New limits for the $2\nu\beta\beta$ decay of $^{96}$Zr to excited nuclear states of $^{96}$Mo

The final results from our search for the $2\nu\beta\beta$ decay of $^{96}$Zr to excited $0^+$ and $2^+$ states of $^{96}$Mo are presented. Such measurements provide valuable test cases for $2\nu\beta\beta$-decay nuclear matrix element calculations, which in turn are used to tune $0\nu\beta\beta$-decay nuclear matrix element calculations. After undergoing double-$\beta$ decay to an excited state, the excited daughter nucleus decays to the ground state, emitting two coincident $\gamma$ rays. These two $\gamma$ rays are detected in coincidence by two HPGe detectors sandwiching the $^{96}$Zr sample, with a NaI veto in anti-coincidence. This experimental apparatus, located at the Kimballton Underground Research Facility (KURF), has previously measured the $2\nu\beta\beta$ decay of $^{100}$Mo and $^{150}$Nd to excited nuclear states. Experimental limits on the $T_{1/2}$ and corresponding nuclear matrix element are presented for each of these decays. As a byproduct of this experiment, limits were also set on the single-$\beta$ decay of $^{96}$Zr.