Mass and Lifetime of Unstable Particles

The relationship between the properties of the propagator of an unstable particle and the observation of mass and lifetime is considered. For illustrative purposes a model of a scalar (or pseudoscalar) particle ($\theta$) weakly coupled to two pions is treated. The propagator is shown to have a simple pole on the second (unphysical) Riemann sheet and it is assumed, as suggested by Peierls, that this is generally the case. By analysis of a prototype experiment in terms of wave packets, it is shown that the measured mass and lifetime are determined by the real and imaginary parts of the pole, respectively. Nonexponential terms occur in the life-time curve, as is well known. These are shown to be related to the uncertainty in the time of the production or detection event under normal circumstances. This conclusion is similar to those of Lévy and of Schwinger, but more closely related to experimental conditions. In particular it is found that the wave packets introduce a "mass filter" in a somewhat different manner from that suggested by Schwinger.