This paper extends the shell-model formalism for calculating the isospin-symmetry-breaking correction $\left({\delta }_C\right)$, which has been highly successful in studies of superallowed $0^{+}\to 0^{+}$ Fermi $\beta$ decays [Phys. Rev. C 77, 025501 (2008)], to the nuclear matrix elements of Gamow-Teller (GT) transitions. Despite the relaxation of the selection rules for GT transitions, it remains possible to recast ${\delta }_C$ into the isospin-mixing term $\left({\delta }_{C1}\right)$, induced by the effective Coulomb and nuclear charge-dependent forces, and the radial mismatch term $\left({\delta }_{C2}\right)$, which arises from differences between proton and neutron realistic wave functions. Consequently, all the refinement strategies for the individual correction terms, as considered in the aforementioned reference for superallowed $0^{+}\to 0^{+}$ Fermi $\beta$ decays, are also applicable to GT transitions. We demonstrate that the introduction of ${\delta }_C$ significantly improves the convergence with respect to spectator-nucleus states, compared to direct calculations of GT matrix elements. Additionally, when initial and final states belong to different isospin multiplets, only spectator states with lesser isospin contribute to ${\delta }_{C2}$, thereby disabling the counter term presented in Fermi transitions. Another unique aspect of the axial-vector process is the influence of the spin-orbit potential on ${\delta }_{C2}$ when spin-flip transitions, such as those connecting spin-aligned and -antialigned orbits, are present within the chosen model space. Furthermore, in the limit of two-level mixing, the dependence of ${\delta }_{C1}$ for GT transitions on the isospin-mixing amplitude begins at first order, whereas for Fermi transitions, it starts at second order. Therefore, both ${\delta }_{C1}$ and ${\delta }_{C2}$ could be distinctively substantial in GT transitions, and subsequently the higher-order correction terms [Phys. Rev. C 109, 014317 (2024)] which account for interference effects, are generally not negligible. These behaviors are in good agreement with our numerical calculations for various $p$- and $sd$-shell nuclei using phenomenological effective Hamiltonians and realistic Woods-Saxon radial wave functions.

Close

PrevNext

View figure in article

• Nuclear structure & decays
• Nuclear tests of fundamental interactions
• Shell model