5) & {\rm nucleon~production\,(the~case~of~gamma~projectile)} \\
2.1        & {\rm charged~pion/kaon~production} 
\end{array} \right.
\end{equation}\vspace*{2mm}\\
%%
%%
$\bullet$~DW and DA are dominantly determined by the secondary energy and 
emission angle, individually. \\ [2mm]
%%
%%
$\bullet$~V1 is a true double-differential production cross-section (divided by the total
inelastic cross-section)~\cite{MARS98}, calculated in {\it G4Mars5GeV::D2N2}
according to the projectile type and energy, target atomic mass, and simulated secondary 
energy, emission angle and particle type. \\

\section{Status of this document}
11.06.2002 created by N. Kanaya. \\
20.06.2002 modified by N.V.Mokhov. \\


\begin{latexonly}

\begin{thebibliography}{599}
\bibitem{MARS98} N.V.~Mokhov, {\it The MARS Code System User's Guide, Version 13(98)},
                 Fermilab-FN-628.

\bibitem{MARSWWW} {\it http://www-ap.fnal.gov/MARS/}
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item N.V.~Mokhov, {\it The MARS Code System User's Guide, Version 13(98)},
                 Fermilab-FN-628.

\item {\it http://www-ap.fnal.gov/MARS/}
\end{enumerate}

\end{htmlonly}