Lifetimes

Clearly the lifetime and the width of a particle are inversely related. For practical applications, however, any particle with a non-negligible width decays too close to its production vertex for the lifetime to be of any interest. In the program, the two aspects are therefore considered separately. Particles with a non-vanishing nominal proper lifetime $\tau_0 = \langle \tau \rangle$ are assigned an actual lifetime according to

$${\cal P}(\tau) \, \d\tau \propto \exp(- \tau / \tau_0 ) \, \d\tau ~,$$ (266)

i.e. a simple exponential decay is assumed. Since the program uses dimensions where the speed of light $c \equiv 1$, and space dimensions are in mm, then actually the unit of $c \tau_0$ is mm and of $\tau_0$ itself mm $/c \approx 3.33\times10^{-12}$ s.

If a particle is produced at a vertex $v = (\mathbf{x}, t)$ with a momentum $p = (\mathbf{p}, E)$ and a lifetime $\tau$, the decay vertex position is assumed to be

$$v' = v + \tau \, \frac{p}{m} ~,$$ (267)

where $m$ is the mass of the particle. With the primary interaction (normally) in the origin, it is therefore possible to construct all secondary vertices in parallel with the ordinary decay treatment.

The formula above does not take into account any detector effects, such as a magnetic field. It is therefore possible to stop the decay chains at some suitable point, and leave any subsequent decay treatment to the detector simulation program. One may select that particles are only allowed to decay if they have a nominal lifetime $\tau_0$ shorter than some given value or, alternatively, if their decay vertices $\mathbf{x}'$ are inside some spherical or cylindrical volume around the origin.