Mirror and triplet energy differences in $sd$-shell nuclei using microscopic interactions with isospin-symmetry breaking effects

In this study, we developed and tested two different isospin symmetry-breaking (ISB) versions of the microscopic DJ16A interaction. Starting with the isospin symmetric DJ16A interaction, we introduced two different Coulomb interactions- Coulomb-CD and Coulomb-w/SRC- along with phenomenological charge symmetry breaking (CSB) and charge independence breaking (CIB) effects. Then, we employed these interactions to calculate $b$- and $c$-parameters of the isobaric multiplet mass equation for $|T_z| = 1/2$ and $|T_z| = 1$ nuclei across the $sd$-shell. Our results indicate that the DJ16A$^\dagger$ interaction provides the most accurate $b$-parameter predictions between the two DJ16A-based interactions. Additionally, we explored mirror energy differences (MEDs) in low-energy spectra around $A = 20$ and demonstrated that large MEDs are primarily associated with high occupancies of the $1s_{1/2}$ orbital. Furthermore, $E2$ transition strengths were calculated using both DJ16A-based ISB interactions agreed with the experimental data, with minimal ISB effects observed on these transitions. Overall, the DJ16A$^\dagger$ interaction serves as a complementary set to the newly developed USD-family interactions, USDC, and USDCm and can be further tested for other mirror nuclei across the $sd$-shell to study nuclear structure properties and ISB effects in nuclear $\beta$-decay.