[1211] | 1 | \chapter{Decay} |
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| 2 | \noindent |
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| 3 | The decay of particles in flight and at rest is simulated by the |
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| 4 | {\it G4Decay} class. |
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| 5 | |
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| 6 | \section{Mean Free Path for Decay in Flight} |
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| 7 | \noindent |
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| 8 | The mean free path $\lambda$ is calculated for each step using |
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| 9 | \begin{eqnarray*} |
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| 10 | \lambda = \gamma \beta c \tau |
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| 11 | \end{eqnarray*} |
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| 12 | where $\tau$ is the lifetime of the particle and |
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| 13 | \begin{eqnarray*} |
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| 14 | \gamma = \frac{1}{\sqrt{1 - \beta^2}}. |
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| 15 | \end{eqnarray*} |
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| 16 | $\beta$ and $\gamma$ are calculated using the momentum at the beginning of |
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| 17 | the step. The decay time in the rest frame of the particle (proper time) |
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| 18 | is then sampled and converted to a decay length using $\beta$. |
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| 19 | |
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| 20 | \section{Branching Ratios and Decay Channels} |
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| 21 | \noindent |
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| 22 | {\it G4Decay} selects a decay mode for the particle according to branching |
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| 23 | ratios defined in the {\it G4DecayTable} class, which is a member of the |
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| 24 | {\it G4ParticleDefinition} class. Each mode is implemented as a class |
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| 25 | derived from {\it G4VDecayChannel} and is responsible for generating |
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| 26 | the secondaries and the kinematics of the decay. In a given decay channel |
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| 27 | the daughter particle momenta are calculated in the rest frame of the parent |
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| 28 | and then boosted into the laboratory frame. Polarization is not currently |
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| 29 | taken into account for either the parent or its daughters. |
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| 30 | |
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| 31 | A large number of specific decay channels may be required to simulate an |
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| 32 | experiment, ranging from two-body to many-body decays and $V-A$ to |
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| 33 | semi-leptonic decays. Most of these are covered by the five decay channel |
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| 34 | classes provided by Geant4:\\ |
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| 35 | \begin{tabular}[t]{ll} |
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| 36 | G4PhaseSpaceDecayChannel & : phase space decay \\ |
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| 37 | G4DalitzDecayChannel & : dalitz decay \\ |
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| 38 | G4MuonDecayChannel & : muon decay \\ |
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| 39 | G4TauLeptonicDecayChannel & : tau leptonic decay \\ |
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| 40 | G4KL3DecayChannel & : semi-leptonic decays of kaon .\\ |
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| 41 | & \\ |
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| 42 | \end{tabular} |
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| 43 | \\ |
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| 44 | |
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| 45 | \subsection{G4PhaseSpaceDecayChannel} |
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| 46 | \noindent |
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| 47 | The majority of decays in Geant4 are implemented using the |
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| 48 | {\it G4PhaseSpaceDecayChannel} class. It simulates phase space decays with |
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| 49 | isotropic angular distributions in the center-of-mass system. Three private |
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| 50 | methods of {\it G4PhaseSpaceDecayChannel} are provided to handle two-, three- |
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| 51 | and N-body decays:\\ |
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| 52 | \begin{tabular}[t]{ll} |
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| 53 | TwoBodyDecayIt() & \\ |
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| 54 | ThreeBodyDecayIt() & \\ |
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| 55 | ManyBodyDecayIt() & \\ |
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| 56 | \end{tabular} |
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| 57 | |
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| 58 | \noindent |
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| 59 | Some examples of decays handled by this class are: |
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| 60 | \begin{eqnarray*} |
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| 61 | \pi^{0} \rightarrow \gamma \gamma , |
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| 62 | \end{eqnarray*} |
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| 63 | \begin{eqnarray*} |
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| 64 | \Lambda \rightarrow p \pi^- |
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| 65 | \end{eqnarray*} |
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| 66 | \noindent |
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| 67 | and |
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| 68 | \begin{eqnarray*} |
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| 69 | {K^0}_L \rightarrow \pi^0 \pi^+ \pi^- . |
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| 70 | \end{eqnarray*} |
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| 71 | |
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| 72 | |
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| 73 | \subsection{G4DalitzDecayChannel} |
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| 74 | \noindent |
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| 75 | The Dalitz decay |
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| 76 | \begin{eqnarray*} |
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| 77 | \pi^{0} \rightarrow \gamma + e^{+} + e^{-} |
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| 78 | \end{eqnarray*} |
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| 79 | and other Dalitz-like decays, such as |
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| 80 | \begin{eqnarray*} |
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| 81 | {K^0}_L \rightarrow \gamma + e^{+} + e^{-} |
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| 82 | \end{eqnarray*} |
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| 83 | and |
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| 84 | \begin{eqnarray*} |
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| 85 | {K^0}_L \rightarrow \gamma + \mu^{+} + \mu^{-} |
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| 86 | \end{eqnarray*} |
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| 87 | are simulated by the {\it G4DalitzDecayChannel} class. In general, it handles |
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| 88 | any decay of the form |
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| 89 | |
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| 90 | \begin{eqnarray*} |
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| 91 | P^{0} \rightarrow {\gamma} + l^{+} + l^{-} , |
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| 92 | \end{eqnarray*} |
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| 93 | |
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| 94 | \noindent |
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| 95 | where $P^{0}$ is a spin-0 meson of mass $M$ and $l^{\pm}$ are leptons of |
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| 96 | mass $m$. The angular distribution of the $\gamma$ is isotropic in the |
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| 97 | center-of-mass system of the parent particle and the leptons are generated |
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| 98 | isotropically and back-to-back in their center-of-mass frame. The magnitude |
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| 99 | of the leptons' momentum is sampled from the distribution function |
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| 100 | |
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| 101 | \begin{eqnarray*} |
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| 102 | f(t) = {(1-\frac{t}{M^2})}^3(1+\frac{2m^2}{t})\sqrt{1-\frac{4m^2}{t}} , |
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| 103 | \end{eqnarray*} |
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| 104 | |
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| 105 | \noindent where $t$ is the square of the sum of the leptons' energy in their |
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| 106 | center-of-mass frame. |
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| 107 | |
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| 108 | |
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| 109 | \subsection{Muon Decay} |
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| 110 | \noindent |
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| 111 | {\it G4MuonDecayChannel} simulates muon decay according to $V-A$ theory. |
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| 112 | Neglecting the electron mass, the electron energy is sampled from the |
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| 113 | following distribution: |
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| 114 | \begin{eqnarray*} |
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| 115 | d\Gamma = \frac{{G_F}^2{m_{\mu}}^5}{192\pi^3}2\epsilon^2(3 - 2\epsilon)\, |
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| 116 | \end{eqnarray*} |
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| 117 | where: \quad |
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| 118 | \begin{tabular}[t]{ll} |
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| 119 | $\Gamma$ & : decay rate \\ |
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| 120 | $\epsilon$ & : $= E_e / E_{max} $\\ |
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| 121 | $E_e$ & : electron energy \\ |
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| 122 | $E_{max}$ & : maximum electron energy $ = m_{\mu}/2$\\ |
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| 123 | & \\ |
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| 124 | \end{tabular} |
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| 125 | \\ |
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| 126 | |
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| 127 | \noindent |
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| 128 | The momenta of the two neutrinos are not sampled from their $V-A$ |
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| 129 | distributions. Instead they are generated back-to-back and isotropically in |
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| 130 | the neutrinos' center-of-mass frame, with the magnitude of the neutrino |
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| 131 | momentum chosen to conserve energy in the decay. The two neutrinos are then |
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| 132 | boosted opposite to the momentum of the decay electron. This approximation is |
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| 133 | sufficient for most simulations because the neutrino is usually not observed |
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| 134 | in any detector. |
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| 135 | |
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| 136 | Currently, neither the polarization of the muon or the electron is considered |
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| 137 | in this class. |
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| 138 | |
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| 139 | |
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| 140 | \subsection{Leptonic Tau Decay} |
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| 141 | \noindent |
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| 142 | {\it G4TauLeptonicDecayChannel} simulates leptonic tau decays according to |
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| 143 | $V-A$ theory. This class is valid for both |
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| 144 | \begin{eqnarray*} |
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| 145 | {\tau}^{\pm} \rightarrow e^{\pm} + {\nu}_{\tau} + {\nu}_e |
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| 146 | \end{eqnarray*} |
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| 147 | and |
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| 148 | \begin{eqnarray*} |
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| 149 | {\tau}^{\pm} \rightarrow {\mu}^{\pm} + {\nu}_{\tau} + {\nu}_{\mu} |
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| 150 | \end{eqnarray*} |
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| 151 | modes. |
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| 152 | |
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| 153 | The energy spectrum is calculated without neglecting lepton mass as follows: |
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| 154 | \begin{eqnarray*} |
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| 155 | 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{ |
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| 156 | m_{\tau}}{m_l}^2) |
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| 157 | \end{eqnarray*} |
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| 158 | where: \quad |
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| 159 | \begin{tabular}[t]{ll} |
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| 160 | $\Gamma$ & : decay rate \\ |
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| 161 | $E_l$ & : daughter lepton energy (total energy) \\ |
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| 162 | $p_l$ & : daughter lepton momentum \\ |
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| 163 | $m_l$ & : daughter lepton mass \\ |
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| 164 | & \\ |
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| 165 | \end{tabular} |
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| 166 | \\ |
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| 167 | \noindent |
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| 168 | As in the case of muon decay, the energies of the two neutrinos are not |
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| 169 | sampled from their $V-A$ spectra, but are calculated so that energy and |
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| 170 | momentum are conserved. Polarization of the ${\tau}$ and final state leptons |
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| 171 | is not taken into account in this class. |
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| 172 | |
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| 173 | |
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| 174 | \subsection{Kaon Decay} |
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| 175 | The class {\it G4KL3DecayChannel} simulates the following four semi-leptonic |
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| 176 | decay modes of the kaon:\\ |
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| 177 | |
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| 178 | \begin{tabular}[t]{ll} |
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| 179 | ${K^{\pm}}_{e3}$ & : $ K^{\pm} \rightarrow {\pi}^0 + e^{\pm} + {\nu} $ \\ |
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| 180 | ${K^{\pm}}_{{\mu}3}$ & : $ K^{\pm} \rightarrow {\pi}^0 + {\mu}^{\pm} + {\nu} $ \\ |
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| 181 | ${K^0}_{e3}$ & : $ K^0_L \rightarrow \pi^{\pm} + e^{\mp} + {\nu} $ \\ |
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| 182 | ${K^0}_{{\mu}3}$ & : $ K^0_L \rightarrow \pi^{\pm} + {\mu}^{\mp} + {\nu} $\\ |
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| 183 | & \\ |
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| 184 | \end{tabular} |
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| 185 | \\ |
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| 186 | \noindent |
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| 187 | Assuming that only the vector current contributes to |
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| 188 | $K \rightarrow l{\pi}{\nu}$ decays, the matrix element can be described by |
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| 189 | using two dimensionless form factors, $f_+$ and $f_-$, which depend only on |
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| 190 | the momentum transfer $t = ( P_K - P_\pi )^2$.\\ |
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| 191 | The Dalitz plot density used in this class is as follows \cite{Chounet72}:\\ |
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| 192 | \begin{eqnarray*} |
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| 193 | \rho\,(E_\pi , E_\mu ) \propto f^2_+\,(t) [ A + B \xi\,(t) + C{\xi\,(t)}^2 ] |
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| 194 | \end{eqnarray*} |
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| 195 | where: \quad |
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| 196 | \begin{tabular}[t]{ll} |
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| 197 | $ A = m_K (2E_\mu E_\nu - m_K E'_\pi) |
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| 198 | + {m_{\mu}}^2 ( \frac{1}{4} E'_\pi - E_\nu) $\\ |
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| 199 | $ B = {m_{\mu}}^2 (E_\nu- \frac{1}{2} E'_\pi ) $\\ |
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| 200 | $ C = \frac{1}{4} {m_{\mu}}^2 E'_\pi$\\ |
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| 201 | $ E'_\pi = {E_\pi}^{max} - E_\pi $\\ |
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| 202 | & \\ |
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| 203 | \end{tabular} |
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| 204 | \\ |
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| 205 | Here $\xi\,(t)$ is the ratio of the two form factors \\ |
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| 206 | \begin{eqnarray*} |
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| 207 | \xi\,(t) = f_-\,(t) / f_+\,(t) . |
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| 208 | \end{eqnarray*} |
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| 209 | $f_+\,(t)$ is assumed to depend linearly on t, i.e. |
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| 210 | \begin{eqnarray*} |
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| 211 | f_+\,(t) = f_+\,(0) [ 1 + \lambda_+ (t/{m_\pi}^2) ] |
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| 212 | \end{eqnarray*} |
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| 213 | and $f_-\,(t)$ is assumed to be constant due to time reversal invariance.\\ |
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| 214 | \\ |
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| 215 | Two parameters, $\lambda_+$ and $\xi\,(0)$ are then used for describing the |
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| 216 | Dalitz plot density in this class. The values of these parameters are taken |
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| 217 | to be the world average values given by the Particle Data Group \cite{PDG}. |
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| 218 | |
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| 219 | \section{Status of this document} |
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| 220 | 10.04.02 re-written by D.H. Wright \\ |
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| 221 | 02.04.02 editing by Hisaya Kurashige \\ |
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| 222 | 14.11.01 editing by Hisaya Kurashige \\ |
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| 223 | |
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| 224 | \begin{latexonly} |
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| 225 | |
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| 226 | \begin{thebibliography}{99} |
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| 227 | \bibitem{Chounet72} L.M. Chounet, J.M. Gaillard, and M.K. Gaillard, |
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| 228 | Phys. Reports 4C, 199 (1972). |
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| 229 | \bibitem{PDG} |
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| 230 | {\em Review of Particle Physics} |
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| 231 | The European Physical Journal C, 15 (2000). |
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| 232 | \end{thebibliography} |
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| 233 | |
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| 234 | \end{latexonly} |
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| 235 | |
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| 236 | \begin{htmlonly} |
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| 237 | |
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| 238 | \section{Bibliography} |
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| 239 | |
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| 240 | \begin{enumerate} |
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| 241 | \item L.M. Chounet, J.M. Gaillard, and M.K. Gaillard, Phys. Reports 4C, 199 (1972). |
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| 242 | \item {\em Review of Particle Physics} The European Physical Journal C, 15 (2000). |
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| 243 | \end{enumerate} |
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| 244 | |
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| 245 | \end{htmlonly} |
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