Additional resonant contribution to the potential model for the 12C(alpha,gamma)16O reaction

The additional resonant contribution to the potential model is examined in $\alpha$+$^{12}$C elastic scattering and the low-energy $^{12}$C($\alpha$,$\gamma$)$^{16}$O reaction. The excitation function of elastic scattering below $E_{c.m.}= 5$ MeV seems to be reproduced by the potential model satisfactorily, and it is not profoundly disturbed by the additional resonances. The weak coupling is good enough to describe the $^{16}$O structure in the vicinity of the $\alpha$-particle threshold, especially below $E_{c.m.}= 8$ MeV, corresponding to the excitation energy $E_x \approx 15$ MeV. The additional resonances give the complement of the astrophysical $S$-factors from the simple potential model. The $S$-factor of $^{12}$C($\alpha$,$\gamma$)$^{16}$O at $E_{c.m.}=300$ keV is dominated by the $E$2 transition, which is enhanced by the subthreshold 2$^+_1$ state at $E_x= 6.92$ MeV. The contribution from the subthreshold 1$^-_1$ state at $E_x= 7.12$ MeV is predicted to be small. The additional resonances do not give the large contribution to the thermonuclear reaction rates of $^{12}$C($\alpha$,$\gamma$)$^{16}$O at helium burning temperatures.