Time evolution of cascade decay
We study non-perturbatively the time evolution of cascade decay for generic fields $\pi \rightarrow \phi_1\phi_2\rightarrow \phi_2\chi_1\chi_2$ and obtain the time dependence of amplitudes and populations for the resonant and final states. We analyze in detail the different time scales and the manifestation of unitary time evolution in the dynamics of production and decay of resonant intermediate and final states. The probability of occupation (population) "flows" as a function of time from the initial to the final states. When the decay width of the parent particle $\Gamma_\pi$ is much larger than that of the intermediate resonant state $\Gamma_{\phi_1}$ there is a "bottleneck" in the flow, the population of resonant states builds up to a maximum at $t^* = \ln[\Gamma_\pi/\Gamma_{\phi_1}]/(\Gamma_\pi-\Gamma_{\phi_1})$ nearly saturating unitarity and decays to the final state on the longer time scale $1/\Gamma_{\phi_1}$. As a consequence of the wide separation of time scales in this case the cascade decay can be interpreted as evolving sequentially $\pi \rightarrow \phi_1\phi_2; ~ \phi_1\phi_2\rightarrow \phi_2\chi_1\chi_2$. In the opposite limit the population of resonances ($\phi_1$) does not build up substantially and the cascade decay proceeds almost directly from the initial parent to the final state without resulting in a large amplitude of the resonant state. An alternative but equivalent non-perturbative method useful in cosmology is presented. Possible phenomenological implications for heavy sterile neutrinos as resonant states and consequences of quantum entanglement and correlations in the final state are discussed.