The second measurable quantity we will consider is the lifetime, or equivalently the decay rate, of unstable particle species. An unstable particle has a finite probability of decaying during any given time interval, and the decay rate, ฮ, is the probability of decay in a given unit time, just as in the case of radioactive decay. The lifetime, ฯ, of a particle species is a measure of the average lifetime of a particle of this type before it decays, and is related to the decay rate by ฯ = 1/ฮ. Equivalently, the lifetime is the time it takes for the size of a sample of such particles to decrease by a factor of e. A related quantity is the half-life: the time it would take a similar sample to halve in size. The half-life, t1/2 is related to the lifetime by t1/2 = ฯ ln 2. These quantities may be measured experimentally in one of two ways. First, if the lifetime is sufficiently long for the particle to be observed before decay, and if the decay products are observable, then the decay rate may be measured directly by counting the number of decays per unit time. More commonly in particle physics, however, the lifetime of an unstable particle is so short that the particle is not directly observed before decay. Instead, only the products of its decay are observed. Such a particle is known as resonance. In this case, the decay rate can be measured experimentally through statistical analysis of the invariant mass for groups of ยญ particles. If the same final states appear many times in collisions, and the invariant masses are found to peak around a particular value, this is evidence of resonance with that mass.
One may expect that the invariant mass measurements should form an infinitesimally thin spike around the resonance mass rather than a broad peak, but remember that, since the particle is short-lived, there is considerable uncertainty in its energy. The width of the resonance peak at half maximum height is equal to the decay rate of the resonance (in natural units).
Robert Purdy, Particle Physics: An Introduction
