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 nonexponential. Thus, the “threshold cusp” is shown to be a new type of unstable mode, which is found only in coupled channels.

Figure 1

Schematic figure of the uniformized $z$ plane. The red line shows the physical region. th1 (th2) is the lower (upper) threshold. $A$ ($A^{*}$), $B$ ($B^{*}$), and $C$ ($C^{*}$) correspond to the poles specified in Table 1.

Figure 2

Energy spectrum $|{\mathcal{G}}_n{|}^2$ (left), and the survival probability $|{\mathcal{A}}_n{|}^2$ (middle, right), of cases A, B, and C. Labels, “Exact,” “BW,” and “DD” in the left figures, correspond to ${\mathcal{G}}_n$ of Eqs. (21), (24), and (25), respectively. Labels, “Exact,” “Effective Single Channel,” and “Inverse Power” in the right figures, correspond to ${\mathcal{A}}_n$ of Eqs. (22), (27), and (29), respectively.

Figure 3

Energy spectrum (left) and the survival probability (center, right) for $m=1.1$, ${\gamma }_1=0.1$, ${\gamma }_2=0.9$. “Exact,” “Effective Single Channel,” and “Inverse Power” correspond to $\mathcal{A}$ of Eqs. (22), (27), and (29), respectively.