Particle Decay

The decay of particles in flight and at rest is simulated by the G4Decay class.

Mean Free Path for Decay in Flight

The mean free path \(\lambda\) is calculated for each step using

\[\lambda = \gamma \beta c \tau\]

where \(\tau\) is the lifetime of the particle and

\[\gamma = \frac{1}{\sqrt{1 - \beta^2}}.\]

\(\beta\) 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 \(\beta\).

Branching Ratios and Decay Channels

G4Decay selects a decay mode for the particle according to branching ratios defined in the G4DecayTable class, which is a member of the G4ParticleDefinition class. Each mode is implemented as a class derived from 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:

G4PhaseSpaceDecayChannel	phase space decay
G4DalitzDecayChannel	dalitz decay
G4MuonDecayChannel	muon decay
G4TauLeptonicDecayChannel	tau leptonic decay
G4KL3DecayChannel	semi-leptonic decays of kaon

G4PhaseSpaceDecayChannel

The majority of decays in Geant4 are implemented using the G4PhaseSpaceDecayChannel class. It simulates phase space decays with isotropic angular distributions in the center-of-mass system. Three private methods of G4PhaseSpaceDecayChannel are provided to handle two-, three- and N-body decays: TwoBodyDecayIt(), ThreeBodyDecayIt(), ManyBodyDecayIt().

Some examples of decays handled by this class are:

\[\begin{split}\pi^{0} & \rightarrow \gamma \gamma , \\ \Lambda & \rightarrow p \pi^-\end{split}\]

and

\[{K^0}_L \rightarrow \pi^0 \pi^+ \pi^- .\]

G4DalitzDecayChannel

The Dalitz decay

\[\pi^{0} \rightarrow \gamma + e^{+} + e^{-}\]

and other Dalitz-like decays, such as

\[{K^0}_L \rightarrow \gamma + e^{+} + e^{-}\]

and

\[{K^0}_L \rightarrow \gamma + \mu^{+} + \mu^{-}\]

are simulated by the G4DalitzDecayChannel class. In general, it handles any decay of the form

\[P^{0} \rightarrow {\gamma} + l^{+} + l^{-} ,\]

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

\[f(t) = \left( 1-\frac{t}{M^2} \right)^3 \left( 1+\frac{2m^2}{t} \right) \sqrt{1-\frac{4m^2}{t}} ,\]

where \(t\) is the square of the sum of the leptons’ energy in their center-of-mass frame.

Muon Decay

G4MuonDecayChannel simulates muon decay according to V-A theory. The electron energy is sampled from the following distribution:

\[d\Gamma = \frac{{G_F}^2{m_{\mu}}^5}{192\pi^3}2\epsilon^2(3 - 2\epsilon)\,\]

where:

\[\begin{split}\Gamma &= \mbox{decay rate} \\ \epsilon &= E_e / E_{max} \\ E_e &= \mbox{electron energy} \\ E_{max} &= \mbox{maximum electron energy } = m_{\mu}/2\end{split}\]

The magnitudes of the two neutrino momenta are also sampled from the V-A distribution and constrained by energy conservation. The direction of the electron neutrino is sampled using

\[\cos(\theta) = 1 - 2/E_e - 2/E_{\nu e} + 2/E_e/E_{\nu e}\]

and the muon anti-neutrino momentum is chosen to conserve momentum. Currently, neither the polarization of the muon nor the electron is considered in this class.

Leptonic Tau Decay

G4TauLeptonicDecayChannel simulates leptonic tau decays according to V-A theory. This class is valid for both

\[{\tau}^{\pm} \rightarrow e^{\pm} + {\nu}_{\tau} + {\nu}_e\]

and

\[{\tau}^{\pm} \rightarrow {\mu}^{\pm} + {\nu}_{\tau} + {\nu}_{\mu}\]

modes.

The energy spectrum is calculated without neglecting lepton mass as follows:

\[d\Gamma = \frac{{G_F}^2{m_{\tau}}^3}{24\pi^3}{p_l}{E_l} \left(3E_l{m_{\tau}}^2 - 4{E_l}^2{m_{\tau}} - 2{ m_{\tau}}{m_l}^2 \right)\]

where:

\[\begin{split}\Gamma &= \mbox{decay rate} \\ E_l &= \mbox{daughter lepton energy (total energy)} \\ p_l &= \mbox{daughter lepton momentum} \\ m_l &= \mbox{daughter lepton mass}\end{split}\]

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.

Kaon Decay

The class G4KL3DecayChannel simulates the following four semi-leptonic decay modes of the kaon:

\[\begin{split}{K^{\pm}}_{e3} &: K^{\pm} \rightarrow {\pi}^0 + e^{\pm} + {\nu} \\ {K^{\pm}}_{{\mu}3} &: K^{\pm} \rightarrow {\pi}^0 + {\mu}^{\pm} + {\nu} \\ {K^0}_{e3} &: K^0_L \rightarrow \pi^{\pm} + e^{\mp} + {\nu} \\ {K^0}_{{\mu}3} &: K^0_L \rightarrow \pi^{\pm} + {\mu}^{\mp} + {\nu}\end{split}\]

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 [LMCG72]:

\[\rho\,(E_\pi , E_\mu ) \propto f^2_+\,(t) \left[ A + B \xi\,(t) + C{\xi\,(t)}^2 \right]\]

where:

\[\begin{split}A &= m_K (2E_\mu E_\nu - m_K E'_\pi) + {m_{\mu}}^2 \left( \textstyle \frac{1}{4} E'_\pi - E_\nu \right) \\ B &= {m_{\mu}}^2 \left( E_\nu- \textstyle \frac{1}{2} E'_\pi \right) \\ C &= \textstyle \frac{1}{4} {m_{\mu}}^2 E'_\pi \\ E'_\pi &= {E_\pi}^{max} - E_\pi\end{split}\]

Here \(\xi\,(t)\) is the ratio of the two form factors

\[\xi\,(t) = f_-\,(t) / f_+\,(t) .\]

\(f_+\,(t)\) is assumed to depend linearly on t, i.e.,

\[f_+\,(t) = f_+\,(0) [ 1 + \lambda_+ (t/{m_\pi}^2) ]\]

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 [eal00].

Bibliography

eal00

D.E. Groom et al. Review of Particle Physics. The European Physical Journal, C15:1+, 2000. URL: http://pdg.lbl.gov.

LMCG72

L.M. Chounet, J.M. Gaillard and M.K. Gaillard. Phys. Reports 4C, 199, 1972.