Asymptotic normalization coefficients of resonant and bound states from the phase shifts for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mi>α</mml:mi></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mmultiscripts><mml:mi mathvariant="normal">C</mml:mi><mml:mprescripts/><mml:none/><mml:mn>12</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>scattering
Аннотация
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 $\mathrm{nucleon}+\ensuremath{\alpha}$ and $\ensuremath{\alpha}+^{12}\mathrm{C}$ systems. The SMP results are compared with the effective-range expansion method (EFE) results. In the present paper the SMP and EFE plus the Pad\'e 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 $\ensuremath{\alpha}\ensuremath{\alpha}$ 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 $\ensuremath{\alpha}+^{12}\mathrm{C}$ states ${1}^{\ensuremath{-}}$ and ${3}^{\ensuremath{-}}$ 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 $\mathrm{O}$ ground ${0}^{+}$ state and for the excited ${1}^{\ensuremath{-}}$ and ${2}^{+}$ levels, which are situated near the threshold of $\ensuremath{\alpha}+^{12}\mathrm{C}$ channel.