Fusion and reactions of $\alpha$+${}^8$Be in the Hoyle resonance and associated resonances region

The fusion of $\alpha$ and ${}^8$Be to produce a ${}^{12}$C nucleus is a crucial process in nucleosynthesis. In the laboratory, this process can only be studied theoretically as a ${}^8$Be target or projectile cannot be prepared experimentally. We use the potential scattering theory in the coupled-channel formalism to study such a process in terms of the collision between the $\alpha$ particle on a deformed ${}^8$Be nucleus, both on resonance and off resonance in the Hoyle resonance and associated resonances region. The experimental ${}^{12}$C energy levels and widths constrain the nuclear potential to suggest the need to include a parity-dependent surface potential component that is more attractive for even-$L$ positive-parity partial waves than for odd-$L$ negative-parity partial waves. As a consequence, the radial dependence of the total potentials for the set of {0${}^{+}$, 2${}^{+}$, 4${}^{+}$} resonances of ${}^{12}$C exhibit a double-hump behavior, possessing two local energy minima and a doublet of each of the ${}^{12}$C {0${}^{+}$, 2${}^{+}$, 4${}^{+}$} resonances in the Hoyle and associated resonances region. We examine the approximate agreement of the theoretical results with experiment and suggest the search for the as-yet unobserved lower-energy 2${}_2^{+}$ and 4${}_1^{+}$ resonances to test the double-hump potential description. In addition, for practical astrophysical applications, we evaluate and estimate the astrophysical $S(E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}})$-factor for the $\alpha$+${}^8$Be $\to$ ${}^{12}$C$(0^{+\ast })$ $\to$ ${}^{12}$C$(2_1^{+})$ + $\gamma$ reaction for $E_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}$ $<$ 1.0 MeV.