The mirror energy difference (MED) of the mirror state, particularly for states exhibiting the Thomas–Ehrman shift, serves as a sensitive probe of mirror symmetry breaking. We employ the Gamow shell model, which includes the inter-nucleon correlation and continuum coupling, to investigate the MED for sd-shell nuclei by taking \(^{18}\text{Ne}/{}^{18}\text{O}\) and \(^{19}\text{Na}/{}^{19}\text{O}\) as examples. Our GSM provides good descriptions of the excitation energies and MEDs for the \(^{18}\text{Ne}/{}^{18}\text{O}\) and \(^{19}\text{Na}/{}^{19}\text{O}\) . Furthermore, our calculations also reveal that the large MED of the mirror states is caused by the significant occupation of the weakly bound or unbound \(s_{1/2}\) waves, giving the radial density distribution of the state in the proton-rich nucleus greater extension than that of mirror states in deeply bound neutron-rich nuclei. In addition, our GSM calculation shows that the contribution of the Coulomb interaction is different for the low-lying states in proton-rich nuclei, which significantly contributes to the MEDs of mirror states, as is well recognized. Finally, our GSM calculation indicates that the contributions of the nucleon–nucleon interaction are different for the mirror state, especially for the state of proton-rich nuclei bearing the Thomas–Ehrman shift, which also contributes to the significant mirror symmetry breaking with a large MED.
