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

TL;DR

This paper investigates how variations in the electromagnetic fine-structure constant $\alpha$ would affect the astrophysical $S$-factor for the radiative capture $^{12}$C$(\alpha,\gamma)^{16}$O at stellar energies. Using cluster effective field theory, the authors derive cross sections for $E1$ and $E2$ transitions and show that the energy dependence is governed by the inverse propagator and interference among amplitudes, leading to a stringent bound on $\alpha$ variation: $|\delta\alpha/\alpha| \leq 0.2$ per mille for both channels. In the $E1$ case, a single dominant effect via the inverse propagator drives the sensitivity, while in $E2$ strong cancellations among amplitudes yield a comparable bound through interference. This bound improves upon comparable constraints from Hoyle-state and Big Bang nucleosynthesis, and the work suggests future cross-checks with nuclear lattice EFT and notes that assessing quark-mass dependence remains challenging.

Abstract

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.

Figures (7)

• Figure 1: Diagrams of amplitudes for radiative $\alpha$ capture on ${}^{12}\textnormal{C}$. A wavy line denotes the outgoing photon, the thin dashed line the ${}^{4}\textnormal{He}$ and the solid line the ${}^{12}\textnormal{C}$ state. The double thin-dashed / solid lines represent the dressed propagation of the ${}^{16}\textnormal{O}$ dimer in the intermediate and final state. The shaded ellipses represent the Coulomb-interaction. Diagrams (a) and (b) are initial state radiation contributions. The vertex in diagram (c) is a counter term proportional to $h^{(\ell)}_r$ to renormalize the infinities from the loop diagrams in (d), (e), (f). Figure adopted from Ref. Ando:2025ibj.
• Figure 2: Fine-structure constant ($\alpha = (1+\delta)\,\alpha_0)$ variation of the astrophysical $S$-factor of the ${}^{4}\textnormal{He}+{}^{12}\textnormal{C} \to {}^{16}\textnormal{O}(1^-) \to {}^{16}\textnormal{O}(0^+)$ radiative capture. The result for the nominal value $\alpha_0$ is displayed in black. The blue (dashed) curves correspond to $\delta = -0.0002 (-0.0001)$; the red (dashed) curves to $\delta=0.0002 (0.0001)$ . The position of the Gamow energy at $E_G \simeq 0.3\,\unit{MeV}$ is indicated by a vertical green line. The data are from Refs. Plag:2012zzMakii:2009zzAssuncao:2006vyKunz:2001zzRoters:1999xyzKremer:1988zzRedder:1987xbaGialanella:2001Ouellet:1996zz.
• Figure 3: Fine-structure constant ($\alpha = (1+\delta)\,\alpha_0)$ variation of the astrophysical $S$-factor of the ${}^{4}\textnormal{He}+{}^{12}\textnormal{C} \to {}^{16}\textnormal{O}(2^+) \to {}^{16}\textnormal{O}(0^+)$ radiative capture. The result for the nominal value $\alpha_0$ is displayed in black. The blue (dashed) curves correspond to $\delta = -0.0002 (-0.0001)$; the red (dashed) curves to $\delta=0.0002 (0.0001)$ . The position of the Gamow energy at $E_G = 0.3\,\unit{MeV}$ is indicated by a vertical green line. The data are from Refs. Plag:2012zzMakii:2009zzKunz:2001zzRoters:1999xyzOuellet:1996zzFey:2004.
• Figure 4: Ratio of the $S$-factor for radiative $E1$-capture accounting only for the change in the amplitude $A^{(1)}$ of Eq. (\ref{['eq:AmpA1']}). The red (dashed) curve is the ratio for $\delta=0.0002 (0.0001)$, the blue (dashed) curve for $\delta=-0.0002 (-0.0001)$.
• Figure 5: Fine structure constant ($\alpha = (1+\delta)\,\alpha_0)$ variation of the astrophysical $S$-factor of the ${}^{4}\textnormal{He}+{}^{12}\textnormal{C} \to {}^{16}\textnormal{O}(1^-) \to {}^{16}\textnormal{O}(0^+)$ radiative capture for two alternative values of the coordinate space cutoff: $r_c=0.05\,\unit{fm}$ (left), $r_c=0.10\,\unit{fm}$ (right). For the parameters, see Table \ref{['tab:parE1']}. Also see the caption to Fig. \ref{['fig:SalphavarE1001']}.