Asymptotic normalization coefficients of resonant and bound states from the phase shifts for $αα$ and $α^{12}\rm C$ scattering

Abstract: Recently we have published a paper [Irgaziev, Phys. Rev. C 91, 024002 (2015)] where the $S$-matrix pole method (SMP) which is only valid for resonances has been developed to derive a new explicit expression for the asymptotic normalization coefficient (ANC), and is applied to the low-energy resonant states of nucleon$+\alpha$ and $\alpha+<sup>{12}\rm{C}$ systems. The SMP results are compared with the effective-range expansion method (EFE) results. In the present paper the SMP and EFE plus the Pad\'e-approximation are applied to study the excited 2$<sup>+$ resonant states of $<sup>{8}\rm{Be}$. A contradiction is found between descriptions of the experimental phase shift data for $\alpha\alpha$ scattering and of the $<sup>{8}\rm{Be}$ resonant energy for 2$<sup>+$ state. Using the EFE method, we also calculate the ANC for the $<sup>{8}\rm{Be}$ ground 0$<sup>+$ state with a very small width. This ANC agrees well with the value calculated using the known analytical expression for narrow resonances. In addition, for the $\alpha+<sup>{12}\rm{C}$ states1$<sup>-$ and 3$<sup>-$ the SMP results are compared with the Pad\'e-approximation results. We find that the Pad\'e-approximation improves a resonance width description compared with the EFE results. The EFE method is also used to calculate the ANCs for the bound $<sup>{16}\rm{O}$ ground 0$<sup>+$ state and for the excited 1$<sup>-$ and 2$<sup>+$ levels which are situated near the threshold of $\alpha+<sup>{12}\rm{C}$ channel.