Coupled-channels treatment of ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in effective field theory

The $E1$ and $M1$ contributions to ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ at low energies are calculated in halo effective field theory. The excited ${}^7\mathrm{Be}^{★}$ core is included as an explicit degree of freedom in a coupled-channels calculation. The $E1$ transition is calculated up to next-to-next-to-leading order. The leading contribution from $M1$ transition that gives significant contribution in a narrow energy region around the $1^{+}$ resonance state of ${}^8\mathrm{B}$ is included. We compare our results with previous halo effective field theory calculations that also included the ${}^7\mathrm{Be}^{★}$ as an explicit degree of freedom. We disagree with these previous calculations in both the formal expressions and also in the analysis. Bayesian inference of the data gives $S_{17}\left(0\right)=21.0\left(7\right)$ eV b when combined with the expected theory error.

Figure 1

$E1$ capture without initial-state strong interactions. Double solid line represents a ${}^7\mathrm{Be}$ or ${}^7\mathrm{Be}^{★}$ core as appropriate, single solid line a proton, wavy line a photon, and $\otimes$ represents the final ${}^8\mathrm{B}$ bound state. The gray rectangles represent Coulomb interactions between the charged particles. The photons are minimally coupled to the charged particles.

Figure 2

$E1$ capture with initial state strong interactions. The dashed line with the gray blog represents initial-state $s$-wave strong and Coulomb interactions. The rest of the notation is the same as in Fig. 1.

Figure 3

$M1$ capture through the resonant $1^{+}$ state. The dashed line with the gray blog represents $p$-wave interaction in the $1^{+}$ resonance channel, represents magnetic photon coupling and $\blacksquare$ the two-body current. The rest of the notation is the same as in Fig. 1.

Figure 4

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in ${\mathrm{EFT}}_{\text{gs}}$. The grid lines are at the ${}^7\mathrm{Be}^{★}$ inelasticity ${\gamma }_{\mathrm{\Delta }}=26.5$ MeV and the $1^{+}$ resonance momentum $p_R=32.2$ MeV. The dot-dashed (black) curve includes diagrams with only $s$-wave contributions and no initial-state strong interactions, the short-dashed (blue) curve includes all diagrams except $d$-wave contributions, and the long-dashed (red) curve includes full contributions from all diagrams. Results were fitted to ANCs as explained in the text.

Figure 5

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in ${\mathrm{EFT}}_{\text{gs}}$. The grid lines are explained in Fig. 4. The LO, NLO, and NNLO curves are identified in the legends. Results were fitted to ANCs in perturbation as explained in the text.

Figure 6

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in ${\mathrm{EFT}}_{★}$. The plots used $r_0^{\left(1\right)}=-30\mathrm{fm}$ and $a_{22}=-30\mathrm{fm}$ that were varied by 95% and had negligible effect on the final result. A positive $a_{22}=30\mathrm{fm}$ also had a negligible effect. The curves follow the same notation as in Fig. 4.

Figure 7

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in ${\mathrm{EFT}}_{\text{gs}}$ fitted in the momentum region I, $p\le 28.6$ MeV ($E\le 500\mathrm{keV}$). The LO, NLO, and NNLO curves are identified in the legends. At LO and NLO, EFT I and ${\mathrm{EFT}}_{★}$ I have the same momentum dependence. The inset shows the posterior distribution for $S_{17}\left(0\right)$ in eV b units at NNLO.

Figure 8

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in ${\mathrm{EFT}}_{★}$. Notation is the same as in Fig. 7.

Figure 9

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ in ${\mathrm{EFT}}_{★}$ fitted in the momentum region II, $p\le 40.5$ MeV. Notation is analogous to the one in Fig. 7. The bottom panel shows scaled data which is to be interpreted carefully as explained in the text.

Figure 10

$S$ factor for ${}^7\mathrm{Be}(p,\gamma ){}^8\mathrm{B}$ from ${\mathrm{EFT}}_{★}$ II fits at NNLO. The top panel compares the EFT result with numbers from Descouvemont [71] scaled by a factor of 0.859. The dot-dashed (black) curve is the EFT $E1$ contribution and the solid (red) curve is the total EFT $E1+M1$ contribution at NNLO. The bottom panel compares the EFT $M1$ numbers with the R-matrix calculation from Junghans et al. [40].

Figure 11

Taylor series approximation of $S_{17}\left(E\right)$ up to the quadratic term in $E$. The dot-dashed (black) curve is the ${\mathrm{EFT}}_{\text{gs}}$ I NNLO fit normalized to 1.1 times the threshold value, the dashed (blue) curve is the ${\mathrm{EFT}}_{★}$ I NNLO fit normalized to 1.05 times the threshold value, and the long-dashed (red) curve is the ${\mathrm{EFT}}_{★}$ II NNLO fit normalized to the threshold value. The data points (red circles) are the numerical evaluations scaled appropriately for the three fits.