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

TL;DR

This paper compares methods for calculating asymptotic normalization coefficients and resonance parameters in alpha-cluster nuclear systems, highlighting discrepancies and improvements in describing phase shifts and resonance widths.

Contribution

It applies and compares the S-matrix pole, effective-range expansion, and Padé-approximation methods to alpha scattering data, revealing their relative strengths and inconsistencies.

Findings

Padé-approximation improves resonance width description.

Contradiction between phase shift data and 8Be 2+ resonance energy.

ANC for 8Be ground state agrees with narrow resonance formula.

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 $\alpha+^{12}\rm{C}$ 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$^+$ resonant states of $^{8}\rm{Be}$. A contradiction is found between descriptions of the experimental phase shift data for $\alpha\alpha$ scattering and of the $^{8}\rm{Be}$ resonant energy for 2$^+$ state. Using the EFE method, we also calculate the ANC for the $^{8}\rm{Be}$ 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 $\alpha+^{12}\rm{C}$ states1$^-$ and 3$^-$ 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 $^{16}\rm{O}$ ground 0$^+$ state and for the excited 1$^-$ and 2$^+$ levels which are situated near the threshold of $\alpha+^{12}\rm{C}$ channel.