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

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 an explicit expression for the asymptotic normalization coefficient (ANC) and is applied to the low-energy resonant states of $\mathrm{nucleon}+\alpha$ and $\alpha +{}^{12}\mathrm{C}$ systems. The SMP results are compared with the effective-range expansion method (EFE) results. In the present paper the SMP and EFE plus the Padé approximation are applied to study the excited $2^{+}$ resonant states of $\mathrm{Be}$. A contradiction is found between descriptions of the experimental phase shift data for $\alpha \alpha$ scattering and of the $\mathrm{Be}$ resonant energy for $2^{+}$ state. Using the EFE method, we also calculate the ANC for the $\mathrm{Be}$ ground $0^{+}$ 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 $\alpha +{}^{12}\mathrm{C}$ states $1^{-}$ and $3^{-}$ the SMP results are compared with the Padé approximation results. We find that the Padé approximation improves a resonance width description compared with the EFE results. The EFE method is also used to calculate the ANCs for the bound $\mathrm{O}$ ground $0^{+}$ state and for the excited $1^{-}$ and $2^{+}$ levels, which are situated near the threshold of $\alpha +{}^{12}\mathrm{C}$ channel.

Figure 1

Comparison of the fitted effective-range function for the $s$ wave $\alpha \alpha$ elastic scattering with the experimental values. The experimental data are taken from Ref. [21]. The energy is given in the c.m.s. frame.

Figure 2

Comparison of the fitted effective-range function for the $d$ wave $\alpha \alpha$ elastic scattering (red color line) with the experimental values (points). The dashed line represents the result of fitting with the Padé approximant. The experimental data are taken from Ref. [21]. The energy is given in the c.m.s. frame.

Figure 3

Comparison of the fitted phase shifts for the $d$ wave $\alpha \alpha$ elastic scattering obtained by the $S$-matrix pole method with the experimental data taken from Ref. [21]. The solid line shows when the complex momentum $k_r$ of the resonance is added to the parameters of fitting. The dashed line indicates when the energy and width of resonance is fixed as in Eq. (21). The energy is given in the laboratory frame.