Decomposition of scattering phase shifts of coupled-channels systems in the complex scaling method

The decomposition of scattering phase shifts in single-channel systems is extended to coupled-channels systems in the complex scaling method. The present method describes the contribution of the resonant states to the phase shift separately from the background in each channel phase shift of the coupled-channels system. We apply this method to the (${}^3\mathrm{H}+p)+({}^3\mathrm{He}+n)$ model for ${}^4\mathrm{He}$, and show that the decomposition of phase shifts in coupled-channels systems provides useful information about the structure of scattering states.

Figure 1

The schematic eigenvalue distribution of $H\left(\theta \right)$ for a single-channel system. Continuum states on the $2\theta$-line are discretized in the finite basis function method as solid circles.

Figure 2

The schematic eigenvalue distribution of $H\left(\theta \right)$ for a coupled-channels system in the complex energy plane. Continuum states on the $2\theta$ lines ($L_1,L_2,\cdots ,L_{n_c}$) are discretized in the finite basis function method as solid circles.

Figure 3

Distribution of the energy eigenvalues in the complex energy plane obtained by solving the coupled-channels equation Eq. (34) with $\theta ={30}^{\circ }$. The filled black circles represent the calculated eigenvalues and dashed lines are $2\theta$ lines. Left (a): ${}^{3}P_0$ state. Middle (b): ${}^{3}P_1$ state. Right (c): ${}^{3}P_2$ state.

Figure 4

The CLD of the ${}^{3}P_1$ state for $\theta ={20}^{\circ }, {30}^{\circ }$, and ${35}^{\circ }$ measured from the ${}^3\mathrm{H}+p$ energy. (a) Left: the total CLD. (b) Middel; the ${}^3\mathrm{H}+p$ channel CLD. (c) Right: the ${}^3\mathrm{He}+n$ channel CLD. The vertical dashed line indicates the threshold energy of ${}^3\mathrm{He}+n$ channel.

Figure 5

The phase shifts of $P$ waves of ${}^3\mathrm{H}+p$ and ${}^3\mathrm{He}+n$ channels. The experimental data are taken from Ref. [18]. The observed phase shifts are displayed by stars, circles, and triangles for the ${}^{3}P_0, {}^{3}P_1$, and ${}^{3}P_2$ states, respectively. The calculated results are given by dotted (${}^{3}P_0$), solid (${}^{3}P_1$), and open (${}^{3}P_2$) curves, respectively. Left (a): the phase shifts of ${}^3\mathrm{H}+p$ channel. Right (b): the phase shifts of ${}^3\mathrm{He}+n$ channel.

Figure 6

The diagonal potential $\left[V_D^q\left(r\right)\right]$, the coupling potential $\left[V_C^q\left(r\right)\right]$, and the state dependence of the $3N-N$ potential for the ${}^3{P}_0$ state. The diagonal and the coupling potentials are presented by thick and thin solid curves. The potential for $T=0$ and $T=1$ states are presented by thin and thick dashed curves.

Figure 7

The eigenvalue distribution on the complex energy plane (a) and the calculated phase shifts (b) of the ${}^{3}P_0$ state calculated without the coupling potential for $\theta ={30}^{\circ }$. Left panel (a) displays the calculated single-channel eigenvalue distributions for ${}^3\mathrm{H}+p$ (open circles) and ${}^3\mathrm{He}+n$ (open triangles) channels. Right panel (b) shows the calculated singe-channel phase shifts. The contributions of the resonant poles are expressed by thin and thick dashed curves, the continuum parts are displayed by thin and thick dotted curves, and the summations of the resonant pole and continuum parts are implied by thin and thick solid curves for ${}^3\mathrm{H}+p$ and ${}^3\mathrm{He}+n$ channels, respectively.

Figure 8

The eigenvalue distribution on the complex energy plane (a) and the calculated phase shifts (b) of the ${}^{3}P_0$ state with the coupling potential Eq. (36) for $\theta ={30}^{\circ }$. In (a) the calculated coupled-channels eigenvalue distribution is expressed by solid diamonds. The open circles and open triangles present the first and second resonant poles obtained by changing the strength factor $f$ for the coupling potential, respectively. The arrows show the pole trajectory of the two resonant poles at different strength factor $f$ for the coupling potential. In (b) the decomposition of the calculated coupled-channels phase shifts is presented. The contributions of the resonant poles are expressed by thin and thick dotted dashed and dashed curves, the continuum parts are displayed by thin and thick dotted curves, and the summations of the resonant poles and continuum parts are implied by thin and thick solid curves for ${}^3\mathrm{H}+p$ and ${}^3\mathrm{He}+n$ channels, respectively.