Time evolution of unstable particle decay seen with finite resolution

Time evolution of the decay process of unstable particles is investigated in field theory models. We first formulate how to renormalize the nondecay amplitude beyond perturbation theory and then discuss the short-time behavior of very long-lived particles. Two different formalisms, one that does and one that does not assume the existence of the asymptotic field of unstable particles, are considered. The nondecay amplitude is then calculated by introducing a finite time resolution of the measurement, which makes it possible to discuss both renormalizable and nonrenormalizable decay interactions including nucleon decay. In ordinary circumstances the onset of the exponential decay law starts at times as early as at roughly the resolution time, but with an enhanced amplitude which may be measurable. It is confirmed that the short-time formula [Formula Presented] of the exponential decay law may be used to set limits on the nucleon decay rate in underground experiments. On the other hand, an exceptional example of S-wave decay of a very small Q value is found, which does not have the exponential period at all.