Shadow poles in coupled-channel problems calculated with the Berggren basis

Background: In coupled-channels models the poles of the scattering $S$ matrix are located on different Riemann sheets. Physical observables are affected mainly by poles closest to the physical region but sometimes shadow poles have considerable effect too.

Purpose: The purpose of this paper is to show that in coupled-channels problems all poles of the $S$ matrix can be located by an expansion in terms of a properly constructed complex-energy basis.

Method: The Berggren basis is used for expanding the coupled-channels solutions.

Results: The locations of the poles of the $S$ matrix for the Cox potential, constructed for coupled-channels problems, were numerically calculated and compared with the exact ones. In a nuclear physics application the $J^{\pi }=3/2^{+}$ resonant poles of ${}^5\mathrm{He}$ were calculated in a phenomenological two-channel model. The properties of both the normal and shadow resonances agree with previous findings.

Conclusions: We have shown that, with an appropriately chosen Berggren basis, all poles of the $S$ matrix including the shadow poles can be determined. We have found that the shadow pole of ${}^5\mathrm{He}$ migrates between Riemann sheets if the coupling strength is varied.

Figure 1

Illustration of the contours $L$ (thick lines) on the $k$ and $p$ planes for the second Riemann sheet. Resonance states can be determined with contours similar to those in parts (a) and (b). Both virtual and resonance states can be calculated with contours similar to those in parts (c) and (d). The double roles of the shaded areas are explained in the text.

Figure 2

Similar to Fig. 1 but for the third Riemann sheet.

Figure 3

The wave function of the first resonance state located in the second Riemann sheet and calculated with the Berggren expansion method. The upper (a) and lower (b) parts show the first and second channel components, respectively. The imaginary part of the second component is practically zero.

Figure 4

Trajectories of the physical and shadow poles as functions of the interaction strength $V_{12}$. Circles (squares) denote the physical (shadow) poles. The numbers next to the symbols indicate the values of $V_{12}$ in MeV. Arrows point to the positions of the physical and shadow poles at the experimental strength $V_{12}=14.7$ MeV (see details in the text).