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 the decay occurs between $(0,t)$ in the $i$-th channel is a relevant object. The general form of the function $w_i(t)$ was recently presented in PLB $\mathbf{\text{831}}$ (2022), 137200. 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 also not such. In particular, the ratio $w_i\mathrm{/}{w}_j$ (for $i\mathrm{\ne }j)$ 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.