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The elastic $\alpha$-$^{12}$C scattering for $l=2$ at low energies is studiedin an effective Lagrangian approach. We explicitly include two resonant $2_2^+$and $2_3^+$ states of $^{16}$O in the scattering amplitudes and construct an$S$ matrix assuming that three amplitudes, the non-resonant part of theamplitude which has the sub-threshold $2_1^+$ state of $^{16}$O and those ofthe two resonant states, are represented by the summation of correspondingparts of the phase shift. Then, we fit the parameters in the $S$ matrix to thephase shift data by imposing three conditions at very low energies where thephase shift data are not available. By using the fitted parameters, wecalculate the asymptotic normalization coefficients (ANC) of the $2_1^+$ stateof $^{16}$O and find that the previously reported small and large values of theANC can be reproduced depending on the imposed conditions, but we obtain largeerror bars for the large ANC values which are reported from the $\alpha$transfer reactions.

Elastic $\alpha$-$^{12}$C scattering at low energies with the resonant $2_2^+$ and $2_3^+$ states of $^{16}$O