Ab initio calculation of the $\beta$ decay from ${}^{11}\mathrm{Be}$ to a ${}^{10}\mathrm{Be}+p$ resonance

The exotic $\beta$-delayed proton emission is calculated in ${}^{11}\mathrm{Be}$ from first principles using chiral two- and three-nucleon forces. To investigate the unexpectedly large branching ratio measured in Ayyad et al. [Phys. Rev. Lett. 123, 082501 (2019)], we calculate the proposed $(1/2^{+},1/2)$ proton resonance in ${}^{11}\mathrm{B}$ using the no-core shell model with continuum. This calculation helps to address whether this enhancement is caused by unknown dark decay modes or an unobserved proton resonance. We report a branching ratio of $b_p=(1.3\pm 0.5)\times {10}^{-6}$, suggesting that its unexpectedly large value is caused by an unobserved proton resonance in ${}^{11}\mathrm{B}$.

Figure 1

Calculated and experimental levels of ${}^{11}\mathrm{Be}$. Only states corresponding to experimentally bound states with respect to the ${}^{10}\mathrm{Be}+n$ threshold (horizontal red dashed line) are shown. The left column shows the original ab initio NCSMC calculation in $N_{\max }=7$ space. The phenomenological adjusted calculation is presented in the middle column. See the text for details.

Figure 2

Calculated and experimental levels in ${}^{11}\mathrm{B}$. Selected states corresponding to experimentally bound states with respect to the ${}^{10}\mathrm{Be}+p$ threshold (horizontal red dashed line) are shown. The SRG evolved $NN\text{−}{\mathrm{N}}^4\mathrm{L}\mathrm{O}\left(500\right)+3N_{\mathrm{lnl}}$ interaction was used. The left column shows the original ab initio NCSMC calculation in $N_{\max }=7$ space. The phenomenolgically adjusted calculation is presented in the middle column. See the text for details. The isospin of all shown states is $T=1/2$.

Figure 3

NCSMC-calculated ${}^{10}\mathrm{Be}+p$ diagonal phase shifts. The solid lines correspond to the ${}^{2}S_{1/2}$ (or ${}^{2}P_{1/2}$) channels while the dashed and dotted curves correspond to the ${}^{4}D_{1/2}$ and ${}^{6}D_{1/2}$ channels, respectively. The vertical dashed line indicates the experimentally predicted location of the $(1/2^{+},1/2)$ resonance at 197 keV. (a) NCSMC-calculated phase shifts using $N_{\max }=5$ basis size. (b) NCSMC-calculated phase shifts using $N_{\max }=7$ basis size. (c) Phenomenologically adjusted, ${\mathrm{NCSMC}}_{\mathrm{pheno}}$, phase shifts such that the $(1/2^{+},1/2)$ resonance coincides with the experimentally predicted resonance at 197 keV. The $(3/2^{+},1/2)$ resonance is the result of shifting the degenerate states to the corresponding experimental levels (see Fig. 2). See the text for details. $E_{\mathrm{c}.\mathrm{m}.}$ is the kinetic energy of ${}^{10}\mathrm{Be}+p$ in the center-of-mass frame.

Figure 4

NCSMC-calculated ${}^{10}\mathrm{Be}+p$ eigenphase shifts. The vertical dashed line indicates the experimentally predicted location of the $(1/2^{+},1/2)$ resonance at 197 keV. (a) NCSMC-calculated phase shifts using $N_{\max }=5$ basis size. (b) NCSMC-calculated phase shifts using $N_{\max }=7$ basis size. The solid lines correspond to the first eigenphase shift while the dashed and dotted curves correspond to the second and the third eigenphase shift, respectively. The sharp resonance in the $(1/2^{+},1/2)$ solid line in panel (b) is not shifted by $\pi$ for clarity. $E_{\mathrm{c}.\mathrm{m}.}$ is the kinetic energy of ${}^{10}\mathrm{Be}+p$ in the center-of-mass frame.

Figure 5

Energy-dependent $B_{\mathrm{GT}}\left(E\right)$ calculated from the ground state of ${}^{11}\mathrm{Be}$ to the $(1/2^{+},1/2)$ proton continuum of ${}^{11}\mathrm{B}$ using the NCSMC.