\_l\^l (|-|\_[k\^2=-\_l\^2]{})\^[-12]{}    , \[eq;Cb\] where $\eta_b=\kappa/k_b$.

[**3. Fitting the parameters to phase shifts data** ]{}

Four excited states of $^{16}$O exist below the $\alpha$-$^{12}$C threshold, which we include in the parameter fitting in the present study. The binding energies, $B_i(l^\pi)$, of the $i$-th excited bound states of $^{16}$O in $l^\pi$ states from the $\alpha$-$^{12}$C threshold energy are $B_1(0^+) = 1.113$, $B_2(3^-) = 1.032$, $B_3(2^+) = 0.245$, $B_4(1^-) = 0.045$ MeV. Thus, the binding momenta, $\gamma_l=\sqrt{2\mu B_i(l^\pi)}$, are \_l = 79.843,15.860,37.007,   75.954  , for the $0_2^+$, $1_1^-$, $2_1^+$, $3_1^-$ states, respectively, where $\mu = m_\alpha m_C/(m_\alpha + m_C) = 2795.079\,$MeV with $m_\alpha = 3727.379$ MeV and $m_C = 11174.862$ MeV. As mentioned above, the first effective range term, $a_l$, is constrained by using the binding momenta.

To fix the other effective range parameters, the phase shift data for each $l$-th partial wave state are used. In the present work, we employ the phase shit data from the Tischhauser [*et al.*]{}’s paper [@tetal-prc09]. The reported energies of the $\alpha$ particle in the lab. frame are $T_\alpha=2.6$-6.6 MeV, and corresponding momenta in the C.M. frame are $k=105$-166 MeV (i.e., $k= \sqrt{1.5\mu T_\alpha}$). Because our large momentum scale of the theory is $\Lambda_H\sim 150$ 