Nuclei in the vicinity of driplines have been receiving a lot of attention in nuclear structure studies. In the nuclei, the continuum coupling is crucial in reproducing weakly bound and unbound phenomena. To calculate observables of the nuclei as open quantum systems, we have developed valence-space effective operators in the complex-energy Berggren basis using many-body perturbation theory. We focus on the Gamow-Teller $\beta$ decay in the $\mathit{sd}$ shell. The two- plus three-nucleon force from the chiral effective field theory, named EM1.8/2.0, has been used. The Gamow shell model which takes the continuum coupling into account can properly reproduce experimental observations of weakly bound and unbound states. The $\beta$-decay isospin asymmetry between the dripline nucleus ${}^{22}\mathrm{Si}$ and its mirror partner ${}^{22}\mathrm{O}$ is reproduced, in which the $s_{1/2}$ continuum plays a key role. Significant Thomas-Ehrman shift is seen through mirror energy differences between the mirror daughters ${}^{22}\mathrm{Al}$ and ${}^{22}\mathrm{F}$, in which the continuum effect plays an important role.

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• Beta decay
• Effective field theory
• Lifetimes & widths
• Nuclear forces
• Nuclear many-body theory
• Nuclear structure & decays
• Shell model
• 20 ≤ A ≤ 38