Fine-tunings in radiative $\alpha$-particle capture on ${}^{12}$C at astrophysical energies

We investigate the fine-tuning of radiative alpha-particle capture on carbon, $\alpha {(}^{12}\mathrm{C}{,}^{16}\mathrm{O})\gamma$, at astrophysical energies. Utilizing results from cluster effective field theory for this reaction, we find that the low-energy data of the astrophysical S-factor allow for only very small variations in the electromagnetic fine-structure constant $\alpha$, namely $\mathrm{\mid }\delta \alpha \mathrm{/}\alpha \mathrm{\mid }\le 0.2$ per mille, in both the $E1$ and the $E2$ radiative capture.