$J=0,T=1$ pairing-interaction selection rules

Wave functions arising from a pairing Hamiltonian $E\left(0\right),$ i.e., one in which the interaction is only between $J=0^{+},T=1$ pairs, lead to magnetic dipole and Gamow-Teller (GT) transition rates that are much larger than those from an interaction $E\left(J_{\text{max}}\right)$ in which a proton and a neutron couple to $J=2j$. With realistic interactions the results are between the two extremes. In the course of this study we found that certain $M1$ and GT matrix elements vanish with $E\left(0\right)$. These are connected to seniority and reduced isospin selection rules. We also relate our results to the single $j$ scissors mode.