Multichannel Decay: Alternative Derivation of the i-th Channel Decay Probability

In the study of decays, it is quite common that an unstable quantum state/particle has multiple distinct decay channels. In this case, besides the survival probability p(t), also the probability w(i)(t) that a decay occurs between (0, t) in the i-th channel is a relevant object. The general form of the function wi(t) was recently presented in PLB 831, 137200 (2022). Here, we provide a novel and detailed "joint" derivation of both p(t) and w(i)(t). As it is well known, p(t) is not an exponential function; similarly, w(i)(t) is not one either. In particular, the ratio w(i)=w(j) (for i not equal j) is not a simple constant as it would be in the exponential limit. The functions w(i)(t) and their mutual ratios may therefore represent a novel tool to study the non-exponential nature of the decay law.