W\_l() = \^[l]{}\_[n=0]{}( 1+ ),     H\_l(k) = W\_l() H(), with H() = (i) + -(i). \[eq;H\] $\psi(z)$ is the digamma function, $P_l(x)$ are the Legendre polynomials, and $\sigma_l$ are the Coulomb phase shifts. When $\eta$ goes to zero, the factor $C_\eta^2$ is normalized to one whereas, when $\eta$ becomes large, the Gamow factor, $P=\exp(-2\pi\eta)$, appears from the factor $C_\eta^2\propto  P$. We note that the function, $-2\kappa H_l(k)$, in the denominator of the amplitude is obtained from the Coulomb bubble diagram for the dressed propagator of $^{16}$O in Fig. \[fig;dressed-propagator\], and the factor, $e^{2i\sigma_l}W_l(\eta)C_\eta^2$, in the numerator is from the initial and final state Coulomb interactions between $\alpha$ and $^{12}$C in Fig. \[fig;scattering-amplitude\].

The function $K_l(k)$ represents the interaction due to the short range nuclear force (compared with the long range Coulomb force), which is obtained in terms of the effective range parameters as [^2] K\_l(k) = - + 12 r\_l k\^2 - 14 P\_l k\^4 + Q\_l k\^6 - R\_l k\^8 + . \[eq;Kl\] Because the UV divergence comes out of the loop integrals in the Coulomb self-energy terms (from the diagrams in Fig. \[fig;dressed-propagator\]), we need to introduce counter terms for renormalization. We employ the dimensional regularization and the 