of four and eight-nucleon system. $T_{(4)}$ and $T_{(12)}$ are the first excited energies of $\alpha$ and $^{12}$C, respectively. By including the terms up to next-to-next-to-leading order, therefore, one may obtain about 10% theoretical uncertainty for the process.

![Diagrams for dressed $^{16}$O propagator. A thick (thin) dashed line represents a propagator of $^{12}$C ($\alpha$), and a thick and thin double dashed line with and without a filled blob represent a dressed and bare $^{16}$O propagator, respectively. A shaded blob represents a set of diagrams consisting of all possible one-potential-photon-exchange diagrams up to infinite order and no potential-photon-exchage one. []{data-label="fig;dressed-propagator"}](fig-16c-propagator.eps){width="14cm"}

![Diagram of the scattering amplitude. See the caption of Fig. 1 as well. []{data-label="fig;scattering-amplitude"}](fig-Amplitude.eps){width="3.5cm"}

The amplitudes of the elastic scattering are calculated from diagrams depicted in Figs. \[fig;dressed-propagator\] and \[fig;scattering-amplitude\]. In our previous works, we have obtained the scattering amplitudes for $l$-th partial wave states as [@a-epja16; @aetal-prc07; @a-epja07] A\_l &=& , \[eq;Al\] where $k$ is the magnitude of relative momentum between $\alpha$ and $^{12}$C and $\theta$ is the scattering angle in the C.M. frame. In addition, $\eta$ is the Sommerfeld parameter, $\eta = \kappa/k$, where $\kappa$ is the inverse of the Bohr radius, $\kappa = Z_2Z_6\mu \alpha$, and C\_\^2 = ,     