Survival Probability of Unstable States in Coupled-Channels -- nonexponential decay of "threshold-cusp"

We investigate the survival probability of unstable states, the time-dependence of an initial state, in coupled channels. First, we extend the formulation of the survival probability from single channel to coupled channels (two channels). We derive an exact general expression of the two-channel survival probability using uniformization, a method which makes the coupled-channel S matrix single-valued, and the Mittag-Leffler expansion, i.e. a pole expansion. Second, we calculate the time dependence of the two-channel survival probability by employing the derived expression. It is the minimal distance between the pole and the physical region in the complex energy plane, not the imaginary part of the pole energy, which determines not only the energy spectrum of the Green's function but also the survival probability. The survival probability of the "threshold-cusp" caused by a pole on the unusual complex-energy Riemann sheet is shown to decay, not grow in time though the imaginary part of the pole energy is positive. We also show that the decay of the "threshold-cusp" is non-exponential. Thus, the "threshold-cusp" is shown to be a new type of unstable mode, which is found only in coupled channels.