structure of the UV divergence from the Coulomb self-energy terms is given in Appendix. Thus, we need to include one counter term, $-1/a_0$ term in the effective range expansion for $l=0$, two counter terms, $-1/a_1$ and $r_1$ for $l=1$, three counter terms, $-1/a_2$, $r_2$, and $P_2$ for $l=2$, and four counter terms, $-1/a_3$, $r_3$, $P_3$, and $Q_3$ for $l=3$, to remove the UV divergence and make the terms finite. The other higher order terms in the effective range expansion are introduced as finite terms in the conventional renormalization method and supposed to obey counting rules in which higher order terms are less important than lower order terms. We find that the expression obtained in Eq. (\[eq;Al\]) reproduces well the previous results reported in Refs. [@klh-jpg13; @hot-npb73; @h-jmp77; @scb-prc10].

At the binding energies of excited states of $^{16}$O, the amplitudes should have a pole at $k_b = i\gamma_l$ where $\gamma_l$ are the binding momenta [^3], $\gamma_l = \sqrt{2\mu B_l}$; $B_l$ denote the binding energies of excited states of $^{16}$O. Thus the denominator of the scattering amplitude, $D_l(k)$, should vanish at $k_b$; D\_l(k\_b) = K\_l(k\_b) - 2H\_l(k\_b) = 0. \[eq;flb\] Using this condition, the first effective range parameter, $a_l$, is related to 