\chapter{Decay}
\noindent
The decay of particles in flight and at rest is simulated by the 
{\it G4Decay} class.

\section{Mean Free Path for Decay in Flight}
\noindent
The mean free path $\lambda$ is calculated for each step using
\begin{eqnarray*}
\lambda = \gamma \beta c \tau
\end{eqnarray*}
where $\tau$ is the lifetime of the particle and
\begin{eqnarray*}
\gamma = \frac{1}{\sqrt{1 - \beta^2}}.
\end{eqnarray*}
$\beta$ and $\gamma$ are calculated using the momentum at the beginning of 
the step.  The decay time in the rest frame of the particle (proper time)
is then sampled and converted to a decay length using $\beta$.

\section{Branching Ratios and Decay Channels}
\noindent 
{\it G4Decay} selects a decay mode for the particle according to branching 
ratios defined in the {\it G4DecayTable} class, which is a member of the
{\it G4ParticleDefinition} class.  Each mode is implemented as a class
derived from {\it G4VDecayChannel} and is responsible for generating 
the secondaries and the kinematics of the decay.  In a given decay channel 
the daughter particle momenta are calculated in the rest frame of the parent
and then boosted into the laboratory frame.  Polarization is not currently 
taken into account for either the parent or its daughters.

A large number of specific decay channels may be required to simulate an
experiment, ranging from two-body to many-body decays and $V-A$ to 
semi-leptonic decays.  Most of these are covered by the five decay channel 
classes provided by Geant4:\\
\begin{tabular}[t]{ll}
 G4PhaseSpaceDecayChannel    & : phase space decay \\
 G4DalitzDecayChannel        & : dalitz decay \\
 G4MuonDecayChannel          & : muon decay \\
 G4TauLeptonicDecayChannel   & : tau leptonic decay \\
 G4KL3DecayChannel           & : semi-leptonic decays of kaon .\\
             & \\
\end{tabular}
\\

\subsection{G4PhaseSpaceDecayChannel}
\noindent
The majority of decays in Geant4 are implemented using the  
{\it G4PhaseSpaceDecayChannel} class.  It simulates phase space decays with 
isotropic angular distributions in the center-of-mass system.  Three private 
methods of {\it G4PhaseSpaceDecayChannel} are provided to handle two-, three- 
and N-body decays:\\
\begin{tabular}[t]{ll}
 TwoBodyDecayIt()  & \\
 ThreeBodyDecayIt() & \\
 ManyBodyDecayIt() & \\
\end{tabular}

\noindent 
Some examples of decays handled by this class are:
\begin{eqnarray*}
  \pi^{0} \rightarrow \gamma \gamma ,
\end{eqnarray*}
\begin{eqnarray*}
  \Lambda \rightarrow p \pi^- 
\end{eqnarray*}
\noindent
and 
\begin{eqnarray*}
  {K^0}_L \rightarrow  \pi^0 \pi^+ \pi^- . 
\end{eqnarray*}


\subsection{G4DalitzDecayChannel}
\noindent
The Dalitz decay 
\begin{eqnarray*}
  \pi^{0} \rightarrow \gamma + e^{+} + e^{-}  
\end{eqnarray*}
and other Dalitz-like decays, such as 
\begin{eqnarray*}
  {K^0}_L \rightarrow \gamma + e^{+} + e^{-}
\end{eqnarray*}
and  
\begin{eqnarray*}
  {K^0}_L \rightarrow \gamma + \mu^{+} + \mu^{-}  
\end{eqnarray*}
are simulated by the {\it G4DalitzDecayChannel} class.  In general, it handles 
any decay of the form 

\begin{eqnarray*}
 P^{0} \rightarrow {\gamma} + l^{+} + l^{-} ,
\end{eqnarray*}

\noindent 
where $P^{0}$ is a spin-0 meson of mass $M$ and $l^{\pm}$ are leptons of 
mass $m$.  The angular distribution of the $\gamma$ is isotropic in the 
center-of-mass system of the parent particle and the leptons are generated 
isotropically and back-to-back in their center-of-mass frame.  The magnitude 
of the leptons' momentum is sampled from the distribution function

\begin{eqnarray*}
 f(t) = {(1-\frac{t}{M^2})}^3(1+\frac{2m^2}{t})\sqrt{1-\frac{4m^2}{t}} ,
\end{eqnarray*}

\noindent where $t$ is the square of the sum of the leptons' energy in their 
center-of-mass frame.


\subsection{Muon Decay}
\noindent
{\it G4MuonDecayChannel} simulates muon decay according to $V-A$ theory. 
Neglecting the electron mass, the electron energy is sampled from the 
following distribution:
\begin{eqnarray*}
  d\Gamma = \frac{{G_F}^2{m_{\mu}}^5}{192\pi^3}2\epsilon^2(3 - 2\epsilon)\,
\end{eqnarray*}
where: \quad
\begin{tabular}[t]{ll}
 $\Gamma$    & : decay rate \\
 $\epsilon$  & : $= E_e / E_{max} $\\
 $E_e$       & : electron energy \\
 $E_{max}$   & : maximum electron energy $ = m_{\mu}/2$\\
             & \\
\end{tabular}
\\

\noindent
The momenta of the two neutrinos are not sampled from their $V-A$ 
distributions.  Instead they are generated back-to-back and isotropically in
the neutrinos' center-of-mass frame, with the magnitude of the neutrino 
momentum chosen to conserve energy in the decay.  The two neutrinos are then 
boosted opposite to the momentum of the decay electron.  This approximation is
sufficient for most simulations because the neutrino is usually not observed 
in any detector.

Currently, neither the polarization of the muon or the electron is considered
in this class.


\subsection{Leptonic Tau Decay}
\noindent
{\it G4TauLeptonicDecayChannel} simulates leptonic tau decays according to 
$V-A$ theory.  This class is valid for both
\begin{eqnarray*}
 {\tau}^{\pm} \rightarrow  e^{\pm} + {\nu}_{\tau} + {\nu}_e
\end{eqnarray*}
 and
\begin{eqnarray*}
 {\tau}^{\pm} \rightarrow  {\mu}^{\pm} + {\nu}_{\tau} + {\nu}_{\mu}
\end{eqnarray*}
modes.

The energy spectrum is calculated without neglecting lepton mass as follows:
\begin{eqnarray*}
  d\Gamma = \frac{{G_F}^2{m_{\tau}}^3}{24\pi^3}{p_l}{E_l}(3E_l{m_{\tau}}^2 - 4{E_l}^2{m_{\tau}} - 2{
m_{\tau}}{m_l}^2)
\end{eqnarray*}
where: \quad
\begin{tabular}[t]{ll}
 $\Gamma$    & : decay rate \\
 $E_l$       & : daughter lepton energy (total energy) \\
 $p_l$       & : daughter lepton momentum \\
 $m_l$       & : daughter lepton mass \\
             & \\
\end{tabular}
\\
\noindent
As in the case of muon decay, the energies of the two neutrinos are not 
sampled from their $V-A$ spectra, but are calculated so that energy and
momentum are conserved.  Polarization of the ${\tau}$ and final state leptons
is not taken into account in this class.


\subsection{Kaon Decay}
The class {\it G4KL3DecayChannel} simulates the following four semi-leptonic 
decay modes of the kaon:\\ 

\begin{tabular}[t]{ll}
 ${K^{\pm}}_{e3}$    & : $ K^{\pm} \rightarrow {\pi}^0 + e^{\pm} + {\nu} $ \\ 
 ${K^{\pm}}_{{\mu}3}$  & : $ K^{\pm} \rightarrow {\pi}^0 + {\mu}^{\pm} + {\nu} $ \\
 ${K^0}_{e3}$    & : $ K^0_L \rightarrow \pi^{\pm} + e^{\mp} + {\nu} $ \\ 
 ${K^0}_{{\mu}3}$  & : $ K^0_L \rightarrow \pi^{\pm} + {\mu}^{\mp} + {\nu} $\\ 
             & \\
\end{tabular}
\\
\noindent
Assuming that only the vector current contributes to 
$K \rightarrow l{\pi}{\nu}$ decays, the matrix element can be described by 
using two dimensionless form factors, $f_+$ and $f_-$, which depend only on 
the momentum transfer $t = ( P_K - P_\pi )^2$.\\
The Dalitz plot density used in this class is as follows \cite{Chounet72}:\\
\begin{eqnarray*}
  \rho\,(E_\pi , E_\mu ) \propto f^2_+\,(t) [ A + B \xi\,(t) + C{\xi\,(t)}^2 ]
\end{eqnarray*}
where: \quad
\begin{tabular}[t]{ll}
 $ A = m_K (2E_\mu E_\nu - m_K E'_\pi) 
       + {m_{\mu}}^2 ( \frac{1}{4} E'_\pi - E_\nu) $\\
 $ B = {m_{\mu}}^2 (E_\nu- \frac{1}{2} E'_\pi ) $\\
 $ C =  \frac{1}{4}  {m_{\mu}}^2 E'_\pi$\\
 $ E'_\pi =  {E_\pi}^{max} -  E_\pi $\\
            & \\
\end{tabular}
\\
Here $\xi\,(t)$ is the ratio of the two form factors \\
\begin{eqnarray*}
  \xi\,(t) = f_-\,(t) / f_+\,(t) .
\end{eqnarray*}
$f_+\,(t)$ is assumed to depend linearly on t, i.e. 
\begin{eqnarray*}
  f_+\,(t)  = f_+\,(0) [ 1 + \lambda_+ (t/{m_\pi}^2) ] 
\end{eqnarray*}
and $f_-\,(t)$ is assumed to be constant due to time reversal invariance.\\
\\
Two parameters, $\lambda_+$ and $\xi\,(0)$ are then used for describing the 
Dalitz plot density in this class.  The values of these parameters are taken
to be the world average values given by the Particle Data Group \cite{PDG}.

\section{Status of this document}
 10.04.02   re-written by D.H. Wright \\
 02.04.02   editing by Hisaya Kurashige \\
 14.11.01   editing by Hisaya Kurashige \\

\begin{latexonly}

\begin{thebibliography}{99}
\bibitem{Chounet72} L.M. Chounet, J.M. Gaillard, and M.K. Gaillard, 
   Phys. Reports 4C, 199 (1972).
\bibitem{PDG} 
   {\em Review of Particle Physics}
   The European Physical Journal C, 15 (2000).
\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item L.M. Chounet, J.M. Gaillard, and M.K. Gaillard, Phys. Reports 4C, 199 (1972).
\item {\em Review of Particle Physics} The European Physical Journal C, 15 (2000).
\end{enumerate}

\end{htmlonly}
