Shell-Model Description of the Isospin-Symmetry-Breaking Correction to Gamow-Teller $\beta$-Decay Rates and Their Mirror Asymmetries

The isospin-symmetry breaking correction, denoted as $\delta_C$, is introduced for the first time within the shell-model framework to the nuclear matrix element of Gamow-Teller transitions. $\delta_C$ is separated into two components: the isospin mixing term, $\delta_{C1}$, induced by the Coulomb and nuclear charge-dependent forces in the effective Hamiltonian, and the radial mismatch term, $\delta_{C2}$, arising from differences between proton and neutron realistic wave functions. Consequently, the refinement strategy developed for superallowed $0^+\rightarrow 0^+$ Fermi transitions is applied to Gamow-Teller transitions as well. It is demonstrated that, to a given precision level, the shell model calculation of $\delta_C$ converges much faster than the calculation of the transition matrix element. Furthermore, higher-order correction terms are investigated and considered for consistent study of our works. Various interesting properties of the leading correction terms are discovered within the two-level model and parentage expansion of the one-body transition densities in angular momentum and isospin spaces. One such property is the dependence of $\delta_{C1}$ on the isospin admixture amplitude, $\alpha$, starting from the first order, while the same model yields $\delta_{C1} = \alpha^2$ for Fermi transitions. The calculated $\delta_C$ values are then utilized to evaluate the mirror asymmetry of Gamow-Teller transition strengths, which are compared with available experimental data and other theoretical calculations. Due to the refined fitting procedure of the Woods-Saxon potential parameters and the improved convergence as a function of intermediate state number, our results show better agreement on average compared to those of Smirnova and Volpe [Nucl. Phys. {\bf A 714}, 441 (2003)].