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

We investigate the fine-tuning of radiative alpha-particle capture on carbon, $α(^{12}{\rm C},^{16}{\rm O})γ$, 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 $α$, namely $|δα/α| \leq 0.2\,$ per mille, in both the $E1$ and the $E2$ radiative capture.