Resonance properties including asymptotic normalization coefficients deduced from phase-shift data without the effective-range function

Recently, a new $\mathrm{\ensuremath{\Delta}}$ method for the calculation of asymptotic normalization coefficients (ANC) from phase-shift data has been formulated, proved, and used for bound states. This method differs from the conventional one by fitting only the nuclear part of the effective-range function which includes a partial phase shift. It should be applied to large-charge nuclei when the conventional effective-range expansion or the Pad\'e approximations using the effective-range function ${K}_{l}({k}^{2})$ fitting do not work. A typical example is the nucleus vertex $\ensuremath{\alpha}+^{12}\mathrm{C}\phantom{\rule{4pt}{0ex}}\ensuremath{\longleftrightarrow}^{16}\mathrm{O}$. Here we apply the $\mathrm{\ensuremath{\Delta}}$ method, which totally excludes the effective-range function, to isolated resonance states. In fact, we return to the initial renormalized scattering amplitude with a denominator which defines the well-known pole condition. Concrete calculations are made for the resonances observed in the $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$, $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\alpha}$, and $\ensuremath{\alpha}\text{\ensuremath{-}}^{12}\mathrm{C}$ collisions. We use the experimental phase-shift and resonant energy data including their uncertainties and find the ANC variations for the states considered. The corresponding results are in a good agreement with those for the $S$-matrix pole method which uses the differing formalism. The simple formula for narrow resonances given in the literature is used to check the deduced results. The related ANC function clearly depends on the resonance energy (${E}_{0}$) and width ($\mathrm{\ensuremath{\Gamma}}$), which is used to find the ANC uncertainty ($\mathrm{\ensuremath{\Delta}}\mathrm{ANC}$) through the energy ($\mathrm{\ensuremath{\Delta}}{E}_{0}$) and the width ($\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Gamma}}$) uncertainties.

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