Combining ab initio calculations and low-energy effective field theory for halo nuclear systems: The case of ${}^7\mathrm{Be}+p\to {}^8\mathrm{B}+\gamma$

We report a leading-order (LO) calculation of ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in a low-energy effective field theory. ${}^{8}B$ is treated as a shallow $\text{proton}+{}^7\mathrm{Be}$ core and $\text{proton}+{}^7\mathrm{Be}*$ (core excitation) $p$-wave bound state. The couplings are fixed using measured binding energies and proton-${}^7\mathrm{Be}$ $s$-wave scattering lengths, together with ${}^{8}B$ asymptotic normalization coefficients from ab initio calculations. We obtain a zero-energy $S$-factor of $18.2\pm 1.2\left(\mathrm{ANC}\mathrm{only}\right)$ eV b. Given that this is a LO result it is consistent with the recommended value $S\left(0\right)=20.8\pm 1.6$ eV b. Our computed $S\left(E\right)$ compares favorably with experimental data on ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ for $E<0.4$ MeV. We emphasize the important role of proton-${}^7\mathrm{Be}$ scattering parameters in determining the energy dependence of $S\left(E\right)$, and demonstrate that their present uncertainties significantly limit attempts to extrapolate these data to stellar energies.

Figure 1

Diagrams for capture. The line assignments are designated by the spin indices; the shaded blobs denote the full Coulomb-Green functions; the bold vertex in the last two diagrams means $S$-wave multiple scattering with the Coulomb interaction included to all orders in ${\alpha }_{\mathrm{EM}}$ [18, 23]. Diagrams with the photon coupled to the proton are not shown but are included in our calculation.

Figure 2

$E1$ $S$ factor for proton capture on ${}^7\mathrm{Be}$. The green shaded band is the range of $S\left(E\right)$ in the LO calculation with “No + A” input (cf. Table 3), varying scattering lengths within their $1\sigma$ errors. The (nearly energy-independent) impact of the ANC error bar on the prediction is indicated by the error bar on the “No + A” band at 20 keV. The data are from Refs. [32, 33, 34, 35]. The solid black curve is the NCSM/RGM calculation of Ref. [4], shown for comparison with its LO EFT approximation (dashed), computed using the “Na” parameters given in Table 3.