employ a theoretical model to extrapolate the reaction rate down to $T_G$ by fitting model parameters to experimental data typically measured at a few MeV. During a last half century, a lot of experimental and theoretical studies for the reaction have been carried out. See Refs. [@bb-npa06; @chk-epja15; @bk-ppnp16; @detal-17] for review.

The elastic $\alpha$-$^{12}$C scattering at low energies is an important reaction to fix some parameters of a model for the study. Accurate measurements of the elastic scattering have been reported in Refs. [@petal-npa87; @tetal-prc09], and those data provide indispensable input for the parameter fittings. Elastic scattering data at low energies in general can be used for deducing an asymptotic normalization constant (ANC), which determines an overall strength of a nuclear reaction involving bound states [@mt-prc99; @by-pan08; @tetal-rpp14].

The ANC of deuteron, for example, where the deuteron is a simple system consisting of loosely bound proton and neutron, leads to an overall factor of the reactions at low energies, such as radiative neutron capture on a proton at BBN energies [@r-npa00; @aetal-prc06] and proton-proton fusion in the Sun [@kr-npa99; @bc-plb01; @aetal-plb08; @cetal-plb13]. The ANC of deuteron is accurately determined by two effective range parameters: the deuteron binding momentum and 