Generalization of the sum rule for double Gamow-Teller operators

We show that for Pauli-blocked nuclei, in the closure approximation, the summed strength for the double Gamow-Teller $0^{+}$→$0^{+}$ transitions is less than the value 6(N-Z)(N-Z+1) and that the difference can be related to sum rules for magnetic dipole transitions, which in turn can be related to the double Gamow-Teller matrix element for the transition from the $0^{+}$ initial state to its double isobaric analog state. A ‘‘wrong isospin phase sum rule’’ is found that is valid in limited cases.