Mass and Lifetime of Unstable Particles

Abstract

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 ($\ensuremath{\theta}$) 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 L\'evy 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.Under special conditions a ${t}^{\ensuremath{-}\frac{3}{2}}$ term may occur in the amplitude but would be unimportant in magnitude for, say, the decay of a strange particle. It is noted that such nonexponential decay curves might occur for certain low-energy nuclear processes.Consideration is also given to the treatment of two degenerate, unstable particles, such as the neutral $K$ mesons. The general method for handling the problem leads, in the weak-coupling limit, to the same result as the Wigner-Weisskopf method.